Rationality of Varieties 3030754200, 9783030754204

Table of contents :
Contents
Rationality of Algebraic Varieties
On Geometry of Fano Threefold Hypersurfaces
1. Introduction
2. Non-solid Fano threefolds
3. Birationally non-rigid Fano threefolds
3.1. How to read off the equation of Z?
4. Evidence for Conjecture 1.4
Acknowledgement
References
On the Image of the Second l-adic Bloch Map
Introduction
0.1. Mazur's question with Q-coefficients
0.2. Mazur's question with Q-coefficients in positive characteristic
0.3. Mazur's question with Z-coefficients
0.4. Universal cycles and the image of the second l-adic Bloch map
0.5. Decomposition of the diagonal and the image of the second l-adic Bloch map
0.6. Stably rational vs. geometrically stably rational varieties over finite fields
0.7. Notation and conventions
1. On various notions of coniveau filtrations
1.1. Recalling the geometric coniveau filtrations
1.2. p-adic coniveau filtrations
2. The image of the l-adic Bloch map and the coniveau filtration
2.1. The image of the -adic Bloch map
2.2. The image of the p-adic Bloch map
3. Decomposition of the diagonal, algebraic representatives,and miniversal cycles
3.1. Decomposition of the diagonal
3.2. Surjective regular homomorphisms and algebraic representatives
3.3. Miniversal cycles and miniversal cycles of minimal degree
3.4. Decomposition of the diagonal and algebraic representatives
4. Miniversal cycles and the image of the second l-adic Bloch map
5. Decomposition of the diagonal and the image of the second -adic Bloch map
6. Modeling cohomology via correspondences
6.1. Modeling Q-cohomology via correspondences
6.2. Modeling Z-cohomology via correspondences: Theorem 15
7. The image of the -adic Bloch map in characteristic 0
Appendix: A review of the l-adic Bloch map
A.1. Conventions for -adic and p-adic cohomology
A.1.1. -adic cohomology
A.1.2. p-adic cohomology
A.2. The -adic Bloch map
A.2.1. The Abel–Jacobi map on torsion
A.2.2. Bloch's preliminaries
A.2.3. The -Bloch map
A.2.4. Bloch's Key Lemma
A.2.5. The -adic Bloch map
A.2.6. The Bloch map
A.3. Suwa's construction of the l-adic Bloch map
A.3.1. Structure of abelian l-primary torsion groups
A.3.2. -adic cohomology from cohomology with torsion coefficients
A.3.3. Suwa's -adic Bloch map
A.3.4. The -adic Bloch map and Suwa's construction
A.3.5. Gross–Suwa's p-adic Bloch map
A.4. Properties of the Bloch maps
A.5. Restriction of the Bloch map to algebraically trivial cycle classes
Acknowledgment
References
Rational Curves and MBM Classes on Hyperkähler Manifolds: A Survey
1. Introduction
2. MBM classes: equivalent definitions and basic properties
2.1. Deforming rational curves: first remarks
2.2. Parameter spaces for hyperkähler manifolds
2.3. MBM classes
3. Results on MBM classes and applications
3.1. Markman's Torelli theorem and the birational cone conjecture
3.2. The cone conjecture via ergodic theory
3.3. Uniform boundedness and an appication
4. Contractibility and deformations
5. Classification of MBM classes in low dimension for K3 type
6. Some open questions
Acknowledgement
References
Unirationality of Certain Universal Families of Cubic Fourfolds
1. Introduction
2. The existence of the universal cubic fourfold, and some properties of scrolls and associated K3 surfaces
3. Unirationality for C26,1 and C42,1 via universal K3 surfaces
4. Unirationality through rational special surfaces
4.1. Special cubics in Cd in the range 8d 38
4.2. Special cubics in C42
4.3. Unirationality of Cd,n
5. Some results of non-unirationality
5.1. Open questions
Acknowledgement
References
A Categorical Invariant for Geometrically Rational Surfaces with a Conic Bundle Structure
1. Introduction
Notations
2. Basics on geometrically rational surfaces
2.1. Elementary links
3. Basics on derived categories
3.1. Categorical representability
3.2. Conic bundles
4. Links of type I/III and the definition ofthe Griffiths–Kuznetsov component
5. Links of type II
6. Links of type IV
Acknowledgment
References
Marked and Labelled Gushel–Mukai Fourfolds
1. Introduction
2. Gushel–Mukai fourfolds
2.1. Cohomology and period domain of Gushel–Mukai fourfolds
2.2. Hodge-special Gushel–Mukai fourfolds
3. Marked and labelled Gushel–Mukai fourfolds
4. Gushel–Mukai fourfolds with associated K3 surface
4.1. Rational maps to moduli spaces of K3 surfaces
4.2. Fibers of Fourier–Mukai partners
5. Gushel–Mukai fourfolds and twisted K3 surfaces
5.1. Moduli and periods of twisted K3 surfaces
5.2. Twisted K3 surfaces associated to GM fourfolds
5.3. Fourier-Mukai partners in the twisted case
Acknowledgment
References
Supersingular Irreducible Symplectic Varieties
1. Introduction
2. Generalities on the notion of supersingularity
3. Supersingular symplectic varieties
4. Moduli spaces of stable sheaves on K3 surfaces
5. Moduli spaces of twisted sheaves on K3 surfaces
6. Moduli spaces of sheaves on abelian surfaces
References
Symbols and Equivariant Birational Geometry in Small Dimensions
1. Brief history of previous work
2. Equivariant birational types
2.1. Antisymmetry
2.2. Multiplication and co-multiplication
2.3. Birational invariant
3. Computation of invariants on surfaces
3.1. Sample computations of B2(Cp)
3.2. Examples for noncyclic groups
3.3. Linear actions yield torsion classes
3.4. Algebraic structure in dimension 2
4. Reconstruction theorem
5. Refined invariants
5.1. Encoding fixed points
5.2. Encoding points with nontrivial stabilizer
5.3. Examples of blowup relations
5.4. Examples
5.5. Limitation of the birational invariant
5.6. Reprise: Cyclic groups on rational surfaces
6. Cubic fourfolds
7. Nonabelian invariants
7.1. The equivariant Burnside group
7.2. Resolution of singularities
7.3. The class of XG
7.4. Elementary observations
7.5. Dihedral group of order 12
7.6. Embeddings of S3C2 into the Cremona group
Acknowledgment
References
Rationality of Fano Threefolds of Degree 18 over Non-closed Fields
1. Introduction
2. Projection constructions
2.1. Projection from lines
2.2. Projection from conics
2.3. Projection from points
3. Unirationality constructions
3.1. Using a point
3.2. Using a point and a conic
4. Rationality results
5. Analysis of principal homogeneous spaces
5.1. Proof of Theorem 1
5.2. A corollary to Theorem 1
5.3. Generic behavior
5.4. Connections with complete intersections?
Acknowledgment
References
Rationality of Mukai Varieties over Non-closed Fields
1. Introduction
2. A birational transformation given by a family of quadrics
2.1. The statement
2.2. The proof
2.3. Grassmannians of lines
2.4. Orthogonal Grassmannian
2.5. Grassmannian of the group G2
3. Mukai varieties of genus 7, 8, and 10
3.1. Forms of linear sections
3.2. Rationality of Mukai varieties
4. Mukai varieties of genus 9
4.1. The statement
4.2. The proof
4.3. Implications for genus 9 Mukai varieties
5. Fano threefolds of genus 12
5.1. Vector bundles and Grassmannian embedding
5.2. Birational transformation for `39`42`"613A``45`47`"603AGr(3,7)
5.3. The induced transformation of threefolds
Appendix: Application to cylinders
Acknowledgment
References
A Refinement of the Motivic Volume, and Specialization of Birational Types
1. Introduction
Terminology
2. The Grothendieck ring of varieties graded by dimension
2.1. Reminders on the Grothendieck ring of varieties
2.2. The graded Grothendieck ring
2.3. Birational types
2.4. A refinement of Bittner's presentation
2.5. A refinement of the theorem of Larsen & Lunts
3. Dimensional refinement of the motivic volume
3.1. The motivic volume
3.2. Strictly toroidal models
3.3. Construction of the motivic volume
4. Applications to rationality problems
4.1. Specialization of birational types
4.2. Obstruction to stable rationality
4.3. Examples
5. The monodromy action
5.1. The equivariant Grothendieck ring
5.2. The monodromy action on the motivic volume
Acknowledgement
References
Explicit Rationality of Some Special Fano Fourfolds
Introduction
1. Rationality via linear systems of hypersurfaces of degree 3e-1 having points of multiplicity e along a surface
1.1. Linear systems of quintics with double points along a general Sd
2. Birational maps to P4 for cubics in C14, C26 and C38
3. Birational maps to linear sections of G(1,3+k) for cubics in C(14+12k) for k

Citation preview

Progress in Mathematics 342

Gavril Farkas Gerard van der Geer Mingmin Shen Lenny Taelman Editors

Rationality of Varieties

Progress in Mathematics Volume 342

Series Editors Antoine Chambert-Loir , Université Paris-Diderot, Paris, France Jiang-Hua Lu, The University of Hong Kong, Hong Kong SAR, China Michael Ruzhansky, Ghent University, Belgium and Queen Mary University of London, UK Yuri Tschinkel, Courant Institute of Mathematical Sciences, New York, USA

More information about this series at http://www.springer.com/series/4848

Gavril Farkas • Gerard van der Geer Mingmin Shen • Lenny Taelman Editors

Rationality of Varieties

Editors Gavril Farkas Institut für Mathematik Humboldt-Universität zu Berlin Berlin, Germany

Gerard van der Geer Korteweg-de Vries Instituut Universiteit van Amsterdam Amsterdam, The Netherlands

Mingmin Shen Korteweg-de Vries Instituut Universiteit van Amsterdam Amsterdam, The Netherlands

Lenny Taelman Korteweg-de Vries Instituut Universiteit van Amsterdam Amsterdam, The Netherlands

ISSN 0743-1643 ISSN 2296-505X (electronic) Progress in Mathematics ISBN 978-3-030-75420-4 ISBN 978-3-030-75421-1 (eBook) https://doi.org/10.1007/978-3-030-75421-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents Rationality of Algebraic Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

H. Abban, I. Cheltsov and J. Park On Geometry of Fano Threefold Hypersurfaces . . . . . . . . . . . . . . . . . . . . .

1

J.D. Achter, S. Casalaina-Martin and C. Vial On the Image of the Second l-adic Bloch Map . . . . . . . . . . . . . . . . . . . . . .

15

E. Amerik and M. Verbitsky Rational Curves and MBM Classes on Hyperk¨ ahler Manifolds: A Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

H. Awada and M. Bolognesi Unirationality of Certain Universal Families of Cubic Fourfolds . . . . .

97

M. Bernardara and S. Durighetto A Categorical Invariant for Geometrically Rational Surfaces with a Conic Bundle Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

E. Brakkee and L. Pertusi Marked and Labelled Gushel–Mukai Fourfolds . . . . . . . . . . . . . . . . . . . . . .

129

L. Fu and Z. Li Supersingular Irreducible Symplectic Varieties . . . . . . . . . . . . . . . . . . . . . .

147

B. Hassett, A. Kresch and Y. Tschinkel Symbols and Equivariant Birational Geometry in Small Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201

B. Hassett and Y. Tschinkel Rationality of Fano Threefolds of Degree 18 over Non-closed Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

237

A. Kuznetsov and Y. Prokhorov Rationality of Mukai Varieties over Non-closed Fields . . . . . . . . . . . . . . .

249

J. Nicaise and J.C. Ottem A Refinement of the Motivic Volume, and Specialization of Birational Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

291 v

vi

Contents

o F. Russo and G. Staglian` Explicit Rationality of Some Special Fano Fourfolds . . . . . . . . . . . . . . . .

323

S. Schreieder Unramified Cohomology, Algebraic Cycles and Rationality . . . . . . . . . .

345

M. Shen Vanishing Cycles under Base Change and the Integral Hodge Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

389

A. Verra The Igusa Quartic and the Prym Map, with Some Rational Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

403

Rationality of Algebraic Varieties An algebraic variety of dimension d is called rational if a non-empty Zariski open subset of it can be parametrized by rational functions of d free variables. In other words, if the variety is birational to projective space of dimension d. Such parametrizations, when available, were of great use in the evaluation of abelian integrals and the search for such parametrizations goes back to the beginnings of algebraic geometry. In some sense rational varieties are the simplest varieties. A weaker notion is that of unirationality, where we require only that there is a dominant map of a projective space to our variety. In 1876 L¨ uroth proved that a unirational curve is rational. The question in how far this holds in higher dimension stimulated the early development of the theory of algebraic surfaces at the end of the 19th century. The problem whether unirationality implies rationality is called the L¨ uroth problem. Sometimes one can deduce non-rationality from numerical invariants of a variety, but deducing rationality from numerical invariants is considerably more difficult. Castelnuovo established a criterion for varieties of dimension 2 over the complex numbers in 1894 that says that the vanishing of the second plurigenus P2 and the first Betti number of a smooth projective surface characterize rational surfaces. This was a big step forward and provided an answer to the L¨ uroth problem in dimension 2 and characteristic 0. The question whether one could extend it to dimension 3 remained open for a long time, though, as documented for instance by Roth’s book from 1955 on algebraic threefolds, a negative outcome was suspected. Fano worked extensively on this problem and studied the varieties now named after him. But one had to wait till the early 1970s for a breakthrough, even several almost simultaneous breakthroughs that produced counterexamples to the L¨ uroth problem. The first one was due to Iskovskikh and Manin who showed that a quartic threefold is not rational using the birational automorphism group of the variety in question. Since Segre had shown that some quartic threefolds are unirational this provided a negative answer to L¨ uroth’s problem. The second breakthrough was by Clemens and Griffiths who used Hodge theory to see that the intermediate Jacobian of a rational threefold is a direct sum of Jacobians of curves and using delicate degeneration arguments they showed that every smooth cubic threefold vii

viii

Rationality of Algebraic Varieties

is not rational. A reinterpretation of this work by Mumford opened the way to a systematic geometric study of Prym varieties and of their theta divisors. The third breakthrough at the time was the use of torsion in the middle cohomology as a birational invariant by Artin and Mumford following a suggestion of Ramanujam. This invariant was then generalized in the notion of unramified cohomology and unramified Brauer group by Colliot-Th´el`ene and Ojanguren. It is interesting uroth to note that in 1967 Manin had sketched three possible approaches to the L¨ problem that fitted these approaches rather precisely. The moduli spaces of curves and abelian varieties are among the most studied algebraic varieties that we know and the question whether these are rational or unirational is a fundamental one, amounting to whether one can write down explicitly the general curve (or abelian variety) in a family depending on free parameters. Severi proved in 1915 that the moduli space Mg of curves of genus g is unirational for g ≤ 10 and even conjectured that these are rational for all g. In the 1970s advances in the theory of moduli, especially of curves and abelian varieties drew again attention to the question whether the moduli space Mg of curves of given genus g or the moduli space Ag of principally polarized abelian varieties of dimension g are rational. Here the results by Harris and Mumford for Mg and by Freitag and Tai for Ag showed in the early 1980s that for almost all values of g these are of general type. The question whether the remaining ones are unirational or rational is still the focus of attention. More recently, the fast development of efficient computer algebra systems, like Macaulay2 or Singular also led to a change of perspective in the question whether a moduli space is (uni)rational, for in that case one can try to write down algorithmically its random element over a (finite) ground field. These techniques have been put to work by Schreyer and collaborators. Beginning with the 1980s, the new methods of algebraic geometry were employed with great success to the classification of threefolds and varieties of higher dimension. It eventually led to enormous progress on the classification of varieties in the early 21st century, now known by the term Minimal Model Program. These advances led to a new perspective on rationality questions, to be found for instance in Koll´ar’s book Rational curves on algebraic varieties. The question how close a variety is to being rational led to several intermediate notions of rationality, like rational connectedness and stable rationality. A variety is said to be rationally connected if two of its general points can be connected by a rational curve. In dimension one or two, the class of rationally connected varieties coincides with that of rational (or unirational) varieties, though for higher dimension these classes are different. Despite this fact, to this day, not one example of a rationally connected variety that is not unirational is known. In many ways, rationally connected varieties behave much better than rational varieties. For instance, they define open and closed conditions in families and, as shown by Koll´ar, Miyaoka and Mori, rational connectedness can be detected by negativity properties of the canonical bundle.

Rationality of Algebraic Varieties

ix

A variety is stably rational if a product of the variety and a projective space is rational. This notion sits between rationality and unirationality and leads to an uroth’s problem, called Zariski’s problem. Apart from this one can analogue of L¨ consider rationality questions over fields that are not algebraically closed. An early result here is that of Segre who proved that a smooth cubic surface with Picard number 1 over a non-closed field k is never rational over k. Such questions have found renewed interest in recent years. The last years have seen dramatic progress on rationality questions by Voisin who employs the universal triviality of the Chow group of zero cycles and uses of integral decompositions of the diagonal in 2015. This method has been strengthened by Colliot-Th´el`ene and Pirutka and provides a stable birational invariant and has led to numerous results of non-stably birationality results, like for very general hypersurfaces of suffiently high degree d of in projective space of dimension n. Another new development is the use of derived categories, precisely the question to what extent rationality can be captured by the derived category of the variety in question. This point of view has been used extensively to study the rationality of cubic fourfolds, where a set of independent conjectures of Kuznetsov and Hassett predict that rationality is essentially a Noether–Lefschetz phenomenon, detected at the level of non-trivial middle-dimensional cycles. All these new developments, involving vastly different perspectives and techniques made it tempting to organize a conference in the tradition of conferences on the Dutch islands. The present volume on Rationality of Varieties appears on the occasion of the conference ‘Rationality of Varieties’ that was held on the island of Schiermonnikoog in the spring of 2019. It contains contributions by experts in the field and offers an overview of current developments and new results. We would like to take the opportunity to thank the participants and the speakers who made the conference to a success. We also like to thank NWO, KNAW and the Foundation Compositio Mathematica for supporting the conference.

Gavril Farkas Gerard van der Geer Mingmin Shen Lenny Taelman

On Geometry of Fano Threefold Hypersurfaces Hamid Abban, Ivan Cheltsov and Jihun Park Abstract. We prove that a quasi-smooth Fano threefold hypersurface is birationally rigid if and only if it has Fano index one. Mathematics Subject Classification (2010). 14E05, 14E08, 14E30. Keywords. Fano varieties, hypersurfaces, Mori fibre spaces, Sarkisov theory, birational rigidity.

1. Introduction End points of Minimal Model Program are either Mori fibre spaces or minimal models. In dimension three, Mori fibre spaces form three classes: Fano threefolds with Picard number 1, del Pezzo fibrations over curves, and conic bundles over surfaces. They are Q-factorial with at worst terminal singularities, and with relative Picard number one. The focus of this article is on Fano threefolds. They lie in finitely many deformation families (see [5] and [13]), and studying birational relations among them, as well as birational maps to other Mori fibre spaces is fundamental, as it sheds light to birational classification of rationally connected threefolds in general. For a Fano threefold X with terminal Q-factorial singularities, let A be a Weil divisor for which −KX = ιX A with maximum ιX ∈ Z>0 . This integer ιX is known as the Fano index of X. It follows from [4] and [6] that there are precisely 130 families of Fano threefolds such that the Z-graded ring    R(X, A) = H 0 X, −mA m0

has 5 generators. Denote these generators by x, y, z, t, w, respectively in degrees a0 , a1 , a2 , a3 , a4 , and let the algebraic relation amongst them be f (x, y, z, t, w) = 0 of weighted degree d. This redefines X as a hypersurface in the weighted projective space P(a0 , a1 , a2 , a3 , a4 ) that is given by the quasi-homogeneous equation f (x, y, z, t, w) = 0, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Farkas et al. (eds.), Rationality of Varieties, Progress in Mathematics 342, https://doi.org/10.1007/978-3-030-75421-1_1

1

2

H. Abban, I. Cheltsov and J. Park

 and the Fano index of X is computed via ιX = 4i=0 ai − d. Among these 130 families, exactly 95 have index one and their birational geometry is well studied. Notably, the following theorem holds. Theorem 1.1 ([7, 10]). All index one quasi-smooth Fano threefold hypersurfaces are birationally rigid. In particular, all index one quasi-smooth Fano threefold hypersurfaces are irrational. The remaining families of Fano threefold hypersurfaces with ιX  2 are listed in Table 1, and their rationality has been studied by several people: the irrationality of every smooth cubic threefold (family № 96) has been proved by Clemens and Griffiths [8], the irrationality of all smooth threefolds in family № 97 has been proved by Voisin [19], and the irrationality all smooth threefolds in family № 98 has been proved by Grinenko [11, 12]. Recently, the complete classification of families whose very general member is irrational was carried out by Okada [16]. Whether a very general Fano threefold in a given family is rational or not is also indicated in Table 1. It is natural to expect more rigidity in lower indices and rationality in higher indices. For example, it was proven recently by Prokhorov that all Fano threefolds with index 8 or above are rational [17]. We are interested in irrational Fano threefolds, and their birational geometry. In this paper, we prove that, contrary to the index one case, there are no birationally rigid Fano threefold hypersurfaces with ιX  2. To be precise, our main result is Theorem 1.2. Let X be a quasi-smooth Fano threefold in family № n in Table 1. Then (i) there exists a birational map to a Mori fibration over a curve or a surface if n ∈ {100, 101, 102, 103, 110} (see Section 2), (ii) there exists an elementary Sarkisov link from X to another Fano threefold if n ∈ {100, 101, 102, 103, 110} (see Section 3). In particular, if ιX  2, then X is not birationally rigid. Corollary 1.3. A quasi-smooth Fano threefold hypersurface with terminal singularities is birationally rigid if and only if it has index one. Our results go beyond the proof of birational non-rigidity. They also provide evidence for the following conjecture. Conjecture 1.4. A quasi-smooth Fano hypersurface X with ιX  2 is solid if and only if it belongs to one of the families № 100, № 101, № 102, № 103, № 110. Recall from [2, Definition 1.4] that a Fano variety (with Picard number 1) is called solid if it is not birational to a Mori fibre space with a positive dimensional base. In Section 2, we prove one direction of this conjecture, i.e., we show that if a quasi-smooth Fano threefold hypersurface X with ιX  2 is solid, then it must

On Geometry of Fano Threefold Hypersurfaces № Xd ⊂ P(a0 , a2 , a2 , a3 , a4 ) ιX Rational

3

№ Xd ⊂ P(a0 , a2 , a2 , a3 , a4 ) ιX Rational

96

X3 ⊂ P(1, 1, 1, 1, 1)

2

No

114

X6 ⊂ P(1, 1, 2, 3, 4)

5

Yes

97

X4 ⊂ P(1, 1, 1, 1, 2)

2

No

98

X6 ⊂ P(1, 1, 1, 2, 3)

2

No

115

X6 ⊂ P(1, 2, 2, 3, 3)

5

Yes

116

X10 ⊂ P(1, 2, 3, 4, 5)

5

No

99

X10 ⊂ P(1, 1, 2, 3, 5)

2

No

117

X15 ⊂ P(1, 3, 4, 5, 7)

5

No

100

X18 ⊂ P(1, 2, 3, 5, 9)

2

101

X22 ⊂ P(1, 2, 3, 7, 11)

2

No

118

X6 ⊂ P(1, 1, 2, 3, 5)

6

Yes

No

119

X6 ⊂ P(1, 2, 2, 3, 5)

7

Yes

102

X26 ⊂ P(1, 2, 5, 7, 13)

2

No

120

X6 ⊂ P(1, 2, 3, 3, 4)

7

Yes

103 X38 ⊂ P(2, 3, 5, 11, 19)

2

No

121

X8 ⊂ P(1, 2, 3, 4, 5)

7

Yes

104

X2 ⊂ P(1, 1, 1, 1, 1)

3

Yes

122

X14 ⊂ P(2, 3, 4, 5, 7)

7

No

105

X3 ⊂ P(1, 1, 1, 1, 2)

3

Yes

123

X6 ⊂ P(1, 2, 3, 3, 5)

8

Yes

106

X4 ⊂ P(1, 1, 1, 2, 2)

3

Yes

124

X10 ⊂ P(1, 2, 3, 5, 7)

8

Yes

107

X6 ⊂ P(1, 1, 2, 2, 3)

3

No

125

X12 ⊂ P(1, 3, 4, 5, 7)

8

Yes

108

X12 ⊂ P(1, 2, 3, 4, 5)

3

No

126

X6 ⊂ P(1, 2, 3, 4, 5)

9

Yes

109

X15 ⊂ P(1, 2, 3, 5, 7)

3

No

127

X12 ⊂ P(2, 3, 4, 5, 7)

9

Yes

110

X21 ⊂ P(1, 3, 5, 7, 8)

3

No

128

X12 ⊂ P(1, 4, 5, 6, 7)

11

Yes

111

X4 ⊂ P(1, 1, 1, 2, 3)

4

Yes

129

X10 ⊂ P(2, 3, 4, 5, 7)

11

Yes

112

X6 ⊂ P(1, 1, 2, 3, 3)

4

Yes

130

X12 ⊂ P(3, 4, 5, 6, 7)

13

Yes

113

X4 ⊂ P(1, 1, 2, 2, 3)

5

Yes

Table 1: Fano threefold hypersurfaces of index ιX  2 and their rationality data. belong to one of the families №100, №101, №102, №103, №110. In Section 4, we indicate how to prove the other direction of our Conjecture 1.4. Unfortunately, the technical difficulties in the final step of the indicated proof do not allow us to finish the proof. We encourage the reader to fill this gap.

2. Non-solid Fano threefolds Let X be a quasi-smooth Fano threefold in family № n in Table 1. Then X is a hypersurface in the weighted projective space P(a0 , a1 , a2 , a3 , a4 ) of degree d. Suppose also that n ∈ {100, 101, 102, 103, 110}. Then the Fano index iX of X satisfies a0 a1 < iX = a0 + a1 + a2 + a3 + a4 − d. Here, we assume that a0  a1  a2  a3  a4 as in Table 1.

4

H. Abban, I. Cheltsov and J. Park

Let ψ : X  P1 be the map given by the projection P(a0 , a1 , a2 , a3 , a4 )  P(a0 , a1 ). Then there exists a commutative diagram  X  @@@  @@φ π  @@  @   X _ _ _ _ _ _ _ / P1 ψ

 is a smooth projective threefold, π is a birational map, and φ is a surjective where X morphism. Let S be a general fibre of the rational map ψ, and let S be its proper  If n ∈ {122, 127, 129, 130}, then S is a hypersurface transform on the threefold X. in the weighted projective space P(a0 , a2 , a3 , a4 ) of degree d. One can check that S has at most isolated singularity and it is normal. By adjunction, −KS is ample, hence S is uniruled. Similarly, if n ∈ {122, 127, 129, 130}, then the surface S is a complete intersection in P(a0 , a1 , a2 , a3 , a4 ) of two hypersurfaces: the hypersurface X of degree d, and the hypersurface of degree a0 a1 that is given by y a0 = λxa1 for some λ ∈ C, where x and y are coordinates on P(a0 , a1 , a2 , a3 , a4 ) of weights a0 and a1 , respectively. Note that S → S is a resolution of singularities on S. Let g : S¯ → S be the normalisation of S, and let f : Sˆ → S¯ be its minimal resolution of singularities. Then Kodaira dimensions of Sˆ and S˜ are the same since they are both smooth. Denoting numerical Q-equivalence of divisors by ∼Q , we have that KSˆ ∼Q f ∗ (KS¯ ) − A and KS¯ ∼Q g ∗ ((KX + S)|S ) − B, where A and B are some effective divisors. Note that the first assertion follows from the fact that f is a minimal resolution, and the second follows from subadjunction lemma [15, Lemma 5.1.9]. Given the above relations, and our choice of S, as a divisor on X, it is clear that KSˆ is not numerically equivalent to any effective Q-divisor. Hence, the Kodaira dimension of Sˆ is negative. Thus, in all cases, the Kodaira dimension of the surfaces S is negative by the adjunction formula, so that  over S is uniruled. Now we can apply the (relative) Minimal Model Program to X P1 . One possibility is that we obtain a commutative diagram  _ _ _χ _ _ _/ V X φ

 P1

η

 P1

where χ is a birational map, V is a threefold with terminal Q-factorial singularities, rk Pic(V ) = 2, and η is a fibration into del Pezzo surfaces. Another possibility is

On Geometry of Fano Threefold Hypersurfaces

5

that we obtain a commutative diagram  _ _ _χ _ _ _/ V X η

φ

 P1 o

υ

 Z

where χ is a birational map, V is a threefold with terminal Q-factorial singularities, Z is a normal surface, η is a conic bundle, rk Pic(V ) = rk Pic(Z) + 1, and υ is a surjective morphism with connected fibres. Corollary 2.1. Let X be a quasi-smooth Fano threefold in family № n in Table 1 such that n ∈ {100, 101, 102, 103, 110}. Then X is not solid. In particular, if n ∈ {100, 101, 102, 103, 110}, then X is not birationally rigid.

3. Birationally non-rigid Fano threefolds In this section, we will complete the proof of Theorem 1.2. In each case, we regard the threefold as a hypersurface X inside the weighted projective ambient space P = P(a0 , a1 , a2 , a3 , a4 ), and distinguish a quotient singularity p ∈ X; for families № 100, № 101, № 102, № 103 this point is p = p3 = (0 : 0 : 0 : 1 : 0) and for family № 110 it is p = p4 = (0 : 0 : 0 : 0 : 1). The extremal extraction from this point is prescribed by Kawamata [14] in a local analytic neighbourhood of p. We proceed by identifying a projective toric variety T with Picard rank 2 together with a birational morphism Φ : T → P. We then demand that ϕ : Y → X viewed locally near p is the Kawamata blow up at the point p, where Y = Φ−1 ∗ (X) and ϕ is simply the restriction of Φ to Y . Now, by construction, Y has Picard number two, and hence admits a 2-ray game. We proceed by running the 2-ray game on T and restricting it to Y , and each time we recover the 2-ray game of Y , which ends with a divisorial contraction to a Fano threefold not isomorphic to X. See [3] for a comprehensive explanation of explicit 2-ray games and the links obtained from them. We treat the first four families together, and then we consider family № 110 separately as it has a higher index and behaves slightly differently. Let X ⊂ P be a quasi-smooth member in one of the four families above with Fano index 2, and denote the variables by x0 , . . . , x4 . Note that in each case the defining polynomial f (x0 , . . . , x4 ) contains two monomial x24 and x33 xk with nonzero coefficient, where k = 2 in families № 100, № 102, № 103, and k = 0 in family № 101. This shows that p is locally described by the quotient singularity 1 1 (a0 , a1 , a4 − a3 ) in the first three cases, and (a1 , a2 , a4 − a3 ) in family № 101. a3 a3 In each case a0 + (a4 − a3 ) ≡ 0 or a2 + (a4 − a3 ) ≡ 0 mod a3 , and the third local coordinate weight is 2, the index of X. In order to consider the Kawamata

6

H. Abban, I. Cheltsov and J. Park

blow up of a 3-fold quotient singularity, the local description is expected to be of type 1 (1, a, b), where a + b ≡ 0 mod r. r For this, we can multiply the local weights above by a23 . If we denote the local coordinates of 1r (1, a, b) with x, y, z, then the Kawamata blow up, with the new variable u, is given by 1 a b u r x, u r y, u r z. This is because if the germ 1r (1, a, b) is viewed as an affine toric variety with rays ρx , ρy and ρw , then the Kawamata blow up of this toric variety is realised by adding a primitive new ray ρu with relation r · ρu = 1 · ρx + a · ρy + b · ρz . We take advantage of this. We view P as a projective toric variety defined by 5 primitive rays in Z4 and one relation amongst them, namely 4 

ai · ρi = 0,

i=0

where ρi is the primitive ray corresponding to the toric divisor Di = {xi = 0}. Using this, we define a rank two toric variety T with coordinate weights given by the matrix ⎞ ⎛ u y3 y4 y2 y1 y0 ⎝ 0 a3 a4 a2 a1 a0 ⎠ (3.1) −a3 0 b4 b2 b1 b0 where bi is the smallest positive integer congruent to a23 · ai modulo a3 when xi appears as a local coordinate of the singular point p ∈ X, otherwise bi = a4 − a3 . Let us explain this by an example. Suppose X is in family № 100. Then the defining equation of X is of the form x33 x2 = x24 + x91 + x32 x4 + · · · . In an analytic neighbourhood of p we can eliminate the (tangent) variable x2 , which gives local coordinates x0 , x1 , x4 with a finite group action of Z5 by 1 1 (a0 , a1 , a4 − a3 ) = (1, 2, 4), a3 5 which can be seen as 1 1 (b0 , b1 , b4 ) = (3, 1, 2). a3 5 Clearly, the Kawamata blow up is locally given by 3

1

2

(u, y0 , y1 , y4 ) → (u 5 y0 , u 5 y1 , u 5 y4 ) = (x0 , x1 , x4 ). Plugging these into the equation of X above shows that the multiplicity of u in the right-hand side of the equation is exactly 45 , precisely due to the presence of x24 in the equation. Other cases are similar.

On Geometry of Fano Threefold Hypersurfaces

7

The map Φ : T → P is defined by the restriction to the y3 -wall, which defines b0

b1 b2 b4 (u, y3 , y4 , y2 , y1 , y0 ) → u a3 y0 : u a3 y1 : u a3 y2 : y3 : u a3 y4 = (x0 : x1 : x2 : x3 : x4 ). Consequently, Y ⊂ T is defined by the vanishing of the polynomial g(u, y0 , · · · , y4 ) =

1

b0

a3 a4 −a3

b1

b2

b4

· f (u a3 x0 , u a3 x1 , u a3 x2 , x3 , u a3 x4 ).

u The nonzero determinants of the 2 × 2 minors of the matrix above are all divisible by a3 , which makes this particular description of T “not well-formed” (see [1] for a general treatment). However, this can easily be fixed by an action of the matrix   1/a3 −2/a3 , 0 1 which, for example, transforms ⎛ u ⎝ 2 −5

(3.1) in the case of family № 100 into ⎞ y 3 y4 y1 y2 y0 1 1 0 −1 −1 ⎠ 0 2 1 4 3

Note that we have also rearranged the columns of the matrix (easily tractable using the variables order). This is because we will run the 2-ray game on T according to the GIT chambers of the action of (C∗ )2 . Restricting the wall-crossing to Y over the y4 -wall is an isomorphism on Y as g includes the term y42 . Crossing the next wall on T is locally the flip (2, 1, 1, −1, −1), read-off from the first row of the wellformed (3.1) after setting yi = 1, the variable corresponding to the next wall (i = 1 for families № 100, 101, 102 and i = 0 for family № 103). This flip, as the weights suggest, contracts a P(1, 1, 2) inside the toric variety and replace it with a P(−1, −1) ∼ = P1 . This restricts to an Atiyah flop (1, 1, −1, −1) as g includes the monomial uyiα with nonzero coefficient, where α = deg2 f . Note that this coefficient is nonzero as a consequence of X being quasismooth. The last wall, that is the y2 -wall, contracts the divisor {x = 0} on T (or {x = 0} in the case of № 103). The restriction to Y is always a divisorial contraction to a Fano 3-fold with a singular point at the image of the contraction. Table 2 contains the information on the image in each case. Now we turn our attention to Family № 110. A quasi-smooth member in this family is a hypersurface X21 ⊂ P(1, 3, 5, 7, 8) and contains a singular point of type 1 2 8 (1, 3, 7) as the defining polynomial contain the monomial x4 x2 . In Kawamata 1 format this singularity is of type 8 (3, 2, 5). Similar to the computations above, the toric variety T is given by the matrix ⎛

u ⎝ 0 −8

y4 8 0

y1 3 1

y3 7 5

y2 5 7

⎞ y0 1 ⎠, 3

8

H. Abban, I. Cheltsov and J. Park № 100

p3 ∈ Xd ⊂ P(a0 , a2 , a2 , a3 , a4 ) 1 (3, 1, 2) 5

101

1 (1, 5, 2) 7

102

1 (4, 1, 3) 7

103

1 (1, 7, 4) 11

New Model

∈ X18 ⊂ P(1, 2, 3, 5, 9)

cE6 ∈ Z10 ⊂ P(1, 1, 1, 3, 5)

∈ X22 ⊂ P(1, 2, 3, 7, 11) ∈ X26 ⊂ P(1, 2, 5, 7, 13)

cE7 ∈ Z12 ⊂ P(1, 1, 1, 4, 6) 1 (1, 1, 1, 0; 0) 2

∈ X38 ⊂ P(2, 3, 5, 11, 19)

∈ Z14 ⊂ P(1, 1, 2, 4, 7)

cE8 ∈ Z22 ⊂ P(1, 1, 3, 7, 11)

Table 2: Birational models for Fano hypersurfaces in families № 100, 101, 102, 103. which can be wellformed to ⎛

u y4 ⎝ 5 7 1 3

y1 2 1

y3 3 2

y2 0 1

⎞ y0 −1 ⎠ , 0

The 2-ray game of T restricts to Y via an isomorphism followed by a flip of type (5, 1, −3, −2), then it ends with a divisorial contraction to a singular point of type cE7 on a hypersurface Z7 ⊂ P(1, 1, 1, 2, 3). 3.1. How to read off the equation of Z? The final model Z sits inside a weighted projective space, determined by the 2ray game of the ambient toric variety. To find its weights, it suffices to rearrange the rank two matrix, with an action of a 2 × 2 matrix in SL(2, Z), so that the second row of the matrix takes the form of five positive numbers followed by a 0. Those positive numbers are the weights of the projective space that contains Z. For instance, in the case of Family № 100 one could multiply the matrix by   1 0 , 3 1 ⎛

⎞ u y3 y4 y1 y2 y0 ⎝ 2 1 1 0 −1 −1 ⎠ . 1 3 5 1 1 0 The equation of Z is simply read off by “removing” the variable that represents the last column. For example, in the case of Family № 100 above we set y0 = 1 in g to find the equation of Z. The singular point is at py2 in this case, which can be further analysed.

to obtain

4. Evidence for Conjecture 1.4 Let us first prove that no smooth point or curve on any quasi-smooth member in families № 100, 101, 102, 103, 110 can be a centre of maximal singularities (see [18] for an introduction to the theory of maximal singularities).

On Geometry of Fano Threefold Hypersurfaces

9

Lemma 4.1. Let X be a quasi-smooth Fano hypersurface of degree d in the weighted projective space P(a0 , a1 , a2 , a3 , a4 ) that belongs to one of the families № 100, 101, 102, 103, or 110. Let M be a mobile linear subsystem in | − nKX | for some positive integer n. Then any smooth point in X cannot be a centre of non-canonical singularities of the pair (X, n1 M). Proof. Let p be a smooth point of X and suppose that p is a centre of non-canonical singularities of the pair (X, n1 M). We then obtain multp (M2 ) > 4n2 from [9, Corollary 3.4]. We now seek for a contradiction case by case. For the families № 100, 101, 102, 103 we consider a general member H in |OX (a1 a2 a3 )| that passes through the point p. Since the linear subsystem of |OX (a1 a2 a3 )| consisting of members passing through the point p has a zero-dimensional base locus, H contains no one-dimensional components of the base locus of M. Then 4a1 a2 a3 n2 d = H · M2  multp (H)multp (M2 ) > 4n2 , a0 a1 a2 a3 a4 which yields an absurd inequality 2 > a0 . For the family № 110 we consider a general member H in |OX (15)| that passes through the point p. It is easy (but a bit tedious) to check that the base locus of the linear subsystem of |OX (15)| consisting of members passing through the point p is zero-dimensional. Then 27n2 = H · M2  multp (H)multp (M2 ) > 4n2 , 8 

which is absurd.

Lemma 4.2. Under the same condition as Lemma 4.1, any curve contained in the smooth locus of X cannot be a centre of non-canonical singularities of the pair (X, n1 M). Proof. Let C be a curve that is contained in the smooth locus of X and suppose that C is a centre of non-canonical singularities of the pair (X, n1 M). We then obtain multC (M) > n. Choose two general members H1 and H2 from the mobile linear system M. We then obtain 3

2

n2 (−KX ) = −KX · H1 · H2  (multC (M)) (−KX · C) > n2 (−KX · C) . Since the curve C is contained in the smooth locus of X, the inequality above implies that 3 (−KX ) > 1. However, the anticanonical classes of the families № 100, 101, 102, 103, 110 have self-intersection numbers less than 1. This completes the proof. 

H. Abban, I. Cheltsov and J. Park

10

Together with [14, Theorem 5], the lemma above shows that a curve on X cannot be a centre of non-canonical singularities of the pair (X, n1 M). In conclusion, it follows from Lemmas 4.1 and 4.2 that only singular points of X can be centres of non-canonical singularities of the pair (X, n1 M). So far we have excluded smooth points and curves from being centres of maximal singularity. It remains to exclude the singular points on X. This will be done in what follows. We show that the unique extremal extraction from any singular point in these 5 families does not yield a Sarkisov link. Recall from Section 3 that in each family there exists a birational map to another Fano threefold constructed via an elementary Sarkisov link starting from the Kawamata blow up of the highest index quotient singularity. We now concentrate on lower index singularities and show that in each case of Fano index 2 these singular points produce no elementary link, and in the index 3 case we obtain a new birational map to a Fano threefold. Let us proceed with the latter. A quasi-smooth X21 ⊂ P(1, 3, 5, 7, 8) contains two singular points of type 1 1 8 (1, 3, 7) and 5 (3, 2, 3). A typical equation for this hypersurface is f = x42 x0 + x24 x2 + x33 + x71 + · · · . Indeed, the tangent space to p = p2 is determined by x0 and the local coordinates of the tangent space are x1 , x3 , x4 . In Kawamata format this singularity is of type 1 5 (1, 4, 1) and the local blow up is given by 1

4

1

(u, y1 , y3 , y4 ) → (u 5 x1 , u 5 x3 , u 5 x4 ). So we can define the rank two toric variety T ⎛ u y2 y4 y1 ⎝ 0 5 8 3 −5 0 1 1

via the matrix ⎞ y3 y0 7 1 ⎠ 4 2

together with the map 2

1

4

1

Φ(u, y2 , y4 , y1 , y3 , y0 ) = (u 5 x0 : u 5 x1 : x2 : u 5 x3 : u 5 x4 ) ∈ P(1, 3, 5, 7, 8). Note that the blow up weight of the variable x0 is determined by the multiplicity of f , which is 25 . It is easy to see that Y does not follow the 2-ray game of T . This is because g, the defining equation of Y , belong to the irrelevant ideal (u, y2 ) ∩ (y4 , y1 , y3 , y0 ) as g = u(y17 + uy33 + · · · ) + y2 (y42 + y23 y0 + · · · ). This can be resolved by replacing T with a higher-dimensional toric variety (see [3] for an explanation and examples), using an unprojection defined by the cross-ratio y=

y17 + uy33 + · · · y 2 + y23 y0 + · · · , = 4 −y2 u

which defines an isomorphism between Y and the complete intersection of yy2 + y17 + uy33 + · · · = −uy + y42 + y23 y0 + · · · = 0

On Geometry of Fano Threefold Hypersurfaces inside the toric variety defined by ⎛ u y2 y4 ⎝ 0 5 8 −5 0 1 which can be wellformed into ⎛ u y2 ⎝ 3 1 21 7

y4 1 8

y1 3 1 y1 0 1

y y3 16 7 7 4

11

⎞ y0 1 ⎠ 2

⎞ y y3 y0 −1 −1 −1 ⎠ . 0 −3 −5

The 2-ray game of the toric variety restricts to a 2-ray game on Y by two isomorphisms followed by a flip of type (8, 1, −3, −5) and ends in a divisorial contraction into a cE7 /2 singular point on the complete intersection Z6,7 ⊂ P(1, 1, 2, 2, 3, 5). The exclusion in all other quotient singularities for the 4 families of index 2 follow the same principle. We first identify the quotient singularity, with a “key monomial”, which defines the tangent space. Then we construct the rank two toric variety with weights prescribed by Kawamata blow up and one weight dictated by the tangent multiplicity. If needed, we perform an unprojection as above. In each case we either obtain no link, when −KY is outside the closure of the cone of movable divisors, or we obtain a bad link, when −KY is in the boundary of the cone of movable divisors (see [3] for justification of exclusions). So, neither case results in an elementary Sarkisov link. Note that a bad link occurs when the final map of the 2-ray game corresponds to a K-trivial contraction or a fibration with K-trivial fibres. This does not provide a Sarkisov link on its own, but it is a particular case of a “no-link”, which carries other interesting geometric information. We capture all these in Table 3, with the essential data needed to carry on the computation. Instead of writing the full matrix of T in each case we only write the blow up weights indexed by the corresponding variable in anticlockwise order of the GIT chambers. Whenever an unprojection is needed there is an extra variable y in the list of weight-variables denoted by my (n), where m is weight corresponding to the blow up and n is the other weight (in relation to the weighted projective space P). Remark 4.3. Note that the movable cone of T does not necessarily determine the movable cone of Y . However, as Y has Picard number 2, we need only two ingredients to verify that the movable cone of Y is determined by that of T : (1) each small modification (for example, a flip) in the wall crossing of the toric space restricts to a small modification or an isomorphism on the threefold, and (2) the final contraction or fibration at the toric level, which corresponds to the boundary of the movable cone, restricts to a contraction or fibration (in no particular order) on the threefold. We have now excluded all possible points and curves in all 5 families, except the singular points that provide a link to Z. A proof of Conjecture 1.4 requires one to exclude all points and curves on Z and their extremal extractions, except those that provide the link (in reverse direction) to X. Unless another method

H. Abban, I. Cheltsov and J. Park

12 №

singularity

key monomial

blow up

exclusion

100

2 × 13 (1, 2, 2)

x24 + x4 x22

−3u , 0y2 , 1y3 , 3y4 , 6y (15), 1y1 , 2y2

bad link

101

1 (1, 2, 2) 3

102

1 (2, 2, 3) 5

103

103

1 (1, 1, 1) 3

1 (2, 1, 4) 5

x32 x4

x72 x0

−3u , 0y2 , 1y4 , 2y4 , 1y1 , 2y0

bad link

x52 x0

−5u , 0y2 , 1y3 , 4y4 , 8y (21), 1y1 , 3y0

bad link

x91 x4

−3u , 0y1 , 2y4 , 1y2 , 4y3 , 1y0

no link

x11 1 x2

−3u , 0y1 , 1y3 , 2y4 , 1y0 , 4y2

bad link

x13 1 x0

−3u , 0y1 , 1y3 , 2y4 , 1y2 , 4y0

no link

x72 x1

−5u , 0y2 , 2y4 , 3y3 , 1y0 , 4y1

bad link

or

Table 3: Exclusion of links in families № 100, 101, 102, 103. is invented, this requires knowing the classification of extremal extractions from singular points that occur on Z, which in some cases here are not known. Acknowledgement This project was initiated during a Research-in-Teams programme at the Erwin Schr¨ odinger Institute for Mathematics and Physics during August 2018. The authors would like to express their gratitude to the ESI and its staff for their support and hospitality. The first author has been supported by EPSRC (grant EP/T015896/1), and the third author has been supported by IBS-R003-D1, Institute for Basic Science in Korea. We are grateful to the anonymous referees for their useful comments.

References [1] Hamid Ahmadinezhad, On pliability of del Pezzo fibrations and Cox rings, J. Reine Angew. Math. 723 (2017), 101–125. [2] Hamid Ahmadinezhad and Takuzo Okada, Birationally rigid Pfaffian Fano 3-folds, Algebr. Geom. 5 (2018), no. 2, 160–199. [3] Hamid Ahmadinezhad and Francesco Zucconi, Mori dream spaces and birational rigidity of Fano 3-folds, Adv. Math. 292 (2016), 410–445. [4] Selma Altınok, Gavin Brown, and Miles Reid, Fano 3-folds, K3 surfaces and graded rings, Topology and geometry: commemorating SISTAG, Contemp. Math., vol. 314, Amer. Math. Soc., Providence, RI, 2002, pp. 25–53. [5] Caucher Birkar, Singularities of linear systems and boundedness of Fano varieties, arXiv:1609.05543 (2016), 33 pages. [6] Gavin Brown and Kaori Suzuki, Fano 3-folds with divisible anticanonical class, Manuscripta Math. 123 (2007), no. 1, 37–51. [7] Ivan Cheltsov and Jihun Park, Birationally rigid Fano threefold hypersurfaces, Mem. Amer. Math. Soc. 246 (2017), no. 1167, v+117.

On Geometry of Fano Threefold Hypersurfaces

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[8] Herbert Clemens and Phillip Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356. [9] Alessio Corti, Singularities of linear systems and 3-fold birational geometry, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 259–312. [10] Alessio Corti, Aleksandr Pukhlikov, and Miles Reid, Fano 3-fold hypersurfaces, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 175–258. [11] Mikhail Grinenko, On the double cone over the Veronese surface, Izv. Ross. Akad. Nauk Ser. Mat. 67 (2003), 5–22. [12]

, Mori structures on a Fano threefold of index 2 and degree 1, Tr. Mat. Inst. Steklova 246 (2004), 116–141.

[13] Yujiro Kawamata, Boundedness of Q-Fano threefolds, Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989), Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 439–445. [14]

, Divisorial contractions to 3-dimensional terminal quotient singularities, Higher-dimensional complex varieties (Trento, 1994), de Gruyter, Berlin, 1996, pp. 241–246. MR 1463182

[15] Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. [16] Takuzo Okada, Stable rationality of orbifold Fano threefold hypersurfaces, J. Algebraic Geom. 28 (2019), 99–138. [17] Yuri Prokhorov, Rationality of Q-Fano threefolds of large Fano index, arXiv: 1903.07105 (2019), 21 pages. [18] Aleksandr Pukhlikov, Essentials of the method of maximal singularities, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 73–100. [19] Claire Voisin, Unirational threefolds with no universal codimension 2 cycle, Invent. Math. 201 (2015), no. 1, 207–237.

Hamid Abban Department of Mathematical Sciences Loughborough University Loughborough LE11 3TU, UK e-mail: [email protected] Ivan Cheltsov School of Mathematics The University of Edinburgh Edinburgh, UK and

14

H. Abban, I. Cheltsov and J. Park

Laboratory of Algebraic Geometry and its applications Higher School of Economics Moscow, Russia e-mail: [email protected] Jihun Park Centre for Geometry and Physics Institute for Basic Science 77 Cheongam-ro, Nam-gu, Pohang Gyeongbuk, 37673, Korea and Department of Mathematics, POSTECH 77 Cheongam-ro, Nam-gu, Pohang Gyeongbuk, 37673, Korea e-mail: [email protected]

On the Image of the Second l-adic Bloch Map Jeffrey D. Achter, Sebastian Casalaina-Martin and Charles Vial Abstract. For a smooth projective geometrically uniruled threefold defined over a perfect field we show that there exists a canonical abelian variety over the field, namely the second algebraic representative, whose rational Tate modules model canonically the third l-adic cohomology groups of the variety for all primes l. In addition, there exists a rational correspondence inducing these identifications. In the case of a geometrically rationally chain connected variety, one obtains canonical identifications between the integral Tate modules of the second algebraic representative and the third l-adic cohomology groups of the variety, and if the variety is a geometrically stably rational threefold, these identifications are induced by an integral correspondence. Our overall strategy consists in studying – for arbitrary smooth projective varieties – the image of the second l-adic Bloch map restricted to the Tate module of algebraically trivial cycle classes in terms of the “correspondence (co)niveau filtration”. This complements results with rational coefficients due to Suwa. In the appendix, we review the construction of the Bloch map and its basic properties. Mathematics Subject Classification (2010). 14C25, 14K30, 14E08, 14K15, 14G17, 14C15. Keywords. Algebraic cycles, Bloch map, coniveau filtration, intermediate Jacobian, regular homomorphism.

Introduction 0.1. Mazur’s question with Q -coefficients In the context of the generalized Hodge conjecture, it is natural to ask the following question [Voi14, Que. 2.43] : Let X be a smooth complex projective manifold, and let ν, i be natural numbers. Given a weight-i Hodge structure L ⊆ H i (X, Q) such that the Tate twist L(ν) is effective (i.e., Lp,q = 0 unless p, q ≥ ν), does there exist a complex projective manifold Y and an inclusion of Hodge structures L(ν) → H i−2ν (Y, Q) ? Essentially by definition, one can rephrase this question, via • the Hodge coniveau filtration NH H i (X, Q), as asking whether for a given ν, there © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Farkas et al. (eds.), Rationality of Varieties, Progress in Mathematics 342, https://doi.org/10.1007/978-3-030-75421-1_2

15

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J.D. Achter, S. Casalaina-Martin and C. Vial

ν exists a smooth projective manifold Y such that NH H i (X, Q(ν)) ⊆ H i−2ν (Y, Q). The generalized Hodge conjecture predicts that the Hodge coniveau filtration coincides with the so-called geometric coniveau filtration, N• H i (X, Q) ; in this sense one can rephrase the question in terms of the geometric coniveau filtration, and it is this version we will focus on in this paper. As a motivating example for this work, consider the case where i = 2n − 1 is odd and where ν = n. Setting Ja2n−1 (X) to be the algebraic intermediate Jacobian, i.e., the image of the Abel–Jacobi map AJ : An (X) → J 2n−1 (X) restricted to algebraically trivial cycles, it is well known that Nn−1 H 2n−1 (X, Q) = H 1 (Ja2n−1 (X), Q)  H 1 (Ja2n−1 (X), Q), answering the question in the case of the geometric coniveau filtration.

For smooth projective varieties over arbitrary fields, one can rephrase the Hodge theoretic question above by replacing Betti cohomology with -adic cohomology. Mazur [Maz14, Maz11] asked the following : Question 1 (Mazur’s question with Q -coefficients). Let X be a smooth projective variety over a field K with separable closure K. Given a natural number n, does there exist an abelian variety A/K such that for all primes  = char(K) there is an isomorphism of Galois modules 

V A −−−−→ Nn−1 H 2n−1 (XK , Q (n)) ?

(0.1)

In fact, one can pose an analogue of Mazur’s question for any Weil cohomology H(·). In positive characteristic p, much of our work also extends to the case of cohomology H(−, Qp ), which can be recovered as the F -invariants of the crystalline cohomology. In this introduction, where possible, we phrase statements uniformly in a prime l, although we remind the reader that, for example, dim H i (X, Qp ) is typically smaller than dim H i (X, Q ) ; we reserve  for primes distinct from the characteristic of the base field. From the motivic perspective, it is natural to ask that the isomorphism (0.1) be induced by a correspondence. Note that given the isomorphism (0.1), the Tate conjecture provides for each  a correspondence Γ ∈ CHn (A ×K X) ⊗ Q inducing the isomorphism for that . One might expect to find a correspondence Γ with integral coefficients, and that it be independent of l : Question 2 (Mazur’s motivic question with Ql -coefficients). Does there exist a correspondence Γ ∈ CHn (A ×K X) inducing for all primes l the above isomorphisms (0.1) ? Our first observation is that our results in [ACMV20] provide an affirmative answer to Questions 1 and 2 for any field K of characteristic zero, and for any n. Recall that in [ACMV20], it is shown that the algebraic intermediate Jacobian Ja2n−1 (XC ) attached to a smooth projective variety X defined over K ⊆ C admits 2n−1 over K in the sense that this model makes the Abel– a distinguished model Ja,X/K Jacobi map AJ : An (XC ) → Ja2n−1 (XC ) restricted to algebraically trivial cycles an Aut(C/K)-equivariant map.

On the Image of the Second l-adic Bloch Map

17

Theorem 3 ([ACMV20, Thm. 2.1]). Let X be a smooth projective variety over 2n−1 a field K ⊆ C. Given a natural number n, let Ja,X/K denote the distinguished model of the intermediate Jacobian J 2n−1 (XC ). Then there exists a correspondence 2n−1 ×K X) inducing for all primes  an inclusion of Galois modules Γ ∈ CHn (Ja,X/K 2n−1   Γ∗ : V Ja,X/K

/ H 2n−1 (X , Q (n)) K

with image Nn−1 H 2n−1 (XK , Q (n)). Thus we turn next to Questions 1 and 2 in the case where K has positive characteristic. As a first partial result, we establish in Proposition 6.1 a positive answer to Question 1 and 2 under the further assumptions that K is perfect, 2n − 1 ≤ dX := dim X, and H 2n−1 (XK , Ql (n)) has geometric coniveau n − 1 (this was established in [ACMV17, Thm. 2.1(d)] for K ⊆ C). While the condition that H 2n−1 (XK , Ql (n)) have geometric coniveau n − 1 is in general quite restrictive, it does not impose a condition for n = 1 or n = dX , and therefore Proposition 6.1 establishes an affirmative answer to Mazur’s Questions 1 and 2 for n = 1, dX and K perfect (see Remark 6.2 for the case n = dX ). Thus, moving forward, we will essentially be focusing on the case n = 2 in positive characteristic. 0.2. Mazur’s question with Q -coefficients in positive characteristic We focus now on Question 1 in positive characteristic, and set aside the issue of the correspondence in Question 2. We attempt to use algebraic representatives and the Bloch map as replacements for intermediate Jacobians and Abel–Jacobi maps. 2n−1 More precisely, recall that over an arbitrary field a replacement for Ja,X/K is the algebraic representative AbnX/K . If it exists (as in the case n = 1, 2, dX ), it comes with a Gal(K)-equivariant morphism φnX

K /K

: An (XK ) −→ AbnX/K (K),

where as in the case K = C we set

 An (XK ) := {α ∈ CHn (XK )  α is algebraically trivial}.

For n = 1, dX , the algebraic representative is the reduced Picard variety with the Abel–Jacobi map, and the Albanese variety with the Albanese map, respectively. For the case n = 2, the existence was proved for smooth projective varieties over an algebraically closed field in [Mur85]. This was extended to smooth projective varieties defined over a perfect field (e.g., a finite field) in [ACMV17], and to smooth projective varieties defined over any field in [ACMVb]. In characteristic 0, this agrees with the distinguished model of Ja3 (XC ) of [ACMV20]. We refer to §3.2 for more details. At the same time, recall that for a smooth projective variety X over a field K with separable closure K, and a prime l, Bloch [Blo79] in the case l = char(K) and later Gros–Suwa [GS88] in the case l = char(K) defined a map λn : CHn (XK )[l∞ ] −→ H 2n−1 (XK , Ql /Zl (n))

(0.2)

J.D. Achter, S. Casalaina-Martin and C. Vial

18

on l-primary torsion, extending the Abel–Jacobi map on homologically trivial ltorsion cycle classes (see (A.6)) in the case K = C. Suwa [Suw88] in the case l = char(K) and Gros–Suwa [GS88] in the case l = char(K) then defined an l-adic Bloch map Tl λn : Tl CHn (XK ) −→ H 2n−1 (XK , Zl (n))τ (0.3) n by taking the Tate module of the Bloch map λ . Here the subscript τ indicates the quotient by the torsion subgroup ; recall that a result of Gabber states that for a given X, the cohomology groups are torsion-free for all but finitely many l. Tensoring by − ⊗Zl Ql defines a map Vl λn : Vl CHn (XK ) −→ H 2n−1 (XK , Ql (n)). In the appendix, we give a more direct construction of the -adic Bloch map for primes  = char(K), following Bloch’s original construction, but taking an inverse limit rather than a direct limit. We show these two constructions agree, and that the -adic Bloch map agrees with the -adic Abel–Jacobi map in the case K = C, when one restricts to homologically trivial cycle classes. We then review in §A.4 a few properties of the l-adic Bloch map that we use in the body of the paper. It is well known that one can use these maps to model H 2n−1 (XK , Ql (n)) for n = 1, dX via the Picard and Albanese : in those cases, we have isomorphisms (see Propositions A.25 and A.26) (Vl φ1X

Vl (Pic0X/K red )

(Vl φ

Vl AlbX/K

K

/K

 dX X /K K



)−1

/ Vl A1 (X ) K

Vl CH1 (XK )

)−1

/ Vl AdX (X ) K

Vl CHdX (XK )

Vl λ1 

Vl λdX 

/ H 1 (X , Ql (1)) K

/ H 2dX −1 (X , Ql (dX )). K

Therefore, we focus here on the cases n = 1, dX , and in particular, the case n = 2. While in positive characteristic, the relationship between the Tate module of the algebraic representative and the coniveau filtration is not known in general, a result of Suwa relates the coniveau filtration to the image of the l-adic Bloch map (restricted to algebraically trivial cycles). Proposition 4 (Suwa [Suw88, Prop. 5.2]). Let X be a smooth projective variety over a perfect field K and let n be a natural number. For all prime numbers l, the image of the composition n  / Vl CHn (X ) Vl λ / H 2n−1 (X , Ql (n)) Vl An (XK )  (0.4) K K is equal to Nn−1 H 2n−1 (XK , Ql (n)). We refer to Proposition 2.1 for a more precise statement. As a minor technical point, we mention that Suwa proves this result for a slightly different coniveau filtration, which we call the correspondence coniveau filtration ; he shows this filtration agrees with the usual coniveau filtration in characteristic 0, and we extend this result to perfect fields in Proposition 1.1.

On the Image of the Second l-adic Bloch Map

19

Since Tl λ2 is an inclusion (see Proposition A.27), as an immediate corollary of our results above, and Suwa’s proposition, one obtains : Corollary 5 (Modeling Ql -cohomology). Let X be a smooth projective variety over a perfect field K and let l be a prime number. If Vl φ2X /K : Vl A2 (XK ) → Vl Ab2X/K K

is an isomorphism, e.g., if X is a geometrically uniruled threefold (Proposition 3.8(3)), then the composition Vl Ab2X/K

(Vl φ2X

K

/K

)−1

/ Vl A2 (X )   K



/ Vl CH2 (X )   K

Vl λ2

/ H 3 (X , Ql (2)) K (0.5)

is an inclusion of Gal(K)-modules with image N1 H 3 (XK , Ql (2)). The assumption that Vl φ2X

K /K

be an isomorphism is implied (see Lemma 3.6)

by the, possibly vacuous, assumption that φ2X /K : A2 (XK ) → Ab2X/K be an K isomorphism on l-primary torsion. It turns out that this assumption also plays a crucial role in our work [ACMVa], so we single it out : Definition 6 (Standard assumption at l). Let X be a smooth projective variety over a field K and let l be a prime number. We say that φ2X /K (or by abuse, X) K satisfies the standard assumption at the prime l if φ2X

K /K



[l∞ ] : A2 (XK )[l∞ ] −−−−→ Ab2X/K [l∞ ]

(0.6)

is an isomorphism. We say that φ2X /K (or by abuse X) satisfies the standard K assumption if it satisfies the standard assumption at l for all primes l. In Proposition 3.8, we give sufficient conditions for the standard assumption to be satisfied ; in particular, the standard assumption holds if char(K) = 0 [Mur85, Thm. 10.3] or if X is geometrically rationally connected with K perfect. We are unaware of an example of a smooth projective variety for which the standard assumption at a prime l fails. We mention here that both Proposition 4 and Corollary 5 hold with Ql /Zl coefficients so long as one replaces the coniveau filtration (1.1) with the correspondence niveau filtration (1.4), thus providing an answer to Mazur’s Question 1 with Ql /Zl -coefficients and with the geometric coniveau filtration N• replaced with the  •. correspondence niveau filtration N As a final note, we mention that in principle, the technique used to prove Corollary 5 would work for any n, assuming that there exists an algebraic representative in codimension-n, that Vl φnX /K is an isomorphism, and that Vl λn is an K inclusion ; however, unlike the case n = 1, 2, dX , in general, for n = 1, 2, dX , one does not expect these conditions to hold. Nevertheless, in the body of the paper, we explain the general case, and indicate where special assumptions are needed.

J.D. Achter, S. Casalaina-Martin and C. Vial

20

0.3. Mazur’s question with Z -coefficients Next we consider Mazur’s question in the case of Zl -coefficients (while acknowledging that, in positive characteristic p and with l = p, it might be more natural to seek an isomorphism of F -crystals than an isomorphism of cohomology groups H • (−, Zp )). To start with, as in the case of Ql -coefficients, it is well known that one can model H 2n−1 (XK , Zl (n))τ for n = 1, dX via the Picard and Albanese : in those cases, we have isomorphisms (see Propositions A.25 and A.26) (Tl φ1X /K )−1 K 0 Tl (PicX/K )red  / Tl A1 (XK ) (Tl φ

Tl AlbX/K

dX X /K K



Tl CH1 (XK )

)−1

/ Tl AdX (X ) K

Tl CHdX (XK )

Tl λ1  Tl λdX 

/ H 1 (X , Zl (1)) K

/ H 2dX −1 (X , Zl (dX ))τ K

Motivated by the discussion in §0.1 leading to Corollary 5, we proceed in a similar way, focusing on the case n = 2. The starting point is again to assume that φ2X /K [l∞ ] : A2 (XK )[l∞ ] → Ab2X/K [l∞ ] is an isomorphism ; i.e., we assume the soK

called standard assumption for n = 2 (Definition 6) ; e.g., we assume char(K) = 0 or X is geometrically rationally chain connected. By taking Tate modules one has (Lemma 3.6) that Tl φ2X

K /K

: Tl A2 (XK ) −→ Tl Ab2X/K

(0.7)

is an isomorphism as well. Consequently, one can consider the composition (Tl φ2X /K )−1   Tl λ2 K / Tl CH2 (X )  / Tl A2 (X )  / H 3 (X , Zl (2))τ . Tl Ab2X/K K K K  (0.8) That Tl λ2 is an inclusion is reviewed in Proposition A.27. In Proposition 2.1, we show that the image of the map Tl λ2 : Tl A2 (XK ) → H 3 (XK , Zl (2))τ contains  • is the correspondence niveau filtration defined  n−1 H 2n−1 (X , Zl (n))τ ; here N N K in (1.4). Combined with Suwa’s Proposition 4 (together with Proposition 1.1 com • with N• ), we find that the image contains N  n−1 H 2n−1 (X , Zl (n))τ as a paring N K finite index subgroup. With the expectation that the standard assumption should be true in general, we are thus led to ask : Question 7 (Mazur’s question with Zl -coefficients). Let X be a smooth projective variety over a field K. Does there exist an abelian variety A/K such that for almost all primes l there is an isomorphism of Galois modules   n−1 H 2n−1 (X , Zl (n))τ ? Tl A −−−−→ N K

(0.9)

The main technical result of this paper is Theorem 4.2, a particular instance of which takes the form below. Theorem 8 (Image of the l-adic Bloch map). Let X be a smooth projective variety over a perfect field K. Assume that φ2X /K [l∞ ] : A2 (XK )[l∞ ] → Ab2X/K [l∞ ] is K an isomorphism for all but finitely many primes l ; i.e., X satisfies the standard assumption at all but finitely many primes l (Definition 6). Then, for all but finitely

On the Image of the Second l-adic Bloch Map

21

many prime numbers l, the image of the composition 

Tl A2 (XK )

/ Tl CH2 (X )   K

Tl λ2

/ H 3 (X , Zl (2))τ K

 1 H 3 (X , Zl (2))τ . is equal to N K We obtain the following immediate corollary providing a partial answer to Question 7 : Corollary 9 (Modeling Zl -cohomology). Under the hypotheses of Theorem 8, the inclusion (0.8) induces an isomorphism of Galois modules   1 H 3 (X , Zl (2))τ Tl Ab2X/K −−−−→ N K

for all but finitely many prime numbers l. Theorem 8 is proved in §4. In fact, we can control the primes for which Theorem 8 and Corollary 9 might fail ; this is related to miniversal cycle classes, as well as decomposition of the diagonal (see Theorem 4.2). Moreover, in a way we make precise in Lemma 4.3 and Proposition 4.4, to prove Theorem 8 and Corollary 9, if one knows the standard assumption holds for all varieties over finite fields, then one does not need to assume the standard assumption for X. We note again that in principle, the technique used to prove the Corollary 9 would work for any n, assuming that there exists an algebraic representative in codimension n, that φnX /K [l∞ ] is an isomorphism, and that Tl λn is an inclusion ; K but in general, for n = 1, 2, dX , one does not expect these conditions to hold. Nevertheless, in the body of the paper, we explain the general case, and indicate where special assumptions are needed. 0.4. Universal cycles and the image of the second l-adic Bloch map Still under the standard assumption that φ2X

K /K



[l∞ ] : A2 (XK )[l∞ ] −−−−→ Ab2X/K [l∞ ]

is an isomorphism for all primes l, we show that a sufficient condition for the  1 H 3 (X , Zl (2))τ for all l is provided composition (0.8) to have image equal to N K by the existence of a so-called universal cycle for φ2X /K ; see §3.3 for a definition. K As before, we start by determining the image of the second l-adic Bloch map under such conditions : Theorem 10 (Universal cycles and the image of the second l-adic Bloch map). Let X be a smooth projective variety over a perfect field K. Assume that X satisfies the standard assumption for all primes l. If φ2X /K : A2 (XK ) → Ab2X/K (K) admits a K universal cycle, then for all prime numbers l the image of the composition Tl A2 (XK ) 



 1 H 3 (X , Zl (2))τ . is equal to N K

/ Tl CH2 (X )  K



Tl λ2

/ H 3 (X , Zl (2))τ K

J.D. Achter, S. Casalaina-Martin and C. Vial

22

Theorem 10 is a particular instance of our main Theorem 4.2. As an immediate consequence, we obtain : Corollary 11 (Universal cycles and modeling Zl -cohomology). Under the hypotheses of Theorem 10, the inclusion (0.8) induces for all prime numbers l an isomorphism of Galois modules  2  1 H 3 (X , Zl (2))τ . Tl AbX/K −−−−→ N K

Due to the connection with universal cycles and decomposition of the diagonal (see Proposition 3.10), this is connected with the notion of rationality, as we discuss in the next section. 0.5. Decomposition of the diagonal and the image of the second l-adic Bloch map We next turn our focus to the case of smooth projective varieties X over a perfect field K with CH0 (XK ) ⊗ Q universally supported in dimension 2 (see Definition 3.1). It is well known, via a decomposition of the diagonal argument [BS83], that in this case we have N1 H 3 (XK , Ql (2)) = H 3 (XK , Ql (2)), but also that Vl φ2X /K is an isomorphism (Proposition 3.8(3)), for all primes l. As a consequence, K

we see that in this case (0.5) induces an isomorphism Vl Ab2X/K  H 3 (XK , Ql (2)). We show the following result for cohomology with Zl -coefficients : Theorem 12 (Decomposition of the diagonal and the image of the second l-adic Bloch map). Let X be a smooth projective variety over a perfect field K of characteristic exponent p.   1. Assume CH0 (XK ) ⊗ Z N1 is universally supported in dimension 2 for some N > 0, e.g., X is a geometrically uniruled threefold. Then, for all primes  1 H 3 (X , Z (2)) ⊆ H 3 (X , Z (2)) is an equality and   N p, the inclusion N K K the second -adic Bloch map restricted to algebraically trivial cycles T A2 (XK )



/ T CH2 (X )   K

T λ2

/ H 3 (X , Z (2))τ K

is an isomorphism of Gal(K)-modules.   2. Assume CH0 (XK ) ⊗ Z N1 is universally supported in dimension 1 for some N > 0, e.g., X is a geometrically rationally chain connected. Then, for all primes l, the second l-adic Bloch map restricted to algebraically trivial cycles Tl A2 (XK ) 



/ Tl CH2 (X )   K

Tl λ2

/ H 3 (X , Zl (2))τ K

is an isomorphism of Gal(K)-modules. Moreover, for all primes   N p, H 3 (XK , Z (2)) is torsion-free. A slight generalization of Theorem 12 that deals with the prime p in case resolution of singularities holds over K in dimensions < dim X is proved in Proposition 5.2. See §3.1, and in particular Remark 3.4, for the classical link between stable rationality, rational connectedness, and decomposition of the diagonal.

On the Image of the Second l-adic Bloch Map

23

Again, from Theorem 12 (combined with Proposition 3.8), we have the following corollary. Corollary 13 (Decomposition of the diagonal and modeling Z -cohomology). Let X be a smooth projective variety over a perfect field K of characteristic exponent p.   1. Assume CH0 (XK ) ⊗ Z N1 is universally supported in dimension 2 for some N > 0, e.g., X is a geometrically uniruled threefold. Then, for all primes   N p, T φ2X /K : T A2 (XK ) −→ T Ab2X/K is an isomorphism and the K

canonical inclusion (0.8) induces an isomorphism of Galois modules 

T Ab2X/K −−−−→ H 3 (XK , Z (2))τ . 2. Assume CH0 (XK ) ⊗ Q is universally supported in dimension 1, e.g., X is a geometrically rationally chain connected. Then, for all primes l, Tl φ2X /K : K

Tl A2 (XK ) −→ Tl Ab2X/K is an isomorphism and the canonical inclusion (0.8) induces an isomorphism of Galois modules 

Tl Ab2X/K −−−−→ H 3 (XK , Zl (2))τ . 0.6. Stably rational vs. geometrically stably rational varieties over finite fields We now turn to the motivic question : Question 14 (Mazur’s motivic question with Z -coefficients). For which smooth projective varieties X over a field K do there exist an abelian variety A/K and a correspondence Γ ∈ CH2 (A ×K X) such that for all primes  = char(K)   1 H 3 (X , Z (2))τ Γ∗ : T A −−−−→ N K

is an isomorphism of Gal(K)-modules ? In other words, we turn now to the issue of addressing the existence of a correspondence Γ ∈ CH2 (Ab2X/K ×K X) inducing the isomorphisms (0.9). As already mentioned, it is easy to establish a positive answer under the further assumption that 2n − 1 ≤ dim X and H 2n−1 (XK , Q (n)) has geometric coniveau n − 1 ; see Proposition 6.1. In case X is a smooth projective geometrically uniruled threefold, then Proposition 6.3 establishes more precisely the existence of a correspondence Γ ∈ CH2 (Ab2X/K ×K X) ⊗ Q such that the induced morphism of Galois modules 

Γ∗ : Vl Ab2X/K −→ H 3 (XK , Ql (2)) coincides with the canonical map (0.5) and is an isomorphism for all primes l. On the other hand, due to the failure of the integral Tate conjecture over finite fields [Ant16, Kam15, PY15], an isomorphism as in (0.9) might not be induced by some correspondence Γ ∈ CH2 (Ab2X/K ×K X). However, using the -adic Bloch map, we provide a positive answer for the third -adic cohomology groups of smooth projective stably rational varieties over finite or algebraically closed fields, thereby addressing Question 14 :

24

J.D. Achter, S. Casalaina-Martin and C. Vial

Theorem 15 (Modeling Z -cohomology via correspondences). Let X be a smooth projective stably rational variety over a field K that is either finite or algebraically closed. Then there exists a correspondence Γ ∈ CH2 (Ab2X/K ×K X) inducing for all primes  = char K the isomorphisms (0.8) 

Γ∗ : T Ab2X/K −−−−→ H 3 (XK , Z (2))

(0.10)

of Gal(K)-modules. Moreover, if char(K) = 0, the correspondence Γ induces an isomorphism  (0.11) Γ∗ : H1 (J 3 (XC ), Z) −−−−→ H 3 (XC , Z(2)). The proof of Theorem 15 is given in Theorem 6.4, via a decomposition of the diagonal argument. There we also explain how the conclusion of Theorem 15 holds at l = p in case dim X ≤ 4, due to the existence resolution of singularities in dimensions ≤ 3. There are two reasons for restricting to algebraically closed fields or finite fields in Theorem 15. First, in order to use alterations, we restrict to the case of perfect fields. Second, in order to obtain the existence of the universal line-bundle, we use that K is finite or separably closed (see [ACMVb, §7.1.2]). 0.7. Notation and conventions a For a field K, we will denote by K a separable closure, and by K an algebraic closure. If X is a scheme of finite type over a field K, we denote by CHi (X) its Chow group of codimension-i cycle classes, and by Ai (X) ⊂ CHi (X) the subgroup consisting of algebraically trivial cycle classes (see [Ful98, §10.3]). If X is puredimensional, we denote dX its dimension. In case X is smooth over K, we still denote dX its dimension, which should be thought of as a locally constant function on X. A variety over K is a separated geometrically reduced scheme of finite type over K. The symbol l is allowed to denote an arbitrary prime, whereas  is always assumed invertible in the base field K. The phrase “for almost all” means “for all but finitely many”. Let M be an abelian group, let l be a prime, and let ν be an integer. We denote : Mtors := TorZ1 (Q/Z, M ) = the torsion subgroup of M ; Mcotors := the quotient of M by its largest divisible subgroup ; Mτ := M/Mtors = the quotient of M by its torsion subgroup. M [lν ] := TorZ1 (Z/lν Z, M ) = the lν -torsion subgroup of M ; M [l∞ ] := limν M [lν ] = the l-primary torsion subgroup of M ; −→ Tl M := limν M [lν ] = HomZ (Ql /Zl , M ) = the Tate module of M ; ←− Vl M := Tl M ⊗Zl Ql . In the definition of Tl M , the transition maps are given by the multiplication ·l by l morphisms M [lν+1 ] → M [lν ] and the equality limν M [lν ] = HomZ (Ql /Zl , M ) ←− can be found, e.g., in [Mil06, Prop. 0.19]. Note that Tl M = Tl (M [l∞ ]). Note also that for a Zl -module M , we have Mtors = TorZ1 l (Ql /Zl , M ).

On the Image of the Second l-adic Bloch Map

25

the l-adic valuation by vl , so that for a natural number r, we have Wevdenote l (r) l . l For a smooth projective variety X over an algebraically closed field K = a K = K, we denote by Hi (XK , Z ) the -adic homology. This will be primarily an indexing convention that is useful from the motivic perspective, since in the case where X is smooth and projective of pure dimension dX , the cap product with the fundamental class of X induces for all i (e.g., [Lau76, p.173]) an isomorphism  − ∩ [X] : H 2dX −i (XK , Z (dX )) → Hi (XK , Z ). r=

1. On various notions of coniveau filtrations Given a smooth projective variety X over a field K, one obtains coniveau filtrations on cohomology with various coefficients. In (1.1), (1.3) and (1.4) below we recall the definitions of the (classical) geometric coniveau filtration N• , of the (less classical) correspondence coniveau filtration N• and of the (still less classical)  •. correspondence niveau filtration N Although the filtrations might not agree in general, in Proposition 1.1 below we recall that they are related by  • ⊆ N• ⊆ N• , N where, over a perfect field K and with Q -coefficients, the second inclusion is an equality while the first is conjecturally an equality. In this section, the ring of coefficients Λ denotes either Z, Q, Z/r Z, Z , Q , or Q /Z . Cohomology groups H • (−, Λ) are computed in the corresponding topology : e.g., H • (−, Z) is computed in the analytic topology, while H • (−, Z/r Z) is computed in the ´etale topology. 1.1. Recalling the geometric coniveau filtrations Let X be a smooth projective variety over a field K with separable closure K. We recall the νth piece of the geometric coniveau filtration

 Nν H i (XK , Λ) := im HZi (XK , Λ) → H i (XK , Λ) (1.1) Z⊂XK



=



ker H i (XK , Λ) → H i (XK  Z, Λ)

(1.2)

Z⊂XK

where the sum runs through all Zariski closed subsets Z of XK of codimension ≥ ν. (The equivalence of (1.1) and (1.2) comes from the long exact sequence of a pair.) For our purpose, we will have to work with the following variant of the coniveau filtration. We recall the νth piece of the correspondence coniveau filtration

 ν N H i (XK , Λ) := im Γ∗ : H i−2ν (Z, Λ(−ν)) → H i (XK , Λ) , (1.3) Γ:ZXK

26

J.D. Achter, S. Casalaina-Martin and C. Vial

where the sum is over all smooth projective varieties Z over K and all correspondences Γ ∈ CHdX −ν (Z ×K XK ). Let us also introduce a related filtration, which was considered in [FM94] and in [Via13]. The νth piece of the correspondence niveau filtration is defined as

  ν H i (X , Λ) := im Γ∗ : Hi−2ν (Z, Λ(ν − i)) → H i (XK , Λ) , (1.4) N K Γ:ZXK

where the sum is over all smooth projective varieties Z over K and all correspondences Γ ∈ CHdX −ν (Z ×K XK ). As we will see in the proof of Proposition 1.1 below, one may restrict the sum in (1.4) to smooth projective varieties Z over K of pure dimension i − 2ν. Here, as outlined in the Notation and Conventions §0.7, we use homology as a convenience ; for Z of pure dimension dZ , we set Hi−2ν (Z, Λ(ν − i)) := H 2dZ −(i−2ν) (Z, Λ(dZ − ν − i)). By considering fields of definitions of Z (and Γ) that are finite Galois over K and by considering Galois orbits, we note that in the definitions of all three filtrations above, we could have restricted the sums to those Z (and Γ) defined over K. The above three filtrations admit the following already-known containment relation : Proposition 1.1. Let X be a smooth projective variety over a field K. Suppose that Λ is one of Z/r Z, Z , Q , or Q /Z or that K = C and Λ is one of Z, Q or Q/Z. Then there are natural inclusions  ν H i (X , Λ) ⊆ N ν H i (X , Λ) ⊆ Nν H i (X , Λ). N K

K

K

In case (ν, i) = (n, 2n) or (n−1, 2n−1), the first inclusion is an equality. Moreover, assuming Λ is either Q or Q, if K is perfect then the second inclusion is an equality, and if Grothendieck’s Lefschetz standard conjecture holds then the first inclusion is an equality.  ν H i (X , Λ) ⊆ Nν H i (X , Λ) is explained in [Via13, Proof. The containment N K K §1.1] in the case K ⊆ C and Λ = Q. The same argument applies here and we spell it out for the sake of completeness. First, note that up to replacing Z with Z ×K PnK , we can assume dim Z ≥ i − 2ν. Second, if ι : Y → Z is a smooth linear intersection of Z of dimension i − 2ν, then the push-forward ι∗ : Hi−2ν (Y, Λ) → Hi−2ν (Z, Λ) is surjective by the Lefschetz hyperplane theorem (e.g., [SGA72, Exp. XIV, Cor. 3.3]), and the image of Γ∗ : Hi−2ν (Z, Λ(i − ν)) → H i (XK , Λ) coincides thus with the image of Γ∗ ◦ι∗ . Therefore, one may restrict the sum in (1.4) to those smooth projective varieties Z over K of pure dimension i−2ν. In particular, since Hi−2ν (Z, Λ) = H i−2ν (Z, Λ(i − 2ν)) when dim Z = i − 2ν, we get the asserted containment. As outlined in the argument in [Via13, Prop. 1.1], a sufficient condition for the first inclusion to be an equality is the following : if for all smooth projective varieties Z over K there exists a smooth projective variety Z  over K and a correspondence  L ∈ CHi−2ν (Z  ×K Z) inducing an isomorphism L∗ : Hi−2ν (Z  , Λ(i − ν)) −→

On the Image of the Second l-adic Bloch Map

27

H i−2ν (Z, Λ(ν)), then the image of Γ∗ : H i−2ν (Z, Λ(−ν)) → H i (XK , Λ) coincides  ν H i (X , Λ) ⊆ with the image of Γ∗ ◦ L∗ and it follows that the containment N K Nν H i (XK , Λ) is an equality. Now, in case (ν, i) = (n, 2n), the class of Z ×K Z  induces an isomorphism H0 (Z, Λ) → H 0 (Z, Λ), while in case (ν, i) = (n−1, 2n−1), using the identification of torsion line bundles and ´etale covers, the universal line bundle L on Pic0Z/K ×K Z induces natural identifications H1 (Pic0Z/K , Z ) = H 2g−1 (Pic0Z/K , Z (g)) = T Pic0Z/K = H 1 (Z, Z (1)). In case Λ = Q or Q , a correspondence L as above exists with Z  = Z for all Z and all (ν, i) provided Grothendieck’s standard conjecture holds. We now turn to the containment N H i (XK , Λ) ⊆ Nν H i (XK , Λ). Let Γ ∈ CHdX −ν (Z ×K XK ) be a correspondence with Z a smooth projective variety over K. By refined intersection, the image of Γ∗ is supported on the closed subscheme Z := pXK (Γ) of dimension ≤ dX − ν, where pXK : Γ → XK is the natural projection. In other words, the composition ν

H i−2ν (Z, Λ(−ν))

Γ∗

/ H i (X , Λ) K

/ H i (X  Z, Λ) K

(1.5)

vanishes, thereby giving the asserted containment. For the statement of equality when Λ denotes Q or Q and when K is perfect, since Jannsen [Jan94] only asserts this for fields of characteristic 0, as de Jong’s results were not available at the time, here we reproduce the argument of [Jan94, p. 265–266] to show how the argument can be extended to fields of positive characteristic. Consider a closed embedding ι : Z → XK with dim Z = dX − ν and use the theory of alterations to produce a diagram  ι π / / Z /X f : Z K with Z  smooth of pure dimension dX − ν. We get a commutative diagram (using -adic homology) f∗





H2dX −i (Z  , Q (dX ))

* / H i (X , Q ) K

/ H i (X , Q ) Z K

H i−2ν (Z  , Q (−ν))

π∗

 / H2dX −i (Z, Q (dX ))



ι∗

 / H2dX −i (X , Q (dX )) K

and we then argue using weights (see [Jan90, §6]). To utilize weights, we must assume that K is finitely generated over its prime field ; however, since all the varieties are of finite type, they are all defined over a base field K  ⊆ K that is finitely generated over its prime field, and we may work over K  , and then base change to K. In other words, we may assume that K is finitely generated over its prime field and that X, Z, Γ and π are defined over K. Now, since H i (XK , Q ) is pure of weight i, the image of ι∗ equals the image of Wi H2dX −i (ZK , Q (dX )). On

28

J.D. Achter, S. Casalaina-Martin and C. Vial

 the other hand, it is shown in [Jan90, Rem. 7.7] that H2dX −i (ZK , Q (dX )) surjects  onto this space via π∗ .

1.2. p-adic coniveau filtrations If char(K) = p > 0 and if K is perfect, we also allow Λ to variously denote Wn (K), W(K) or K(K), in which case H • (−, Λ) denotes a crystalline cohomology group. (See §A.1.2 for our notations concerning p-adic cohomology theories in characteristic p.) One defines Nν H i (XK , Λ) and

Nν H i (XK , Λ)

exactly as in (1.1) (note that in this setting (1.2) is not well behaved) and (1.3), respectively. Since we will only need homology for smooth projective varieties X, we simply define Hi (X, Λ) = H 2dX −i (X, Λ(dX )). With this notation we then define  exactly as in (1.4). the correspondence niveau filtration N In contrast to the crystalline cohomology groups, the groups H i (XK , Qp ) no longer have a useful theory of weights or Poincar´e duality. Since H i (XK , Zp ) can be recovered as (H i (X/W))F , the suitably-defined F -invariants of H i (X/W)  N, N by [Gro85, §I.3], we simply define the p-adic coniveau filtrations N = N, N ν H i (XK , Zp ) = (N ν H i (X/W))F N ν H i (XK , Zp /pr ) = (N ν H i (X/Wr ))F N ν H i (XK , Qp /Zp ) = lim N ν H i (XK , Zp /pr ). −→ Then Proposition 1.1 holds in this context, too : Proposition 1.1(bis). Let X be a smooth projective variety over a perfect field K. Suppose that Λ is one of Z/lr Z, Zl , Ql , or Ql /Zl , or that char(K) = p > 0 and Λ is one of Wr (K), W(K) or K(K), or that K = C and Λ is one of Z, Q or Q/Z. Then there are natural inclusions  ν H i (X , Λ) ⊆ N ν H i (X , Λ) ⊆ Nν H i (X , Λ). N K K K In case (ν, i) = (n, 2n) or (n − 1, 2n − 1), the first inclusion is an equality. If Λ is a field of characteristic zero, then the second inclusion is an equality, and if in addition Grothendieck’s Lefschetz standard conjecture holds, then the first inclusion is an equality. Proof. It only remains to prove the assertions for the various p-adic coefficients in characteristic p > 0. If Λ = Wn (K), W(K) or K(K), the argument is identical ; the key point in the case Λ = K(K) is that like ´etale cohomology, rigid cohomology has a good theory of weights and Poincar´e duality. (The second map in (1.5) simply needs to be replaced with H i (X, Λ) → H i (X, Λ)/HZi (X, Λ), which makes sense in crystalline cohomology.) The results for Λ = Zp /pn , Zp , Qp or Qp /Zp follow by taking F -invariants. 

On the Image of the Second l-adic Bloch Map

29

2. The image of the l-adic Bloch map and the coniveau filtration Again, for brevity, in this section, the ring of coefficients Λ denotes either Zl , Ql , or Ql /Zl . The subscript τΛ on H • (−, Λ)τΛ indicates that when Λ = Zl , we take the quotient by the torsion subgroup. For each of these Λ and for M = CHn (XK ) or An (XK ), we have the corresponding groups MΛ : MZl := Tl M , MQl := Vl M and MQl /Zl := M [l∞ ]. We now consider the l-adic Bloch map constructed by Suwa [Suw88] in the case l = char(K) and by Gros–Suwa [GS88] in the case l = char(K) ; see the appendix for a review of these maps, especially §A.3.3 and §A.3.5. Our main result in this section extends a result of Suwa [Suw88, Prop. 5.2], originally stated for Λ = Q with  = char(K), and gives a preliminary description of the image of the l-adic Bloch map restricted to the Tate module of algebraically trivial cycles in terms of the correspondence coniveau filtration (1.3) : Proposition 2.1 ([Suw88, Prop. 5.2]). Let X be a smooth projective variety over a perfect field K, and let l be a prime number. The image of the composition An (XK )Λ



/ CHn (X )Λ K

λn

/ H 2n−1 (X , Λ(n))τΛ K

(2.1)

n−1

 H 2n−1 (XK , Λ(n))τΛ , with equality if Λ = Ql /Zl or Ql . Recall that contains N the subscript τΛ indicates that when Λ = Zl , we take the quotient by the torsion subgroup. Remark 2.2. Note that in the case where Λ = Ql /Zl , this implies that  n−1 H 2n−1 (X , Ql /Zl (n)) ⊆ H 2n−1 (X , Zl (n)) ⊗Z Ql /Zl N l K K ⊆ H 2n−1 (XK , Ql /Zl (n)) ; see §A.5. We split the proof of Proposition 2.1 in two, depending on whether the prime l is invertible in K. Key to the proof is the following lemma of Suwa [Suw88, Lem. 3.2] originally stated for primes l = char(K) ; see Remark 2.5 below, where we verify that Suwa’s argument works even when l = char(K). Lemma 2.3 ([Suw88, Lem. 3.2]). Let C be a smooth projective irreducible curve (resp. abelian variety) over a field K and let Γ ∈ CHn (C ×K X). Let α ∈ An (XK )[lν ], and assume there exists β ∈ A0 (CK ) such that Γ∗ β = α. Then there exists γ ∈ A0 (CK )[lμ ] for some μ ≥ ν such that λn (Γ∗ γ) = λn (α).  Example 2.4. In Lemma 2.3, it may be that one must take μ > ν. For instance, take X = C = E to be an elliptic curve over K, let Γ = Γf be the graph of the multiplication by  map f : C → X, and let α ∈ A1 (XK )[]. Then there does not exist γ ∈ A1 (CK )[] such that λ1 (Γ∗ γ) = λ1 (α).

J.D. Achter, S. Casalaina-Martin and C. Vial

30

2.1. The image of the -adic Bloch map Here we review the argument of Suwa [Suw88] yielding the proof of Proposition 2.1 in the case Λ = Q and extend it to other rings of coefficients. Proof of Proposition 2.1, prime-to-char(K). Suppose l =  = char(K). Consider the diagram    λ0 = / A0 (Z)Λ CH0 (Z)Λ  / H1 (Z, Λ)τΛ (2.2) Γ:ZXK

Γ:ZXK Γ∗

Γ:ZXK Γ∗

  An (XK )Λ

 / CHn (X )Λ K

λn



Γ∗

/ H 2n−1 (X , Λ(n))τΛ K

where the direct sums are over all smooth projective varieties Z over K and all correspondences Γ ∈ CHn (Z ×K XK ). The image of the right vertical arrow is by  n−1 H 2n−1 (X , Λ(n))τΛ . We conclude from commutativity of the diadefinition N K

 n−1 H 2n−1 (X , Λ(n)). gram that the image of the bottom row of (2.2) contains N K We now show that the inclusion is an equality in case Λ = Q /Z . By Lemma 2.3, for any α ∈ An (XK )[∞ ], there exists an element  γ∈ A0 (ZK )[∞ ] Γ:ZX

such that λn (Γ∗ γ) = λn (α). It readily follows from a diagram chase that the image  n−1 H 2n−1 (X , Q /Z (n)). of the bottom row of (2.2) is contained in N K Finally we show equality in the case Λ = Q . Taking Tate modules in (2.2) with Q /Z -coefficients, and using the previous case, we have that the image of  n−1 H 2n−1 (X, Q /Z (n)). Since the bottom row of (2.2) with Z -coefficients is T N H 1 (Z, Q /Z (1)) is a divisible group for any smooth projective variety Z over K, and since an increasing chain of divisible abelian -torsion subgroups of a finite corank abelian -torsion group is stationary [Suw88, Lem. 1.2], then by adding connected components to Z, one can conclude there exist a smooth projective curve Z over K and a correspondence Γ ∈ CHn (Z ×K XK ) such that

 n−1 H 2n−1 (X ,Z (n)) = im Γ∗ : H 1 (Z,Z (1)) → H 2n−1 (X ,Z (n)) N K K

 n−1 H 2n−1 (X ,Q /Z (n)) = im Γ∗ : H 1 (Z,Q /Z (1)) → H 2n−1 (X ,Q /Z (n)) . N K K Then the image of the bottom row of (2.2) with Z -coefficients is  n−1 H 2n−1 (X, Q /Z (n)) = T Γ∗ H1 (Z, Q /Z ) T N ⊇ Γ∗ T H1 (Z, Q /Z ) = Γ∗ H1 (Z, Z )  n−1 H 2n−1 (X , Z (n))τ . =N K From Lemma A.16, the inclusion above has torsion cokernel. This gives the assertion with Q -coefficients. 

On the Image of the Second l-adic Bloch Map

31

2.2. The image of the p-adic Bloch map Now let K be a perfect field of characteristic p > 0. Here we secure Proposition 2.1 for p-adic cohomology groups. Remark 2.5. Lemma 2.3 ([Suw88, Lem. 3.2]) holds as well for p-power torsion. The proof is essentially identical. Briefly, suppose K is finite ; then A1 (CK ) = Pic0C/K (K) is a torsion group. Write the order of β as pm M with M relatively prime to p, and choose N with N M ≡ 1 mod pν . Then γ := N M β has order pm , and Γ∗ γ = N M α = α. To account for arbitrary K, we may assume that K is the perfection of a field K0 which is finitely generated over Fp , and then spread out the data X, Γ, C, α and β to an irreducible scheme S of finite type over Fp , with function field K0 ; let k be the algebraic closure of Fp in K0 . Consider the finite flat group scheme G := Pic0C/S [pν ] → S, and let Q ∈ S be a closed point. Because the residue field of Q is finite, possibly after base-change by a finite extension of k, there exists some γQ ∈ GQ (k) = A1 (CQ ) such that ΓQ,∗ γQ = αQ . Possibly after replacing S with an open neighborhood of Q, there exists a finite surjective T → S such that GT → T admits a section γ ∈ G(T ) passing through some pre-image of γQ in GQ ×S T [Gro66, 14.5.10] ; the generic fiber of γ is the sought-for class. Proof of Proposition 2.1, char(K)-torsion. The assertion of the inclusion in Proposition 2.1 with Λ = Qp /Zp (for arbitrary n) follows immediately from diagram (2.2). For the opposite inclusion, the argument is identical. For the inclusion of Proposition 2.1 with Zp -coefficients, one uses diagram (2.2) with Λ = W, and the fact that taking F -invariants commutes with push-forward [Gro85, Cor. I.3.2.7]. Tensoring with Qp gives the inclusion with Qp -coefficients. One then argues identically that the inclusion with Zp -coefficients has torsion quotient. Indeed, taking Tate modules of (2.2) with Qp /Zp -coefficients, one sees that the image of the  n−1 H 2n−1 (X , Qp /Zp (n)). As bebottom row of (2.2) with Zp -coefficients is N K fore, since H 1 (Z, Qp /Zp (1)) is a divisible group for any smooth projective variety Z over K, and since an increasing chain of divisible abelian p-torsion subgroups of a finite corank abelian p-torsion group is stationary [Suw88, Lem. 1.2], then by adding connected components to Z, one can conclude there is a smooth surface Z/K, possibly disconnected but of finite type, equipped with a morphism  n−1 H 2n−1 (X , Qp /Zp (n)) and f : Z → X such that f∗ H 1 (Z, Qp /Zp (1)) = N K

n−1

 f∗ H 1 (Z/W(1)) = N n−1

 Tp N

K

H 2n−1 (XK /W(n)). We then have

H 2n−1 (XK , Qp /Zp (n)) = Tp f∗ H 1 (Z, Qp /Zp (1)) ⊇ f∗ Tp H 1 (Z, Qp /Zp (1))

= f∗ H 1 (Z, Zp (1)) = f∗ (H 1 (Z/W(1))F ) = (f∗ H 1 (Z/W(1)))F by [Gro85, Cor. I.3.2.7]  n−1 H 2n−1 (X /W(n)))F = N  n−1 H 2n−1 (X , Zp (n))τ . = (N K K Since the inclusion has torsion cokernel (see Lemma A.16 and Remark A.20), we are done. 

32

J.D. Achter, S. Casalaina-Martin and C. Vial

3. Decomposition of the diagonal, algebraic representatives, and miniversal cycles In [ACMVa] we consider decomposition of the diagonal and algebraic representatives in detail. These ideas also come into play in the proofs of Theorems 8, 12 and 15, and so we review these notions in this section. We refer the reader to [ACMVa] for more details. 3.1. Decomposition of the diagonal The aim of this subsection is to fix the notation for decomposition of the diagonal, and to recall the existence of decompositions of the diagonal for various flavors of rational varieties. Let K be a field and let  be a prime not equal to char K. Let R be a commutative ring, let X be a smooth projective variety over a field K and let W1 and W2 be two closed subschemes of X not containing any component of X. A cycle class Z ∈ CHdX (X ×K X) ⊗Z R is said to admit a decomposition of type (W1 , W2 ) if Z = Z1 + Z2 ∈ CHdX (X ×K X) ⊗Z R, where Z1 ∈ CHdX (X ×K X)⊗Z R is supported on W1 ×K X and Z2 ∈ CHdX (X ×K X) ⊗Z R is supported on X ×K W2 . If Z = ΔX and if one can choose W2 with dim W2 = 0, we simply say that ΔX ∈ CHd (X ×K X) ⊗ R has a Chow decomposition. Definition 3.1 (Universal support of CH0 ⊗R). Let X be a smooth projective variety over a field K. We say that CH0 (X) ⊗ R is universally supported in dimension d if there exists a closed subscheme W2 ⊆ X of dimension ≤ d such that CH0 (X) ⊗ R is universally supported on W2 , i.e., if the push-forward map CH0 ((W2 )L ) ⊗Z R → CH0 (XL ) ⊗Z R is surjective for all field extensions L/K. The following proposition is classical and goes back to Bloch and Srinivas [BS83], and relates decomposition of the diagonal to the universal support of CH0 : Proposition 3.2 (Bloch–Srinivas [BS83]). The diagonal ΔX ∈ CHdX (X ×K X)⊗Z R of a smooth projective variety X over K admits a decomposition of type (W1 , W2 ) if and only if CH0 (X) ⊗Z R is universally supported on W2 . In particular, taking R = Z, the diagonal ΔX ∈ CHdX (X ×K X) of a smooth projective, stably rational, variety X admits an decomposition of type (W1 , P ) for any choice of K-point  P ∈ X(K). Remark 3.3. The existence of a K-point on a stably rational variety over K is ensured by the Lang–Nishimura theorem [Nis55] ; see [RY00, Prop. A.6] for a modern treatment. Remark 3.4 (Varieties admitting decompositions of the diagonal). Let X and Y be smooth projective varieties over K of respective dimension dY ≤ dX . If X and Y

On the Image of the Second l-adic Bloch Map

33

are stably rationally equivalent, i.e., if there exists a nonnegative integer n such that n+dX −dY n Y × K PK is birational to X ×K PK , then CH0 (X) is universally supported in dimension dY . In particular, if X is stably rational, then CH0 (X) is universally supported in dimension 0, and in fact on a point by the Lang–Nishimura theorem. Moreover, if X is only assumed to be geometrically rationally chain connected, then CH0 (X) ⊗Z Q is universally supported in dimension 0 (see e.g., [ACMVa, Rem. 2.8]). Notation 3.5 (Decomposition of the diagonal and alterations). Let X be a puredimensional smooth projective variety over a perfect field K of characteristic exponent p. Suppose we have a cycle class Z1 +Z2 in CHdX (X ×K X) with Z1 supported on W1 ×K X and Z2 supported on X ×K W2 with W1 and W2 two closed subschemes of X not containing any component of X such that dim W1 ≤ n1 and 2 → W2 of 1 → W1 and W dim W2 ≤ n2 . By [Tem17], there exist alterations W   degree some power of p such that W1 and W2 are smooth projective over K. The cycle classes Z1 and Z2 , seen as self-correspondences on X, factor up to inverting 2 , respectively. Precisely, there exists a nonnegative integer e 1 and W p through W such that pe Z1 = r1 ◦ s1

in CHdX (X ×K X)

pe Z2 = r2 ◦ s2

and

1 ×K X), s2 ∈ CHdX (X ×K W 1 ), r1 ∈ CHdX (W 2 ) and for some s1 ∈ CHdX (X ×K W dX  2 with 1 and W r2 ∈ CH (W2 ×K X). Note that by replacing each component of W a product with projective space of an appropriate dimension, we may assume that 1 and W 2 are of pure dimension n1 and n2 , respectively. We refer to [ACMVa, W §3.1] for more details. Since resolution of singularities exists for threefolds over a perfect field [CP09], if each dim Wi ≤ 3, then we may take e = 0. 3.2. Surjective regular homomorphisms and algebraic representatives The aim of this subsection is to fix notation for algebraic representatives. We start by reviewing the definition of an algebraic representative (i.e., [Mur85, Def. 1.6.1] or [Sam60, 2.5]). Let X be a smooth projective variety over a perfect field K and let n be a nonnegative integer. For a smooth separated scheme T of finite type over n K, we define AX/K (T ) to be the abelian group consisting of those cycle classes n Z ∈ CH (T ×K X) such that for every t ∈ T (K) the Gysin fiber Zt is algebraically trivial. For Z ∈ AXn /K (T ) denote by wZ : T (K) → An (XK ) the map defined by K

wZ (t) = Zt . Given an abelian variety A/K, a regular homomorphism (in codimension n) φ : An (XK )

/ A(K)

n (T ) the composition is a homomorphism of groups such that for every Z ∈ AX/K

T (K)

wZ

/ An (X ) K

φ

/ A(K)

34

J.D. Achter, S. Casalaina-Martin and C. Vial

is induced by a morphism of varieties ψZ : TK → A. An algebraic representative (in codimension n) is a regular homomorphism φnXK : An (XK ) −→ AbnX

K /K

(K)

that is initial among all regular homomorphisms. For n = 1, an algebraic representative is given by (Pic0X /K )red together with the Abel–Jacobi map. For n = dX , K an algebraic representative is given by the Albanese variety and the Albanese map. For n = 2, it is a result of Murre [Mur85, Thm. A] that there exists an algebraic representative for XK , which in the case K = C is the algebraic intermediate Jacobian Ja3 (X) ; i.e., the image of the Abel–Jacobi map restricted to algebraically trivial cycle classes. The main result of [ACMV17] (see also [ACMVb]) is that if there exists an algebraic representative φnXK : An (XK ) → AbnX /K (K), then AbnX /K adK

K

mits a canonical model over K, denoted AbnXK /K , such that φnXK is Gal(K/K)equivariant and such that for any Z ∈ AXnK /K (T ) the morphism ψZK : TK → AK descends to a morphism ψZ : T → A of K-schemes. In particular, the algebraic representative Ab2X /K of [Mur85] admits a canonical model over K, denoted K

Ab2XK /K . In the case K ⊆ C, the abelian variety Ab2X/K is the distinguished 3 model Ja,X/K of the algebraic intermediate Jacobian, as defined in [ACMV20]. We include the following lemma for clarity ; we also note that it is clear from the definitions that an algebraic representative φnX : An (XK ) → AbnX /K (K) is K K a surjective regular homomorphism. Lemma 3.6. Let φ : An (XK ) → A(K) be a surjective regular homomorphism. 1. Let l be prime. Then : (a) φ[l∞ ] : An (XK )[l∞ ] → A[l∞ ] is surjective. (b) The following are equivalent : (i) φ[l∞ ] : An (XK )[l∞ ] → A[l∞ ] is an isomorphism. (ii) φ[l∞ ] : An (XK )[l∞ ] → A[l∞ ] is an inclusion. (iii) φ[lν ] : An (XK )[lν ] → A[lν ] is an inclusion for all natural numbers ν. (iv) φ[l] : An (XK )[l] → A[l] is an inclusion. (c) If any of the equivalent conditions in (1)(b) hold, then Tl φ : Tl An (XK ) → Tl A is an isomorphism. 2. There exists a natural number e (independent of l) such that for all natural numbers ν the image of the map φ[lν+e ] : An (XK )[lν+e ] → A[lν+e ] contains A[lν ] ⊆ A[lν+e ]. 3. For all but finitely many primes l the map φ[lν ] : An (XK )[lν ] → A[lν ] is surjective. Proof. (1)(a) is [ACMV20, Rem. 3.3]. (Even though it is only claimed there for K of characteristic zero, the arguments of the references cited there are valid for arbitrary torsion in arbitrary characteristic.) The equivalence of (1)(b)(i) and

On the Image of the Second l-adic Bloch Map

35

(1)(b)(ii) is obvious. To show the equivalence of (1)(b)(ii) and (1)(b)(iii) we argue as follows. For a group G, there is an inclusion G[lν ] ⊆ G[lν+1 ], so that in our situation, we have An (XK )[lν ] ⊆ An (XK )[l∞ ] and A[lν ] ⊆ A[l∞ ]. The equivalence of (1)(b)(ii) and (1)(b)(iii) then follows by a diagram chase ; the equivalence of (1)(b)(iii) and (1)(b)(iv) is elementary. One obtains (1)(c) by applying the Tate module to the isomorphism (1)(b)(i). Item (3) follows from Item (2), which in turn is [ACMV20, Rem. 3.3].  Remark 3.7. It is worth noting that if there exists a regular homomorphism φ : An (XK ) → A(K) such that φ[∞ ] is injective, then there is an algebraic representative in codimension-n ; this follows directly from Saito’s criterion ([Mur85, Prop. 2.1]). 3.3. Miniversal cycles and miniversal cycles of minimal degree Let X be a smooth projective variety over a field K and let φ : An (XK ) → A(K) be a regular homomorphism. A miniversal cycle for φ is a cycle Z ∈ AXnK /K (A) such that the homomorphism ψZ : A → A is given by multiplication by r for some natural number r, which we call the degree of the cycle. A miniversal cycle is called universal if ψZ : A → A is given by the identity. In case φ is an algebraic representative for codimension-n cycles on X, we call a universal cycle for φ a universal cycle in codimension-n for X. Clearly there is a minimal r such that there exists a miniversal cycle of degree r ; taking linear combinations of miniversal cycles, one can see this minimum is achieved by the GCD of all of the degrees of miniversal cycles. If K is algebraically closed, it is a classical and crucial fact [Mur85, 1.6.2 & 1.6.3] that a miniversal cycle exists if and only if φ is surjective ; this also holds without any restrictions on the field K by [ACMVb, Lem. 4.7]. In particular, since an algebraic representative is always a surjective regular homomorphism [ACMVb, Prop. 5.1], it always admits a miniversal cycle. However, the existence of a universal cycle is restrictive. For example, both over the complex numbers [Voi15] and over a field of characteristic at least three [ACMVa], the standard desingularization of the very general double quartic solid with 7 nodes does not admit a universal cycle in codimension-2. 3.4. Decomposition of the diagonal and algebraic representatives We now recall a result due to [Mur85] and [BS83] : Proposition 3.8 (Murre [Mur85], Bloch–Srinivas [BS83]). Let X be a smooth projective variety over a perfect field K of characteristic exponent p. 1. If char(K) = 0, then φ2XK [∞ ] : A2 (XK )[∞ ] −→ Ab2X/K [∞ ](K). is an isomorphism of Gal(K)-modules for all prime numbers .

(3.1)

36

J.D. Achter, S. Casalaina-Martin and C. Vial

2. Assume that the diagonal ΔXK ∈ CHdX (XK ×K XK ) ⊗ Q admits a decomposition of type (W1 , W2 ) with dim W2 ≤ 1. Then φ2XK : A2 (XK ) −→ Ab2X/K (K) is an isomorphism of Gal(K)-modules. 3. Assume that the diagonal N ΔXK ∈ CHdX (XK ×K XK ) admits a decomposition of type (W1 , W2 ) with dim W2 ≤ 2 for some positive integer N . Then Vl φ2XK : Vl A2 (XK ) −→ Vl Ab2X/K is an isomorphism of Gal(K)-modules for all primes l, and T φ2XK : T A2 (XK ) −→ T Ab2X/K is an isomorphism of Gal(K)-modules for all prime numbers  not dividing N p. 4. In the setting of (3), further assume that p ≥ 2, resolution of singularities holds in dimensions < dX , and p  N . Then Tp φ2XK : Tp A2 (XK ) −→ Tp Ab2X/K is an isomorphism of Gal(K)-modules. Proof. First recall from [ACMV17] that φ2XK is Gal(K)-equivariant. Item (2) is [BS83, Thm. 1(i)], while Item (1) reduces via [ACMV17] to the case K = C, which is covered by [Mur85, Thm. 10.3]. We now prove Item (3) and assume that char(K) = p > 0. With Notation 3.5, we have a commutative diagram with composition of horizontal arrows being multiplication by N pe : A2 (XK ) φ2X

 Ab2X/K (K)

r1∗ ⊕r2∗

1 ) ⊕ A2 (W 2 ) / A1 (W 2 φ1W  ⊕φW  1

2

 / Pic0 (K) ⊕ Alb  (K) W2 W 1

∗ s∗ 1 +s2

/ A2 (X ) K

(3.2)

φ2X

 / Ab2X/K (K).

where the bottom horizontal arrows are the K-homomorphisms induced by the 2 = 2 we universal property of algebraic representatives. (Note that since dim W 2 1 have identified φW 2 with the Albanese map.) Since φ is an isomorphism and since the Albanese morphism is an isomorphism on torsion by Rojtman [Blo79, GS88, Mil82], a simple diagram chase establishes that φ2X : A2 (XK ) → Ab2X (K) is an isomorphism on prime-to-N pe torsion. It follows that φ2X is an isomorphism on l-primary torsion for all primes l not dividing N pe . The statement about T φ2X K then ensues by passing to the inverse limit. Alternately, since the middle vertical arrow is an isomorphism on torsion, it is an isomorphism on Tate modules. By applying Tl to (3.2), and since Tl A2 (XK ) and Tl Ab2X/K are finite free Zl -modules (Tl λ2 : Tl A2 (XK ) → Tl Ab2X/K is injective, Proposition A.27), we directly see that

On the Image of the Second l-adic Bloch Map

37

2 Tl φX is an isomorphism for l  N p and we also see, after tensoring with Ql , that K 2 Vl φXK is an isomorphism for all primes l. For (4), it suffices to observe that, if W1 and W2 admit resolutions of singularities (which is the case if dX ≤ 4 by [CP09]), then we may take e = 0 above. 

Remark 3.9. By rigidity, the same results in Proposition 3.8 hold if the separable a closure K is replaced with the algebraic closure K . Proposition 3.10 (Decomposition of the diagonal and miniversal cycle classes). Let X be a smooth projective variety over a field K of characteristic exponent p that is either finite or algebraically closed, and let N be a natural number. Assume that N ΔXK ∈ CHdX (X ×K X) admits a decomposition of type (W1 , W2 ) with dim W2 ≤ 1. Then Ab2X/K admits a miniversal cycle of degree pe N for some nonnegative integer e that may be chosen to be zero if dim X ≤ 4. In particular, if dim X ≤ 4 and if CH0 (X) is universally supported in dimension 1, then Ab2X/K admits a universal cycle. Proof. Similarly to (3.2), we have with Notation 3.5 a commutative diagram with composition of horizontal arrows being multiplication by N pe (where, due to resolution of singularities in dimensions < 4, e can be chosen to be zero if dim X ≤ 4) : r1∗

A2 (XK )

/ A2 (X ) K

φ1W 

φ2X

φ2X

1

 Ab2X/K (K) r1∗

s∗ 1

1 ) / A1 (W

r1∗

 / Pic0 (K)  W

 / Ab2 (K), X/K

s∗ 1

1

s∗1

2 where : → and : (Pic0W 1 /K )red → AbX/K denote the Khomomorphisms induced by the correspondences r1 and s1 . Since we are assuming K to be either finite or algebraic closed, the Abel–Jacobi map φ1W 1 admits a 0 1  universal divisor D ∈ A ((Pic )red ) (see, e.g., [ACMVb, §7.1]), meaning

Ab2X/K

(Pic0W 1 /K )red

1 /K W

W1 /K

that the induced morphism ψD : (Pic0W 

1 /K

)red → (Pic0W 

1 /K

)red

is the identity. It is then clear that the homomorphism associated to the cycle class  ◦ r∗ ∈ A n (Abn ). Z := s∗ ◦ D 1

is given by N pe IdAb2X/K .

1

X/K

X/K



4. Miniversal cycles and the image of the second l-adic Bloch map In this section, we prove our main Theorem 4.2. As an immediate consequence of the existence of an algebraic representative for codimension-2 cycles (see §3.2), we obtain proofs of Theorem 8 and of Theorem 10. We start with a lemma that parallels Lemma 2.3.

J.D. Achter, S. Casalaina-Martin and C. Vial

38

Lemma 4.1. Let X be a smooth projective variety over an algebraically closed field a K = K = K and let l be a prime. Let φ : An (XK ) → A(K), be a surjective n regular homomorphism, and let Γ ∈ AX/K (A) be a miniversal cycle of degree r. If φ[l∞ ] : An (XK )[l∞ ] → A[l∞ ] is an inclusion, then

  lvl (r) · Tl An (XK ) ⊆ im Γ∗ : Tl A0 (A) → Tl An (XK ) ,

where vl (r) is the l-adic valuation of r. Proof. By the definition of a miniversal cycle and its degree, the composition A(K)

/ A0 (A)

Γ∗

/ An (X ) K

/ A(K)

φ

is multiplication by r. Here, the map A(K) → A0 (A) is the map of sets a → [a]−[0]. Recall the general fact about abelian varieties, due to Beauville, that the map of sets A(K) → A0 (AK ), a → [a] − [0] is an isomorphism on torsion (see [ACMV20, Lem. 3.3] for references, and recall that Beauville’s argument works for arbitrary torsion in arbitrary characteristic). Therefore, restricting to l-primary torsion, we get a composition of homomorphisms A[l∞ ]

/ A0 (A)[l∞ ]



Γ∗

/ An (X )[l∞ ] K

φ[l∞ ] 

/ A[l∞ ]

which is given by multiplication by r. Here φ[l∞ ] is an isomorphism due to Lemma 3.6. Passing to the inverse limit, we obtain a composition of homomorphisms Tl A



/ Tl A0 (A)

Γ∗

/ Tl An (X ) K

Tl φn 

/ Tl A

which is given by multiplication by r. It immediately follows that lvl (r) Tl An (XK ) lies in the image of Γ∗ .  Theorem 4.2. Let X be a smooth projective variety over an algebraically closed a field K = K = K and let l be a prime. Let φ : An (XK ) → A(K), be a surjective n regular homomorphism, and let Γ ∈ AX/K (A) be a miniversal cycle of minimal degree r (see §3.3). Then the morphisms Tl A

Γ∗

/ Tl An (X ) K

Tl λn

/ H 2n−1 (X , Zl (n))τ K

induce inclusions n−1

 im(Tl λn ◦ Γ∗ ) ⊆ N

H 2n−1 (XK , Zl (n))τ ⊆ im(Tl λn ).

Moreover, if φ[l∞ ] : An (XK )[l∞ ] → A[l∞ ] is an inclusion, then in addition we have lvl (r) im(Tl λn ) ⊆ im(Tl λn ◦ Γ∗ ). In other words, if φ[l∞ ] is an inclusion, then im(Tl λn ) is an extension of  n−1 H 2n−1 (X , Zl (n))τ N K

On the Image of the Second l-adic Bloch Map

39

by a finite l-primary torsion group killed by multiplication by lvl (r) . In particular, if l does not divide r, then  n−1 H 2n−1 (X , Zl (n))τ = im(Tl λn ). N K  n−1 H 2n−1 (X , Zl (n))τ ⊆ im(Tl λn ) is Proposition 2.1 (due Proof. The inclusion N K to Suwa).  n−1 H 2n−1 (X , Zl (n))τ . For that purpose, Let us now show im(Tl λn ◦Γ∗ ) ⊆ N K consider the commutative diagram   / Tl A0 (AK )  Tl A0 (ZK ) LLL  :ZX Γ LLL LLL Γ∗ LLL Γ∗ L&   Tl An (XK ) 

=

/



Tl CH0 (ZK )

Γ :ZX

Tl λ0 / 



H1 (ZK , Zl )τ

Γ :ZX

Γ∗

Γ∗

 / H 2n−1 (X , Zl (n))τ K (4.1) where the direct sums run through all smooth projective varieties Z over K and all correspondences Γ ∈ CHdX −n+1 (Z ×K X). By definition, the image of the  n−1 H 2n−1 (X , Zl (n))τ , completing the proof via right vertical arrow consists of N K a diagram chase. Finally, under the assumption that φ[l∞ ] : An (XK )[l∞ ] → A[l∞ ] is an isomorphism, the assertion lvl (r) im(Tl λn ) ⊆ im(Tl λ◦Γ∗ ) follows from Lemma 4.1.   / Tl CHn (X ) K

Tl λn

Proof of Theorems 8 and 10. Recall from §3.2 that an algebraic representative for codimension-2 cycles φ2XK : A2 (XK ) → Ab2X /K (K) always exists. Both Theorems K 8 and 10 are then a special case of Theorem 4.2.  Since an algebraic representative always exists for codimension-2 cycles (see §3.2), in order to prove Theorem 4.2 unconditionally for the algebraic representative in codimension-2, it suffices to show the standard assumption holds. The following lemma allows us to reduce to assuming the standard assumption holds for varieties over finite fields : Lemma 4.3 (Standard assumption and generization). Let S be the spectrum of a discrete valuation ring with generic point η = Spec K and closed point ◦ = Spec κ. Let X/S be a smooth projective scheme, and let Γ ∈ AX2η /K (Ab2Xη /K ) be a miniversal cycle of minimal degree r. For all primes   r · char(K), if X◦ satisfies the standard assumption at  (i.e., φ2X◦ /κ [∞ ] is an isomorphism), then Xη satisfies the standard assumption at  (i.e., φ2Xη /K [∞ ] is an isomorphism). Proof. By [ACMVb, Thm. 8.3] we have (Ab2X/S )η  Ab2Xη /η . Let ΓX/S ∈ A2X/S (Ab2X/S )

40

J.D. Achter, S. Casalaina-Martin and C. Vial

be a miniversal cycle of minimal degree r induced by the one in the assumption of the lemma (see [ACMVb, Lem. 4.7]). Its specialization induces a group homomorphism wΓX/S ,◦ : Ab2X/S (κ) → A2 (X◦,κ ), and thus a homomorphism ψΓX/S,◦ : (Ab2X/S )◦ → Ab2X◦ /◦ . On -primary torsion, we have a commutative diagram Ab2X/S [∞ ](K)



wΓX/S



A2 (XK )[∞ ] 

/ (Ab2 )◦ [∞ ](κ) X/S 



(4.2)

wΓX/S ,◦

/ A2 (X0 )[∞ ] _

ψΓX/S ,◦

φ2X◦ /κ [ ∞ ]

 v Ab2X◦ /κ [∞ ](κ)

Both the top and bottom horizontal arrows are the specialization maps ; the fact that the specialization map on torsion cycle classes is injective in codimension 2 follows from the fact, due to Merkurjev–Suslin [MS82] (see also Propositions A.27 and A.30), that the second Bloch map is an inclusion. Choose a prime  = char(K) relatively prime to r. Then wΓX/S and is injective, and by commutativity, this implies wΓX/S,◦ is injective. Then since we assume that φ2X◦ /κ [∞ ] is injective, it follows that all arrows in (4.2) are injective. We now complete diagram (4.2) by introducing a second copy of the isogeny ψΓX/S,◦ . Note that the bottom square does not commute, and the outer rectangle fails to commute by a factor of r : Ab2X/S [∞ ](K) 



/ (Ab2X/S )◦ [∞ ](κ)

wΓX/S

A2 (XK )[∞ ] φ2X

η /K

 (Ab2X/S )[∞ ](K) 

wΓX/S ,◦





/ A2 (X0 )[∞ ]



(4.3)



ψΓX/S ,◦

φX◦ /κ

v ∞ / Ab2 [ ](κ) 6 X◦ /κ lll l l ll lll lll ψΓX/S ,◦

 Ab2X/S [∞ ](κ)

If   r, then the injectivity of φ2X◦ /κ implies the injectivity of φ2X

η /K

.



Proposition 4.4. Assume that for any finite field F and any smooth projective variety Y of dimension d over F we have that for all primes  = char(F), φ2YF [∞ ] : A2 (YF )[∞ ] → Ab2Y

F /F

(F)[∞ ]

On the Image of the Second l-adic Bloch Map

41

is an isomorphism ; i.e., assume the standard assumption holds for varieties of dimension d over finite fields. Let X be a smooth projective variety of dimension d over an algebraically a 2 closed field K = K = K, and let Γ ∈ AX/K (Ab2X /K ) be a miniversal cycle of K minimal degree r. Then for all primes   r · char(K), we have  1 H 3 (X , Z (1))τ = im(T λ2 : T A2 (X ) → H 3 (X , Z (2))τ ). N K K K Proof. By Proposition 3.8, we need only treat the case where K has characteristic p > 0. From Theorem 4.2 we only need to establish that X satisfies the standard assumptions for   r · char(K). Since X is of finite type over K, we may and do replace K with a field of finite transcendence degree n over the prime field Fp . By spreading out and then taking successive hypersurface sections, we may conclude from our hypothesis on finite fields and Lemma 4.3. We provide details for the sake of completeness. Spread X to a smooth scheme over Spec R, where R is a smooth Fp -algebra of Krull dimension n with Frac(R) = K. Let D ⊂ Spec(R) be a prime divisor with generic point ηD . Then D defines a discrete valuation on K, whose valuation ring RηD ⊃ R has residue field isomorphic to the function field of D. By iterating this construction we obtain a sequence of ring surjections R = Rn  Rn−1 · · ·  R0 where dim Rj = j, and for j ≥ 1 the field Kj := Frac(Rj ) admits a discrete valuation with valuation ring Sj ⊃ Rj and residue field isomorphic to Kj−1 . By hypothesis, X ×Spec R Spec R0 satsifies the standard assumption at . By repeated invocation of Lemma 4.3 for X ×Spec R Spec Sj with j = 1, 2, . . . , n, we find that XK satisfies the standard assumption at . 

5. Decomposition of the diagonal and the image of the second -adic Bloch map The aim of this section is to establish Theorem 12. First, we have the following proposition that extends [BW20, Prop. 2.3(ii)] to the  = p case. Proposition 5.1 ([BW20, Prop. 2.3(ii)]). Let X be a smooth projective variety over a perfect field K. Assume that N ΔXK ∈ CHdX (XK ×K XK ) admits a decomposition of type (W1 , W2 ) with dim W2 ≤ 1. Then, for all primes l, the second Bloch map λ2 : A2 (XK )[l∞ ] −→ H 3 (XK , Zl (2)) ⊗Zl Ql /Zl and the second l-adic Bloch map Tl λ2 : Tl A2 (XK ) −→ H 3 (XK , Zl (2))τ are isomorphism of Gal(K)-modules. Proof. The following argument is due to Benoist–Wittenberg for l = char(K). We check that it holds for l = char(K) as well. For X smooth and projective, we have

J.D. Achter, S. Casalaina-Martin and C. Vial

42

a diagram with exact row (A.28) : ∞ CH2 (XK  _ )[l R ] R

R R R R R)  / H 4 (X , Zl (2)) / H 3 (X , Ql /Zl (2)) K K λ2

0

/ H 3 (X , Zl (2)) ⊗ Ql /Zl K

where the dashed arrow is, up to sign, the cycle class map ([CTSS83, Cor. 4], [GS88, Prop. III.1.16 and Prop. III.1.21]). Since algebraically trivial cycles are homologically trivial, it follows that the image of A2 (XK )[l∞ ] under λ2 is contained in H 3 (XK , Zl ) ⊗Zl Ql /Zl ⊆ H 3 (XK , Ql /Zl ). In particular, the cokernel of λ2 : A2 (XK )[l∞ ] → H 3 (XK , Zl ) ⊗Zl Ql /Zl is divisible. Now suppose N ΔXK ∈ CHdX (XK ×K XK ) admits a decomposition of type (W1 , W2 ) with dim W2 ≤ 1. With Notation 3.5 we obtain by the naturality of the Bloch map (Proposition A.23) a commutative diagram A2 (XK )[l∞ ] λ2

 H 3 (XK , Zl (2)) ⊗ Ql /Zl

r1∗

1 )[l∞ ] / A1 (W

s∗ 1

 λ1

/ A2 (X )[l∞ ] K λ2

 1 , Zl (1)) ⊗ Ql /Zl / H 1 (W

 r1∗ s∗ 1 / H 3 (X , Zl (2)) ⊗ Ql /Zl . K (5.1) 2 term above for reasons of codimension. The middle vertical Note there is no W arrow in (5.1) is an isomorphism by Proposition A.28, while the composition of the horizontal arrows in (5.1) is multiplication by N pe . It follows that coker λ2 is torsion, annihilated by N pe , and consequently that this cokernel is trivial, i.e., λ2 : A2 (XK )[l∞ ] → H 3 (XK , Zl (2)) ⊗Zl Ql /Zl is an isomorphism. Taking the Tate module of this isomorphism gives the result for the l-adic Bloch map, since H 3 (XK , Zl ) is a finitely generated Zl -module, and so it is elementary to check that there is an identification Tl (H 3 (XK , Zl ) ⊗Zl Ql /Zl )  H 3 (XK , Zl )τ .  The following proposition establishes Theorem 12. Proposition 5.2 (Theorem 12). Let X be a smooth projective variety over a perfect field K of characteristic exponent p and let N be a natural number. Assume that N ΔXK ∈ CHdX (XK ×K XK ) admits a decomposition of type (W1 , W2 ). 1. Assume dim W2 ≤ 2. Suppose l is a prime such that l  N , and such that either l = char(K) or resolution of singularities holds in dimensions < dX .  1 H 3 (X , Zl (2)) ⊆ H 3 (X , Zl (2)) is an equality, the Then the inclusion N K

K

inclusion Tl A2 (XK ) → Tl CH2 (XK ) is an equality, and the second l-adic Bloch map Tl λ2 : Tl CH2 (XK ) −→ H 3 (XK , Zl (2))τ is an isomorphism of Gal(K)-modules.

On the Image of the Second l-adic Bloch Map

43

2. Assume dim W2 ≤ 1. Let l be any prime. Then the inclusion Tl A2 (XK ) → Tl CH2 (XK ) is an equality, and the second l-adic Bloch map Tl λ2 : Tl CH2 (XK ) −→ H 3 (XK , Zl (2))τ is an isomorphism of Gal(K)-modules. Moreover, if l  N and if either l = char(K) or resolution of singularities holds in dimensions < dX , then H 3 (XK , Zl ) is torsion-free. Proof. We first assume that l =  = p. That T λ2 is a morphism of Gal(K)-modules is Proposition A.22 and that T λ2 is injective in general is Proposition A.27. Concerning item (1), with Notation 3.5 we obtain by the naturality of the -adic Bloch map (Proposition A.23) a commutative diagram r1∗ ⊕r2∗

T A2 (XK ) T λ2X

 H 3 (XK , Z (2))τ

∗ s∗ 1 +s2

1 ) ⊕ T A2 (W 2 ) / T A1 (W

/ T A2 (X ) K

1 2  T λW  ⊕T λW  1

2

 r1∗ ⊕r2∗ 1 , Z (1)) ⊕ H 3 (W 2 , Z (2))τ / H 1 (W

∗ s∗ 1 +s2



T λ2X

/ H 3 (X , Z (2))τ . K (5.2) The middle vertical arrow in (5.2) is an isomorphism by Propositions A.25 and A.26, while the composition of the horizontal arrows in (5.2) is multiplication by N pe . In particular, the latter are bijective if  does not divide N pe . A diagram chase then establishes the surjectivity of T λ2X restricted to algebraically trivial cycles, and hence the bijectivity of T A2 (XK ) → T CH2 (XK ) and of T λ2 : T CH2 (XK ) −→ H 3 (XK , Z (2))τ . Finally, since the composition H 3 (XK , Z (2))

r1∗ ⊕r2∗

∗

∗

1 , Z (1)) ⊕ H 3 (W 2 , Z (2)) s1 +s2 / H 3 (X , Z (2)) / H 1 (W K

2 ≤ 2, we obtain from the equality is multiplication by N pe and since dim W 1 1 3 3  H = N H of Proposition 1.1, for   N pe , the inclusion H 3 (X , Z (2)) ⊆ N K  1 H 3 (X , Z (2)). N K

In case (2), by Proposition 5.1, it suffices to see that H 3 (XK , Z (2)) is torsionfree for  not dividing N pe . This follows simply from the factorization of the multiplication by N pe map as H 3 (XK , Z (2))

r1∗

1 , Z (1)) / H 1 (W

s∗ 1

/ H 3 (X , Z (2)) K

1 , Z (1)) is torsion-free. and the fact that H 1 (W Now suppose l = char(K) = p > 0. Bearing in mind the properties of the padic Bloch map summarized in §A.4, we see that the composition of the horizontal arrows in (5.2) is again multiplication by N pe . If resolution of singularities holds in dimension at most dX − 1, then we may take e = 0. Under this hypothesis, if p  N , we again see that Tp λ2 is an isomorphism of Gal(K)-modules. 

44

J.D. Achter, S. Casalaina-Martin and C. Vial

Remark 5.3. Note that Proposition 5.2(2), together with Theorem 4.2, implies that CH0 (XK ) ⊗ Q is universally supported in dimension 1, then the primes  for  1 H 3 (X , Z (2))τ ⊆ H 3 (X , Z (2))τ might fail to be an equality are the which N K K primes dividing the minimal degree of a miniversal cycle. Due to Proposition 3.10, this is in this case a priori finer than the conclusion of Proposition 5.2(1).

6. Modeling cohomology via correspondences In this section, we prove Theorems 12 and 15. The starting point is that a geometrically rationally chain connected variety (resp. stably rational variety) has universally trivial Chow group of zero-cycles with Q-coefficients (resp. Z-coefficients) ; see §3.1 and specifically Remark 3.4. We then combine the existence of the -adic Bloch map with the existence of an algebraic representative for codimension-2 cycles to establish Proposition 5.2 (which implies Theorem 12) and the main Theorem 6.4 (which implies Theorem 15). Along the way we establish related results concerning the third -adic cohomology group of uniruled threefolds (Proposition 6.1). 6.1. Modeling Q -cohomology via correspondences The aim of this section is to show Mazur’s Questions 1 and 2, which are with Q -coefficients, can be easily answered positively under some assumption on the coniveau of H 2n−1 (XK , Q (n)). The following Proposition extends [ACMV17, Thm. 2.1(d)] in the positive characteristic case. Note that it applies to smooth projective geometrically uniruled threefolds. Proposition 6.1. Let X be a smooth projective variety over a perfect field K. Assume H 2n−1 (XK , Q 0 (n)) = Nn−1 H 2n−1 (XK , Q 0 (n)) for some prime 0 = char(K) and for some integer n such that 2n − 1 ≤ dX . Then there exist an abelian variety A over K and a cycle class Γ ∈ CHn (A ×K X) such that the induced morphism / H 2n−1 (X , Ql (n)) Γ∗ : Vl A (6.1) K

is an isomorphism of Gal(K)-modules for all primes l. (In particular, we have H 2n−1 (XK , Ql (n)) = Nn−1 H 2n−1 (XK , Ql (n)).) Moreover, if K has positive characteristic, then Γ induces an isomorphism of F-isocrystals Γ∗ : H 2dA −1 (A/K)(dA )

/ H 2n−1 (X/K)(n) .

Proof. By Proposition 1.1, there is a smooth projective variety W over K of dimension dX − n + 1 and a K-morphism f : W → X inducing a surjection f∗ : H 1 (WK , Q 0 (1))  H 2n−1 (XK , Q 0 (n)). Let ZWK ∈ CH1 ((Pic0WK )red ×K WK ) be the universal divisor on WK ; it induces an isomorphism of Z 0 -modules T 0 Pic0WK → H 1 (WK , Z 0 (1)). Let L/K be

On the Image of the Second l-adic Bloch Map

45

a finite field extension over which ZWK is defined. By pushing forward, we obtain a cycle ZW ∈ CH1 ((Pic0W )red ×K WK ) inducing an isomorphism V 0 Pic0W → H 1 (WK , Q 0 (1)) of Gal(K)-modules. Let us set B := (Pic0W )red . Composing f∗ with ZW we obtain a correspondence γ ∈ CHn (B ×K X) inducing a surjection γ∗ : V 0 B  H 2n−1 (XK , Q 0 (n)) of Gal(K)-modules. Consider now the cycle Δ∗ (c1 (OX (1))d−2n+1 ) ∈ CH2d−2n+1 (X ×K X), where Δ : X → X ×K X is the diagonal embedding. By the Hard Lefschetz Theorem, this cycle induces an isomorphism L : H 2n−1 (XK , Q 0 (n)) → H 2d−2n+1 (XK , Q 0 (d − n + 1)) of Gal(K)-modules and we obtain a homomorphism V 0 B

γ∗

/ / H 2n−1 (X , Q 0 (n)) K

L 

/ H 2d−2n+1 (X , Q 0 (d − n + 1))   K

γ∗

/ (V 0 B)∨

induced by the correspondence γ ∗ ◦L◦γ∗ ∈ CH1 (B ×K B). In particular, the above homomorphism, which is Gal(K)-equivariant, is induced by a K-homomorphism ϕ : B → B ∨ . It is clear that ker γ∗ = ker ϕ∗ . By Poincar´e reducibility, there exist an abelian variety A and ψ ∈ Hom(A, B) ⊗ Q such that γ∗ ◦ ψ∗ : V 0 A → H 2n−1 (XK , Q 0 (n)) is an isomorphism. In addition, there exists an idempotent θ ∈ Hom(B ∨ , B ∨ ) ⊗ Q with image A such that θ ◦ ϕ = ϕ. Setting Γ = γ ◦ ψ, it follows from the independence of  of the -adic Betti numbers that Γ∗ : V A → H 2n−1 (XK , Q (n)) is an isomorphism for all primes  = char(K). Now suppose char(K) = p > 0. Since Γ∗ ◦ Γ∗ : V A → V A is an isomorphism, this same cycle induces an automorphism of the F -isocrystal H 1 (A/K) ([KM74], after a spread and specialization argument to reduce to K finite). Because crystalline and -adic Betti numbers coincide, Γ∗ : H 2dA −1 (A/K)(dA )

/ H 2n−1 (X/K)(n)

is an isomorphism of crystals. Taking F -invariants shows that (6.1) holds for l = p, too.  Remark 6.2 (2n − 1 > dX ). We note that using the hard Lefschetz theorem, with the notation and assumptions of Proposition 6.1, there also exists a cycle class Γ ∈ CHdX −n (A ×K X) such that for all primes l Γ∗ : Vl A

/ H 2dX −2n+1 (X , Ql (dX − n + 1)) K

is an isomorphism of Gal(K)-modules. In case n = 2 and under the assumption that Vl φ2XK : Vl A2 (XK ) −→ Vl Ab2X/K is an isomorphism for all primes l, one can make Question 2 more precise and ask whether there exists a correspondence Γ ∈ CH2 (Ab2X/K ×K X) ⊗ Q inducing for all primes l the canonical identifications (0.5). We provide a positive answer for geometrically uniruled threefolds:

J.D. Achter, S. Casalaina-Martin and C. Vial

46

Proposition 6.3. Let X be a smooth projective variety over a perfect field K and assume CH0 (XK ) ⊗ Q is universally supported in dimension 2, e.g., X is a geometrically uniruled threefold. (In particular, due to Proposition 3.8, Vl φ2X : K Vl A2 (XK ) −→ Vl Ab2X/K is an isomorphism for all primes l.) Then there exists a correspondence Γ ∈ CH2 (Ab2X/K ×K X) ⊗ Q inducing for all primes l the canonical identifications (0.5) Vl Ab2X/K

(Vl φ2X

K

/K

)−1



/ Vl A2 (X )  K



/ Vl CH2 (X )  K



Vl λ2



/ H 3 (X , Ql (2)). K 2

Γ∗

Proof. First we note that the assumption that CH0 (XK ) ⊗ Q is universally supported in dimension 2 implies that im(Vl λ2 ) = N1 H 3 (XK , Ql (2)) for all primes l. Together with Proposition 2.1, this implies that Vl λ2 is an isomorphism for all l, so that (0.5) is an isomorphism for all l. Second, let Z ∈ CH2 (Ab2X/K ×K X) be a miniversal cycle and let r denote its degree. Then, by [ACMVa, Cor. 11.8] with Ql -coefficients, we have that Z∗ = r · Vl λ2 ◦ (Vl φ2X /K )−1 for all primes l, and it follows that Γ = 1r Z induces (0.5) for all primes l.

K



6.2. Modeling Z -cohomology via correspondences : Theorem 15 Combining Proposition 3.8 with Proposition 5.2, we see that, under the assumption that ΔXK ∈ CHdX (XK ×K XK ) admits a decomposition of type (W1 , W2 ) with dim W2 ≤ 2, we obtain for all primes  = char(K) canonical isomorphisms (T φ2X )−1

T λ2

 T Ab2X/K −−−−−K−−→ T A2 (XK ) −−−−→ T CH2 (XK ) −−− −→ H 3 (XK , Z (2))τ .







(6.2) Under the further assumption that dim W2 ≤ 1, H 3 (XK , Z (2)) is torsion-free by Proposition 5.2 and the main result below establishes that the isomorphisms (6.2) are induced by a correspondence defined over K. In view of Proposition 3.2 and Remark 3.4, the theorem below establishes Theorem 15. Theorem 6.4 (Theorem 15). Let X be a smooth projective variety over a field K of characteristic exponent p, which is assumed to be either finite or algebraically closed. Assume that there is a natural number N such that N ΔX ∈ CHdX (X ×K X) admits a decomposition of type (W1 , W2 ) with dim W2 ≤ 1. Then there exists a correspondence Γ ∈ CH2 (Ab2X/K ×K X) inducing for all primes l not dividing N p isomorphisms  (6.3) Γ∗ : Tl Ab2X/K −−−−→ H 3 (XK , Zl (2)) of Gal(K)-modules. If p ≥ 2, if p  N , and if resolution of singularities holds in dimensions < dX , then (6.3) holds with l = p. Finally, if K ∈ C, the correspondence Γ induces an isomorphism      (6.4) Γ∗ : H1 ((Ab2X/K )C , Z) ⊗ Z N1 −−−−→ H 3 (XC , Z(2)) ⊗ Z N1 .

On the Image of the Second l-adic Bloch Map

47

Proof. First we focus on (6.3). With Notation 3.5, we have the diagram r1∗ ⊕r2∗

A2 (XK )

∗ s∗ 1 +s2

1 ) ⊕ A2 (W 2 ) / A1 (W 2 φ1W  ⊕φW 

/ A2 (X ) K

φ2X

1

 Ab2X/K (K)

(6.5)

φ2X

2

 / Ab2 (K). X/K

 / Pic0 (K) ⊕ Alb  (K)  W2 W 1

Assuming dim W2 ≤ 1, the diagram (6.5) takes the simpler form r1∗

A2 (XK ) 

s∗ 1

1 ) / A1 (W 1  φW 

φ2X g

Ab2X/K (K)

1



/ Pic0 (K) W

/ A2 (X ) K 

φ2X

/ Ab2X/K (K)

f

1

where the composition of the horizontal arrows is multiplication by N pe . Note that the homomorphisms f and g are in fact induced by K-homomorphisms Ab2X/K → 2 0 (Pic0W 1 )red and (PicW 1 )red → AbX/K by [Mur85] in the case K algebraically closed and by the main result of [ACMV17] in the case K perfect. By Proposition 5.2 with Proposition 3.8, we get for all  not dividing N p a commutative diagram H 3 (XK , Z (2)) O

r1∗

 T λ2

T A2 (XK ) O

s∗ 1

/ H 3 (X , Z (2)) KO

 T λ1 r1∗

 T λ2 s∗ 1

) / T A1 (W O 1

/ T A2 (X ) O K

1 −1  (T φW  )

 (T φ2X )−1

T Ab2X/K

1 , Z (1)) / H 1 (W O

 (T φ2X )−1

1

g∗

/ T Pic0 W

1

f∗

/ T Ab2X/K

where the vertical arrows are isomorphisms and the composition of the horizontal arrows is multiplication by N pe . Now, the condition that K be either finite or algebraically closed ensures 1 admits a universal divisor Z  ∈ CH1 (W 1 ×K (Pic0 )red ), meaning that that W  W1 W 1

0 the K-homomorphism (Pic0W 1 is the identity. In 1 )red → (PicW 1 )red induced by ZW addition, the homomorphism 1 , Z (1)) T λ1 ◦ (T φ1 )−1 : T Pic0 → H 1 (W 1 W

W1

coincides with the action of ZW 1 ; this follows from Kummer theory, see, e.g., [ACMVa, §11]. We then define a codimension-2 cycle Z ∈ CH2 (X ×K Ab2X/K ) as the composition t s1 ◦ t ZW 1 ◦ g. By a simple diagram chase, its induced action 2 ∗ 3 Z∗ = s∗1 ◦ ZW  ◦ g∗ : T AbX/K → H (XK , Z (2)) 1

is equal to N pe T λ2 ◦ (T φ2X )−1 , which is an isomorphism.

J.D. Achter, S. Casalaina-Martin and C. Vial

48

If K has positive characteristic p > 0 and if resolution of singularities holds in dimension < dX , then we may take e = 0. Consequently, under this hypothesis, the same argument shows that if p  N , then (6.3) is an isomorphism for l = p, too. Finally for the case K ⊆ C and (6.4), one uses essentially the same argument, but with the canonical identification of the cohomology modulo torsion of a smooth complex projective variety with the first homology of the intermediate Jacobian. 

7. The image of the -adic Bloch map in characteristic 0 In this section we show that we can model the third integral cohomology with an abelian variety for any smooth projective variety liftable to a smooth projective rationally chain connected variety in characteristic 0. We start with the following proposition, which essentially shows that the strategy of the proof of [ACMV17, Thm. B] works with Z -coefficients for rationally chain connected varieties. Proposition 7.1. Let X be a smooth projective variety over a field K ⊆ C such that N1 H 3 (XC , Q) = H 3 (XC , Q) (e.g., X is geometrically rationally chain connected, or dim X = 3 and X is geometrically uniruled). Then for any prime  the morphisms T φ2X

K /K

: T A2 (XK ) → T Ab2X/K

and T λ2 : T A2 (XK ) → H 3 (XK , Z (2))τ

are isomorphisms, so that the composition (T φ2X /K )−1 2 K 2 / T A2 (X ) T λ T AbXK /K K

/ H 3 (X , Z (2))τ K

(7.1)

is an isomorphism. Proof. The fact due to Murre that φ2X

K /K

[∞ ] : A2 (XK )[∞ ] → Ab2X/K [∞ ] is

an isomorphism was explained in Proposition 3.8(1) ; now one simply applies Tate modules to get that T φ2X /K is an isomorphism. K

We now consider the Bloch map. The first observation is that φ2XC /C agrees with the Abel–Jacobi map AJ : A2 (XC ) → Ja3 (XC ). Thus, applying Tate modules to the Abel–Jacobi map, we have that T φ2XC /C = T AJ, so that, by the above, T AJ is an isomorphism. At the same time, the assumption N1 H 3 (XC , Q) = H 3 (XC , Q) implies that the Abel–Jacobi map AJ : A2 (XC ) → J 3 (XC ) to the full intermediate Jacobian is surjective (e.g., [Voi07, Thm. 12.22]) ; i.e., Ja3 (XC ) = J 3 (XC ) = H 1,2 (XC )/H 3 (XC , Z)τ . We conclude using the fact that T AJ = T λ2 (see Remark A.19).  Remark 7.2. Recall that for a smooth projective variety X the condition N1 H 3 (XC , Q) = H 3 (XC , Q)

On the Image of the Second l-adic Bloch Map

49

is implied by the diagonal ΔXC ∈ CHdX (XC ×C XC )Q admitting a decomposition of type (W1 , W2 ) with dim W2 ≤ 2. The diagonal of XC has such a decomposition if X is geometrically rationally chain connected, or dim X = 3 and X is geometrically uniruled, since in these cases CH0 (XK )Q is universally supported on a surface (see Proposition 3.2 and Remark 3.4). Remark 7.3. We note here that Proposition 7.1 strengthens Theorem 3 in the case n = 2 in that it allows for Z -coefficients ; on the other hand, Proposition 7.1 is weaker than Theorem 3 in the sense that it does not provide a correspondence giving the isomorphism (7.1). Similarly, Proposition 7.1 strengthens Theorem 6.4 in the case char(K) = 0 in the sense that it gives isomorphisms for all primes without assuming a decomposition of the diagonal with Z-coefficients, and it allows for a weaker form of the decomposition (see the previous remark). For example, in characteristic 0, Theorem 6.4 implies (7.1) is an isomorphism for all primes if X is geometrically stably rational, whereas Proposition 7.1 implies the same if X is just assumed to be geometrically rationally chain connected (or even a geometrically uniruled threefold). On the other hand, Proposition 7.1 is weaker than Theorem 6.4 in the sense that it does not provide a correspondence giving the isomorphism (7.1). We now use Proposition 7.1 to prove the following result on varieties liftable to characteristic 0 : Corollary 7.4. Let X◦ be a smooth projective variety over a field κ, and suppose that X◦ lifts to a smooth projective variety Xη over a subfield of C; i.e., there is a DVR with spectrum S, generic point η = Spec K with K ⊂ C, closed point ◦ = Spec κ, and a smooth projective scheme X/S, with special fiber over ◦ equal to X◦ , and generic fiber over η equal to Xη . If N1 H 3 ((Xη )C , Q) = H 3 ((Xη )C , Q), e.g., if Xη is a geometrically rationally chain connected variety, or a smooth geometrically uniruled threefold, then for any prime  = char(κ) the morphism T λ2 : T A2 (Xκ ) → H 3 (Xκ , Z (2))τ is an isomorphism. Proof. We consider the diagram 

T λ2



2

T A2 (XK ) 

T A2 (Xκ )

T λ

/ H 3 (X , Z )τ K 



/ H 3 (Xκ , Z )τ

The vertical arrows are specialization maps. The result holds by commutativity and by Proposition 7.1. 

50

J.D. Achter, S. Casalaina-Martin and C. Vial

Appendix: A review of the l-adic Bloch map The aim of this appendix is to review the original construction [Blo79] of the Bloch map on -primary torsion and show it in fact yields a direct construction of the -adic Bloch map. We also review Suwa’s construction [Suw88] of the adic Bloch map and show it coincides with our direct construction. For future referencing purposes we list in §A.4 the properties of the -adic Bloch map that can be directly derived from the corresponding properties of the original Bloch map via Suwa’s construction. In addition, the construction of Gros–Suwa [GS88] of the p-adic Bloch map is briefly reviewed. Finally, we study the restriction of the Bloch map to the subgroup of algebraically trivial cycles. A.1. Conventions for -adic and p-adic cohomology A.1.1. -adic cohomology. We fix a variety X over a field K, and consider the ´etale cohomology groups of XK := X ×K K with values in prime-to-char(K) torsion sheaves. For each n, r, and ν, we use the convention H n (XK , Z/ν Z(r)) := H´ent (XK , μ ⊗r

ν ), the ´etale cohomology of the ´etale sheaf μ ν of ν -roots of unity. Note that since ´etale cohomology is invariant under purely inseparable extensions, we can replace a K with K throughout this subsection. There are maps H n (XK , Z/ν Z(r)) → H n (XK , Z/ν+1 Z(r)) induced from the natural map Z/ν Z → Z/ν+1 Z, [x] → [x], or more precisely, from the natural map μ ν → μ ν+1 , ζ → ζ , as well as maps H n (XK , Z/ν+1 Z(r)) → H n (XK , Z/ν Z(r)) induced from the natural quotient map Z/ν+1 Z  Z/ν Z, or more precisely the μ  μ ν The -adic cohomology groups of X are natural map μ ν+1  μ ν+1 /μ defined as follows : H n (XK , Z (r)) := lim H n (XK , Z/ν Z(r)) ←− ν

H n (XK , Q (r)) := H n (XK , Z (r)) ⊗Z Q . The cohomology groups of X with -torsion coefficients are defined as follows : H n (XK , Q /Z (r)) := lim H n (XK , Z/ν Z(r)). −→ ν We denote by Z (r) := lim μ ⊗r ←− ν Q (r) := Z (r) ⊗Z Q

Q /Z (r) := lim μ ⊗r −→ ν

On the Image of the Second l-adic Bloch Map

51

and we can obtain the various twists in cohomology as : H n (XK , Z (r)) = H n (XK , Z ) ⊗Z Z (r) H n (XK , Q (r)) = H n (XK , Q ) ⊗Q Q (r) H n (XK , Q /Z (r)) = H n (XK , Q /Z ) ⊗Q /Z Q /Z (r) . For the sake of completeness, we recall the following basic fact and its proof (see also e.g., [Mil80, Ch. III, Rmk. 3.6(d)]) : Proposition A.1. Viewing Q /Z as a torsion ´etale sheaf on X, there is a natural isomorphism H´ent (XK , Q /Z ) ⊗Q /Z Q /Z (r) = H n (XK , Q /Z (r))

:= lim H n (XK , Z/ν Z(r)) . −→ ν Proof. The Snake Lemma applied to / Z  _

0

 / Q

0

· ν

· ν

/ Z  _

/ Z/ν Z

/0

 / Q

 /0

/0

gives a short exact sequence of ´etale sheaves 0

/ Z/ν Z

· ν

/ Q /Z

/ Q /Z

/ 0.

The map on the left can be written explicitly as [x] → [x/ν ], where we are viewing Z ⊆ Z in the natural way. This gives a long exact sequence in cohomology ···

/ H n (X , Z /ν Z ) K

/ H n (X , Q /Z ) K

· ν

/ H n (X , Q /Z ) K

/ ··· (A.2)

In fact we have a diagram 0

/ Z/ν Z 

0

/ Z/ν+1 Z

/ Q /Z

· ν

/ Q /Z

/0

·

/ Q /Z

·

ν+1

 / Q /Z

/0

which is commutative since given [x] ∈ Z/ν Z we have [x/ν ] = [x/ν+1 ]. Taking direct limits is exact, and so we obtain an exact sequence ···

/0

/ lim H n (X , Z /ν Z ) K −→ ν

thereby settling the proposition.

/ H n (X , Q /Z ) K

/0

/ ··· , 

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52

A.1.2. p-adic cohomology. For K perfect of characteristic p > 0, let W(K) be the ring of Witt vectors of K, with field of fractions K(K). For X/K, we adopt the standard notation [GS88, §I.3.1], [Mil86, §1] r H n (X, Z/pν Z(r)) := H´en−r t (X, Wν ΩX,log )

H n (XK , Zp (r)) := lim H n (XK , Z/pν Z(r)) ←− ν

H n (XK , Qp (r)) := H n (XK , Zp (r)) ⊗Zp Qp H n (XK , Qp /Zp (r)) := lim H n (XK , Z/pν Z(r)) −→ ν

With these conventions, H´ent (XK , Qp (r)) := lim H´ent (XK , μ ⊗r pν ) ⊗Zp Qp ← − ν and H n (X, Qp (r)) coincide [Ill79, (5.2.1)], [Mil86, Prop. 1.15], but if n > 1 then the corresponding statement for integral coefficients need not hold. For X/K smooth and projective, we let n H n (X, W(K)(r)) := Hcris (X/W(K))(r)

denote the crystalline cohomology group, and let H n (X, K(K)(r)) := H n (X, W(K)(r)) ⊗W(K) K(K). n This group may also be computed as the rigid cohomology group Hrig (X/K(K))(r). ν Note that if we set Wν (K) := W(K)/p W(K), then n H n (X, Wν (K)(r)) = Hcris (X/Wν (K)(r)).

A.2. The -adic Bloch map In [Blo79], Bloch constructed a map λn : CHn (XK a )[∞ ] → H 2n−1 (XK a , Q /Z (n)) for smooth projective varieties over a field K. In this section, we review his construction, showing how it also defines a map T λn : T CHn (XK a ) → H 2n−1 (XK a , Z (n))τ on Tate modules. A.2.1. The Abel–Jacobi map on torsion. We start by recalling the definition of the Abel–Jacobi map on torsion. This is rather elementary from the definition of the Abel–Jacobi map, but gives some motivation for Bloch’s approach to his algebraic construction of the map, as well as some motivation for our interest in what we call the ν -Bloch maps (Definition A.2). We also explain in Remark A.19 that the -adic Abel–Jacobi map (A.4) agrees with the -adic Bloch map on homologically trivial cycle classes. For a complex projective variety X one has the Abel–Jacobi map AJ

CHn (X)hom −→ J 2n−1 (X) = F n \H 2n−1 (X, C)/H 2n−1 (X, Z)τ

On the Image of the Second l-adic Bloch Map

53

from the group of homologically trivial cycle classes of codimension-n to the (2n − 1)-st intermediate Jacobian J 2n−1 (X). Here, F • refers to the Hodge filtration on H 2n−1 (X, C) ; in the situation above and in terms of the Hodge decomposition we have F n H 2n−1 (X, C) = H 2n−1,0 (X) ⊕ H 2n−2,1 (X) ⊕ · · · ⊕ H n,n−1 (X). We can identify the torsion J 2n−1 (X)[ν ] as follows. For any complex torus A = V /Λ, we have A[ν ] = 1ν Λ/Λ. If we consider the commutative diagram of short exact sequences, / ΛQ / ΛQ /Λ /0 /Λ 0  · ν

· ν

 /Λ

0



 / ΛQ

· ν

/ ΛQ /Λ

/ ···

the snake lemma gives an identification A[ν ] = Λ/ν Λ. In our situation with A = J 2n−1 (X), we have Λ = H 2n−1 (X, Z)τ . In other words, J 2n−1 (X)[ν ] = H 2n−1 (X, Z)τ /ν H 2n−1 (X, Z)τ . We then consider the diagram 0

/ H 2n−1 (X, Z)tors

0

 / H 2n−1 (X, Z)tors

/ H 2n−1 (X, Z)

· ν



/ H 2n−1 (X, Z)τ

· ν

/0

· ν

 / H 2n−1 (X, Z)τ

/ H 2n−1 (X, Z)

/0

2n−1

H (X,Z)tors For brevity, we denote δ an ν =

ν H 2n−1 (X,Z)tors the cokernel of the vertical map on the left. The snake lemma, and the long exact sequence in cohomology associated · ν to 0 → Z → Z → Z/ν Z → 0, together give a diagram

0

/ δ an

ν

0

/ δ an

ν

/

H 2n−1 (X,Z)

ν H 2n−1 (X,Z)

_

 / H 2n−1 (X, Z/ν Z)

H 2n−1 (X,Z)τ

ν H 2n−1 (X,Z)τ

/0

 / H 2n−1 (X, Z/ν Z)/δ an

ν

/0

/

Thus we obtain maps CHn (X)hom [ν ]

AJ[ ν ]

/ J 2n−1 (X)[ν ]

/ H 2n−1 (X, Z/ν Z)/δ an

ν . (A.3)

2n−1 It is easy to see that for sufficiently large ν, we have δ an (X, Z) -tors , ν = H 2n−1 2n−1 (X, Z ) = H (X, Z) ⊗Z Z , we have that and via the isomorphism H H 2n−1 (X, Z) -tors = H 2n−1 (X, Z )tors . It follows that lim δ an = H 2n−1 (X, Z )tors . ←− ν 2n−1 ν an 2n−1 (X, Z/ Z)/δ ν = H (X, Z )τ . For this we We claim now that lim H ←− consider the short exact sequence 2n−1 0 → δ an (X, Z/ν Z) → H 2n−1 (X, Z/ν Z)/δ an ν → H ν → 0, 1 an and use the fact that since the δ an ν are finite, we have lim δ ν = 0. ←−

J.D. Achter, S. Casalaina-Martin and C. Vial

54

Taking the inverse limit of the maps (A.3), we therefore obtain a map T CHn (X)hom

T AJ

/ T J 2n−1 (X)

/ H 2n−1 (X, Z )τ

(A.4)

/ H 2n−1 (X, Q ).

(A.5)

and then tensoring with − ⊗Z Q , we obtain a map V CHn (X)hom

V AJ

/ V J 2n−1 (X)

The next claim is that lim δ an ν = 0. This also follows from the fact that for −→ an sufficiently large ν, we have δ ν = H 2n−1 (X, Z) -tors , since the latter group is finite, and is therefore killed by multiplication by N for some sufficiently large N . As a consequence, taking the direct limit in (A.3) we obtain a map CHn (X)hom [∞ ]

AJ[ ∞ ]

/ J 2n−1 (X)[∞ ]

/ H 2n−1 (X, Q /Z ).

(A.6)

A.2.2. Bloch’s preliminaries. The set-up in [Blo79] is from the paper [BO74]. It μ ⊗n is described in [Blo79] in the following way. One sets Hq (μ

ν ) to be the Zariski q ⊗n sheaf on XK a associated to the pre-sheaf U → H´et (U, μ ν ). In other words, this is a et via the morphism of sites the derived push-forward of the sheaf μ⊗n

ν on (XK )´ a a π : (XK )´et → (XK )zar , from the ´etale site to the Zariski site : ⊗n q μ ⊗n Hq (μ

ν ) := R π∗μ ν . a

Consider the composition of morphisms of sites (XK a )´et → (XK a )zar → Spec K . The Leray spectral sequence is p+q ⊗n p a μ⊗n E2p,q = Hzar (XK a , Hq (μ et (XK , μ ν ).

ν )) =⇒ H´

μ⊗n The main tool is the existence of a particular flasque resolution of Hq (μ

ν ) [Blo79, (1.3)], 0 1 μ⊗n 0 → Hq (μ

ν ) → F → F → · · · , p μ ⊗n which of course computes Hzar (XK a , Hq (μ

ν )) in the pth place. This resolution has two nice properties. First, it turns out to be easy to read off from the resolution that p μ⊗n Hzar (XK a , Hq (μ

ν )) = 0

for p > q, and consequently, from the shape of the spectral sequence, one obtains so-called boundary maps for the spectral sequence [Blo79, Cor. 1.4] 2n−1 n−1 μ⊗n Hzar (XK a , Hn (μ (XK a , μ ⊗n

ν )) → H´ et

ν ).

(A.7)

Second, the precise description of the flasque resolution in [Blo79, (1.3)] shows n−1 μ⊗n that the group Hzar (XK a , Hn (μ

ν )) on the left in (A.7) is the cohomology of the complex [Blo79, Cor. 1.5]    ν ∂ν 2 HGal (Q(W ), μ ⊗2 Q(V )∗ /Q(V )∗ → Z/ν Z,

ν ) → W n−2 ⊆XK a

V n−1 ⊆XK a

T n ⊆XK a

On the Image of the Second l-adic Bloch Map

55

where the sums are taken over irreducible subvarieties of the indicated codimensions, and Q(−) indicates the function field. The map ∂ ν is obtained from the standard exact sequence below after reduction modulo ν : 

Q(V )∗ → ∂



Z → CHn (XK a ) → 0,

(A.8)

T n ⊆XK a

V n−1 ⊆XK a

where ∂ sends a rational function and to its divisor of zeros and poles on V , and then one pushes forward via the inclusion V ⊆ XK a to cycles on XK a . In particular, we have surjections n−1 μ⊗n ker ∂ ν  Hzar (XK a , Hn (μ

ν )).

(A.9)

A.2.3. The ν -Bloch map. To get the construction started, we simply consider the commutative diagram of short exact sequences of groups [Blo79, (2.1)] : 0

/



Q(V )∗ /K

a∗

(−) 

/

n−1

V ⊆XK a

Q(V )∗ /K

a∗

n−1

ν

Z

T n ⊆XK a

/

/



Q(V )∗ /Q(V )∗

ν

/0

n−1

V ⊆XK a

∂

 /

0

ν

V ⊆XK a

∂



Z

T n ⊆XK a

/

∂ν

 

Z/ν Z

/0

T n ⊆XK a

(A.10) The snake lemma yields via (A.8) the long exact sequence

ν

0 → ker ∂ → ker ∂ → ker ∂ ν → CHn (XK a )

ν

→ CHn (XK a ) → CHn (XK a )/ν CHn (XK a ) → 0. We obtain a diagram where the top row is a short exact sequence [Blo79, (2.2)] : 0

/



ker ∂ ν  ker ∂



/ ker ∂ ν

/ CHn (X a )[ν ] K

/0

(A.9)



n−1 μ⊗n (XK a , Hn (μ Hzar

ν )) ρν

(A.7)

(  H´e2n−1 (XK a , μ ⊗n t

ν ) (A.11)

J.D. Achter, S. Casalaina-Martin and C. Vial

56

where ρ ν is defined as the indicated composition in the diagram. In fact, we find it convenient to define δ ν to be the image of ρ ν , to obtain the diagram :   ker ∂ / ker ∂ ν /0 / CHn (X a )[ν ] / 0 K ν ker ∂ (A.9)



n−1 μ ⊗n (XK a , Hn (μ Hzar

ν ))



/ δ ν

ρν

(A.7)

(  / H 2n−1 (X a , μ ⊗n K ´ et

ν )

 / H 2n−1 (X a , μ ⊗n K ´ et

ν )/δ ν

/0 (A.12) We now give a name to the vertical arrow on the right ; this map is used tacitly by Bloch in many places, and it will be convenient for us to give this a name : 0

Definition A.2 (The ν -Bloch map). The map λn [ ν ]

CHn (XK a )[ν ] −−−−→ H´e2n−1 (XK a , μ ⊗n t

ν )/δ ν which is the negative of the map defined from (A.12) is the ν -Bloch map in codimension n. Remark A.3. As explained on [Blo79, p. 112], the choice of the negative sign is for compatibility, in the case n = 1, with the natural map coming from the Kummer sequence, as in Proposition A.25. A.2.4. Bloch’s Key Lemma. Consider as in [Blo79, (2.3)] the map ρ : ker ∂

/ H 2n−1 (X a , Z (n)) K

(A.13)

defined as the composition lim ρ

−−ν−−→ H 2n−1 (X a , Z (n)). − ρ : ker ∂ −−−−→ lim (ker ∂/ν ker ∂) −←

K ←− The following lemma, whose proof uses the Weil conjectures (via specialization to finite fields), is key to constructing the Bloch map on CHn (XK a )[∞ ] and the -adic Bloch map on T CHn (XK a ). ν

Lemma A.4 (Bloch’s Key Lemma [Blo79, Lem. 2.4]). The image of ρ is torsion.



What is left tacit by Bloch, but is used in his construction of the Bloch map, is that Lemma A.4 implies : Lemma A.5. The image of the map lim ρ ν : lim (ker ∂/ν ker ∂) → H 2n−1 (XK a , Z (n)) ←− ←− is torsion.

On the Image of the Second l-adic Bloch Map

57

Proof. Since ker ∂ ⊆ lim (ker ∂/ν ker ∂) is a dense subset, and the map lim ρ ν ←− ←− on completions is continuous, the image of lim (ker ∂/ν ker ∂) (i.e., the image of ←− lim ρ ν ) is contained in the closure of the image of ker ∂ (i.e., the image of ρ) ; the ←−

image of the closure of a set is contained in the closure of the image. By Bloch’s Key Lemma A.4, the image of ρ is contained in Tors H´e2n−1 (XK a , Z (n)). Now use t 2n−1 r a , Z (n)) ⊆ H a , Z (n)) is closed. Indeed, find N =  (X (X that Tors H´e2n−1 K K t ´ et which kills the torsion ; multiplication by N is continuous, and the torsion is the inverse image of 0 under this continuous map.  A.2.5. The -adic Bloch map. The -adic Bloch map is defined using the inverse limit of (A.12). We obtain a diagram 0

/ lim

←−



ker ∂ ν ker ∂



/ lim ker ∂ν

/ T CHn (XK a )

←−

/0

(A.9)



n−1 μ ⊗n (XK a , Hn (μ lim Hzar ν )) ←−

0

lim ρν ν

← −

(A.7)

/ lim δν

/ H 2n−1 (XK a , Z (n))

 / H 2n−1 (XK a , Z (n))/ lim δν

/0



 / H 2n−1 (XK a , Z (n))τ

 / H 2n−1 (XK a , Z (n))τ /(lim δν )τ

/0



←−

(lim δν )τ ←−

)



←−

←−

(A.14)

The top row remains short exact after taking the inverse limit, since lim1 (ker ∂/ν ker ∂) = 0. ←− ν Indeed, the system (ker ∂/ ker ∂) is clearly surjective, by virtue of the fact that the terms are defined by quotients of an increasing chain of subgroups. The δ ν , being contained in H´e2n−1 (XK a , μ ⊗n t

ν ), are finite, and thus the middle row, which is obtained as the inverse limit of the bottom row of (A.12), remains a short exact sequence, as well. The bottom row is obtained from the middle row by taking the quotient by torsion subgroups in the left and middle entries. Lemma A.5 and the commutativity of the diagram (A.14) yield (lim δ ν )τ = 0, ←− allowing us to define the -adic Bloch map :

(A.15)

Definition A.6 (-adic Bloch map). The map T λn

 T CHn (XK a ) −−− −→ H 2n−1 (XK a , Z (n))τ ,

which is the negative of the map defined from (A.14) and Lemma A.5, is defined as the -adic Bloch map in codimension n.

J.D. Achter, S. Casalaina-Martin and C. Vial

58

Remark A.7 (-adic Bloch map over the separable closure). Using the fact that ´etale cohomology is invariant under purely inseparable extensions and the fact that the prime-to-char(K) torsion of Chow groups is also invariant under purely inseparable extensions (see e.g., [ACMVb, Lem. 4.10]), the -adic Bloch map over the algebraic closure induces a well-defined map T λn : T CHn (XK ) → H 2n−1 (XK , Z (n))τ . A.2.6. The Bloch map. Bloch defines his map by considering the direct limit of (A.12) :   ker ∂ / / lim ker ∂ ν /0 / CHn (X a )[∞ ] 0 lim ν K −→  ker ∂ −→ (A.9)



n−1 μ⊗n lim Hzar (XK a , Hn (μ

ν )) −→ lim ρν

− →

(A.7)

 2n−1 (XK a , Q /Z (n)) / /0 0 H lim δ ν −→ (A.16) Bloch’s observation is that [Blo79, Lem. 2.4] implies the following : 

)

/ lim δ ν −→



/ H 2n−1 (X a , Q /Z (n)) K

Lemma A.8 ([Blo79, p. 112]). The map lim ρ ν : lim (ker ∂/ν ker ∂) → H 2n−1 (XK a , Q /Z (n)) −→ −→ is the zero map. Proof. To quote Bloch verbatim, the assertion follows from Lemma A.4 using the 2n−1 fact that the image in H´e2n−1 (XK a , μ ⊗n (XK a , Z (n)) is t

ν ) of the torsion in H 2n−1 (XK a , Q /Z (n)). zero in H In a little more detail, we consider the commutative diagram     ker ∂ ker ∂ ker ∂ / / lim ν lim ν −→  ker ∂ ←−  ker ∂ ν ker ∂ lim ρ

← −   H´e2n−1 (XK , Z (n)) t (XK , μ ⊗n = lim H´e2n−1

ν ) ←− t ν

ρν

 / H 2n−1 (X , μ ⊗n K ´ et

ν )

lim ρν

− →  / lim H 2n−1 (X , μ ⊗n K

ν ) −→ ´et

where the horizontal maps are the natural maps. To show the direct limit lim ρ ν is the zero map, it suffices to show that −→ ∂ for all ν, the image of νker H´e2n−1 (XK , μ ⊗n t

ν ) is zero. To this end, let ker ∂ in lim − → ker ∂ ker ∂ α ∈ ν ker ∂ . As we observed before, ν ker ∂ forms a surjective system, so we may ∂ , and then send β to γ ∈ H´e2n−1 (XK , Z (n)). By Lemma lift α to β ∈ lim νker t ←− ker ∂

On the Image of the Second l-adic Bloch Map

59

A.5, γ is torsion. Now we use that the image of torsion, under the composition in the bottom row, is zero. Since this last assertion is not immediately obvious, we sketch a proof here. Let (α1 , α2 , . . . ) be a torsion element of H´e2n−1 (XK , Z ), say of order r . One t can show that in order for (α1 , α2 , . . . ) to be a consistent system, and also be r torsion, one must have a Z/r Z summand of the group H´e2n−1 (XK , Z/ν Z) for all t 2n−1 r ν sufficiently large ν, with αν ∈ Z/ Z ⊆ H´et (XK , Z/ Z). Now since r αν = 0, by definition, this means that in the directed system for the direct limit the image of αν in H´e2n−1 (XK , Z/ν+r Z) is zero (at each step, we multiply by ). Thus the t (XK , Z/ν Z) is zero.  image of αν in lim H´e2n−1 −→ t As a consequence of Lemma A.8, we can define the Bloch map ; we also refer to [CT93] where a construction of the Bloch map is discussed and to the recent [Sch20, §10] where a construction of the Bloch map that avoids Bloch–Ogus theory [BO74] is given. Definition A.9 (Bloch map [Blo79, (2.7)]). The map λn

CHn (XK a )[∞ ] −−−−→ H 2n−1 (XK a , Q /Z (n)), which is the negative of the map defined from (A.16) and Lemma A.8, is the Bloch map in codimension n. In some cases we will write λn [∞ ] for clarity. Remark A.10 (Bloch map over the separable closure). As in Remark A.7, the -adic Bloch map induces a well-defined map T λn : T CHn (XK ) → H 2n−1 (XK , Z (n)). Remark A.11 (Abel–Jacobi map on torsion). Bloch shows in [Blo79, Prop. 3.7] that for a complex projective manifold X, the Bloch map (Definition A.9) agrees with the Abel–Jacobi map on homologically trivial -primary torsion (A.6). We explain below, in Remark A.19, that this implies that the -adic Bloch map (Definition A.6) agrees with the Abel–Jacobi map on homologically trivial cycle classes (A.4). A.3. Suwa’s construction of the l-adic Bloch map In [Suw88], Suwa has given a construction of the -adic Bloch map by simply taking the Tate module associated to the standard Bloch map. We review the construction here, and show it agrees with the construction given in §A.2. This construction is quite convenient in may cases. Later Gros–Suwa [GS88] constructed an extension of the Bloch map to p-torsion ; taking the Tate module gives a p-adic Bloch map. We review this in §A.3.5. A.3.1. Structure of abelian l-primary torsion groups. Let M be an abelian lprimary torsion group for some prime number l. The set of divisible elements Mdiv forms a divisible abelian subgroup, and since divisible abelian groups are injective, we see that M splits as a direct sum M = Mdiv ⊕ (M/Mdiv ). It is a basic fact (e.g., [Fuc70, Ch. IV]) that every divisible abelian l-primary torsion group is a  direct sum of factors of the form Ql /Zl , so that M = ( Ql /Zl ) ⊕ (M/Mdiv ). We

J.D. Achter, S. Casalaina-Martin and C. Vial

60

say that M has finite corank if there in an injective homomorphism M → (Ql /Zl )r for some integer r ≥ 0. It is elementary to show that the following are equivalent : • M has finite corank. • M [l] is finite. • M  (Ql /Zl )r ⊕ A for some integer r ≥ 0 and some finite l-primary torsion group A. A.3.2. -adic cohomology from cohomology with torsion coefficients. In this subsection, we recall a crucial point used in Suwa’s construction of the -adic Bloch map (see Proposition A.13). We start with a general statement about cohomology. Proposition A.12. There is a natural long exact sequence · · · −→ H n−1 (XK , Z ) −→ H n−1 (XK , Q ) −→ H n−1 (XK , Q /Z ) −→ H n (XK , Z )−→ · · ·

(A.17)

In particular, if X is proper, H n (XK , Q /Z )  (Q /Z )r ⊕ A

(A.18)

for some integer r and some finite -primary torsion abelian group A. Proof. Taking the inverse limit over μ of the long exact sequence associated to the short exact sequence of ´etale sheaves

ν

0 −−−−→ Z/μ Z −−−−→ Z/μ+ν Z −−−−→ Z/ν Z −−−−→ 0

(A.19)

gives a long exact sequence ···

/ H n (X , Z ) K

/ H n (X , Z ) K

ν

/ H n (X , Z/ν Z) K

/ H n+1 (X , Z ) K

/ ···

Taking the direct limit over ν of the system of long exact sequences ···

/ H n (X , Z ) K

ν /

···

/ H n (X , Z )

K

ν+1

H n (XK , Z )

 / H n (X , Z ) K

/ H n (X , Z/ν Z) K

/ H n+1 (X , Z ) K

/ ···

/ H n+1 (X , Z ) K

/ ···



 / H n (X , Z/ν+1 Z) K

together with Proposition A.1 provides the desired long exact sequence (A.17). Alternately, this can be obtained from the short exact sequence of sheaves 0 → Z → Q → Q /Z → 0 in the pro-´etale topology ; see [BS15]. The finiteness property (A.18) follows immediately from the finiteness property of H n (XK , Z ) when X is proper. More elementarily, the finiteness property (A.18) is a consequence of the finiteness of H´ent (XK , Z/Z) and the fact (§A.3.1) that any -primary torsion abelian group M such that M [] is finite is isomorphic to (Q /Z )r ⊕A for some integer r and some finite abelian group A. 

On the Image of the Second l-adic Bloch Map

61

From the long exact sequence (A.2), we obtain a short exact sequence / H n−1 (X , Q /Z )/ν K

/ 0. (A.20) Now with the finiteness property (A.18) and the notation therein, (Q /Z )r is the maximal divisible subgroup of H n (XK , Q /Z ), and A = H n (XK , Q /Z )cotors . As Q /Z is divisible, the ν -torsion in Q /Z forms a surjective system. Thus, since A is finite, the associated lim1 for the direct sum vanishes, giving [Suw88, (2.6.2)] : ←−  n−1  / H n (X , Z ) / T H n (X , Q /Z ) / 0. 0 / lim H (XK , Q /Z )/ν K K ← − ν

0

/ H n (X , Z /ν Z ) K

/ H n (X.K , Q /Z )[ν ]

(A.21) Proposition A.13. Assume that X is a proper variety over a field K. Then the morphism H n (XK , Z ) → T H n (XK , Q /Z ) from (A.21), obtained as the inverse limit of the morphisms H n (XK , Z /ν Z ) → H n (XK , Q /Z )[ν ] from (A.20), factors through the quotient H n (XK , Z ) → H n (XK , Z )τ , giving a natural isomorphism : / T H n (X , Q /Z ) H n (XK , Z ) (A.22) KO RRR RRR RRR RRR  R( H n (XK , Z )τ Proof. It is a basic fact (e.g., [Mil06, Prop. 0.19]) that T H n (XK , Q /Z ) is torsionfree. Thus, considering (A.21), we clearly have   lim H n−1 (XK , Q /Z )/ν ⊇ Tors H n (XK , Z ). ← − ν As asserted in [Suw88, (2.6.3)], we claim we have equality, completing the proof. Specifically :   lim H n−1 (XK , Q /Z )/ν = H n−1 (XK , Q /Z )cotors (A.23) ←− ν

= Tors H n (XK , Q /Z ).

(A.24)

Here is an explanation of the claim from Michael Spieß. To prove (A.23), we start with (A.18), that H n−1 (XK , Q /Z ) = (Q /Z )r ⊕ A, with A a finite -primary torsion abelian group A. Then we consider the short exact sequence 0 → ν H n−1 (XK , Q /Z ) → H n−1 (XK , Q /Z ) → H n−1 (XK , Q /Z )/ν → 0. Its limit is

0 → lim ν H n−1 (XK , Q /Z ) → H n−1 (XK , Q /Z ) ←− ν

→ lim H n−1 (XK , Q /Z )/ν → 0. ← − ν To see that this stays short exact in the limit, we argue as above. More precisely, as Q /Z is divisible, ν (Q /Z ) forms a surjective system. We also have that ν A = 0

62

J.D. Achter, S. Casalaina-Martin and C. Vial

for ν sufficiently large. Thus the associated lim1 for the direct sum vanishes. Now, ←− the image of the inclusion is exactly (Q /Z )r , so that lim H n−1 (XK , Q /Z )/ν ←− identifies with A, i.e., with the cotorsion. This completes the proof of (A.23). For (A.24), the long exact sequence i

· · · → H n−1 (XK , Z ) → H n−1 (XK , Q ) → H n−1 (XK , Q /Z ) → H n (XK , Z )→H n (XK , Q ) → · · · of Proposition A.12 provides an identification Tors H n (XK , Z ) = H n−1 (XK , Q /Z )/ Im(i). The image of i, being the image of a divisible group, is divisible in H n−1 (XK , Q /Z ), and since Tors H n (XK , Z ) is finite, Im(i) must be the maximal divisible subgroup. This means that we have H n−1 (XK , Q /Z )/ Im(i) = H n−1 (XK , Q /Z )cotors .  Lemma A.14 (Functoriality of (A.22)). Let f : X → Y be a morphism of smooth proper varieties over K. Then there are commutative diagrams : H n (XK , Z (dY − dX ))τ f∗



/ T H n (X , Q /Z (dX − dY )) K f∗



H n+2(dX −dY ) (YK , Z )τ



H n (XK , Z )τ O



f∗

H n (YK , Z )τ

 / T H n+2(dY −dX ) (Y , Q /Z ) K / T H n (X , Q /Z ) KO f∗



/ T H n (Y , Q /Z ) K

where the horizontal arrows are the isomorphisms from (A.22). More generally, given a correspondence Γ : X  Y , we obtain the corresponding commutative diagrams for Γ∗ and Γ∗ . Proof. The commutativity of the diagrams follows from the definitions.



Example A.15. We note that while there are natural inclusions f∗ T H n (XK , Q /Z ) ⊆ T f∗ H n (XK , Q /Z ),

(A.25)

f ∗ T H n (YK , Q /Z ) ⊆ T f ∗ H n (YK , Q /Z ),

(A.26)

these need not be equalities. For instance, consider the case where X = Y = E is an elliptic curve over K, and f : X → Y is the multiplication by  map. Then the containment f∗ T H 1 (XK , Q /Z ) ⊆ T f∗ H 1 (XK , Q /Z ) is the containment (Z )2  (Z )2 . In accordance with the example above, the inclusions (A.25) and (A.26) have torsion cokernels in the case where n = 1 :

On the Image of the Second l-adic Bloch Map

63

Lemma A.16. Let f : X → Y be a morphism of smooth proper varieties over K. Then we have isomorphisms 

(f∗ T H 1 (XK , Q /Z )) ⊗Z Q −→ T f∗ H 1 (XK , Q /Z ) ⊗Z Q , 

f ∗ T H 1 (YK , Q /Z ) ⊗Z Q −→ T f ∗ H 1 (YK , Q /Z ) ⊗Z Q . More generally, given a correspondence Γ : X  Y , we obtain the corresponding commutation relations for Γ∗ and Γ∗ . Proof. We will establish the first isomorphism regarding the push-forward. The second is similar, as are the cases of correspondences. From (A.25), we only need to show that the morphism is surjective. We consider the morphism f∗ : H 1 (XK , Q /Z ) → f∗ H 1 (XK , Q /Z ) ⊆ H 1+2(dY −dX ) (YK , Z /Q ). Then, using that H 1 (XK , Q /Z ) is divisible (it is the cohomology of the abelian variety Pic0X/K ), and that the image of a divisible group is divisible, we apply [Suw88, Lem. 1.4] that given a surjective homomorphism φ : M → N of divisible abelian -primary torsion groups, with N of finite corank, the associated map T φ ⊗ 1 : T M ⊗Z Q → T N ⊗Z Q is surjective.  A.3.3. Suwa’s -adic Bloch map. Definition A.17 (Suwa’s -adic Bloch map [Suw88, (2.6.5)]). Assume X is a smooth projective variety over K and  is a prime not equal to char K. The -adic Bloch map for codimension-n cycles is the map T λn

 T CHn (XK ) −−− −→ H 2n−1 (XK , Z (n))τ

obtained by applying T to the Bloch map λn : CHn (XK )[∞ ] → H 2n−1 (XK , Q /Z (n)) and making the identification T H n (XK , Q /Z ) = H n (XK , Z )τ from Proposition A.13. A.3.4. The -adic Bloch map and Suwa’s construction. Here we show that Suwa’s construction of the -adic Bloch map agrees with the direct construction. Proposition A.18. Let X be a smooth projective variety over K. Then the -adic Bloch maps of Definition A.17 and Definition A.6 coincide. Proof. We first observe that we have the following commutative diagram CHn (XK )[ν ]  (CHn (XK ) [∞ ])[ν ] _ )  CHn (XK )[∞ ]

λn [ ν ]

/ H 2n−1 (X , Z/ν Z)/δ ν K  / H 2n−1 (X , Q /Z )[ν ] K _ u  / H 2n−1 (X , Q /Z ) K

(λn [ ∞ ])[ ν ]

λn [ ∞ ]

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J.D. Achter, S. Casalaina-Martin and C. Vial

The bottom arrow, λn [∞ ] = λn is the Bloch map (Definition A.9), and the bottom square describes the map induced on the ν -torsion for the Bloch map. In any case, the bottom square in the diagram is commutative by definition. The top map in the diagram is the ν -Bloch map (Definition A.2). The vertical arrows from the top row to the bottom row are the canonical morphisms from the definition of a direct limit. Thus the outer square is commutative. It is only left to describe the vertical arrows from the top to the middle row. These are defined by the fact that CHn (XK )[ν ] and H i (XK , Z/ν Z)/δ ν are ν -torsion groups, so that the images of the vertical arrows from the top row to the bottom row are contained in the ν -torsion. Taking the inverse limit in the top square, we obtain a commutative diagram T CHn (XK ) T CHn (XK )

T (λn [ ν ])

T (λn [ ∞ ])

/ H i (X , Z )τ K

 / T H i (X , Q /Z ) K

where the top row is the definition of -adic Bloch map from Definition A.6, while the bottom row is Suwa’s definition of the -adic Bloch map (Definition A.17). The only thing to check is that the vertical arrow in the diagram is the canonical isomorphism from Proposition A.13. But this is clear from the construction of this isomorphism in Proposition A.13 via the limit of the maps on the finite levels.  Remark A.19 (-adic Abel–Jacobi map). The same argument as in the proof of Proposition A.18 shows that for a complex projective manifold X, the -adic Abel– Jacobi map T AJ (A.4) is equal to the Tate module of the Abel–Jacobi map on torsion AJ[∞ ] (A.6) ; i.e., T AJ = T (AJ[∞ ]). Now using the fact that the Bloch map (Definition A.9) agrees with the Abel–Jacobi map on homologically trivial primary torsion (A.6) ([Blo79, Prop. 3.7], Remark A.11), it follows that the -adic Bloch map T λ2 (Definition A.6) agrees with the -adic Abel–Jacobi map T AJ on homologically trivial cycle classes (A.4). A.3.5. Gross–Suwa’s p-adic Bloch map. Now suppose that K is a perfect field of characteristic p > 0, and recall the notation concerning p-adic cohomology groups. With these conventions, Gros and Suwa have secured p-adic versions of the results reviewed above. So, Proposition A.1 holds by definition ; the proof of Proposition A.12 is valid at p, provided one replaces (A.19) with the exact sequence of ´etale sheaves / Wμ Ωr / Wμ+ν Ωr / Wν Ωr /0 0 X,log

X,log

X,log

(see also [GS88, Prop. I.4.18] for (A.18)). Gros and Suwa construct a group homomorphism λn = λnp : CHn (X)[p∞ ] → H 2n−1 (X, Qp /Zp (n)) [GS88, Def. III.1.25], and, as in Definition A.17, by applying the Tate module obtain a map Tp CHn (XK )

Tp λn

/ H 2n−1 (X , Zp (n))τ . K

On the Image of the Second l-adic Bloch Map

65

Remark A.20. Moreover, Lemma A.14 holds for p-adic coefficients, as well. Finally, Lemma A.16 holds with Qp /Zp -coefficients since, as we have seen, H 1 (XK , Qp /Zp ) is p-divisible of finite corank. A.4. Properties of the Bloch maps In this section we fix a field K. The aim of this section consists simply, for future reference, in restating known results due to Bloch [Blo79] concerning the usual Bloch map λn : CHn (XK )[l∞ ] → H 2n−1 (XK , Q /Zl (n)) in the setting of the l-adic Bloch map Tl λn : T CHn (XK ) → H 2n−1 (XK , Zl (n))τ , as well as the finite level Bloch maps. All statements for the l-adic Bloch map are direct consequences of the fact that Tl λn is simply obtained from λn by applying the Tate module functor. Alternatively for  = char(K), using the direct definition of the -adic Bloch map via the ν -Bloch maps, the proofs in [Blo79] carry over directly. In fact, the proofs in [Blo79] regarding the Bloch map go directly through the corresponding assertions about the ν -Bloch maps. Proposition A.21 (Flat pull back and proper push forward). The Bloch map, the l-adic Bloch, and for all primes  = char(K), the ν -Bloch maps, are functorial for flat pull back and proper push forward. Proof. The case of the Bloch map is [Blo79, Prop. 3.3] for  = char(K) and [GS88, III. Prop. 2.3] in the p-adic case. The proposition for the l-adic Bloch map can be obtained simply obtained by applying Tl to the case of the Bloch map. For the ν -Bloch maps, this follows directly from the proof of [Blo79, Prop. 3.3].  Proposition A.22 (Bloch maps and Galois-equivariance). The Bloch map, the l-adic Bloch map, and for  = char(K), the ν -Bloch maps, are Aut(K/K)-equivariant. Proof. Fix  = char(K). Let L/K be a finite Galois extension, and let X/K be a smooth projective variety. Then for each σ ∈ Gal(L/K), applying the previous proposition to the morphisms σ : XL → XL given by the Galois descent data, shows that each of the various Bloch maps is Gal(L/K)-equivariant. The general case follows by passing to the limit over all finite Galois extensions. For the p-adic case, this is [GS88, III. Prop. 2.1].  Proposition A.23 (Bloch maps and correspondences). The Bloch map, l-adic Bloch map, and for all primes  = char(K), the ν -Bloch maps, are compatible with the action of correspondences. Precisely, in the case of the l-adic Bloch map, let X and Y be smooth projective varieties over K and let Γ be a cycle on X ×K Y of codimension dim Y + n − m. Then the following natural diagram Tl CHm (YK ) Tl λm



H 2m−1 (YK , Zl (m))τ is commutative.

Γ∗

/ Tl CHn (X ) K Tl λn

Γ∗

 / H 2n−1 (X , Zl (n))τ K

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Proof. The case of the Bloch map is [Blo79, Prop. 3.5] for  = char(K) and [GS88, III. Prop. 2.9] for the p-adic case. The proposition for the l-adic Bloch map is simply obtained by applying Tl to the case of the Bloch map. For the ν -Bloch  maps, this follows directly from the proof of [Blo79, Prop. 3.5]. Proposition A.24 (Bloch maps and specialization). The Bloch map, -adic Bloch map, and ν -Bloch maps are compatible with specialization. Precisely, in the case of the -adic Bloch map, given a local ring R with fraction field K and residue field K0 , and a smooth projective morphism X → Spec R with generic fiber X and special fiber X0 , the following diagram / T CHn (X 0 )

T CHn (X) T λn

 H 2n−1 (X, Z (n))τ

T λn



 / H 2n−1 (X 0 , Z (n))τ

is commutative for  prime to char(X0 ). Here, X and X 0 denote the base-changes of X and X0 to K and K 0 , respectively. The top horizontal arrow is obtained by applying T to the specialization map ( e.g., [Ful98, Ex. 20.3.5]), while the bottom horizontal arrow is obtained from smooth proper base-change. Proof. The case of the Bloch map is [Blo79, Prop. 3.8]. The proposition in the case of the -adic Bloch map is simply obtained by applying T to the case of the Bloch map. For the ν -Bloch maps, this follows directly from the proof of [Blo79, Prop. 3.8].  Proposition A.25 (Bloch maps and Kummer sequence). Let X be a smooth projective variety over K. The l-adic Bloch map Tl λ1 : Tl CH1 (XK ) → H 1 (XK , Zl (1)) is the natural isomorphism arising from the Kummer sequence lν

0 −−−−→ Z/lν Z(1) −−−−→ Gm −−−−→ Gm −−−−→ 0 and the identification CH1 (XK ) = H 1 (XK , Gm ). Proof. The case of the Bloch map is [Blo79, Prop. 3.6] in the case  = char(K) and [GS88, III. Prop. 3.1, Cor. 3.2] in the p-adic case. The proposition is simply obtained by applying Tl to the case of the Bloch map, and noting that H 1 (XK , Zl (1)) is torsion-free.  Proposition A.26 (Bloch maps and Albanese morphism, Roitman’s theorem). Let X be a smooth projective variety of dimension d over K ; if l = char(K), assume

On the Image of the Second l-adic Bloch Map

67

K is perfect. Then the following diagram Tl λd

/ H 2d−1 (X , Zl (d))τ Tl CHd (XK ) K RRR RRR RRR RRR alb R( Tl AlbX is commutative, where alb is obtained by applying Tl to the map CHd (XK ) → AlbX (K) mapping a zero cycle on XK to the sum of the corresponding points on the Albanese. Moreover, Tl λd is an isomorphism. Proof. The case of the Bloch map is [Blo79, Prop. 3.9] for  = char(K) and [GS88, III. Prop. 3.14, Cor. 3.17] in the p-adic case. The commutativity of the diagram is simply obtained by applying Tl to the case of the Bloch map. Finally, that alb : Tl CHd (XK ) → Tl AlbX is an isomorphism is due to Roitman [Blo79, Thm. 4.1]. Note that in loc. cit. it is stated that alb : CHd (XK ) → AlbX (K) is an isomorphism on torsion for K algebraically closed ; this implies the needed fact that CHd (XK ) → AlbX (K) is an isomorphism on prime-to-char(K) torsion for K separably closed. Indeed, this follows from the general fact that the prime-tochar(K) torsion in the Chow group of a scheme of finite type over K is invariant under purely inseparable extensions (see, e.g., [ACMVb, Lem. 4.10]), and the fact that the prime-to-char(K) torsion of an abelian variety is invariant under extension of separably closed fields.  Finally, we have the following l-adic analogue of a result of due to Merkurjev– Suslin [MS82]. Proposition A.27 (Injectivity of the second Bloch map). Let X be a smooth projective variety over K ; if l = char(K), assume K is perfect. The second l-adic Bloch map Tl λ2 : Tl CH2 (XK ) −→ H 3 (XK , Zl (2))τ is injective, as are the second ν -Bloch maps λ2 [ν ] : CH2 (XK )[ν ] → H 3 (XK , μ ⊗2

ν )/δ ν for  = char(K). Proof. The injectivity of the Bloch map λ2 in the case  = char(K) is due to Merkurjev–Suslin [MS82] ; see also [Mur85, Prop. 9.2] and [CTR85, Prop. 3.1 and Rmk. 3.2]. For the p-adic case this is [GS88, III. Prop. 3.4]. The injectivity of the second l-adic Bloch map follows via applying Tl to the Bloch map. For  = char(K), the fact that the second ν -Bloch maps are injective is [Mur85, Prop. 6.1]. Indeed, we consider diagram (A.12). Using K-theoretic tech-

J.D. Achter, S. Casalaina-Martin and C. Vial

68

niques, Murre showed that (A.12) can be factored as follows [Mur85, Rem. 6.3] :   ker ∂ / ker ∂ ν / /0 / CHn (X a )[ν ] 0 K ν ker ∂

0

 / ker αν

 n−1 / Hzar (XK a , Kn /ν )

αν

/ CHn (X

K

a

)[ν ]

/0

∼ = βν

 n−1 μ ⊗n (XK a , Hn (μ Hzar

ν )) 

/ δ ν

γν

 / H 2n−1 (X a , μ ⊗n K ´ et

ν )

−λn [ ν ]

 / H 2n−1 (X a , μ ⊗n K ´ et

ν )/δ ν

/0 (A.27) Here Kn is the sheaf on XK a associated to the pre-sheaf that assigns to each Zariski open subset U ⊆ XK a the algebraic K-group Kn (OXK a (U )) (see, e.g., [Mur85, §4.1]). The maps αν and βν are defined in [Mur85, Rem. 6.3], and the map γν is Murre’s notation for the map defined in (A.7). The fact that βν is an isomorphism is explained in [Mur85, Rem. 6.3] using [MS82], and this isomorphism then defines the top vertical map in the center of (A.27). The fact that the top right square of the diagram is commutative comes from the construction of αν in [Mur85, Rem. 6.3] ; in fact, the commutativity of (A.27) is tacitly asserted in [Mur85], so that the construction of the Bloch map given there agrees with that given in [Blo79]. The rest of (A.27) can be established via a diagram chase. The key point in the case n = 2 is that Murre shows in [Mur85, Prop. 6.1, Cor. 5.4(c)] that γν is an inclusion. Applying the snake lemma to the bottom half of (A.27) shows that the second ν -Bloch maps are injective. Note that taking the direct limit of the injective ν -Bloch maps gives Murre’s proof that λ2 is injective, while taking the inverse limit gives another proof that T λ2 is injective.  0

A.5. Restriction of the Bloch map to algebraically trivial cycle classes Let X be a smooth projective variety over a field K. From (A.17), we have a diagram with exact row (see, e.g., [GS88, (3.33)] for the p-adic case) CHn (XK )[l∞ ] RR RR λn RR R(  0 / H 2n−1 (XK , Zl (n)) ⊗Zl Ql /Zl / H 2n−1 (XK , Ql /Zl (n)) / H 2n (XK , Zl (n)) (A.28) where the dashed arrow is, up to sign, the cycle class map ([CTSS83, Cor. 4], [GS88, Prop. III.1.16 and Prop. III.1.21]). Since algebraically trivial cycles are ho-

On the Image of the Second l-adic Bloch Map

69

mologically trivial, it follows that the image of An (XK )[l∞ ] under λn is contained in H 2n−1 (XK , Zl (n)) ⊗Zl Ql /Zl ⊆ H 2n−1 (XK , Ql /Zl (n)). In other words, when we restrict the Bloch map to algebraically trivial cycle classes we obtain a map λn : An (XK )[l∞ ] −→ H 2n−1 (XK , Zl (n)) ⊗Zl Ql /Zl ⊆ H 2n−1 (XK , Ql /Zl (n)). (A.29) Proposition A.28 (Codimension-1). Let X be a smooth projective variety over K ; if l = char(K), assume K is perfect. The Bloch map (A.29) λ1 : A1 (XK )[l∞ ] −→ H 1 (XK , Zl (1)) ⊗Zl Ql /Zl is an isomorphism, and taking Tate modules yields an isomorphism Tl λ1 : Tl A1 (XK ) −→ H 1 (XK , Zl (1)). Proof. This follows from Proposition A.25. Indeed, we start with the fact that CH1 (XK )/ A1 (XK )  PicX/K (K)/ Pic0X/K (K) = NS(XK ) is a finitely generated Z-module. From this we can conclude that Tl A1 (XK ) = Tl CH1 (XK ). This gives the result for Tl λ1 . Then from the identification A1 (XK ) = Pic0X/K (K), the torsion and Tate modules are free of the same rank, and so we have A1 (XK )[l∞ ] = Tl A1 (XK ) ⊗Zl Ql /Zl . This gives the result for the Bloch map.  Proposition A.29 (Bloch maps and Albanese morphisms, Roitman’s theorem). Let X be a smooth projective variety of dimension d over K. Then the following diagram  λd / H 2d−1 (X , Ql /Zl (d)) Ad (XK )[l∞ ] ZZZZ / H 2d−1 (XK , Zl (d)) ⊗Zl Ql /Zl  K ZZZZZZZ ZZZZZZZ ZZZZZZZ ZZZZZZZ alb ZZZZZZZ ZZZ, AlbX [l∞ ] (A.30) is commutative, where alb is obtained by restricting to Ad (XK ) the map CHd (XK ) → AlbX (K) mapping a zero cycle on XK to the sum of the corresponding points on the Albanese. Moreover, λd , as well as the inclusion H 2d−1 (XK , Zl (d)) ⊗Zl Ql /Zl → H 2d−1 (XK , Ql /Zl (d)), are isomorphisms. Taking Tate modules, we obtain a commutative diagram d

Tl λ / H 2d−1 (X , Zl (d))τ Tl Ad (XK ) K QQQ QQQ QQQ alb QQQQ ( Tl AlbX

Moreover, Tl λd is an isomorphism.

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J.D. Achter, S. Casalaina-Martin and C. Vial

Proof. This just follows from Proposition A.26 and the fact that Ad (XK )[l∞ ] = CHd (XK )[l∞ ]. Indeed, the commutativity of (A.30) follows from this and the discussion above. Then Proposition A.26 and the fact that Ad (XK )[l∞ ] = CHd (XK )[l∞ ] implies that the composition of the top row is an isomorphism. This forces λd to be an isomorphism. The result for Tate modules follows immediately.  Finally, we have the following l-adic analogue of a result due to Merkurjev– Suslin [MS82]. Proposition A.30 (Injectivity of the second Bloch map). Let X be a smooth projective variety over K. The second Bloch map λ2 : A2 (XK )[l∞ ] −→ H 3 (XK , Zl (2)) ⊗Zl Ql /Zl , the second l-adic Bloch map Tl λ2 : Tl A2 (XK ) −→ H 3 (XK , Zl (2))τ , and for all primes  = char(K), the second ν -Bloch maps λ2 [ν ] : A2 (XK )[ν ] → H 3 (XK , μ ⊗2

ν )/δ ν , are injective. Proof. This just follows from Proposition A.27 and from the inclusion of A2 (XK ) ⊆  CH2 (XK ). Acknowledgment The first- and second-named authors were partially supported by grants 637075 and 581058, respectively, from the Simons Foundation.

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Jeffrey D. Achter Colorado State University Department of Mathematics Fort Collins, CO 80523, USA e-mail: [email protected] Sebastian Casalaina-Martin University of Colorado Department of Mathematics Boulder, CO 80309, USA e-mail: [email protected] Charles Vial Universit¨ at Bielefeld Fakult¨ at f¨ ur Mathematik Postfach 100131 D-33501 Bielefeld, Germany e-mail: [email protected]

Rational Curves and MBM Classes on Hyperk¨ahler Manifolds: A Survey Ekaterina Amerik and Misha Verbitsky Abstract. This paper deals with rational curves and birational contractions on irreducible holomorphically symplectic manifold. We survey some recent results about minimal rational curves, their deformations, extremal rays asahler cone. sociated with these curves, and the geometry of the K¨ Mathematics Subject Classification (2010). 14E30, 32J27. ahler manifolds, rational curves, MBM classes. Keywords. Hyperk¨

1. Introduction This paper deals with rational curves and birational contractions on hyperk¨ahler manifolds. Here “hyperk¨ ahler holomorphically symplecahler” means compact K¨ tic manifold. We survey some recent results about minimal rational curves, their deformations, extremal rays associated with these curves, and the geometry of the K¨ahler cone. It is known since Shafarevich and Pyatetski-Shapiro’s work [PS-S] in the algebraic case and Looijenga-Peters [LP] in the compact K¨ahler case that some important aspects of the geometry of K3 surfaces are governed by smooth rational curves on these surfaces. By adjunction, the square of such a curve is always equal to −2. One calls it a (−2)-curve. A fundamental theorem directly concerned with the subject of this paper states that a line bundle on a K3 surface is ample if and only if it is of positive square and positive on all (−2)-curves. We also have an analogue in the non-algebraic case, with an ample line bundle replaced by a K¨ahler class. In other words, the orthogonal hyperplanes to the (−2)-curves bound the ahler) cone inside the positive cone in N S(X) ⊗ R (resp. HR1,1 (X)). ample (resp. K¨ The number of (−2)-curves on a K3 surface S is not necessarily finite. However H. Sterk [St] proved that there are finitely many of them up to the action of Aut(S). In the projective case he actually proved a more precise result: the group Aut(S) has a polyhedral fundamental domain on the “rational hull of the ample © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Farkas et al. (eds.), Rationality of Varieties, Progress in Mathematics 342, https://doi.org/10.1007/978-3-030-75421-1_3

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cone” Nef + (X) (which by definition is the ample cone to which one attaches the part of the boundary defined over the rationals). In this survey we consider the irreducible holomorphic symplectic manifolds ahler manifolds, which are higher-dimensional (IHSM), also known as simple hyperk¨ analogues of K3 surfaces. Definition 1.1. An irreducible holomorphic symplectic manifold is a simply-connected compact K¨ ahler manifold X whose space of holomorphic two-forms H 2,0 (X) is generated by a symplectic (that is, nowhere degenerate) form σ. In particular such an X has even dimension 2n and trivial canonical class. There are, at present, two infinite series of families and two “sporadic” families of IHSM known, and so far we are unable to answer the question whether there are other examples. Example. If S is a K3 surface, then the nth punctual Hilbert scheme S [n] is an IHSM, see [Be]. The deformations of S [n] are, at this date, the most studied hyperk¨ahler manifolds. One calls them IHSM of K3 type. The analogy with a surface is especially striking because on the second cohomology H 2 (X, Z) of such a manifold X there is an integral quadratic form q, the Beauville–Bogomolov form of signature (1, b2 − 3) on H 1,1 (X, R) (and (3, b2 − 3) on the whole second cohomology). This form is defined by integrating differential forms against appropriate powers of σ and σ ¯ in [Be], but in fact is of topological nature by [F]. Example. Let S (n) be the nth symmetric power of a K3 surface S and HC : S [n] → S (n) the Hilbert-Chow map. The cohomology class of the exceptional divisor of HC is divisible by 2, and the second cohomology group of S [n] is naturally isomorphic to the direct sum of H 2 (S, Z) and Ze, where e is one half of the exceptional divisor. This direct sum turns out to be orthogonal with respect to q. Moreover q restricts to the intersection form on H 2 (S, Z), and q(e) = −2(n − 1). The second cohomology lattice remains the same under deformations, so this description can be used to study the IHSM of K3 type. The difference between S [n] and an arbitrary deformation of it is, roughly speaking, that the class e does not have to be a Hodge class on the latter. All K¨ahler classes on an IHSM X have positive Beauville–Bogomolov square. The set of classes with positive square in H 1,1 (X, R) has two connected components. We call the positive cone the component containing the K¨ahler classes. The following analogue of the result of Shafarevich and Pyatetski-Shapiro and that of Looijenga and Peters is available. ahler classes on an Theorem 1.2 (Huybrechts, Boucksom; [Bou, Hu2]). The K¨ IHSM X are the elements of the positive cone Pos(X) which are strictly positive on all rational curves on X.

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Thus the K¨ ahler cone is cut out within the positive cone by the orthogonal hyperplanes to the classes of the rational curves. In the same way in the projective case, the ample classes are elements of the Neron-Severi group which have positive square and are positive on all the rational curves. If we pursue the analogy with the surface case, the next natural question is: what can be said about the possible Beauville–Bogomolov squares of these curves? Note that using the Beauville–Bogomolov form, the curve classes are naturally viewed as classes in H 2 (X, Q), that is why we can talk of the Beauville–Bogomolov square of such a class. It is a rational number. One might also wonder about a possible connection between the precise value of this square and the geometry of the corresponding rational curves on X. Hassett and Tschinkel initiated the study of this question in [HT1], dealing with projective IHS fourfolds, especially those which are of K3 type. In the projective case, the classes we want to understand are exactly those generating the “extremal rays” of the Mori cone in birational geometry. Each curve class has two numerical invariants: its Beauville–Bogomolov square and its “denominator”, that is, the image in the discriminant group L∗ /L, where L = H 2 (X, Z), and H2 (X, Z) is viewed as its dual embedded in H 2 (X, Q), just as we have mentioned above. For IHS manifolds of K3 type of dimension 2n, the discriminant group is cyclic of order 2n − 2. Hassett and Tschinkel have observed that in the fourfold examples one sees three types of extremal rays: integral with square (−2), half-integral with square −1/2 and half-integral of square −5/2. They conjectured that these are indeed all the extremal rays of the cone of curves. Equivalently, the ample cone consists of such classes in N S(X) ⊗ R which have positive square and are positive on all such curves. By the Kawamata–Shokurov base-point-freeness theorem (see, e.g., [KM]), extremal rays on a projective IHS manifold can be contracted. Hassett and Tschinkel, describing several examples of such contractions, have also conjectured that one could read the geometry of the contraction locus from the numerical invariants of the ray. Namely a half-integer class of square −5/2 should give rise to a Lagrangian plane (contracted to a point) and every other class to a family of rational curves over a K3 surface (contracted to the K3 surface). Part of these conjectures has been proved in the subsequent papers by Hassett and Tschinkel; also an effort has been made to state and prove similar conjectures in higher dimension. The initial conjecture of Hassett and Tschinkel on the numerical description of the ample cone on an IHS manifold of K3 type, however, turned out to be false. A correct formula in terms of the Mukai lattice has been given by Bayer and Macri [BM] for punctual Hilbert schemes and more generally, moduli spaces of stable sheaves on a K3 surface. The extension from a punctual Hilbert scheme to its deformation can be found in [BHT]. Bayer and Macri’s work is highly involved and is certainly out of the scope of the present survey; we shall see, however, that in low dimensions one can recover their results by fairly elementary means.

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Concerning the finiteness property, the extension of Sterk’s theorem to the higher-dimensional case is known as the Morrison–Kawamata cone conjecture and has attracted considerable attention since early 1990s. Conjecture 1.3 ([Ka, Mor2]). Let X be a projective manifold with trivial canonical class. Then Aut(X) acts with finitely many orbits on the set of faces of its ample cone. Moreover denote by Nef + (X) the “rational hull” of the ample cone, that is, the smallest convex cone containing the ample cone and all rational points of its boundary. Then Aut(X) has a rational polyhedral fundamental domain on Nef + (X). In this generality, the conjecture is surprisingly difficult and still wide open even for Calabi-Yau threefolds (though there are some partial results, notably by Kawamata [Ka] and Totaro [T]). Apart from Sterk’s original result, little has been known until recently. For holomorphic symplectic manifolds, the “birational” version of this conjecture has been established by Markman in [Mar]; later Markman and Yoshioka in [MY] have proved the cone conjecture for IHS manifolds of K3 or generalized Kummer type. The cone conjecture for IHS manifolds in general has been proved in [AV2], [AV3]. In this survey we summarize this contribution and some of our other ones to the field in recent years. Our techniques are largely based on deformations, in particular to the non-algebraic manifolds which often have fairly simple geometry. Deformation theory in the hyperk¨ahler context is well understood and many computations are easy and elementary. The definition of MBM classes arises naturally from the urge to put the notion of an extremal rational curve in the deformationinvariant context. The crucial ingredient which made it possible to prove the cone conjecture for IHS manifolds was ergodic theory, more particularly Ratner’s theorems about orbits of actions of unipotent groups on homogeneous spaces. These techniques have been used initially by the second-named author in order to prove the density of orbits of the mapping class group action on the Teichm¨ uller space ([V2]), and can be applied successfully to the cone conjecture setting in [AV2]. Unless otherwise specified, by a rational curve we mean the image of a generically injective map from P1 , so that it can be singular, but not reducible.

2. MBM classes: equivalent definitions and basic properties 2.1. Deforming rational curves: first remarks The Riemann-Roch formula implies that any divisor of square −2 on a K3 surface is effective, up to a ± sign. However not all Hodge classes of square −2 on a K3 surface, up to a ± sign, are represented by (−2)-curves. Indeed, the corresponding effective divisor can be reducible. Nevertheless, for any (−2)-class z on a K3 surface X there is a deformation X  where H 1,1 (X  ) ∩ H 2 (X  , Q) = z, or, equivalently, the Picard group is cyclic

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and generated by z. Indeed, the locus where a class α remains of type (1, 1) is equal to α⊥ in the space of local deformations Def(X), identified with a neighbourhood 1 of zero in H 1 (X, ΩX ) ∼ = H 1 (X, TX ), and a general element of the hyperplane α⊥ is not orthogonal to any other integral class. Also the same is true globally in the period domain, interpreted as an open subset of a quadric in PH 2 (X, C) (Definition 2.7). The orthogonality here is taken with respect to the intersection form; see Subsection 2.2 for the relevant definitions. On such a deformation, z (up to a ± sign) is necessarily represented by a smooth rational curve, and there are no other curves on X. The notion of an MBM class is inspired by this observation, which works in the same way in higher dimension. Indeed, similar arguments can be applied to a higher-dimensional IHSM X. The space of deformations of X where a class α ∈ H 2 (X, Z) is of type (1, 1) can be described as the hyperplane α⊥ (where the orthogonality is taken with respect to the Beauville–Bogomolov form). A general deformation X  of this type satisfies H 1,1 (X  ) ∩ H 2 (X  , Q) = α. We can now distinguish two cases: it can happen that on such a generic deformation X  , some multiple of α is represented by a curve (recall that we view curve classes as rational (1, 1)-classes, via the Beauville–Bogomolov form), or that no multiple of α is represented by a curve1 . By a theorem of Huybrechts-Boucksom (Theorem 1.2), in the second case every class in the positive cone of X  is K¨ahler. In the first case, a multiple of α is represented by a rational curve and α⊥ defines the unique wall of the K¨ahler cone of X  . Let us suppose from now on that we are in the first case. The crucial observation is that by deformation theory of rational curves in IHS manifolds, this is also the case for any deformation X  with Picard group generated by α over the rationals. Let us briefly summarize the basic facts on the deformations of rational curves (see, e.g., [AV1] for details). Proposition 2.1. Let C ∈ X be a rational curve on an IHSM X of dimension 2n. Then • C deforms within X in a family of dimension at least 2n − 2. • If, moreover, C is minimal (that is, C cannot be bent-and-broken2 ), then C deforms within X in a family of dimension exactly 2n − 2. • Finally, if X → Def(X) denotes the local universal family, then a minimal rational curve C deforms exactly to those fibers Xt that keep the cohomology class of C of type (1, 1). Remark. The first part is due to Ran [Ran], Cor. 5.1, with the proof below communicated to us by Markman. 1 As we are dealing with manifolds which are not necessarily projective, this second case is indeed possible. 2 This holds if C is of minimal degree, with respect to a K¨ ahler class, among the rational curves in the uniruled subvariety of X covered by the deformations of C.

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The idea of proof of the proposition is that the deformation spaces of rational curves on IHS manifolds have a dimension greater (by one) than what we can expect from the Riemann-Roch formula. An extra parameter comes from the hyperk¨ ahler structure in the differential-geometric setting, the twistor family. In fact, it follows from the Calabi-Yau theorem that for any IHS X = (M, I), where M denotes the underlying differentiable manifold and I the complex structure defining X, there is a hyperk¨ahler metric g on M inducing a whole 2-sphere (or complex projective line) of complex structures aI +bJ +cK, where I, J, K multiply like quaternions and a2 + b2 + c2 = 1. Every IHS X thus comes as a member of a “twistor family” Tw(X) → P1 . We can consider the deformations of C in Tw(X): they are the same as those of C in X since the class of C is no longer Hodge in the neighbouring complex structures (most of which carry no curves at all). Since the dimension of Tw(X) is greater by one, we also get an extra parameter by Riemann-Roch (which involves the rank of the pull-back of the tangent bundle of the ambient manifold). In the proof of the second assertion, the rational quotient fibration, also known as MRC fibration, of the uniruled subvariety Z ⊂ X covered by deformations of C is involved as a main tool. One obtains the following proposition as a byproduct. Proposition 2.2. The codimension of Z in X is equal to the relative dimension of its MRC fibration and to the corank of the restriction of the symplectic form σ to Z at a general point. In particular, Z is co-isotropic, and if Z is a divisor then it is birationally a P1 -bundle. To get the statement on the walls of the K¨ahler cone from proposition 2.1, one applies it to the minimal rational curves which always have class proportional to α on a generic deformation X  preserving α as a Hodge class: indeed on such X  there are simply no other Hodge classes. One gets that the minimal rational curves deform together with their cohomology class α and thus define a wall of the K¨ ahler cone on X  . On X itself, the group of Hodge classes is greater and rational curves in the class α are not necessarily minimal. But though α⊥ does not necessarily define a wall of the K¨ahler cone of X, it turns out that it is closely related to such walls. We call α an MBM (“monodromy birationally minimal”) class. We shall give the precise definition 2.10, and we shall also formulate a theorem explaining this name. In order to do this, we need some discussion concerning the parameter spaces for IHS manifolds. Before dealing with the parameter spaces in the next subsection, we recall one important source of rational curves on IHS manifolds. Theorem 2.3 (Boucksom, [Bou, Prop. 4.7]). Every prime divisor with negative Beauville–Bogomolov square on an IHS manifold is uniruled. The converse is not true in general. However in this survey we are particularly interested in the description of the Kahler cone inside the positive cone; if we want

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to describe its walls as orthogonals to some classes, then by the Hodge index theorem these classes necessarily have negative square. In Markman’s terminology [Mar], the prime divisors of negative Beauville– Bogomolov square are called prime exceptional divisors. Indeed, by a result of Druel [D] they can be contracted, possibly after a sequence of flops. We shall return to contractibility issues later. 2.2. Parameter spaces for hyperk¨ahler manifolds Let M be the underlying differentiable manifold of an IHS X: we view X as a pair (M, I) where I is a complex structure on M . We consider the space Comp ahler type on M . By the Kodaira-Spencer stability of all complex structures of K¨ theorem ([KS]), this is an open subset in the space of all complex structures. The group Diff of diffeomorphisms of M and its subgroup of isotopies Diff 0 act on Comp. uller space Teich = Teich(M ) is Comp / Diff 0 , the quoDefinition 2.4. The Teichm¨ tient of Comp by isotopies. The mapping class group is Γ = Diff / Diff 0 . It turns out that the Teichm¨ uller space is a possibly non-Hausdorff finitedimensional smooth complex space. One would like to consider the quotient Teich /Γ as the “moduli space” of IHS. Unfortunately this makes little sense. Indeed Verbitsky proved in [V2] that most of the orbits of the action of Γ on Teich are dense. Also the space Teich may have infinitely many connected components. This is the case exactly when the subgroup K ⊂ Γ acting trivially on H 2 (M ), called the Torelli subgroup, is infinite. In fact, this group acts on Teich by permuting its connected components ([V1bis]) and the space Teich /K already has finitely many connected components which can be identified to those of Teich. In a more algebraic setting, Teich /K is also known as the moduli space of marked IHS manifolds. We denote by TeichI the connected component of Teich containing our given complex structure I. The monodromy group MonI is the subgroup of Γ preserving TeichI . By [V1, Corollary 7.3], its intersection with K is finite. Sometimes we drop the index I, if no risk of confusion arises. The Hodge monodromy group MonHdg is the part of the monodromy which preserves the Hodge type. The crucial facts about the monodromy group are as follows. Theorem 2.5 ([V1bis, Thm. 2.6]). MonI has a finite index image (as well as a finite kernel) under the natural map Φ to O(H 2 (M, Z), q). Consequently, the Hodge monodromy maps to a subgroup of finite index in O(N S(X), q). Theorem 2.6. The monodromy group acts “ergodically” on TeichI , that is, most of its orbits are dense (namely the union of non-dense orbits is of measure zero). More precisely, using Ratner’s theory on unipotent group actions on homogeneous spaces, Verbitsky gave a classification of orbit closures of this action. In order to explain it we first need to introduce the period map for hyperk¨ahler manifolds.

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Definition 2.7. The period domain is the space P = {l ∈ PH 2 (M, C)|q(l) = 0, q(l, ¯l) > 0}. The period map Per : Teich → P assigns H 2,0 (X) = H 2,0 (M, I) to a complex structure I. Verbitsky proved in [V1] that the period map is an isomorphism on each connected component TeichI of Teich, provided that one glues together the nonseparated points of TeichI (a global Torelli theorem for hyperk¨ ahler manifolds). Some more precise versions of this result have later been given by Markman in [Mar]. In particular, Markman gives the following description of the non-separated points glued together by the period map. For a complex IHS manifold X = (M, I) the space H 1,1 (X, R) only depends on the period point. The positive cone Pos(X) ⊂ H 1,1 (X, R) has a chamber decomposition in the birational K¨ ahler cone3 Bir(X) and its monodromy transforms γ(Bir(X)), γ ∈ Mon. The walls of this decomposition are given precisely by hyperplanes orthogonal to the classes of prime divisors of negative Beauville–Bogomolov square on X (prime exceptional divisors; see end of Section 2.1). The chamber Bir(X) in its turn is subdivided into K¨ahler cones of the birational models of X, and analogously for γ(Bir(X)). The resulting smaller chambers are called the K¨ahler chambers. Theorem 2.8 (Markman [Mar]). The points of Per−1 (Per(I)) are in one-to-one ahler chambers in Pos(X) ⊂ H 1,1 (X, R). correspondence with the K¨ Just in the same way as for the local deformations, the complex structures where a given class z ∈ H 2 (M, Z) is of type (1, 1) are parametrized by Per−1 (z ⊥ ), and their period points by the hyperplane section z ⊥ ⊂ P. For a very general complex structure X = (M, I), there are no integral (or rational) classes of type (1, 1). In this case the K¨ ahler cone Kah(X) is equal to the positive cone and the Teichm¨ uller space is separated at I. The same holds if I does have rational classes of type (1, 1) but they are all of non-negative Beauville–Bogomolov square, or are not represented by curves. For other IHS the chamber decomposition can be rather complicated, in particular Teich is not separated even for K3 surfaces. For K3 surfaces, the chamber decomposition is given by the orthogonals to (−2)-classes and Bir coinsides with Kah (see the beginning of this text for some finiteness results in this case). It is often useful to view P as a real variety of Grassmannian type rather than an open subset of a complex quadric. Namely by taking the real and the 3 Our terminology here is non-standard, to save space and keep notations simple. Markman calls this chamber the fundamental exceptional chamber whereas by the birational K¨ ahler cone of X one usually means the union of the K¨ ahler cones of the IHS birational models of X, i.e., the complement of a certain number of walls in the fundamental exceptional chamber. We prefer not to separate the two notions and add the “interior” walls to the union of the K¨ ahler cones of the birational models.

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imaginary part of a complex line l one observes that P is identified to the space of oriented positive 2-planes in H 2 (M, R): P = Gr++ (H 2 (M, R)) ∼ = SO(3, b2 − 3)/SO(2) × SO(1, b2 − 3). The monodromy orbit classification by Verbitsky, provided that b2 ≥ 5, is now as follows ([V2bis]): 1. If the period plane P is rational, the monodromy orbit of the complex structure is closed; 2. If the period plane P contains no rational vectors, the orbit is dense; 3. If the period plane contains a single rational vector v up to proportionality, the image of the orbit closure by the period map is the set of all period planes containing a monodromy image of v. So it has, in general, countably many irreducible components. Each component (consisting of the planes containing a fixed rational vector) is a totally real submanifold of the period domain, that is, a fixed point set of an antiholomorphic involution. In particular, looking at its tangent space we see that the orbit closure is not contained in a proper complex submanifold, even locally, nor contains any positivedimensional complex subvarieties. This classification is an essential ingredient for the results on contractibility and deformation invariance in Section 4. 2.3. MBM classes Let z ∈ H 2 (M, Z) be a negative class, that is, q(z) < 0. Denote by Teichz the uller space formed by the complex structures in which z is a part of the Teichm¨ Hodge class (so that Teichz = Per−1 Pz , where Pz = z ⊥ ). The following theorem is proved in [AV1]. Theorem 2.9. The following properties of z are equivalent: 1. Teichz contains no twistor lines; 2. For all I ∈ Teichz such that X = (M, I) has Picard rank one, X contains a rational curve. 3. For generic I ∈ Teichz , X = (M, I) contains a rational curve. 4. For I ∈ Teichz , z ⊥ contains a wall of a K¨ ahler chamber, that is, for some γ ∈ MonHdg , γ(z)⊥ contains a wall of the K¨ ahler cone of an IHS birational model X  of X = (M, I). Definition 2.10. An MBM class is a negative class in H 2 (M, Z) which has these equivalent properties. If z is MBM, then up to a rational multiple it is represented by a rational curve on a general X = (M, I) with I ∈ Teichz . On a special X, such a curve can become reducible and the classes of the components might be not proportional to z. In particular this means that z ⊥ no longer contains a wall of the K¨ahler cone, though by definition it remains MBM. The classes of the irreducible components

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are also MBM, provided they have negative Beauville–Bogomolov square. Clearly, at least one of them necessarily has. So the walls of K¨ ahler chambers are parts of the orthogonal hyperplanes to the MBM classes, whereas, by Markman, the walls of the (larger) birational K¨ahler chambers are parts of the orthogonal hyperplanes to the classes of the prime uniruled divisors with negative square (“prime exceptional divisors”). As we have seen in the previous section (proposition 2.2), every prime exceptional (in fact, even just uniruled) divisor is birationally a P1 -bundle, and we may consider the class of its generating P1 in H 2 (X, Q). Lemma 2.11. This class is MBM, proportional to the class of the divisor itself. Indeed, both survive as (1, 1)-classes on the same deformations of X. In this way, our concept of an MBM class is an extension of that of the prime exceptional divisor: in a suitable complex structure X = (M, I), an MBM class is represented by a rational curve whose deformations within X may cover a divisor or a lower-dimensional submanifold. This dimension is invariant by small deformations of X, see [AV1]. G. Mongardi in [Mon] has introduced the notion of a wall divisor which has turned out to be the same as our MBM class (but this was not immediately clear to us because of a somewhat misleading terminology, and also because the earlier versions of his paper dealt specifically with the K3 type case). The birational K¨ahler cone Bir(X) is a connected component of the complement in Pos(X) to the union of hyperplanes D⊥ , where D runs through the prime exceptional divisor classes. In other words the hyperplanes D⊥ within Pos(X) are entirely made of the walls of birational K¨ ahler chambers. Subdividing Bir(X) into K¨ahler cones of birational models of X is more delicate: indeed a given rational curve C may disappear on a given birational model, so it is not obvious that the hyperplane [C]⊥ continues through Bir(X). We have however proved that this is true. ahler cone is a connected component of Theorem 2.12 ([AV1, Thm. 6.2]). The K¨ the complement to the union of hyperplanes z ⊥ in Pos(X), where z runs through the set of MBM classes of type (1, 1). Remark 2.13. It follows easily from the definition that the monodromy group preserves the set of MBM classes and that the Hodge monodromy group4 MonHdg preserves the set of MBM classes of type (1, 1).

4 The Hodge monodromy group MonHdg of an IHS manifold is the subgroup of the monodromy group (Subsection 2.2) preserving the Hodge decomposition.

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3. Results on MBM classes and applications 3.1. Markman’s Torelli theorem and the birational cone conjecture The following corollary of the global Torelli theorem due to Markman (see [Mar], Th. 1.3, Cor. 5.7) is crucial for understanding the (birational) automorphisms of an IHS manifold. Theorem 3.1. a) Let γ ∈ O(H 2 (X, Z)) be an element of the monodromy group preserving the ahler cone. Then there exists f ∈ Aut(X) Hodge decomposition and the K¨ such that f ∗ = γ (i.e., inducing γ on cohomology). b) Let γ ∈ O(H 2 (X, Z)) be an element of the monodromy group preserving the ahler cone. Then there exists a biHodge decomposition and the birational K¨ rational automorphism f of X such that f ∗ = γ. In particular, in this way questions about automorphisms can be reduced to questions about the monodromy. With a little bit of linear algebra work ([AV1], sections 3, 6) one sees that 1. To prove that Aut(X) acts with finitely many orbits on the walls of the K¨ahler cone, it suffices to prove that the Hodge monodromy group MonHdg acts with finitely many orbits on the set of MBM classes of type (1, 1); 2. To prove that the group of birational automorphisms acts with finitely many orbits on the walls of the birational K¨ahler cone, it suffices to prove that MonHdg acts with finitely many orbits on the set of divisorial MBM classes of type (1, 1). The second part has been settled by Markman quite some time before we have settled the first one in [AV2] and it was in fact a huge source of inspiration for all our work. Hopefully, our point of view brought some simplifications to Markman’s original arguments, however the crucial step (reflections with respect to the divisorial MBM classes are in the monodromy) remains unchanged. The main steps of proof are as follows. Step 1. We know that the Hodge monodromy is of finite index in O(N S(X), q), therefore it suffices to prove that O(N S(X)) acts with finitely many orbits on the set of prime exceptional divisor classes (divisorial MBM classes). Step 2. By classical results on quadratic forms ([Kn], Satz 30.2, and a variant [AV1], Theorem 3.14) this follows once we show that the divisorial MBM classes have bounded square. Step 3. Markman shows that for a divisorial MBM class E, the reflexion RE (x) = x − 2q(E, x)/q(E, E) is in the (Hodge) monodromy group. Step 4. By definition, the monodromy group is a subgroup of O(H 2 (X, Z), q), in particular RE are integral. Hence q(E, E) must be bounded in terms of the discriminant of q.

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The statement about the finiteness of the orbits is often referred to as the weak Morrison–Kawamata cone conjecture, and the statement about the rational polyhedral fundamental domain in conjecture 1.3 as the strong Morrison–Kawamata cone conjecture. The latter has a chance to be valid only in the algebraic setting, i.e., for the ample/movable cone rather than the K¨ahler/birational K¨ahler one. ahler cone is round (it is equal to Pos(X)) Indeed, for a very general IHS X, the K¨ whereas the group of (birational) automorphisms is trivial. In Section 6 of [Mar] Markman has also proved the following birational version of the strong Morrison–Kawamata cone conjecture. Theorem 3.2. The group of birational automorphisms of X has a rational polyhedral fundamental domain on the cone Mov+ (X), where Mov(X) is the rational part of Bir(X) (that is, the intersection of Bir(X) and N S(X) ⊗ R) and Mov+ (X) stands for keeping only the rationally defined part of the boundary. A few years later Markman and Yoshioka [MY] have essentially proved the equivalence of the weak and the strong version for IHS, namely that the boundedness of squares of primitive MBM classes implies the strong Morrison–Kawamata cone conjecture. Here is the original formulation of their result. Theorem 3.3 ([MY, Thm. 1.3]). Let X be an IHS manifold. Assume that the Beauville–Bogomolov square of any integral primitive and extremal class in the Mori cone of any IHS birational model Y is bounded from below by a constant, which depends only on the birational equivalence class of X. Then the strong Morrison–Kawamata conjecture holds for X. As Markman and Yoshioka also remarked, the cone conjecture implies that the number of isomorphism classes of IHS birational models of an IHS manifold is finite (cf. [AV2, Theorem 1.9]). 3.2. The cone conjecture via ergodic theory As we have mentioned already, the weak version of the cone conjecture is equivalent to the fact that the set of primitive MBM classes of type (1, 1), (or their orthogonal hyperplanes in H 1,1 (X, R)) is finite modulo the Hodge monodromy action. This in turn is equivalent to the boundedness of the Beauville–Bogomolov square of primitive MBM classes of type (1, 1) (Subsection 3.1, Step 2). To prove this boundedness assertion, it suffices to do the case when X has maximal Picard rank (indeed it is easy to see that we can deform X to a manifold of maximal Picard rank in such a way that all MBM classes of type (1, 1) remain of type (1, 1)). Such an X satisfies H 1,1 (X, R) = N S(X) ⊗ R, the Beauville–Bogomolov form is of signature (1, b2 − 3) on N S(X), in particular X is projective (see [Hu1]). It turns out that using ergodic theory one can prove the following general statement. Theorem 3.4. Let V = VZ ⊗ R be a vector space with integral structure, equipped with an integral quadratic form q of signature (1, n). We suppose n ≥ 3. Consider the projectivization PV + of the positive cone of q, non-canonically isomorphic to

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SO+ (1, n)/SO(n), as a model for the hyperbolic n-space Hn . Let Γ be a discrete subgroup of finite index in SO+ (VZ ). Let Σ be a Γ-invariant union of hyperplanes Si , i ∈ I. Then either Σ is dense in Hn or the set I is finite modulo Γ. The key point is that one can view the hyperplanes Si as the projections ∼ SO+ (1, n − 1) (it is of orbits of a single subgroup H ⊂ G = SO+ (1, n), H = especially visible when n = 2, though this case is excluded from the theorem: G interprets as the unit tangent bundle to the hyperbolic plane H, and by lifting the hyperplanes to G tautologically we obtain a foliation, known as “the geodesic flow”). When n > 2, the subgroup H is generated by unipotents, which is the case studied in Ratner’s theory. (Ratner’s original papers are [Ra1], [Ra2]; see also [Morr] for an accessible exposition.) Ratner gives a classification of H-invariant, H-ergodic measures on G/Γ, where G is a non-compact simple Lie group with finite center, Γ is a lattice (i.e., a discrete subgroup of finite index in G) and H is a subgroup generated by unipotents. Her main result is that such a measure is algebraic, i.e., is the pushforward of the Haar measure of an orbit Sx of a closed subgroup S, such that xΓx−1 is a lattice in S. Moreover, a combination of results by Mozes–Shah and Dani-Margulis states that the weak limit of such measures either goes to infinity or is again a measure of the same type. The following formulation of these is Corollary 4.6 from [AV2]. Theorem 3.5. Let G be a connected Lie group, Γ a lattice, P(Y ) the space of all probability measures on Y = G/Γ, and Q(Y ) ⊂ P(Y ) the space of all algebraic probability measures. Then Q(Y ) is closed in P(Y ) with respect to weak topology. Moreover, let Y ∪ ∞ denote the one-point compactification of Y , so that P(Y ∪ ∞) is compact. If for a sequence μi ∈ Q(Y ) one has μi → μ ∈ P(Y ∪ ∞), then either μ ∈ Q(Y ), or μ is supported at infinity. We associate measures μi to our hyperplanes Si , by taking the closures of the corresponding orbits in G/Γ (in fact, they are closed already since the hyperplanes are rational). A purely geometric argument yields that no subsequence of μi can tend to infinity; one easily reduces the problem to the case of geodesics on a hyperbolic Riemann surface, where it amounts to the statement that there are no closed geodesics around cusps. Any sequence of μi should then have a subsequence weakly converging to an algebraic measure. We remark that there is no intermediate closed subgroup between H = SO+ (1, n − 1) and G = SO+ (1, n). The case when the limit is supported on an orbit of H amounts to finiteness (the converging subsequences must be constant in this case) and the case when the limit is supported on the whole of G/Γ amounts to density. This is of course a very rough sketch; see [AV2] for details. If V = H 1,1 (X, R), Γ is the Hodge monodromy and Σ is the set of orthogonal hyperplanes to MBM classes, the first option of this alternative is clearly impossible, indeed the hyperplanes must bound the ample cone. Hence the cone conjecture holds:

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Theorem 3.6 ([AV2]). Let X be an IHS manifold with b2 (X) > 5, then the Hodge monodromy group acts with finitely many orbits on the set of primitive MBM classes of type (1, 1). The Beauville–Bogomolov square of a primitive MBM class on X is bounded by a constant c = c(X). 3.3. Uniform boundedness and an appication The following theorem is a more precise version of Theorem 3.6 which also allows to settle the case b2 = 5. Theorem 3.7 ([AV3]). Let X be an IHS manifold with b2 (X) ≥ 5, then the monodromy group acts with finitely many orbits on the set of primitive MBM classes. The Beauville–Bogomolov square of a primitive MBM class on X is bounded by a constant c depending only on the deformation type of X (or equivalently, on the underlying differentiable manifold M ). The idea behind the proof is similar, but one works with the homogeneous space Gr+++ (H 2 (M, R)), isomorphic to SO+ (3, b2 −3)/SO(3)×SO(b2 −3), instead of the hyperbolic space Hb2 −3 (as usually, M denotes the underlying differentiable manifold). There are two steps in the proof. In the first step one observes that the subsets z ⊥ of Gr+++ (H 2 (M, R)) (formed by positive 3-spaces orthogonal to an MBM class z) are orbits of conjugated subgroups Pz = Stab(z) of G (we call them “orbits of hyperplane type”). It turns out that the orthogonality condition implies that all these orbits are pro∼ SO+ (3, b2 − 4). The subgroup P jections of right orbits of a single subgroup P = is generated by unipotents when b2 ≥ 5, so Ratner’s theory and the Mozes–Shah theorem apply as soon as b2 ≥ 5. One derives that the subsets z ⊥ are either dense in Gr+++ (H 2 (M, R)), or account to finitely many up to the monodromy action. This part is independent of the hyperk¨ ahler context, just as Theorem 3.4. The geometric statement analogous to Theorem 3.4 which we prove is as follows. If L is a lattice of signature (p, q) with p, q ≥ 2, Γ is a finite index subgroup of O(L), and Σ an infinite union of Γ-orbits of negative vectors, then the set of positive p-planes orthogonal to a member of Σ is dense in the positive grassmannian. The second step is to show that the density is impossible in the hyperk¨ahler context, and therefore the set of the negative vectors modulo Γ-action (that is, of the monodromy orbits of MBM classes) must be finite. To show this we remark that the complement to the union of all z ⊥ is identified with the Teichm¨ uller space of hyperk¨ ahler structures on M , and therefore must have non-empty interior. This is the main content of the paper [AV4]. In this paper we consider the space Hypm ahler metrics of volume 1 on a differentiable manifold M , and we of all hyperk¨ uller space of hyperk¨ ahler structures as the quotient of Hypm by define the Teichm¨ isotopies. Each metric comes with a hyperk¨ ahler triple I, J, K, inducing a triple of 2-forms ωI , ωJ , ωK . The corresponding triples of second cohomology classes generate positive three-planes, and it turns out that a positive three-plane corresponds to an actual hyperk¨ ahler structure if it is not orthogonal to any MBM class; see [AV4] for details.

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Note that Theorem 3.7 is really a strengthening of Theorem 3.6: if we know that the monodromy acts with finitely many orbits on the set of primitive MBM classes, we derive that the Beauville–Bogomolov square of a primitive MBM class is bounded, in particular this holds for MBM classes of type (1, 1) and so the Hodge monodromy must act with finitely many orbits on the set of MBM classes of type (1, 1). In this way it is easy to construct IHS without MBM classes of type (1, 1). It suffices to take a period point such that the Picard lattice of the corresponding complex structures does not represent small negative numbers (smaller than c in absolute value). Of course one has to make sure that such points exist. But there are lattice-theoretic results on primitive embeddings from which one derives that any lattice of small rank r and signature (1, r − 1) embeds primitively into H 2 (M, Z). Moreover, it follows from surjectivity of Torelli map that every primitive sublattice of H 2 (M, Z) of signature (1, r − 1) is the Picard lattice of some complex structure (see [Mor1] for details in the case of K3 surfaces; the general IHSM case is completely analogous, [AV5]). An example of what one can prove in this way is the following result from [AV5]. Theorem 3.8. Let X be an IHS manifold with b2 (X) ≥ 5. Then X admits a projective deformation X  with infinite group of symplectic automorphisms and Picard rank 2. Indeed, it is possible to take a deformation X  with Picard lattice Λ of signature (1, 1), not representing negative numbers of small absolute value nor zero. When Λ does not represent zero, it is well known that O(Λ) is infinite. On the other hand, the Hodge monodromy group surjects onto a subgroup of finite index in O(Λ), so it is infinite. Since there are no MBM classes of type (1, 1), the K¨ahler cone is equal to the positive cone and Markman’s Hodge-theoretic Torelli theorem (Theorem 3.1 of the present survey) applies to conclude that every element of Hodge monodromy is induced by an automorphism.

4. Contractibility and deformations For an MBM class z, the space Teichz is naturally split in two identical pieces, both isomorphic to Pz = z ⊥ ⊂ P after the Hausdorff reduction, according to whether z takes positive or negative values on the K¨ ahler cone. This can be easily seen from Verbitsky’s global Torelli theorem and the description of non-Hausdorff ahler chambers (Markman’s theorem 2.8). We call these two points in terms of K¨ pieces Teichz+ resp. Teichz− . Both are separated at a general point of Teichz (that is, where z is the only MBM class of type (1, 1)) but not at points where many other MBM classes are of type (1, 1). It is natural to consider the subspace Teichmin of z ⊥ where z generates an extremal ray, in other words, z defines a wall of the Teich+ z K¨ahler cone: over each period point where there are several K¨ahler chambers in

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Teichz+ , one keeps those which are adjacent to the hyperplane z ⊥ and discards the rest. Along Teichzmin , a positive rational multiple of z is represented by an extremal rational curve. By the Kawamata–Shokurov base-point-freeness theorem, when X is projective, extremal rational curves can be contracted. Indeed in this case the wall of ahler cone contained in z ⊥ has integral points. An integral point is the class the K¨ of a line bundle L which is nef and big. Since X has trivial canonical class, the theorem of Kawamata–Shokurov implies that some power L⊗n is base-point-free. The morphism defined by the sections of L⊗n is birational and contracts exactly the curves whose cohomology class is proportional to z. A more recent result, essentially due to Bakker and Lehn, is that extremal rational curves also contract in the non-projective case. ahler manifold with b2 > 5 and z an extremal Theorem 4.1. Let X be a hyperk¨ class on X, that is, z is MBM and the complex structure I of X = (M, I) is in Teichmin z . Then there exists a bimeromorphic contraction f : X → Y contracting exactly the curves of cohomology class proportional to z. The proof has two ingredients: local deformations and monodromy action. To describe the first one, take a projective IHS X and let f : X → Y be a birational morphism contracting the class z. According to Namikawa ([N] Section 3, p. 104) there is a commutative diagram extending f : X ⏐ ⏐ 

Φ

−−−−→

Y ⏐ ⏐ 

G

Def(X) −−−−→ Def(Y ) where G is finite. This diagram in itself does not carry much information on contractions because the fiber of the family Y over a general t ∈ Def(Y ) is smooth and the map Φt is an isomorphism even though Φ0 = f . Bakker and Lehn have shown in [BL] that G induces an isomorphism between the subspace of “locally trivial” deformations Def lt (Y ) ⊂ Def(Y ) and the subspace Def z (X) ⊂ Def(X) of deformations preserving z as a Hodge class. For t ∈ Def lt (Y ), Φt is a birational morphism contracting z. Here “locally” in the definition of local triviality means “locally on Y” (every point of Y has a small neighbourhood which decomposes into a product, see [FK] for precise definition and more), so this is an equisingularity condition. The upshot is that if z can be contracted on a projective X, then it also can be contracted on its sufficiently close non-projective neighbours. The second ingredient is as follows. On each connected component of Teichmin z , there is an action of the group Γz , defined as the subgroup of the mapping class group preserving the component and the class z. One shows exactly in the same way as in [V2], [V2bis], that there are three types of Γz -orbits according to the rationality properties of the period plane. Here it is important to assume that

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b2 (X) > 5 (Verbitsky’s description is valid for b2 (X) ≥ 5, but Pz and Teichz are hyperplanes in P resp. in Teich). So the local picture globalizes as follows. For a non-projective X  we can find a projective X such that the monodromy transform of X  (by some element of Γz ) lands in an arbitrarily small neighbourhood of X: we can take any X if the orbit of X  is dense, and a projective X from the closure of a non-dense orbit otherwise. So z contracts on this monodromy transform, and consequently on X  itself, since the action of the monodromy is the transport of complex structure which moreover preserves z. Remark 4.2. This result yields some information on the structure of the rational quotient map already mentioned in Proposition 2.2. Namely, the rational quotient of any component of the exceptional locus Z is a regular map. Indeed, this is the restriction of the contraction map to Z. In [AV6], we prove that Bakker and Lehn’s local triviality implies the local triviality in the usual sense of the word (local on the base, that is, any fiber Yt of the family Y lt has a neighbourhood which decomposes into a product) in the real analytic category. This implies in particular that the contraction loci are diffeomorphic as we move X along Teichzmin . Using ergodic actions, one can prove, with some restrictions, a stronger result. Namely, the fibers of the rational quotient map keep their biholomorphism type as X varies in Teichmin z , under the condition that they are normal and assuming that the Picard number of X is not maximal. See [AV6] for details. Remark 4.3. As an obvious example of the exceptional divisor of the Hilbert square of a K3 surface S shows, one cannot expect that the entire contraction loci keep their biholomorphism type. Indeed, this exceptional divisor is a P1 -bundle over S, so its biholomorphism type does vary with S.

5. Classification of MBM classes in low dimension for K3 type On K3 surfaces, primitive MBM classes are exactly the classes of square −2 in H 2 (M, Z). On IHS fourfolds of K3 type, there are three “obvious” types of extremal curves/MBM classes, described by Hassett and Tschinkel in [HT1]. Indeed first of all there are “exceptional” projective lines contracted by the Hilbert-Chow map. Since the restriction of the exceptional divisor to such a curve has degree −2 (blow-up of double points), one computes that the class of such a line l1 is half-integral and its Beauville–Bogomolov square equals −1/2. To see the other two, take a K3 surface Y with a (−2)-curve C and let X be the Hilbert square of Y . Then X carries at least two MBM classes: that of a line l2 in the Lagrangian plane S 2 C ⊂ S [2] Y (where S 2 means the symmetric square) and that of C itself viewed as a ruling of the image of C × Y . The class C is integral of square −2, and l2 is half-integral of square −5/2 (see, e.g., [HT2]). Note that the class of C is not extremal: contracting C, one automatically contracts l2 . However it is MBM, and in fact becomes extremal when we perform the Mukai flop of X centered at S 2 C.

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Using the deformation-invariance properties of MBM classes and MBM loci, we provide an easy proof that there are no other MBM classes (up to monodromy) in [AV7]. The key point of our argument is that, given an IHS fourfold X of K3 type and a negative integral (1, 1)-class η on X, we can deform X, keeping η of type (1, 1), to an IHS X  with negative definite Picard lattice of rank two, which also happens to be the Hilbert scheme of a nonprojective K3 surface. Indeed, one can detect whether a given IHS manifold of K3 type is a Hilbert scheme of a K3 surface from its periods. It is a Hilbert scheme if it is a monodromy transform of a manifold X such that e is of type (1,1), where e is, as usual, one half of the class of the exceptional divisor (under some fixed identification of H 2 (M, Z) with the second cohomology of a Hilbert scheme). Therefore, the periods of Hilbert schemes are a union of countably many divisors in Per realized as γ(e)⊥ , for all γ in the monodromy group. It turns out that one can find an element γ in the monodromy such that the sublattice in the second cohomologies generated by η and γ(e) is negative definite. Then for X  as above, one can take a general complex structure in the intersection of Teichη and Teichγ(e) . On X  there are not many rational curves: indeed, X  is the Hilbert square of a K3 surface Y  which contains a single (−2)-curve and no other curves. All MBM classes and their contraction loci are therefore as described in the previous paragraph. Clearly η must be one of them, and if η is extremal, its contraction locus is either the projective plane or a P1 -bundle over a K3 surface. The same is true on X by deformation invariance properties. An interesting phenomenon occurs when we study the MBM classes on an IHS sixfold X of K3 type. It turns out that one can not always deform X, preserving η, to the Hilbert cube of a K3 surface with negative cyclic Picard group. However, one can always deform it to the Hilbert cube of a K3 surface with non-positive cyclic Picard group. On such manifolds, it is still easy to describe all rational curves. In this way, one gets a “new” MBM class (with respect to the four types obtained from deformation to the Hilbert cube of a K3 surface with Picard group generated by a (−2)-curve, as we did for Hilbert square). This is also the class which was missing from the tables of Hassett and Tschinkel in [HT2] and which is present in their later elaboration of Bayer and Macri’s computations for low-dimensional IHS of K3 type [HT3]. We refer the reader to [AV7] for details and further analysis.

6. Some open questions All known examples of IHS manifolds carry MBM classes. Can one prove that this must always be the case? Markman’s result that the reflections in divisorial MBM classes are elements of the monodromy group is extremely important. One may ask how the nondivisorial MBM classes contribute to the monodromy. Whether a reflection in a

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non-divisorial MBM classes is or is not in the monodromy, what geometric information does this give? Though one has a good idea about the geometry of subvarieties covered by extremal rational curves, many details are still to be verified. For example a Lagrangian subvariety covered by extremal rational curves tends to be isomorphic to the projective space. Wisniewski and Wierzba proved that on a holomorphic symplectic fourfold it is actually a plane ([WW]). In higher dimension one can observe that for a Lagrangian submanifold Y ⊂ X, the normal bundle NY,X is isomorphic to the cotangent ΩY1 and speculate that a subvariety contractible to a point should have “negative” normal bundle, so that Ω1Y is “negative” and TY is “positive”. If “positive” meant ample, then we would conclude that Y is Pn , at least when smooth ([Mo]). However, this does not work literally, because the ∗ contractibility of Y does not mean that NY,X is ample, unless Y is of codimension one. By a result of Ancona and Vo Van Tan, the contractibility of Y implies that there exists a scheme structure Y  on Y such that the conormal sheaf of Y  is ample (see [AT] for details; we are grateful to Andreas H¨oring for indicating this result and the reference). In their setting, Y does not have to be smooth. The question about the possible isomorphism type of contractible Lagrangian subvarieties remains open. Acknowledgement Partially supported by HSE University Basic Research Program.

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Ekaterina Amerik Universit´e Paris-Saclay Laboratoire de Math´ematiques d’Orsay Campus d’Orsay, Bˆ atiment 307 F-91405 Orsay, France and

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Laboratory of Algebraic Geometry National Research University HSE Department of Mathematics 6 Usacheva Str. Moscow, Russia e-mail: [email protected] Misha Verbitsky Instituto Nacional de Matem´ atica Pura e Aplicada (IMPA) Estrada Dona Castorina, 110 anico, CEP 22460-320 Jardim Botˆ Rio de Janeiro, RJ – Brasil and Laboratory of Algebraic Geometry National Research University HSE Department of Mathematics 6 Usacheva Str. Moscow, Russia e-mail: [email protected]

Unirationality of Certain Universal Families of Cubic Fourfolds Hanine Awada and Michele Bolognesi Abstract. The aim of this short note is to define the universal cubic fourfold over certain loci of their moduli space. Then, we propose two methods to prove that it is unirational over the Hassett divisors Cd , in the range 8 ≤ d ≤ 42. By applying inductively this argument, we are able to show that, in the same range of values, Cd,n is unirational for all integer values of n. Finally, we observe that for explicit infinitely many values of d, the universal cubic fourfold over Cd can not be unirational. Mathematics Subject Classification (2010). 14E05, 14E08, 14D22. Keywords. Birational geometry, rationality questions, universal families, moduli spaces, cubic hypersurfaces.

1. Introduction Since the celebrated result of Clemens and Griffiths [CG72] on the irrationality of smooth cubic threefolds, at least three generations of algebraic geometers have been working hard on the problem of rationality of cubic fourfolds. The search for the good invariant that would detect (ir)rationality has involved Hodge theoretical [Has16], homological [Kuz10], K-theoretic [GS14] and motivic [BP20] methods. While the general suspicion is that the generic cubic fourfold should be non rational as well, no explicit example of irrational cubic fourfold is known at the moment. A little more is known, and the expectation are slightly more precise, about rationality. Let C denote the moduli space of smooth cubic fourfolds and Cd ⊂ C the Hassett divisor of discriminant d, then a well-known conjecture due to Harris-Hassett-Kuznetsov (see [Kuz10, Has16, AT14]) states that there should be a countable infinity of Hassett divisors that parametrize rational cubic fourfolds. In particular these divisors should correspond to cubic fourfolds that have an associated K3 surface, in a Hodge theoretical or derived categorical sense (see [Kuz10, Has00]). In the second decade of this century, the search for rational examples of cubic fourfolds has concentrated more on cubics containing some special © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Farkas et al. (eds.), Rationality of Varieties, Progress in Mathematics 342, https://doi.org/10.1007/978-3-030-75421-1_4

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surfaces [ABBVA14, Nue17, AHTVA19, Awa19]. This has been developed further in the works of Russo-Staglian`o, and at the moment the HHK conjecture is known to be true for four low values of d [BD85, BRS19, RS19b, RS19a]. In all these cases, the special cubic fourfolds involved contain rational surfaces. In this note we are also interested in the birational geometry of cubic fourfolds, but under a slightly different point of view. Universal families are well known and studied objects in the realm of moduli spaces of curves, where forgetful maps are the algebraic geometer’s everyday tools. Also in the context of K3 surfaces several results about these objects have appeared in the literature, also very recently [FV18, FV19, Ma19]. Probably because the birational geometry of a single cubic fourfold is already so difficult to understand, the problem of studying the same problems for universal families of cubics seems to have been overtaken. Thanks to the recent advances in the study of rationality of a single cubic fourfold, it seems however natural to ask what can be said in families, in particular over certain relevant divisors Cd in the moduli space. In this paper we prove the following fact: Theorem 1.1. The universal cubic fourfolds Cd,1 are unirational for 8 ≤ d ≤ 42. More precisely, we propose two different methods, since each one seems to have its own interest, concerning different aspects of the geometry of cubic fourfolds. On one hand, we use some recent results of Farkas–Verra about universal K3 surfaces and use the relation with cubic fourfolds given by associated K3 surfaces. On the other hand, we use the presentation of Hassett divisors as cubics containing certain rational surfaces, as done by Nuer and Russo-Staglian` o, and Koll´ar’s Theorem on unirationality of cubics. The second argument, applied inductively, also allows us to prove the following fact: Theorem 1.2. The universal families Cd,n are unirational for all values of n if 8 ≤ d ≤ 42. Of course, though the intuitive definition is clear, this requires a few checks that actually the universal family exists, which we make in Sect. 2. In the same Section, we also collect some important properties of cubic fourfolds, the rational scrolls they contain, their associated K3 surfaces and their Fano varieties of lines, that are then necessary in Section 3. Section 3 is devoted to the proof of unirationality of C26,1 and C42,1 via universal K3 surfaces, which is based on the construction of a couple of rational/unirational moduli quotients involving the families of scrolls inside our cubic fourfolds. In Sect. 4 we develop the second argument, which is slightly more general, and allows us to complete the missing cases of Thm. 1.1. We use thoroughly the recent theory of rational surfaces contained in special cubic fourfolds of low discriminants, and combine this with Koll´ar’s Theorem on unirationality of cubic hypersurfaces. In Section 5 we underline the special nature of the cases we are considering, by comparing them with preceding results of Gritsenko-Hulek-Sankaran [GHS13] and V´arilly-Alvarado-Tanimoto [TVA19].

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In fact, in [GHS13] it is proven that the moduli space of polarized K3 surfaces of degree 2m has positive Kodaira dimension or is even of general type for an infinity of values (see Prop. 5.2 for details). Furthermore one expects that the gaps in Prop. 5.2 can be filled in by using automorphic form techniques. V´ arilly-Alvarado and Tanimoto [TVA19] have started pursuing this project obtaining another relevant range of values where the Kodaira dimension of Cd is positive (see Prop. 5.3). These results give us negative information about the divisors Cd where the universal cubic Cd,1 can not be unirational. Proposition 1.3. The universal cubic fourfold Cd,1 is not unirational if: 1. d > 80, d ≡ 2 (mod 6), 4  |d and such that for any odd prime p, p|d implies p ≡ 1 (mod 3); 2. d = 6n + 2 for n > 18 and n = 20, 21, 25; 3. d = 6n + 2, n > 13 and n = 15; 4. d = 6n, and n > 18, n = 20, 22, 23, 25, 30, 32; 5. d = 6n for n > 16 and n = 18, 20, 22, 30.

2. The existence of the universal cubic fourfold, and some properties of scrolls and associated K3 surfaces As it is customary when discussing moduli spaces, one of the first questions one considers is the actual existence of a universal family. The invariant theory of cubic fourfolds, with respect to the natural P GL(6)-action, is known by the work of Laza [Laz09, Laz10]. Theorem 2.1 ([Laz09, Thm. 1.1]). A cubic fourfold X ⊂ P5 is not GIT stable if and only if one of the following conditions holds: 1. X is singular along a curve C spanning a linear subspace of P5 of dimension at most 3; 2. X contains a singularity that deforms to a singularity of class P8 , X9 or J10 . In particular, if X is a cubic fourfold with isolated singularities, then X is stable if and only if X has at worst simple singularities. The GIT quotient that we are considering is slightly different than the one constructed in [Laz09], since we only consider smooth cubic fourfolds. That is, we consider the complement U ⊂ |OP5 (3)| of the discriminant and take the quotient C := U//P GL(6). The resulting quotient is a quasi-projective variety of dimension 20 and Thm. 2.1 assures that all points are stable in the GIT sense. We are in particular concerned by certain divisors Cd inside C, with 8 ≤ d ≤ 42. By the seminal work of Hassett [Has00], and further developments [Nue17, RS19b, RS19a, FV18, FV19], they can be characterized in terms of special surfaces – not homologous to linear sections – contained in the generic cubic fourfold in each divisor. Recall in fact that for the general cubic fourfold H 2,2 (X, Z) = Z, generated

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by a linear section, and the locus where the rank of this group is bigger than one is the union of a countable infinity of divisors Cd ⊂ C. These irreducible divisors Cd are nonempty if and only if d ≥ 8 and d ≡ 0, 2 (mod 6). In particular, for a special cubic fourfold, one can associate a labelling Kd , which is a rank two saturated lattice generated by the linear section and the special surface. The integer d is actually the discriminant (or determinant) of the intersection form on Kd . For example, we will need the following descriptions (where the closure is intended to be taken inside the moduli space of cubic fourfolds): Definition 2.2. We have: • C26 := {Cubic fourfolds containing a 3 − nodal septic scroll} • C42 := {Cubic fourfolds containing a 8 − nodal degree 9 scroll}. Moreover, cubic fourfolds X inside these two divisors have an associated K3 surface1 . As it is well known, this means that the Hodge structure of the nonspecial cohomology of the cubic fourfold is essentially the Hodge structure of the primitive cohomology of certain polarized K3 surface T of degree d, i.e, Kd⊥ 2 is Hodge-isometric to Hprim (T, Z)(−1). Equivalently, this means that Kuznetsov component of the bounded derived category of X is equivalent to the derived category of a K3 surface T , i.e, A(X) ∼ = Db (T ), where A(X) is the right orthogonal to {OX , OX (1), OX (2)}. We refrain to give more details about this, since these notions are now well known and the references in the literature are very good [Kuz10, Has00, Has16]. The associated K3 surfaces for cubics from C26 have genus 14 and degree 26, whereas for cubics from C42 they have genus 22 and degree 42. More generally, thanks in particular to the work of Nuer [Nue17] (see also [RS19b, Sect. 4]), it is known that if d ≤ 42, then the cubics in each Cd can be defined as the cubics containing certain rational surfaces. These rational surfaces, that are introduced in [Nue17, Sect. 3], are obtained as images of P2 via linear systems with prescribed fixed locus. For each value of d we have a different linear system. We will see more about this in Section 4. Let Fg denote the moduli space of genus g K3 surfaces. For X ∈ Cd with discriminant d = 2(n2 + n + 1), with n ≥ 2 (remark that 26 and 42 all verify this equality), Hassett shows [Has00, Sect. 6] that there is an isomorphism F (X) ∼ = S [2]

(2.3)

between the Fano variety of lines F (X) and the Hilbert scheme S [2] of couple of points of (S, H) a polarized K3 surface with H 2 = d, the K3 surface associated to X. The moduli spaces of the corresponding K3 surfaces have dimension 19, the same dimension as divisors in C, and this assignment induces a rational map F d+2  Cd 2

1 Actually

(2.4)

cubics in C42 have two associated K3 surfaces, we will see more details about this later

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that is birational for d ≡ 2 (mod 6) and degree two for d ≡ 0 (mod 6). In particular, for C26 and C42 , we have rational maps ∼

F14

 C26 ;

F22

 C42 .

2:1

Another upshot of the isomorphism (2.3) is that, once we fix a smooth cubic fourfold in C26 or C42 , the family of scrolls contained inside X is precisely parametrized by the associated K3 surface. The construction, roughly speaking, goes as follows. For each p ∈ S, one defines a rational curve   Δp := y ∈ S [2] : {p} = Supp(y) . The image of Δp inside F (X) then defines a (possibly singular) scroll Rp ⊂ X. Proposition 2.5 ([FV18, FV19]). 1. The family of septimic 3-nodal scrolls inside a generic cubic fourfold X in C26 is the genus 14 K3 surface associated to X. 2. The family of 8-nodal degree 9 scrolls inside a generic cubic fourfold X in C42 is the genus 22 K3 surface associated to X. Recall that the generic K3 surface in F14 and F22 has no automorphism, hence universal families exist: f14 : F14,1

→ F14 ;

f22 : F22,1

→ F22 .

at least over an open subset of each moduli space. This has allowed the study of F14,1 and F22,1 by Farkas–Verra. They considered universal K3 surfaces as moduli spaces for couples (X, R), where X is a cubic fourfold, and R is a scroll – belonging to a given class of surfaces – contained in X. Proposition 2.6 ([FV18, FV19]). The universal K3 surfaces F14,1 and F22,1 are unirational. Let us now come to the definition of the object that we will study in Sect. 3. Similarly to the case of curves and K3 surfaces we give the following definition. Definition 2.7. By universal cubic fourfold over a divisor Cd we mean the moduli space Cd,1 of 1-pointed cubic fourfolds. As it is customary over moduli spaces, the mere existence of a universal cubic fourfold needs a little bit of justification. Proposition 2.8. The generic cubic fourfold in any divisor Cd does not have projective automorphisms, hence a universal family of cubic fourfolds exists over an open subset of each divisor in the moduli space C.

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Proof. In [GAL11, Thm 3.8] it is proven that the families of cubic fourfolds with a non-trivial projective automorphism have dimension at most 14. This means that these families cannot contain any divisor of the moduli space. Hence the generic element of any divisor has no non-trivial automorphism. The existence of a universal family is clear on the fine moduli space, which is a Deligne-Mumford stack. On the other hand, it is well know that this universal family descends to the coarse moduli space wherever the cubic fourfolds have non-trivial automorphism. 

3. Unirationality for C26,1 and C42,1 via universal K3 surfaces By the results of the preceding section, at least over a dense subset of C26 and C42 , there exists a universal family of cubic fourfolds. For simplicity, and since we are however working in the birational category, we will still denote the two universal families by C26,1 → C26 and C42,1 → C42 , without stressing the fact that they may not be defined everywhere. In this section we will show that: Theorem 3.1. The universal cubic fourfolds C26,1 and C42,1 are unirational. The strategy will be the same for the two cases, hence we will resume here below shortly the properties, that hold for both divisors, that we will need. In order to keep the notation not too tedious, when we will say that a certain property holds for Cn we will assume that n = 26, 42. Let X be a generic smooth cubic fourfold in Cn . We will denote by F (X) the associated K3 surface and by S(X) the family of scrolls (as defined in Def. 2.2) contained in X ⊂ P5 . For n = 26 septimic 3-nodal scrolls, for n = 42 8-nodal degree 9 scrolls. One can rephrase Prop. 2.5 by saying that F(X) is isomorphic to S(X). More generally, we will denote by Sn the Hilbert scheme of scrolls contained in cubics in Cn (those appearing in Def. 2.2), and by Sn the P GL(6)-quotient of Sn . These moduli quotients have been also considered in [FV18, FV19, Lai17]. Taking example from these papers let us give the following definition. Definition 3.2. Let us denote by Hn = {(X, R) : R ⊂ X, [X] ∈ Cn , R ∈ S(X)} the ”nested” Hilbert scheme given by the couples (X cubic fourfold whose class lives in Cn ) + (R rational normal scroll in S(X)); and by Hn := Hn //P GL(6) the corresponding moduli quotient. From [FV18, FV19] we have the following result. Proposition 3.3. The universal K3 surfaces Fn,1 are birational to Hn .

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In particular, in [FV18, FV19] the authors show that Sn is unirational and Hn is birational to a projective bundle over Sn , and hence unirational as well. We add just one more character to the plot, that is: Definition 3.4. We denote by Hn,1 the Hilbert scheme of the triples (X, R, p) as follows: Hn,1 = {(X, R, p)) : R ⊂ X, [X] ∈ Cn , R ∈ S(X), p ∈ X}. We will denote by Hn,1 the corresponding moduli quotient by P GL(6). Of course, as a consequence of Prop. 2.8, this is generically a fibration in cubic hypersurfaces over Hn . Finally, we will denote by Un the open subset complementary to the discriminant inside |OP5 (3)|, and we denote by Un,1 the universal family over Un . Theorem 3.5. The Hilbert scheme Hn,1 is unirational. Proof. Let us try to give a global picture of the situation. (R, X, p) ∈

H.S. level

rr rrr r r y rr r

Hn,1

P5 × SL n

LLL LLL LLL L&

//P GL(6)



M.S. level

Sn

forget scroll

GG GG GG GG G#

ww ww w ww {w w





s sss s s s y ss s

Hn,1

Hn

forget scroll

FF FF FF FF F"  ∼

(P5 × Sn )//P GL(6)

Hn = F n+2 ,1 KKK 2 x KKK xx x KKK KK {xxxx % Sn

/ Un,1  (X, p) II II II II I$ univ. K3 / Un  X

 / Cn,1

HH HH HH HH HH  # univ. K3 / Cn

(3.6)

Here above all vertical arrows shall be intended as quotients by the P GL(6)action. On the upper level of the diagram all spaces are Hilbert schemes, whereas on the lower they are moduli quotients (H.S and M.S. for short). By the unirationality of Sn we have the unirationality of P5 ×Sn . Then we observe that there is a natural forgetful map φn : Hn,1

→

(X, R, p) →

P5 × Sn ;

(3.7)

(p, R).

(3.8)

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Namely, the fiber over a given couple (R, p) is exactly the linear system PH 0 (P5 , IR∪p (3)) of cubics in P5 containing R and p. Over an open subset of P5 × Sn (i.e., the locus where p ∈ R) the rank does not change and it is 11, and 5 respectively for d = 26 and 42. This in turn implies that Hn,1 is unirational.  Proposition 3.9. The universal cubic fourfold Cn,1 → Cn is unirational. Proof. Recall from Proposition 2.8 that the generic cubic fourfold in the divisors Cn has no projective automorphism. Call V ⊂ Cn the dense locus where cubics have no automorphism. It is straightforward to see – and we have already implicitly used this in Diag. 3.6 – that, over V , the universal cubic has a natural quotient structure Cn,1 = Un,1 //P GL(6). (3.10) Hence Cn,1 is the natural moduli space for couples (X, p), up to the action of P GL(6). The upshot is that there exists a natural rational surjective forgetful map (up to P GL(6)-action) ϕR : Hn,1 → Cn,1 ; (X, R, p) → (X, p);

(3.11) (3.12)

that forgets the scrolls contained in X. By Thm. 3.5, the variety Hn,1 is unirational, and it dominates Cn,1 , thus also Cn,1 is unirational.  Remark 3.13. Using the same method and a recent result of Di Tullio [DT20], one can prove the unirationality of the universal family C14,1 over C14 , the (closure of the) locus of cubic fourfolds containing a smooth quartic scroll. In fact, shortly after this paper came out, Di Tullio [DT20] showed that the universal family F8,1 is rational. Recall that the isomorphism 2.3 holds for cubics in C14 and we have as well a birational map F8  C14 . Nevertheless, in the next Section we obtain the same result by different methods.

4. Unirationality through rational special surfaces 4.1. Special cubics in Cd in the range 8 ≤ d ≤ 38 The goal of this Section is to prove the unirationality of the non-empty families of universal cubics Cd,1 for 8 ≤ d ≤ 38 by generalizing some results of Nuer, and using a celebrated result of Koll´ar [Kol02]. Then, by applying inductively the same argument, we will show that Cd,n – the universal cubic with n marked points – is also unirational in the same range. Let us recall shortly the results we need. In his paper [Kol02], extending a result of Segre that held only over Q, Koll´ar shows the following Theorem 4.1. Let k be a field and X ⊂ Pn+1 be a smooth cubic hypersurface of dimension n ≥ 2 over k. Then the following statements are equivalent: 1. X is unirational (over k); 2. X has a k-point.

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We are going to use this in a relative setting. In order to do this we recall some results due to Nuer. In [Nue17], the author studies the birational geometry of the divisors Cd , for 12 ≤ d ≤ 38, via rational surfaces contained in cubic fourfolds. In particular, let S be the rational surface obtained as the blow-up of P2 along the p points in general position x1 , . . . , xp , and let us embed it into P5 via the linear series |aL − (E1 + · · · + Ei ) − 2(Ei+1 + · · · + Ei+j ) − 3(Ei+j+1 + · · · + Ep )|, where Em , for m = 1, . . . , p, are the exceptional divisors and L the pull-back of the hyperplane class on P2 ; the parameters (a, i, j, p) of the linear system are those displayed in [Nue17, Table 1]. They are chosen appropriately so that the surface image of the linear system is contained in P5 and the cubics in its ideal all belong to Cd , for a fixed value of d. More precisely, certain cohomological invariants of the surface S ⊂ P5 define an open subset Ud ⊂ (P2 )p parametrizing generic p-tuples of distinct points such that S has such invariants (see [Nue17, Table 2]) and the cubics in its ideal live in Cd . The main theorem [Nue17, Thm. 3.1] establishes the unirationality of the divisors Cd for 18 ≤ d ≤ 38 and d = 44, and it is based on some deformation theory of Hilbert schemes flags and semicontinuity arguments. In particular it shows that there exists a vector bundle Vd → Ud such that the fiber over (x1 , . . . , xp ) is the space of global sections H 0 (IS|P5 (3)). The associated projective bundle is clearly unirational. The natural classifying morphism P(Vd ) → Cd , obtained by the universal property of Cd is dominant and this implies the unirationality of Cd . Let us now denote by Xd,1 the universal cubic hypersurface over P(Vd ), and by πp+1 : (P2 )p+1 → (P2 )p the forgetful map that forgets the last point. Now we can claim the following result: Theorem 4.2. The universal cubic fourfolds Cd,1 are unirational if 12 ≤ d ≤ 38. Proof. By pulling back through π, we have the following commutative diagram. / Xd,1 / Cd,1 π ∗ Xd,1 (4.3) I s

 P(π ∗ Vd )  (P2 )p+1

 / P(Vd )

π

 / Cd

 / Ud

Actually the LHS column is only defined on π −1 (Ud ) but for simplicity we will not write this. Working in the birational category, this does not affect our results. We observe that the pulled-back family π ∗ Xd,1 , seen as universal family over ∗ P(π Vd ) has a tautological rational section s, that is the image in S of the (p +

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1)th point from (P2 )p+1 . This means that π ∗ Xd,1 has a rational point over the function field of P(π ∗ Vd ). By Koll´ar’s Theorem π ∗ Xd,1 is then unirational over this field. This in turn is equivalent to the existence of a P4 -bundle on P(π ∗ Vd ) that dominates π ∗ Xd,1 . The P4 -bundle is rational, π ∗ Xd,1 dominates Xd,1 and this in turn dominates Cd,1 , which is then unirational.  Remark 4.4. An easy adaptation of this argument allows us to show the unirationality of C8,1 as well. Of course, all the planes in P5 are projectively equivalent, and the linear system of cubics through a given 2-plane P is |IP/P5 (3)| ∼ = P45 . The plane P has rational points over C and the universal cubic X8,1 ⊂ P45 × P5 contains the ”constant” plane P × P45 . The variety X8,1 also dominates C8,1 , by definition. Hence X8,1 has rational points over the function field of P45 and hence is unirational over this field. By the same argument as in Thm. 4.2, this implies the unirationality of C8,1 . 4.2. Special cubics in C42 The unirationality of universal cubics C42,1 goes along the same lines as in the previous section, but we need to extract a couple of quite subtle results from [RS19b] about rational surfaces contained in cubics belonging to these divisor. In [RS19b], the authors construct a 48-dimensional unirational family S42 of 5-nodal surfaces of degree 9 and sectional genus 2, such that the divisor C42 can be described as the locus of cubics containing surfaces from this family. We reconstruct briefly their argument. The authors start from p1 , . . . , p5 , five general points on the projective plane. The image of the rational map defined by the linear system |H 0 (I(p21 ,,p2 ,p3 ,p4 ,p5 ) (4))| of quartic curves having a point of multiplicity two at p1 and simple base points at p2 , p3 , p4 , p5 is a smooth surface T of degree 8 and sectional genus 2 contained in P7 . This surface, in particular, can be embedded into the Segre product P1 × P3 ⊂ P7 via the linear system |H 0 (I(p1 ,p3 ,p4 ,p5 ) (2))| × |H 0 (I(p1 ,p2 ) (2))|. Then we consider a general secant line L to T, H a general hyperplane con∼ P6 taining L, and p ∈ L a general point. The projection from p of H ∩ T ⊂ H = 5 5 to P , since p lies on the secant line L, gives a one-nodal curve C ⊂ P . This has degree 8 and arithmetic genus 3 and it is contained in a singular quartic threefold scroll B ⊂ P5 . Now, consider the quadrics through C. Their linear system defines a birational map β : P5  A, where A is a quartic hypersurface in P6 . The closure of the exceptional locus of β contains a Segre threefold Y = P1 × P2 ⊂ P5 , which in turn is sent onto a smooth quadric surface Q ⊂ P6 . Of course, there are two lines contained in Q that pass through a general point of Q. Their inverse images via β are: a plane of the ruling of Y , and a smooth quintic del Pezzo surface D ⊂ P5 such that D ∩ B = C. Now let us consider the map α : P5  P8

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107

defined by cubics through B. The image of such a map is a del Pezzo fivefold F (that is, a hyperplane section of Gr(2, 5)), and morover α induces an isomorphism between D and a smooth surface R of degree 9 and sectional genus 2. The fivefold F contains a rational, three-dimensional family of planes, whose members all have class (2, 2) inside the Chow ring CH • (Gr(2, 5)) of the Grassmannian. The projection with center any one of these planes defines a birational morphism between R and some 5-nodal S ∈ S42 . As observed in [RS19b], this construction implies that S42 is unirational. We observe in fact that the construction of such surfaces depends on certain choices (5 points on P2 , a line secant to a particular rational surface, etc.) which all depend on free, rational parameters. This implies that the family S42 is unirational. Let us denote by S the rational parameter space that dominates S42 . Other relevant work about C42 has been done in [Lai17] and [HS20]. Remark 4.5. We observe that the construction above implies that any S ∈ S42 is birational to a del Pezzo quintic over C(S), the function field of S. This means that on the same field there exist a birational map P2  S, as we explain in the following Lemma. Lemma 4.6. Let S be any surface in S42 , then S is rational over the function field of S. Proof. As we have observed in Remark 4.5, S is birational to a del Pezzo quintic, over C(S). By a well-known result of Enriques [Enr97] (see also [SB92] for a modern  account), a del Pezzo quintic surface is rational over any field. Proposition 4.7. The universal cubic C42,1 is unirational. Proof. Let us denote by P(V42 ) the relative linear system over S42 of cubics through S ∈ S42 , and by X42,1 the universal family over P(V42 ), that dominates C42,1 . Then the situation is the following. ρ∗ π ∗ X42,1 I s

 P(ρ∗ π ∗ V42 )  P2 × S

ρ

/ π ∗ X42,1

/ X42,1

 / P(π ∗ V42 )

 / P(V42 )

 /S

π

/ C42,1 ϕ

 / C42

(4.8)

 / S42

Here π : S → S42 is the unirational parametrization described above, ρ the second projection and ϕ is the natural classifying map. Now, all cubics that are the fibers of the fibration X42,1 → P(V42 ) contain by construction a surface S ∈ S42 , and since these surfaces are rational over C(S) (they are birational to del Pezzo quintics), they have rational points over the same field. Summing up, there exists a section s : P(ρ∗ π ∗ V42 ) → ρ∗ π ∗ X42,1 , that – thanks to Koll´ar’s Theorem – makes ρ∗ π ∗ X42,1 unirational over P2 × S and hence implies the unirationality of C42,1 . 

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Remark 4.9. As the reader may observe, the universal cubic C44,1 is missing from our description. Cubics in this divisor contain a Fano model of an Enriques surface [Nue17, Theorem 3.2]. The divisor C44 is unirational, so in order to apply Koll´ ar’s Theorem and show that C44,1 is unirational, one should show that the generic Enriques surface has a rational point. 4.3. Unirationality of Cd,n In this section we are going to use the inductive structure apparent in the construction of Cd,n – the moduli space of cubic fourfolds in Cd with n marked points – and Koll´ar’s Theorem, in order to establish the unirationality of Cd,n for all n. Recall in fact that Cd,n is just the universal cubic fourfold over Cd,n−1 . This allows us to use our machinery to prove inductively the following. Theorem 4.10. The moduli spaces Cd,n are unirational for all n, if 8 ≤ d ≤ 42. Proof. We will work inductively on n. For n = 0 (or respectively 1) the claim is true thanks to [Nue17] and [RS19b] (or respectively Sections 4.1 and 4.2). We will denote by Sd the unirational family of rational surfaces contained in cubics in Cd , P(Vd ) → Sd the relative ideal of cubic hypersurfaces through each surface and Xd,1 → P(Vd ) the universal cubic over the relative linear system. As seen in Theorem 4.2 and Proposition 4.7, the variety Xd,1 is unirational and dominates Cd,1 through the natural classifying map of the coarse moduli space Cd,1 . The situation is described by the following diagram: π ∗ Xd,1 I s

 P(π ∗ Vd )  P2 × Sˆd

π

/ / Xd,1

σ

/ /, Cd,1

 / / P(Vd )

 / / Cd ,

(4.11)

 / Sd

where P2 × Sˆd is the appropriate rational space that assures the existence of the section s. As seen in Theorems 4.2 and 4.7, this boils down to taking P2 × (P2 )p for 8 ≤ d < 42, and the rational parameter space P2 × S for d = 42. We remark that s is the section that makes π ∗ Xd,1 unirational following Koll´ar’s Theorem. Call σ the classifying map to Cd,1 . Then, we can add one point and consider the universal cubic Cd,2 → Cd,1 , that fits in the following diagram τ

s

 P2 × π ∗ Xd,1  P × Sˆd 2

/ /, Cd,2

/ / σ ∗ Cd,2

γ ∗ σ ∗ Cd,2 I γ

 / / π ∗ Xd,1  / Sˆd

σ

 / / Cd,1 ,

(4.12)

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109

where γ is the second projection and σ the dominant map from Diagram 4.11. Recall that π ∗ Xd,1 is unirational, and dominates Sˆd . We observe that, exactly as we have done in Diag 4.11 for Cd,1 , since π ∗ Xd,1 dominates Sˆd,1 , we are able to define a natural rational section s : P2 × π ∗ Xd → γ ∗ σ ∗ Cd,2 . Hence the universal cubic has rational points over P2 × π ∗ Xd,1 and this space is unirational. By Koll´ar’s Theorem and since γ ∗ σ ∗ Cd,2 dominates Cd,2 , we get that Cd,2 is unirational. Now it is straightforward to see how to continue the argument inductively; we just draw the diagram for the following step for clarity. Recall that γ ∗ σ ∗ Cd,2 = τ ∗ Cd,2 . ω

/ /, Cd,3

/ / τ ∗ Cd,3

λ∗ τ ∗ Cd,3 I s

 P2 × τ ∗ Cd,2  P × Sˆd 2

λ

 / / τ ∗ Cd,2  / Sˆd

τ

 / / Cd,2

(4.13)



5. Some results of non-unirationality In this last section we collect some recent results about the Kodaira dimension of moduli spaces of K3 surfaces and of the Hassett divisors Cd . These allow us to show that for an infinite range of values of d, the universal cubic fourfold over Cd can not be unirational. Let us first prove the following straightforward Lemma, in which we need to assume that d is even. Lemma 5.1. Suppose that the Kodaira dimension of F d+2 or of Cd is positive, then 2 the universal cubic fourfold Cd,1 is not unirational. We will apply Lemma 5.1 to the following two Propositions, due to GritsenkoHulek-Sankaran and V´ arilly-Alvarado-Tanimoto. Prop. 5.2 was initially conceived for moduli of K3 surfaces; following [Nue17] we write its “translation” in terms of cubic fourfolds via the rational map of (2.4). Proposition 5.2. Let d > 80, d ≡ 2 (mod 6), 4  |d be such that for any odd prime p, p|d implies p ≡ 1 (mod 3). Then the Kodaira dimension of Cd is non-negative. If moreover d > 122, then Cd is of general type. Proposition 5.3. The divisor C6n+2 is of general type for n > 18 and n = 20, 21, 25 and has nonnegative Kodaira dimension for n > 13 and n = 15. Moreover, C6n is

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of general type for n > 18, n = 20, 22, 23, 25, 30, 32 and has nonnegative Kodaira dimension for n > 16 and n = 18, 20, 22, 30. By combining Propositions 5.3 and 5.2 with Lemma 5.1 we obtain the following result: Proposition 5.4. Under the hypotheses on d of Proposition 5.3 and 5.2, the universal cubic fourfold Cd,1 is not unirational. The full, lenghty statement contained in Proposition 1.3 in the Introduction. 5.1. Open questions In light of the results of this paper, it would be interesting to start a systematic study of the birational geometry of universal cubic fourfolds. We formulate some very natural questions, which are of course related to the classical birational geometry of cubic fourfolds. Questions: 1. What is the Kodaira dimension of the universal cubic fourfold over Cd , for different values of d. 2. In particular, is C44,1 unirational? 3. Is there a relation between the Kodaira dimension of the universal cubic fourfold over Cd and that of the universal associated K3? 4. What is the Kodaira dimension of the universal cubic fourfold over the full moduli space C? 5. Are there divisors Cd , where the geometry of special rational surfaces inside cubic fourfolds can help to describe the birational geometry of universal associated K3 surfaces? Acknowledgement o and Sandro Verra for lots We wish to thank Francesco Russo, Giovanni Staglian` of discussions on related topics in the last few months. Thanks to Shouhei Ma and Zhiwei Zheng for giving us precise references and information. The authors are members of the research groups GAGC and GRIFGA, whose support is acknowledged.

References arilly-Alvarado, Cubic four[ABBVA14] A. Auel, M. Bernardara, M. Bolognesi, and A. V´ folds containing a plane and a quintic del Pezzo surface, Algebr. Geom. 1 (2014), no. 2, 181–193. [AHTVA19] Nick Addington, Brendan Hassett, Yuri Tschinkel, and Anthony V´ arillyAlvarado, Cubic fourfolds fibered in sextic del Pezzo surfaces, Amer. J. Math. 141 (2019), no. 6, 1479–1500. [AT14] Nicolas Addington and Richard Thomas, Hodge theory and derived categories of cubic fourfolds, Duke Math. J. 163 (2014), no. 10, 1885–1927.

Unirationality of Certain Universal Families of Cubic Fourfolds [Awa19] [BD85]

[BP20] [BRS19] [CG72] [DT20] [Enr97]

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[HS20] [Kol02] [Kuz10]

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Hanine Awada, Rational fibered cubic fourfolds with nontrivial Brauer classes, arXiv preprint (2019), 1–20. A. Beauville and R. Donagi, La vari´et´e des droites d’une hypersurface cubique de dimension 4, C. R. Acad. Sci. Paris S´er. I Math. 301 (1985), no. 14, 703–706. Michele Bolognesi and Claudio Pedrini, The transcendental motive of a cubic fourfold, J. Pure Appl. Algebra 224 (2020), no. 8, 106333. o, Some loci of Michele Bolognesi, Francesco Russo, and Giovanni Staglian` rational cubic fourfolds, Math. Ann. 373 (2019), no. 1-2, 165–190. C. Herbert Clemens and Phillip A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356. MR 302652 Daniele Di Tullio, Rationality of the universal k3 surface of genus 8, 2020. Federigo Enriques, Sulle irrazionalit` o farsi dipendere la risolua da cui pu` zione d’un’ equazione algebrica f (xyz) = 0 con funzioni razionali di due parametri, Math. Ann. 49 (1897), no. 1, 1–23. Gavril Farkas and Alessandro Verra, The universal K3 surface of genus 14 via cubic fourfolds, J. Math. Pures Appl. (9) 111 (2018), 1–20. , The unirationality of the moduli space of K3 surfaces of genus 22, arXiv preprint (2019), 1–17. V´ıctor Gonz´ alez-Aguilera and Alvaro Liendo, Automorphisms of prime order of smooth cubic n-folds, Arch. Math. (Basel) 97 (2011), no. 1, 25–37. V. Gritsenko, K. Hulek, and G.K. Sankaran, Moduli of K3 surfaces and irreducible symplectic manifolds, Handbook of moduli. Vol. I, Adv. Lect. Math. (ALM), vol. 24, Int. Press, Somerville, MA, 2013, pp. 459–526. S. Galkin and E. Shinder, The fano variety of lines and rationality problem for a cubic hypersurface, arxiv preprint https://arxiv.org/abs/1405.5154 (2014). B. Hassett, Special cubic fourfolds, Comp. Math. 120 (2000), no. 1, 1–23. Brendan Hassett, Cubic fourfolds, K3 surfaces, and rationality *questions, Rationality problems in algebraic geometry, Lecture Notes in Math., vol. 2172, 2016, pp. 29–66. M. Hoff and G. Staglian` o, New examples of rational Gushel–Mukai fourfolds, Math. Zeit. (2020). J´ anos Koll´ ar, Unirationality of cubic hypersurfaces, J. Inst. Math. Jussieu 1 (2002), no. 3, 467–476. A. Kuznetsov, Derived categories of cubic fourfolds, Cohomological and Geometric Approaches to Rationality Problems, Progress in Mathematics, vol. 282, Birkh¨ auser Boston, 2010, pp. 219–243. Kuan-Wen Lai, New cubic fourfolds with odd-degree unirational parametrizations, Algebra Number Theory 11 (2017), no. 7, 1597–1626. Radu Laza, The moduli space of cubic fourfolds, J. Algebraic Geom. 18 (2009), no. 3, 511–545. , The moduli space of cubic fourfolds via the period map, Ann. of Math. (2) 172 (2010), no. 1, 673–711.

112 [Ma19] [Nue17] [RS19a]

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H. Awada and M. Bolognesi Shouhei Ma, Mukai models and borcherds products, arXiv preprint (2019), 1–37. Howard Nuer, Unirationality of moduli spaces of special cubic fourfolds and K3 surfaces, Algebr. Geom. 4 (2017), no. 3, 281–289. Francesco Russo and Giovanni Staglian` o, Congruences of 5-secant conics and the rationality of some admissible cubic fourfolds, Duke Math. J. 168 (2019), no. 5, 849–865. , Trisecant Flops, their associated K3 surfaces and the rationality of some Fano fourfolds, arXiv preprint (2019), 1–30. N.I. Shepherd-Barron, The rationality of quintic Del Pezzo surfaces – a short proof, Bull. London Math. Soc. 24 (1992), no. 3, 249–250. MR 1157259 Sho Tanimoto and Anthony V´ arilly-Alvarado, Kodaira dimension of moduli of special cubic fourfolds, J. Reine Angew. Math. 752 (2019), 265–300.

Hanine Awada and Michele Bolognesi Institut Montpellierain Alexander Grothendieck Universit´e de Montpellier CNRS Case Courrier 051 – Place Eug`ene Bataillon F-34095 Montpellier Cedex 5, France e-mail: [email protected] [email protected]

A Categorical Invariant for Geometrically Rational Surfaces with a Conic Bundle Structure Marcello Bernardara and Sara Durighetto Abstract. We define a categorical birational invariant for minimal geometrically rational surfaces with a conic bundle structure over a perfect field via components of a natural semiorthogonal decomposition. Together with the similar known result on del Pezzo surfaces, this provides a categorical birational invariant for geometrically rational surfaces. Mathematics Subject Classification (2010). 14E08, 14F08, 14D10, 14J26. Keywords. Conic bundles, rational surfaces, derived categories, semiorthogonal decompositions.

1. Introduction In recent years the study of the derived category of an algebraic variety has been widely developed. It is clear now that semiorthogonal decompositions can provide a useful tool in order to detect the geometrical structure of a variety. Particularly relevant and interesting is the research of a birational invariant to be used, for example, to study the rationality of the variety. In this context, the first author and M. Bolognesi [6] introduced the concept of categorical representability and formulated the following question: is a rational variety always categorically representable in codimension 2? Analogously, is it possible to characterize obstruction to rationality via natural components of some semiorthogonal decomposition which cannot be realized in codimension 2? Over the complex field, for example, if we consider a V14 Fano threefold X, its derived category admits a semiorthogonal decomposition with only one nontrivial component AX . For a smooth cubic threefold Y we can also find a decomposition with only one nontrivial component AY , and Kuznetsov showed that AX is equivalent to AY if Y is the unique cubic threefold birational to X [13]. This suggests that one could consider AX to be a birational invariant. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Farkas et al. (eds.), Rationality of Varieties, Progress in Mathematics 342, https://doi.org/10.1007/978-3-030-75421-1_5

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In the case of complex conic bundles π : X → S over a minimal rational surface the situation is quite well known. A necessary condition for rationality of X is that the intermediate Jacobian J(X), as principally polarized Abelian variety, is the direct sum of the intermediate Jacobian of smooth projective curves. It follows for example that smooth cubic threefolds are not rational [7]. From a categorical point of view it is possible to characterize the rationality of the conic bundle from the semiorthogonal decomposition of the derived category. By Kuznetsov [12] we have Db (X) = ΦDb (S, B), π ∗ Db (S), (1.1) where B is the sheaf of even parts of Clifford algebras associated to the quadratic form defining the fibration and Φ : Db (S, B) → Db (X) is a fully faithful functor from the derived category of B-algebras over S. If S is rational the only nontrivial part for this semiorthogonal decomposition must then be contained in the component Db (S, B). If S is minimal, the first author and Bolognesi proved that X is rational if and only if Db (S, B) has a decomposition whose components are derived categories of smooth curves or exceptional objects [5]. All those results hold on the complex field C, but we want to study the problem over an arbitrary perfect field k. Auel and the first author worked out the case of del Pezzo surfaces [2]1 . Given a minimal del Pezzo surface S of degree d and Picard rank 1, a natural subcategory AS ⊂ Db (S) is defined by the right orthogonal complement to the structure sheaf. In [2], a category GKS is defined, roughly speaking, as the product of all components of AS which are not representable in dimension 0, and it is a birational invariant. Such Griffiths–Kuznetsov component, where it is defined, is then the suitable birational invariant to detect the rationality of the given variety. Our aim is to extend this approach to the other class of geometrically rational minimal surfaces, that is, conic bundles. The precise definition of such an invariant is given in Definition 12. Roughly speaking, we define the Griffiths–Kuznetsov component to be the direct sum of subcategories of Db (S) which are not representable in dimension 0. However, unlike in the case of del Pezzo surfaces, there is no, to the best of the authors’ knowledge, argument to prove that the (natural) decomposition we choose to define GKS is unique up to mutations. This motivates the involved case-by-case definition, and a fundamental part of this work is to prove that GKS is indeed well defined. Our main result is the following. Theorem 1. Let k be a perfect field and S be a geometrically rational surface birational to a conic bundle over k. The Griffiths–Kuznetsov component GKS is well defined and is a birational invariant. Recall the classification of minimal geometrically rational surfaces over an arbitrary field (see, e.g., [9]): minimal conic bundles are one of the two possible 1 The

results in [2] are claimed to hold over general fields, but perfection is required, as we will show in Remark 4, to ensure that every birational map can be factored into Sarkisov links as in [10].

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classes of such surfaces, namely the ones with Picard rank two, the other being del Pezzo surfaces with Picard rank one. Combining Theorem 1 with the results from [2], we obtain the following result. Theorem 2. Let S be a geometrically rational surface over a perfect field k. Then the Griffiths–Kuznetsov component GKS is well defined and it is a birational invariant. Notations Functors of geometric origin between derived categories will be denoted underived (i.e., f ∗ instead of Lf ∗ for the pull-back via a morphism). Given a k-algebra A, the notation Db (k, A) stands for the k-linear bounded derived category of coherent A-modules.

2. Basics on geometrically rational surfaces In this section we will introduce some useful and known results. Let k be a perfect field and k an algebraic closure. Let us consider S, a smooth projective geometrically integral surface over k. We say that S is geometrically rational if S := S ×k k is k-rational. A field extension l of k is a splitting field for S if S ×k l is birational to P2l through a sequence of monoidal transformations centered at closed l-points. A smooth projective surface S is minimal over k if every birational morphism φ : S → Y , defined over k, to a smooth variety Y is an isomorphism. If k is algebraically closed, the only minimal rational surfaces are the projective plane and projective bundles over P1 . Over a general field, we have the following classification (see, e.g., [9]). Proposition 3. Let S be a minimal geometrically rational surface over k. Then S is one of the following: (i) S = P2k is the projective plane, so Pic(S) = Z, generated by the hyperplane section O(1); (ii) S ⊂ P3k is a smooth quadric and Pic(S) = Z, generated by the hyperplane section O(1); (iii) S is a del Pezzo surface with Pic(S) = Z, generated by the canonical class ωS ; (iv) S is a conic bundle π : S → C over a geometrically rational curve, with Pic(S)  Z ⊕ Z. 2.1. Elementary links We recall some elements of the Sarkisov program describing the factorization of a birational map between minimal rational surfaces in elementary links [10]. Let S be a minimal geometrically rational surface with an extremal contraction π : S → Y . Then either Y is a point and S is a minimal surface with Picard rank 1 or Y is a curve and S is a conic bundle with Picard rank 2. If π : S → Y and π  : S  → Y 

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are two such extremal contractions (but S and S  not necessarily minimal), an elementary link is a birational map φ : S  S  of one of the following types: Type I)

There is a commutative diagram So

σ

S

 Y o

ψ

 Y

where φ = σ −1 , the morphism σ : S  → S is a Mori divisorial elementary contraction and ψ : Y  → Y is a morphism. In this case, Y = Spec(k), ρ(S) = 1, S is a minimal del Pezzo surface, and S  → Y  is a conic bundle over a geometrically rational curve. Type II) There is a commutative diagram So

σ

 Y o

τ

X

/ S  Y

∼ =

where φ = τ ◦ σ −1 , and σ : X → S and τ : X → S  are Mori divisorial elementary contractions. In this case, S and S  have the same Picard number, and are hence either both minimal del Pezzo surfaces (and Y is a point), or both conic bundles of Picard rank two (and Y is a geometrically rational curve). Type III) There is a commutative diagram S  Y

σ

/ S

ψ

 / Y

where φ = σ, σ : S → S  is a Mori divisorial elementary contraction, S → Y is a conic bundle, Y is a geometrically rational curve, S  is minimal del Pezzo surface, Y  = Spec(k) and ψ : Y → Y  is the structural morphism. Links of type III can be seen as inverses of links of type I. Type IV) There is a commutative diagram φ S _ _ _ _ _ _ _ _ _/ S 

  Y G Y GG ww GGψ ψ  ww GG w w GG w # {ww Spec(k)

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where S  S  are isomorphic minimal conic bundles, Y and Y  are geometrically rational curves and ψ and ψ  are the structural morphisms. Then both S and S  are conic bundles and the link amounts to a change of conic bundle structure on S. Any birational map φ : S  S  between minimal geometrically rational surfaces can be factored through elementary links, and Iskovskikh gives the complete list of all possible such links [10]. We note that the Picard rank is invariant under links of type II and IV, while it changes under links of type I and III. Moreover, if we suppose that S is not rational the list of links of type I (and hence of their inverses of type III) is very limited: either S is of degree 8 with a point of degree 2, and S  is of degree 6 and the curve Y  can be rational (according to S being a quadric or not), or S is of degree 4, has a rational point and S  is of degree 3 and Y  is a rational curve. Remark 4. If k is not perfect, then a birational map may not be decomposable in a finite sequence of elementary links centered at closed points. An example of such map was given in [14, Rmk. 1.3]: if k = (Z/2Z)[t], the birational map φ of Pk2 given by [x0 : x1 : x2 ]  [x0 x2 : x1 x2 : x02 + tx12 ] √ has [ t : 1 : 0] as a base point, and such a point is never defined over a separable field extension of k.

3. Basics on derived categories 3.1. Categorical representability Using semiorthogonal decompositions, one can define a notion of categorical representability for triangulated categories. In the case of smooth projective varieties, this is inspired by the classical notions of representability of cycles, see [6]. We refrain here from recalling standard notions of semiorthogonal decompositions, exceptional objects, and mutations. The interested reader can refer to [1]. Let us just recall a nonstandard definition of exceptional object. Definition 5. Let A be a division (not necessarily central) simple k-algebra (i.e., the center of A could be a field extension of k), and A a k-linear triangulated category. An object V of A is called A-exceptional if Hom(V, V ) = A

and Hom(V, V [r]) = 0

for r = 0.

An exceptional object in the classical sense of the term [8, Def. 3.2] is a kexceptional object. By exceptional object, we mean an A-exceptional object for some division k-algebra A. Remark 6. Note that, if E is an A-exceptional object of A, then the (full triangulated) category E generated by E is equivalent to Db (k, A), the category of bounded complexes of coherent A-algebras.

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Example 7. Let A be a central simple algebra over k and X := SB(A) the SeveriBrauer variety associated to it, and let n = dim X. The Quillen vector bundle V is a rank n + 1 indecomposable vector bundle whose base change to a splitting field is O(1)⊕n+1 , and is in particular an A-exceptional object [15]. Definition 8. A triangulated category A is representable in dimension m if it admits a semiorthogonal decomposition A = A1 , . . . , Ar , and for each i = 1, . . . , r there exists a smooth projective k-variety Yi with dim Yi ≤ m, such that Ai is equivalent to an admissible subcategory of Db (Yi ). The above definition is motivated by the following question: Question 9. Let X be a smooth projective k-variety of dimension n. Does X rational imply Db (X) categorically representable in dimension n − 2? In this work, we consider the above question for surfaces, and we are hence interested in characterizing categories which are representable in dimension 0. These were fully described in [2]. Lemma 10. A triangulated category A is representable in dimension 0 if and only if there exists a semiorthogonal decomposition A = A1 , . . . , Ar , such that for each i, there is a k-linear equivalence Ai  Db (Ki ) for a separable field extension Ki /k. 3.2. Conic bundles We recall a natural semiorthogonal decomposition of the derived category of a surface with the structure of a conic bundle, following the work of Kuznetsov [12] and its generalization to general fields [3]. Let S be a surface over the field k with a structure of conic bundle π : S → C over a geometrically rational curve. Such conic bundle is associated to a quadratic form q : E → L on a locally free OC module E of rank 3. Denote by OS/C (1) the restriction to S of the line bundle OPE/C (1), and let B be the even Clifford algebra associated to the form q, which is a locally free OC -algebra, well defined up to isomorphism. Under these conditions, we have that π ∗ : Db (C) → Db (S) is fully faithful, and there exist a fully faithful functor Φ : Db (C, B) → Db (S) such that Db (S) = π ∗ Db (C), ΦDb (C, B). Notice that one can choose fully faithful functors from Db (C, B) to Db (S) to realize either the left (as in the introduction) or the right (as above) orthogonal complement to π ∗ Db (C). The two functors are different (the differ by a twist by the canonical bundle), but they anyway give equivalent subcategories of Db (S). Here and further we choose the functor giving the above semiorthogonal decomposition for practical computational reasons that will be clear in the proofs.

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Moreover, since C is a geometrically rational curve, there is a central simple k-algebra A (trivial if and only if C  Pk1 ) such that C = SB(A). In particular, there is an A-exceptional object V , which is either O(1) if C  Pk1 or the Quillen bundle V as in example 7 if C is not rational, such that (as proved in [4]) Db (C) = OC , V  = V ∗ , OC . It follows that we can refine the semiorthogonal decomposition of S (by abuse of notation, we set V := π ∗ V ) as follows: Db (S) = OS , V, ΦDb (C, B).

(3.1)

P1k

Now, let π : S → be the base change of the conic bundle to the algebraic closure. Such a conic bundle is not necessarily a Hirzebruch surface and can indeed be not minimal, and have a finite number, say r, of singular fibers which are given by two lines meeting in a point. We can pick one line in each such fiber and denote such set of lines by F1 , . . . , Fr . The Picard rank of S is then 2 + r, and there is a semiorthogonal decomposition obtained by considering the k-minimal model S → S0 , which is a Hirzebruch surface: Db (S) = OS , OS (F ), OS (Σ), OS (Σ + F ), OF1 , . . . , OFr ,

(3.2)

where F is the general fiber of π, and Σ is a general section of π. We finally notice that the base change of the semiorthogonal decomposition (3.1) is exactly the semiorthogonal decomposition (3.2): indeed, either C is rational and we already have V = OS (F ), or C is not rational, V has rank 2 and we have V = OS (F )⊕2 . The latter generates the same category as OS (F ) since we are considering thick subcategories. It follows that the base change of ΦDb (C, B) to S is the subcategory OS , OS (F )⊥ = OS (Σ), OS (Σ + F ), OF1 , . . . , OFr .

(3.3)

4. Links of type I/III and the definition of the Griffiths–Kuznetsov component We are going to construct a birational invariant for geometrically rational surfaces with a conic bundle structure π : S → C as the collection of subcategories in the semiorthogonal decomposition (3.1) which are not representable in dimension 0. Such an invariant will match the one constructed in [2] in the case where S is birational to a minimal del Pezzo surface. Based on results of Karpov and Nogin [11], the authors have shown uniqueness of semiorthogonal decompositions for such del Pezzo surfaces. We are unfortunately not able to show this result for conic bundles, and the definition has to be more ad hoc. We start by recalling [2, Def. 1]. Definition 11. Let S be a minimal del Pezzo surface over k, and AS = OS ⊥ . We define the Griffiths–Kuznetsov component GKS of S as follows: if AS is representable in dimension 0, set GKS = 0. If not, GKS is either the product of all

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indecomposable components of AS of the form Db (l, α) with l/k a field extension and α ∈ Br(l) nontrivial or, if AS does not admit such subcategories, GKS = AS . If S is not minimal, then we set GKS = GKS  for a minimal model S → S  . Note that the term component is slightly abused here, since there are cases where GKS is not a component of Db (S) but rather the direct sum of some components. Since we can operate mutations on semiorthogonal decompositions (and we will indeed do so to prove the main theorem), we cannot in general give any canonical gluing of components contributing to GKS . The same abuse of terminology will appear in Definition 12. One of the main results of [2] is that GKS is well defined, unless S is nonrational and has either degree 8 and a point of degree 2, or degree 4 and a rational point. In this case, S is birational to a minimal conic bundle (by blowing up the given point), so these cases will be included in Definition 12. Let us analyze them first. Let S  be a minimal non-rational del Pezzo surface of degree 8 with a point of degree 2. Then (see, e.g., [2, §7]), S  is an involution surface in a Severi-Brauer threefold SB(B), and there is an associated even Clifford algebra C, which is a simple algebra whose center is a degree two field extension of k, and a semiorthogonal decomposition Db (S  ) = Db (k), Db (k, B), Db (k, C), where the first category is generated by OS  and the second one either by OS  (1) (in the case B = 0 and S  is a quadric) or by the restriction of the Quillen bundle of SB(B) to S  (in the case where S  is not a quadric). It follows that the Griffiths–Kuznetsov component for S  should be GKS  := Db (k, C) if S  is a quadric and GKS  := Db (k, C) ⊕ Db (k, B) if S  is not a quadric. In this case, such a category is not shown to be a birational invariant in [2], and this is due to the existence of a link of type I, from which follows that the birational class of S  contains minimal conic bundles of degree 6. Indeed, the blow-up of a degree 2 point S → S  is a conic bundle π : S → C, with C either rational if S  is a quadric or non-rational if S  is not a quadric. In [2, §B], it is proved that the component we want to construct is indeed related to the standard semiorthogonal decomposition of the conic bundle as follows: writing C = SB(A) we have that A and B are Morita-equivalent (note that B has order dividing 2 since it has an involution defining S  ), and that there is a degree 2 extension l/k and a semiorthogonal decomposition: Db (C, B) = Db (l), Db (k, C), where we keep the previous notations for the Clifford algebra associated to the conic bundle S → C. It follows that the components which are (potentially) not representable in dimension 0 in the standard decomposition (3.1) are exactly Db (k, C) and Db (k, B) as suggested by Definition 11 for the del Pezzo surface S  of degree 8.

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Let S  be a minimal del Pezzo surface of degree 4 with a rational point. Note that S  is not rational. In particular, there is a semiorthogonal decomposition Db (S  ) = OS  , AS  , and AS  (or rather, its non-zero-dimensional component) is expected to be the good candidate for the birational invariant we are looking for. This case neither was treated in [2], since, again, the existence of a link of type I implies that the birational class of S  contains minimal conic bundles of degree 3. Indeed, the blowup of a rational point S → S  is a conic bundle π : S → P1 . In [2, A.2] (see also [3]), it is proved that the component we want to construct is indeed related to the standard semiorthogonal decomposition of the conic bundle since there is an equivalence Db (P1 , B)  AS  , where we keep the previous notations for the Clifford algebra associated to the conic bundle S → P1 . It follows that the components which are (potentially) not representable in dimension 0 in the standard decomposition (3.1) are the ones suggested by Definition 11 for the del Pezzo surface S  of degree 4. Definition 12. Let S be a surface admitting a structure of conic bundle π : S → C over a geometrically rational smooth curve C. If S is minimal, the Griffiths– Kuznetsov component GKS of S is defined as follows: (i) if S is rational, GKS = 0; (ii) if S = C1 ×C2 where Ci is a geometrically rational curve with associated Azumaya algebra Ai , then GKS is the sum of those between Db (k, A1 ), Db (k, A2 ) and Db (k, A1 ⊗ A2 ) which are not equivalent to Db (k) (equivalently, the algebra is not Brauer-trivial); (iii) if C  P1 and S is birational to a non-rational quadric with associated even Clifford algebra C, then GKS = Db (k, C); (iv) if C  P1 , and S is neither rational nor birational to a quadric, GKS = Db (P1 , B); (v) if C is irrational with associated Azumaya algebra A, and S is birational to a quadric with associated even Clifford algebra C, then GKS = Db (k, A) ⊕ Db (k, C); (vi) if C is irrational with associated Azumaya algebra A, and S is not birational to a quadric, then GKS = Db (k, A) ⊕ Db (C, B); If S is not minimal, the Griffiths–Kuznetsov component is GKS := GKS  for a minimal model S → S  . The rest of the paper is dedicated to the proof of Theorem 1. Note that we can restrict to minimal models. If π : S → C and π  : S  → C  be are minimal conic bundle structures and φ : S  S  is a birational morphism, then φ can be decomposed in a finite number of links. By the discussion above on links of type I/III, we are left to prove the invariance under links of type II (Theorem 13) and type IV (Corollary 16), in the non rational cases.

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5. Links of type II Links of type II between conic bundles are the most common birational transformation. They are given by an elementary transformation along a closed fiber of the conic bundle structure. Let π : S → C be a conic bundle. To define a link of type II, pick a closed point x ∈ S of degree d such that the geometric points {xi }di=1 = x ⊗ k are in general position on S, and denote by Sx the fiber of π containing x. Then perform the blow-up of S in x followed by the subsequent contraction of the fiber Sx . This gives a conic bundle π  : S  → C and a commutative diagram: Z@  @@@ q   @@  @   φ S _ _ _ _ _ _ _/ S  p

π

π

 C

id

 /C

Theorem 13. In the above setting, we have GKS  GKS  . Proof. First note that S and S  can be of any type except (i) from Definition 12. However, if the degree KS2 (which is constant under the link) is 8, the invariance has been proven in [2, §C]. This rules out all of the case (ii). In the other cases, note that it will be sufficient to show two equivalences: between the categories V  and V   generated by the pull-back of the Quillen bundle from C (which is redundant if C  P1 , cases (iii) and (iv)), and between the categories Db (C, B) and Db (C, B  ) of sheaves over the Clifford algebras associated to the conic bundle structures. The latter is evident in cases (iv) and (vi), and follows from the discussion preceeding definition 12 for in cases (iii) and (v). We proceed to show the above equivalences for links of type II. Let E be the exceptional divisor of p and E  the exceptional divisor of q. We denote by f (resp. f  ) the pullback of the generic fiber of π (resp. π  ) via p (resp. q) in Z. Recall [10] that p∗ (−KS ) = q ∗ (−KS  ) + df − 2E  f = f

(5.1) 



∗

∗



E = df − E = p (−KS ) − q (−KS  ) + E , up to linear equivalence. Since the isomorphism class of C is preserved under this link, so is the algebra A (which is trivial if and only if C  P1 ), and therefore the category Db (k, A). It is left to prove that the equivalence class of the category AS := Db (C, B) is also preserved. Over the algebraic closure, the category AS ⊂ Db (S) admits the following semiorthogonal decomposition: AS = OS (Σ), OS (Σ + F ), OF1 , . . . , OFr ,

(5.2)

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as in (3.2). Note that we use here F to denote the class of the general fiber of S, while over k we used the notation f as in (5.1). This is because, if C is not rational, then f = F , but rather f = 2F . We have a semiorthogonal decomposition AS  = OS  (Σ ), OS  (Σ + F  ), OF1 , . . . , OFr .

(5.3)

Over k, we have that S is the blow up of r points on a Hirzebruch surface Fn . Ifwe denote by D1 , . . . , Dr the exceptional divisors of such blow up, and by r D = i=1 Di , we have that −KS = 2Σ+(n+2)F −D. Similarly, S  is the blow-up of r points on a Hirzebruch surface Fm , with exceptional divisors D1 , . . . , Dr , and r  we use the notation D = i=1 Di , so that −KS  = 2Σ + (m + 2)F  − D . By abuse of notations, we will drop p∗ (resp. q ∗ ) when dealing with divisors on Z that  are pull-back of divisors of S (resp. S ). For example, Σ will denote p∗ Σ and Σ will denote q ∗ Σ and so on. One can see that the map φ is obtained by lifting to S the composition of d elementary transformations on Fn , in particular m = n − d, along fibers that do not contain the divisors Di . It follows that the divisors Di are preserved by φ and so D is sent to D . We conclude with the following equalities of (rational equivalence classes of) divisors in Z: D = D , F = F  . Using the above equalities and the above explicit descriptions of the canonical bundles, the first relation in (5.1) gives Σ = Σ −E  as (rational equivalence classes of) divisors in Z. Then we have the following equivalence of subcategories of Db (Z): p∗ AS ⊗ O(E  ) = q ∗ AS  . Indeed, first note the singular fibers are preserved under the birational transformation φ, and we can choose Fi such that p∗ Fi = q ∗ Fi for i = 1, . . . , r. Moreover Oq∗ Fi does not change under tensoring with O(E  ), since the exceptional divisor is not supported on singular fibers. Secondly, using the above relation Σ = Σ − E  , it is not difficult to see that O(Σ), O(Σ + F ) ⊗ O(E  ) = O(Σ ), O(Σ + F  ) We can now conclude since the autoequivalence ⊗O(E  ) descends to an autoequivalence of Db (Z), since E  is defined over k. It follows, that AS is equivalent to AS  . 

6. Links of type IV A link of type IV is a birational self-transformation of a minimal irrational surface S exchanging two conic bundle structures πi : S → Ci , for i = 1, 2. In particular,

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the birational map φ : S  S fits a commutative diagram φ S _ _ _ _ _ _ _ _ _ _/ S

π1

π2

  C2 C1 G GG ww GG ww GG w GG ww # {ww Spec(k)

For i = 1, 2, we will denote by Vi the Quillen bundle of Ci and by Ai the associated Azumaya algebra, and by Bi the even Clifford algebra of the conic bundle πi and by Φi : Db (Ci , Bi ) → Db (S) the functor described in [12] giving the semiorthogonal decomposition (3.1). We will also use the shorter notation Ai := Φi Db (Ci , Bi ) ⊂ Db (S). As proved in [10], the degree KS2 of S must be 8, 4, 2 or 1, and the list of birational maps is quite limited. In this case, we need to prove that the Griffiths– Kuznetsov component is well defined, namely that it does not depend on the choice of semiorthogonal decomposition given by the different conic bundle structures. We proceed by a case by case analysis. Proposition 14. In the situation above, let S have degree 8 and φ a link of type IV. Then GKS is invariant under φ. Proof. In this case, S = C1 × C2 , and we are in case (ii) of Definition 12. The fact that GKS is well defined is proved in [2, §C].  Proposition 15. In the situation above, let S have degree 4, 2, or 1, and φ a link of type IV. Then GKS is invariant under φ. Proof. First note that such an S cannot be birational to a quadric: indeed, the only type I/III link relating a non-rational quadric to a conic bundle ends up in a degree 6 conic bundle, and links of type II do not change the degree. Similarly, S is not birational to a del Pezzo of degree four with a rational point, and hence S is not birational to any minimal del Pezzo surface. In particular, we are either in case (iv) (if C1  C2  P1 ) or in case (vi) (if C1 , C2 are not rational) of Definition 12. For i = 1, 2, consider the semiorthogonal decomposition Db (S) = OS , Vi , Ai ,

(6.1)

and recall that Vi  = OS (Fi ), for Fi a geometric fiber of πi . Indeed, either we are in case (iv), Ci is rational and Vi = OS (Fi ) is already a line bundle (and does not contribute to the Griffiths–Kuznetsov components), or we are in case (vi) Ci is not rational and Vi = OS (Fi )⊕2 . Now we proceed by case by case analysis following the possibilities given by [10]. Degree 4. Assume S has degree 4. We have F1 = −KS − F2 (see [10, Page 611]), so that V1 = V2∗ ⊗ωS∗ . It follows that V1   V2∗   V2 , first via the autoequivalence

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⊗ωS∗ and secondly since A2op and A2 are Brauer equivalent. It follows that A1 and A2 are Brauer equivalent. It is left to prove that A1  A2 . To this end, consider the semiorthogonal decompositions: Db (S) = V2∗ , OS , A2  = V1 , ωS∗ , A2 ⊗ ωS∗  = OS , A2 , V1 , where the second equality is given by the autoequivalence ⊗ωS∗ on Db (S) and the third is the mutation of ωS∗ , A2 ⊗ ωS∗  to the left with respect to V1 . We can then mutate A2 to the right with respect to V1 and obtain the right orthogonal complement of OS , V1  which is A1 , as in (6.1) for i = 1. This last mutation gives then the required equivalence between A2 and A1 . Degree 2. Assume S has degree 2. We have F1 = −2KS − F2 (see [10, Page 611]), so that V1 = V2∗ ⊗ (ω ∗ )⊗2 . It follows that V1   V2∗   V2 , first via the autoequivalence ⊗(ωS∗ )⊗2 and secondly since Aop 2 and A2 are Brauer equivalent. It follows that A1 and A2 are Brauer equivalent. It is left to prove that A1  A2 . To this end, consider the semiorthogonal decompositions: Db (S) = OS , V1 , A1  = V1 , A1 , ωS∗  = A1 , V1 , ωS∗ , where the second equality is the mutation of OS to the right with respect to its orthogonal complement and the third one is the mutation of A1 to the left with respect to V1 , so that A1 = ⊥ V1 , ωS∗  is equivalent to A1 . Now consider the second conic bundle structure and the semiorthogonal decompositions Db (S) = V2∗ , OS , A2  = V1 , (ωS∗ )⊗2 , A2 ⊗ (ωS∗ )⊗2  = ωS∗ , A2 ⊗ (ωS∗ ), V1  = A2 , ωS∗ , V1 , where we first tensor with (ωS∗ )⊗2 , then mutate (ωS∗ )⊗2 , A2 ⊗ (ωS∗ )⊗2  to the left with respect to its orthogonal complement, then mutate A2 ⊗ ωS∗ to the left with respect to ωS∗ . It follows in particular that A2 = ⊥ ωS∗ , V1  is equivalent to A2 . Finally, the two semiorthogonal decompositions give the full orthogonality between V1 and ωS∗ , so that ωS∗ , V1  = V1 , ωS∗ . This implies that A1 = A2 and the proof follows. Degree 1. Assume S has degree 1. Then F1 = −4KS − F2 and Ci are rational (see [10, Page 611]), so that Vi = OS (Fi ) are k-exceptional and we are in case (iv) of Definition 12. In particular, we only need to prove that the categories A1 and A2 are equivalent. Let us consider the first conic bundle structure and the semiorthogonal decompositions: Db (S) = O(−F1 ), OS , A1  = O(F2 ), (ωS∗ )⊗4 , A1 ⊗ (ωS∗ )⊗4  = (ωS∗ )⊗3 , A1 ⊗ (ωS∗ )⊗3 , O(F2 ) = (ωS∗ )⊗3 , O(F2 ), A1 ,

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where we first tensor by (ωS∗ )⊗4 , then mutate (ωS∗ )⊗4 , A1 ⊗ (ωS∗ )⊗4  to the left with respect to its orthogonal complement, then mutate A1 ⊗ (ωS∗ )⊗3 to the right with respect to O(F2 ). The category A1 is the result of this last mutation and is then equivalent to A1 . Now we need to mutate O(F2 ) to the left with respect to (ωS∗ )⊗3 . To this end, let us calculate : Homi ((ωS∗ )⊗3 , O(F2 )) = H i (S, O(3KS + F2 )). First of all, note that (3KS + F2 ).F2 < 0, which implies that H 0 (S, O(3KS + F2 )) = 0. Similarly, by Serre duality we have that H 2 (S, O(3KS + F2 )) = H 0 (S, O(2KS + F1 )) = 0 since (2KS + F1 ).F1 < 0. Finally we are left with dim H 1 (S, O(3KS + F2 )) = −χ(OS , O(3KS + F2 )). The latter can be calculated by Riemann-Roch: 1 χ(OS , O(3KS + F2 )) = (3KS + F2 ) · (2KS + F2 ) + 1 = −1, 2 since KS · F2 = −2 and S has degree 1. It follows that there is a unique extension 0 −→ O(F2 ) −→ F −→ (ωS∗ )⊗3 −→ 0, which has rank 2 and first Chern class F2 − 3KS . Moreover, F is the result of the mutation of O(F2 ) to the left with respect to (ωS∗ )⊗3 , so that we end up with the decomposition Db (S) = F, (ωS∗ )⊗3 , A1 . (6.2) Now consider the second conic bundle structure and the semiorthogonal decompositions: Db (S) = OS , O(F2 ), A2  = O(F2 ), A2 , ωS∗  = O(F2 ), ωS∗ , A2 , where the first equality is the mutation of OS to the right with respect to its right orthogonal, and A2 is the mutation of A2 to the left with respect to ωS∗ and hence equivalent to A2 . We mutate now O(F2 ) to the right with respect to ωS∗ . A calculation similar to the above one shows that there is exactly one nontrivial extension 0 −→ ωS∗ −→ G −→ O(F2 ) −→ 0, which has rank 2 and first Chern class F2 − KS . Moreover, G is the result of the mutation of O(F2 ) to the right with respect to ωS∗ , and is an exceptional object. Thanks to Gorodentsev [8], we know that exceptional bundles on S are characterized by their rank and their first Chern class. Note that F and G both have rank 2, while the first Chern class of G is the first Chern class of F ⊗ ωS . It follows that G  F ⊗ ωS is the mutation of O(F2 ) to the right with respect to ωS∗ . We hence end up with the decompositions Db (S) = ωS∗ , F ⊗ ωS , A2  = (ωS∗ )⊗2 , F , A2 ⊗ ωS∗  = F, A2 ⊗ ωS∗ , (ωS∗ )⊗3  = F, (ωS∗ )⊗3 , A2 ,

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where first we tensor by ωS∗ , then mutate (ωS∗ )⊗2 to the right with respect to its right orthogonal, and A2 is the left mutation of A2 ⊗ (ωS∗ )⊗2 to the right with respect to (ωS∗ )⊗3 and is therefore equivalent to A2 . The proof follows then by comparison with 6.2.  Corollary 16. The Griffiths–Kuznetsov component is invariant under links of type IV and hence well defined for minimal conic bundles. Acknowledgment The authors are grateful to St´ephane Lamy for providing and discussing the example in Remark 4, and to J´er´emy Blanc and Michele Bolognesi for fruitful conversations. We thank moreover the anonymous referee for his/her careful work and advices which helped us improve the exposition.

References [1] A. Auel, and M. Bernardara, Cycles, derived categories, and rationality, in Surveys on Recent Developments in Algebraic Geometry, Proceedings of Symposia in Pure Mathematics 95, 199–266 (2017). [2] A. Auel and M. Bernardara, Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields, Proc. London Math. Soc. (3) 117 (2018) 1–64. [3] A. Auel, M. Bernardara, and M. Bolognesi, Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems, Journal de Math. Pures et Appliqu´ees 102, 249–291 (2014) [4] M. Bernardara, A semiorthogonal decomposition for Brauer–Severi schemes. Math. Nachr. 282 (2009), no. 10. [5] M. Bernardara and M. Bolognesi, Derived categories and rationality of conic bundles, Compositio Math. 149 (2013), no. 11, 1789–1817. [6] M. Bernardara and M. Bolognesi, Categorical representability and intermediate Jacobians of Fano threefolds. EMS Ser. Congr. Rep., Eur. Math. Soc., 2013. [7] C. Clemens and P. Griffiths, The intermediate Jacobian of the cubic threefold. Ann. Math. 95 (1972), 281–356. [8] A.L. Gorodentsev, Exceptional bundles on surfaces with a moving anticanonical class, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 4, 740–757, 895; translation in Math. USSR-Izv. 33 (1989), no. 1, 67–83. [9] B. Hassett, Rational surfaces over nonclosed fields, in Arithmetic geometry 155–209, Clay Math. Proc., 8, Amer. Math. Soc., Providence, RI, 2009. [10] V.A. Iskovskikh, Factorization of birational mappings of rational surfaces from the point of view of Mori theory (Russian) Uspekhi Mat. Nauk 51 (1996), no. 4(310), 3–72; translation in Russian Math. Surveys 51 (1996), no. 4, 585–652. [11] B.V. Karpov, and D.Yu. Nogin, Three-block exceptional sets on del Pezzo surfaces, Izv. Ross. Akad. Nauk Ser. Mat. 62 (1998), no. 3, 3–38; translation in Izv. Math. 62 (1998), no. 3, 429–463.

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[12] A. Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics, Adv. Math. 218 (2008), no. 5, 1340–1369. [13] A. Kuznetsov, Derived category of cubic and V14 threefold, Proc. V.A. Steklov Inst. Math. 246 (2004), 183–207. [14] S. Lamy and S. Zimmermann, Signature morphisms from the Cremona group over a non-closed field, to appear in Journal of the European Mathematical Society. [15] D. Quillen, Higher algebraic K-theory. I, Algebraic K-theory I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math., vol. 341, Springer, Berlin, 1973, pp. 85–147. Marcello Bernardara and Sara Durighetto Institut de Math´ematiques de Toulouse Universit´e Paul Sabatier 118 route de Narbonne F-31062 Toulouse Cedex 9, France e-mail: [email protected] [email protected]

Marked and Labelled Gushel–Mukai Fourfolds Emma Brakkee and Laura Pertusi Abstract. We prove that the moduli stacks of marked and labelled Hodgespecial Gushel–Mukai fourfolds are isomorphic. As an application, we construct rational maps from the stack of Hodge-special Gushel–Mukai fourfolds of discriminant d to the moduli space of (twisted) degree-d polarized K3 surfaces. We use these results to prove a counting formula for the number of fourdimensional fibers of Fourier–Mukai partners of very general Hodge-special Gushel–Mukai fourfolds with associated K3 surface, and a lower bound for this number in the case of a twisted associated K3 surface. Mathematics Subject Classification (2010). 14J35, 14J45, 14J28, 14D22. Keywords. Gushel–Mukai varieties, K3 surfaces, Fano fourfolds, Period domain.

1. Introduction In the last 20 years the study of cubic fourfolds has been a central research topic, due for instance to their rich associated hyperk¨ahler geometry and the still open question about their rationality. One foundational work is [Has00], where Hassett studied special cubic fourfolds, i.e., cubic fourfolds containing a surface which is not a complete intersection. Special cubic fourfolds form divisors in the moduli space of cubic fourfolds parametrized by a positive even integer d called the discriminant. Moreover, depending on the value of d, the cubic fourfold is related to a degree-d polarized K3 surface via Hodge theory. In order to study this relation on the level of period domains and moduli spaces, Hassett introduced the notions of marked and labelled special cubic fourfolds. Depending on d, the moduli space of discriminant d marked cubic fourfolds is either isomorphic to or a two-to-one covering of the moduli space of discriminant d labelled cubic fourfolds. Moreover, this is used to construct an either generically injective or degree-two rational map from the moduli space of degree-d polarized K3 surfaces to the divisor of discriminant-d special cubic fourfolds having an associated K3 surface. This difference was further investigated in [Bra18], where © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Farkas et al. (eds.), Rationality of Varieties, Progress in Mathematics 342, https://doi.org/10.1007/978-3-030-75421-1_6

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the geometry of the covering involution arising in the second case is completely described. In this paper, we deal with similar questions in the case of Gushel–Mukai fourfolds. These are smooth Fano fourfolds obtained generically as quadric sections of linear sections of the Grassmannian Gr(2, 5), and they share many similarities with cubic fourfolds. After defining marked and labelled GM fourfolds of discriminant d and their associated moduli stacks in Section 3, we show that they provide equivalent notions in this case. Theorem 1.1 (Corollary 3.6). The moduli stacks of labelled and marked Hodgespecial GM fourfolds are isomorphic. Like for cubic fourfolds, this result is particularly interesting when we specify to GM fourfolds with Hodge-associated K3 surfaces, as defined in [DIM15]. Recall that a GM fourfold has an associated K3 surface if and only if its discriminant satisfies the following numerical condition: d ≡ 2, 4 mod 8 and p | d for every prime p ≡ 3 mod 4.

(∗∗)

For these values of the discriminant, applying Theorem 1.1, we interpret the condition of having an associated K3 surface on the level of moduli stacks as follows. Theorem 1.2. Let d be a positive integer satisfying condition (∗∗). Then there exists a dominant rational map from the moduli stack of Hodge-special GM fourfolds with discriminant d to the moduli space of degree-d polarized K3 surfaces that sends a GM fourfold to a Hodge-associated K3 surface. The rational map of Theorem 1.2 is defined in (4). As an application, we can count fibers of the period map for GM fourfolds whose elements are Fourier–Mukai partners. By [KP18] the bounded derived category of a GM fourfold X has a semiorthogonal decomposition of the form ∗ ∗ Db (X) = Ku(X), OX , UX , OX (1), UX (1), ∗ is the restriction to X of the tautological rank-2 vector bundle on Gr(2, 5) where UX and Ku(X), defined as the orthogonal complement to the exceptional collection ∗ ∗ OX , UX (1), is a subcategory of K3 type. We say that a GM fourfold , OX (1), UX ∼  → Ku(X  ) X is a Fourier–Mukai partner of X if there is an equivalence Ku(X) − of Fourier–Mukai type. As shown in [DIM15, Theorem 4.4] the period map of GM fourfolds has smooth four-dimensional fibers, so we cannot expect a finite number of Fourier–Mukai partners as in the case of K3 surfaces [BM01] or cubic fourfolds [Huy17, Theorem 1.1]. Indeed, by [KP19, Theorem 1.6] GM fourfolds in the same fiber of the period map are Fourier–Mukai partners. Nevertheless, Theorem 1.2 allows to prove a counting formula to the number of period points of Fourier– Mukai partners for very general GM fourfolds with Hodge-associated K3 surface. See [Ogu02] and [Per16] for the analogous statements for K3 surfaces and cubic fourfolds, respectively.

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Theorem 1.3 (Proposition 4.3). Let X be a very general Hodge-special GM fourfold with discriminant d satisfying (∗∗). Let m be the number of non-isomorphic Fourier–Mukai partners of its Hodge-associated K3 surface. Then when d ≡ 4 mod 8 (resp. d ≡ 2 mod 8), there are m (resp. 2m) fibers of the period map of GM fourfolds such that, when non-empty, their elements are Fourier–Mukai partners of X. Moreover, all Fourier–Mukai partners of X are obtained in this way. We end with proving the analogue of Theorem 1.2 for GM fourfolds with associated twisted K3 surface. Recall that by [Per19, Theorem 1.1] this is equivalent to having discriminant of the form d = dr2 with d satisfying (∗∗) – see Section 5.2. On the other hand, the moduli space of polarized twisted K3 surfaces with fixed degree and order was recently constructed in [Bra20]. Theorem 1.4 (Corollary 5.4). Let d be a positive integer such that a very general GM fourfold of discriminant d admits an associated polarized twisted K3 surface of degree d and order r. There is a dominant rational map from the moduli stack of Hodge-special GM fourfolds of discriminant d to a component of the moduli space of twisted K3 surfaces of degree d and order r, sending a GM fourfold of discriminant d to an associated twisted K3 surface. Finally, as in the untwisted setting, we apply Theorem 1.4 to study Fourier– Mukai partners of a very general GM fourfold with associated twisted K3 surface. Theorem 1.5 (Proposition 5.7). Let d be a positive integer such that a very general GM fourfold of discriminant d admits an associated polarized twisted K3 surface (S, α) of degree d and order r. Let m be the number of non-isomorphic Fourier– Mukai partners of (S, α) of order r. Then when d ≡ 0 mod 4 (resp. d ≡ 2 mod 8), there are at least m (resp. 2m ) fibers of the period map of GM fourfolds such that, when non-empty, their elements are Fourier–Mukai partners of X. Plan of the paper. In Section 2 we recall the definition of (Hodge-special) GM fourfolds and some results concerning their Hodge theory. In Section 3 we define marked and labelled Hodge-special GM fourfolds and we prove Theorem 1.1. Section 4 is devoted to the construction of the rational map of Theorem 1.2 and the proof of Theorem 1.3. Finally, in Section 5 we recall the construction of moduli spaces of twisted K3 surfaces with fixed order and degree, and we prove Theorems 1.4 and 1.5. Notation. Given a lattice L, we denote by Disc L := L∨ /L its discriminant group  := Ker(O(L) → O(Disc L)). For any integer m = 0 we denote by and we set O(L) L(m) the lattice L with the intersection form multiplied by m. ⊕r ⊕s We denote by I1 the lattice Z with bilinear  (1), Ir,s := I1 ⊕ I1 (−1) ,  ⊕2  0form 1 and E8 is the unique even A1 := I1 (2), U is the hyperbolic plane Z , 1 0 unimodular lattice of signature (8, 0). For 3 ≥ i ≥ j ≥ 0 the Schubert cycles on the Grassmannian Gr(2, 5) are denoted by σi,j ∈ H2(i+j) (Gr(2, 5), Z) and we set σi := σi,0 .

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2. Gushel–Mukai fourfolds In this section, we review the definition of Gushel–Mukai fourfolds and some known results concerning their Hodge theory. Our main references are [DIM15, DK19]. We assume the base field is C. 2.1. Cohomology and period domain of Gushel–Mukai fourfolds Let V5 be a five-dimensional C-vector space and denote by CGr(2, V5 ) the cone over  ∼ the Grassmannian Gr(2, V5 ) with vertex ν := P(C), embedded in P(C ⊕ 2 V5 ) = 2 10 ucker embedding of Gr(2, V5 ) ⊂ P( V5 ). P via the Pl¨ Definition 2.1. A Gushel–Mukai (GM) fourfold is a smooth four-dimensional intersection X := CGr(2, V5 ) ∩ Q where Q ⊂ P(W ) is a quadric hypersurface in a linear space P(W ) ∼ = P8 ⊂ P(C ⊕ 2 V5 ). Since X is smooth, the linear projection γX : X → Gr(2, V5 ) from the vertex 2 V5 ) defines ν is a regular map. The restriction of the hyperplane class on P(C ⊕ ∗ a natural polarization H = γX σ1 on X with degree H 4 = 10. By the adjunction formula, the canonical divisor is KX = −2H, so X is a Fano fourfold of degree 10 and index 2. The moduli stack M4 of GM fourfolds is a smooth, irreducible Deligne– Mumford stack of finite type over C of dimension 24 [KP18, Prop. 2.4]. By [IM11, Lemma 4.1] the Hodge diamond of X is 1 0 0 0 0

0 1

0 1

0 0

22

0 1

0.

By [DIM15, Proposition 5.1] there is an isomorphism of lattices H 4 (X, Z) ∼ = Λ := I22,2 . ∗ . The Note that the rank-2 lattice H4 (Gr(2, V5 ), Z) embeds into H4 (X, Z) via γX vanishing lattice of X is the sublattice   ∗ H4 (X, Z)00 := x ∈ H4 (X, Z) | x · γX (H4 (Gr(2, V5 ), Z)) = 0 .

By [DIM15, Proposition 5.1] it is isomorphic to Λ00 := E8⊕2 ⊕ U ⊕2 ⊕ A1⊕2 . ∗ (H4 (Gr(2, V5 ), Z)) with respect to the basis Note that the intersection form on γX  2 2 ∗ ∗ . Fixing a primitive embedding σ2 is represented by the matrix σ1,1 , γX γX 2 4

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of Λ00 into Λ, we set ΛG := Λ⊥ 00 ⊂ Λ and  find two generators λ1 and λ2 of we can 2 0 . ΛG such that the intersection matrix is 0 2 The period domain of GM fourfolds is the complex manifold Ω(Λ00 ) := {w ∈ P(Λ00 ⊗ C) | w · w = 0, w · w ¯ < 0}.  00 ) acts properly discontinuously on Ω(Λ00 ) and it is Note that the group O(Λ isomorphic to Γ := {g ∈ O(Λ) | g|ΛG = idΛG }. The quotient  00 ) D := Ω(Λ00 )/O(Λ is an irreducible quasi-projective variety of dimension 20 and by [DIM15, Theorem 4.4] the period map p : M4 → D is dominant as a map of stacks with smooth fourdimensional fibers. The period point of X is p(X) ∈ D. 2.2. Hodge-special Gushel–Mukai fourfolds A very general GM fourfold X satisfies rk H2,2 (X, Z) = 2. We call X Hodge-special if H 2,2 (X, Z) contains a rank-three primitive sublattice containing ∗ γX (H 4 (Gr(2, V5 ), Z)).

Period points of Hodge-special GM fourfolds lie in codimension-1 Noether– Lefschetz loci in D. Indeed, let Ld ⊂ Λ be a primitive rank-three positive definite sublattice containing ΛG , with discriminant d. By [DIM15, Lemma 6.1] we have d ≡ 0, 2 or 4 mod 8. Consider the codimension-1 locus Ω(Ld⊥ ) := P(Ld⊥ ⊗ C) ∩ Ω(Λ00 ) where Ld⊥ is the orthogonal complement of Ld in Λ. Let DLd ⊂ D Ω(L⊥ d)

be the image of under the map Ω(Λ00 ) → D. Then the period of any Hodge-special GM fourfold lies in DLd for some Ld . By [DIM15, Proposition 6.2], the lattice Ld only depends on the discriminant d, and depending on d, there are one or two embeddings of Ld into Λ up to  00 ),  00 ). To be precise, up to the action of O(Λ composition with elements of O(Λ there exists τ ∈ Ld such that λ1 , λ2 , τ is a basis for Ld with intersection matrix given by ⎞ ⎞ ⎛ ⎞ ⎛ ⎛ 2 0 0 2 0 1 2 0 0 ⎝0 2 0 ⎠ if d = 8k, ⎝0 2 0 ⎠ or ⎝0 2 1 ⎠ if d = 2 + 8k, 0 1 2k 1 0 2k 0 0 2k ⎞ ⎛ 2 0 1 ⎝0 2 1 ⎠ if d = 4 + 8k. (1) 1 1 2k In the case d = 2 + 8k, denote by Dd and Dd the irreducible divisors DLd corresponding to the first and second embedding of Ld , respectively. It follows from

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[DIM15, Corollary 6.3] that the periods of Hodge-special GM fourfolds are contained in the union of (i) the irreducible hypersurfaces Dd := DLd ⊂ D for all d ≡ 0 mod 4; (ii) the unions Dd := Dd ∪ Dd for all d ≡ 2 mod 8.  00 ), inducing Moreover, there exists an involution r ∈ O(Λ00 ) which is not in O(Λ   an involution rD on D which exchanges Dd and Dd when d ≡ 2 mod 8. We say that a Hodge-special GM fourfold X has discriminant d if its period point belongs to Dd . The moduli stack of Hodge-special GM fourfolds of discriminant d is M4 ×D Dd . Note that a very general X ∈ M4 ×D Dd satisfies rk H2,2 (X, Z) = 3. It is known that each of the irreducible divisors DLd intersects the image of p for d > 8 [DIM15, Theorem 8.1], so DLd ∩ Im(p) contains an open dense subset of DLd . It follows that the restriction p : M4 ×D Dd → Dd is still dominant when d > 8.

3. Marked and labelled Gushel–Mukai fourfolds In analogy to [Has00, Definition 3.1.3] for cubic fourfolds, we give the following definition. Let Ld be a rank-3 positive definite lattice containing ΛG and fix an embedding ΛG ⊂ Ld . Definition 3.1. A marked Hodge-special GM fourfold is a GM fourfold X together with a primitive embedding ϕ : Ld → H2,2 (X, Z) preserving the classes λ1 and λ2 . A labelled Hodge-special GM fourfold is a GM fourfold X together with a primitive sublattice Ld ⊂ H2,2 (X, Z). So a labelling of a GM fourfold is the image of a marking. Two marked GM fourfolds (X, ϕ : Ld → H2,2 (X, Z))

and (X  , ϕ : Ld → H2,2 (X  , Z))

are isomorphic if there is an isomorphism f : X → X  such that f ∗ : H4 (X  , Z) → H4 (X, Z) satisfies f ∗ ◦ϕ = ϕ . Two labelled GM fourfolds (X, Ld ⊂ H2,2 (X, Z)) and (X  , Ld ⊂ H2,2 (X  , Z)) are isomorphic if there exists an isomorphism f : X → X  such that f ∗ preserves Ld . Remark 3.2. Consider the sets of isomorphism classes of marked and labelled GM fourfolds. There is a map { marked GM 4-folds }/∼ = → { labelled GM 4-folds }/∼ = sending (X, ϕ : Ld → H2,2 (X, Z)) to (X, ϕ(Ld ) ⊂ H2,2 (X, Z)). It is surjective but need, a priori, not be injective: The lattice Ld could have non-trivial automorphisms fixing the λi .

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Fix an embedding Ld → Λ. Recall that DLd is the image of Ω(L⊥ d ) under Ω(Λ00 ) → D = Ω(Λ00 )/Γ. Let G(Ld ) := {g ∈ Γ : g(Ld ) = Ld } H(Ld ) := {g ∈ G(Ld ) : g|Ld = idLd } and define lab := Ω(Ld⊥ )/G(Ld ) DL d mar DL := Ω(L⊥ d )/H(Ld ). d

Then we have surjective maps lab mar → DLd ⊂ D. → DL DL d d lab mar When d ≡ 0 mod 4, we set Ddlab := DL . When d ≡ 2 mod 8, and Ddmar := DL d d ∼ =

∼ =

→ Dd ⊂ D and DLd − → Dd ⊂ D; let (Dd )lab and we have two embeddings DLd − lab   lab (Dd ) be the corresponding spaces DLd over Dd and Dd , respectively. Note that if x ∈ Dd ∩ Dd , then there are two embeddings of Ld into the (2,2)-part of the  00 )-orbits. So x has corresponding Hodge structure on Λ00 that are in different O(Λ two labellings, giving rise to one point in (Dd )lab and one in (Dd )lab . Accordingly, we let Ddlab be the disjoint union  Ddlab := (Dd )lab (Dd )lab .  Analogously, define Ddmar := (Dd )mar (Dd )mar . Then the moduli stacks of labelled and marked Hodge-special GM fourfolds of discriminant d are M4 ×D Ddlab and M4 ×D Ddmar , respectively. In the rest of this section, we analyze the natural surjective morphisms mar lab → DL → DLd . DL d d lab  DLd is a normalization. Lemma 3.3. The natural map ν : DL d

Proof. The argument is the same as in the case of cubic fourfolds in [Bra18, Sec tion 2.3]. Note that a non-normal point in DLd has two different labellings by Ld . In particular, the integral (2,2)-part of the corresponding Hodge structure has rank bigger than 3. mar lab Proposition 3.4. The map DL is an isomorphism.  DL d d

It follows that Ddmar → Ddlab is an isomorphism. In order to prove Proposition 3.4, we will show that G(Ld )/H(Ld ) ∼ = Z/2Z and it is generated by an element that restricts to − id on L⊥ ⊥ induces d . As − idLd the trivial action on Ω(L⊥ d ), we deduce the proof of Proposition 3.4.

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Let G (Ld ) := {g ∈ O(Ld ) : g(λi ) = λi for i = 1, 2}. By [Bra18, Lemma 1.1] the map G(Ld ) → G (Ld ) given by restriction to Ld is surjective, since Λ is uni∼ modular and O(Ld⊥ ) → O(Disc L⊥ d ) = O(Disc Ld ) is surjective by [Nik80, Theorem 1.14.2]. It follows that G(Ld )/H(Ld ) is isomorphic to G (Ld ). Lemma 3.5. The group G (Ld ) is Z/2Z, generated by an element that acts on Disc(Ld ) as − id. Proof. Let g ∈ G (Ld ). Assume d = 8k, so Ld has a basis λ1 , λ2 , τ with corresponding intersection matrix ⎞ ⎛ 2 0 0 ⎝0 2 0 ⎠ 0 0 d/4 (see (2.2)). Then either g(τ ) = τ , so g = idLd , or g(τ ) = −τ . In the second case, g acts on the discriminant group ! λ1 λ2 τ , , Z/2Z ⊕ Z/2Z ⊕ Z/(d/4)Z = 2 2 d/4 of Ld by



λ1 λ2 τ , , 2 2 d/4



 →

τ λ1 λ2 , ,− 2 2 d/4

Next, assume d = 2+8k, so Ld and intersection matrix ⎛ 2 ⎝0 0



 ≡−

λ1 λ2 τ , , 2 2 d/4

 .

is isomorphic to the lattice with basis λ1 , λ2 , τ ⎞ 0 0 ⎠ 2 1 1 (d + 2)/4

Write g(τ ) = aλ1 + bλ2 + cτ . It follows from g(λi ) = λi that a = 0 and c = 1 − 2b, and solving (g(τ ))2 = (τ )2 gives (b − b2 )d = 0. Hence we either have b = 0, so g = idLd , or b = 1 and c = −1, so g(τ ) = λ2 − τ . In the second case, the action on the discriminant group ! λ1 λ2 − 2τ , Z/2Z ⊕ Z/(d/2)Z = 2 d/2 of Ld is given by       λ1 λ2 − 2τ λ1 −λ2 + 2τ λ1 λ2 − 2τ → ≡− . , , , 2 d/2 2 d/2 2 d/2 Finally, assume d = 4 + 8k, there is a basis λ1 , λ2 , τ for Ld with intersection matrix ⎞ ⎛ 2 0 1 ⎠ ⎝0 2 1 1 1 (d + 4)/4 and write g(τ ) = aλ1 + bλ2 + cτ . Now g(λi ) = λi , implies a = b and c = 1 − 2a, and solving (g(τ ))2 = (τ )2 gives (a − a2 )d = 0. Hence we either get a = 0, so g = idLd ,

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or a = 1 and c = −1, so g(τ ) = λ1 + λ2 − τ . In the second case, the action on the discriminant group ! λ1 + λ2 − 2τ Z/dZ = d of Ld is given by λ1 + λ2 − 2τ λ1 + λ2 − 2(λ1 + λ2 − τ ) λ1 + λ2 − 2τ → =− . d d d



Proof of Proposition 3.4. By Lemma 3.5, the generator γ  of G (Ld ) acts as − id on Disc Ld . Then − idL⊥ ⊕γ  extends to an element γ of O(Λ) by [Nik80, Corollary d 1.5.2 and Proposition 1.6.1], which generates G(Ld )/H(Ld ). Since by definition γ ⊥ restricts to − id on L⊥ d , we conclude that γ acts trivially on Ω(Ld ). This implies the statement.  As a direct consequence of Proposition 3.4, we get the following identification between moduli stacks of marked and labelled Hodge-special GM fourfolds. Corollary 3.6. We have an isomorphism M4 ×D Ddmar ∼ = M4 ×D Ddlab .

4. Gushel–Mukai fourfolds with associated K3 surface In this section we prove Theorem 1.2 and Theorem 1.3. 4.1. Rational maps to moduli spaces of K3 surfaces The aim of this section is to construct the rational map of Theorem 1.2. Let X be a Hodge-special GM fourfold whose period lies in Dd , that is, there are a rank-3 positive definite lattice Ld of discriminant d containing ΛG and a primitive embedding Ld → H2,2 (X, Z). As in [DIM15, Section 6.2], we say that a quasipolarized K3 surface (S, l) is Hodge-associated to X if there is a Hodge isometry 4 H2 (S, Z) ⊃ l⊥ ∼ = L⊥ d ⊂ H (X, Z)

up to a sign and a Tate twist. In particular, since H2 (S, Z) is unimodular, the degree (l)2 of (S, l) is d. By [DIM15, Prop. 6.5] X has a Hodge–associated quasipolarized K3 surface if and only if d satisfies d ≡ 2, 4 mod 8 and p | d for every prime p ≡ 3 mod 4.

(∗∗)

Moreover, when the period does not lie in Dd ∩ D8 , then the quasi-polarized K3 surface is actually polarized. Denote by Λd := E8 (−1)⊕2 ⊕ U ⊕2 ⊕ I1 (−d) the lattice isomorphic to the primitive middle cohomology H2 (S, Z)pr := l⊥ ⊂ H2 (S, Z) of a polarized K3 surface (S, l) of degree d. Then condition (∗∗) on d is equivalent to the existence ∼ of an isomorphism of lattices L⊥ d = Λd (−1). Under this isomorphism, the group

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∼ H(Ld ). Fix an embedding Ld → Λ. By  ⊥) =  d (−1)) is identified with O(L O(Λ d Proposition 3.4, we obtain the following commutative diagram: ∼ = / Ω(L⊥ )   / Ω(Λ00 ) Ω(Λd (−1)) (2) d   d (−1)) Ω(Λd (−1))/O(Λ

∼ =

 / Ω(L⊥ )/H(Ld ) = Dlab Ld d

ν

 / DL  d

 /D

By Lemma 3.3, the map ν is birational. It follows from the diagram above that there exists a birational map  d (−1)) ∼  d ). D ⊃ DL  Ω(Λd (−1))/O(Λ = Ω(Λd )/O(Λ d

In particular, we obtain a rational map  d) Dd  Ω(Λd )/O(Λ

(3)

which is birational when d ≡ 4 mod 8 and generically two-to-one when d ≡ 2 mod 8. Indeed, for a generic x ∈ Dd , note that for some g ∈ O(L⊥ d ), x and (g ◦ r)D (x)  d ). The quotient Ω(Λd )/O(Λ  d ) can are mapped to the same point in Ω(Λd )/O(Λ be viewed as the moduli space of degree-d quasi-polarized K3 surfaces [HP13, Section 5]. The map above induces a rational map  d) M4 ×D Dd  Ω(Λd )/O(Λ sending a GM fourfold to an associated quasi-polarized K3 surface. Now denote by Md the moduli space of polarized K3 surfaces of degree d. The  d ). When restricted to period map induces an open immersion Md → Ω(Λd )/O(Λ points outside M4 ×D D8 , the image of the above rational map lies in Md . We obtain a dominant rational map γd : M4 ×D Dd  Md

(4)

sending a GM fourfold to an associated polarized K3 surface. This proves Theorem 1.2. Note that γd (X) is defined whenever rk H2,2 (X, Z) = 3.  d) Remark 4.1. Note that γd is not unique, since the map DLd  Ω(Λd )/O(Λ ∼ depends on the choice of an isomorphism L⊥ (−1). To be precise, γd is Λ = d d 2   unique up to O(Λd )/O(Λd ) when d ≡ 4 mod 8, and up to (O(Λd )/O(Λd )) when d ≡ 2 mod 8. 4.2. Fibers of Fourier–Mukai partners We now apply the results in the previous sections to study Fourier–Mukai partners of GM fourfolds. In analogy to [Huy17, Per16] for cubic fourfolds, we say that a Fourier–Mukai partner of a GM fourfold X is a GM fourfold X  such that there ∼ → Ku(X  ) of Fourier–Mukai type, i.e., the exists an exact equivalence Ku(X) − ∼ b → Ku(X  ) → Db (X  ) has a Fourier–Mukai kernel. composition D (X) → Ku(X) − Note that by [KP19, Theorem 1.6] GM fourfolds in the same fiber of the period map are Fourier–Mukai partners.

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Before proving Theorem 1.3, we need to make the following remark. In analogy to [AT12], the Mukai lattice for Ku(X) has been defined in [Per19, Section 3.1] as the abelian subgroup ∗  H(Ku(X), Z) := {κ ∈ K(X)top : χ([OX (i)], κ) = χ([UX (i)], κ) = 0 for i = 0, 1}

of the topological K-theory of X, with the Euler form χ with reversed sign and the weight-2 Hodge structure induced by pulling back via the isomorphism  H(Ku(X), Z) ⊗ C → H• (X, C) "  given by the Mukai vector v(−) = ch(−). td(X). As a lattice, H(Ku(X), Z) ∼ = ⊕4 ⊕2 U ⊕ E8 (−1) by [DK19, Theorem 1.2]. We set  1,1 (Ku(X)) ∩ H(Ku(X),   1,1 (Ku(X), Z) := H Z). H  1,1 (Ku(X), Z) spanning a lattice By [KP18, Lemma 2.27] there are two classes in H ⊕2 A1 . By [Per19, Proposition 3.1], there is a Hodge isometry ∼ A⊕2⊥ = H4 (X, Z)00 , 1 up to a sign and a Tate twist, where the orthogonal complement is taken in  H(Ku(X), Z). In particular, a very general Hodge-special GM fourfold X with  1,1 (Ku(X), Z) = 3 and Disc H  1,1 (Ku(X), Z) = d. Moreover, discriminant d has rkH we have the following property. ∼

Lemma 4.2. Every equivalence Ku(X) − → Ku(X  ) of Fourier–Mukai type induces    ), Z). a Hodge isometry H(Ku(X), Z) ∼ = H(Ku(X Proof. Apply a similar argument as in [Huy17, Proposition 3.3].



By the above lemma, every Fourier–Mukai partner of a very general Hodgespecial GM fourfold with discriminant d is a very general Hodge-special GM fourfold of the same discriminant. We are now ready to prove the next proposition which implies Theorem 1.3. Denote by τ (d) the number of distinct primes that divide d/2. Proposition 4.3. Let d be a positive integer satisfying condition (∗∗). If X is a very general Hodge-special GM fourfold with discriminant d ≡ 4 mod 8 (resp. d ≡ 2 mod 8), then there are 2τ (d)−1 (resp. 2τ (d)) fibers of the period map p such that, when non-empty, their elements are Fourier–Mukai partners of X. Moreover, all Fourier–Mukai partners of X are obtained in this way. Proof. We fix a choice of the rational map γd : M4 ×D Dd  Md of Section 4.1. Let X be a GM fourfold as in the statement and consider γd (X) = (S, l), a degree-d polarized K3 surface associated to X. Note that S has Picard rank 1. Moreover, by [Per19, Theorem 3.6] and [PPZ19, Theorem 1.9] there exists an exact ∼ equivalence Ku(X) − → Db (S). By [Ogu02, Proposition 1.10], S has m := 2τ (d)−1 non-isomorphic Fourier–Mukai partners.

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Choose m K3 surfaces S1 := S, S2 , . . . , Sm as representatives for each isomorphism class of Fourier–Mukai partners endowed with the unique degree-d polarizations l1 := l, . . . , lm , respectively. These polarized K3 surfaces determine m  d ), which we still denote by (Si , li ) for 1 ≤ i ≤ m. As distinct points in Ω(Λd )/O(Λ summarized in diagram (2), their image via (3) defines m (resp. 2m) period points in Dd if d ≡ 4 mod 8 (resp. d ≡ 2 mod 8). We denote by xi ∈ Dd the period point defined by (Si , li ) if d ≡ 4 mod 8, and by xi ∈ Dd , xi ∈ Dd those defined by (Si , li ) if d ≡ 2 mod 8. Assume that xi (resp. xi or xi ) is in the image of the period map p and consider a GM fourfold X  in the fiber of p over this point. Then ∼

∼

∼

→ Ku(X). → Db (S) − → Db (Si ) − Ku(X  ) − Finally, by Lemma 4.2, if X  is a Fourier–Mukai partner of X, then X  is a very general Hodge-special GM fourfold of the same discriminant. Thus γd (X  ) is a well-defined element in Md . But then γd (X  ) is a degree-d polarized K3 surface which is a Fourier–Mukai partner of (S, l), hence isomorphic to (Si , li ) for a certain 1 ≤ i ≤ m. This implies the statement.  Remark 4.4. Note that the image of the period map is not known [DIM15, Question 9.1]. Thus some of the fibers of Proposition 4.3 could a priori be empty.

5. Gushel–Mukai fourfolds and twisted K3 surfaces In this section we recall the definition of the moduli spaces of twisted polarized K3 surfaces with fixed order and degree introduced in [Bra20], and then we prove Theorem 1.4 and Theorem 1.5. 5.1. Moduli and periods of twisted K3 surfaces We summarize the relevant results of [Bra20]. Recall that for a complex K3 surface S, the Brauer group Br(S) is isomorphic to the cohomological Brauer group ∼ H2 (S, O∗ )tors = ∼ (Q/Z)⊕22−ρ(S) . H´2et (S, Gm ) = S Let T (S) := NS(S)⊥ ⊂ H2 (S, Z) be the transcendental lattice of S. Then there is an isomorphism ∼ Hom(T (S), Q/Z). Br(S) = We denote by Br(S)[r] the group of elements in Br(S) whose order divides r. There exists a surjection  := Hom(H2 (S, Z)pr , Z/rZ)  Hom(T (S), Z/rZ) ∼ Br(S)[r] = Br(S)[r] which is an isomorphism if and only if ρ(S) = 1. Theorem 5.1 ([Bra20, Theorem 1]). There exists a scheme Md [r] which is a coarse moduli space for triples (S, l, α) consisting of a polarized K3 surface (S, l) of degree  There exists a subscheme Mrd ⊂ Md [r] which is a d and an element α ∈ Br(S)[r]. coarse moduli space for those triples for which α has order r.

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The spaces Md [r] and Mrd are constructed as follows. Let Mmar be the (fine) d moduli space of triples (S, l, ϕ) where (S, l) is as before and ϕ is an isomorphism H2 (S, Z)pr ∼ = Λd . Note that ϕ induces an isomorphism ϕr : Br(S)[r] →  d ) induces an action on Hom(Λd , Z/rZ), and on Hom(Λd , Z/rZ). The group O(Λ the product Mdmar [r] := Mdmar × Hom(Λd , Z/rZ) by g(S, l, ϕ, α) = (S, l, g ◦ ϕ, ϕr−1 gϕr (α)).  The space Md [r] is the quotient Mmar d [r]/O(Λd ).  d ) its stabilizer unFor w ∈ Hom(Λd , Z/rZ), denote by Stab(w) ⊂ O(Λ   Mw where [w] ∈ der the action of O(Λd ). Then Md [r] is a disjoint union  d ) and Hom(Λd , Z/rZ)/O(Λ

[w]

Mw = (Mmar ×{w})/ Stab(w). d Each component Mw is an irreducible quasi-projective variety with at most finite quotient singularities [Bra20, Corollary 2.3]. It parametrizes triples (S, l, α) that admit a marking ϕ such that ϕr (α) = w. The space Mrd is the union of those Mw for which w has order r. For a K3 surface S, we will denote by H∗ (S, Z) the full cohomology with the  := E8 (−1)⊕3 ⊕ lattice structure given by the cup product. It is isometric to Λ 2 2 1 ∨ ∼  U ⊕4 . Given (S, l) ∈ Md and α ∈ Br(S)[r] = r H (S, Z)∨ pr / H (S, Z)pr , there is an ∗  associated Hodge structure H(S, α, Z) of K3 type on H (S, Z). Namely, fix a lift 2  of α to 1r H2 (S, Z)∨ pr ⊂ H (S, Q), that we will also denote by α. Then H(S, α, Z) is defined by  2,0 (S, α) := C[σ + α ∧ σ] ⊂ H∗ (S, Z), H where σ is a non-degenerate holomorphic 2-form on S. If α maps to α under   Br(S)[r]  Br(S)[r], then H(S, α, Z) is isomorphic to the Hodge structure    H(S, α , Z) defined by α as in [Huy05, Section 4]. Using the above, one can define period maps for the components Mw as   : H∗ (S, Z) ∼ follows. Any marking ϕ : H2 (S, Z)pr ∼ = Λ. = Λd extends naturally to ϕ 1 ∨ 1 ∨ ∨ ∼ Let w ∈ Hom(Λd , Z/rZ) = r Λd /Λd and let u ∈ r Λd ⊂ Λd ⊗ Q be a representative of w. Denote by Tw the finite-index sublattice Ker(w) ⊂ Λd . The map exp(u) ∈  One shows that  ⊗ Q) (see [Bra20, Section 3.1]) embeds Tw primitively into Λ. O(Λ there is a holomorphic, injective map Mmar ×{w} → Ω (exp(u)Tw ) ∼ = Ω(Tw ) d

 2,0 (S, ϕ−1 (w) . (S, l, ϕ, w) → ϕ  H r It induces an algebraic embedding Mw → Ω(Tw )/ Stab(w). For later use, we define the Picard group of a twisted K3 surface as  1,1 (S, α) ∩ H(S,  Pic(S, α) := H α, Z)

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and its transcendental lattice T (S, α) as the orthogonal complement of Pic(S, α)  α, Z). When α is trivial, we have Pic(S, α) = H0 (S, Z) ⊕ H1,1 (S, Z) ⊕ in H(S, 4 H (S, Z) and T (S, α) = T (S). One can show that there is an isomorphism of ∼ Ker(α : T (S) → Q/Z). lattices T (S, α) = 5.2. Twisted K3 surfaces associated to GM fourfolds Recall [Per19, Definition 3.11] that if X is a Hodge-special GM fourfold, a twisted K3 surface (S, α) is said to be associated to X when there is a Hodge isometry ∼ H(S,   H(Ku(X), α, Z). Z) = Note that if d is the degree and r the order of (S, α), then X has discriminant dr2 . One can show [Per19, Theorem 1.1] that X has an associated twisted K3 surface if and only if the period point of X lies in Dd for some d satisfying # pni i with ni ≡ 0 mod 2 for pi ≡ 3 mod 4 (∗∗ ) d = i

where the pi are distinct primes. Note that this is equivalent to the following: d is of the form dr2 for some integers d and r, where d satisfies (∗∗). This decomposition d = dr2 is however not unique. We will prove that (∗∗ ) is equivalent to a condition on the lattice Ld⊥ , where ⊥ Ld is the orthogonal complement of Ld in Λ. We need the following lemma. Let w ∈ Hom(Λd , Z/rZ) of order r, and let u ∈ r1 Λ∨ d be a representative of w as in Section 5.1.  The canonical map O(Sw ) → O(Disc Sw ) Lemma 5.2. Let Sw := (exp(u)Tw )⊥ ⊂ Λ. is surjective. Proof. Note that Sw has rank 3 and its discriminant group is isomorphic to Disc Tw . By [DIM15, Proposition 6.5], this group is either cyclic or isomorphic to (Z/2Z)2 × Z/(d /4)Z. In the first case, the statement follows from [Nik80, Theorem 1.14.2]. In the second case, it follows from [MM09, Corollary VIII.7.8].  Corollary 5.3. Consider an integer d > 8 with d ≡ 0, 2, 4 mod 8. Then d satisfies (∗∗ ) if and only if for some decomposition d = dr2 with d satisfying (∗∗), there ∼ is a w ∈ Hom(Λd , Z/rZ) and a lattice isometry L⊥ d (−1) = Tw . Proof. Suppose d satisfies (∗∗ ). Let X be a very general GM fourfold with period point in Dd , which exists by [DIM15, Theorem 8.1], and let (S, α, Z) be a twisted K3 surface associated to X. Let d be the degree of S and r the order of α, so   d = dr2 . Then the Hodge isometry H(Ku(X), Z) ∼ α, Z) induces a lattice = H(S, isometry of the transcendental parts: L⊥ (−1) ∼ = T (S, α) ∼ = Ker(α : H2 (S, Z)pr → Z/rZ). d

∼ Now any marking H2 (S, Z)pr ∼ = Λd induces an isometry L⊥ d (−1) = Tw for some w ∈ Hom(Λd , Z/rZ). ∼ Vice versa, assume we have an isometry L⊥ d (−1) = Tw as above. Then the ⊥ ⊥ ∼ associated period domains Ω(Ld (−1)) = Ω(Ld ) and Ω(Tw ) are also isomorphic.

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It follows that if X is a very general GM fourfold of discriminant d , then Ld⊥ ⊂ H4 (X, Z) is Hodge isometric, up to a sign and a Tate twist, to T (S, α) for some twisted K3 surface (S, α) of degree d and order r. By Lemma 5.2, this can be   α, Z). Since X is very general in extended to a Hodge isometry H(Ku(X), Z) ∼ = H(S,  Dd , so its period lies in De if and only if e = d , it follows that d satisfies (∗∗ ).  Assume we are in the situation of Corollary 5.3, so d satisfies (∗∗ ) and we have a fixed w ∈ Hom(Λd , Z/rZ) such that Ld⊥ (−1) is isomorphic to Tw . This lab  w) ∼ induces an isomorphism Ω(Tw )/O(T . Next, note that = Ω(Ld⊥ )/H(Ld ) = DL d  O(Tw ) is a subgroup of Stab(w) [Bra20, Lemma 5.1]. Summarized in a diagram, we have ∼ = / Ω(L⊥ )   / Ω(Λ00 ) Ω(Tw ) (5) d

  w) Ω(Tw )/O(T

∼ =

 / Dlab L d

ν

/ DL   d



 /D

π

Mw



 / Ω(Tw )/ Stab(w)

 w ) → Ω(Tw )/ Stab(w) is a finite map. As in Section 4.1, we where π : Ω(Tw )/O(T obtain a finite dominant rational map DLd  Mw . Corollary 5.4. There is a dominant rational map δd : M4 ×D Dd  Mw which sends a very general Hodge-special GM fourfold X of discriminant d to a polarized twisted K3 surface associated to X. The map is defined whenever rk H2,2 (X, Z) = 3. When rk H2,2 (X, Z) > 3 and the map is defined at X, then the image of X is a triple (S, l, α) ∈ Mw such that  if α ∈ Br(S)[r] is the image of α ∈ Br(S)[r], then (S, l, α ) is associated to X. Namely, by Lemma 5.2, the Hodge isometry ∼  H4 (X, Z)00 ⊃ L⊥ d = Tw ⊂ H(S, α, Z)    extends to a Hodge isometry H(Ku(X), Z) ∼ α, Z) = H(S, α , Z). = H(S, 5.3. Fourier-Mukai partners in the twisted case In this section we apply Corollary 5.4 to construct Fourier–Mukai partners of a very general GM fourfold with a twisted associated K3 surface. First, we need the following lemma, which is the analogue of [Huy17, Lemma 2.3] in the case of cubic fourfolds.  Lemma 5.5. The Mukai lattice H(Ku(X), Z) of a GM fourfold X has an orientation reversing Hodge isometry.

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 := U ⊕4 ⊕ E8 (−1)⊕2 by Proof. Note that A1⊕2 has a primitive embedding in Λ [Nik80, Corollary 1.12.3], which is unique up to isomorphism by [Nik80, Theorem 1.14.4]. If we denote by λ1 and λ2 the standard generators of A⊕2 and by u1 , v1 1 (resp. u2 , v2 ) the standard basis of the first (resp. the second) hyperbolic plane U , then this embedding is given by λ1 → u1 + v1 ,

λ2 → u2 + v2

(see [Per19, Section 3.1]). Consider the isometry g ∈ O(A1⊕2 ) defined by g(λ1 ) = −λ1 and g(λ2 ) = λ2 . Since g acts trivially on the discriminant group of A⊕2 1 , by [Nik80, Prop. 1.6.1  extending g and acting trivially on and Cor. 1.5.2] there is an isometry g˜ of Λ ⊥ A1⊕2 . By definition g˜ reverses the orientation of the two positive directions in A⊕2 1 and preserves the orientation of the two positive directions in A⊕2⊥ . Moreover, 1 . This implies the g˜ preserves the Hodge structure as it acts trivially on A⊕2⊥ 1 statement.  Remark 5.6. Note that there is an autoequivalence of Ku(X) which induces the Hodge isometry described in Lemma 5.5. Indeed, consider the composition LOX ,UX∗ ,OX (1) ◦ (D(−) ⊗ OX (1)), where D(−) := RHom(−, OX ) and LOX ,UX∗ ,OX (1) is the left mutation functor ∗ , OX (1). One can check that this composition induces an autoethrough OX , UX quivalence when restricted to Ku(X), acting on the Mukai lattice as required. We can now prove the following proposition which implies Theorem 1.5. Denote by ϕ(r) the Euler function evaluated in r. Recall that a Fourier–Mukai partner of order r of a twisted K3 surface (S, α) is a twisted K3 surface (S  , α ) with α of ∼ → Db (S  , α ). order r such that there is an equivalence Db (S, α) − Proposition 5.7. Let d = dr2 be a positive integer such that a very general GM fourfold of discriminant d admits an associated polarized twisted K3 surface of degree d and order r. If d ≡ 0 mod 4 (resp. d ≡ 2 mod 8), then there are m (resp. 2m ) fibers of the period map p such that, when non-empty, their elements are Fourier–Mukai partners of X, where $ ϕ(r)2τ (d)−1 if r = 2 or d > 2  m = (6) ϕ(r)/2 if r > 2 and d = 2. Proof. We fix a rational map δd : M4 ×D Dd  Mw as in Corollary 5.4. Let X be a GM fourfold as in the statement and consider the twisted degree-d polarized K3 surface δd (X) = (S, α) with ord(α) = r. Note that S has Picard rank 1 and by ∼ → [Per19, Theorem 1.1] and [PPZ19, Theorem 1.9] there is an equivalence Ku(X) − b  D (S, α). Let m be the number of Fourier–Mukai partners of (S, α) of order r. By Lemmas 5.2 and 5.5, arguing as in [Per16, Proposition 4.4], one can show that m

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is equal to the upper bound given in [Ma10, Proposition 4.3]. Moreover, by [Per16, Proposition 4.7] this number is given by (6) as in the statement. Then using diagram (5) and arguing as in Proposition 4.3, we deduce the statement.  Remark 5.8. Note that a GM fourfold as in Proposition 5.7 could have other Fourier–Mukai partners. Indeed, they could be obtained from Fourier–Mukai partners of (S, α) with order different from r. Remark 5.9. The construction of the rational map in [Bra20, Section 5] can be used in the case of cubic fourfolds to simplify the proof of [Per16, Theorem 1.2]. More precisely, the rational map allows to skip the computation in [Per19, Section 4.1]. As a consequence, it is possible to remove the assumption in [Per16, Theorem 1.2] that 9 does not divide the discriminant, giving a more complete statement. Acknowledgment We thank Gerard van der Geer and Mingmin Shen for their interest, and Thorsten Beckmann for useful discussions. We are grateful to Daniel Huybrechts, Alex Perry and Paolo Stellari for suggestions on the preliminary version of this work. Finally, we thank the referee for careful reading of the manuscript and helpful suggestions. This work started when the second author was visiting the Max-PlanckInstitut f¨ ur Mathematik in Bonn whose hospitality is gratefully acknowledged. The first author is supported by NWO Innovational Research Incentives Scheme 016.Vidi.189.015. The second author is supported by the ERC Consolidator Grant ERC-2017-CoG-771507, Stab-CondEn.

References [AT12] Nicolas Addington and Richard Thomas. “Hodge theory and derived categories of cubic fourfolds”. In: Duke Math. J. 163.10 (2012), pp. 1885–1927. [Bra18] Emma Brakkee. “Two polarised K3 surfaces associated to the same cubic fourfold”. In: Math. Proc. Cambridge Philos. Soc., to appear (2018). eprint: arXiv:1808.0117. [Bra20] Emma Brakkee. “Moduli spaces of twisted K3 surfaces and cubic fourfolds”. In: Math. Ann. 377.3-4 (2020), pp. 1453–1479. eprint: arXiv:1910.03465. [BM01] Tom Bridgeland and Antony Maciocia. “Complex surfaces with equivalent derived categories”. In: Math. Z. 236.4 (2001), pp. 677–697. [DIM15] Olivier Debarre, Atanas Iliev, and Laurent Manivel. “Special prime Fano fourfolds of degree 10 and index 2”. In: Recent Advances in Algebraic Geometry. Vol. 417. London Math. Soc. Lecture Note Ser. Cambridge University Press, 2015, pp. 123–155. [DK19] Olivier Debarre and Alexander Kuznetsov. “Gushel–Mukai varieties: Linear spaces and periods”. In: Kyoto J. Math. 59.4 (2019), pp. 897–953. [Has00] Brendan Hassett. “Special cubic fourfolds”. In: Compositio Math. 120.1 (2000), pp. 1–23.

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[HP13] Klaus Hulek and David Ploog. “Fourier–Mukai partners and polarised K3 surfaces”. In: Arithmetic and geometry of K3 surfaces and Calabi–Yau threefolds. Vol. 67. Fields Inst. Commun. Springer, New York, 2013, pp. 333–365. [Huy05] Daniel Huybrechts. “Generalized Calabi–Yau structures, K3 surfaces, and Bfields”. In: Internat. J. Math. 16 (2005), pp. 13–36. [Huy17] Daniel Huybrechts. “The K3 category of a cubic fourfold”. In: Compos. Math. 153.3 (2017), pp. 586–620. [IM11] Atanas Iliev and Laurent Manivel. “Fano manifolds of degree ten and EPW ´ Norm. Sup´ er. (4) 44.3 (2011), pp. 393–426. sextics”. In: Ann. Sci. Ec. [KP18] Alexander Kuznetsov and Alexander Perry. “Derived categories of Gushel–Mukai varieties”. In: Compos. Math. 154.7 (2018), pp. 1362–1406. [KP19] Alexander Kuznetsov and Alexander Perry. “Categorical cones and quadratic homological projective duality”. Preprint, arXiv:1902.09824. 2019. [Ma10] Shouhei Ma. “Twisted Fourier–Mukai number of a K3 surface”. In: Trans. Amer. Math. Soc. 362.1 (2010), pp. 537–552. [MM09] Rick Miranda and David Morrison. “Embeddings of integral quadratic forms”. https://web.math.ucsb.edu/~ drm/manuscripts/eiqf.pdf, 2009. [Nik80] Vyacheslav V. Nikulin. “Integral symmetric bilinear forms and some of their applications”. In: Math. USSR Izvestija 14 (1980), pp. 103–167. [Ogu02] Keiji Oguiso. “K3 surfaces via almost-primes”. In: Math. Res. Lett. 9.1 (2002), pp. 47–63. [PPZ19] Alexander Perry, Laura Pertusi, and Xiaolei Zhao. “Stability conditions and moduli spaces for Kuznetsov components of Gushel–Mukai varieties”. Preprint, arXiv:1912.06935. 2019. [Per16] Laura Pertusi. “Fourier–Mukai partners for very general special cubic fourfolds”. In: Math. Res. Lett., to appear (2016). eprint: arXiv:1611.06687. [Per19] Laura Pertusi. “On the double EPW sextic associated to a Gushel–Mukai fourfold”. In: J. Lond. Math. Soc. (2) 100.1 (2019), pp. 83–106. Emma Brakkee Korteweg–de Vries Institute University of Amsterdam, P.O. Box 94248 1090 GE Amsterdam, Netherlands e-mail: [email protected] URL: https://staff.fnwi.uva.nl/e.l.brakkee/ Laura Pertusi Dipartimento di Matematica “F. Enriques” Universit` a degli Studi di Milano Via Cesare Saldini 50 I-20133 Milano, Italy e-mail: [email protected] URL: http://www.mat.unimi.it/users/pertusi

Supersingular Irreducible Symplectic Varieties Lie Fu and Zhiyuan Li Abstract. In complex geometry, irreducible holomorphic symplectic varieties, also known as compact hyper-K¨ ahler varieties, are natural higher-dimensional generalizations of K3 surfaces. We propose to study such varieties defined over fields of positive characteristic, especially the supersingular ones, generalizing the theory of supersingular K3 surfaces. In this work, we are mainly interested in the following two types of symplectic varieties over an algebraically closed field of characteristic p > 0, under natural numerical conditions: (1) smooth moduli spaces of semistable (twisted) sheaves on K3 surfaces, (2) smooth Albanese fibers of moduli spaces of semistable sheaves on abelian surfaces. Several natural definitions of the supersingularity for symplectic varieties are discussed, which are proved to be equivalent in both cases (1) and (2). Their equivalence is expected in general. On the geometric side, we conjecture that unirationality characterizes supersingularity for symplectic varieties. Such an equivalence is established in case (1), assuming the same is true for K3 surfaces. In case (2), we show that rational chain connectedness is equivalent to supersingularity. On the motivic side, we conjecture that algebraic cycles on supersingular symplectic varieties are much simpler than their complex counterparts: its rational Chow motive is of supersingular abelian type, the rational Chow ring is representable and satisfies the Bloch–Beilinson conjecture and Beauville’s splitting property. As evidence for this, we prove all these conjectures on algebraic cycles for supersingular varieties in both cases (1) and (2). Mathematics Subject Classification (2010). 14J28, 14J42, 14J60, 14C15, 14C25, 14M20, 14K99. ahler variKeywords. K3 surfaces, irreducible symplectic varieties, hyper-K¨ eties, supersingularity, moduli spaces, unirationality, motives, Bloch–Beilinson conjecture, Beauville splitting conjecture.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Farkas et al. (eds.), Rationality of Varieties, Progress in Mathematics 342, https://doi.org/10.1007/978-3-030-75421-1_7

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1. Introduction This paper is an attempt to generalize to higher dimensions the beautiful theory of supersingular K3 surfaces. 1.1. Supersingular K3 surfaces Let S be a K3 surface over an algebraically closed field k of characteristic p > 0. On the one hand, S is called Artin supersingular if its formal Brauer group  a (see [5], [6] or § 3.2), or equivalently, the % is the formal additive group G Br(S) 2 Newton polygon associated to the second crystalline cohomology Hcris (S/W (k)) is a straight line (of slope 1). On the other hand, Shioda [87] has introduced another notion of supersingularity for K3 surfaces by considering the algebraicity of the -adic cohomology classes of degree 2, for any  = p: we say that S is Shioda supersingular if the first Chern class map c1 : Pic(S) ⊗ Q → H´e2t (S, Q (1)) is surjective; this condition is independent of , as it is equivalent to the maximality of the Picard rank, i.e., ρS = b2 (S) = 22. It is easy to see that Shioda supersingularity implies Artin supersingularity. Conversely, the Tate conjecture [90] for K3 surfaces over finite fields, solved in [71], [70], [62], [19], [58], [20] and [45], implies that these two notions actually coincide for any algebraically closed fields of positive characteristic, cf. [57, Theorem 4.8]: Shioda supersingularity ⇔ Artin supersingularity.

(1)

Supersingularity being essentially a cohomological notion, it is natural to look for its relation to geometric properties. Unlike complex K3 surfaces, there exist unirational K3 surfaces over fields of positive characteristic; the first examples were ˇ c in [82]; then Artin [5] constructed by Shioda in [87] and by Rudakov–Safareviˇ and Shioda [87] observed that unirational K3 surfaces must have maximal Picard rank 22, hence are supersingular. Conversely, one expects that unirationality is a geometric characterization of supersingularity for K3 surfaces: ˇ Conjecture 1.1 (Artin [5], Shioda [87], Rudakov–Safareviˇ c [82]). A K3 surface is supersingular if and only if it is unirational. This conjecture has been confirmed over fields of characteristic 2 by Rudakov– ˇ c [82] via the existence of quasi-elliptic fibration. See also [56], [16] and Safareviˇ [17] for recent progress. Let us also record that by [27], the Chow motive of a supersingular K3 surface is of Tate type and in particular that the Chow group of 0-cycles is isomorphic to Z, which contrasts drastically to the situation over the complex numbers, where CH0 is infinite dimensional by Mumford’s celebrated observation in [67]. 1.2. Symplectic varieties We want to investigate the higher-dimensional analogues of the above story for K3 surfaces. In the setting of complex geometry, the natural generalizations of K3

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surfaces are the irreducible holomorphic symplectic manifolds or equivalently, comahler manifolds. Those are by definition the simply connected compact pact hyper-K¨ K¨ahler manifolds admitting a Ricci flat metric such that the holonomy group is the compact symplectic group. Together with abelian varieties (or more generally complex tori) and Calabi–Yau varieties, they form the fundamental building ahler manifolds with vanishing first Chern class, thanks to blocks for compact K¨ the Beauville–Bogomolov decomposition theorem [11, Th´eor`eme 2]. Holomorphic symplectic varieties over the complex numbers have been studied extensively from various points of view in the last decades. We refer to the huge literature that exists for more details [11], [39], [36, Part III]. In positive characteristic, there seems to be no accepted definition of symplectic varieties (see however [20] and [96]). In this paper, we define them as smooth projective varieties X defined over k with trivial ´etale fundamental group and such that there exists a symplectic (i.e., nowhere-degenerate and closed) algebraic 2-form (Definition 3.1). When k = C, our definition corresponds to products of irreducible holomorphic symplectic varieties. The objective of this paper is to initiate a systematic study of supersingular symplectic varieties: we will discuss several natural definitions for the notion of supersingularity, propose some general conjectures and provide ample evidence for them. 1.2.1. Notions of supersingularity. The notion(s) of supersingularity, which is subtle for higher-dimensional varieties, can be approached in essentially two ways (see § 2): via formal groups and F -crystal structures on the crystalline cohomology as Artin did [5], or via the algebraicity of -adic or crystalline cohomology groups as Shioda did [87]. More precisely, a symplectic variety X is called 2 • 2nd -Artin supersingular, if the F -crystal Hcris (X/W (k)) is supersingular, that is, its Newton polygon is a straight line. Artin supersingularity turns out to be % is isomorphic equivalent to the condition that the formal Brauer group Br(X)  to Ga , provided that it is formally smooth of dimension 1 (Proposition 3.6). • 2nd -Shioda supersingular, if its Picard rank ρ(X) is equal to its second Betti number b2 (X). Or equivalently, the first Chern class map is surjective for the -adic cohomology for any  = p.

The two notions are equivalent assuming the crystalline Tate conjecture for divisors (note that one can reduce to the finite field case by using the crystalline variational Tate conjecture for divisors proved by Morrow [64]). The reason of using the second cohomology in both approaches is the general belief that for a symplectic variety, its second cohomology group equipped with its enriched structure should capture a significant part of the information of the variety up to birational equivalence, as is partially justified by the Global Torelli Theorem over the field of complex numbers [92, 60, 41]. We will introduce nevertheless the notions of full Artin supersingularity and full Shioda supersingularity which concern the entire

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cohomology ring of X. Roughly speaking, the former means that the Newton polygons of all crystalline cohomology groups are straight lines; the latter says that any cohomology group of even degree is spanned rationally by algebraic classes and any cohomology group of odd degree is isomorphic as F -crystal to the first cohomology of a supersingular abelian variety. See Definitions 2.1 and 2.3 respectively. Inspired by the results for K3 surfaces discussed before in § 1.1, we propose the following conjecture, generalizing the equivalence (1) as well as Conjecture 1.1. As will be explained later in § 2 and § 3, all the listed conditions below are a priori stronger than the 2nd -Artin supersingularity. Conjecture A (Characterizations of supersingularity). Let X be a symplectic variety defined over an algebraically closed field k of positive characteristic. If it is 2nd -Artin supersingular, then all the following conditions hold: • • • •

(Equivalence conjecture) X is fully Shioda (hence fully Artin) supersingular. (Unirationality conjecture) X is unirational. (RCC conjecture) X is rationally chain connected. (Supersingular abelian motive conjecture) The rational Chow motive of X is a direct summand of the motive of a supersingular abelian variety.

The Tate conjecture implies the equivalence between the 2nd -Artin supersingularity and 2nd -Shioda supersingularity; while the equivalence conjecture predicts more strongly that the supersingularity of the second cohomology implies the supersingularity of the entire cohomology. The unirationality conjecture and the RCC conjecture can be viewed as a geometric characterization for the cohomological notion of supersingularity. The supersingular abelian motive conjecture comes from the expectation that the algebraic cycles on a supersingular symplectic variety are “as easy as possible”. By the work of Fakhruddin [27], having a supersingular abelian motive implies that the variety is fully Shioda supersingular (see Corollary 2.12). Therefore we can summarize Conjecture A as follows: the notions in the diagram of implications in Figure 1 below are all equivalent for symplectic varieties. Some of the implications in Figure 1 are not obvious and will be explained in § 2 and § 3. 1.2.2. Cycles and motives. Carrying further the philosophy that the cycles and their intersection theory on a supersingular symplectic variety are as easy as possible, we propose the following conjecture, which is a fairly complete description of their Chow rings. Recall that CHi (X)alg is the subgroup of the Chow group i CHi (X) consisting of algebraically trivial cycles. Denote also by CH (X)Q the ith rational Chow group modulo numerical equivalence. There is a natural epimor∗ phism CH∗ (X)Q  CH (X)Q . Conjecture B (Supersingular Bloch–Beilinson–Beauville Conjecture). Let X be a supersingular symplectic variety defined over an algebraically closed field k. Then the following conditions hold:

Supersingular irreducible symplectic varieties

Supersingular abelian motive

RCC

Unirational

151

Fully Shioda supersingular

Fully Artin supersingular

2nd -Shioda supersingular

2nd -Artin supersingular

Figure 1. Characterizations of supersingularity for symplectic varieties (i) Numerical equivalence and algebraic equivalence coincide on CH∗ (X)Q : ∗

CH∗ (X)alg,Q = ker(CH∗ (X)Q  CH (X)Q ). In particular, the Griffiths groups of X are of torsion. (ii) For any 0 ≤ i ≤ dim(X), there exists a regular surjective homomorphism νi : CHi (X)alg → Abi (X)(k) to the group of k-points of a supersingular abelian variety Abi (X) of dimension 12 b2i−1 (X), called the algebraic representative, such that νi has finite kernel and it is universal among regular homomorphisms from CHi (X)alg to abelian varieties. (iii) There is a multiplicative decomposition CH∗ (X)Q = DCH∗ (X) ⊕ CH∗ (X)alg,Q ,

(2)

∗

where DCH (X), called the space of distinguished cycles, is a graded Qsubalgebra of CH∗ (X)Q containing all the Chern classes of X. In particular, ∗ the natural map DCH∗ (X) → CH (X)Q is an isomorphism. (iv) The intersection product restricted to the subring CH∗ (X)alg is zero. In particular, it forms a square zero graded ideal. In other words, the Q-algebra CH∗ (X)Q is the square zero extension of a graded ∗ subalgebra isomorphic to CH (X)Q by a graded module Ab∗ (X)(k)Q . In particular, if all the odd Betti numbers vanish, then the natural map CH∗ (X)Q → CH∗ (X)Q is an isomorphism and the rational Chow motive of X is of Tate type. As a consequence of Conjecture B, the space DCH∗ (X) provides a Q-vector space such that for all  = p, the restriction of the cycle class map DCH∗ (X) ⊗Q  Q − → H 2∗ (X, Q ) is an isomorphism. Moreover, the decomposition (2) is expected ∗ to be canonical and the CH (X)Q -module structure on Ab∗ (X)(k)Q should be determined by, or be at least closely related to, the H 2∗ (X)-module structure on H 2∗−1 (X), where H is any Weil cohomology theory. Remark 1.2. The following remarks will be developed with more details later.

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• Conjecture B can be seen as the supersingular analogue of Beauville’s splitting property conjecture, which was formulated over the field of complex numbers in [13]. See § 3.5 for the details. • Conjecture B is implied by the combination of the equivalence conjecture, the Bloch–Beilinson conjecture (see Conjecture 2.7 for its fully supersingular version) and the section property conjecture 3.11 proposed in Fu–Vial [31]. • We will also show in § 2.4 that the Bloch–Beilinson conjecture for supersingular symplectic varieties is a consequence of the supersingular abelian motive conjecture discussed above. To summarize: Supersingular abelian motive conjecture

Equivalence conjecture

Fully supersingular Bloch–Beilinson conjecture

Section property conjecture

The ultimate description (Conjecture B): Supersingular Bloch–Beilinson–Beauville conjecture Figure 2. Conjectures on algebraic cycles of supersingular symplectic varieties

1.3. Main results Let k be an algebraically closed field of characteristic p > 0. As in characteristic zero, two important families of examples of (simply connected) symplectic varieties are provided by the moduli spaces of stable sheaves on K3 surfaces and the Albanese fibers of the moduli spaces of stable sheaves on abelian surfaces. As evidence for the aforementioned conjectures, we establish most of them for most of the varieties in these two series. The key part is that we relate these moduli spaces birationally to punctual Hilbert schemes of surfaces, in the supersingular situation. Our first main result is the following one concerning the moduli spaces of sheaves on K3 surfaces. Theorem 1.3. Let S be a K3 surface defined over k. Let H be an ample line bundle on S and X the moduli space of H-semistable sheaves on S with Mukai vector v = (r, c1 , s) satisfying v, v ≥ 0, r > 0 and v is coprime to p.

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(i) If H is general with respect to v, then X is a symplectic variety of dimension 2n = v, v + 2 and deformation equivalent to the nth Hilbert scheme of points of S. Moreover, X is 2nd -Artin supersingular if and only if S is supersingular. (ii) If X is 2nd -Artin supersingular, then X is irreducible symplectic and it is birational to the Hilbert scheme S [n] , where n = v,v+2 . In particular, the 2 unirationality conjecture holds for X provided that S is unirational. (iii) If X is 2nd -Artin supersingular, the rational Chow motive of X is of Tate type and the natural epimorphism CH∗ (X)Q  CH∗ (X)Q is an isomorphism. In particular, the supersingular Bloch–Beilinson–Beauville conjecture holds for X and the cycle class maps induce isomorphisms ∗ CH∗ (X)K  Hcris (X/W )K

and

CH∗ (X)Q  H´e∗t (X, Q )

for all  = p. Here, the notion of generality for H and that of v being coprime to p are explained in Definitions 4.3 and 4.7 respectively. In fact, these conditions are the natural ones to ensure the smoothness of X. The coprime condition is automatically satisfied if p  ( 12 dim X − 1), see Remark 4.9. Moreover, our results also hold for moduli spaces of twisted sheaves on supersingular K3 surfaces with Artin invariant at most 9. See Theorem 5.9 and Corollary 5.10 in § 5. The second main result is for moduli spaces of sheaves on abelian varieties. Theorem 1.4. Let A be an abelian surface defined over k. Let H be an ample line bundle on A and X the Albanese fiber of the projective moduli space of Hsemistable sheaves on A with Mukai vector v = (r, c1 , s) satisfying v, v ≥ 2 and r > 0. Denote 2n := v, v − 2. (i) If p  (n + 1) and if H is general with respect to v, then X is a smooth projective symplectic variety of dimension 2n and deformation equivalent to the nth generalized Kummer variety. Moreover, X is 2nd -Artin supersingular if and only if A is supersingular. (ii) Assume A is supersingular and p  (n + 1). Then X is a 2nd -Artin supersingular irreducible symplectic variety and it is birational to the nth generalized Kummer variety associated to some supersingular abelian surface A . (iii) Under the assumption of (ii), the supersingular abelian motive conjecture (hence the equivalence conjecture), the RCC conjecture and the supersingular Bloch–Beilinson–Beauville conjecture all hold for X. In our subsequent work [28], the construction of O’Grady’s six-dimensional symplectic varieties [74, 79], which are symplectic resolutions of certain singular moduli spaces of sheaves on abelian surfaces, is extended to positive characteristics, and Conjecture A and Conjecture B are proved for those varieties.

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Conventions: Throughout this paper, k is an algebraically closed field of characteristic p > 0, W = W (k) is its ring of Witt vectors, Wi = W/pi W is the ith truncated Witt ring of k, and K = K(W ) = W [1/p] is the field of fractions of W . If X is a variety defined over a field F and let L be a field extension of F , we write XL = X ⊗F L for the base change.

2. Generalities on the notion of supersingularity In this section, we introduce several notions of supersingularity for cohomology groups of algebraic varieties in general and discuss the relations between them. 2.1. Supersingularity via F -crystals Let X be a smooth projective variety of dimension n defined over k. The ring of Witt vectors W = W (k) comes equipped with a morphism σ : W → W induced i by the Frobenius morphism x → xp of k. For any i ∈ N, we denote by Hcris (X/W ) the ith integral crystalline cohomology group of X, which is a W -module whose rank is equal to the ith Betti number1 of X. We set i i (X/W )/torsion, H i (X)K = Hcris (X/W ) ⊗W K. H i (X) := Hcris

Then H i (X) is a free W -module and it is endowed with a natural σ-linear map ϕ : H i (X) → H i (X) induced from the absolute Frobenius morphism F : X → X by functoriality. Moreover, by Poincar´e duality, ϕ is injective. The pair (H i (X), ϕ) (resp. (H i (X)K , ϕK )) forms therefore an F -crystal (resp. F -isocrystal), associated to which we have the Newton polygon Nwti (X) and the Hodge polygon Hdgi (X). According to Dieudonn´e–Manin’s classification theorem (cf. [59]), the F -crystal (H i (X), ϕ) is uniquely determined, up to isogeny, by (the slopes with multiplicities of) the Newton polygon Nwti (X). Following [59], we say that an F -crystal (M, ϕ) is ordinary if its Newton polygon and Hodge polygon agree and supersingular if its Newton polygon is a straight line. Definition 2.1 (Artin supersingularity). For a given integer i, a smooth projective variety X over k is called ith -Artin supersingular, if the F -crystal (H i (X), ϕ) is supersingular. X is called fully Artin supersingular if it is ith -Artin supersingular for all integer i. Example 2.2 (Supersingular abelian varieties). Let A be an abelian variety over k. We say that A is supersingular if the F -crystal (H 1 (A), ϕ) is supersingular, i.e., A is 1st -Artin supersingular. When g = 1, this definition is equivalent to the 1 By

definition, the ith Betti number of X, denoted by bi (X), is the dimension (over Q ) of the -adic cohomology group H´eit (X, Q ). The Betti number bi (X) is independent of the choice of the prime number  different from p ([24], [25], [44]).

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classical notion of supersingularity for elliptic curves: for instance, the group of ptorsion points is trivial. We will show in § 2.4 that a supersingular abelian variety is actually fully Artin supersingular. 2.2. Supersingularity via cycle class map We discuss the supersingularity in the sense of Shioda [87]. Let X be a smooth projective variety defined over k. For any r ∈ N, there is the crystalline cycle class map 2r clr : CHr (X) ⊗Z K −→ H 2r (X)K := Hcris (3) (X/W ) ⊗W K, whose image lands in the eigenspace of eigenvalue pr with respect to the action of ϕ on H 2r (X). Definition 2.3 (Shioda supersingularity). A smooth projective variety X defined over k is called th • (2r) -Shioda supersingular, if (3) is surjective; • even Shioda supersingular 2 if (3) is surjective for all r; th • (2r + 1) -Shioda supersingular, if there exist a supersingular abelian variety A and an algebraic correspondence Γ ∈ CHdim X−r (X × A) such that the 2r+1 1 (X/W )K → Hcris (A/W )K is an cohomological correspondence Γ∗ : Hcris isomorphism; • odd Shioda supersingular, if it is (2r + 1)th -Shioda supersingular for all r; • fully Shioda supersingular, if it is even and odd Shioda supersingular. Remark 2.4 (“Shioda implies Artin”). Each notion of Shioda supersingularity is stronger than the corresponding notion of Artin supersingularity. More precisely, (i) For even-degree cohomology, the (2r)th -Shioda supersingularity implies that the Frobenius action ϕ is the multiplication by pr on the crystalline cohomology of degree 2r, hence we have the (2r)th -Artin supersingularity (i.e., the F -crystal (H 2r (X), ϕ) is supersingular). The converse is implied by the combination of the crystalline Tate conjecture over finite fields (cf. [3, 7.3.3.2]) and the crystalline variational Tate conjecture (see [64, Conjecture 0.1]). (ii) For odd-degree cohomology, the idea of Shioda supersingularity is “of niveau 1” and supersingular. The implication from (2r+1)th -Shioda supersingularity to (2r + 1)th -Artin supersingularity follows from the definition and Example 2.2. The converse follows again from the crystalline Tate conjecture and its variational analogue. (iii) Moreover, one can try to define the (even) Shioda supersingularity using other Weil cohomology theory. For instance, for any prime number  different from p, consider the -adic cycle class map cl r : CHr (X) ⊗ Q → H´e2rt (X, Q (r)). Note that the standard conjecture implies that the surjectivities of clr and clr are equivalent for all  = p, as the images of the cycle class maps for various 2 This

is called fully rigged in [91, Definition 5.4].

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Weil cohomology theories share the same Q-structure, namely CH (X)Q , the rational Chow group modulo numerical equivalence. When r = 1, the 2nd -Shioda supersingularity does not depend on the choice of the Weil cohomology theory, as it is equivalent to say that the Picard rank is maximal. 2.3. Full supersingularity and algebraic cycles We explore in this subsection the conjectural implications on the Chow groups of the full (Artin or Shioda) supersingularity discussed in § 2.1 and § 2.2. Recall that for a smooth projective variety, a cohomology group is called of (geometric) coniveau at least i, if it is supported on a closed algebraic subset of codimension i. Let X be a smooth projective variety that is fully Shioda supersingular (Definition 2.3), which should be equivalent to being fully Artin supersingular assuming the (usual and variational) crystalline Tate conjecture [64]. Then by definition, • All the even cohomology groups are of Tate type. In particular, H 2i (X) is of coniveau i for all 0 ≤ i ≤ dim X. • All the odd cohomology groups are of abelian type. In particular, H 2i+1 (X) is of coniveau ≥ i for all 0 ≤ i ≤ dim X − 1. In particular, for any 2 ≤ j ≤ i ≤ dim X, the coniveau of H 2i−j (X) is at least i − j + 1. By the general philosophy of coniveau, the conjectural Bloch–Beilinson filtration F · on the rational Chow group satisfies that (cf. [3, § 11.2], [93, Conjecture 23.21]) GrjF CHi (X)Q = 0, ∀j ≥ 2, ∀i. Therefore by the conjectural separatedness of the filtration F · , one expects that F 2 CHi (X)Q = 0 in this case. It is generally believed that F 2 is closely related, if not equal, to the Abel–Jacobi kernel. Hence one can naturally conjecture that for fully supersingular varieties, all rational Chow groups are representable ([67]), or better, their homologically trivial parts are represented by abelian varieties. The precise statement is Conjecture 2.7 below. Before that, let us explain the notion of representability with some more details. Recall first the notion of algebraic representatives ([9, 68]) for Chow groups, developed by Murre. Let CH∗ (X)alg be the group of algebraically trivial cycles modulo rational equivalence. Definition 2.5 (Regular homomorphism [68, Definition 1.6.1]). Let X be a smooth projective variety and A an abelian variety. A homomorphism φ : CHi (X)alg → A(k) is called regular, if for any family of algebraic cycles, that is, a connected pointed variety (T, t0 ) together with a cycle Z ∈ CHi (X × T ), the map T

→

t →

A(k) φ(Zt − Zt0 )

gives rise to a morphism of algebraic varieties T → A.

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Definition 2.6 (Algebraic representative [68]). Let X be a smooth projective variety and i a positive integer. An algebraic representative for cycles of codimension i, is a couple (νi , Abi ), where Abi is an abelian variety and νi : CHi (X)alg → Abi (k) is a regular homomorphism (Definition 2.5), and it is universal in the following sense: for any regular homomorphism to the group of k-points of an abelian variety φ : CHi (X)alg → A(k), there exists a unique homomorphism of abelian varieties φ¯ : Abi → A such that φ = φ¯ ◦ νi . It is easy to see that an algebraic representative, if exists, is unique up to unique isomorphism and νi is surjective ([68, § 1.8]). As examples, the algebraic representative for the Chow group of divisors (resp. 0-cycles) is the Picard variety (resp. the Albanese variety). The main result of [68] is the existence of an algebraic representative for codimension-2 cycles and its relation to the algebraic part of the intermediate Jacobian. However, the understanding of the kernel of νi , when it is not zero, seems out of reach. In the spirit of the Bloch–Beilinson Conjecture, we can make the following speculation on the algebraic cycles on fully supersingular varieties. Conjecture 2.7 (Fully supersingular Bloch–Beilinson Conjecture). Let X be a smooth projective variety over k which is fully Shioda supersingular. Then for any 0 ≤ i ≤ dim(X), • Numerical equivalence and algebraic equivalence coincide on CHi (X)Q . In particular, the rational Griffiths group Griff i (X)Q = 0. • There exists a regular surjective homomorphism νi : CHi (X)alg → Abi (X)(k) to the group of k-points on an abelian variety Abi (X), which is the algebraic representative in the sense of Murre. • The kernel of νi is finite and dim Abi (X) = 21 b2i−1 (X). • Abi is a supersingular abelian variety. • The intersection product restricted to CH∗ (X)alg is zero. ∗

In particular, the kernel of the algebra epimorphism CH∗ (X)Q  CH (X)Q is a square zero graded ideal  given by the group of k-points on supersingular abelian varieties Ab∗ (X)(k)Q := i Abi (X)(k) ⊗Z Q. Thanks to the work of Fakhruddin [27], Conjecture 2.7 is known for supersingular abelian varieties as well as other varieties with supersingular abelian motives. We will give an account of this aspect in § 2.4. 2.4. Supersingular abelian varieties and their motives In this subsection, we illustrate the previous discussions in the special case of abelian varieties and establish some results on their motives for later use. Let A be a g-dimensional abelian variety defined over  k. For any n ∈ N, n H n (A) := Hcris (A/W ) is a torsion free W -module of rank 2g n and there exists a

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canonical isomorphism of F -crystals induced by the cup-product ∼ H n (A) =

n &

H 1 (A).

(4)

By definition, A is (1st -Artin) supersingular if the F -crystal H 1 (A) is supersingular. In this case, by (4), all slopes of the F -crystal (H n (A), ϕ) are the same (= n2 ), and hence H n (A) is as well a supersingular F -crystal for all n ∈ N, that is, A is fully Artin supersingular (Definition 2.1). Before moving on to deeper results on cycles and motives of supersingular abelian varieties, let us recall some basics on their motivic decomposition. Denote by CHM(k)Q the category of rational Chow motives over k. Building upon earlier works of Beauville [10] and [12] on Fourier transforms of algebraic cycles of abelian varieties, Deninger–Murre [26] produced a canonical motivic decomposition for any abelian variety (actually more generally for any abelian scheme; see [47] for explicit formulae of the projectors): h(A) =

2g 

hi (A) in CHM(k)Q ,

i=0

such that the Beauville component CHi (A)(s) is identified with CHi (h2i−s (A)). i 1 Furthermore, K¨ unnemann [47] showed that hi (A) = h (A) for all i, h2g (A)  i 1 h (A) = 0 for i > 2g (see also Kings [46]). 1(−g) and Lemma 2.8. Let E be a supersingular elliptic curve over k. Then for any natural number j, we have in CHM(k)Q , Sym2j h1 (E)  1(−j)⊕j(2j+1) Sym2j+1 h1 (E)  h1 (E)(−j)⊕(j+1)(2j+1) . Proof. As dim End(E)Q = 4, we have h1 (E) ⊗ h1 (E)  1(−1)⊕4 . Hence as a direct summand, Sym2 h1 (E) is also of Tate type, actually isomorphic to 1(−1)⊕3 .  ⊗j j Since Sym2j h1 (E) is a direct summand of Sym2 h1 (E) = 1(−j)⊕3 , it is also of the form 1(−j)⊕m . The rank can be obtained by looking at the realization. As for the odd symmetric powers, Sym2j+1 h1 (E) is a direct summand of Sym2j h1 (E) ⊗ h1 (E) = h1 (E)(−j)⊕j(2j+1) , therefore Sym2j+1 h1 (E)(j) is a direct summand of h1 (E j(2j+1) ). This corresponds to (up to isogeny) a sub-abelian variety of the supersingular abelian variety E j(2j+1) , which must be itself supersingular, hence isogenous to a power of E by [77, Theorem 4.2]. In other words, Sym2j+1 h1 (E)(j)  h1 (E m ), for some m ∈ N, which can be identified by looking at the realization.  We can now compute the Chow motives of supersingular abelian varieties. Theorem 2.9. Let A be a g-dimensional supersingular abelian variety over   defined an algebraically closed field k of positive characteristic p. Let bi = 2g be the ith i

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Betti number of A. Then in the category CHM(k)Q , we have for any i, h2i (A)  1(−i)⊕b2i ; h

2i+1

⊕ 21 b2i+1

(A)  h (E)(−i) 1

(5) ,

(6)

where E is a/any supersingular elliptic curve. In particular, A is fully Shioda supersingular. Proof. By Oort’s result [77, Theorem 4.2], an abelian variety A is supersingular if and only if it is isogenous to the self-product of a/any supersingular elliptic curve E. In CHM(k)Q , we denote Λ := 1⊕g and then h1 (A)  h1 (E g ) = h1 (E) ⊗ Λ, since isogenous abelian varieties have isomorphic rational Chow motives. Standard facts on tensor operations in idempotent-complete symmetric mono¨ıdal categories (cf. [33, Lecture 6]) yields that hi (A) 

i &    1 Sλ h1 (E) ⊗ Sλ Λ, h (E) ⊗ Λ 

(7)

λi

where λ runs over all partitions of i, λ is the transpose of λ and Sλ is the Schur functor associated to λ. Since Sλ Λ is a direct sum of copies of the unit motive 1, we take a closer look at Sλ h1 (E). Recall that (cf. [4] for example3 ) • Sλ h1 (E) = 0 if λ has length at least 3; • if λ has length at most 2, say λ = (a + b, a) with a, b ≥ 0 and 2a + b = i, then 2 1 since h (E)  1(−1) is a ⊗-invertible object, we have '2 (⊗a & 1 1 S(a+b,a) h (E) = ⊗ Symb h1 (E) = 1(−a) ⊗ Symb h1 (E). h (E) Combining this with (7) and using Lemma 2.8, we see that hi (A) is a direct sum of some copies of 1(− 2i ) if i is even and a direct sum of some copies of h1 (E)(− i−1 2 ) if i is odd. The numbers of copies needed are easily calculated by looking at their realizations.  Definition 2.10 (Supersingular abelian motives). Let CHM(k)Q be the category of rational Chow motives over k. Let Mssab be the idempotent-complete symmetric mono¨ıdal subcategory of CHM(k)Q generated by the motives of supersingular abelian varieties. A smooth projective variety X is said to have supersingular abelian motive if its rational Chow motive h(X) belongs to Mssab . Remark 2.11. Mssab contains the Tate motives by definition. Thanks to Theorem 2.9, Mssab is actually generated, as idempotent-complete tensor category, by the Tate motives together with h1 (E) for a/any supersingular elliptic curve E. It can be shown that any object in Mssab is a direct summand of the motive of some 3 The convention in [4] is the graded/super one, which is the reason why the symmetric product and exterior product are switched from ours when applied to an “odd” object h1 in loc. cit. We prefer to stick to the ungraded convention so the comparison to the corresponding facts from classical cohomology theory is more transparent.

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supersingular abelian variety. Therefore, for a smooth projective variety X, the condition of having supersingular abelian motive is exactly Fakhruddin’s notion of “strong supersingularity” in [27]. The following result confirms in particular the Fully Supersingular Bloch– Beilinson Conjecture 2.7 and the full Shioda supersingularity for varieties with supersingular abelian motives. The results (ii)–(v) are due to Fakhruddin [27]; while our proof presented below is somehow different and emphasizes the more fundamental result Theorem 2.9; (i) and (vi) are new according to authors’ knowledge. Corollary 2.12 (cf. Fakhruddin [27]). Let X be an n-dimensional smooth projective variety defined over an algebraically closed field k of characteristic p > 0, such that X has supersingular abelian motive (Definition 2.10). Let bi be the ith Betti number of X. Then we have the following: (i) In the category CHM(k)Q , we have h(X) 

n  i=0

1(−i)⊕b2i ⊕

n−1 

1

h1 (E)(−i)⊕ 2 b2i+1 ,

(8)

i=0

where E is a supersingular elliptic curve. (ii) X is fully Shioda supersingular (Definition 2.3). (iii) Numerical equivalence and algebraic equivalence coincide. In particular, for any integer i, the Griffiths group is of torsion: Griff i (X)Q = 0. (iv) CHi (X)Q = CHi (X)(0) ⊕ CHi (X)(1) with CHi (X)(0)  Q⊕b2i providing a 1 Q-structure for cohomology and CHi (X)(1)  E(k) 2 b2i−1 ⊗Z Q is the algebraically trivial part. (v) CHi (X)alg has an algebraic representative (νi , Abi ) with ker(νi ) finite and Abi a supersingular abelian variety of dimension 12 b2i−1 . (vi) The intersection product restricted to CH∗ (X)alg is zero. Proof. For (i), using Theorem 2.9, the motive of X must be a direct sum of Tate motives and Tate twists of h1 (E). Then the precise numbers of Tate twists and copies are easily determined by looking at the realization. (ii) follows immediately from (8) in (i). For (iii), using (i), it suffices to observe that numerical equivalence and algebraic equivalence are the same on elliptic curves and all the Griffiths groups of a point and an elliptic curve are trivial. For (iv), it is an immediate consequence of (i) together with the simple fact that CH1 (h1 (E)) = CH1(1) (E)  Pic0 (E)(k) ⊗Z Q  E(k) ⊗Z Q. 1

For (v), we argue similarly as in [27]: denote B = E 2 b2i−1 , then in (8), the only term contributes non-trivially to CHi (X)alg is h1 (E)(1−i)⊕b2i−1 . Now the isomorphisms (8) of rational Chow motives can be interpreted as follows: there exist a positive integer N and two correspondences Z1 , Z2 ∈ CHn−i+1 (X × B), such that the two

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compositions of Z1∗ : CH1 (B)alg = Pic0 (B)(k) → CHi (X)alg and Z2,∗ : CHi (X)alg → CH1 (B)alg = Pic0 (B)(k) are both multiplication by N . By the divisibility of CHi (X)alg and Pic0 (B)(k) (cf. [15, Lemma 1.3]), both Z1∗ and Z2,∗ are surjective and the kernel of Z2,∗ is finite. The surjectivity of Z2,∗ implies the representability of CHi (X)alg and in particular (by [83] for example) there exists an algebraic representative νi : CHi (X)alg → Abi (k). By the universal property of algebraic representative, the (regular) surjective homomorphism Z2,∗ is factorized through νi , hence the kernel of νi is finite and Pic0 (B) is dominated by Abi . On the other hand, the surjectivities of Z1∗ and νi show that Abi is also dominated by Pic0 (B). Therefore, Abi is isogenous to Pic0 (B), hence is supersingular of dimension 21 b2i−1 . Finally for (vi), recall first that any algebraically trivial cycle on X is a linear combination of cycles of the form Γ∗ (α), where Γ ∈ CH(C ×X) is a correspondence from a connected smooth projective curve C to X and α is a 0-cycle of degree 0 on C. Therefore we only need to show that for any two connected smooth projective curves C1 , C2 , Γi ∈ CH(Ci × X) and αi ∈ CH0 (Ci )deg 0 for i = 1, 2, then Γ1,∗ (α1 ) · Γ2,∗ (α2 ) = 0 in CH(X). Indeed, let Γ be the correspondence from C1 × C2 to X given by the composition δX ◦ (Γ1 × Γ2 ), then Γ1,∗ (α1 ) · Γ2,∗ (α2 ) = Γ∗ (α1 × α2 ), here × is the exterior product. Now on one hand, the Albanese invariant of the cycle α1 × α2 ∈ CH0 (C1 × C2 )alg is trivial. On the other hand, by the universal property of algebraic representative, whose existence is proved in (v), we have the commutative diagram CH0 (C1 × C2 )alg

alb=ν2

/ Alb(C1 × C2 )(k)

ν∗

 / Ab∗ (X)(k).

Γ∗

 CH∗ (X)alg

Hence Γ∗ (α1 ×α2 ) belongs to ker(ν∗ ), which is a finite abelian group by (v). In other words, the image of the intersection product CH∗ (X)alg ⊗CH∗ (X)alg → CH∗ (X)alg is annihilated by some integer. However, this image is also divisible, hence must be zero.  Remark 2.13 (Beauville’s conjectures on supersingular abelian varieties). Let A be an abelian variety of dimension g. Recall Beauville’s decomposition [12] on rational Chow groups: for any 0 ≤ i ≤ g, i

CH (A)Q =

i  s=i−g

CHi (A)(s) ,

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where CHi (A)(s) is the common eigenspace of the multiplication-by-m map (with eigenvalue m2i−s ) for all m ∈ Z. As is pointed out in [27], when A is supersingular, Corollary 2.12 confirms all the conjectures proposed in [10] and [12], and the corresponding Beauville decomposition takes the following form: CH∗ (A)Q = CH∗ (A)(0) ⊕ CH∗ (A)(1) , ∗

∗

(9) ∗

with CH (A)(0) injects into the cohomology and CH (A)(1) = CH (A)alg,Q . A remarkable feature of the Chow rings of supersingular abelian varieties that is not shared by other varieties with supersingular abelian motives in general is that the decomposition (9) is multiplicative, that is, in addition to the properties (i)–(vi) in Corollary 2.12, we have that the subspace CH∗ (X)(0) is closed under the intersection product. We will see in § 3.5 how this supplementary feature is extended to another class of supersingular varieties, namely the supersingular symplectic varieties. Remark 2.14. There is more precise information on the Griffiths group of codimension-2 cycles on a supersingular abelian variety if the base field is the algebraic closure of a finite field of characteristic p: in [34], Gordon–Joshi showed that it is at most a p-primary torsion group.

3. Supersingular symplectic varieties We now start to investigate the various notions of supersingularity introduced in the previous section, as well as their relations, for a special class of varieties, namely the symplectic varieties. As in the case of K3 surfaces, we expect that all these notions are equivalent in this case. 3.1. Symplectic and irreducible symplectic varieties Definition 3.1. Let X be a connected smooth projective variety defined over k of characteristic p > 0, and let Ω2X/k be the locally free sheaf of algebraic 2-forms over k. The variety X is called symplectic if 1. π1´et (X) = 0; 2. X admits a nowhere degenerate closed algebraic 2-form; In particular, X is even-dimensional with trivial canonical bundle. When dim H 0 (X, Ω2X/k ) = 1, we say that X is irreducible symplectic. Remark 3.2. There is also a definition of K3[n] -type irreducible symplectic varieties in the recent work of Yang [96], which requires the liftability to characteristic 0. In that case, the comparison of two definitions is not clear to us. The construction methods for symplectic varieties over fields of positive characteristic are somehow limited as is in the case over the complex numbers. Let us collect some examples.

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Example 3.3. (i) When p > 2, two-dimensional symplectic varieties are nothing but K3 surfaces. When p = 2, there is an extra class, namely, the supersingular Enriques surfaces, which are also irreducible symplectic but with dim H 1 (O) = 1; (ii) the Hilbert scheme of length-n subschemes on a K3 surface ([11]); (iii) smooth moduli spaces of stable sheaves (or more generally stable complexes with respect to some Bridgeland stability condition) on a K3 surface ([66], [72], [39], [7], [8]), under some mild numeric conditions on the Mukai vector and the polarization (see Proposition 4.4); (iv) the generalized Kummer varieties Kn (A) associated to an abelian surface A ([11]), provided that it is smooth over k (the smoothness condition does not always hold, see [84]). By definition, it is the fiber of the isotrivial fibration s : A[n+1] → A, where s sends a subscheme to the summation of its support (with multiplicities); (v) the Albanese fiber of a smooth moduli space of stable sheaves (or more generally Bridgeland stable objects in the derived category) on an abelian surface ([66], [97]), under some mild numerical conditions on the Mukai vector and the polarization, provided that it is smooth over k (see Proposition 6.9); (vi) the Fano varieties of lines of smooth cubic fourfolds ([14]); (vii) O’Grady provided in [73] and [74] other examples by symplectic resolutions of some singular moduli space of semistable sheaves on K3 and abelian surfaces. His construction was over C; see [28] for the adaptation to characteristic p > 2; (viii) as is shown in the recent work [89], the punctual Hilbert schemes of supersingular Enriques surfaces (which only exist in characteristic 2) are symplectic varieties but not irreducible symplectic. Some nice properties for irreducible symplectic varieties defined over the complex numbers, such as the Beauville–Bogomolov quadratic form on the second cohomology and Torelli theorems, can be expected to hold in positive characteristics. We refer the readers to [50] for such an attempt via the theory of displays. 3.2. Artin supersingularity and formal Brauer group As indicated by the global Torelli theorem over the complex numbers, we expect that the second cohomology of a symplectic variety should control most of its geometry, up to birational equivalence. This motivates us to single out the following most important piece in Definition 2.1: Definition 3.4. Let X be a symplectic variety of dimension 2n over k. The variety X is called 2nd -Artin supersingular if the F -crystal (H 2 (X), ϕ) is supersingular, i.e., the Newton polygon Nwt2 (X) is a straight line (of slope 1). For a K3 surface, Artin defined his supersingularity originally in [5] by looking % which turns out to be equivalent to the supersinat its formal Brauer group Br, gularity of the F -crystal (H 2 (X), ϕ) introduced above. More generally, Artin and

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Mazur [6] observed that the formal Brauer group actually fits into a whole series of formal groups. Recall that if the functor % X : (Artin local k-algebras) → (Abelian groups) Br   R → ker H´e2t (X ×k R, Gm ) → H´e2t (X, Gm ) , is pro-representable by a formal group, then we call it the formal Brauer group % Note that by [6], we have the pro-representability if of X, denoted by Br(X). H 1 (X, OX ) = 0. Thus we can make the following definition. % % X is Definition 3.5. A symplectic variety X is called Artin Br-supersingular if Br  pro-represented by the formal additive group Ga . One can obtain the following consequence which guarantees that the two notions of supersingularity in Definitions 3.4 and 3.5 coincide under mild condition, which presumably always holds for symplectic varieties. % X is pro-repreProposition 3.6. Let X be a symplectic variety. If the functor Br % sentable, formally smooth and H 2 (X, OX )  k, then X is Artin Br-supersingular if and only if the F -crystal (H 2 (X), ϕ) is supersingular, that is, X is 2nd -Artin supersingular. % Proof. Under the hypothesis, Br(X) is a one-dimensional formal Lie group as 2 dim H (X, OX ) = 1, hence is classified by its height. It is the formal additive group if and only if the Dieudonn´e–Cartier module D(Φ2 (X)) ⊗ K is zero. Then we can conclude by [6, Corollary 2.7], since the Newton polygon of (H 2 (X), ϕ) is a straight line if and only if there is no sub-F -isocrystals with slopes strictly less  than 1. Furthermore, similar to the case of K3 surfaces [5, Theorem 1.1], the Pi% card number behaves very well for families of Artin Br-supersingular symplectic varieties: % Corollary 3.7. Let π : X → B be a smooth projective family of Artin Br-supersingular symplectic varieties over a connected base scheme B over k. Write Xb = 2 π −1 (b) for b ∈ B. Assume either Picτ (X/B) is smooth or Hcris (Xb /W ) are torsion free for any b ∈ B. Then the Picard number ρ(Xb ) is constant for all b ∈ B. In particular, all the fibers of π are 2nd -Shioda supersingular if and only if one of them is 2nd -Shioda supersingular. Proof. The proof is similar to [5, Theorem 1.1]. It suffices to show ρ(Xη ) = ρ(X0 ) for any family over B = Spec k[[t]], where Xη is the generic fiber and X0 is the special fiber. The supersingular assumption indicates that the F -crystal H 2 (Xb /W ) is constant in b. Then the result in [5, 64, 23] implies that the cokernel of the specialization map NS(Xη ) → NS(X0 ) is finite and annihilated by powers of p. It follows that ρ(Xη ) = ρ(X0 ). 

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3.3. Artin supersingularity vs. Shioda supersingularity As mentioned in Remark 2.4, the 2nd -Artin supersingularity is a priori weaker than the 2nd -Shioda supersingularity. In the other direction, we know the Tate conjecture holds for certain symplectic varieties. Theorem 3.8 (Charles [19]). Let Y be a symplectic variety of dimension 2n over C and let L be an ample line bundle on Y and d = c1 (L)2n . Assume that p is coprime to d and that p > 2n. Suppose that Y can be defined over a finite unramified extension of Qp and that Y has good reduction at p. If the Beauville–Bogomolov form of Y induces a non-degenerate quadratic form on the reduction modulo p of the primitive lattice in the second cohomology group of Y , then the reduction of Y at p, denoted by X, satisfies the Tate conjecture for divisors. This yields the following consequence: Corollary 3.9. Suppose X is a symplectic variety defined over k satisfying all the conditions in Theorem 3.8, then X is 2nd -Artin supersingular if and only if X is 2nd -Shioda supersingular. The more difficult question is to go beyond the second cohomology and ask whether X is fully Shioda supersingular (Definition 2.3) and hence fully Artin supersingular (Definition 2.1), if X is 2nd -Artin supersingular; that is, whether the notions in the following diagram of implications are equivalent. Fully Shioda supersingular

+3 Fully Artin supersingular

 2nd -Shioda supersingular

 +3 2nd -Artin supersingular

This is the equivalence conjecture in the introduction (see Conjecture A). Later we will verify this for smooth moduli spaces of semistable sheaves on K3 surfaces and abelian surfaces. A fundamental reason to believe this conjecture comes from a motivic consideration (the supersingular abelian motive conjecture in Conjecture A), which will be explained in § 3.5. 3.4. Unirationality vs. supersingularity In the direction of looking for a geometric characterization of supersingularity for symplectic varieties, we proposed in the introduction to generalize Conjecture 1.1 for K3 surfaces to the unirationality conjecture and the RCC conjecture for symplectic varieties, which respectively says that the 2nd -Artin supersingularity is equivalent to the unirationality and the rational chain connectedness; see Conjecture A in the introduction. One can get many examples of 2nd -Shioda supersingular varieties from varieties with “sufficiently many” rational curves. The result below, which says that the rational chain connectedness implies the algebraicity of H 2 , is well known in characteristic 0, and it holds in positive characteristics as well.

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Theorem 3.10 (cf. [35, Theorem 1.2]). Let X be smooth projective variety over k. If X is rationally chain connected, then the first Chern class map induces an ∼ H 2 (X, Q (1)) for all  = p. In particular, ρ(X) = isomorphism Pic(X) ⊗ Q = ´ et nd b2 (X) and X is 2 -Shioda supersingular. Theorem 3.10 gives the following implications, which is part of Figure 1 in the introduction: Unirational =⇒ RCC =⇒ 2nd -Shioda supersingular =⇒ 2nd -Artin supersingular. As evidence, we will verify the unirationality conjecture for most smooth moduli spaces of semistable sheaves on K3 surfaces, assuming the unirationality of supersingular K3 surfaces (Theorem 1.3), and prove the RCC conjecture for most smooth Albanese fibers of moduli spaces of semistable sheaves on abelian surfaces (Theorem 1.4). However, the unirationality conjecture remains open even for generalized Kummer varieties of dimension at least four. In a sequel to this paper [28], we investigate these conjectures for some symplectic varieties of O’Grady type. 3.5. Algebraic cycles and supersingularity Over the field of complex numbers, thanks to the Kuga–Satake construction (cf. [48]), one expects that the Chow motive of any projective symplectic variety is of abelian type. In positive characteristic, it is also natural to conjecture that any supersingular symplectic variety has supersingular abelian Chow motive (Definition 2.10), in the sense that its Chow motive belongs to the idempotent-complete additive tensor subcategory of CHMQ generated by the motive of supersingular abelian varieties. See the supersingular abelian motive conjecture in the introduction (Conjecture A) for the statement. Thanks to Theorem 2.9 and Corollary 2.12, the supersingular abelian motive conjecture implies the equivalence conjecture (i.e., 2nd -Shioda supersingular implis fully Shioda supersingular) and the fully supersingular Bloch–Beilinson conjecture 2.7 for symplectic varieties: +3 Fully supersingular Bloch–Beilinson Supersingular abelian motive  Fully Shioda supersingular which is part of the diagrams in Figure 1 and Figure 2. One can summarize differently Corollary 2.12: for any fully supersingular variety having supersingular abelian motive, there is a short exact sequence of graded vector spaces ∗

0 → Ab∗ (X)(k)Q → CH∗ (X)Q → CH (X)Q → 0, ∗

∗

∗

∗

∗

(10)

where CH := CH / ≡ and Ab (X)(k)Q = CH (X)alg,Q = CH (X)num,Q . However so far the only thing we can say in general about the ring structure on CH∗ (X)Q given by the intersection product is that Ab∗ (X)(k)Q forms a square zero graded ideal. It is the insight of Beauville [13] that reveals a supplementary structure on

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CH∗ (X)Q : if X is moreover symplectic, then its rational Chow ring should have a multiplicative splitting of the Bloch–Beilinson filtration. In the fully supersingular case, the Bloch–Beilinson filtration is precisely (10) and Beauville’s splitting conjecture reduces to the supersingular case of the following conjecture. Conjecture 3.11 (Section property conjecture [31]). Let X be a symplectic variety over an algebraically closed field. Then the algebra epimorphism CH∗ (X)Q  ∗ CH (X)Q admits a (multiplicative) section whose image contains all the Chern classes of the tangent bundle of X. The study of this conjecture in [31] was inspired by O’Sullivan’s theory on symmetrically distinguished cycles on abelian varieties [78], which provides such an ∗ algebra section for the epimorphism CH∗ (A)Q  CH (A)Q for any abelian variety A. This is an important step towards Beauville’s general conjecture in [10] and [12]. As a result, we call the elements in the image of the section in Conjecture 3.11 distinguished cycles and denote the corresponding subalgebra by DCH∗ (X). By ∗ definition, the natural composed map DCH∗ (X) → CH (X)Q is an isomorphism. Remark 3.12. It is easy to deduce the supersingular Bloch–Beilinson–Beauville conjecture ( = Conjecture B in the introduction) from the other aforementioned conjectures, thus completing Figure 2. First of all, for a symplectic variety, the equivalence conjecture allows us to go from the 2nd -Artin supersingularity to the full Shioda supersingularity. Now items (i), (ii), (iv) of Conjecture B are included in the fully supersingular Bloch–Beilinson conjecture 2.7. The only remaining condition is in item (iii) that DCH∗ (X) is closed under the intersection product and contains Chern classes of X; this is exactly the content of the section property conjecture 3.11. To make the link to Beauville’s original splitting property conjecture [13] more transparent, we record that in his notation, CH∗ (X)(0) := DCH∗ (X) and CH∗ (X)(1) = CH∗ (X)alg,Q . The fully supersingular Bloch–Beilinson conjecture 2.7 gives that CH∗ (X)(0) ⊕ CH∗ (X)(1) = CH∗ (X)Q and CH∗ (X)(1) is a square zero ideal; while the “splitting” says simply that CH∗ (X)(0) is closed under the product. 3.6. Birational symplectic varieties In this subsection, we compare the Chow rings and cohomology rings of two birationally equivalent symplectic varieties. The starting point is the following result of Rieß, built on Huybrechts’ fundamental work [39]. Actually her proof yields the following more precise result. We denote by ΔX ⊂ X × X the diagonal and δX = {(x, x, x) | x ∈ X} ⊂ X × X × X the small diagonal for a variety X. Theorem 3.13 (Rieß [81, § 3.3 and Lemma 4.4]). Let X and Y be d-dimensional projective symplectic varieties over k = C. If they are birational, then there exists a correspondence Z ∈ CHd (X × Y ) such that (i) (Z × Z)∗ : CHd (X × X) → CHd (Y × Y ) sends ΔX to ΔY ;

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(ii) (Z × Z × Z)∗ : CHd (X × X × X) → CHd (Y × Y × Y ) sends δX to δY . (iii) Z∗ : CH(X) → CH(Y ) sends ci (X) to ci (Y ) for any i ∈ N; (iv) Z induces an isomorphism of algebra objects h(X) → h(Y ) in CHM(k) with inverse given by t Z. In particular, Z induces an isomorphism between their Chow rings and cohomology rings. Note that (iv) is a reformulation of (i) and (ii). Our objective is to extend Rieß’s result to other algebraically closed fields under the condition of liftability. More precisely, Proposition 3.14. Let X and Y be two birationally equivalent projective symplectic varieties defined over an algebraically closed field k. If the characteristic of k is positive, we assume moreover that X and Y are both liftable to X and Y over a characteristic zero base W with geometric generic fibers Xη W and Yη W being birational symplectic varieties. Then the same results as in Theorem 3.13 hold. Proof. We first treat the case where char(k) = 0. Without loss of generality, we can assume that k is finitely generated over its prime field Q, and fix an embedding k → C. As XC and YC are birational complex symplectic varieties, we have a cycle ZC ∈ CHd (XC ×C YC ) verifying the properties in Theorem 3.13. Let L be a finitely generated field extension of k, such that ZC , as well as all rational equivalences involved, is defined over L. Take a smooth connected k-variety B whose function field k(B) = κ(ηB ) = L and choose a closed point b ∈ B(k). Let Z ∈ CHd (XηB ×L YηB ) with Z ⊗L C = ZC and such that Z satisfies the properties in Theorem 3.13 for XηB and YηB . Now the specialization of Z from the generic point ηB to the closed point b gives rise to a cycle sp(Z) ∈ CHd (X × Y ), which satisfies all the properties of Theorem 3.13 because specialization respects compositions of correspondences, diagonals and small diagonals (cf. [32, § 20.3]). In the case where char(k) > 0, let W , X and Y be as in the statement. Denote by K = Frac(W ), w the closed point of W with residual field k. By hypothesis, XK and YK are geometrically birational, hence the result in characteristic zero proved in the previous paragraph implies that there exist a finite field extension L/K, an algebraic cycle Z ∈ CHd (XL ×L YL ) which satisfies all the properties of Theorem 3.13 for XL and YL . Take any W -scheme B with κ(ηB ) = L and choose a closed point b of B in the fiber of w, then κ(b) = k since k is algebraically closed. Then as before, the specialization of Z from the generic point ηB to the closed point b yields a cycle sp(Z) ∈ CHd (X ×k Y ) which inherits all the desired properties from Z.  In view of Proposition 3.14, the following notions are convenient. Definition 3.15 ((Quasi-)liftably birational equivalence). Two symplectic varieties X and Y are called liftably birational if they are both liftable to X and Y over some base W of characteristic zero with geometric generic fibers being birationally equivalent symplectic varieties.

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Two symplectic varieties X and Y are called quasi-liftably birational if there exists a (finite) sequence of symplectic varieties X = X0 , X1 , . . . , Xm = Y, such that for any 0 ≤ i ≤ m − 1, Xi and Xi+1 are liftably birational. Remark 3.16. Two quasi-liftably birational symplectic varieties are indeed birationally equivalent. To see this, note that all symplectic varieties, having trivial canonical bundle, are non-ruled. Then it follows from [61, Theorem 1] that the birational equivalence between the geometric generic fibers of X and Y implies that X and Y are birationally equivalent after possibly taking a field extension of the residue field κ. As k is algebraically closed, X and Y are birationally equivalent over k as well. Corollary 3.17. Let X and Y be two quasi-liftably birational symplectic varieties. Then X is 2nd -Artin (resp. 2nd -Shioda, fully Artin, fully Shioda) supersingular if and only if Y is so. Proof. By Proposition 3.14 (iv), the rational Chow motives of X and Y are isomorphic. By realization, the crystalline cohomology groups of X and Y are isomorphic as F -isocrystals, hence we have the statements for 2nd and full Artin supersingularities. Similarly, the algebraicity of cohomology is controlled by the motive, hence the statements on the 2nd and full Shioda supersingularities follow. 

4. Moduli spaces of stable sheaves on K3 surfaces 4.1. Preliminaries on K3 surfaces over positive characteristic Some useful facts needed later on K3 surfaces are collected here. We start with the N´eron–Severi lattices of supersingular K3 surfaces. Let S be a smooth projective K3 surface defined over k, an algebraically closed field of positive characteristic p. As the Tate conjecture holds for K3 surfaces over finite fields of any characteristic ([19], [58], [20], [45]), S is Artin supersingular if and only if it is Shioda supersingular (cf. the argument in [57, Theorem 4.8]), hence the notion of supersingularity has no ambiguity for K3 surfaces. By Artin’s work [5], if S is supersingular, the discriminant of the intersection form on NS(S) is disc NS(S) = ±p2σ(S) for some integer 1 ≤ σ(S) ≤ 10, which is called the Artin invariant of S. The lattice NS(S) is uniquely determined, up to isomorphism, by its Artin invariant σ(S). Such lattices are completely classified ˇ c in [82, § 2] and when p > 2, there is a refinement due to by Rudakov–Safareviˇ Shimada [86]. We summarize their results as follows. Given a lattice Λ and an integer n, we denote by Λ(n) the lattice obtained by multiplying its bilinear form by n. Proposition 4.1 (Supersingular K3 lattices). Let S be a supersingular K3 surface defined over an algebraically closed field of positive characteristic p. The lattice NS(S) is isomorphic to −Λσ(S) . When p > 2, the intersection form of Λσ(S) is

Lie Fu and Zhiyuan Li

170 given as below

$

(i) σ(S) < 10, Λσ(S) =

(p)

U ⊕ V20,2σ(S) ,

if p ≡ 3 mod 4, and 2  σ(S)

(p) U ⊕ H ⊕ V16,2σ(S) , (p) U (p) ⊕ H (p) ⊕ V16,16 , (p)

otherwise

(ii) σ(S) = 10, Λ10 = Here, U is the hyperbolic plane, ⎛ 2 1 0 0 ⎜ 1 (q + 1)/2 γ 0 (p) ⎜ H =⎝ p(q + 1)/2 0 0 p γ p 2(p + γ 2 )/q 0

⎞ ⎟ ⎟ ⎠

(11)

2 satisfying that the prime q ≡ 3 mod 8, ( −q p ) = −1, γ + p ≡ 0 mod q, and ( ' m 1 (p) Vm,n = V0 ∪ ei + V0 2 1

where + V0 =

m  i=1

ai e i |



, ai ≡ 0

mod 2

⎧ ⎪ ⎨0, ⊆ ⊕Zei ; ei ej = 1, ⎪ ⎩ p,

if i = j, if i = j > n, otherwise.

(12)

When p = 2, Λσ(S) has been explicitly classified in [82, § 2, P.157], we will only use the fact that it contains the hyperbolic lattice U as a direct summand when σ(S) is odd and U (2) when σ(S) is even. As a consequence, Liedtke [56, Proposition 3.9] showed the existence of elliptic fibrations on supersingular K3 surfaces (p > 3). As we need elliptic fibrations with more special properties on supersingular K3 surfaces, we will prove a strengthening of Liedtke’s result later using Proposition 4.1. Let us turn to the liftability problem of K3 surfaces. Ogus [75, Corollary 2.3] (attributed also to Deligne) showed that every polarized K3 surface admits a projective lift. We will need the following stronger result. Proposition 4.2 (Lifting K3 surfaces with line bundles [53, Corollary 4.2], [55, Appendix A.1 (iii)], [20, Proposition 1.5]). Let S be a smooth K3 surface over an algebraically closed field k of characteristic p > 0. Let Σ ⊆ NS(S) be a saturated subgroup of rank < 11 containing an ample class. Then there exist a complete discrete valuation ring W  of characteristic 0 with fraction field K  and residue field κ containing k, a relative K3 surface S → Spec(W  )

(13) ∼ with special fiber Sκ = S ×k κ, such that the specialization map NS(SK  ) → NS(Sκ )  NS(S) induces an isomorphism 

→ Σ, NS(SK  ) − where SK  is the geometric generic fiber over the algebraic closure K  .

(14)

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4.2. Moduli spaces of stable sheaves One important source of examples of K3[n] -type symplectic varieties is formed by the moduli spaces of stable sheaves on a K3 surface ([66], [72], [39]). Given a K3 surface S, we denote by  = Z · 1 ⊕ NS(S) ⊕ Z · ω H(S) the algebraic Mukai lattice of S, where 1 is the fundamental class of S and ω is the  with r, s ∈ Z and L ∈ NS(S), is class of a point. An element r ·1+L+s·ω of H(S),  is given by the following often denoted by (r, L, s). The lattice structure on H(S) Mukai pairing −, −: (r, L, s), (r , L , s ) = L · L − rs − r s ∈ Z.

(15)

For a coherent sheaf F on S, its Mukai vector is defined by "  v(F ) := ch(F ) td(S) = (rk(F), c1 (F), χ(F) − rk(F)) ∈ H(S).  Definition 4.3 (General polarizations). Let v ∈ H(S) be a primitive4 Mukai vector. We say a polarization H is general with respect to v if every H-semistable sheaf with Mukai vector v is H-stable. For instance, if v = (r, c1 , s), one can easily show that H is general with respect to v if it satisfies the following numerical condition (cf. [20]): gcd(r, c1 · H, s) = 1. (16)  Given a primitive element v = (r, c1 , s) ∈ H(S) such that r > 0 and v, v ≥ 0, together with a general ample line bundle H (with respect to v) on S, we consider the moduli space of Gieseker–Maruyama H-stable sheaves on S with Mukai vector v, denoted by MH (S, v). According to the work of Langer [49], MH (S, v) is a quasiprojective scheme over k. When char(k) = 0, the works of Mukai [66], O’Grady [72] and Huybrechts [39] show that MH (S, v) is an irreducible symplectic variety of dimension 2n = v, v + 2 and is of K3[n] -deformation type. Over fields of positive characteristic, the following analogous properties of MH (S, v) hold. Proposition 4.4 (cf. [20, § 2]). Let S be a smooth projective K3 surface, H an ample  be a primitive Mukai vector with r > 0 and line bundle and v = (r, c1 , s) ∈ H(S) v, v > 0. If H is general with respect to v, then (i) MH (S, v) is a smooth projective, symplectic variety of dimension 2n = v, v+ 2 over k and it is deformation equivalent to the nth Hilbert scheme S [n] . (ii) When  = p, there is a canonical quadratic form on H 2 (MH (S, v), Z (1)). Let v ⊥ be the orthogonal complement of v in the -adic Mukai lattice of S. There is an injective isometry  θv : v ⊥ ∩ H(S) → NS(MH (S, v)), (17)  whose cokernel is a p-primary torsion group. Here v ⊥ ∩H(S) is the orthogonal  complement of v in H(S). 4 that

 is, v cannot be written as mv for an integer m ≥ 2 and v another element in H(S).

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(iii) There is an isomorphism of Galois representations (resp. isocrystals) v ⊥ ⊗ K → H 2 (MH (S, v), K)

(18)

between the rational ´etale (resp. crystalline) cohomology groups, where K = Q (resp. the fraction field of W ). Here v ⊥ ⊗ K is the orthogonal complement of v in H ∗ (S, K). It is compatible with the isometry (ii) via the ´etale (resp. crystalline) cycle class map. Proof. All the assertions are well known over fields of characteristic zero (cf. [66], [72]). When char(k) > 0 and assuming condition (16), the statements were proved in [20, Theorem 2.4] via lifting to characteristic zero. But there is no difficulty to extend them under the more general assumption that H is general with respect to v. Here we just sketch the proof of the missing parts in [20]: In part (i), it remains to prove that MH (S, v) has trivial ´etale fundamental group. One can use Proposition 4.2 to find a projective lift S → Spec W  ,

(19)

of S over some discrete valuation ring W  such that both H and v lift to S, denoted by H ∈ Pic(S) and v, respectively. We consider the relative moduli space of semistable sheaves on S over W  MH (S, v) → Spec W  .

(20)

Let Sη be the generic fiber of (19) and let Hη be the restriction of H to Sη . Observe that the ample line bundle Hη on the generic fiber Sη remains general with respect to v (cf. [16, Theorem 4.19 (1)]), so the geometric generic fiber of (20) is a smooth projective symplectic variety. Now one can conclude by the general fact that the ´ specialization morphism for ´etale fundamental group is surjective [1, Expos´ e X, Corollaire 2.3]. In part (ii) and (iii), [20] only deals with the ´etale cohomology groups, but  the same argument holds for crystalline cohomology groups as well. Remark 4.5. One can also consider the moduli space of semistable purely onedimensional sheaves and a similar result still holds. In particular, if v = (0, c1 , s) satisfying v, v = 0 and gcd(H · c1 , s) = 1, then every H-semistable coherent sheaf is H-stable and MH (S, v) is a (non-empty) smooth K3 surface (cf. [16, Proposition 4.20]). % is of height 1), Remark 4.6. When the K3 surface S is ordinary (i.e., Br(S) MH (S, v) is indeed irreducible symplectic, i.e., h2,0 (MH (S, v)) = 1, thanks to Charles [20, Proposition 2.6]. His method of computing h2,0 can not be applied to K3 surfaces whose formal Bauer group is of height > 1, since in general we can only lift (S, H, v) to a finite cover of W .

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4.3. Moduli spaces of sheaves on supersingular K3 surfaces Now we specialize the previous discussion to supersingular K3 surfaces. Our first goal is to show in Lemma 4.8 that in the supersingular case, the condition (16) can always be achieved, up to twisting by line bundles, except an extreme case. Let us first recall two standard auto-equivalences of the derived category Db (S) of bounded complexes of coherent sheaves, as well as the corresponding cohomological transforms (cf. [40]). For simplicity, we will use the same notation for a line bundle and its first Chern class. 1. Let L be a line bundle, then tensoring with L gives an equivalence 

− ⊗ L : Db (S) − → Db (S). The corresponding map on cohomology   expL : H(S) → H(S) 2

sends a Mukai vector v = (r, c1 , s) to eL · v = (r, c1 + rL, s + rL2 + c1 · L). 2. Let E ∈ Db (S) be a spherical object, that is, dim Ext∗ (E, E) = dim H ∗ (S2 , Q), where S2 is the two-dimensional sphere, whence the name. There is the spherical twist with respect to E (see [85], [40, § 8.1]) 

TE : Db (S) − → Db (S), and we may still use TE to denote the corresponding cohomological transform on the algebraic Mukai lattice. In particular, when E = OC (−1) with C ∼ = P1 a (−2)-curve on S, we have TE (r, c1 , s) = (r, s[C] (c1 ), s) where s[C] is the Mukai reflection with respect to the (−2)-class [C] ∈ NS(S). Definition 4.7. Let Λ be a lattice. Given an element x ∈ Λ and m ∈ Z>0 , we say that x · Λ is divisible by m and denote it by m | x · Λ if m | (x, y) for all y ∈ Λ. We denote by m  x · Λ if x · Λ is not divisible by m. When Λ = NS(S), a Mukai  vector v = (r, c1 , s) ∈ H(S) is called coprime to p, if p  r or p  s or p  c1 · NS(S). The following observation says that we can always reduce to the case where the numerical condition (16) holds, if the K3 surface is supersingular.  Lemma 4.8. Let S be a supersingular K3 surface. If v ∈ H(S) is a Mukai vector coprime to p, then up to a derived equivalence of tensoring with a line bundle, the numerical condition (16) and hence the conclusions in Proposition 4.4 holds for any ample line bundle H. Proof. As tensoring a sheaf with a line bundle L induces an isomorphism of moduli spaces MH (S, v) ∼ = MH (S, expL (v)),

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it suffices to prove the existence of a line bundle L, such that expL (v) verifies the condition (16), i.e.,   rL2 + c1 · L = 1 gcd r, (c1 + rL) · H, s + (21) 2 We will make use of Proposition 4.1 on the structure of the lattice NS(S). Set q = gcd(r, c1 · H) and we have   rL2 + c1 · L = gcd(q, s + c1 · L). gcd r, (c1 + rL) · H, s + 2 Let q1 , . . . , qr be the distinct prime numbers dividing q. If for any i, we could manage to find a line bundle Li ∈ NS(S) such that gcd(qi , s + c1 · Li ) = 1, i.e., qi  s + c1 · Li , then the line bundle ⎛ ⎞ r  # ⎝ ai qj ⎠ · Li L := i=1

j =i

satisfies that qi  s + c1 · L  for any  i, or equivalently, gcd(q, s + c1 · L) = 1, where r ai ’s are integers such that i=1 ai j =i qj = 1. As a result, one can assume that q = gcd(r, c1 · H) is a prime number. Now, if q = p, then by the hypothesis that the Mukai vector v is coprime to p, we know that either p  s or there exists L0 ∈ NS(S) such that p  c1 · L0 . Then we can take L to be the trivial bundle or L0 respectively. If q = p and assume by contradiction that gcd(q, s + c1 · L) = q for all L ∈ NS(S). This means q | s and q | c1 · NS(S). Then, note that NS(S) is a p-primary lattice, the class c1 has to be divisible by q. This is invalid because the vector v = (r, c1 , s) is primitive by our assumption.  Remark 4.9. The exceptional case where the Mukai vector is not coprime to p possibly can only happen when p | 12 v, v. This is obvious when p > 2. When p = 2, this follows from the fact that NS(S) is an even lattice of type I in the ˇ sense of Rudakov–Safareviˇ c [82, § 2]. In other words, MH (S, v) is a smooth projective symplectic variety for any polarization H whenever MH (S, v) = ∅ and p  12 (dim MH (S, v) − 2). Next, we discuss the birational equivalences between moduli spaces MH (S, v) when varying the stability condition. The following result is based on Bayer– Macr`ı’s wall-crossing principle in characteristic 0 (see [7]). Theorem 4.10. Let S be a supersingular K3 surface over an algebraically closed field  k of characteristic p > 0. Let v1 , v2 ∈ H(S) be two Mukai vectors that are coprime  to p (Definition 4.7). Let H and H be two ample line bundles on S. Suppose that v1 and v2 differ by a cohomological transform induced by an auto-equivalence of Db (S) of the following form: 1. tensoring with a line bundle;

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2. spherical twist associated to a line bundle or to a sheaf of the form OC (−1) for some smooth rational curve C on S; Then MH (S, v1 ) is liftably birational to MH  (S, v2 ) in the sense of Definition 3.15. Proof. This is indeed a special case of the wall-crossing principle for the moduli space of stable complexes on K3 surfaces, which is proved in characteristic zero. Over positive characteristic fields, the wall-crossing principle for K3 surfaces is not completely known. But we can easily prove the assertion by lifting to characteristic zero. Here we briefly sketch the proof: using the argument in Proposition 4.4, we can find a projective lift (S, H, H , v1 , v2 ) of the tuple (S, H, H  , v1 , v2 ) over a discrete valuation ring W  of characteristic zero whose residue field κ contains k. This gives rise to projective liftings MH (S, v1 ) → Spec W 

and

MH (S, v2 ) → Spec W 

(22)

of MH (Sκ , v1 ) and MH  (Sκ , v2 ) respectively as the moduli space of relative stable sheaves on S over W  with the given Mukai vectors. Let K  be the fraction field of W  and denote by SK  the generic fiber of S → Spec W  . In our case, we can assume that the auto-equivalences of type (1) or (2) can be lifted to W  by lifting additional line bundles. By [7, Corollary 1.3], after possibly taking a finite extension of K  , there is a birational map MH (SK  , v1 )  MH (SK  , v2 ) between the generic fibers of (22) defined over K  . This completes the proof.

(23) 

Remark 4.11. 1. The result [7, Corollary 1.3] is originally stated over complex numbers, but it is valid for any algebraically closed field of characteristic 0. 2. In general, one may consider the moduli space of Bridgeland stable complexes on K3 surfaces. According to the wall-crossing principle, they are expected to be birationally equivalent to the moduli space of stable sheaves considered above. See [63] for more details. 4.4. Relating moduli spaces to Hilbert schemes The goal of this subsection is to establish our main result Theorem 1.3 (ii). The basic strategy is to exploit the elliptic fibration structure on supersingular K3 surfaces and to use the following result of Bridgeland. Theorem 4.12 (Bridgeland [18]). Let π : X → C be a smooth relatively minimal elliptic surface over an algebraically closed field K. We denote by f ∈ NS(X) the  satisfying r > 0 and gcd(r, c1 · f ) = 1, fiber class of π. Given v = (r, c1 , s) ∈ H(X) then there exists an ample line bundle H and a birational morphism MH (X, v)  Pic0 (Y ) × Y [n] where Y is an smooth elliptic surface.

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Moreover, if X is a K3 surface or an abelian surface, so is Y . The map (24) can be chosen liftable (Definition 3.15) when char(K) > 0. In this case, the assertions still hold even when π is only a quasi-elliptic fibration. Proof. This theorem is due to Bridgeland [18] over the field of complex numbers and his proof can be easily generalized to any algebraically closed field of characteristic 0. See also [99, Appendix] for a proof over arbitrary characteristic for elliptic surfaces without non-reduced fibers. We sketch here a simplified proof when X is a K3 surface (resp. an abelian surface), by using the wall-crossing results in [7] and [63], and we refer the reader to [18, 99] for more details. As gcd(r, c1 · f ) = 1, we can set (a, b) to be the unique pair of integers satisfying br − a(c1 · f ) = 1

(25)

and 0 < a < r. As in [18], there exists an ample line bundle H ∈ Pic(X) such that a torsion-free sheaf F with Mukai vector v is stable if and only if its restriction to the general fiber of π is stable. Let Y := MH  (X, (0, af, b)) be the fine moduli space of stable sheaves of pure dimension 1 on X with Mukai vector (0, af, b), where H  is a generic ample line bundle defined over K. Then Y is a K3 surface (resp. an abelian surface) as well. Bridgeland’s work [18, Theorem 5.3] essentially shows that the universal Poincar´e sheaf on X × Y induced an equivalence Φ from Db (Y ) to Db (X) with Φ∗ (1, 0, n) = v. This enables us to conclude the assertion by using [7, Corollary 1.3]. When char(K) > 0, assume first X is a K3 surface with a (quasi-)elliptic fibration. By the lifting result Proposition 4.2, there are a complete discrete valuation ring W  of characteristic zero with residue field κ containing K, a K3 surface X over W  with special fiber Xκ such that the geometric generic fiber has an elliptic fibration. Hence, up to a finite extension of W  , there is a birational map Pic0 (Y) × Y [n]

MH (X , v)

(26) Spec W  where Y = MH (X , (0, af, b)). The special fiber Yκ of Y → Spec W  is exactly MH  (Xκ , (0, af, b)), which is a smooth K3 surface. Then by taking their reduc[n] tion, one can get MH (Xκ , v) and Pic0 (Yκ ) × Yκ are birationally equivalent. This proves the assertion. For abelian surfaces, as the lifting method is still valid (see Proposition 6.4), the same argument allows us to conclude.  As mentioned before, we need a more refined analysis of (quasi-)elliptic fibrations on supersingular K3 surfaces. The second key result is the following.

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Theorem 4.13. Let S be a supersingular K3 surface defined over an algebraic closed  field k with characteristic p > 0. Let v = (r, c1 , s) ∈ H(S) be a Mukai vector that is coprime to p. Then up to changing the Mukai vector v via the auto-equivalences listed in Theorem 4.10, there exists a (quasi-)elliptic fibration π : S → P1 such that gcd(r, c1 · E) = 1, where E ∈ NS(S) is the fiber class of π. Proof. We carry out the proof in several steps. Step 0. We claim that it suffices to show that there exists a primitive element x ∈ NS(S) such that (27) x2 = 0 and gcd(r, c1 · x) = 1. This is because the solution x in (27) gives an effective divisor E on S with E 2 = 0 and c1 (E) = ±x. If the linear system |E| is base point free, then |E| defines a (quasi-)elliptic fibration S → P1 as desired. If E is not base point free, one can find a base point free divisor E  with (E  )2 = 0 obtained by taking finitely many Mukai reflections of E with respect to a sequence of (−2) curves C1 , C2 , . . . , Cn on S, i.e., E  = s[C1 ] ◦ s[C2 ] · · · ◦ s[Cn ] (E). Then we set v  = TOC1 (−1) ◦ TOC2 (−1) · · · TOCn (−1) (v). The Mukai vector v  and the (quasi-)elliptic fibration S → P1 induced by |E  | thus satisfy the desired condition. Step 1. We reduce to the case c1 is primitive. First, if the vector (r, c1 ) ∈ Z⊕NS(S) is primitive and c1 is not primitive, then c1 + rL will be primitive for some L ∈ NS(S). This means that we can replace v by the Mukai vector v  = expL (v). Similarly, if (c1 , s) is primitive, one can get v  = TL (v) satisfying the condition for some line bundle L. Suppose now the vectors (r, c1 ) and (c1 , s) are both non-primitive, we can write r = q1 r , c1 = q1 q2 c1 , s = q2 s , (28) such that (r , q2 c1 ) and (q1 c1 , s ) are primitive, where r , qi , s ∈ Z are non-zero integers and c1 ∈ NS(S). Then what we need is the following: find L ∈ NS(S) and L2 = 0 such that the vector (c1 + rL, s +

rL2 + c1 · L) = (q1 (q2 c1 + r L), s + c1 · L) 2

(29)

is primitive because one can replace v by expL (v) and the same argument as above works. To see the existence of such an L, note that (q1 , s + c1 · L) = (q1 , q2 (s + q1 c1 · L)) is always primitive, so it suffices to find a square zero element L such that the class q2 c1 + r L is primitive. This can be easily achieved because (r , q2 c1 ) ∈ Z ⊕ NS(S) is primitive by our assumption and the natural basis of NS(S) contains square zero element (See Proposition 4.1).

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Step 2. Let us first assume that p  r. Suppose r is a prime number. Recall that NS(S) is isomorphic to the lattice −Λσ(S) given in Proposition 4.1, we may have two cases as below: (1) If Λσ(S) contains a hyperbolic lattice U , we denote by {f1 , f2 } the natural basis of U satisfying f12 = f22 = 0 and f1 · f2 = 1. We could find a basis of NS(S) of the form the f1 , f2 , v −

v, v f1 + f2 v 2

(30)

for some v ∈ U ⊥ . Note that every element in (30) is square zero, so if there is no solution of (27), we must have r | c1 · NS(S). However, as c1 is primitive and the lattice NS(S) is p-primary, it forces r = p which contradicts to our assumption. (2) If Λσ(S) contains U (p) with basis {f1 , f2 } satisfying fi2 = 0, f1 · f2 = p, the primitive square zero elements p v, v f1 + f2 2 can also form a basis of the p-primary sublattice f1 , f2 , pv −

(31)

U (p) ⊕ p(U (p)⊥ ) ⊆ NS(S). Then the same argument shows that there must exists x ∈ U (p) ⊕ p((U (p)⊥ ) satisfying (27). Combining (1) and (2), we prove the case when r is a single prime. In general, for any positive integer r, let Ξ ⊆ Z be the collection of all prime factors of r. Let X be the projective quadric hypersurface over Z defined by x2 = 0, which is geometrically integral. We can view the desired element x ∈ NS(S) as a rational point in X(Z). By weak approximation on quadrics (cf. [80, § 7.1 Corollary 1]), the diagonal map # X(Z) → X(Zq ) (32) q∈Ξ

is dense if X(Z) = ∅. Set Uq = {x ∈ X(Zq )| x · c1 ∈ Z× q } ⊆ X(Zq ). and we know that Uq is an open subset of X(Zq ) if Uq = ∅. From the discussion above, Uq is non-empty for each q ∈ Ξ and hence it is anopen dense subset. Thus, there exists x ∈ X(Z) whose image via (32) lies in Uq . Then we have q∈Ξ

gcd(x · c1 , q) = 1 for all q ∈ Ξ from the construction, which proves the assertion. If p  s, we can replace v by TOS (v). Then the same argument applies to TOS (v). Step 3. Finally, if p | gcd(r, s), then we have p  c1 · NS(S) by (∗). As in Step 2, we only need to show U (Zq ) is not empty for all q ∈ Σ. When q = p, it is easy to see U (Zq ) = ∅ via the same analysis.

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When q = p, note that p  c1 · NS(S), the argument in Step 2 actually shows that there exists an element x in (30) or (31) such that p  c1 · x. This implies that Up is non-empty and hence proves the assertion.  4.5. Proof of Theorem 1.3 The assertion (i) follows from Proposition 4.4 (i) and (iii). To prove (ii), assume that S is supersingular. By Theorem 4.10 and Theo rem 4.13, there exists v  = (r , c1 (L), s ) ∈ H(S) for some r , s ∈ Z and L ∈ Pic(S), 1 and a (quasi-)elliptic fibration π : S → P such that X = MH (S, v) is quasi-liftably birational to MH (S, v  ) and gcd(r , c1 (L) · E) = 1, where E is the fiber class of π. Then one can invoke Bridgeland’s result Theorem 4.12 to see that X is quasiliftably birational to the nth Hilbert scheme of some K3 surface Y that is derived equivalent to S. As supersingular K3 surfaces have no Fourier–Mukai partners (cf. [53, Theorem 1.1]), actually Y  S. Therefore X is quasi-liftably birational to the Hilbert scheme S [n] . Since S [n] is irreducible symplectic (Beauville’s proof [11, § 6] goes through in positive characteristics), and unirational if S is, the same properties hold for X. For (iii), thanks to Proposition 3.14, two quasi-liftably birational symplectic varieties have isomorphic Chow motives, hence it suffices to show that the Chow motive of the Hilbert scheme S [n] is of Tate type for a supersingular K3 surface S. To this end, we invoke the following result of de Cataldo–Migliorini [21] on the motivic decomposition of S [n] : in the category of rational Chow motives,

Sλ

 (|λ|), h S |λ| h S [n] (n)  λn

where h is the Chow motive functor, the direct sum is indexed by all partitions of n and for a given partition λ = (1a1 2a2 · · · nan ), its length |λ| := i ai and Sλ := Sa1 × · · · × San . As a result, we only need to show that the Chow motive of S is of Tate type. This was proved for p ≥ 5 by Fakhruddin [27]. In general, if S is unirational, 2 → S, where P 2 is a successive blow-up there exists a surjective morphism P 2  2 of P at points. Therefore h(P ) is of Tate type by the blow-up formula. Hence 2 ), must be of Tate type, too. Since the h(S), being a direct summand of h(P unirationality of supersingular K3 surfaces is only know for Kummer surfaces and remains open in general, we use the result proved by Bragg and Lieblich instead. By [16, Proposition 5.15], every supersingular K3 surfaces S with Artin invariant σ(S) is derived equivalent to a twisted K3 surface S  with Artin invariant σ(S  ) < σ(S) (see also § 5). Thanks to [42, Theorem 2.1] (extended in [30, Theorem 1]), the Chow motives of S and S  are isomorphic. Now by induction on the Artin invariant, one can show that the Chow motive of S is isomorphic to the Chow motive of a supersingular K3 surface of Artin invariant 1, which is a Kummer surface. The Chow motive of a supersingular Kummer surface is known to be of Tate type, since it is unirational by Shioda [88] (or one simply uses Corollary 2.12).

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5. Moduli spaces of twisted sheaves on K3 surfaces In this section, we extend our results in § 4 to the moduli spaces of twisted sheaves on K3 surfaces. 5.1. Twisted sheaves on K3 surfaces We mainly follow [54] and [16] to review the basic facts of twisted sheaves on K3 surfaces. Let S be a K3 surface over k and let S → S be a μm -gerbe over S. This corresponds to a pair (S, α) for some α ∈ Hf2l (S, μm ), where the cohomology group is with respect to the flat topology. For any integer m, there is a Kummer exact sequence x→xm 1 → μm → Gm −−−−→ Gm → 1 in flat topology and it induces a surjective map Hf2l (S, μm ) → Br(S)[m].

(33)

Definition 5.1 (B-fields). For a prime  = p, an -adic B-field on S is an element B ∈ H´e2t (S, Q (1)). It can be written as α/n for some α ∈ H´e2t (S, Z (1)) and n ∈ Z. We associate to it a Brauer class [Bα ], defined as the image of α under the following composition of natural maps H´e2t (S, Z (1)) → H 2 (S, μ n ) → Br(S)[n ], if   p. 2 2 A crystalline B-field is an element B = pαn ∈ Hcris (S/W )⊗K with α ∈ Hcris (S/W ), 2 so that the projection of α in Hcris (S/Wn (k)) lies in the image of the map d log

2 (S/Wn (k)). Hf2l (S, μpn ) −−−→ Hcris

(34)

See [43, I.3.2, II.5.1] for the details of the map (34). Then we can associate a pn -torsion Brauer class [Bα ] via the map (33). Let us write B = αr as either a -adic or crystalline B-field of S → S, we define the twisted Mukai lattice as $  ⊗ Z ); if p  m ea/r (H(S)  H(S ) = (35) a/r  e (Hcris (S/W )); if m = pn under the Mukai pairing (15). Definition 5.2. An S -twisted sheaf F on S is an OS -module compatible with the μm -gerbe structure (cf. [52, Def 2.1.2.4]). With the notation as above, the Mukai vector of F is defined as " v α/r (F) = ea/r chS (F) tdS ∈ CH∗ (S, Q). 2

where ea/r = (1, a/r, (a/r) 2 ) and chS (F ) is the twisted Chern character of F (cf. [54, 3.3.4]). It can be also viewed as an element in the twisted Mukai lattice  H(S ) via the corresponding cycle class map. Similarly as in the case of untwisted sheaves, we say that H is general with respect to v if every H-semistable twisted sheaf is H-stable.

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5.2. Moduli spaces of twisted sheaves Definition 5.3 (cf. [52]). Fix a polarization H on a K3 surface S, the moduli stack MH (S , v) of S -twisted sheaves with Mukai vector v is the stack whose objects over a k-scheme T are pairs (F , φ), where F is a T -flat quasi-coherent twisted sheaf of finite presentation and φ : det F → O(D) is an isomorphism of invertible sheaves on X, such that for every geometric point t → T , the fiber sheaf Ft has Mukai vector v and endomorphism ring k(t). Theorem 5.4. Assume that char(k) = p > 0 and v = (r, c1 , s) satisfies v, v ≥ 0 and r > 0. If H is general with respect to v, the moduli stack MH (S , v) has a smooth and projective coarse moduli space MH (S , v) of dimension v,v + 1 if 2 non-empty. The coarse moduli space MH (S , v) is a symplectic variety and there exists a canonical quadratic forms on the N´eron–Severi group NS(MH (S , v)) such that there is an injective isometry  (v ⊥ ) ∩ H(S ) → NS(MH (S , v)) ⊗ R.

(36)

where R = Z or W depending on S as in (35). Moreover, there is an isomorphism 2 (MH (S , v)/K) v ⊥ ⊗ K → Hcris

(37)

as F -isocrystals. Here we regard v as an element in H ∗ (S/K) and v ⊥ ⊗ K is the orthogonal complement with respect to the Mukai pairing. Proof. Everything is known in characteristic 0 by Yoshioka [98, Theorem 3.16; Theorem 3.19]. Similar to the untwisted case, as H is general, all the assertions can be proved by lifting to characteristic 0 (see [20, Theorem 2.4]). We also refer to [16] for the case where S is supersingular.  Let us consider the case where S is supersingular. By [5], we know that the Brauer group Br(S) is of p-torsion. In this case, there is an explicit description of twisted Mukai lattice in [16]. This enables us to give a sufficient condition for MH (S , v) to be smooth and projective. The following result is analogous to Lemma 4.8. Proposition 5.5. Suppose S is supersingular. If the Mukai vector v is coprime to p with r > 0 and v, v ≥ 0, the coarse moduli space MH (S , v) is a symplectic variety for any ample polarization H. Proof. The non-emptyness again follows from [98, Theorem 3.16] (see also [16, Proposition 4.20]). It suffices to show that H is general with respect to v. Since v is coprime to p, up to change c1 by twisting a line bundle, we know that H and v satisfy the numerical condition (16) by using the same argument in the proof of Lemma 4.8.  5.3. From twisted sheaves to untwisted ones We show all our conjectures in the introduction for most moduli spaces of twisted sheaves on K3 surfaces. The key result is Theorem 5.9, which shows that the

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moduli space MH (S , v) of twisted sheaves on S is quasi-liftably birational to some moduli space of untwisted sheaves. To start, we have Lemma 5.6. Assume that MH (S , v) is a symplectic variety. Then it is 2nd -Artin supersingular if and only if S is supersingular. Proof. Similar to the untwisted case, this follows from the isomorphism (37).



By Lemma 5.6, we are reduced to consider the moduli space of (semi)-stable twisted sheaves on supersingular K3 surfaces. Let D(1) (S ) be the bounded derived category of twisted sheaves on S . If the gerbe S → S is trivial, then D(1) (S ) = Db (S). The following result shows that every twisted supersingular K3 surface is derived equivalent to a untwisted supersingular K3 surface, provided that the Artin invariant is less than 10. Theorem 5.7 (Untwisting). If σ(S) < 10, there is a Fourier–Mukai equivalence from D(1) (S ) to Db (S  ) for some supersingular K3 surface S  .   ) and w ∈ H(S Proof. First, we claim that there exists primitive vectors τ ∈ H(S ) satisfying the condition τ 2 = 0 and p  τ · w. (38) Assuming this, we consider the moduli space MH (S , τ ) for an ample line bundle H. By [16, Theorem 4.19], MH (S , τ ) is a Gm -gerbe over a supersingular K3 surface S  = MH (S , τ ) if non-empty. The universal sheaf induces a Fourier-Mukai equivalence Φ : D(1) (S ) → D(−1) (MH (S , τ )). So it suffices to show that the gerbe MH (S , τ ) → S  is trivial. By [16, Theorem 4.19 (3)], this is equivalent to find  ) such that τ · w is coprime to p. Thus we can conclude our a vector w ∈ H(S assertion. To prove the claim, write α = α0 + L for some L ∈ NS(S) and [α0 ] the image of α0 in Br(S). There are two possibilities: (i) If L · NS(S) is not divisible by p, then we can find a primitive element E ∈ NS(S) such that E 2 = 0 and p  L · E. Then we take  1  τ = (0, [E], s) and w = p, L, L2 2p . It follows that τ · w = L · E − sp is coprime to p. (ii) If p | L · NS(S), then we can take τ = (p, c1 , s) with c21 = 2ps and choose w = (0, D, D·L p ) for some D with p  c1 · D. It follows that τ · w = c1 · D − D · L is coprime to p. The existence of c1 and D can be also deduced from Proposition 4.1. When p > 2, as σ(S) < 10 and NS(S) contains a hyperbolic lattice U , we can let fi , i = 1, 2 be the standard basis of U and take c1 = f1 + pf2 and D = f2 as desired. When p = 2, this is more complicated. The lattice NS(S) contains U only when σ(S) is odd. If σ(S) is even, NS(S) contains U (2) as a direct summand

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instead. We claim that there exist an element y ∈ (U (2))⊥ such that y 2 = −4 and 2  y · NS(S). Assuming this, we can pick x ∈ U (2) with x2 = 4 and c1 ∈ NS(S) with 2  c1 · y, one can easily see that the classes D = x − y and c1 are as desired. Then one can use the explicit description of NS(S) to prove the claim. For instance, ˇ c [82] showed that Λσ(S) is isomorphic to if σ(S) = 8, Rudakov and Safareviˇ U (2) ⊕ D4⊕3 ⊕ E8 (2), where D4 and E8 are root lattices defined by the corresponding Dynkin diagram. There certainly exists some element y  ∈ D4 with (y  )2 = 2 and 2  y  · D4 . Then we can select y = (y  , y  ) ∈ D4⊕2 with y 2 = 4 and 2  y · D4⊕2 which automatically satisfies the conditions. Similar analysis holds when σ(S) = 2, 4 and 6. Here we omit the details and left it to the readers.  Remark 5.8. Theorem 5.7 can be viewed as a converse of [16, Proposition 5.15], which says that every supersingular K3 surface is derived equivalent to a supersingular twisted K3 surface with Artin invariant less than 10. Theorem 5.9. Let v be a Mukai vector which is coprime to p. If the Artin invariant σ(S) < 10, then MH (S , v) is quasi-liftably birational to MH  (S  , v  ) for some   ) a Mukai vector which is coprime to p and supersingular K3 surface S  , v  ∈ H(S   H ∈ Pic(S ) an ample line bundle.  ) with Proof. By assumption, as in the proof of Theorem 5.7, we can find τ ∈ H(S 2 τ = 0 such that there is a Fourier-Mukai equivalence Φ : D(1) (S ) → Db (S  ), where S  = MH (S , τ ). As before, we prove the birational equivalence from the wall-crossing principle. As this remain unknown over positive characteristic fields, we proceed the proof by lifting to characteristic 0. As in Theorem 5.4 (See also [16, Theorem 4.19]), we take a projective lift  W ), SW → SW , H ∈ Pic(S), τW ∈ H(S

(39)

of the triple (SW → S, H, τ ) over W for some discrete valuation ring W . Consider the relative moduli space MH (S , τW ) → Spec(W ),

(40)

whose special fiber is the (untwisted) supersingular K3 surface MH (S, τ ) shown in Theorem 5.7. By [16, Theorem 4.19], we have a Fourier-Mukai equivalence Φ : D(1) (Sη ) → D(1) (MH (Sη , τη )),

(41)

for the generic fiber of (39) and (40). Due to the wall-crossing theorem, there is a birational map between the coarse moduli spaces MH (Sη , v)  MH (Sη , v  ),

(42) 

after possibly taking a finite extension of W , where v = Φ∗ (v). Then we can conclude the assertion by taking the reduction.

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Finally, we have to check that the Mukai vector v  is coprime to p. This is clear as the Fourier–Mukai transform does not change the divisibility of Mukai vectors.  Corollary 5.10. If the Artin invariant σ(S) < 10, then the moduli space MH (S , v) is 2nd -Artin supersingular and all the same statements in Theorem 1.3 hold for MH (S , v) as well. Proof. As quasi-liftably birational symplectic varieties have isomorphic Chow motives by Proposition 3.14, Theorem 5.9 allows us to reduce to the untwisted case, namely, Theorem 1.3. 

6. Moduli spaces of sheaves on abelian surfaces 6.1. Preliminaries on abelian surfaces in positive characteristics Let A be an abelian surface defined over an algebraically closed field k of positive characteristic p. For n ∈ Z, consider the multiplication-by-n map nA : A → A, which is separable if and only if p  n. When p | n, the inseparable morphism factors through the absolute Frobenius map F : A → A. Denote by A[n] the kernel of nA . The Newton polygon of H 1 (A) can be computed via the k-rational points of A[p] and in particular, ∼ (Z/pZ)⊕2 ; • A is ordinary if A[p](k) = • A is supersingular if A[p](k) = {0}. Note that the above characterizations no longer hold for higher-dimensional abelian varieties. When A is a supersingular abelian surface, the N´eron–Severi group NS(A) equipped with the intersection form, is an even lattice of rank 6 with discriminant equal to p2 or p4 (one could say that the Artin invariant is 1 or 2). As before, we have the following description of the NS(A), in a parallel way to Proposition 4.1. Proposition 6.1 ([82, § 2], [86]). Let A be a supersingular abelian surface defined over an algebraically closed field of positive characteristic p. If disc(NS(A)) = p2 , then the lattice NS(A) is isomorphic to −Λ1 , with ⎧ ⎪U ⊕ D4 if p = 2, ⎨ (p) (43) Λ1 = U ⊕ V4,2 , if p ≡ 3 mod 4, ⎪ ⎩ (p) U ⊕ H , if p ≡ 1 mod 4, If disc(NS(A)) = p4 , then the lattice NS(A) is isomorphic to −Λ2 , with $ if p = 2, U (p) ⊕ D4 Λ2 = (p) U (p) ⊕ H , otherwise. (p)

Here, U, V4,2 and H (p) are the lattices defined in Proposition 4.1.

(44)

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We have the following simple observation. Lemma 6.2. Any supersingular abelian surface admits a principal polarization. Proof. Let A be a supersingular abelian surface. By Proposition 6.1, we know that the N´eron–Severi lattice NS(A) has two possibilities −Λ1 or −Λ2 given by (43) and (44). In any case, we have a line bundle L with (L2 ) = 2. Replacing L by its 2 inverse if necessary, we see that L is ample. Since h0 (A, L) = χ(A, L) = (L2 ) = 1, it is a principal polarization.  Similarly to § 4.3, we need the following two types of auto-equivalences of the derived category Db (A). Again, the same notation is used for a line bundle and its first Chern class. 1. For any line bundle L on A, tensoring with L is an auto-equivalence 

− ⊗ L : Db (A) − → Db (A). The corresponding cohomological transform   → H(A) expL : H(A) 2

sends a Mukai vector v = (r, c1 , s) to eL · v = (r, c1 + rL, s + rL2 + c1 · L). 2. Choose any principal polarization of A (Lemma 6.2), which allows us to  Let P be the Poincar´e line bundle on A × A. Mukai’s identify A and A. Fourier transform [65] gives an auto-equivalence: Φ : Db (A)

− →



Db (A)

E

→

Rp2,∗ (p∗1 (E) ⊗ P),

where p1 , p2 are natural projections from A × A to the factors. The cohomological Fourier transform was computed by Beauville [10, Proposition 1]: it sends a Mukai vector (r, c1 , s) to (s, −c1 , r). As an analogue of Theorem 4.13 in the case of supersingular abelian surfaces, the following existence theorem of elliptic fibrations is a crucial step in the proof of Theorem 1.4.  Theorem 6.3. Given a supersingular abelian surface A, we denote by H(A) the  (algebraic) Mukai lattice. Let v = (r, c1 , s) ∈ H(A) be a Mukai vector that is coprime to p. Then up to changing the Mukai vector v via the two types of autoequivalences recalled above, there exists an elliptic fibration π : A → E such that gcd(r, c1 · f ) = 1 where f ∈ NS(A) is the fiber class of π and E is an elliptic curve. Proof. The existence of elliptic fibration is equivalent to the existence of a square zero element in NS(A) satisfying the coprime condition (note that there is no (−2) curve on an abelian surface). The argument is similar to that of Theorem 4.13 with the following changes: • replace NS(S) by NS(A); • skip Step 0;

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• in Step 2 of the proof of Theorem 4.13, we used spherical twists to “changing the roles” of r and s, more precisely, to switch between the Mukai vectors (r, c1 , s) and (s, −c1 , r). Here for abelian surfaces, the structural sheaf is no longer spherical; instead, we use Mukai’s Fourier transform Φ recalled above, which has the same effect on Mukai vectors.  Next, we need to slightly generalize the lifting results of Mumford, Norman and Oort (cf. [69]) for the liftability of supersingular abelian surfaces together with line bundles. The following result is analogous to Proposition 4.2. Proposition 6.4. Let A be an abelian surface over a perfect field k of characteristic p > 0. Suppose L1 , L2 are two line bundles on A with L1 a separable polarization. Then there exists a complete discrete valuation ring W  of characteristic zero which is finite over W (k), and an abelian scheme that is a projective lift of A A → Spec(W  ),

(45)

such that rank NS(Aη ) = 2 and the image of the specialization map NS(Aη ) → NS(A)

(46)

contains L1 , L2 , where Aη is the generic fiber of A over W  . In particular, every supersingular abelian surface A admits a projective lift over W such that A is isogenous to the product of elliptic curves. Proof. Similar to the proof of Proposition 4.2, we first consider the case that A is not supersingular. As in [53], let Def(A; L1 , L2 ) be the deformation functor parametrizing deformations of A together with L1 and L2 . For simplicity, we can assume that L2 is a separable polarization. By deformation theory of abelian varieties, the formal deformation space Def(A) of A over W is isomorphic to Spf(W [[t1 , t2 , t3 , t4 ]]). As inspired by Grothendieck and Mumford, each Li imposes one equation fi on W [[t1 , t2 , t3 , t4 ]] (cf. [76, § 2.3–2.4]) and the forgetful functor Def(A; L1 , L2 ) → Def(A) can be identified as the quotient map W [[t1 , t2 , t3 , t4 ]] → W [[t1 , t2 , t3 , t4 ]]/(f1 , f2 ). Then it is easy to see that Def(A; L1 , L2 ) is formally smooth over W (cf. [76, 2.4.1] and [53, Proposition 4.1]). Since W is Henselian, the k-valued point (A; L1 , L2 ) extends to a W -valued point, giving a formal lifting. Moreover, the formal family is formally projective. By the Grothendieck Existence Theorem, this lifting is therefore algebraizable as a projective scheme, as desired. If A is supersingular, as in [55, Lemma A.4], it suffices to show each triple (A, L1 , L2 ) is the specialization of an object (A , L1 , L2 ) of finite height along a local ring. In other words, we can deform an abelian surface in equal-characteristic to a non-supersingular abelian surface. Consider the formal universal deformation space ΔA = Spec k[[x1 , . . . , x4 ]]

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of A over k. Similarly as above, each line bundle Li determines a divisor in ΔA . The complete local ring of ΔA at (X, L1 , L2 ) is given by two equations f1 , f2 . Moreover, since L1 is a separable polarization, the universal deformation (A, L1 , L2 ) of the triple (A, L1 , L2 ) is Spf k[[t1 , . . . , t4 ]]/(f1 , f2 ) and it is algebraizable. Now, it is known that the supersingular locus in the deformations of A has dimension 1 (cf. [51]). But the deformation space T := Spec k[[t1 , . . . , t4 ]]/(f1 , f2 ) has dimension at least 2 and hence can not lie entirely in the supersingular locus. The generic point of T parametrizes a triple (A , L1 , L2 ) with A of finite height.  This proves the assertion. 6.2. Generalized Kummer varieties Let A be an abelian surface over k and s : A[n+1] → A be the morphism induced by the additive structure on A, which is an isotrivial fibration. By definition, the generalized Kummer variety (see [11]) is its fiber over the origin: Kn (A) := s−1 (OA ), which is an integral variety of dimension 2n with trivial dualizing sheaf. It will be a symplectic variety if it is smooth. We shall remark that different from the case of characteristic zero, the generalized Kummer varieties over positive characteristic fields can be singular and even non-normal (see [84] for some examples in characteristic 2). In [84], Schr¨oer raised the question when Kn (A) is smooth. Here we partially answer this question. Proposition 6.5. Kn (A) is a smooth symplectic variety if p  (n + 1). Proof. We first show the smoothness. After the base change (n + 1)A : A → A, we obtain a trivialization of s : A[n+1] → A, i.e., there is a cartesian commutative diagram A × Kn (A)  A

ψn

/ A[n+1]

(47)

s

×(n+1)

 /A

Since p  (n + 1), the map ψn is ´etale. Hence the smoothness of A[n+1] implies that Kn (A) is smooth as well. To show that Kn (A) is a symplectic variety, we first use the lifting argument as in Proposition 4.4: by lifting A to an abelian scheme A over a base of characteristic zero W , we obtain that the relative generalized Kummer variety Kn (A) is a lifting of Kn (A) over W . The simple connectedness of Kn (A) is obtained from the simple connectedness of the generic fiber of Kn (A) (a result known in characteristic zero) together with the surjectivity of the specialization map of ´etale fundamental groups [1, Expos´e X, Corollaire 2.3].

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Finally, let us construct a symplectic form on Kn (A). Fix a symplectic/canonical form ω on A. We perform the construction of Beauville [11, Proposition 7] to get a symplectic form on A[n+1] , which restricts to an algebraic 2-form ωn on Kn (A). We want to show that if p  (n+1), then ωn is nowhere degenerate, or equivalently, ωn∧n is nowhere vanishing. Since the canonical bundle of Kn (A) is trivial, it suffices to show that ωn∧n is non-zero at some point. At a point ξn ∈ Kn (A) represented by an (n + 1)-tuple of distinct points {x0 , . . . , xn } with i=0 xi = 0 ∈ A,

+

→ T0 A the tangent space Tξ Kn (A) is canonically identified with ker T0 A⊕n+1 −

and the 2-form ωn at ξ is the restriction of the symplectic form (ω, . . . , ω) from T0 A⊕n+1 . The elementary lemma below shows that ωn is non-degenerate at ξ,  when p  (n + 1). Lemma 6.6. Let (V, ω) be a symplectic vector space over a field of characteristic form on V ⊕m . Then the symplectic p and m ∈ N. Let (ω, . . . , ω) be the induced

+

→V restriction of (ω, . . . , ω) to V0m := ker V ⊕m − if p  m.

is non-degenerate if and only

Proof. If p | m, then for any non-zero vector v ∈ V , we have (v, . . . , v) ∈ V0m is in the kernel of the restricted 2-form: for any (v1 , . . . , vm ) ∈ V0m + m , m   (v, . . . , v), (v1 , . . . , vm ) = vi = 0. v, vi  = v, i=1

i=1 +

If p  m, then we have the following section of the surjection V ⊕m − →V, V

→

v

→

V ⊕m   1 1 v, . . . , v , m m

which respects the symplectic forms and gives rise to an orthogonal direct sum decomposition V ⊕m = V ⊕ V0m . In particular, V0m equipped with the restricted 2-form is a symplectic space.  We have the following result concerning the motive of generalized Kummer varieties. Proposition 6.7. Let Kn (A) be a smooth generalized Kummer variety associated to an abelian surface A. Then Kn (A) is 2nd -Artin supersingular if and only if A is supersingular. Moreover, the supersingular abelian motive conjecture (see Conjecture A) and the supersingular Bloch–Beilinson–Beauville conjecture (see Conjecture B) hold for Kn (A). Proof. Since the standard conjecture is known for generalized Kummer varieties, the notion of 2nd -Shioda supersingularity is independent of the cohomology theory used. Let us write X = Kn (A). For the first assertion, since H 2 (X) ∼ = H 2 (A) ⊕ 2 W (−1), the supersingularity of the F -crystal H (X) is equivalent to the supersingularity of the crystal H 2 (A), which is equivalent to the supersingularity of A.

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To prove the supersingular abelian motive conjecture, we use the following motivic decomposition of Kn (A) (see [22], [29, Theorem 7.9] and [95]):  h(Aλ0 )(|λ|), h(X)(n) ∼ = λ(n+1)

where λ runs through all partitions of n+1; for a partition λ = (λ1 , . . . , λl ), |λ| := l denotes its length and    A0λ := (x1 , . . . , xl ) ∈ Al | λi xi = OA . i

A0λ

is isomorphic, as algebraic varieties, to the disjoint union of Observe that gcd(λ)4 copies of the abelian variety Al−1 , where gcd(λ) is the greatest common divisor of λ1 , . . . , λl . As a result, the motive of X is a direct sum of the motives of some powers of A with Tate twists, precisely:  h(X)(n) ∼ h(Ali −1 )(li ). = i

Since A is supersingular, h(X) is by definition a supersingular abelian motive. The fully supersingular Bloch–Beilinson conjecture for X follows from Corollary 2.12. To establish the supersingular Bloch–Beilinson–Beauville conjecture for X, it remains to show the section property conjecture for it. But this is done in Fu–Vial  [31, § 5.5.2], where the argument works equally in positive characteristics. Finally, we show the RCC Conjecture for the generalized Kummer varieties: Proposition 6.8. Let A be a supersingular abelian surface and n a natural number, then the generalized Kummer variety Kn (A) is rationally chain connected. Proof. The main input is Shioda’s theorem [88] that the Kummer K3 surface K1 (A) is unirational. Let us first consider the singular model of Kn (A), namely Kn (A) := A0n+1 /Sn+1 ,    where A0n+1 := (x0 , . . . , xn ) ∈ An+1 | equipped with the natural i xi = O Sn+1 -action by permutation. A typical element of Kn (A) is thus denoted by {x0 , . . . , xn }. Now we show that any two points of Kn (A) can be connected by a chain of unirational surfaces K1 (A) = A/−1. Indeed, for any {x0 , . . . , xn } ∈ Kn (A) (hence  i xi = 0), let us explain how it is connected to {O, . . . , O}. Firstly, the image of the map ϕ1 : A/−1 →



Kn (A)

→ {t, −t, O, . . . , O}

t

connects {O, O, O, . . . , O} and{x0 , −x0 , O, . . . , O}. Next, one can choose any u ∈ A such that 2u = x0 , then the surface A/−1 → t



Kn (A)

→ {x0 , t − u, −t − u, O, . . . , O}

Lie Fu and Zhiyuan Li

190 connects the two points:

{x0 , −x0 , O, . . . , O}

and {x0 , x1 , −x0 − x1 , O, . . . , O}

by taking t = −u and t = u + x1 respectively. Continuing this process n times, we connect {O, . . . , O} to {x0 , . . . , xn } in Kn (A) by the unirational surfaces K1 (A). In conclusion, Kn (A) is rationally chain connected. At last, note that we have a birational morphism Kn (A) → Kn (A)

(48)

as the crepant resolution. As the exceptional divisors of (48) are rationally chain connected (see also Remark 6.13), it follows that Kn (A) is rationally chain con nected as well. 6.3. Moduli spaces of stable sheaves on abelian surfaces Now we turn to the study of moduli spaces of sheaves on abelian surfaces and  their Albanese fibers in general. Given v ∈ H(A) and a general polarization H with respect to v, we denote by MH (A, v) the moduli space of stable sheaves on A with Mukai vector v. If we choose F0 ∈ MH (A, v), the Albanese morphism  × A, alb : MH (A, v) → Pic0 (A) × A = A F → (det(F ) ⊗ det(F0 )−1 , alb(c2 (F ) − c2 (F0 )), is an isotrivial fibration. The Albanese fiber is denoted by KH (v). If KH (v) is smooth, it is of dimension 2n := v, v − 2, and is deformation equivalent to the generalized Kummer variety Kn (A). The following result is similar to Proposition 4.4 for K3 surfaces. Proposition 6.9. Let A be an abelian surface, H a polarization and v = (r, c1 , s) ∈  H(A) a primitive element such that r > 0 and v, v ≥ 2. Denote 2n := v, v − 2. Assume that p  (n + 1). If H is general with respect to v, then (i) MH (A, v) is a smooth projective variety of dimension v, v + 2 over k. Let KH (A, v) be the fiber of its Albanese map  MH (A, v) → A × A.

(49)

Then KH (A, v) is a smooth projective symplectic variety of dimension 2n and deformation equivalent to the generalized Kummer variety Kn (A). (ii) When  = p, there is a canonical quadratic form on H 2 (KH (A, v), Z (1)). Let v ⊥ be the orthogonal complement of v in the -adic Mukai lattice of A. There is an injective isometry  θv : v ⊥ ∩ H(A) → NS(KH (A, v)), whose cokernel is a p-primary torsion group. 2 (iii) There is an isomorphism v ⊥ ⊗ K → Hcris (KH (A, v)/W )K as F -isocrystals.

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Proof. Everything is known in characteristic zero (cf. [66], [97, Theorem 0.1]). For fields of positive characteristic, the proof is the same as in the case of K3 surfaces by the lifting techniques, with the following extra arguments for the smoothness of KH (A, v) and the existence of the symplectic form on it. For the smoothness, note that by Yoshioka [97, P.840], the base change of (49) by the multiplication becomes a trivial product. In other words, we have a by-(n + 1) map on A × A cartesian diagram:  KH (A, v) × A × A

ϕ

pA×A

 A×A

MH (A, v) alb

×(n+1)

(50)

 A × A.

If p  (n + 1), then the multiplication-by-(n + 1) map is ´etale, and so is ϕ by base change. Therefore, the smoothness of MH (A, v) implies the smoothness of KH (A, v). As for the symplectic form, let ω be the symplectic form on MH (A, v) constructed by Mukai [66]. If we know that H 0 (KH (A, v), Ω1 ) = 0, then by the  respectively, K¨ unneth formula, there are 2-forms α and β on KH (A, v) and A × A such that ϕ∗ (ω) = p∗1 (α) + p∗2 (β), (51)   where p1 and p2 are projections from KH (A, v) × A × A to KH (A, v) and to A × A ∗ respectively. Note that α = ω|KH (A,v) . When p  (n + 1), ϕ is ´etale, hence ϕ (ω) is nowhere degenerate. By (51), α (and β) must be nowhere degenerate as well, providing a symplectic form on KH (A, v). As we do not have a proof of the vanishing of H 0 (KH (A, v), Ω1 ) in general, let us provide the following work around using lifting techniques. By Proposition 6.4, one can lift (A, H, v) to (A, H, v) over a characteristic zero base W  . We have the relative moduli spaces of stable sheaves M := MH (A, v) and K := KH (A, v),  : K×W  A×W  A → M, in a way that the diagram together with a W  -morphism ϕ (50) is the reduction of the analogous diagram over W  . Mukai’s construction [66] gives a relative algebraic 2-form ω  ∈ H 0 (M, Ω2M/W  ), which restricts to a symplectic form ωη on the generic fiber and also to a symplectic form ω on the special fiber MH (A, v). When p  (n + 1), ϕ  is ´etale, hence ϕ ∗ ( ω ) gives again a ∗ symplectic form ϕη (ωη ) on the generic fiber Kη × Aη × Aη and a symplectic form  Since there are no non-zero algebraic ϕ∗ (ω) on the special fiber KH (A, v) × A × A. 1-forms on Kη , there exists an algebraic 2-form βη on Aη × Aη such that ϕ∗η (ωη ) = p∗1 (ωη |Kη ) + p∗2 (βη ).  such that By reduction, there is an algebraic 2-form β on A × A, ϕ∗ (ω) = p∗1 (ω|KH (A,v) ) + p∗2 (β). ∗

(52)

As before, since ϕ (ω) is symplectic (by the fact that ϕ is ´etale when p  (n + 1)), (52) implies that ω|KH (A,v) is also symplectic. 

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By the same argument as in Lemma 4.8, we can always reduce to the case where the polarization is general with respect to the Mukai vector, if the abelian surface is supersingular: Lemma 6.10. Let A be a supersingular abelian surface defined over an algebraically  closed field of characteristic p > 0. If v ∈ H(A) is coprime to p, then for any ample line bundle H, by performing a derived equivalence of tensoring with a line bundle, we have gcd(r, c1 · H, s) = 1; hence H is general with respect to v. Theorem 6.11 ([63, Theorem 0.2.11]). Let k be an algebraically closed field of  characteristic p > 0. Let A be a supersingular abelian surface over k. Let v ∈ H(A) be a Mukai vector which is coprime to p. Let H and H  be two ample line bundles on A. Then MH (A, v) is quasi-liftably birational to MH  (A, v). Proof. The birational equivalence is exactly the assertion of [63, Theorem 0.2.11]. To show that this birational equivalence is liftable, we apply the lifting argument in Theorem 4.10. The only thing one has to be careful with is that according to Proposition 6.4, we can only lift a supersingular abelian surface with at most two linearly independent line bundles to characteristic zero. Note that tensoring with a line bundle on c1 would not change the moduli space, therefore, we can assume that c1 = [H  ] for some separably ample line bundle H  after twisting some sufficiently separable ample line bundle to c1 . Using the coprime condition, Lemma 6.10 then ensures gcd(r, c1 · H, s) = 1. Thus we can lift the supersingular abelian surface with line bundles H and H  . In this case, the Mukai vector v can be lifted as well. So the same proof in Theorem 4.10 shows that MH (A, v) is birationally equivalent to MH  (A, v) via some liftably birational map. Similarly, there is a quasi-liftably birational map MH  (A, v)  MH  (A, v) and the assertion follows.  Now we can relate a moduli space of generalized Kummer type to generalized Kummer varieties via birational equivalences as below. Theorem 6.12. Let A be a supersingular abelian surface over an algebraically closed  field k of positive characteristic p. Let v = (r, c1 , s) ∈ H(A) be a Mukai vector coprime to p, with r > 0 and v 2 ≥ 2, then there is a birational map MH (A, v)  Pic0 (A ) × A[n+1] 

(53)

v2 2

− 1. Moreover, when KH (v) for some supersingular abelian surface A and n = is smooth, there is a quasi-liftably birational equivalence KH (A, v)  Kn (A ).

(54)

Proof. Let c1 = c1 (L) for some line bundle L ∈ Pic(A) which we can assume to be ample by tensoring with a sufficiently ample line bundle. Let E ∈ Pic(A) be the line bundle which induces the elliptic fibration in Theorem 6.3. One can easily prove that there exists a separable polarization H = L + nE ∈ Pic(A) for some n ∈ Z≥0 (up to a replacement of L by L + rL1 for some line bundle L1 ).

Supersingular irreducible symplectic varieties

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Now, one can lift the triple (A, H, E) to characteristic 0 by Proposition 6.4. Using the same argument as in Theorem 1.3, one can deduce (53) by using Theorem 6.11, Theorem 4.12 and a specialization argument. To see that A is supersingular, note that MH (A, v) and Pic0 (A )× A[n+1] , being birational, must have isomorphic Albanese varieties: % × A , ×A  A A which are supersingular. It follows that A is supersingular. The birational equivalence (53) yields the following commutative diagram _ _/ Pic0 (A ) × A [n+1] MH (A, v) _ ∼ (id,s)

alb

 Pic0 (A) × A



 / Pic0 (A ) × A

where the vertical maps are the isotrivial albanese fibrations, and the bottom map is an isomorphism thanks to the general fact that a birational equivalence between two smooth projective varieties induces an isomorphism between their Albanese varieties. Now take any non-empty open subsets U in MH (A, v) and V in Pic0 (A ) × (A )[n+1] which are identified under the birational equivalence (53). By restricting to general fibers of the two isotrivial Albanese fibrations, this induces an isomorphism of a non-empty open subset of KH (A, v) to an open subset of Kn (A ), that is, a birational equivalence between them.  Remark 6.13. 1. Similarly as in Remark 4.9, the coprimality assumption on v is automatically satisfied if p  ( 12 dim KH (A, v) + 1). 2. The abelian surface A in Theorem 6.12 is derived equivalent to A. When A is the product of elliptic curves, we know that A has to be isomorphic to A (cf. [37]). In general, the supersingular abelian surface A is expected to be isomorphic to either A or its dual A∨ (cf. [38]). 6.4. Proof of Theorem 1.4 Part (i) follows directly from Proposition 6.9 (i) and (iii). When A is supersingular, as X is quasi-liftably birationally equivalent to the generalized Kummer variety, we deduce that dim H 2,0 (X) = 1, i.e., X is irreducible symplectic. For the RCC conjecture: according to the weak factorization theorem ([94, 2]), a liftably birational transformation can be viewed as the reduction of a sequence of blow-ups and blow-downs along smooth centers. As the exception divisors of the blow-ups and blow-downs are projective bundles over the center, the rational chain connectedness is preserved. Therefore, it follows from Theorem 6.12 that when A is supersingular, X is also rationally chain connected. Another way to prove the rational chain connectedness without using the lifting argument is to exploit the fact that Kn (A ) is “unirational surface chain

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connected”, as is shown in the proof of Proposition 6.8. We leave the details to the interested readers. Finally, the supersingular abelian motive conjecture follows from the combination of Theorem 6.12, Proposition 6.7 and Proposition 3.14. Therefore X satisfies the fully supersingular Bloch–Beilinson conjecture 2.7 by Corollary 2.12. As is explained in § 3.5, to establish the supersingular Bloch–Beilinson–Beauville conjecture for X, it is enough to verify the section property conjecture 3.11 for X. But this follows from that for Kn (A), which is established in Proposition 6.7, because Proposition 3.14 provides an algebraic correspondence inducing an isomorphism of Q-algebras between CH∗ (X)Q and CH∗ (Kn (A))Q which sends Chern classes of TX to the corresponding ones of TKn (A) . 6.5. A remark on the unirationality Theorem 6.12 also provides us an approach to the unirationality conjecture for generalized Kummer type varieties KH (A, v). Corollary 6.14. Notation and assumption are as in Theorem 6.12. If A is supersingular, then KH (A, v) is unirational if the generalized Kummer variety Kn (E × E) 2 is unirational for some supersingular elliptic curve E, where n = v2 − 1. Proof. By Theorem 6.12, KH (A, v) is birational to Kn (A ) for some supersingular abelian surface A . By [77, Theorem 4.2], for any supersingular elliptic curve E, A and E × E are isogenous. Therefore it suffices to see that for two isogenous abelian surfaces A1 and A2 , the generalized Kummer varieties Kn (A1 ) and Kn (A2 ) are   + dominated by each other. To this end, we remark that B1 := ker An+1 − → A1 1   + − → A2 which is compatible with the natural has an isogeny to B2 := ker An+1 2 Sn+1 -actions. Therefore Kn (A1 ) := B1 /Sn+1 has a dominant map to Kn (A2 ) := B2 /Sn+1 . Hence Kn (A1 ) dominates Kn (A2 ). By symmetry, they dominate each other.  For each p and n, Corollary 6.14 allows us to reduce the unirationality conjecture for moduli spaces of sheaves of generalized Kummer type to a very concrete question: Question 6.15. Let E be a supersingular elliptic curve. Is the generalized Kummer variety Kn (E × E) unirational ? This question remains open even in the case n = 2. Acknowledgment The authors want to thank Nicolas Addington, Fran¸cois Charles, Cyril Demarche, Najmuddin Fakhruddin, Daniel Huybrechts, Qizheng Yin and Weizhe Zheng for very helpful discussions, and to thank the referees for their pertinent suggestions. The first author is also grateful to the Hausdorff Research Institute for Mathematics for the hospitality during the 2017 Trimester K-theory and related fields.

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Lie Fu is supported by the Radboud Excellence Initiative program. He was also supported by the Agence Nationale de la Recherche through ECOVA (ANR15-CE40-0002), HodgeFun (ANR-16-CE40-0011), LABEX MILYON (ANR-10LABX-0070) of Universit´e de Lyon, and Projet Inter-Laboratoire 2017, 2018 and 2019 by F´ed´eration de Recherche en Math´ematiques Rhˆone-Alpes/Auvergne CNRS 3490. Zhiyuan Li is supported by National Science Fund for General Program (11771086), Key Program (11731004) and the “Dawn” Program (17SG01) of Shanghai Education Commission.

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Symbols and Equivariant Birational Geometry in Small Dimensions Brendan Hassett, Andrew Kresch and Yuri Tschinkel Abstract. We discuss the equivariant Burnside group and related new invariants in equivariant birational geometry, with a special emphasis on applications in low dimensions. Mathematics Subject Classification (2010). 14L30, 14E07, 14M20. Keywords. Cremona transformation, equivariant geometry, birational invariants, Burnside group, cubic fourfolds.

Let G be a finite group. Suppose that G acts birationally and generically freely on a variety over k, an algebraically closed field of characteristic zero. After resolving indeterminacy and singularities we may assume that G acts regularly on a smooth projective variety X. Classifying such actions X ý G, ∼

up to birational conjugation, especially in cases where X  Pn , is a longstanding problem. This entails understanding realizations of G in the Cremona group BirAut(Pn ). There is an enormous literature on this subject; we will summarize some key results in Section 1. The focus of this survey is a new approach to the analysis of actions via new invariants extracted from the fixed points and stabilizer loci of X. This new approach has its origin in the study of specializations of birational type. Suppose a smooth projective variety specializes to a reduced normal crossing divisor – we seek to gain information about the general fiber from combinatorial structures associated with the special fiber [29, 20]. That circle of ideas, in turn, was inspired by motivic integration – another instance of this philosophy [25]. Actions of finite groups yield formally similar stratifications: we have the open subset, on which the group acts freely, and the locus with nontrivial stabilizers. The combinatorics of the resulting stratification – along with the representations of the stabilizers of the strata on the normal directions – sheds light on the original group action. In many cases, we can extract additional information from the birational type of these strata and the action of the normalizer of the stabilizer. We explain these constructions © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Farkas et al. (eds.), Rationality of Varieties, Progress in Mathematics 342, https://doi.org/10.1007/978-3-030-75421-1_8

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in detail, with many examples of small dimensions, referring to the original papers [19, 22]. Our goal is to illustrate how the layers of this new formalism reveal various levels of structure among G-birational types. In particular, we apply these new obstructions to cyclic actions on cubic fourfolds, including rational ones, and produce examples of nonlinearizable actions. Here is the roadmap of the paper: After briefly recalling several classical results, we discuss the case of finite abelian groups G. We introduce the group of equivariant birational types Bn (G), as a quotient of a Z-module of certain symbols by explicit defining relations and find a simpler presentation of these relations (Proposition 2.1). In Section 3 we explain, in numerous examples, how to compute the invariants on surfaces. In Section 4 we show that for G = Cp , of prime order p, all symbols in Bn (G) are represented by smooth projective varieties with G-action. In Section 5 we introduce and study refined invariants of abelian actions, taking into account not only the representation on the tangent bundle to the fixed point strata, but also birational types of these strata. In Section 6 we exhibit cyclic actions on cubic fourfolds that are not equivariantly birational to linear actions; our main goal is to highlight the applications of the different invariants in representative examples. Finally, in Section 7 we consider nonabelian groups, define the equivariant Burnside group, which encodes new obstructions to equivariant rationality, and show how these obstructions work in a striking example, due to Iskovskikh [18]: distinguishing two birational actions of S3 × C2 on rational surfaces.

1. Brief history of previous work These questions were considered by Bertini, Geiser, De Jonqui`eres, Kantor, etc. over a century ago but continue to inspire new work: • Manin [26, 27] studied G-surfaces both in the arithmetic and geometric context, focusing on the induced G-action on the geometric Picard group, and on cohomological invariants of that lattice; • Iskovskikh [17] laid the groundwork for the G-birational classification of surfaces and their linkings; • Bayle, Beauville, Blanc, and de Fernex [4, 5, 12, 8, 9] classified actions of finite abelian G on surfaces; • Dolgachev and Iskovskikh [13] largely completed the surface case; • Bogomolov and Prokhorov [10, 33] considered the stable conjugacy problem for the surface case using cohomological tools introduced by Manin; • Prokhorov, Cheltsov, Shramov, and collaborators proved numerous theorems for threefolds – both concerning specific groups, such as simple groups [32, 11], as well as general structural properties [34]. Much of this work fits into the Minimal Model Program, using distinguished models to reduce the classification problem to an analysis of automorphisms of a restricted class of objects, e.g., del Pezzo surfaces. With a few exceptions – the application of

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the cohomology on the N´eron-Severi group, by Manin and Bogomolov–Prokhorov, and the ‘normalized fixed curve with action (NFCA)’ invariant of Blanc [9] – invariants play a limited role. A fundamental observation, recorded in [35, App. A], is that the presence of a point fixed by a given abelian subgroup H of G is a birational invariant of a smooth projective variety X with generically free G-action. Furthermore, Reichstein and Youssin showed that the determinant of the action of abelian stabilizers on the normal bundle to the locus with the given stabilizer, up to sign, is also a birational invariant [36]. However, for finite groups this is only useful when the minimal number of generators of the abelian group equals the dimension of the variety [35, Th. 1.1]. For cyclic groups, it is applicable only for curves. The invariants defined in [19, 21, 22] record all eigenvalues for the action of abelian stabilizers, as well as additional information about the action on the components of the fixed locus, and on their function fields. These collections of data are turned into a G-birational invariant, via explicit blowup relations. The groups receiving these invariants, the equivariant Burnside groups, have an elaborate algebraic structure. And they led to new results in birational geometry, some of which will be discussed below.

2. Equivariant birational types Here we restrict to the situation where G is abelian and consider only fixed points of X ý G. In general, there are no such fixed points and we obtain no information. However, large classes of actions do have fixed points, e.g., if G is cyclic and hi (X, OX ) = 0, for each i > 0, then the Atiyah-Bott holomorphic Lefschetz formula [2, Cor. 4.13] yields a fixed point. The vanishing assumption holds for rational and rationally connected X. If G is an abelian p-group (p prime) acting on X without fixed points then every Chern number of X is divisible by p [16, Cor. 1.1.2]. To define an invariant of X ý G, we consider collections of weights for the action of G in the tangent bundle at G-fixed points in X. To formalize this, let A = G∨ = Hom(G, Gm ) be the character group of G, and n = dim X. Let Sn (G) be the free abelian group on symbols [a1 , . . . , an ],

aj ∈ A,

∀j,

subject to the conditions: (G) Generation: {a1 , . . . , an } generate A, i.e., n Zai = A, i=1

thus, n is at least the minimal number of generators of G;

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(S) Symmetry: for each permutation σ ∈ Sn we have [aσ(1) , . . . , aσ(n) ] = [a1 , . . . , an ]. Let Sn (G) → Bn (G) be the quotient, by relations, for all 2 ≤ r ≤ n:

(2.1)

(Br ) Blow-up: for all [a1 , . . . , ar , b1 , . . . , bn−r ] ∈ Sn (G) one has [a1 , . . . , ar , b1 , . . . , bn−r ]  [a1 − ai , . . . , ai , . . . , ar − ai , b1 , . . . , bn−r ]. =

(2.2)

1≤i≤r, ai =ai for i 2. We will work in B2 (CN ) ⊗ Q. We use generators for this space arising from the alternative symbol formalism M2 (CN ) introduced in [19] with the property that [19, Prop. 7] B2 (CN ) ⊗ Q = M2 (CN ) ⊗ Q. For a, b ∈ Hom(CN , Gm ) generating the group we set $ if a, b = 0 [a, b] a, b = 1 [a, 0] if a = 0, b = 0. 2 The advantage of these generators is that the relations are uniformly a, b = a, b − a + a − b, b

(B),

even when a = b. We follow the proof of [19, Th. 14]. For all a, b with gcd(a, b, N ) = 1 we write δ(a, b) := a, b + −a, b + a, −b + −a, −b . We claim this is zero in B2 (CN ) ⊗ Q. First, we check that δ(a, b) is invariant under SL2 (Z/N Z). This has generators     1 −1 0 −1 . , 0 1 1 0 and δ(a, b) = δ(−b, a) by the symmetry of the underlying symbols. We also have δ(a, b − a) = a, b − a + −a, b − a + a, a − b + −a, a − b applying B to second and third terms above = a, b − a + −a, b + −b, b − a + a, −b + b, a − b + −a, a − b applying B to get first and four terms below = a, b + −a, b + a, −b + −a, −b = δ(a, b). Average δ(a, b) over all pairs a, b with gcd(a, b, N ) = 1 to obtain   S := δ(a, b) = 4 a, b . a,b

a,b

Applying the blowup relation (B) to all terms one finds S = 2S, which implies that S = 0 ∈ B2 (CN ) ⊗ Q. We may regard δ(a, b) and S as elements of B2 (CN ). It follows that δ(a, b) is torsion in B2 (CN ), annihilated by the number of summands in S. Substituting b = 0, we obtain that δ(a, 0) = [a, 0] + [−a, 0] = 0 ∈ B2 (CN ) ⊗ Q.



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The invariance of δ(a, 0) shows that [a, 0]+[−a, 0] is independent of the choice of a relatively prime to N . 3.4. Algebraic structure in dimension 2 For reference, we tabulate dim B2 (G) ⊗ Q, for G = CN and small values of N : 2 0

N

3 4 1 1

5 6 2 2

7 8 3 3

9 10 11 12 13 14 15 16 5 4 6 7 8 7 13 10

For primes p ≥ 5 there is a closed formula [19, §11]: dim B2 (Cp ) ⊗ Q =

(p − 5)(p − 7) p − 1 p2 − 1 +1= + , 24 24 2

(3.1)

which strongly suggested a connection with the modular curve X1 (p)! We also have (p − 5)(p − 7) , (3.2) dim B2− (Cp ) ⊗ Q = 24 and, by [19, Prop. 30], B1− (Cp ) ⊗ Q = Ker(B2 (Cp ) → B2− (Cp )) ⊗ Q. Computations in noncyclic cases have been performed by Zhijia Zhang1 ; we summarize the results: for primes p ≥ 5 one has dim B2 (Cp × Cp ) ⊗ Q =

(p − 1)(p3 + 6p2 − p + 6) , 24

dim B2− (Cp × Cp ) ⊗ Q =

(p − 1)(p3 − p + 12) 24

For G = CN1 × CN2 and small values of N1 , N2 , we have: N1 N2 dQ d− Q d2 d− 2 1 see

2 2 0 0 2 2

2 4 2 0 5 3

2 6 3 0 8 5

2 2 2 3 3 3 8 10 16 3 6 9 6 7 21 7 15 37 1 1 9 3 7 19 13 18 36 7 15 37 8 12 24 3 7 19

3 27 235 163 235 163

https://cims.nyu.edu/~ zz1753/ebgms/ebgms.html

4 4 4 5 6 8 16 32 25 36 33 105 353 702 577 17 65 257 502 433 34 106 354 702 578 17 65 257 502 433

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4. Reconstruction theorem The examples offered so far might suggest that very few invariants in Bn (G) are actually realized geometrically by smooth projective varieties X ý G. If one allows nonrational examples far more invariants arise: Proposition 4.1. Let p be a prime. Then Bn (Cp ) is generated as an abelian group by β(X ý Cp ), where X is smooth and projective. Proof. We proceed by induction on n. The case of n = 1 follows from the Riemann existence theorem applied to cyclic branched covers of degree p with the prescribed ramification data. (See also Lemma 4.2 below.) For the symbols [a1 , . . . , an−1 , 0] we construct (n − 1)-dimensional varieties D with the prescribed invariants and D × P1 with trivial action on the second factor. Since [a, a, a3 , . . . , an ] = [0, a, a3 , . . . , an ] we may focus on symbols [a1 , a2 , . . . , an ], 0 < a1 < a2 < · · · < an < p. We are reduced to the following lemma: Lemma 4.2. Any sum



m[a1 ,a2 ,...,an ] [a1 , a2 , . . . , an ]

(4.1)

of symbols [a1 , a2 , . . . , an ],

0 < a1 < a2 < · · · < an < p,

with nonnegative coefficients, is realized as β(X ý Cp ), where X is smooth, projective, and irreducible. For each symbol [a1 , a2 , . . . , an ] appearing in the sum, take an n-dimensional representation V[a1 ,a2 ,...,an ] with the prescribed weights and the direct sum W[a1 ,a2 ,...,an ] = (V[a1 ,a2 ,...,an ] ⊕ C)m[a1 ,a2 ,...,an ] where C is the trivial representation of Cp . Write W = ⊕W[a1 ,a2 ,...,an ] , where the index is over the terms appearing in (4.1), and consider the projectivization P(W ) and the n-planes P[a1 ,a2 ,...,an ],j ⊂ P(W[a1 ,a2 ,...,an ] ),

j = 1, . . . , m[a1 ,a2 ,...,an ]

associated with the summands of W[a1 ,a2 ,...,an ] , each with distinguished fixed point p[a1 ,a2 ,...,an ],j = (0 : 0 : · · · : 0 : 1). The action on Tp[a1 ,a2 ,...,an ] P[a1 ,a2 ,...,an ],j coincides with the action on V[a1 ,a2 ,...,an ] . The fixed  points of P(W ) correspond to the eigenspaces for the Cp action. Write M = m[a1 ,a2 ,...,an ] for the number of summands; each weight occurs at most M times. Thus the fixed point loci are projective spaces of dimension ≤ M − 1.

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Choose a high-degree smooth complete intersection X of dimension n, invariant under the action of Cp , containing the p[a1 ,a2 ,...,an ],j and tangent to P[a1 ,a2 ,...,an ],j for each [a1 , a2 , . . . , an ]. This complete intersection exists by polynomial interpolation applied to the quotient P(W )/Cp ; smoothness follows from Bertini’s Theorem. Since complete intersections of positive dimension are connected, the resulting X is irreducible. It only remains to show that such a complete intersection need not have fixed points beyond those specified. Now X has codimension (M − 1)(n + 1), so we may assume it avoids the fixed point loci – away from the stipulated points p[a1 ,s2 ,...,an ],j – provided (M − 1)(n + 1) ≥ M . It only remains to consider the special case M = 1. Here we take W = (V[a1 ,a2 ,...,an ] ⊕ C)2 , imposing conditions at just one point (0 : 0 : · · · : 0 : 1). Here X ⊂ P(W ) has codimension n + 1 and the fixed point loci are P1 s, so we may avoid extraneous  points of intersection.

5. Refined invariants 5.1. Encoding fixed points Since B2 (C2 ) = 0 this invariant says nothing about involutions of surfaces! Bertini, Geiser, and De Jonqui`eres involutions are perhaps the most intricate parts of the classification, which relies on the study of fixed curves. This leads to a natural refinement of the invariants: For the symbols of type [a, 0], corresponding to curves Fα ⊂ X fixed by CN , we keep track of the (stable) birational equivalence class of Fα and the element of B1 (CN ) associated with [a]. In general, [19] introduced a group combining the purely number-theoretic information encoded in Bn (G) with geometric information encoded in the Burnside group of fields from [20]. Let Birn−1,m (k),

0 ≤ m ≤ n − 1,

be the set of k-birational equivalence classes of (n − 1)-dimensional irreducible varieties over k, which are k-birational to products W ×Am , but not to W  ×Am+1 , for any W  , and put Bn (G, k) := ⊕n−1 m=0 ⊕[Y ]∈Birn−1,m (k) Bm+1 (G).

(5.1)

Let X be a smooth projective variety of dimension n with a regular, generically free, action of an abelian group G. Put  βk (X ý G) := βk,α , α

where, as before, the sum is over components Fα ⊂ X G of the G-fixed point locus, but in addition to the eigenvalues a1 , . . . , an−dim(Fα ) ∈ A in the tangent

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space Txα X one keeps information about the function field of the component Fα . Choosing mα so that [Fα × An−1−dim(Fα ) ] ∈ Birn−1,mα (k) we set 0, . . . , 0 2 34 5

βk,α := [a1 , . . . , an−dim(Fα ) ,

] ∈ Bmα +1 (G),

mα +1−n+dim(Fα )

identified with the summand of (5.1) indexed by [Fα × An−1−dim(Fα ) ]. When Fα is not uniruled we get a symbol in Bcodim(Fα ) (G). Theorem 5.1 ([19, Remark 5]). The class βk (X ý G) ∈ Bn (G, k) is a G-birational invariant. 5.2. Encoding points with nontrivial stabilizer We continue to assume that G is a finite abelian group, acting regularly and generically freely on a smooth variety X. Let H ⊂ G arise as the stabilizer of some point of X, F ⊂ X H an irreducible component of the fixed point locus with generic stabilizer H, and Y the minimal G-invariant subvariety containing G. In Section 7.2, we will explain how to blow up X so that Y is always a disjoint union of translates of F . Additional information about the action of G on X is contained in the action of the quotient G/H, which could act on the function field of F , or by translating F in X. The paper [22] introduced the group Burnn (G) as the quotient by certain relations of the free abelian group generated by symbols (H, G/H ý K, β),

(5.2)

where K is a G/H-Galois algebra over a field of transcendence degree d ≤ n over k, up to isomorphism, and β is a faithful (n − d)-dimensional representation of H (see [22, Def. 4.2] for a precise formulation of conditions on K and relations). Passing to a suitable G-equivariant smooth projective model X – as discussed in Section 7.2 – its class is defined by   [X ý G] := (H, G/H ý k(Y ), βY (X)) ∈ Burnn (G), (5.3) H⊆G Y

where the sum is over all G-orbits of components Y ⊂ X with generic stabilizer H as above, the symbol records the eigenvalues of H in the tangent bundle to x ∈ Y as well as the G/H-action on the total ring of fractions k(Y ).

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Example 5.2. We revisit Example 3.1 using the dual basis of characters χ1 , χ2 of G = C2 × C2 . Here we have [P1 ý G] = (1 , G ý k(t)) + (g1  , G/ g1  ý {(1 : ±i)}, χ1 ) + (g2  , G/ g2  ý {(1 : 0), (0 : 1)}, χ2 ) + (g1 g2  , G/ g1 g2  ý {(1 : ±1)}, χ1 + χ2 ). The action on k(P1 ) = k(t) is by g1 (t) = −t and g2 (t) = −1/t. Blowup relations ensure that [X ý G] is a well-defined G-birational invariant – see Section 5.3 for more details. There is a distinguished subgroup Burnntriv (G) ⊂ Burnn (G) generated by symbols (1, G ý K = k(X)). For ‘bootstrapping’ purposes – where we seek invariants of n-folds in terms of lower-dimensional strata with nontrivial stabilizers – we may suppress these tautological symbols to get a quotient Burnn (G) → Burnnontriv (G). n And there is also a natural quotient group Burnn (G) → BurnG n (G) obtained by suppressing all symbols (H, G/H ý K, β) where H is a proper subgroup of G. By [22, Prop. 8.1 and Prop. 8.2], there are natural surjective homomorphisms BurnG n (G) → Bn (G, k) → Bn (G). (G) is freely generated by nontrivial subgroups Example 5.3. The group Burnnontriv 1 (CN ) ∼ H ⊂ G and injective characters a : H → Gm , e.g., Burnnontriv = ZN −1 and 1 nontriv 3 ∼ (C2 × C2 ) = Z . Burn1 5.3. Examples of blowup relations We illustrate the relations for fixed points of cyclic actions on surfaces. More computations are presented in Section 5.4; the reader may refer to Section 4 of [22] for the general formalism. All the key ideas are manifest in the surface case because the full set of blowup relations follows from those in codimension two – see [22, Prop. 8.1] and the special case Prop. 2.1 above. Suppose that G = CN acts on the surface X with fixed point p and weights  denote the blowup of X a1 and a2 that generate A = Hom(CN , Gm ). Let X at p and E  P1 the exceptional divisor. Let H = ker(a1 − a2 ) ⊂ G denote the generic stabilizer of E which acts faithfully on the normal bundle NE/X via ¯2 ∈ A := Hom(H, Gm ). a ¯1 = a Assume first that a1 and a2 are both prime to N , so p is an isolated fixed point of X. The quotient G/H (when nontrivial) acts faithfully on E = P1 with fixed points p1 and p2 . If H = G then a1 = a2 and we get the relation (G, triv ý k = k(p), (a1 , a1 )) = (G, triv ý k(t) = k(E), (a1 )).

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If triv  H   G then ¯1 = a ¯2 ) (G, triv ý k = k(p), (a1 , a2 )) = (H, G/H ý k(t) = k(E), a + (G, triv ý k = k(p2 ), (a1 , a2 − a1 )) + (G, triv ý k = k(p1 ), (a2 , a1 − a2 ))

(5.4)

where G/H acts on t by a primitive dth root of unity with d = |G/H|. If H is trivial then (G, triv ý k = k(p),(a1 , a2 )) = (G, triv ý k = k(p2 ), (a1 , a2 − a1 )) + (G, triv ý k = k(p1 ), (a2 , a1 − a2 )).

(5.5)

Assume now that a1 = m1 n1 and a2 = m2 n2 where n1 , n2 |N (and are relatively prime modulo N ) and m1 and m2 are prime to N and each other. Then we have p ∈ F1 ∩ F2 for irreducible curves F1 and F2 with generic stabilizers Cn1 and Cn2 respectively. 1 , F 2 ⊂ X  denote the proper transforms of F1 and F2 and p1 , p2 ∈ E their Let F intersections with the exceptional divisors. Thus the contribution to the strata containing p is (G, triv ý k(p),(a1 , a2 )) +(Cn1 , CN /Cn1 ý k(F1 ), a2 ) + (Cn2 , CN /Cn2 ý k(F2 ), a1 )  with the Fi replaced with the latter two terms appearing in the symbol sum on X, by the Fi . Note that H = ker(a1 − a2 )  G because a1 ≡ a2 . Here the blowup formula takes the form (5.4) or (5.5) depending on whether H  is trivial or not. Now suppose that a2 = 0. Let F ⊂ X denote the irreducible component of the fixed locus containing p, so that a1 is the character by which G acts on NF/X .  for the proper transform of F , p1 = F ∩ E, and p2 ∈ E for the other Write F ⊂ X fixed point. Here we get the relation (G, G ý k(F ), a1 ) = (G, G ý k(F ), a1 ) + (G, G ý k(p2 ), (a1 , −a1 )), whence the latter term vanishes. 5.4. Examples We now complement the computations in Section 3, for G = CN , and small N . As before, we work over an algebraically closed base field k. (N = 2) • B2 (C2 ) = 0. 2 • B2 (C2 , k) ∼ = BurnC 2 (C2 ); has a copy of B1 (C2 ) = Z, for every isomorphism class of curves of positive genus. C2 • Burn2 (C2 ) = Burntriv 2 (C2 ) ⊕ Burn2 (C2 ).

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(N = 3) ∼ Z, generated by [1, 1], • B2 (C3 ) = [2, 2] = −[1, 1].

[1, 2] = 0,

3 • B2 (C3 , k) ∼ = BurnC 2 (C3 ), is a direct sum of Z, corresponding to points and ∼ Z2 for every isomorphism class of rational curves, and a copy of B1 (C3 ) = curves of positive genus. C3 • Burn2 (C3 ) = Burntriv 2 (C3 ) ⊕ Burn2 (C3 ).

(N = 4) ∼ Z, generated by [1, 2], • B2 (C4 ) = [1, 1] = 2[1, 2],

[1, 3] = 0,

[2, 3] = −[1, 2],

[3, 3] = −2[1, 2].

• B2 (C4 , k) is a direct sum of Z, corresponding to points and rational curves, ∼ Z2 for every isomorphism class of curves of positive and a copy of B1 (C4 ) = genus. (C4 ) has, for every (C4 ): Burnnontriv • Burn2 (C4 ) = Burn2triv (C4 ) ⊕ Burnnontriv 2 2 isomorphism class of curves of positive genus, a copy of B1 (C4 ) and a copy of B1 (C2 ), with an additional copy of B1 (C2 ) for every isomorphism class of curves of positive genus with involution. We claim points and rational curves contribute Z2 ⊂ Burnnontriv (C4 ), 2 generated by [1, 2] and [2, 3] where [i, j] = (C4 , k, (i, j)),

(i, j) = (1, 1), (1, 2), (1, 3), (2, 3), (3, 3).

Abusing notation, write [1, 0] = (C4 , k(t), 1) [3, 0] = (C4 , k(t), 1). We write down the blowup relations, both orbits of points with special stabilizers and orbits on one-dimensional strata with nontrivial stabilizer: [0, 1] = [0, 1] + [1, 3] [0, 3] = [0, 3] + [1, 3] [1, 1] = [1, 0] [1, 2] = [1, 1] + [2, 3] [1, 3] = [1, 2] + [2, 3] + (C2 , C2 ý k(t), 1) [2, 3] = [1, 2] + [3, 3] [3, 3] = [3, 0] (C2 , C2 ý k , (1, 1)) = (C2 , C2 ý k(t)2 , 1) 2

(C2 , C2 ý k(t), 1) = (C2 , C2 ý k(t), 1) + (C2 , C2 ý k(t)2 , 1) (C2 , C2 ý k(t)2 , 1) = (C2 , C2 ý k(t)2 , 1) + (C2 , C2 ý k(t)2 , 1)

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Thus we find [1, 3] = 0, [0, 3] = [3, 3] = −[1, 2] + [2, 3], [0, 1] = [1, 1] = [1, 2] − [2, 3], (C2 , C2 ý k(t), 1) = −[1, 2] − [2, 3], (C2 , C2 ý k 2 (t), 1) = 0, (C2 , C2 ý k 2 , (1, 1)) = 0. Here k n denotes the total ring of fractions for an orbit of length n and k n (t) the total ring of fractions of the exceptional locus of the blowup of such an orbit. Furthermore, C2 ý k(t) is via t → −t. For example, the linear action on P2 (x : y : z) → (x : iy : −iz) has invariant [1, 3] + [1, 2] + [2, 3] + (C2 , C2 ý k(y/z), 1) = 0 ∈ Burnnontriv (C4 ). 2 We close this section with a noncyclic example: (C2 × C2 ): Write G = C2 × C2 = {1, g1 , g2 , g3 = g1 g2 } and G∨ = {0, χ1 , χ2 , χ3 = χ1 + χ2 }, χ1 (g1 ) = χ2 (g2 ) = −1, χ1 (g2 ) = χ2 (g1 ) = 1 as before. • B2 (C2 × C2 ) = {0, [χ1 , χ2 ], [χ1 , χ3 ], [χ2 , χ3 ]} with the structure of the Klein four group presented in Section 3. • B2 (C2 × C2 , k) ∼ = B2 (C2 × C2 ) as B1 (C2 × C2 ) = 0. (C2 × C2 ) where the second • Burn2 (C2 × C2 ) = Burntriv (C2 × C2 ) ⊕ Burnnontriv 2 term is a direct sum of the subgroup R generated by points and rational curves, a copy of ∼ Z3 Burn1nontriv (C2 × C2 ) = for each curve of positive genus, and another copy for each curve of positive genus equipped with an involution. The group R fits into an exact sequence 0 → Burnnontriv (C2 × C2 ) → R → B2 (C2 × C2 ) → 0 1 obtained by computing generators and relations.

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Generators: Here we take 1 ≤ i < j ≤ 3: [χi , χj ] := (G, triv ý k, (χi , χj )) ¯i = 1) ei := (gi  , G/ gi  ý k(t), χ qi := (gi  , G/ gi  ý k 2 , (χ ¯i ) = (1, 1)) ¯i , χ fi := (gi  , G/ gi  ý k 2 (t), χ¯i = 1) Relations: Here we choose h so that {h, i, j} = {1, 2, 3}: [χi , χj ] = eh + [χh , χi ] + [χh , χj ] (blow up fixed point) qi = fi

(blow up orbit qi )

ei = ei + fi

(blow up general orbit of ei )

Thus the qi and fi are zero and we have R/ e1 , e2 , e3  = B2 (C2 × C2 ). We revisit the actions of C2 × C2 on rational surfaces in Section 3 using these new techniques: (1) the action (x, y) → (±x±1 , y) on P1 × P1 has invariant f1 + f2 + f3 = 0; (2) the action (x, y) → (±x, ±y) on P1 × P1 has invariant 2e1 + 2e2 + 4[χ1 , χ2 ] = 0; (3) the diagonal action on P1 × P1 has invariant 2(q1 + q2 + q3 ) = 0; (4) the action on the conic fibration admits an elliptic curve F = {y12 (x1 − ax2 )(x1 + ax2 ) = y22 (x1 − bx2 )(x1 + bx2 )} that is fixed by g2 and fibers over (x1 : x2 ) = (0 : 1), (1 : 0) fixed by g1 and thus has invariant 4[χ1 , χ2 ] + 2e1 + (g2  , g1  ý k(F ), 1) = 0 where g1 acts on k(F ) by x1 /x2 → −x1 /x2 ; (5) the action on the degree two del Pezzo surface has nontrivial invariant arising from the positive genus curves fixed by g1 and g2 ; (6) the action on the degree one del Pezzo surface has nontrivial invariant arising from the positive genus curves fixed by the involution.

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5.5. Limitation of the birational invariant It is an elementary observation, recorded in [35, App. A], that the presence of a point fixed by a given abelian subgroup H of G is a birational invariant of a smooth projective variety X with generically free G-action. Two smooth n-dimensional projective varieties with generically free G-action might be distinguished in this way but nevertheless have the same class in Burnnnontriv (G). Indeed, letting C2 act on P1 , we consider the corresponding product action of C2 × C2 on P1 × P1 . As well, the action of C2 × C2 on P1 gives rise to a diagonal action on P1 × P1 . The former, but not the latter, has a point fixed by C2 × C2 , so the actions belong to two distinct birational classes. However, both actions give rise to a vanishing class in Burn2nontriv (C2 × C2 ). 5.6. Reprise: Cyclic groups on rational surfaces As already discussed in Section 3, the presence of higher genus curves in the fixed locus of the action of a cyclic group of prime order on a rational surface is an important invariant in the study of the plane Cremona group. For example, for G = C2 , these curves make up entirely the group Burn2G (G) and famously characterize birational involutions of the plane up to conjugation [4]. For more general cyclic groups acting on rational surfaces, we recover the NFCA invariant of Blanc [9], which governs his classification. We recall the relevant definitions: Let g ∈ Cr2 be a nontrivial element of the plane Cremona group, of finite order m. • Normalized fixed curve [12]: 6 isomorphism class of the normalization of the union NFC(g) := of irrational components of the fixed curve. • Normalized fixed curve with action:  m−1 NFCA(g) := (NFC(g r ), g |NFC(gr ) r=1 , where g |NFC(gr ) is the automorphism induced by g on NFC(g r ). One of the main results in this context is the following characterization: Theorem 5.4 ([9]). Two cyclic subgroups G and H of order m of Cr2 are conjugate if and only if NFCA(g) = NFCA(h), for some generators g of G and h of H. It follows immediately from the definition that the information encoded in Burn2 (G), for G = Cm , is equivalent to NFCA(g). Remark 5.5. It would be interesting to use symbol invariants to organize the classification of cyclic group actions on rational threefolds.

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6. Cubic fourfolds In this section we discuss several illustrative examples, showing various aspects of the new invariants introduced above. Equivariant geometry in dimension ≤ 3 is, in principle, accessible via the Minimal Model Program, and there is an extensive literature on factorizations of equivariant birational maps. We focus on dimension four, and in particular, on cubic fourfolds. Let X ⊂ P5 be a smooth cubic fourfold. No examples are known to be irrational! Here we show that there are actions X ý G where G-equivariant rationality fails, including actions on rational cubic fourfolds. We found it useful to consult the classification of possible abelian automorphism groups of cubic fourfolds in [28]. Here is a list of N > 1, such that the cyclic group CN acts on a smooth cubic fourfold: N = 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 18, 21, 24, 30, 32, 33, 36, 48. Note that B4 (CN ) ⊗ Q = 0, for all N < 27, N = 30, 32. We record their dimensions dQ in the remaining cases: N dQ

33 36 48 2 3 7

One can also work with finite coefficients: let dp = dp (N ) := dim B4 (CN ) ⊗ Fp , we have d2 , d3 , d5 = 0, for all N ≤ 15, and N = 18, 21. In the other cases, we find: N

16

24 30 32 33 36 48

d2 d3 d5 d7

1 0 0 0

5 0 0 0

10 12 0 0 0 0 0 0

3 2 2 2

19 50 3 7 3 7 3 7

Thus, to exhibit applications of B4 (CN ) we have to look at large N . Using Bn (G): Consider X ⊂ P5 given by x21 x2 + x22 x3 + x23 x1 + x24 x5 + x25 x0 + x30 = 0.

(6.1)

It carries the action of G = C36 , with weights (0, 4, 28, 16, 9, 18) on the ambient projective space, and isolated G-fixed points P1 = (0 : 1 : 0 : 0 : 0 : 0), . . . , P5 := (0 : 0 : 0 : 0 : 0 : 1). Computing the weights in the corresponding tangent spaces, we find that β(X) equals [4, 24, 31, 22] + [28, 24, 19, 10] + [24, 12, 7, 34] + [9, 5, 17, 29] + [14, 26, 2, 9].

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Solving a system of 443557 linear equations in 82251 variables, we find, by computer, that β(X) = β(P4 ý C36 ) = 0 ∈ B4 (C36 ) ⊗ F2 = F19 2 . This implies that X is not G-equivariantly birational to P4 . Using co-multiplication: The fourfold X ⊂ P5 given by x22 x3 + x23 x4 + x24 x5 + x52 x0 + x03 + x13 = 0

(6.2)

carries an action of C48 with weights (0, 16, 3, −6, 12, −24), and isolated fixed points. We find that β(X) ∈ B4 (C48 ) equals [−3, 13, 9, −27] + [6, 22, 9, −18] + [−12, 4, −9, −18] + [40, 27, 18, 36]. Here, we apply the co-multiplication formula from Section 2.2 to the image β − (X) of β(X) under the projection B4 (C48 ) → B4− (C48 ). Let

G := Z/3Z, G := Z/16Z, We have a homomorphism

n = 1

and

n = 3.

∇− : B4− (C48 ) → B1− (C3 ) ⊗ B3− (C16 ) and we find that ∇− (β − (X)) equals [1]− ⊗ ([−1, 3, −9]− + [2, 3, −6]− + [−4, −3, −6]− + [9, 6, 12]−). Now we are computing in B3− (C16 ), a much smaller group. We have dim B3 (C16 ) ⊗ Q = 3, but dim B3− (C16 ) ⊗ Q = 0, however

dim B3 (C16 ) ⊗ F2 = 8 and dim B3− (C16 ) ⊗ F2 = 7. We find, by computer, that [−4, −3, −6] = [9, 6, 12] ∈ B3 (C16 ) ⊗ F2 ,

so that the sum of these terms does not contribute to ∇− (β − (X)), and check that [−1, 3, −9]− + [2, 3, −6]− = [1, 2, 10]− = 0 ∈ B3− (C16 ). It follows that ∇− (β − (X)) = 0,

thus

β − (X) = 0 ∈ B4− (C48 ),

and this action of C48 on the cubic fourfold is not equivariantly birational to a linear action on P4 , by Proposition 2.3. Using Bn (G, k): There is also another way to analyze the fourfold in (6.2): observe that the divisor Y ⊂ X, a smooth cubic threefold given by x1 = 0, is fixed by C3 ⊂ C48 . This divisor is irrational, and we get a nontrivial contribution to βk (X) ∈ B4 (C3 , k), in the summand labeled by Y ∈ Bir1,0 ; thus X is not even C3 -equivariantly birational to P4 .

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B. Hassett, A. Kresch and Y. Tschinkel The fourfold X ⊂ P5 given by f3 (x0 , x1 , x2 ) + x32 x4 + x42 x5 + x25 f1 (x0 , x1 , x2 ) = 0

carries the action of G = C8 , with weights (0, 0, 0, 1, 6, 4). The fourfold is smooth, e.g., for f3 = x03 + x13 + x23 and f1 = x0 . Here, there are no isolated fixed points, but we find information from fixed point loci in higher dimension. The G-fixed locus contains the degree 3 curve Y given by x3 = x4 = x5 = f3 (x0 , x1 , x2 ) = 0, which is smooth for appropriate f3 . Thus we get a contribution to βk (X) ∈ B4 (G, k), in the summand labeled by Y ∈ Bir3,2 : [7, 2, 4] = 0 ∈ B3 (C8 ) ∼ = F2 . Here, we solve 289 linear equations in 120 variables. This implies that X is not G-equivariantly birational to P4 . Of course, this also follows by observing that the fourth power of the generator fixes a cubic threefold. Using Burnn (G): Consider X ⊂ P5 given by x0 x21 + x02 x2 − x0 x22 − 4x0 x24 + x21 x2 + x32 x5 − x2 x24 − x35 = 0.

(6.3)

It carries the action of G = C6 , which acts with weights (0, 0, 0, 1, 3, 4). The cubic fourfold X is rational, since it contains the disjoint planes x0 = x1 − x4 = x3 − x5 = 0

and

x2 = x1 − 2x4 = x3 + x5 = 0.

Noticing a cubic surface S ⊂ X, with C3 -stabilizer and scalar action on the normal bundle, the fact that the C2 -action on S fixes an elliptic curve on it lets us conclude, by [10], that the cubic surface is not stably C2 -equivariantly rational; the corresponding symbol [C3 , C2 ý k(S), β] = 0 ∈ Burn4 (C6 ), moreover, it does not interact with any other symbols in [X ý G], which implies that X is not G-birational to P4 with linear action. In this case, no subgroup of C6 fixes a hyperplane section. We discuss obstructions of such type in Section 7.4 below – formally, Burn4 (C6 ) admits a projection to Z that distinguishes the equivariant birational class of X from that of P4 with linear action. Kuznetsov [23] conjectures which cubic fourfolds X are rational, in terms of their derived categories Db (X). Consider the line bundles OX , OX (1), OX (2) and the right orthogonal complement ⊥

AX = OX , OX (1), OX (2) .

(6.4)

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∼ Db (S), the bounded derived Conjecturally, X is rational if and only if AX = category of a K3 surface S. However, the K3 surface need not be canonically determined as there are many examples of derived-equivalent but nonisomorphic K3 surfaces. Indeed the following conditions on complex projective K3 surfaces S1 and S2 are equivalent [30] ∼ Db (S2 ); • Db (S1 ) = • the Mukai lattices  1 , Z) ∼  2 , Z) H(S = H(S as Hodge structures; • the transcendental cohomology lattices ∼ H2 H 2 (S1 , Z) = tran

tran (S2 , Z)

as Hodge structures. There is an alternative Hodge-theoretic version of the conjecture: A smooth cubic fourfold X is rational if and only if there exists a K3 surface S and an isomorphism of integral Hodge structures H 4 (X, Z)tran ∼ (6.5) = H 2 (S, Z)tran (−1), where tran denotes the orthogonal complement of the Hodge classes and (−1) designates the Tate twist. Work of Addington and Thomas [1], and recent extensions [3, Cor. 1.7], show that the conditions (6.4) and (6.5) are equivalent. In particular, both are stable under specialization, consisting of an explicit countable collection of divisors in moduli [14]. The main theorem of [20] – that rationality is stable under specializations of smooth projective varieties – gives the equivalence of Kuznetsov’s conjecture with the Hodge-theoretic statement. Suppose then that X admits an action of a finite group G. If X is rational – and the conjectures are true – then G naturally acts on AX and Db (S), for each  ∗ (S, Z) as well. It is surface S arising in (6.4). There is an induced action on H ∼ natural to speculate that a G-equivariant birational map P4  X should imply that we may choose S in its derived equivalence class so that the G-action on the Mukai lattice is induced by a G-action on S. There are several possible obstructions to finding such an S: • if S exists then there exists a sublattice of algebraic classes   0 1 = H 2 (S, Z)⊥ U := 1 0 in the G-invariant part of the abstract Mukai lattice arising from AX ; • the action of G on Pic(S) preserves the ample cone of S. The first condition fails when G permutes various derived equivalent K3 surfaces. The second condition fails if G includes a Picard-Lefschetz transformation associated with a smooth rational curve P1 ⊂ S. Derived equivalent K3 surfaces might have very different automorphism groups [15, Ex. 23]; this paper discusses descent of derived equivalence in the presence of Galois actions.

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We mention some results on when the group action can be lifted to the associated K3 surface [31, §8]: 3 • if G = {1}, G acts on X symplectically, i.e., acts trivially on H1 (ΩX ), then S is unique; • if X is the Klein cubic fourfold

x30 + x21 x2 + x22 x3 + x32 x4 + x24 x5 + x25 x1 = 0 then X admits a symplectic automorphism of order 11 and AX ∼ = Db (S) for a unique K3 surface, which has no automorphism of order 11. We speculate that the Klein example should not be C11 -equivariantly rational, even though β(X ý C11 ) = 0 ∈ B4 (C11 ), as the C11 -action has isolated fixed points and the target group is trivial [19, §8]. Question 6.1. Let X be a smooth cubic fourfold with the action of a (finite) group ∼ Db (S) for a K3 surface S with G-action, compatible with G. Suppose that AX = the isomorphism. Does it follow that [X ý G] = [P4 ý G] ∈ Burn4 (G), for some action of G on P4 ? It is mysterious how the invariants in the Burnside groups interact with the actions on the Hodge structures on the middle cohomology of X. Obstructions to G-equivariant rationality arise from fixed loci in various dimensions but the Hodge theory encodes codimension-two cycles only. The example (6.3), which is rational but not C6 -rational, is particularly striking to us: How is the cubic surface in the fixed locus coupled with the associated K3 surfaces?

7. Nonabelian invariants In this section, G is a finite group, not necessarily abelian. 7.1. The equivariant Burnside group As in Section 5.2, it is defined as the quotient of the Z-module generated by symbols (H, NG (H)/H ý K, β), similar to those in (5.2), by blow-up relations. The required relations are a bit complicated but similar in spirit to what was written above; precise definitions are in [22, Section 4]. The resulting group Burnn (G) carries a rich combinatorial structure, that remains largely unexplored.

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7.2. Resolution of singularities The class of X ý G, a projective variety with a generically free G-action, is computed on a suitable model of the function field k(X). We explain how such a model may be found in practice. While this is a corollary of Bergh’s ‘destackification’ procedure [6], the approach here can be helpful in specific examples. We first review the resolution process of [35, §3]. A variety with group action as above is in standard form with respect to a G-invariant divisor Y if • X is smooth and Y is a simple normal crossings divisor; • the G action on X \ Y is free; • for every g ∈ G and irreducible component Z ⊂ Y either g(Z) = Z or g(Z) ∩ Z = ∅. We recall several fundamental results. First, we can always put actions in standard form: If X is smooth and Y is a G-invariant closed subset such that G acts freely on X \ Y then there exists a resolution  →X π:X obtained as a sequence of blowups along smooth G-invariant centers,  is in standard form with respect to Exc(π) ∪ π −1 (Y ) [35, such that X Th. 3.2]. An action in standard form has stabilizers of special type: Assume that X is in standard form with respect to Y and x ∈ X lies on m irreducible components of Y then the stabilizer H of x is abelian with ≤ m generators [35, Th. 4.1]. The proof of [35, Th. 4.1] (see Remark 4.4) yields ´etale-local coordinates on X about x x1 , . . . , xk , y1 , . . . , yl , z1 , . . . , zm such that • H acts diagonally on all the coordinates; • y1 = · · · = yl = z1 = · · · = zm = 0 coincides with X H , i.e., these are the coordinates on which H acts nontrivially; • y1 · · · ym = 0 coincides with Y and the associated characters of χi : H → Gm generate Hom(H, Gm ) so the induced representation (χ1 , . . . , χm ) : H → Gm m

(7.1)

is injective. Suppose that X is in standard form with respect to Y with irreducible components Y1 , . . . , Ys . For each orbit of these under the action of G, consider the reduced divisor obtained by summing over the orbit. The resulting divisors D(1), . . . , D(r) have smooth support – by the definition of standard form – and D(1) ∪ · · · ∪ D(r) = Y1 ∪ · · · ∪ Ys .

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The line bundles OX (D(i)) are naturally G-linearized and thus descend to line bundles on the quotient stack [X/G] and we obtain ϕ : [X/G] → BGm × · · · × BGm

(r factors).

We claim ϕ is representable. It suffices to check this by showing that the induced homomorphism of stabilizers is injective at each point [38, Tag 04YY]. For x ∈ X, fix the indices i1 , . . . , im so that D(i1 ), . . . , D(im ) are the components of Y containing x, and consider the induced ϕx : [X/G] → BGm × · · · × BGm

(m factors).

The homomomorphism on stabilizers is given by (7.1) which is injective. Thus we have established: Proposition 7.1. Let X be a smooth projective variety with a generically free action by a finite group G, in standard form with respect to a divisor Y . Then Assumption 2 of [22] holds and the invariants constructed there may be evaluated on X. 7.3. The class of X ý G On a suitable model X, we consider each stratum F ⊂ X with nontrivial (abelian) stabilizer H ⊂ G, the action of the normalizer NG (H)/H on (the orbit of) the stratum, and the induced action of H on its normal bundle, and record these in a symbol. Then we define   [X ý G] := (H, NG (H)/H ý k(F ), β) ∈ Burnn (G), (7.2) H⊆G F

as (5.3). The proof that this is a G-birational invariant relies on G-equivariant Weak Factorization and combinatorics [22, Section 5]. 7.4. Elementary observations As we discussed, the presence of higher genus curves in the fixed locus of the action of a cyclic group of prime order on a rational surface is an important invariant in the study of the plane Cremona group; see, e.g., [9]. These make up entirely Z/2Z the group Burn2 (Z/2Z) and entirely characterize birational involutions of the plane up to conjugation [4]. More generally, for any nontrivial cyclic subgroup H of G and birational class of an (n − 1)-dimensional variety Y , not birational to Z × P1 for any variety Z of dimension n − 2, we have a projection from Burnn (G) onto the free abelian group on the NG (H)-conjugacy classes of pairs (H  , a), where H  is a subgroup, H ⊂ H  ⊂ NG (H), and a ∈ H ∨ is a primitive character. This sends N (H)/H

(H, IndHG /H

(k(Y )), a),

for any H  /H ý k(Y ), to the generator indexed by the conjugacy class [(H  , a)] of the pair.

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A more refined version of this observation might also relax the restriction on Y but take into account the action H  /H ý k(Y ). We do not go into details, but only point out, for instance, that for n = 2 and Y = P1 there is a projection  Z. Burn2 (G) → [(H  ,a)] H  /H not cyclic

Taking G to be the dihedral group of order 12 and H the center of G, we may distinguish between the two inclusions of G into the plane Cremona group considered in [18], see Section 7.6 below. 7.5. Dihedral group of order 12 We now compute Burn2 (G), for G = D6 , the dihedral group with generators ρ,

σ,

with

ρ6 = σ 2 = ρσρσ = eD6 .

We list abelian subgroups, up to conjugacy: • order 6: C6 = ρ • order 4: D2 = ρ3 , σ • order 3: C3 = ρ2  • order 2: central C2 = ρ3 , noncentral S := σ, S  := ρ3 σ • order 1: triv The subgroup of order 4 and two noncentral subgroups of order 2 have normalizer D2 , the others are normal. As before, we use k(X) to denote the function field of the underlying surface X, K to denote the algebra of rational functions of a one-dimensional stratum with nontrivial stabilizer, and k n to denote the algebra of functions of a zerodimensional orbit of length n. When we blow up such an orbit, we use k n (t) to denote the total ring of fractions of the exceptional locus. Generators: (C6 , ¯ σ  ý K, (1)) (C6 , ¯ σ  ý k 2 , (1, j)), j = 1, . . . , 4, (C6 , ¯ σ  ý k 2 , (2, 3)) (D2 , triv ý k, (a1 , a2 )), a1 , a2 ∈ F22 , generating F22 ¯  ý K, (1)) ρ, σ (C3 , ¯ (C3 , ¯ ρ, σ ¯  ý k 4 , (1, 1)) (C2 , ¯ ¯  ý K, (1)) ρ, σ (S, C2 ý K, (1)) (S  , C2 ý K, (1)) (triv, D6 ý k(X), triv) Relations: (C6 , ¯ σ  ý k 2 , (1, 1)) = (C6 , ¯ σ  ý k 2 (t), (1)) 2 (C6 , ¯ σ  ý k 2 , (1, 4)) σ  ý k 2 , (1, 1)) + (C6 , ¯ σ  ý k , (1, 2)) = (C6 , ¯ 2 2 (C6 , ¯ σ  ý k , (1, 3)) = (C6 , ¯ σ  ý k , (1, 2)) + (C6 , ¯ σ  ý k 2 , (2, 3)) +(C2 , ¯ ρ, σ ¯  ý k 2 (t), (1)),

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where ρ¯ acts by cube roots of unity on t (C6 , ¯ σ  ý k 2 , (1, 4)) = (C6 , ¯ σ  ý k 2 , (2, 3)) σ  ý k 2 , (1, 3)) + (C6 , ¯ +(C3 , ¯ ρ, σ ¯  ý k 2 (t), (1)), where ρ¯ acts by −1 on t 0 = 2(C6 , ¯ σ  ý k 2 , (1, 4)) + (C2 , ¯ ρ, σ ¯  ý k 2 (t), (1)), where ρ¯ acts by cube roots of unity on t (C6 , ¯ σ  ý k 2 , (2, 3)) = (C6 , ¯ σ  ý k 2 , (1, 2)) + (C6 , ¯ σ  ý k 2 , (1, 3)) (D2 , triv ý k, ((1, 0), (0, 1))) = (D2 , triv ý k, ((1, 0), (1, 1))) +(D2 , triv ý k, ((0, 1), (1, 1))) + (S  , C2 ý k(t), (1)) (D2 , triv ý k, ((1, 0), (1, 1))) = (D2 , triv ý k, ((1, 0), (0, 1))) +(D2 , triv ý k, ((0, 1), (1, 1))) + (C2 , ¯ ¯  ý k 3 (t), (1)), ρ, σ 3 permutation action on k with σ ¯ acting by −1 on t (D2 , triv ý k, ((0, 1), (1, 1))) = (D2 , triv ý k, ((1, 0), (0, 1))) +(D2 , triv ý k, ((1, 0), (1, 1))) + (S, C2 ý k(t), (1)) (C3 , ¯ ρ, σ ¯  ý k 4 , (1, 1)) = (C3 , ¯ ¯  ý k 4 (t), (1)) ρ, σ 0 = 2(C3 , ¯ ρ, σ ¯  ý k 4 , (1, 1)) 0 = (C2 , ¯ ρ, σ ¯  ý k 6 (t), (1)) 0 = (S, C2 ý k 2 (t), (1)) 0 = (S  , C2 ý k 2 (t), (1)) 7.6. Embeddings of S3 × C2 into the Cremona group Iskovskikh [18] exhibited two nonconjugate copies of G = S3 × C2 ∼ = D6 in BirAut(P2 ): • the action on x1 + x2 + x3 = 0 by permutation and reversing signs, with model P2 ; • the action on y1 y2 y3 = 1 by permutation and taking inverses, with model a sextic del Pezzo surface. To justify the interest in these particular actions we observe that G is the Weyl group of the exceptional Lie group G2 , which acts on the Lie algebra of the torus, respectively on the torus itself, and it is natural to ask whether or not these actions are equivariantly birational. It turns out that they are stably G-birational [24, Proposition 9.11], but not G-birational. The proof of failure of G-birationality in [18] relies on the classification of links, via the G-equivariant Sarkisov program. Here we explain how to apply Burn2 (G) to this problem. Note that neither model above satisfies the stabilizer condition required in the Definition (7.2)! We need to replace the surfaces by appropriate models X and Y , in particular, to blow up points: • (x1 , x2 , x3 ) = (0, 0, 0), with G as stabilizer; • (y1 , y2 , y3 ) = (1, 1, 1), with G as stabilizer, and (ω, ω, ω), with S3 as stabilizer.

(ω 2 , ω 2 , ω 2 ),

ω = e2πi/3 ,

Symbols and Equivariant Birational Geometry We describe these actions in more detail, following on P2 , with coordinates (u0 : u1 : u2 ) is given by ⎞ ⎛ ⎞ ⎛ ⎛ 1 0 0 1 1 0 0 3 ⎝0 0 1⎠ , ⎝0 0 1 ⎠ , ρ = ⎝0 0 −1 −1 0 0 1 0

233

closely [18]. The action ⎞ 0 0 −1 0 ⎠ . 0 −1

There is one fixed point, (1 : 0 : 0); after blowing up this point, the exceptional curve is stabilized by the central involution ρ3 , and comes with a nontrivial S3 action, contributing the symbol (C2 , S3 ý k(P1 ), (1))

(7.3)

to [X ý G]. Additionally, the line 0 := {u0 = 0} has as stabilizer the central C2 , contributing the same symbol. There are also other contributing terms, of the shape: • (C6 , C2 ý k 2 , β) • (D2 , triv ý k, β  ) for some weights β, β  . A better model for the second action is the quadric v0 v1 + v1 v2 + v2 v0 = 3w2 , where S3 permutes the coordinates (v0 : v1 : v2 ) and the central involution exchanges the sign on w. There are no G-fixed points, but a conic R0 := {w = 0} with stabilizer the central C2 and a nontrivial action of S3 . There are also: • a G-orbit of length 2: {P1 := (1 : 1 : 1 : 1), P2 := (1 : 1 : 1 : −1)}, exchanged by the central involution, each point has stabilizer S3 – these points have to be blown up, yielding a pair of conjugated P1 , with a nontrivial S3 -action; • another curve R1 := {v0 + v1 + v2 = 0} with effective G-action; • additional points with stabilizers C6 and D2 in R0 and R1 . The essential difference is that the symbol (7.3) appears twice for the action on P2 , and only once for the action on the quadric: the pair of conjugated P1 with S3 -action has trivial stabilizer and does not contribute. Further blow-ups will not introduce new curves of this type. Formally, examining the relations in Section 7.5, we see that the symbol (7.3) is not equivalent to any combination of other symbols, i.e., it is independent of all other symbols. This implies that [X ý G] = [Y ý G] ∈ Burn2 (G), thus X and Y are not G-equivariantly birational. Note, that X and Y are equivariantly birational for any proper subgroup of G. Remark 7.2. One can view the symbol (7.3) as the analog of a curve of higher genus in the fixed locus of an element in the classification of abelian actions on surfaces, as discussed in Section 3.

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Acknowledgment The first author was partially supported by NSF grant 1701659 and Simons Foundation award 546235. The second author was partially supported by the Swiss National Science Foundation. The third author was partially supported by NSF grant 2000099.

References [1] N. Addington and R. Thomas. Hodge theory and derived categories of cubic fourfolds. Duke Math. J., 163(10):1885–1927, 2014. [2] M.F. Atiyah and R. Bott. A Lefschetz fixed point formula for elliptic complexes. II. Applications. Ann. of Math. (2), 88:451–491, 1968. [3] A. Bayer, M. Lahoz, E. Macr`ı, H. Nuer, A. Perry, and P. Stellari. Stability conditions in families, 2019. arXiv:1902.08184. [4] L. Bayle and A. Beauville. Birational involutions of P2 . Asian J. Math., 4(1):11–17, 2000. Kodaira’s issue. [5] A. Beauville and J. Blanc. On Cremona transformations of prime order. C. R. Math. Acad. Sci. Paris, 339(4):257–259, 2004. [6] D. Bergh. Functorial destackification of tame stacks with abelian stabilisers. Compos. Math., 153(6):1257–1315, 2017. [7] J. Blanc. Finite abelian subgroups of the Cremona group of the plane. PhD thesis, Universit´e de Gen`eve, 2006. Th`ese no. 3777, arXiv:0610368. [8] J. Blanc. Linearisation of finite abelian subgroups of the Cremona group of the plane. Groups Geom. Dyn., 3(2):215–266, 2009. [9] J. Blanc. Elements and cyclic subgroups of finite order of the Cremona group. Comment. Math. Helv., 86(2):469–497, 2011. [10] F. Bogomolov and Yu. Prokhorov. On stable conjugacy of finite subgroups of the plane Cremona group, I. Cent. Eur. J. Math., 11(12):2099–2105, 2013. [11] I. Cheltsov and C. Shramov. Three embeddings of the Klein simple group into the Cremona group of rank three. Transform. Groups, 17(2):303–350, 2012. [12] T. de Fernex. On planar Cremona maps of prime order. Nagoya Math. J., 174:1–28, 2004. [13] I.V. Dolgachev and V.A. Iskovskikh. Finite subgroups of the plane Cremona group. In Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, volume 269 auser Boston, Boston, MA, 2009. of Progr. Math., pages 443–548. Birkh¨ [14] B. Hassett. Special cubic fourfolds. Compositio Math., 120(1):1–23, 2000. [15] B. Hassett and Yu. Tschinkel. Rational points on K3 surfaces and derived equivalence. In Brauer groups and obstruction problems, volume 320 of Progr. Math., pages 87–113. Birkh¨ auser/Springer, Cham, 2017. [16] O. Haution. Fixed point theorems involving numerical invariants. Compos. Math., 155(2):260–288, 2019. [17] V.A. Iskovskih. Minimal models of rational surfaces over arbitrary fields. Izv. Akad. Nauk SSSR Ser. Mat., 43(1):19–43, 237, 1979.

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[38] The Stacks Project Authors. Stacks Project. https://stacks.math.columbia.edu, 2020. Brendan Hassett Brown University Box 1917 151 Thayer Street Providence, RI 02912, USA e-mail: [email protected] Andrew Kresch Institut f¨ ur Mathematik Universit¨ urich at Z¨ Winterthurerstrasse 190 CH-8057 Z¨ urich, Switzerland e-mail: [email protected] Yuri Tschinkel Courant Institute 251 Mercer Street New York, NY 10012, USA and Simons Foundation 160 Fifth Avenue New York, NY 10010, USA e-mail: [email protected]

Rationality of Fano Threefolds of Degree 18 over Non-closed Fields Brendan Hassett and Yuri Tschinkel Abstract. We study unirationality and rationality of Fano threefolds of degree 18 over nonclosed fields. Mathematics Subject Classification (2010). 14J45, 14C25, 14E08, 11G10. Keywords. Fano threefold, rational variety, intermediate Jacobian, non-closed field, abelian surface.

1. Introduction Manin [Man93] proposed to study (uni)rationality of Fano threefolds over nonclosed fields, in situations where geometric (uni)rationality is known. In cases where the Picard group is generated by the canonical class, i.e., those of rank and index one, he assigned an ‘Exercise’ [Man93, p. 47] to explore the rationality of degree 12, 16, 18, and 22. See [IP99, p. 215] for a list of geometrically rational Fano threefolds of rank one. We have effective criteria for deciding the rationality of surfaces over nonclosed fields – the relevant invariant is encoded in the Galois action on the geometric Picard group. This invariant is trivial for Fano threefolds considered above. Our main result is: Theorem 1. Let X be a Fano threefold of degree 18 defined over a field k of characteristic zero and admitting a k-rational point. Then X is rational if and only if X admits a conic over k. Here a conic means a geometrically connected curve of degree two – possibly non-reduced or reducible. Kuznetsov and Prokhorov [KP19] complete the study of rationality for geometrically rational Fano threefolds of Picard rank one over non-closed fields. In particular, they address the degree 16 case where – assuming the existence of a rational point – rationality is equivalent to the existence of a twisted cubic curve. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Farkas et al. (eds.), Rationality of Varieties, Progress in Mathematics 342, https://doi.org/10.1007/978-3-030-75421-1_9

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For Fano threefolds of degrees 12 and 22, rationality holds if and only if there is a rational point. A key step in the proof of the only if direction in Theorem 1 is the analysis of torsors over intermediate Jacobians, as presented in [HT19a, BW19]. The other direction uses deformation and specialization techniques. While much recent work on rationality has focused on applications of specialization to show the failure of (stable) rationality [Voi15, CTP16, HKT16, Tot16, HPT18, Sch19, NS19, KT19], here we use it to prove rationality, avoiding complicated case-by-case arguments for special geometric configurations; see Theorem 8. This technique was also used to analyze rationality for cubic fourfolds [RS19].

2. Projection constructions We work over a field k of characteristic zero. Let X ⊂ Pr be a smooth Fano threefold of Picard rank one, embedded by the minimal very ample multiple of the anticanonical divisor. Fix a center consisting  →X of a point x ∈ X or a smooth curve  ⊂ X and consider the blowup σ : X along that center. Assume that • −KX is nef and big; • there are no effective divisors D ⊂ X such that (−KX )2 · D = 0. Then by [IP99, Lem. 4.1.1] there exists an n ≥ 1 such that | − nKX | determines a birational morphism  → X φ : X to a normal variety with (at worst) terminal singularities. The morphism φ is a small resolution and an isomorphism if and only if X  is nonsingular; this happens  is also Fano. precisely when X  with  + of X After flopping rational curves as necessary, we obtain a model X +   X  denote the induced flop, φ : semiample anticanonical class. Let χ : X  + → Y the contraction of the other extremal ray (which need not be birational), X and ψ : X  Y the composed map (cf. [IP99, (4.1.1)]):  _ _ _ _χ _ _ _/ X + X @@ { @@ φ { + φ {{ @@ @@ {{ }{{ σ φ X

(1)

  ψ X _ _ _ _ _ _ _/ Y The contraction φ may often be understood in terms of projections. We suppose that X is anticanonically embedded and n = 1. • Assume the center is a line  ⊂ X. The induced rational map on X may be interpreted as projection from . Other lines incident to  are contracted by φ .

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 corre• Assume the center is a point x ∈ X. The anticanonical system on X sponds to anticanonical divisors on X with multiplicity ≥ 2 at x. The induced map φ may be interpreted as double projection from x, i.e., projection from the tangent space at x. Conics containing x are contracted by φ . When we refer to m-fold projection along a point or curve, this means imposing zeros of multiplicity m along the exceptional divisor of σ. For the remainder of this section, X denotes a smooth Fano threefold of degree 18 over k. 2.1. Projection from lines The variety of lines R1 (X) is nonempty and connected of pure dimension one [IP99, Prop. 4.2.2] and sweeps out a divisor in X with class −3KX [IP99, Th. 4.2.7]. For generic X, R1 (X) is a smooth curve of genus ten [IP99, Th. 4.2.7]. If R1 (X) is smooth then X admits no nonreduced conics [KPS18, Rem. 2.1.7]. Suppose that  ⊂ X is a line. Then double projection along  induces a birational map as in Diagram (1)   X  + → Y, X where Y ⊂ P4 is a smooth quadric hypersurface [IP99, Th. 4.3.3]. This flops the three lines incident to  and contracts a divisor D ∈ | − 2KX + − 3E + | to a smooth curve C ⊂ Y of degree seven and genus two. Since Y admits a k-rational point it is rational over k; the same holds true for X. Proposition 2. If X is a Fano threefold of degree 18 admitting a line over k then X is rational. 2.2. Projection from conics We discuss the structure of the variety R2 (X) of conics on X: • R2 (X) is nonempty of pure dimension two [IP99, Th. 4.5.10]. • R2 (X) is geometrically isomorphic to the Jacobian of a genus two curve C [IM07, Prop. 3] [KPS18, Th. 1.1.1]. • Through each point of X there pass finitely many conics [IP99, Lem. 4.2.6]; indeed, through a generic such point we have nine conics [Tak89, 2.8.1]. • Given a conic D ⊂ X, double projection along D induces a fibration [IP99, Cor. 4.4.3,Th. 4.4.11] φ + → X  X P2

in conics with quartic degeneracy curve.

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2.3. Projection from points  We recall the results of Takeuchi [Tak89] presented in [IP99, Th. 4.5.8]. Let X denote the blowup of X at x, with exceptional divisor E.  denote the blowup of Proposition 3. Suppose we have a point x ∈ X(k) and let X X at x. We assume that • x does not lie on a line in X; • there are no effective divisors D on X such that (KX )2 · D = 0. Then triple-projection from x gives a fibration ∼ φ + → P1 X  X

in sextic del Pezzo surfaces. We offer a more detailed analysis of double projection from a point x ∈ X(k) not on a line. By [IP99, § 4.5] the projection morphism  → P7 φ˜ : X is generically finite onto its image X and the Stein factorization 

φ φ → X X → X

yields a Fano threefold of genus six with canonical Gorenstein singularities. The condition precluding effective divisors D with (KX )2 · D = 0 means that φ admits no exceptional divisors. The nontrivial fibers of φ are all isomorphic to P1 ’s, with the following possible images in X: 1. a conic in X through x; 2. a quartic curve of arithmetic genus one in X, spanning a P3 , with a singularity of multiplicity two at x; 3. a sextic curve of arithmetic genus two in X, spanning a P4 , with a singularity of multiplicity three at x. Moreover, if φ does not contract any surfaces then the exceptional divisor E over x is embedded in P7 as a Veronese surface. The quartic curves on X with node at a fixed point x have expected dimension 0. The sextic curves on X with transverse triple point at a fixed point x have expected dimension −1. Indeed, we have: Proposition 4 ([IP99, Prop. 4.5.1]). Retain the notation above. For a generic x ∈ X • the quartic and sextic curves described above do not occur; • φ is a small contraction; ∼  + factors as follows • the rational map X  X 1. blow up the point x; 2. flop the nine conics through x; • φ restricts to the proper transform E + of E as an elliptic fibration associated with cubics based at nine points.

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3. Unirationality constructions In this section, we consider the following question inspired by [Man93, p. 46]: Question 5. Let X be a Fano threefold of degree 18 over k. Suppose X(k) = ∅. Is X is unirational over k? From our perspective, unirationality is more delicate than rationality as we lack a specialization theorem for smooth families in this context. We cannot apply the theorem of [KT19] – as we do in the proof of Theorem 8 – to reduce to configurations in general position. The geometric constructions below highlight some of the issues that arise. 3.1. Using a point Proposition 6. Let X be a Fano threefold of degree 18 over k admitting a point x ∈ X(k) satisfying the condition in Proposition 3. Then X is unirational over k and rational points are Zariski dense. Proof. We retain the notation from Proposition 3. Note that the proper transforms of lines L ⊂ E + give trisections of our del Pezzo fibration  + → P1 . φ:X Basechanging to L yields  + ×P1 L → L, φL : X a fibration of sextic del Pezzo surfaces with a section. Thus the generic fiber of φL is rational over k(L) by [Man66, p. 77]. Since L  P1 , the total space of the fibration  + , we conclude that X is unirational.  is rational over k. As it dominates X If the rational points are Zariski dense then we can find one where Proposition 3 applies. However, if we are given only a single rational point on X we must make a complete analysis of degenerate cases as partly described in Section 2.3. In addition, we must consider cases where there exist lines over k¯ passing through our given rational point. For instance, consider the case where a single line x ∈  ⊂ X. To resolve the double projection at x, we must take the following steps: • blow up x to obtain an exceptional divisor E1  P2 ; • blow up the proper transform  of the line  with N  = OP1 (−1) ⊕ OP1 (−2) to obtain an exceptional divisor E2  F1 . Let E1 and denote the proper transform of E1 in the second blowups. The linear series resolving the double projection is h − 2E1 − E2

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which takes E1  F1 to a cubic scroll, E2 to a copy of P2 , and the (−1)-curve on E2 to an ordinary singularity on the image. The induced contraction  → X  ⊂ P7 φ : X has degree

(h − 2E1 − E2 )3 = 10. Thus X  admits a ‘degenerate Veronese surface’ consisting of a cubic scroll and a plane meeting along a line coinciding with the (−1)-curve of the scroll; X  has an ordinary singularity along that line. Of course, the most relevant degenerate cases for arithmetic purposes involve multiple lines through x conjugated over the ground field. It would be interesting to characterize the possibilities. 3.2. Using a point and a conic Here is another approach: Let X admit a point x ∈ X(k) and a conic D ⊂ X defined over k. The results recalled in Section 2.2 imply that X is birational over k to  + → P2 , φ:X a conic bundle degenerating over a plane quartic curve B.  + whose image p ∈ P2 is not Suppose there exists a rational point on X contained in the degeneracy curve. Consider the pencil of lines through p. The  + are conic bundles over P1 with four degencorresponding pencil of surfaces on X erate fibers and the resulting fibration admits a section. Such a surface is either isomorphic to a quartic del Pezzo surface or birational to such a surface [KST89, p. 48]. It is a classical fact that a quartic del Pezzo surface with a rational point is unirational. The argument works even when p is a smooth point or node of B. Here we necessarily have higher-order ramification over the nodes – this is because the associated generalized Prym variety is compact – which we can use to produce a section of the resulting pencil of degenerate quartic del Pezzo surfaces. However, there is trouble when p is a cusp of B.

4. Rationality results Our first statement describes the rationality construction under favorable genericity assumptions: Proposition 7. Let X be a Fano threefold of degree 18 over k. Assume that: • there exists an x ∈ X(k) satisfying the conditions of Proposition 3 so X is  + → P1 in sextic del Pezzo surfaces; birational to a fibration φ : X • there exists an irreducible curve M ⊂ X, disjoint from the indeterminacy of ∼  + , with degree prime to three. X  X Then X is rational over k.

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Proof. We saw in the proof of Proposition 6 that the generic fiber S of φ is a sextic del Pezzo surface admitting a rational point of a degree-three extension. Our assumptions imply that S · M = deg(M ) which is prime to three so applying [Man66, p. 77] we conclude that S is rational over k(P1 ) and X is rational over  the ground field. We now show these genericity assumptions are not necessary: Theorem 8. Let X be a Fano threefold of degree 18 over k. Assume that X admits a rational point x and a conic D, both defined over k. Then X is rational. Proof. Let B denote the Hilbert scheme of all triples (X, x, D) of objects described in the statement. This is smooth and connected over the moduli stack of degree 18 Fano threefolds; indeed, we saw in Section 2.2 that the parameter space of conics on X is an abelian surface. The moduli stack itself is a smooth Deligne-Mumford stack since Kodaira vanishing gives H i (TX ) = 0 for i = 2, 3 and H 0 (TX ) = 0 by [Pro90]. The classification of Fano threefolds shows that the moduli stack is connected. Thus B is smooth and connected. Consider the universal family π

(X → B, x : B → X , D ⊂ X ), where π is smooth and projective. The generic fiber of π is rational over k(B) as the genericity conditions of Proposition 7 are tautologically satisfied – see Proposition 4 for details. The specialization theorem [KT19, Thm. 1] implies that every k-fiber of π is rational over k. This theorem assumes the base is a curve. However, our parameter space B is smooth so Bertini’s Theorem implies that each b ∈ B(k)  may be connected to the generic point by a curve smooth at b.

5. Analysis of principal homogeneous spaces 5.1. Proof of Theorem 1 One direction is Theorem 8; we focus on the converse. Suppose that X is rational over the ground field. Let C be the genus two curve whose Jacobian J(C) is isomorphic to the intermediate Jacobian IJ(X) over k. The mechanism of [HT19a, § 5] gives a principal homogeneous space P over J(C) with the property that the Hilbert scheme Hd parametrizing irreducible curves of degree d admits a morphism Hd → Pd descending the Abel–Jacobi map to k, where [Pd ] = d[P ] in the Weil-Chˆatelet group of J(C). By Theorem 22 of [HT19a], if X is rational then P  Pici (C) for i = 0 or 1. In particular, we have R1 (X) → P and by the known results of Section 2.2 R2 (X)  P2  J(C).

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Indeed, since C has genus two we have identifications J(C) = Pic0 (C)  Pic2 (C), which gives the desired interpretation of P2 whether P = Pic0 (C) or Pic1 (C). As a consequence, R2 (X) admits a k-rational point. 5.2. A corollary to Theorem 1 Retain the notation of the previous section. Without assumptions on the existence of points or conics on X defined over k, we know that 18[P ] = 0 and 9[R2 (X)] = 0 in the Weil-Chˆatelet group. This allows us to deduce an extension of our main result: Corollary 9. Let X be Fano threefold of degree of degree 18 over k with X(k) = ∅. Suppose that X admits a curve of degree prime to three, defined over k. Then X is rational. Our assumption means that 2[P ] = 0, whence [R2 (X)] = 0 and X admits a conic defined over k. Hence Theorem 1 applies. 5.3. Generic behavior There are examples over function fields where the principal homogeneous space is not annihilated by two: Proposition 10. Over k = C(P2 ), there exist examples of X such that the order of [P ] is divisible by three. Proof. Let S be a complex K3 surface with Pic(S) = Zh where h2 = 18. Mukai [Muk88] has shown that S arises as a codimension-three linear section of a homogeneous space W ⊂ P13 arising as the closed orbit for the adjoint representation of G2 S = P10 ∩ W. Consider the associated net of Fano threefolds  : X → P2 obtained by intersecting W with codimension-two linear subspaces P10 ⊂ P11 ⊂ P13 . Write X for the generic fiber over C(P2 ). Let R2 (X /P2 ) denote the relative variety of conics. This was analyzed in [IM07, § 3.1]: The conics in fibers of  cut out pairs of points on S, yielding a birational identification and natural abelian fibration ∼

ψ

S [2]  R2 (X /P2 ) → P2 . The corresponding principal homogeneous space has order divisible by nine; its order is divisible by three if it is nontrivial.

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These fibrations are analyzed in more depth in [MSTVA17, § 3.3] and [KR13]. Let T denote the moduli space of rank-three stable vector bundles V on S with c1 (V ) = h and χ(V ) = 6. Then we have • T is a K3 surface of degree two; • the primitive cohomology of S arises as an index-three sublattice of the primitive cohomology of T H 2 (S, Z)prim ⊂ H 2 (T, Z)prim , compatibly with Hodge structures; • the Hilbert scheme T [2] is birational to the relative Jacobian fibration of the degree-two linear series on T J → P2 ; • the relative Jacobian fibration of ψ is birational to J over P2 . The last statement follows from [Saw07, p. 486] or [Mar06, § 4]: The abelian fibration ψ is realized as a twist of the fibration J → P2 ; the twisting data is encoded by an element α ∈ Br(T )[3] annihilating H 2 (S, Z)prim modulo three. Now suppose that ψ had a section. Then J and S [2] would be birational holomorphic symplectic varieties. The Torelli Theorem implies that their transcendental degree-two cohomology – H 2 (T, Z)prim and H 2 (S, Z)prim respectively – are isomorphic. This contradicts our computation above.  5.4. Connections with complete intersections? Assume k is algebraically closed and X a Fano threefold of degree 18 over k. Kuznetsov, Prokhorov, and Shramov [KPS18] have pointed out the existence of a smooth complete intersection of two quadrics Y ⊂ P5 with R1 (Y )  R2 (X),

(2)

Both have intermediate Jacobian isomorphic to the Jacobian of a genus two curve C. Now suppose that X and Y are defined over a non-closed field k with IJ(X)  IJ(Y ). In general, we would not expect R2 (X) and R1 (Y ) to be related as principal homogeneous spaces; for example, we generally have 9[R2 (X)] = 0 and 4[R1 (Y )] = 0 (see [HT19b]). Verra [Ver18] has found a direct connection between complete intersections of quadrics and singular Fano threefolds of degree 18. Suppose we have a twisted cubic curve R ⊂ Y ⊂ P5 , which forces Y to be rational. Consider the linear series of quadrics vanishing along R; the resulting morphism BlR (Y ) → P11 collapses the line residual to R in span(R) ∩ Y . Its image X0 is a nodal Fano threefold of degree 18.

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Acknowledgment The first author was partially supported by NSF grant 1701659 and the Simons Foundation.

References Olivier Benoist and Olivier Wittenberg. Intermediate Jacobians and rationality over arbitrary fields, 2019. arXiv:1909.12668. [CTP16] Jean-Louis Colliot-Th´el`ene and Alena Pirutka. Hypersurfaces quartiques ´ Norm. Sup´ er. (4), de dimension 3: non-rationalit´e stable. Ann. Sci. Ec. 49(2):371–397, 2016. [HKT16] Brendan Hassett, Andrew Kresch, and Yuri Tschinkel. Stable rationality and conic bundles. Math. Ann., 365(3-4):1201–1217, 2016. [HPT18] Brendan Hassett, Alena Pirutka, and Yuri Tschinkel. Stable rationality of quadric surface bundles over surfaces. Acta Math., 220(2):341–365, 2018. Brendan Hassett and Yuri Tschinkel. Cycle class maps and birational in[HT19a] variants, 2019. arXiv:1908.00406. [HT19b] Brendan Hassett and Yuri Tschinkel. Rationality of complete intersections of two quadrics, 2019. arXiv:1903.08979. [IM07] Atanas Iliev and Laurent Manivel. Prime Fano threefolds and integrable systems. Math. Ann., 339(4):937–955, 2007. [IP99] V.A. Iskovskikh and Yu.G. Prokhorov. Algebraic geometry. V: Fano varieties, volume 47 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 1999. A.N. Parshin and I.R. Shafarevich eds. [KP19] Alexander G. Kuznetsov and Yuri G. Prokhorov. Rationality of Fano threefolds over non-closed fields, 2019. arXiv:1911.08949. [KPS18] Alexander G. Kuznetsov, Yuri G. Prokhorov, and Constantin A. Shramov. Hilbert schemes of lines and conics and automorphism groups of Fano threefolds. Jpn. J. Math., 13(1):109–185, 2018. Michal Kapusta and Kristian Ranestad. Vector bundles on Fano varieties [KR13] of genus ten. Math. Ann., 356(2):439–467, 2013. ` Kunyavski˘ı, A.N. Skorobogatov, and M.A. Tsfasman. del Pezzo sur[KST89] B. E. faces of degree four. M´em. Soc. Math. France (N.S.), (37):113, 1989. [KT19] Maxim Kontsevich and Yuri Tschinkel. Specialization of birational types. Invent. Math., 217(2):415–432, 2019. ´ Sci. [Man66] Ju.I. Manin. Rational surfaces over perfect fields. Inst. Hautes Etudes Publ. Math., (30):55–113, 1966. [Man93] Yu.I. Manin. Notes on the arithmetic of Fano threefolds. Compositio Math., 85(1):37–55, 1993. [Mar06] Dimitri Markushevich. Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces. Manuscripta Math., 120(2):131–150, 2006. arilly[MSTVA17] Kelly McKinnie, Justin Sawon, Sho Tanimoto, and Anthony V´ Alvarado. Brauer groups on K3 surfaces and arithmetic applications. In

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Brauer groups and obstruction problems, volume 320 of Progr. Math., pages 177–218. Birkh¨ auser/Springer, Cham, 2017. Shigeru Mukai. Curves, K3 surfaces and Fano 3-folds of genus ≤ 10. In Algebraic geometry and commutative algebra, Vol. I, pages 357–377. Kinokuniya, Tokyo, 1988. Johannes Nicaise and Evgeny Shinder. The motivic nearby fiber and degeneration of stable rationality. Invent. Math., 217(2):377–413, 2019. Yu.G. Prokhorov. Automorphism groups of Fano 3-folds. Uspekhi Mat. Nauk, 45(3(273)):195–196, 1990. Francesco Russo and Giovanni Staglian` o. Congruences of 5-secant conics and the rationality of some admissible cubic fourfolds. Duke Math. J., 168(5):849–865, 2019. Justin Sawon. Lagrangian fibrations on Hilbert schemes of points on K3 surfaces. J. Algebraic Geom., 16(3):477–497, 2007. Stefan Schreieder. Stably irrational hypersurfaces of small slopes. J. Amer. Math. Soc., 32(4):1171–1199, 2019. Kiyohiko Takeuchi. Some birational maps of Fano 3-folds. Compositio Math., 71(3):265–283, 1989. Burt Totaro. Hypersurfaces that are not stably rational. J. Amer. Math. Soc., 29(3):883–891, 2016. Alessandro Verra. Coble cubic, genus 10 Fano threefolds and the theta map, 2018. Lecture at the ‘Differential, Algebraic and Topological Methods in Complex Algebraic Geometry’ conference held at Cetraro, Italy. Claire Voisin. Unirational threefolds with no universal codimension 2 cycle. Invent. Math., 201(1):207–237, 2015.

Brendan Hassett Brown University Box 1917 151 Thayer Street Providence, RI 02912, USA e-mail: [email protected] Yuri Tschinkel Courant Institute 251 Mercer Street New York, NY 10012, USA and Simons Foundation 160 Fifth Avenue New York, NY 10010, USA e-mail: [email protected]

Rationality of Mukai Varieties over Non-closed Fields Alexander Kuznetsov and Yuri Prokhorov Abstract. We discuss birational properties of Mukai varieties, i.e., of higherdimensional analogues of prime Fano threefolds of genus g ∈ {7, 8, 9, 10} over an arbitrary field k of zero characteristic. In the case of dimension n ≥ 4 we prove that these varieties are k-rational if and only if they have a k-point except for the case of genus 9, where the same holds for n ≥ 5. Furthermore, we prove that Mukai varieties of genus g ∈ {7, 8, 9, 10} and dimension n ≥ 5 contain cylinders if they have a k-point. Finally, we prove that the embedding X → Gr(3, 7) for prime Fano threefolds of genus 12 is defined canonically over any field of zero characteristic and use this to give a new proof of the criterion of k-rationality for these threefolds. Mathematics Subject Classification (2010). 14E08, 14J45, 14E05, 14E30. Keywords. Fano variety, Mukai variety, rationality, rational map.

1. Introduction A Mukai variety is a (smooth) Fano variety of geometric Picard number 1 and coindex 3 which is not a form of a complete intersection in a weighted projective space. Over an algebraically closed field of characteristic zero such varieties have been classified by Mukai in [Muk92]. The main invariant of a Mukai variety X is the genus g(X) defined by the formula 2g(X) − 2 = H n , where H is the ample generator of Pic(X¯k ) and n = dim(X). Clearly, a smooth hyperplane section of a Mukai variety of dimension at least 4 is again a Mukai variety of the same genus. The following table lists maximal Mukai varieties X2g−2 over ¯ k, their genera and dimensions: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Farkas et al. (eds.), Rationality of Varieties, Progress in Mathematics 342, https://doi.org/10.1007/978-3-030-75421-1_10

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g

X2g−2

6

CGr(2, 5) ∩ Q ⊂ P10

6

7

OGr+ (5, 10) ⊂ P

15

10

8

Gr(2, 6) ⊂ P

14

8

9

LGr(3, 6) ⊂ P13

6

dim(X2g−2 )

10

G2 Gr(2, 7) ⊂ P

13

12

I3 Gr(3, 7) ⊂ P

13

5 3

Table 1. Maximal Mukai varieties

Here • CGr(2, 5) ∩ Q is a transverse intersection of the cone over Gr(2, 5) with a quadric, • OGr+ (5, 10) is a connected component of the Grassmannian of isotropic fivedimensional subspaces in a ten-dimensional vector space endowed with a non-degenerate quadratic form, • Gr(2, 6) is the Grassmannian of two-dimensional subspaces in a six-dimensional vector space, • LGr(3, 6) is the Grassmannian of Lagrangian (three-dimensional) subspaces in a six-dimensional symplectic vector space, • G2 Gr(2, 7) is the adjoint Grassmannian of the simple algebraic group G2 , and • I3 Gr(3, 7) is the Grassmannian of three-dimensional subspaces in a sevendimensional vector space isotropic for a (sufficiently general) triple of skewsymmetric forms. Note that for g ∈ {7, 8, 9, 10} the maximal varieties are homogeneous (and in particular rigid), while for g ∈ {6, 12} they vary in non-trivial moduli spaces (of dimensions 25 and 6, respectively). Theorem 1.1 ([Muk92]). If X is a Mukai variety of genus g over an algebraically closed field of characteristic zero then g ∈ {6, 7, 8, 9, 10, 12} and there is an embedding X → X2g−2 into one of the maximal varieties listed in Table 1 such that X is a transverse linear section of X2g−2 . An easy consequence of the Mukai’s theorem is that over an arbitrary field k of characteristic zero any Mukai variety is a k-form of a linear section of one of the maximal varieties. The goal of this paper is to investigate birational properties of Mukai varieties over arbitrary fields of zero characteristic. The case of Mukai threefolds has been studied in [KP19], where in the case g ∈ {7, 9, 10, 12} we established for them criteria of unirationality and rationality over an arbitrary field k of characteristic

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zero. Note that for g ∈ {6, 8} Mukai threefolds are known to be irrational even over an algebraically closed field. In this paper we consider Mukai varieties of dimension n ≥ 4. We denote by X(k) the set of k-points of X. The main result of the paper is the following Theorem 1.2. Let k be an arbitrary field of characteristic zero. Let X be a Mukai variety of genus g = g(X) and dimension n = dim(X) such that • either g ∈ {7, 8, 10} and n ≥ 4, • or g = 9 and n ≥ 5. Then the following conditions are equivalent: (i) X is k-rational; (ii) X is k-unirational; (iii) X(k) =  ∅. In the case (g, n) = (9, 4) we have the equivalence (ii) ⇐⇒ (iii). Implications (i) =⇒ (ii) =⇒ (iii) of the theorem are evident, so the content of the paper is in the implication (iii) =⇒ (i) (or (iii) =⇒ (ii) for (g, n) = (9, 4)). This implication is proved in Theorem 3.3 for g ∈ {7, 8, 10}, and Corollary 4.10 for g = 9. To prove the implication we use the Sarkisov link starting with the blowup of a point for each of the maximal Mukai varieties X2g−2 of genus g ∈ {7, 8, 9, 10} over ¯k. These links are constructed in a uniform way for g ∈ {7, 8, 10} in Theorem 2.2 (see also Propositions 2.10, 2.12, and 2.15), and separately for g = 9 in Theorem 4.4. For g ∈ {7, 8, 10} the links end with projective bundles over smooth fourdimensional Fano varieties X+ (P4 , a quadric, and a quintic del Pezzo fourfold, respectively). We check in Theorem 3.1 that after passing to a smooth linear section X ⊂ X2g−2 of dimension n we obtain a birational transformation (in ˜+ general this is an example of a so-called “bad link”) between X and a variety X n−4 + with a morphism to X and general fiber P¯k . We prove in Theorem 3.3 that if we start with a Mukai variety X defined over k and a k-point, this transformation is ˜ + → X+ is isomorphic also defined over k and the general fiber of the morphism X n−4 to Pk ; this implies k-rationality of X. For g = 9 we use the link constructed in Theorem 4.4 in a similar way to show that any Mukai variety X of genus 9 and dimension n with X(k) = ∅ is birational to a complete intersection X + ⊂ P6 of 6 − n quadrics containing a distinguished Veronese surface S ⊂ X + such that the divisor X + ∩S = X + ∩P5 is k-rational, see Theorem 4.9. For n ≥ 5 this implies k-rationality of X and for n = 4 this implies its k-unirationality, see Corollary 4.10. We expect that k-unirational Mukai fourfolds of genus 9 are not k-rational in general, however we establish for them a sufficient condition of k-rationality, Corollary 4.11. In the Appendix we use the above results to prove that Mukai varieties of genus g ∈ {7, 8, 9, 10} and dimension n ≥ 5 have cylinders, see Proposition A.1.

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The constructed birational transformations can be also applied to Mukai threefolds. In the case (g, n) = (7, 3) we obtain a birational transformation from X to a Springer-type resolution of a special singular quartic threefold in P4 ; considering another Springer resolution we extend this to a birational map between X and a (possibly singular) quintic del Pezzo threefold. The latter is k-rational as soon as it has a k-point. This construction gives a more direct than in [KP19] proof of k-rationality of X, see Remark 3.4 for details. Analogously, in the case (g, n) = (8, 3) we obtain a birational transformation between X and a cubic threefold (Remark 3.5) and in the case (g, n) = (10, 3) a birational transformation between X and a sextic del Pezzo fibration over P1 (Remark 3.6). The last section of the paper is concerned with the case (g, n) = (12, 3). In k this case we prove that the embedding X → Gr(3, 7) constructed by Mukai over ¯ is defined over any field of characteristic zero and is canonical (Corollary 5.6). Using this we construct in Theorem 5.1 a birational map between X and P3 which looks very similar to the general construction of Theorem 2.2. Similarly to the case (g, n) = (7, 3) this gives an alternative and more direct than in [KP19] proof of k-rationality of prime Fano threefolds of genus 12 with k-points. We also apply these results to the derived category of coherent sheaves on X, see Corollary 5.7. To finish the introduction we remind what is known about the case of Mukai varieties of genus 6 (so-called Gushel–Mukai varieties) and dimension n ≥ 4 which are not covered by Theorem 1.2. The case of Gushel–Mukai fourfolds is very hard and interesting already ¯ it is expected that a very general Gushel–Mukai fourfold is not ¯k-rational over k; and there is a countable union of divisorial families (in the moduli space) of Gushel–Mukai fourfolds which are ¯k-rational, see [Pro93, DIM15, KP18, Deb20], analogously to the case of cubic fourfolds. In the remaining cases with g = 6 and n ∈ {5, 6}, it is known that each Gushel–Mukai variety of dimension n ∈ {5, 6} is ¯k-rational (see [DK18, Proposition 4.2]), but the question of its k-rationality is completely unclear. We would like to thank the organizers of the conference “Rationality of Algebraic Varieties” on the Schiermonnikoog Island, where the idea of this paper was born. We are grateful to the referee for valuable comments.

2. A birational transformation given by a family of quadrics Assuming that X is a smooth projective variety “birationally covered” (in the sense described below) by a family of quadrics passing through a fixed point x0 ∈ X, we construct in §§2.1–2.2 a birational transformation of X into a projective bundle. This transformation is, essentially, a relative version of the birational isomorphism between a quadric and a projective space Qm  Pm

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induced by the linear projection from a point x0 ∈ Qm ; it blows up the point x0 and then contracts all lines on Qm passing through x0 . In §§2.3–2.5 we show that this construction applies to the Mukai varieties X12 = OGr+ (5, 10), X14 = Gr(2, 6), and X18 = G2 Gr(2, 7) of genus g = 7, 8, and 10, respectively. 2.1. The statement Let X ⊂ P(W ) be a smooth projective variety and let x0 ∈ X be a point. We denote by H the restriction to X of the hyperplane class of P(W ) and consider it as a polarization. Assume a projective scheme X+ and a subscheme Q ⊂ X × X+ are given such that Q is a flat over X+ family of quadrics in X containing x0 , i.e., for each point x+ ∈ X+ the fiber Qx+ ⊂ X ⊂ P(W ) of Q is a quadric in X containing the point x0 . We denote by pQ : Q −−−−→ X

and

qQ : Q −−−−→ X+

the natural projections and by sx0 : X+ −−−−→ Q

(2.1.1)

the section of qQ given by the point x0 . Let F1 (X, x0 ) be the Hilbert scheme of lines in X (with respect to H) passing through the point x0 and let F1 (Q/X+ , x0 ) be the relative Hilbert scheme of lines in fibers of qQ : Q → X+ passing through the point x0 . Let L (X, x0 ) ⊂ F1 (X, x0 ) × X

L (Q/X+ , x0 ) ⊂ F1 (Q/X+ , x0 ) ×X+ Q

and

be the corresponding universal families of lines and let pL : L (X, x0 ) −−−−→ X

and

p˜L : L (Q/X+ , x0 ) −−−−→ Q

be the natural projections. To state the theorem we will need the following simple observation. Lemma 2.1. The morphism pQ : Q → X induces a morphism F1 (Q/X+ , x0 ) −−−−→ F1 (X, x0 )

(2.1.2)

such that

(2.1.3) L (Q/X+ , x0 ) ∼ = F1 (Q/X+ , x0 ) ×F1 (X,x0 ) L (X, x0 )     + and if morphism (2.1.2) is surjective then pQ p˜L (L (Q/X ,x0 )) = pL L (X,x0 ) as subvarieties in X. Proof. If L ⊂ Qx+ is a line through x0 , then pQ (L) ⊂ X is a line through x0 ; this defines the morphism (2.1.2). Moreover, pQ : L → pQ (L) is an isomorphism; this  proves (2.1.3). The last statement of the lemma is obvious. We denote by ρ(Y ) the Picard number of a variety Y .

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Theorem 2.2. Let X ⊂ P(W ) be a smooth projective variety, and let Q ⊂ X × X+ be an X+ -flat family of quadrics in X passing through a point x0 ∈ X. Set n = dim(X),

m = dim(Q/X+ ).

Assume the following conditions hold: (i) The scheme X+ is smooth, projective, connected, and ρ(X+ ) = ρ(X). (ii) The map pQ : Q → X is birational. (iii) The scheme F1 (X, x0 ) is smooth, the scheme F1 (Q/X+ , x0 ) is smooth over F1 (X, x0 ) and connected, and dim(F1 (X, x0 )) < dim(F1 (Q/X+ , x0 )) = dim(X) − 2. Then there is a commutative diagram  O D qq _ OOOOO q q OOO q OOO qqq q q OO'  xqq  L (X, x0 ) F1 (Q/X + , x0 ) q X OOOO _ _ q q O OOO qq q O q O qq π π+ OOO O'  xqqq  ψ Blx0 (X) _ _ _ _ _ _ _ _ _ _ _ _/ PX+ (E ) MMM o MMMφ φ+ oooo o σ+ σ MMM o o MMM ooo   & wooo X X X+ ,

(2.1.4)

where • • • • • • • • •

σ is the blowup of x0 with the exceptional divisor E ⊂ Blx0 (X), E is a subbundle of rank m + 1 in W ⊗ OX+ , where W = W/x0 , σ+ is the projective bundle morphism, φ is the morphism induced by the linear projection P(W )  P(W ) from x0 , φ+ is the morphism induced by the embedding E → W ⊗ OX+ , X = φ(Blx0 (X)) = φ+ (PX+ (E )) ⊂ P(W ), π is the blowup of L (X, x0 ), π+ is the blowup of F1 (Q/X+ , x0 ),  is the common exceptional divisor of π and π+ , D  ∼ D = L (X, x0 ) ×F1 (X,x0 ) F1 (Q/X+ , x0 ),

• ψ = π+ ◦ π −1 = φ−1 + ◦ φ is a small birational map (flop or antiflip).   If E ⊂ X is the strict transform of E ⊂ Blx0 (X), the morphisms π and π+ induce isomorphisms ∼ E = BlF1 (X,x0 ) (E) where E0 ⊂ E is a rank-m subsheaf.

and

∼ E = PX+ (E0 ),

(2.1.5)

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Furthermore, the maps π, φ, φ+ , and π+ induce isomorphisms φ

π

∼ Blx (X) \ L (X, x0 ) ∼  \D  = X = X \ F1 (X, x0 ) 0 φ+

π+

∼  \ D.  = PX+ (E ) \ F1 (Q/X+ , x0 ) ∼ = X

(2.1.6)

 are the projecOn the other hand, the restrictions of the maps π and π+ to D + tions to the factors L (X, x0 ) and F1 (Q/X , x0 ), respectively, the restriction of φ to L (X, x0 ) is the natural P1 -fibration L (X, x0 ) −−→ F1 (X, x0 ), and the restriction of φ+ to F1 (Q/X+ , x0 ) is the morphism (2.1.2), which is also a projective bundle. Finally, the birational transformation ψ induces an isomorphism E \ F1 (X, x0 ) ∼ = PX+ (E0 ) \ F1 (Q/X+ , x0 ).

(2.1.7)

Remark 2.3. If every quadric Qx+ is smooth at x0 then E0 ⊂ E is a vector subbundle and the morphism PX+ (E0 ) → X+ is a Pm−1 -fibration; otherwise, it has general fiber Pm−1 , cf. Remark 2.16. In the course of proof we will explain the construction of the vector bundle E , its subsheaf E0 , the embeddings −−−− → Blx0 (X), L (X, x0 ) −

F1 (Q/X+ , x0 ) −−−−− → PX+ (E ),

and F1 (X, x0 ) −−−−− →E and other details. Before that we deduce from the theorem the following useful lemma. Lemma 2.4. In the setting of Theorem 2.2, assume KX = −r H for some r ∈ Z. If the divisor class −KX+ + c1 (E ) is very ample on X+ then the map σ+ ◦ ψ : Blx0 (X)  X+ is given by the linear system |(r − m − 1)H − (n − m − 2)E|. Proof. Since ψ is a small birational map, it identifies Pic(Blx0 (X))

and

Pic(PX+ (E )).

The canonical class of Blx0 (X) is equal to −r H + (n − 1)E. On the other hand, the canonical class of PX+ (E ) is equal to KX+ − c1 (E ) − (m + 1)H, where H is the relative hyperplane class of PX+ (E ). Finally, we have H = H − E since φ is induced by the linear projection from x0 and φ+ is induced by the embedding E → W ⊗ OX+ . Combining all this, we obtain the equality −KX+ + c1 (E ) = r H − (n − 1)E − (m + 1)(H − E) = (r − m − 1)H − (n − m − 2)E. If the left side of the equality is very ample, the result follows.



256 2.2. The proof Let

A. Kuznetsov and Y. Prokhorov

E := qQ∗ (p∗Q OX (H))∨ .

Since Q ⊂ P(W ) × X+ → X+ is a flat family of m-dimensional quadrics contained in P(W ), this is a vector subbundle in W ⊗ OX+ of rank m + 2 and there is an embedding ι : Q −−−→ PX+ (E) ⊂ P(W ) × X+ as a divisor of relative degree 2. Note that the pullback to PX+ (E) of the hyperplane class H of P(W ) is a relative hyperplane class. Therefore, there is a divisor class D on X+ such that ι(Q) ∼ 2H + D (2.2.1)  in Pic(PX+ (E )). The section (2.1.1) of Q → X can be also thought of as a section of PX+ (E); it corresponds to an embedding OX+ → E of vector bundles. We denote by E = E/OX+ the quotient bundle. The embedding E → W ⊗ OX+ induces an embedding of vector bundles E −−−→ W ⊗ OX+ , where W = W/x0 , and a morphism φ+ : PX+ (E ) −−−−→ P(W ).

(2.2.2)

We denote by H the hyperplane class of P(W ); its pullback to PX+ (E ) (which we also abusively denote by H) is then a relative hyperplane class. We consider the blowup of PX+ (E) along the section sx0 (X+ ) and denote by EP its exceptional divisor. Lemma 2.5. There is an isomorphism of X+ -schemes Blsx0 (X+ ) (PX+ (E)) ∼ = PPX+ (E ) (O ⊕ O(−H)). Moreover, H is a relative hyperplane class for PPX+ (E ) (O ⊕ O(−H)) and there is a linear equivalence EP ∼ H − H. Finally, EP is equal to the section of the projection πE : PPX+ (E ) (O ⊕ O(−H)) −−−−→ PX+ (E ) that corresponds to the embedding O → O ⊕ O(−H). Proof. This is just a relative version of the isomorphism of the blowup of a projective space at a point and the P1 -bundle over the projective space of dimension by one less. 

Rationality of Mukai Varieties over Non-closed Fields We define

257

 := Bls (X+ ) (Q) X x0

to be the blowup of Q along the image of the section (2.1.1). Lemma 2.6. There exist morphisms π, ˆι, and π+ that make the following diagram commutative / Bls (X+ ) (PX+ (E)) x0 RRR RRRπE π + π RRR ˆ σ RRR RR(  y   , ι  / X PX+ (E ) Blx0 (X) Q XXXXX PX+ (E ) r X S X r S X XXXXX SSS r XXXXXX pQ rrr qQ SSS σ+ XXXXXX σ S rrr XXXXXSXSSSS XXXXSXS),   yrrrr X X+ ,

 X

ι ˆ

(2.2.3)

where σ and σ ˆ are the blowup maps and σ+ is the projective bundle morphism. Moreover, the morphism π is birational. Proof. Since the scheme-theoretic preimage of the point x0 via the map pQ coin is a Cartier divisor, hence cides with sx0 (X+ ), its scheme-theoretic preimage in X  there is a map π : X → Blx0 (X) making the left square commutative. It is birational, because so are the morphisms σ, σ ˆ , and pQ (for pQ this is assumption (ii)).  −−→ Bls (X+ ) (PX+ (E)) in the diSimilarly, there exists a morphism ˆι : X x0 agram such that the central square commutes. The right square has been constructed in Lemma 2.5. Finally, π+ can be defined as the composition πE ◦ ˆι.  We need to identify the maps π and π+ as blowups. We start with π+ . Lemma 2.7. There is an embedding F1 (Q/X+ , x0 ) → PX+ (E ) such that the mor of π+ is phism π+ is the blowup of F1 (Q/X+ , x0 ), and the exceptional divisor D +   isomorphic to L (Q/X , x0 ) over Q. In particular, X and D are smooth. Proof. There is a natural identification of the relative over X+ Hilbert scheme of lines in PX+ (E) passing through x0 as F1 (PX+ (E)/X+ , x0 ) ∼ = PX+ (E ), such that the universal line L (PX+ (E)/X+ , x0 ) is isomorphic to the projective bundle PPX+ (E ) (O ⊕ O(−H)) and the section of L (PX+ (E)/X+ , x0 ) → F1 (PX+ (E)/X+ , x0 ) corresponding to x0 is equal to EP ; this in particular defines a natural embedding F1 (Q/X+ , x0 ) → PX+ (E ). Since ι(Q) is a divisor of type 2H+D (see (2.2.1)) it follows that F1 (Q/X+, x0 ) is the zero locus of a section of πE ∗ O(2H + D − EP ) on PPX+ (E ) (O ⊕ O(−H)). On

258

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 is the strict transform of Q in Bls (X+ ) (PX+ (E)), hence it is the other hand, X x0 the zero locus of the same section of O(2H + D − EP ). By Lemma 2.5 we have O(2H + D − EP ) ∼ = O(H + H + D), so this is a relative hyperplane class for the P1 -bundle PPX+ (E ) (O ⊕ O(−H)). Applying [Kuz16, Lemma 2.1] we conclude that  ∼ X = BlF (Q/X+ ,x ) (PX+ (E )) 1

0

since codimPX+ (E ) (F1 (Q/X , x0 )) = dim(X) − dim(F1 (Q/X+ , x0 )) = 2 by assumption (iii).  of the blowup This argument also proves that the exceptional divisor D +  coincides with the universal line L (Q/X , x0 ). The smoothness of the varieties X + +  and D follows from the smoothness of X and F1 (Q/X , x0 ), assumptions (i) and (iii).  +

Now we can describe the morphism π. Lemma 2.8. There is a natural embedding L (X, x0 ) → Blx0 (X) such that the morphism π is the blowup with center L (X, x0 ). Moreover, E ∩ L (X, x0 ) ∼ (2.2.4) = F1 (X, x0 ), −1 ∼ the intersection is transverse, and π (E) = BlF (X,x ) (E). 1

0

Proof. The scheme-theoretic preimage of the point x0 along the map pL : L (X, x0 ) → X is equal to the image of the section sx0 : F1 (X, x0 ) → L (X, x0 ); in particular, it is a Cartier divisor on L (X, x0 ). Therefore, the morphism pL lifts to a morphism L (X, x0 ) −−→ Blx0 (X), which is clearly a closed embedding. Moreover, the scheme-theoretic preimage of E under this embedding is equal to the image of the section sx0 , which proves (2.2.4). Since the intersection is smooth (by assumption (iii)), it is transverse.  → Blx (X). Its source is smooth by Consider the birational morphism π : X 0 Lemma 2.7 and its target is smooth by the assumptions. Furthermore, from the diagram (2.2.3) and Lemma 2.7 we deduce that  = pQ (ˆ  = pQ (ˆ σ(π(D)) σ (D)) σ (L (Q/X+ , x0 ))) = pQ (˜ pL (L (Q/X+ , x0 ))) = pL (L (X, x0 ))  is irreducible, π(D)  = L (X, x0 ) (the last equality uses Lemma 2.1). Since D follows. Note that codim(L (X, x0 )) = dim(X) − dim(F1 (X, x0 )) − 1 > 1 (again by assumption (iii)). Finally, we have  = ρ(X+ ) + 2 = ρ(X) + 2 = ρ(Blx0 (X)) + 1, ρ(X) hence by [Kuz18, Lemma 2.5] we conclude that π is the blowup with center L (X, x0 ).

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259

Since the divisor E intersects the blowup center transversely along F1 (X, x0 ), its preimage is equal to its strict transform and isomorphic to the blowup with center F1 (X, x0 ).  Finally, we can combine the above results and finish the proof of the theorem. Proof of Theorem 2.2. The morphisms π and π+ are constructed in Lemma 2.6 and they are proved to be the blowups in Lemma 2.7 and 2.8, respectively. The  as the fiber product follows from a combination of description of the divisor D Lemma 2.7 with (2.1.3). The morphism φ is induced by the linear projection from x0 , and the morphism φ+ is defined in (2.2.2). We have φ ◦ π = φ+ ◦ π+ , because the two maps can be rewritten as the compositions y pQ gggg3 X XXXXXXX g XXXXX + ggggg ˆ σ g _ _ _ _ _/3 P(W ), / Q U  w UUU X gg3 P(W ) g UUUU g hhhh g g g U* g ι g hhhhh g PX+ (E) _ _ _ _ _/ PX+ (E ) and the squares commute. The equality φ ◦ π = φ+ ◦ π+ implies that φ(Blx0 (X)) = φ+ (PX+ (E )) and π+ ◦ π −1 = φ−1 + ◦ φ, thus defining the subvariety X ⊂ P(W ) and the birational map ψ, respectively. Since the morphisms π and π+ are contractions with the same exceptional divisor, ψ is small. Assumption (iii) implies that codimBlx0 (X) (L (X, x0 )) ≥ codimPX+ (E ) (F1 (Q/X+ , x0 )), hence ψ is a flop or antiflip. It remains to construct the subsheaf E0 ⊂ E , to prove (2.1.5), (2.1.6), and (2.1.7), and to describe the restrictions of π, π+ , φ, and φ+ to various strata.  Consider the diagram (2.2.3). By construction of π, the preimage of E in X  is the exceptional divisor of σ ˆ , which is also equal X ∩ EP . The divisor EP by Lemma 2.5 is equal to the section of the morphism πE corresponding to the embedding O → O ⊕ O(−H); in particular EP is isomorphic to PX+ (E ) via πE , and the restriction of the hyperplane class H to EP is trivial.  is the divisor of type As it is explained in Lemma 2.7, X H + H + D in PPX+ (E ) (O ⊕ O(−H)),  ∩ EP is equal to the zero locus of a section of the line bundle O(H + D) hence X on EP = PX+ (E ). This section corresponds to a morphism E −−−−→ O(D)  on X . If we define E0 as its kernel, then X∩E P = PX+ (E0 ), and the morphism πE embeds it naturally into PX+ (E ). This proves (2.1.5). The maps π and π+ in (2.1.6) are isomorphisms by Lemma 2.8 and 2.7, respectively. Furthermore, the map φ is an isomorphism because it is induced by −1 the linear projection and the map φ+ is an isomorphism because φ+ = φ ◦ π ◦ π+ . +

260

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 coincides with the projection to the second factor The restriction of π+ to D  coincides with the by Lemma 2.7. Similarly, by Lemma 2.8 the restriction of π to D + map L (Q/X , x0 ) → L (X, x0 ) induced by the morphism (2.1.2), hence it is the projection to the first factor. On the other hand, since π is a smooth blowup, this morphism is a projective bundle, hence so is the morphism (2.1.2). The restriction of φ to L (X, x0 ) is the natural P1 -bundle, because φ is induced by the linear projection from x0 , and the restriction of φ+ to F1 (Q/X+ , x0 ) is the map (2.1.2),  because it is the only map such that φ+ ◦ π+ = φ ◦ π on D. Finally, the isomorphism (2.1.7) follows from (2.1.5) and (2.1.6).  2.3. Grassmannians of lines In this section we show that the assumptions of Theorem 2.2 are satisfied for Grassmannians X = Gr(2, V ). Note that when dim(V ) = 6 the Grassmannian Gr(2, V ) is the maximal Mukai variety X14 of genus g = 8. In this case we show that diagram (2.1.4) exists with X+ = Gr(2, 4). Assume dim(V ) ≥ 5. We consider the Pl¨ ucker embedding X := Gr(2, V ) → P(∧2 V ), so that H is the Pl¨ ucker polarization. Let U0 ⊂ V be a two-dimensional subspace and let x0 ∈ X be the corresponding point. We denote the quotient space by V + := V /U0 . We further denote X+ := Gr(2, V + ),

Q := Fl(2, 4; V ) ×Gr(4,V ) X+ ,

where the embedding X+ → Gr(4, V ) takes a subspace U + ⊂ V + to its preimage under the projection V → V + along U0 . Note that Q → X+ is a Gr(2, 4)-fibration, so its fibers are smooth four-dimensional quadrics on X passing through x0 . Lemma 2.9. There is a commutative diagram F1 (Q/X+ , x0 )

P(U0 ) × Fl(1, 2; V + )

 F1 (X, x0 )

 P(U0 ) × P(V + ),

where the map on the left is the map (2.1.2) and the map on the right is induced by the natural map Fl(1, 2; V + ) → P(V + ). In particular, F1 (X, x0 ) is smooth, F1 (Q/X+ , x0 ) is smooth over F1 (X, x0 ), connected, and dim(F1 (Q/X+ , x0 )) = 2 dim(V ) − 6 = dim(X) − 2. Proof. Recall that any line on Gr(2, V ) has the form (V1 , V3 ) = {[U ] | V1 ⊂ U ⊂ V3 },

(2.3.1)

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261

where V1 ⊂ V3 ⊂ V is a flag with dim(Vi ) = i. Clearly, [U0 ] ∈ (V1 , V3 ) means V1 ⊂ U0 ⊂ V3 ; therefore such lines are determined by the points [V1 ] ∈ P(U0 ) and [V3 /U0 ] ∈ P(V /U0 ) = P(V + ); this proves the bottom equality. The line (V1 , V3 ) lies on the quadric Q[U + ] if and only if V3 /U0 ⊂ U + . Therefore, the fiber of the map (2.1.2) over [(V1 , V3 )] parameterizes all U + ⊂ V + such that V3 /U0 ⊂ U + . This is the space P(V /V3 ), i.e., the fiber of the projection Fl(1, 2; V + ) → P(V + ). This proves the remaining part of the lemma.  Proposition 2.10. For X, X+ , and Q defined above the assumptions of Theorem 2.2 are satisfied. Moreover, the bundles E and E0 constructed in Theorem 2.2 have the form ∼ E0 ⊕ ∧2 U + , E0 ∼ E = = U0 ⊗ U + , where U + is the tautological bundle on Gr(2, V + ), and the map σ+ ◦ ψ : Blx0 (Gr(2, V ))  Gr(2, V + ) from (2.1.4) is given by the linear system |H − 2E|. Proof. Assumption (i) of Theorem 2.2 is evident. The fiber of the map pQ : Q → X over the point [U ] ∈ X = Gr(2, V ) corresponding to a subspace U ⊂ V parameterizes four-dimensional subspaces in V that contain both U and U0 . Generically, this is just one point [U0 + U ], so pQ is birational, hence assumption (ii) is satisfied. Assumption (iii) is proved in Lemma 2.9. Furthermore, we have E = ∧2 (U0 ⊗ O ⊕ U + ) = (∧2 U0 ⊗ O) ⊕ (U0 ⊗ U + ) ⊕ ∧2 U + . The section sx0 corresponds to the first summand and the subbundle E0 is the second summand. To show the last claim note that −KX+ + c1 (E ) = (dim(V ) − 5)H+ and by Lemma 2.4 its pullback to Blx0 (X) is equal to (dim(V ) − 5)(H − 2E), because r = dim(V ), n = 2 dim(V ) − 4, and m = 4. Since the Picard group of Blx0 (X) has no torsion, it follows that if dim(V ) > 5 the pullback of the very ample class H+ is H − 2E, hence the map σ+ ◦ ψ is given by the linear system |H − 2E|. Alternatively, if we choose a basis {vi } in V such that the first two vectors generate U0 (hence all the other vectors project to a basis of V + ), then the map σ+ ◦ ψ is induced by the linear map ∧2 V → ∧2 V + that acts by (vi ∧ vj )1≤i 3 which contradicts Lemma 5.12. On the other hand, if the zero locus of a section of U ∨ is finite, its length is equal to c3 (U ∨ ) = 2, hence the general fiber of σ −1 (X) is a scheme of length 2. Therefore, the morphism σ+ : σ −1 (X) → P3 is generically ˜ 0 has degree 1 finite of degree 2. Furthermore, by Lemma 5.12 the component X 3 ˜ and the components XL have degree 0 over P . This means that the remaining component BlZ (X) has degree 1, hence the map σ+ : BlZ (X) → P3 is birational. 

Appendix: Application to cylinders Recall that a variety X is cylindrical if there is an open subset in X isomorphic to U × A1 . Similarly, for any r ≥ 1 we say that X is r-cylindrical if there is an open subset in X isomorphic to U × Ar . The existence of cylinders on projective varieties is an interesting question related to the study of automorphism groups of affine cones [KPZ13]. k is 4-cylindrical. By [PZ18] every Mukai fourfold of genus g = 10 over ¯ Moreover, there are families of cylindrical Mukai fourfolds of genus g ∈ {7, 8, 9}, see [PZ16, PZ17]. Here we prove the following result. Proposition A.1. Let k be an arbitrary field of characteristic zero. Let X be a Mukai variety of genus g ∈ {7, 8, 9, 10} and dimension n ≥ 5. If X(k) = ∅ then X is (n − 4)-cylindrical.

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Proof. First, let g ∈ {7, 8, 10}. It follows from the proof of Theorem 3.1 that ˜ + \ PX+ (E0 ). X \ pL (L (X, x0 )) = X Let U := X+ \D(ξ0 ) ⊂ X+ be the open subset where the morphism ξ0 (see (3.1.6)) is surjective, so that Ker(ξ|U ) and Ker(ξ0 |U ) are vector bundles of ranks n − 3 and n − 4, respectively. Then −1 σ+ (U ) \ PX+ (E0 )

is isomorphic to the total space of the vector bundle Ker(ξ0 |U ). Restricting to a smaller open subset U  ⊂ U over which this bundle is trivial, we obtain an ˜ + \ PX+ (E0 ), hence in X. (n − 4)-cylinder in X Now let g = 9. If n = 6, the proof of Theorem 4.4 (see Lemma 4.5) shows that X = LGr(3, 6) contains A6 as a Zariski open subset, hence X is 6-cylindrical in this case. Let n = 5, so that X is a hyperplane section of LGr(3, 6). Consider the birational transformation of Theorem 4.9. Let X + ⊂ P6 be the corresponding quadric and let E + ⊂ X + be the strict transform of the exceptional divisor E. Then X + \ E + is isomorphic to an open subset of X, so it is enough to find a cylinder in X + \ E + . But E + is a hyperplane section of the quadric X + containing a smooth k-rational point; projecting from such a point one can easily find the required cylinder.  Acknowledgment The authors were partially supported by the HSE University Basic Research Program.

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Alexander Kuznetsov Steklov Mathematical Institute of Russian Academy of Sciences Moscow, Russia and Laboratory of Algebraic Geometry HSE University, Russian Federation e-mail: [email protected] Yuri Prokhorov Steklov Mathematical Institute of Russian Academy of Sciences Moscow, Russia and Laboratory of Algebraic Geometry HSE University, Russian Federation and Department of Algebra, Moscow State University Moscow, Russia e-mail: [email protected]

A Refinement of the Motivic Volume, and Specialization of Birational Types Johannes Nicaise and John Christian Ottem Abstract. We construct an upgrade of the motivic volume by keeping track of dimensions in the Grothendieck ring of varieties. This produces a uniform refinement of the motivic volume and its birational version introduced by Kontsevich and Tschinkel to prove the specialization of birational types. We also provide several explicit examples of obstructions to stable rationality arising from this technique. Mathematics Subject Classification (2010). 14E08, 14D06, 14E18. Keywords. Rationality of algebraic varieties; Grothendieck ring of varieties; motivic nearby fiber.

1. Introduction In [NS19], Shinder and the first-named author used a refinement of Denef and Loeser’s motivic nearby fiber to construct a motivic obstruction to the stable rationality of very general fibers of a degenerating family of smooth and proper complex varieties. This result implies in particular that stable rationality specializes in smooth and proper families. A variant of this method was developed by Kontsevich and Tschinkel in [KT19] to upgrade the results to rationality instead of stable rationality. The principal aim of the present article is to develop a unified framework for the invariants in [NS19] and [KT19]. The approach in [NS19] relies on the motivic volume, a motivic specialization morphism for Grothendieck rings of varieties which constructs a limit object for the classes of the fibers of degenerating families. It can be viewed as a version of the nearby cycles functor at the level of Grothendieck rings. It is closely related to the motivic nearby fiber of Denef and Loeser [DL01], but with the crucial difference that its construction does not require the inversion of the class of the affine line in the Grothendieck ring. This is essential for the applications to rationality problems. The existence of the motivic © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Farkas et al. (eds.), Rationality of Varieties, Progress in Mathematics 342, https://doi.org/10.1007/978-3-030-75421-1_11

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volume can be deduced from the work of Hrushovski and Kazhdan [HK06], but in [NS19] a direct proof was given based on the weak factorization theorem and logarithmic geometry. It was proved by Larsen and Lunts [LL03] that the stable birational type of a smooth and proper variety over a field of characteristic zero can be read off from its class in the Grothendieck ring of varieties. However, it is an open problem whether this class also determines the birational type, which is why the results in [NS19] only give information about stable birational types. This issue was circumvented in [KT19] by working directly in the free abelian group on birational types instead of the Grothendieck ring of varieties, and constructing an analogous specialization morphism there. Here we introduce a refined version of the Grothendieck ring of varieties, graded by dimension, which detects birational types for trivial reasons. A variant of this graded ring already appeared in [HK06] in the form of the ring !K(RES[∗]). We then show that the arguments in [NS19] can be carried over verbatim to construct the motivic volume at this refined level. The result is an invariant that specializes simultaneously to the motivic volume from [NS19] and to its birational version in [KT19]. All the results for stable birational types in [NS19] can then immediately be upgraded to the level of birational types, yielding a unified treatment of the specialization results in [NS19] and [KT19]. A second new feature is that we present the construction of the motivic volume in a slightly more general setting (this generalization is explained in detail at the beginning of the proof of Theorem 3.3.2). We also replace as much as possible the language of logarithmic geometry by the more widely familiar language of toroidal embeddings. This makes the motivic volume more accessible and userfriendly. The proofs still rely heavily on those in [NS19] and there logarithmic geometry remains by far the most efficient and transparent framework. Still, if the reader is willing to accept the most technical aspects of the construction as black boxes, then they can understand and use the motivic volume without any reference to logarithmic geometry. Denef and Loeser’s motivic nearby fiber and the motivic volume have found many profound applications in singularity theory and motivic Donaldson–Thomas theory; see for instance [GLM06, Lˆe15, NP19]. It would be interesting to investigate what additional information can be extracted from our refined version, beyond the applications to birational types discussed here. The paper is organized in the following way. Section 2 is devoted to the Grothendieck ring of varieties. We first recall the definition of the classical Grothendieck ring of varieties, and we give an overview of its main properties, with an emphasis on the theorems of Bittner (Theorem 2.1.6) and Larsen & Lunts (Theorem 2.1.10) which are the most important structural results in characteristic zero. We explain why the Grothendieck ring detects stable birational types in characteristic zero and why it is unclear whether it also detects birational types. This part is not needed for the remainder of the paper, but it places the results in their proper

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context. We then move on to our refinement of the Grothendieck ring of varieties graded by dimension, and we show that it detects birational types in arbitrary characteristic (Proposition 2.3.1). We prove analogs of the theorems of Bittner (Theorem 2.4.2) and Larsen & Lunts (Theorem 2.5.2). The analog of Bittner’s theorem is an essential ingredient in the construction of the motivic volume; the analog of the theorem of Larsen & Lunts is not used in the sequel and is only included to further clarify the structure of the refined Grothendieck ring. In Section 3, we present the construction of the motivic volume at the level of the refined Grothendieck ring, which is the key invariant for the applications to rationality problems. We define the motivic volume in terms of strictly toroidal models, making the construction as explicit as possible (Theorem 3.3.2). The proof of its existence is essentially the same as for the unrefined version in [NS19]; we explain the general strategy and highlight the new elements of the proof. The main applications to rationality problems are developed in Section 4. We deduce the specialization of birational types in smooth and proper families (Theorem 4.1.1; this result was originally proved in [KT19]) and we explain how every strictly toroidal model gives rise to an obstruction to the (stable) rationality of the geometric generic fiber (Theorem 4.2.1). Some elementary examples of such obstructions are given in Section 4.3; further applications can be found in [NS19] and [NO19]. Finally, in Section 5, we construct the monodromy action on the refined motivic volume (Theorem 5.2.1). This monodromy action is an important feature of the motivic volume and Denef and Loeser’s motivic nearby fiber, as it captures the monodromy action on the nearby cycles complex at the motivic level. Terminology Let F be a field, and let X and Y be reduced F -schemes of finite type. Then X and Y are called birational if there exist a dense open subscheme U of X and a dense open subscheme V of Y such that the F -schemes U and V are isomorphic. This defines an equivalence relation on the set of isomorphism classes of reduced F -schemes of finite type; the equivalence class of X is called its birational type. We say that X is rational if it is birational to the projective space PdF for some d ≥ 0; this implies in particular that X is integral. m We say that X and Y are stably birational if X ×F PF is birational to Y ×F n PF for some m, n ≥ 0. This again defines an equivalence relation on the set of isomorphism classes of reduced F -schemes of finite type; the equivalence class of X is called its stable birational type. We say that X is stably rational if it is n stably birational to Spec F ; equivalently, if X ×F Pm F is birational to PF for some m, n ≥ 0. It is obvious that birational schemes are also stably birational, so that rationality implies stable rationality. The converse is not true: the first counterexample was constructed in [BCTSSD85].

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2. The Grothendieck ring of varieties graded by dimension 2.1. Reminders on the Grothendieck ring of varieties In this section we give a quick overview of the classical Grothendieck ring of varieties. We refer to [CNS18] for further background on the Grothendieck ring and its role in the theory of motivic integration. Let F be a field. The Grothendieck group K(VarF ) of F -varieties is the abelian group with the following presentation. • Generators: isomorphism classes [X] of F -schemes X of finite type; • Relations: whenever X is an F -scheme of finite type, and Y is a closed subscheme of X, then [X] = [Y ] + [X \ Y ]. These relations are often called scissor relations because they allow to cut a scheme into a partition by subschemes. The group K(VarF ) has a unique ring structure such that [X] · [X  ] = [X ×F X  ] for all F -schemes X and X  of finite type. The neutral element for the multiplication is 1 = [Spec F ], the class of the point. We 1 denote by L = [AF ] the class of the affine line in K(VarF ). n into the hyperplane Example 2.1.1. Let n be a positive integer. Partitioning PF at infinity and its complement, we find

[PnF ] = [Pn−1 ] + [AnF ] = [Pn−1 ] + Ln . F F Now it follows by induction on n that n [PF ] = 1 + L + · · · + Ln

in K(VarF ). Remark 2.1.2. The Grothendieck ring K(VarF ) is insensitive to non-reduced structures: if X is an F -scheme of finite type and we denote by Xred its maximal reduced closed subscheme, then the complement of Xred in X is empty, so that [X] = [Xred ]. Thus we would have obtained the same Grothendieck ring by taking as generators the isomorphism classes of reduced F -schemes of finite type (taking the fibered product in this category to define the ring multiplication). In fact, since every F scheme of finite type can be partitioned into regular quasi-projective F -schemes, we could even have taken the isomorphism classes of such schemes as generators (but then some care is required in the definition of the product if F is not perfect). The structure of the ring K(VarF ) is still poorly understood. The main challenge is to characterize geometrically when two F -schemes X and X  of finite type define the same class in the Grothendieck ring of varieties. An obvious sufficient condition is that X and X  be piecewise isomorphic, that is, can be partitioned into subschemes that are pairwise isomorphic; then the scissor relations immediately imply that [X] = [X  ]. Example 2.1.3. Let C ⊂ A2F be the affine plane cusp over F , defined by the equation y 2 − x3 = 0. Then C is piecewise isomorphic to the affine line A1F , 1 because C \ {(0, 0)} is isomorphic to AF \ {0}. It follows that [C] = L in K(VarF ).

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However, this condition is not necessary: L. Borisov has recently given an example of two C-schemes X and X  of finite type such that [X] = [X  ] but X and X  are not birational, and therefore certainly not piecewise isomorphic [Bo18]. This example is due to issues of cancellation: X and X  can be embedded into a common C-scheme W of finite type such that W \X and W \X  can be partitioned into pairwise isomorphic subschemes W1 , . . . , Wr and W1 , . . . , Wr , respectively. It follows that r r   [X] = [W ] − [Wi ] = [W ] − [Wi ] = [X  ], i=1

i=1

even though X and X  are not piecewise isomorphic. The schemes X and X  in Borisov’s example are smooth, but not proper. We do not know any such example where X and X  are smooth and proper, and it is still an open question whether the class in the Grothendieck ring detects the birational type of a smooth and proper F -scheme Y . We will see in Corollary 2.1.11 that, when F has characteristic zero, the class [Y ] determines the stable birational type of Y . The structure of K(VarF ) is particulary obscure when F has positive characteristic. For instance, if F is an imperfect field and F  is a purely inseparable finite extension of F , it is not known whether [Spec F  ] is different from [Spec F ]. The situation is better in characteristic zero, thanks to resolution of singularities and the Weak Factorization Theorem. Let us recall two of the most powerful results in this setting: the theorems of Bittner and Larsen & Lunts. We start with an easy consequence of Hironaka’s resolution of singularities. Lemma 2.1.4. Let F be a field of characteristic zero. Then the group K(VarF ) is generated by the classes of smooth and proper F -schemes. Proof. Let X be a reduced F -scheme of finite type. Since F has characteristic zero, we know that X is birational to a smooth and proper F -scheme X  ; then there exist strict closed subschemes Y and Y  of X and X  , respectively, such that X \ Y is isomorphic to X  \ Y  . It follows from the scissor relations that [X] = [X  ] − [Y  ] + [Y ], and by induction on the dimension, we may assume that [Y ] and [Y  ] can be written as Z-linear combinations of classes of smooth and proper F -schemes. We can also give a slightly more involved argument that is often useful to find a more explicit expression of [X] in terms of classes of smooth and proper F -schemes. We can partition X into separated smooth subschemes X1 , . . . , Xr . Then the scissor relations imply that [X] = [X1 ] + · · · + [Xr ] in K(VarF ). Thus it suffices to show that we can write [X] as a linear combination of classes of smooth and proper F -schemes when X is separated and smooth. By Hironaka’s embedded resolution of singularities, we can find a smooth compactification X of X such that the boundary X \ X is a divisor with strict normal crossings. Let Ei , i ∈ I be the prime components of this divisor. For every subset J of I, we set EJ = ∩j∈J Ej (in particular, E∅ = X). We endow the closed subsets EJ of X with their induced reduced structures; then the schemes EJ are smooth and proper over F . Using the

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scissor relations and an inclusion-exclusion argument, one easily checks that  [X] = (2.1.5) (−1)|J| [EJ ] J⊂I



in K(VarF ).

Theorem 2.1.6 (Bittner 2004). Let F be a field of characteristic zero. We define an abelian group KB (VarF ) by means of the following presentation. • Generators: isomorphism classes [X]B of connected smooth and proper F schemes X; • Relations: [∅]B = 0, and, whenever X is a connected smooth and proper F scheme and Y is a connected smooth closed subscheme of X, [BlY X]B − [E]B = [X]B − [Y ]B

(2.1.7)

where BlY X denotes the blow-up of X along Y , and E is the exceptional divisor. For every smooth and proper F -scheme X, we set [X]B = [X1 ]B + · · · + [Xr ]B where X1 , . . . , Xr are the connected components of X. We endow KB (VarF ) with the unique ring structure such that [X]B · [X  ]B = [X ×F X  ]B for all smooth and proper F -schemes X and X  . Then there exists a unique group morphism KB (VarF ) → K(VarF ) that maps [X]B to [X], for every smooth and proper F -scheme X. This morphism is an isomorphism of rings. Proof. This is the main part of Theorem 3.1 in [Bi04]. Let us briefly sketch the steps of the proof. The uniqueness and existence of the morphism are straightforward: the blow-up relations (2.1.7) are satisfied in K(VarF ) because BlY X \ E is isomorphic to X \ Y . It is also clear that this is a morphism of rings. The surjectivity of the morphism follows from the fact that the group K(VarF ) is generated by the classes of smooth and proper F -schemes, by Lemma 2.1.4. The principal difficulty is showing that KB (VarF ) → K(VarF ) is injective. This is achieved by constructing an inverse of this morphism. Since we can partition every F -scheme of finite type into separated smooth F schemes X of finite type, the essential step is to define the inverse on such schemes X (of course, one needs to check that the final construction is independent of the choice of the partition). Let X be a strict normal crossings compactification of X as in the proof of Lemma 2.1.4. Then a straightforward calculation shows that the element  (−1)|J| [EJ ]B (2.1.8) J⊂I

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in KB (VarF ) is invariant under blow-ups of smooth centers in the boundary of X that have strict normal crossings with all the strata EJ . The Weak Factorization Theorem [AKMW02] implies that any two strict normal crossings compactifications of X can be connected by a chain of such blow-ups and blow-downs. It follows that the element (2.1.8) only depends on X, and not on the chosen strict normal crossings compactification X. Now it is not hard to show that there exists a unique group morphism K(VarF ) → KB (VarF ) that maps [X] to the element (2.1.8) for every separated smooth F -scheme X of finite type. This morphism is inverse to the morphism KB (VarF ) → K(VarF ).  Remark 2.1.9. Bittner’s presentation remains valid we replace “proper” by “projective”, by the same proof. Theorem 2.1.10 (Larsen & Lunts 2003). Let F be a field of characteristic zero. Let SBF be the set of stable birational equivalence classes {X}sb of integral1 F schemes X of finite type, and let Z[SBF ] be the free abelian group on the set SBF . For every F -scheme Y of finite type, we set {Y }sb = {Y1 }sb + · · · + {Yr }sb in Z[SBF ], where Y1 , . . . , Yr are the irreducible components of Y . In particular, {∅}sb = 0. We endow Z[SBF ] with the unique ring structure such that {Y }sb · {Y  }sb = {Y ×F Y  }sb for all F -schemes Y and Y  of finite type. Then there exists a unique group morphism sb : K(VarF ) → Z[SBF ] that maps [X] to {X}sb for every smooth and proper F -scheme X. The morphism sb is a surjective ring morphism, and its kernel is the ideal in K(VarF ) generated by L. Proof. This is a combination of Theorem 2.3 and Proposition 2.7 in [LL03]. The existence of sb is an immediate consequence of Theorem 2.1.62 . Indeed, in the setting of (2.1.7) (and excluding the trivial case Y = X), BlY X is birational to dim(X)−dim(Y )−1 X, and E is birational to Y ×F PF . It is obvious that sb is unique, and that it is a ring morphism. The morphism sb maps L = [P1F ] − [Spec F ] to 0, because Spec F is stably birational to P1F . Thus sb induces a ring morphism sb : K(VarF )/LK(VarF ) → Z[SBF ]. We prove that this is an isomorphism by constructing an inverse. By Hironaka’s resolution of singularities, every class in SBF has a representative X that is a 1 We

follow the convention that integral schemes are non-empty. 2.1.10 slightly predates Theorem 2.1.6, and in [LL03], the existence of sb was deduced directly from the Weak Factorization Theorem. 2 Theorem

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connected smooth proper F -scheme. For every m ≥ 0, we have m 2 [X ×F Pm F ] − [X] = [X](L + L + · · · + L )

in K(VarF ) by the scissor relations. Thus [X ×F Pm F ] and [X] are congruent modulo L. Moreover, the congruence class of [X ×F Pm F ] modulo L is independent under blow-ups of smooth closed subschemes of X ×F Pm F , because the exceptional divisor of such a blow-up is a projective bundle over the center. Now it follows from the Weak Factorization Theorem that the class of X in K(VarF )/LK(VarF ) only depends on the stable birational equivalence class of X. This yields a ring morphism Z[SBF ] → K(VarF )/LK(VarF ) 

that is inverse to sb.

Beware that sb([X]) is usually different from {X}sb when X is not smooth and proper. For instance, if X is a nodal cubic in P2F , then it follows from the scissor relations that [X] = L in K(VarF ). Thus sb([X]) = 0. Corollary 2.1.11. Let F be a field of characteristic zero, and let X and X  be smooth and proper F -schemes. Then X and X  are stably birational if and only if [X] ≡ [X  ] modulo L in K(VarF ). In particular, [X] ≡ c modulo L for some integer c if and only if every connected component of X is stably rational; in that case, c is the number of connected components of X. Proof. By the scissor relations in the Grothendieck ring, we can write [X] = [X1 ] + · · · + [Xr ] where X1 , . . . , Xr are the connected components of X. Now the result follows immediately from Theorem 2.1.10.  Corollary 2.1.11 shows that the Grothendieck ring of varieties detects the stable birational type of smooth and proper schemes over fields of characteristic zero. The analogous question in positive characteristic is open. We are not aware of any example of a pair of smooth and proper schemes X, X  over a field F such that [X] = [X  ] in K(VarF ) and such that X and X  are not birational. Thus, to the best of our knowledge, it remains an open question whether the Grothendieck ring detects birational types of smooth and proper schemes (even in characteristic zero). To overcome this problem, we will introduce in Section 2.2 a finer variant of the Grothendieck ring of varieties, graded by dimension. Remark 2.1.12. Corollary 2.1.11 is false without the assumption that X and X  are smooth and proper. Borisov has constructed an example of two complex CalabiYau threefolds Z and Z  that are not birational and such that [Z × A6C ] = [Z  × A6C ] in K(VarC ) (see [Bo18] and the subsequent refinement in [Ma16] and [CNS18]). Thus Z ×C A6C and Z  ×C A6C are smooth complex varieties that are not stably birational and that define the same class in the Grothendieck ring.

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2.2. The graded Grothendieck ring Let F be a field, and let d be a non-negative integer. We define the Grothendieck group K(Vard F ) of F -varieties of dimension at most d to be the abelian group with the following presentation. • Generators: isomorphism classes [X]d of F -schemes X of finite type and of dimension at most d; • Relations: whenever X is an F -scheme of finite type and of dimension at most d, and Y is a closed subscheme of X, then [X]d = [Y ]d + [X \ Y ]d . The Grothendieck group of varieties graded by dimension is the graded abelian group  dim K(VarF K(Vard )= F ). d≥0

It has a unique structure of a graded ring such that [X]d · [X  ]e = [X ×F X  ]d+e for all F -schemes X and X  of finite type and of dimensions at most d and e, respectively. The identity element for the ring multiplication is the element 1 = [Spec F ]0 . With a slight abuse of notation, we will also use the symbol L to denote 1 the class [AF ]1 of the affine line in degree 1. We set τ = [Spec F ]1 , the class of the 1 dim ) shifts the degree: point in K(VarF ). The multiplicative action of τ on K(VarF for every F -scheme X of finite type of dimension at most d, and for every integer e ≥ 0, we have τ e [X]d = [X]d+e . The graded Grothendieck ring is related to the usual Grothendieck ring K(VarF ) in the following way. Proposition 2.2.1. There exists a unique ring morphism dim ) → K(VarF ) K(VarF

that maps [X]d to [X], for every non-negative integer d and every F -scheme X 1 of finite type and of dimension at most d. In particular, it maps L = [AF ]1 to 1 L = [AF ]. This morphism is surjective, and its kernel is the ideal generated by τ − 1. Proof. It is clear from the definitions of the Grothendieck rings that there is a unique ring morphism mapping [X]d to [X]. It is also obvious that it is surjective, and that its kernel contains τ − 1. Thus this morphism factors through a ring morphism dim K(VarF )/(τ − 1) → K(VarF ). This is an isomorphism: its inverse maps [X] to [X]d for every F -scheme X of finite type, where d is any integer such that d ≥ dim(X). 

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2.3. Birational types Let F be a field, and let d be a non-negative integer. We denote by BirdF the set of birational equivalence classes of integral F -schemes X of finite type of dimension d. The equivalence class of X will be denoted by {X}bir. Let Z[BirdF ] be the free d . This is the set of birational abelian group on the set BirdF . We set BirF = ∪d≥0 BirF equivalence classes of integral F -schemes of finite type. We also introduce the graded abelian group  Z[BirF ] = Z[BirdF ]. d≥0

It has a unique structure of a graded ring such that r  {X}bir · {X  }bir = {Ci }bir i=1

for all integral F -schemes X and X  of finite type, where C1 , . . . , Cr are the irreducible components of X ×F X  (endowed with their induced reduced structures). For a field F of characteristic zero, this graded ring was introduced by Kontsevich and Tschinkel in Section 2 of [KT19]; there it was called the Burnside ring of F . Proposition 2.3.1. There exists a unique morphism of graded rings bir : K(Vardim F ) → Z[BirF ] such that bir([X]d ) = {X}bir when X is an integral F -scheme of finite type of dimension d, and bir([X]d ) = 0 whenever X is an F -scheme of finite type of dimension at most d − 1. This morphism is surjective, and its kernel is the ideal generated by τ . Proof. Let d be a non-negative integer, and let X be an F -scheme of finite type and of dimension at most d. Let X1 , . . . , Xr be the irreducible components of X of dimension d, endowed with their induced reduced structures. We set bir([X]d ) = {X1 }bir + · · · + {Xr }bir . This definition respects the scissor relations in K(Vard F ), and therefore induces a morphism of graded groups dim bir : K(VarF ) → Z[BirF ].

It follows immediately from the definitions of the ring multiplications on the source and the target that bir is a morphism of graded rings. The uniqueness property in the statement follows from the fact that, by the scissor relations, the element [X]d − [X1 ]d − · · · − [Xr ]d d ) K(VarF

can be written as a linear combination of classes [Y ]d where Y is an in F -scheme of finite type of dimension at most d − 1. It is obvious that the morphism bir is surjective, and that its kernel contains the ideal generated by τ . Thus it induces a ring morphism dim K(VarF )/(τ ) → Z[BirF ].

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This is an isomorphism: its inverse maps {X}bir to [X]dim(X) , for every integral F -scheme X of finite type.  Corollary 2.3.2. If X and X  are reduced F -schemes of finite type of pure dimension d, then X and X  are birational if and only if [X]d ≡ [X  ]d modulo τ in dim ). K(VarF Proof. If X1 , . . . , Xr are the irreducible components of X, then bir([X]d ) = {X1 }bir + · · · + {Xr }bir in Z[BirF ], and the analogous property holds for X  . Thus X and X  are birational if and only if bir([X]d ) = bir([X  ]d ). By Proposition 2.3.1, this is equivalent to the  property that [X]d ≡ [X  ]d modulo τ in K(Vardim F ). Corollary 2.3.2 tells us that the graded version of the Grothendieck ring detects birational types; this is its main advantage over the classical Grothendieck ring of varieties. Example 2.3.3. Let Z and Z  be the complex Calabi-Yau threefolds from Borisov’s 6 6 example (see Remark 2.1.12). Then Z ×C AC define the same class in and Z  ×C AC K(VarC ). However, since these varieties are not birational, it follows from Corol9 ) are distinct. Thus the difference of these lary 2.3.2 that their classes in K(VarC two classes lies in the kernel of the map K(Vardim C ) → K(VarC ) defined in Proposition 2.2.1. Remark 2.3.4. A different manifestation of the isomorphism in Proposition 2.3.1 also appears in Proposition 2.2 of [KT21]. 2.4. A refinement of Bittner’s presentation We will now establish an analog of Bittner’s presentation (Theorem 2.1.6) for the dim graded Grothendieck ring K(VarF ), where F is a field of characteristic zero. For every non-negative integer d, we define an abelian group KB (Vard F ) by means of the following presentation. • Generators: isomorphism classes [X]dB of connected smooth and proper F schemes X of dimension at most d; • Relations: [∅]dB = 0, and, whenever X is a connected smooth and proper F -scheme of dimension at most d, and Y is a connected smooth closed subscheme of X, then B [BlY X]dB − [E]dB = [X]B d − [Y ]d

(2.4.1)

where BlY X denotes the blow-up of X along Y , and E is the exceptional divisor. Theorem 2.4.2. Let F be a field of characteristic zero. For every non-negative integer d, there exists a unique group morphism d KB (VarF ) → K(Vard F )

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that maps [X]dB to [X]d , for every connected smooth and proper F -scheme X of dimension at most d. This morphism is an isomorphism. Proof. One can simply copy the proof of Theorem 3.1 in [Bi04]: all the schemes involved in the argument have dimension at most d (see Theorem 2.1.6 for a sketch  of the proof). 2.5. A refinement of the theorem of Larsen & Lunts We can also prove a refinement of the theorem of Larsen and Lunts (Theorem 2.1.10) in the graded setting. This refinement will not be used in the remainder of the paper, because we already know from Corollary 2.3.2 that the graded Grothendieck ring detects birational types. Let F be a field, and let d be a non-negative integer. Let X and X  be irreducible F -schemes of finite type such that dim(X) ≤ d and dim(X  ) ≤ d. d−dim(X)−1 We say that X and X  are d-stably birational if X ×F PF is birational to  d−dim(X )−1 . Here and below, we use the convention that P−1 X  × F PF F = Spec F . See Remark 2.5.4 for a comment on the appearance of the term −1 in the dimensions of the projective spaces. Note that, if d = dim(X) = dim(X  ) or d − 1 = dim(X) = dim(X  ), then X and X  are d-stably birational if and only if they are birational. We denote by SBd F the set of d-stable birational equivalence classes of integral F -schemes X of finite type of dimension at most d. The equivalence class of d X will be denoted by {X}sb,d . Let Z[SBF ] be the free abelian group on the set d SBF . For every F -scheme Y of finite type of dimension at most d, we set {Y }sb,d = {Y1 }sb,d + · · · + {Yr }sb,d d in Z[SBF ], where Y1 , . . . , Yr are the irreducible components of Y (with their induced reduced structures). In particular, {∅}sb,d = 0. We consider the graded abelian group  dim Z[SBF Z[SBd ]= F ]. d≥0

and endow it with the unique structure of a graded ring such that {Y }sb,d · {Y  }sb,e = {Y ×F Y  }sb,d+e whenever d and e are non-negative integers, and Y and Y  are F -schemes of finite type of dimensions at most d and e, respectively. Lemma 2.5.1. For all integers d and n such that d ≥ n ≥ 0 and (d, n) = (1, 1), we n have [PF ]d ≡ τ d modulo [A1F ]2 in K(Vardim F ). Proof. If n = 0 then the assertion is trivial; thus we may assume that n ≥ 1 and n d ≥ 2. The scissor relations imply that [PnF ]d = [Spec k]d + [A1F ]d + · · · + [AF ]d . Thus we can write n−1 n [PF ]d − τ d = [A1F ]2 (τ d−2 + · · · + [AF ]d−2 )

in K(Vardim F ).



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Theorem 2.5.2. Let F be a field of characteristic zero. Then there exists a unique morphism of graded rings dim sbdim : K(Vardim F ) → Z[SBF ]

such that sbdim ([X]d ) = {X}sb,d for every non-negative integer d and every smooth and proper F -scheme X of dimension at most d. The morphism sbdim is surjective, and its kernel is the ideal generated by τ L = [A1F ]2 . Proof. The existence and uniqueness of the morphism sbdim follow immediately from Theorem 2.4.2; note that, in the blow-up relations (2.4.1) (and excluding the trivial case Y = X), the exceptional divisor E is birational to Y ×F Pdim(X)−dim(Y )−1 , so that E and Y are dim(X)-stably birational. The kernel of 1 sbdim contains [A1F ]2 = [P1F ]2 − τ 2 , because PF and Spec F are 2-stably birational. Thus sbdim induces a morphism of graded rings 1 dim sbdim : K(Vardim F )/([AF ]2 ) → Z[SBF ].

We will show that this is an isomorphism by constructing its inverse. For every smooth and proper F -scheme Y and every integer d ≥ dim(Y ), 1 the residue class of [Y ]d in K(Vardim F )/([AF ]2 ) is invariant under any blow-up of a smooth strict closed subscheme Z in Y . Indeed, we may assume that the codimension of Z in Y is at least 2; if we denote by E the exceptional divisor in the blow-up BlZ Y of Y at Z, then the scissor relations imply that [BlZ Y ]d − [Y ]d = [E]d − [Z]d dim(Y )−dim(Z)−1

= [Z]dim(Z) ([PF

]d−dim(Z) − τ d−dim(Z) )

1 in K(Vardim F ), and this element is divisible by [AF ]2 by Lemma 2.5.1. Therefore, by the Weak Factorization Theorem [AKMW02], the residue class of [Y ]d 1 in K(Vardim F )/([AF ]2 ) only depends on the birational equivalence class of Y . Now let X be a non-empty connected smooth and proper F -scheme, and let d d−dim(X)−1 ]d be an integer such that d ≥ dim(X). Then the residue class of [X ×F PF 1 in K(Vardim F )/([AF ]2 ) only depends on the birational equivalence class of X ×F d−dim(X)−1 PF ; in other words, it only depends on the d-stable birational equivalence d−dim(X)−1 1 class of X. But we also have [X ×F PF ]d = [X]d in K(Vardim F )/([AF ]2 ) by Lemma 2.5.1. Thus we obtain a morphism of graded abelian groups dim 1 Z[SBdim F ] → K(VarF )/([AF ]2 ) : {X}sb,d → [X]d

that is inverse to sbdim .



Corollary 2.5.3. Let F be a field of characteristic zero. 1. Let X and X  be connected smooth and proper F -schemes. Let d be an integer such that d ≥ dim(X) and d ≥ dim(X  ). Then X and X  are d-stably birational if and only if [X]d ≡ [X  ]d modulo [A1F ]2 in K(Vardim F ).

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1 2. Let X be a smooth and proper F -scheme. Then [X] ≡ c modulo [AF ]2 for some integer c if and only if, for every connected component C of X, the d−dim(C)−1 scheme C ×F PF is rational; in that case, c is the number of connected components of X.

Proof. This follows immediately from Theorem 2.5.2.



Remark 2.5.4. One can also formulate a weaker version of Theorem 2.5.2, replacing the exponents d − dim(X) − 1 and d − dim(X  ) − 1 in the definition of d-stable birational equivalence by d − dim(X) and d − dim(X  ). With that definition, the morphism sbdim constructed in Theorem 2.5.2 has kernel generated by L = [A1F ]1 , 1 rather than [AF ]2 .

3. Dimensional refinement of the motivic volume 3.1. The motivic volume Let k be a field of characteristic zero, and set 7 7 K= k((t1/n )), R = k[[t1/n ]]. n>0

n>0

The field K is a henselian valued field with valuation ring R with respect to the t-adic valuation ordt : K × → Q. If k is algebraically closed then K is an algebraic closure of the Laurent series field k((t)), but we do not make this assumption. In [NS19], Shinder and the first-named author constructed a ring morphism Vol : K(VarK ) → K(Vark ), 1 called the motivic volume. It maps [AK ] to [Ak1 ] and has the property that for every smooth and proper R-scheme X , one has Vol([XK ]) = [Xk ]. This ring morphism can be viewed as a refinement of the motivic nearby fiber of Denef and Loeser [DL01], where the refinement consists of the fact that we do not need to invert [Ak1 ] in the target; this is crucial for applications to rationality problems. If k is algebraically closed, then the existence of the morphism Vol follows immediately from the work of Hrushovski and Kazhdan [HK06]. 1 The fact that Vol([AK ]) = [A1k ] implies that the morphism Vol factors through a ring morphism K(VarK )/L → K(Vark )/L which, by Theorem 2.1.10, we can identify with a ring morphism

Volsb : Z[SBK ] → Z[SBk ]. If X is a connected smooth and proper R-scheme, then Volsb maps the stable birational equivalence class of XK to that of Xk . Since Vol([Spec K]) = [Spec k], the morphism Volsb maps the class of stably rational K-varieties to the class of stably rational k-varieties. It then follows easily that stable rationality of geometric

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fibers specializes in smooth and proper families, which was one of the main results of [NS19]. The main purpose of the present article is to upgrade the motivic volume to a morphism of graded rings dim Vol : K(VarK ) → K(Vardim k )

that maps [Spec K]1 to [Spec k]1 and fits into a commutative diagram Vol

dim K(Vardim K ) −−−−→ K(Vark ) ⏐ ⏐ ⏐ ⏐  

K(VarK ) −−−−→ K(Vark ) Vol

where the vertical morphims are the forgetful maps from Proposition 2.2.1. The refined volume induces a morphism of graded rings dim K(VarK )/([Spec K]1 ) → K(Vardim k )/([Spec k]1 )

which, by Proposition 2.3.1, can be identified with a morphism of graded rings Volbir : Z[BirK ] → Z[Birk ]. This is precisely the birational specialization morphism from [KT19]. The construction of the motivic volume in [NS19] was phrased in the language of logarithmic geometry. For readers familiar with that language it will be an easy exercise to check that all the arguments in [NS19] still apply to the dimensional refinement of the Grothendieck ring. Therefore, we decided not to reproduce the proofs here, but to explain the main properties in a more explicit way, avoiding as much as possible the language of logarithmic schemes. We hope that this will make the theory more user-friendly. It also makes our formula for the motivic volume a bit more general, because our class of strictly toroidal models (defined below) is slightly more general than the log smooth models that were considered in [NS19], since one does not need to assume that the log structure is defined globally on the model. This will be explained in more detail in the proof of Theorem 3.3.2. There are two further differences in presentation compared to Appendix A of [NS19]: we directly work over K, rather than k((t)), which means that we ignore the monodromy action of the profinite group scheme μ ˆ of roots of unity over k (we will come back to this monodromy action in Section 5). Second, while the formulas in [NS19] were stated in terms of open strata in the special fibers of log smooth models, we will also express our formulas in terms of closed strata. This is more convenient for applications to rationality questions, and for a comparison with the invariant defined in [KT19]. Passing between open and closed strata can always be done by means of basic inclusion-exclusion arguments (see Lemma 3.3.1). Finally, let us also point out a typo in [NS19]: in the formulas in Theorem A.3.9 and Propostion A.4.1, the factor (L − 1)rv (σ) should be (L − 1)rv (σ)−1 , like in the expression for VolK (X) in the middle of page 407.

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3.2. Strictly toroidal models We say that a monoid M is toric if it is sharp3 , finitely generated, integral, saturated and torsion free. Toric monoids are precisely those monoids that are isomorphic to the monoid of lattice points in a strictly convex rational polyhedral cone σ in Rd , for some positive integer d. There is an intrinsic definition of the faces of M and their dimensions; if M = σ ∩ Zd then the faces of M are the submonoids of the form τ ∩ Zd with τ a face of σ, and the dimension of τ ∩ Zd equals the dimension of τ . For any ring A, we denote by A[M ] the monoid A-algebra associated with M . Its elements are the finite A-linear combinations of monomials χm with m in M . Let X be a flat separated R-scheme of finite type, and let x be a point of the special fiber Xk . We say that X is strictly toroidal at x if there exist a toric monoid M , an open neighbourhood U of x in X , and a smooth morphism of R-schemes U → Spec R[M ]/(χm − tq ) (3.2.1) where q is a positive rational number, and m is an element of M such that k[M ]/(χm ) is reduced. This implies in particular that X is normal at x and Xk is reduced at x. We say that X is strictly toroidal if it is so at every point x of Xk . Example 3.2.2. A flat separated R-scheme X of finite type is called strictly semistable if, Zariski-locally, it admits a smooth morphism to a scheme of the form Spec R[z1 , . . . , zn ]/(z1 · · · · · zn − tq )

(3.2.3)

for some n ≥ 1 and some positive rational number q. Then X is also strictly toroidal (take M = Nn and set m = (1, . . . , 1)). Every smooth and proper Kscheme X has a strictly semi-stable proper R-model: we can descend X to k((t1/d )) for some d > 0; by Hironaka’s resolution of singularities, we can then find a proper regular k[[t1/d ]]-model Y whose special fiber is a divisor with strict normal crossings (not necessarily reduced). An elementary local calculation now shows that the normalization of Y ×k[[t1/d ]] R is strictly semi-stable proper R-model of X. Example 3.2.4. The class of strictly toroidal R-models is more flexible than that of strictly semi-stable models. This is useful in applications to rationality problems, as one is allowed to bypass an explicit resolution of singularities over k[[t]]. For instance, one can skip the construction in Lemma 2.2 of Shinder’s paper [Sh19]; other applications are given in Section 4.3 and in [NO19]. Let f0 ∈ k[z0 , . . . , zn ] be a general homogeneous polynomial of positive degree d0 . Let f1 , . . . , fr ∈ k[z0 , . . . , zn ] be general homogeneous polynomials of positive degrees d1 , . . . , dr , respectively, such that d1 + · · · + dr = d0 . Then X = Proj R[z0 , . . . , zn ]/(tf0 − f1 · · · · · fr ) is strictly toroidal, but not strictly semi-stable if r ≥ 2 and n ≥ 3: locally around the singular points of Xk where f0 vanishes, there is no smooth morphism to a scheme of the form (3.2.3). 3A

monoid M is called sharp if the only invertible element of M is the identity.

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To see that X is strictly toroidal, let x be a point on Xk , and let I be the set of indices i ∈ {0, . . . , r} such that fi (x) = 0. After a permuation of the coordinates zj , we may assume that z0 (x) = 0. Since the polynomials fi are general, the regular functions fi /zidi with i ∈ I form a part of a regular system of local parameters in n ,x . Therefore, there exists an open neighbourhood of x in X that admits a OPR smooth morphism to the R-scheme 8  # Y = Spec R[yi | i ∈ I] tw − yi i∈I\{0}

where w = y0 if 0 ∈ I and w = 1 otherwise. In the latter case, Y , and therefore X , are strictly semi-stable and, thus, also strictly toroidal. In the former case, Y is isomorphic to the R-scheme Spec R[M ]/(t − χm ) where M is the quotient of the monoid NI × N by the congruence relation generated by    (e0 , 1) ∼ ei , 0 , i∈I\{0}

where (ei )i∈I is the standard basis of NI , and m is the residue class of (0, 1) ∈ NI × N. If X is a strictly toroidal R-scheme, then a stratum E of Xk is a connected component of the intersection of a non-empty set of irreducible components of Xk . We denote by codim(E) the codimension of E in Xk , and by S(X ) the set of strata in Xk . The interior E o of a stratum E is the complement in E of the union of strictly smaller strata. This interior E o is a connected smooth separated k-scheme of finite type, but E may have singularities along the boundary E \ E o . All of these singularities are strictly toroidal, in the sense that E admits Zariski-locally an ´etale morphism to a toric k-variety. We also attach to E an element in K(Vardim k ) given by the formula  codim(E  ) P (E) = [Gm,k ]codim(E) E  ⊃E

where the sum is taken over all the strata E  in Xk that contain E. If E is contained in precisely codim(E) + 1 irreducible components of Xk , then P (E) = codim(E) [Pk ]codim(E) because the sum in the definition corresponds to the partition of codim(E)

into torus orbits. This happens for instance when the toroidal structure Pk of X at the generic point of E is simplicial. In general, one can write P (E) as the class of a proper toric k-variety of dimension codim(E), which can be computed explicitly from the toroidal structure (that is, the monoid M ) at the generic point of E. The only thing that matters for our applications is that the image of P (E) in Z[Birk ] with respect to the morphism in Proposition 2.3.1 is always equal to codim(E) {Pk }bir ; this follows immediately from the definition of P (E).

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308

3.3. Construction of the motivic volume Lemma 3.3.1. If X is a strictly toroidal R-scheme, then   codim(E) ]e = (−1)codim(E) [E o ×k Gm,k (−1)codim(E) [E]e−codim(E) P (E). E∈S(X )

E∈S(X )

Proof. Every stratum E in Xk is the disjoint union of its open substrata (E  )o , so that  [E]e−codim(E) = [(E  )o ]e−codim(E) E  ⊂E

K(Varkdim ).

in ment as 

Thus we can write the right-hand side of the equality in the state-



(−1)codim(E) [(E  )o ]e−codim(E) P (E)

E∈S(X ) E  ⊂E

=

 E  ∈S(X

'



[(E  )o ]e−codim(E  )

( (−1)codim(E) P (E)τ dim(E)−dim(E



)

.

E⊃E 

)

By the definition of the element P (E), we have for every E  in S(X ) that   (−1)codim(E) P (E)τ dim(E)−dim(E ) E⊃E 

=



'

E⊃E 

=





(−1)codim(E)

( codim(E  ) ]e−dim(E  ) [Gm,k

E  ⊃E

'

codim(E  ) ]e−dim(E  ) [Gm,k

E  ⊃E 



( codim(E)

(−1)

.

E  ⊂E⊂E 

We fix a stratum E  in Xk that contains E  . Since X is strictly toroidal, there exists an inclusion preserving bijective correspondence between the strata E in Xk such that E  ⊂ E ⊂ E  , and the strict faces of a strictly convex rational polyhedral cone σ; the dimension of the face is equal to the dimension of the corresponding stratum minus dim(E  ). It follows that  (−1)codim(E) E  ⊂E⊂E  

is equal to (−1)codim(E ) times the compactly supported Euler characteristic of σ, which is 1 if E  = E  (then σ has dimension 0) and 0 otherwise. Therefore, ( '   codim(E  ) codim(E) ]e−dim(E  ) (−1) [Gm,k E  ⊃E 

E  ⊂E⊂E  codim(E  )

= (−1)

codim(E  ) ]e−dim(E  ) [Gm,k

and this implies the desired equality.



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The dimensional refinement of the motivic volume is characterized by the following theorem. Theorem 3.3.2. There exists a unique ring morphism dim Vol : K(VarK ) → K(Varkdim )

such that for every strictly toroidal proper R-scheme X with smooth generic fiber X = XK , and for every integer e ≥ dim(X), we have  codim(E) ]e (−1)codim(E) [E o ×k Gm,k (3.3.3) Vol([X]e ) = E∈S(X )

=



(−1)codim(E) [E]e−codim(E) P (E).

(3.3.4)

E∈S(X )

Proof. The main difference with the set-up in Appendix A of [NS19] is that we do not have a globally defined log structure on X such that the structural morphism to Spec R with its standard log structure is smooth. In the language of toroidal embeddings [KKMS73], the problem can be described in the following way. Let U be an open subscheme of X as in (3.2.1) and let R0 be a finite extension of k[[t]] in R such that the morphism (3.2.1) is obtained from a smooth morphism of R0 -schemes U0 → Spec R0 [M ]/(tq − χm ) by extension of scalars to R. The pullback of the toric boundary on Spec R0 [M ] is a divisor D on U0 such that U0 \ D → U0 is a toroidal embedding without selfintersection over the discrete valuation ring R0 in the sense of [KKMS73]. But the divisors D do not necessarily glue to a divisor on the whole of X . To resolve this issue, we first define a local variant of the motivic volume. Let Y be a separated flat R-scheme of finite type of pure relative dimension d, with smooth generic fiber YK . By resolution of singularities and the semistable reduction theorem, we can find a positive integer n, a model Y0 for Y over R0 = k[[t1/n ]] and a proper morphism of R0 -schemes h : Z → Y0 such that h is an isomorphism on the generic fibers, Z is regular, and Zk is a reduced divisor with strict normal crossings. Then we define the motivic volume of Y as  codim(E) Vol(Y ) = ]d ∈ K(Varkdim ). (−1)codim(E) [E o ×k Gm,k E∈S(Z )

It can be deduced from the weak factorization theorem in [AT19] and some elementary calculations that this definition only depends on Z , and not on the choices of n, Y0 and Z : the arguments in Appendix A of [NS19] immediately carry over to our setting (see in particular Propositions A.3.5 and A.3.8). It is clear from the definition that Vol(Y ) is local on Y : for every finite open cover {Uα | α ∈ A} of Y , we have  Vol(Y ) = (−1)|B|−1 Vol(∩β∈B Uβ ) ∅ =B⊂A

in

K(Varkdim ).

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The next step is to prove that, when X is a strictly toroidal R-scheme of pure relative dimension d with smooth generic fiber, we still have  codim(E) Vol(X ) = (−1)codim(E) [E o ×k Gm,k ]d E∈S(X )

in K(Varkdim ). Since both sides of the expression are local on X , we may assume that there exists an ´etale morphism X → Spec R[M ]/(tq − χm ) as in (3.2.1), which descends to an ´etale morphism X0 → Spec R0 [M ]/(tq − χm ) over some finite extension R0 of k[[tq ]] in R. Denote by D the pullback to X0 of the toric boundary of Spec R0 [M ]. Then the open embedding X0 \ D → X0 is a toroidal embedding without self-intersections in the sense of [KKMS73], and one can use a suitable subdivision of the associated cone complex to construct, over some finite extension of R0 , a proper morphism Z → X0 that is an isomorphism on the generic fibers and such that Z is regular and Zk is a reduced divisor with strict normal crossings. A toric calculation shows that the expression  codim(E) ]d (−1)codim(E) [E o × Gm,k E∈S(X )

remains invariant under this modification. This calculation is carried out in Section A.2 of [NS19] in the language of logarithmic schemes. Now let X be a connected smooth and proper K-scheme of dimension d, and let X be a strictly toroidal proper R-model of X. Since any pair of proper Rmodels of X can be dominated by a common toroidal proper R-model, the above arguments imply that Vol(X ) only depends on X. Thus for every integer e ≥ d, we may define  codim(E) Vol([X]e ) = Vol(X )τ e−d = ]e . (−1)codim(E) [E o ×k Gm,k E∈S(X )

The equality of the expressions (3.3.3) and (3.3.4) follows from Lemma 3.3.1. The final step of the proof is to show that Vol([X]e ) is multiplicative in [X]e and satisfies the blow-up relations in Bittner’s presentation 2.4.2. Multiplicativity follows easily from the fact that the product of two strictly toroidal R-schemes is again strictly toroidal (see [NS19, A.3.7]). Let Y be a connected smooth strict closed subscheme of X, and let BlY X → X be the blow-up of X along Y , with exceptional divisor E. We can find a proper strictly toroidal R-model X of X such that the schematic closure Y of Y in X has transversal intersections with the special fiber Xk . Then Y is a strictly toroidal proper R-model of Y . Moreover, the blow-up BlY X of X along Y is a strictly toroidal proper R-model of BlY X, and the schematic closure of E in BlY X is a strictly toroidal proper R-model of E. Now a direct calculation shows that Vol([X]e ) − Vol([Y ]e ) = Vol([BlY X]e ) − Vol([E]e ) in

K(Vardim k ),

for all integers e ≥ d.



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Corollary 3.3.5. There exist unique ring morphisms Volbir : Z[BirK ] → Z[Birk ],

Volsb : Z[SBK ] → Z[SBk ]

such that for every strictly toroidal proper R-scheme X with smooth generic fiber X = XK , we have  codim(E) }bir , (3.3.6) (−1)codim(E) {E ×k Pk Volbir ({X}bir) = E∈S(X )

Volsb ({X}sb ) =



(−1)codim(E) {E}sb .

(3.3.7)

E∈S(X )

In particular, if X is smooth and proper over R, then Volbir ({X}bir) = {Xk }bir and Volsb ({X}sb ) = {Xk }sb . Proof. Since Spec R is a strictly toroidal proper R-model of Spec K, the motivic volume Vol maps [Spec K]d to [Spec k]d for every d ≥ 0. It follows from Proposition 2.3.1 that Vol factors through a ring morphism Volbir ; it satisfies the formula in the codim(E) statement because bir(P (E)) = {Pk }bir for every stratum E in Xk . Since n n PR is a strictly toroidal proper R-model of PK for every n ≥ 0, the morphism n n Volbir maps {PK }bir to {Pk }bir . Because Volbir is also multiplicative, it factors through a morphism Volsb as in the statement of the corollary. 

4. Applications to rationality problems 4.1. Specialization of birational types A first application of Corollary 3.3.5 is that it settles the long-standing question of specialization of (stable) birational equivalence. The case of stable birational equivalence was first proved in [NS19]; the stronger result for birational equivalence follows from the results in [KT19]. This application only uses the special case of formulas (3.3.6) and (3.3.7) where X is smooth over R. Theorem 4.1.1. Let S be a Noetherian scheme of characteristic zero, and let X → S and Y → S be smooth and proper S-schemes. For every point s of S, we fix a geometric point s supported at s. We define subsets Sbir (X , Y ) and Ssb (X , Y ) of S in the following way: Sbir (X , Y ) = {s ∈ S | X ×S s ∼bir Y ×S s}, Ssb (X , Y ) = {s ∈ S | X ×S s ∼sb Y ×S s}, where ∼bir and ∼sb denote birational equivalence and stable birational equivalence, respectively. Then Sbir (X , Y ) and Ssb (X , Y ) are countable unions of closed subsets of S. Proof. It follows from a standard Hilbert scheme argument that Sbir (X , Y ) and Ssb (X , Y ) are countable unions of locally closed subsets of S; see for instance Proposition 2.3 in [dFF13], which is stated in a more restrictive setting but whose proof also confirms our more general statement. Therefore, it is sufficient to prove

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that Sbir (X , Y ) and Ssb (X , Y ) are closed under specialization. Now one easily reduces to the case where S = Spec k[[t]] and k is algebraically closed; see the proof of Theorem 4.1.4 in [NS19]. Then X ×S Spec R and Y ×S Spec R are smooth and proper R-schemes, and Corollary 3.3.5 implies that Xk is birational (resp. stably birational) to Yk if XK is birational (resp. stably birational) to YK .  Corollary 4.1.2. Let S be a Noetherian scheme of characteristic zero, and let X → S be a smooth and proper S-scheme. For every point s of S, we fix a geometric point s supported at s. We define subsets Srat (X ) and Ssrat (X ) of S in the following way: Srat (X ) = {s ∈ S | X ×S s is rational}, Ssrat (X ) = {s ∈ S | X ×S s is stably rational}. Then Srat (X ) and Ssrat (X ) are countable unions of closed subsets of S. Proof. This follows immediately from Theorem 4.1.1, because 7 Sbir (X , PnS ) Srat (X ) = n≥0

and Ssrat (X ) = Ssb (X , S). If S is connected, we simply have Srat (X ) = Sbir (X , PnS ) with n the dimension of the fibers of X → S.  Countable unions cannot be avoided in the statements of Theorem 4.1.1 and Corollary 4.1.2: in [HPT18a], Hassett, Pirutka and Tschinkel have constructed a smooth and proper family X → S over a complex variety S such that Srat is dense in S but Ssrat = S. 4.2. Obstruction to stable rationality By contraposition, we can also use Corollary 3.3.5 as an obstruction to rationality or stable rationality of XK . Theorem 4.2.1. Let X be a strictly toroidal proper R-scheme. If  codim(E) (−1)codim(E) {E × Pk }bir = {Spec k}bir E∈S(X )

in Z[Birk ], then XK is not rational. Similarly, if  (−1)codim(E) {E}sb = {Spec k}sb E∈S(X )

in Z[SBk ], then XK is not stably rational. Here the sums are taken over the strata E in Xk . Proof. This follows immediately from Corollary 3.3.5.



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These obstructions are not always easy to use in practice, because one needs to control the cancellations in the alternating sums, and thus understand the (stable) birational equivalences between the individual strata. Let us look at an interesting special case where cancellations do not occur. Corollary 4.2.2. Let X be a strictly toroidal proper R-scheme. Suppose that every connected component of every stratum E of even (resp. odd) codimension in Xk is stably rational, and that at least one connected component of some stratum of odd (resp. even) codimension in Xk is not stably rational. Then XK is not stably rational. Proof. The assumption implies that all the terms appearing with a positive (resp. negative) sign in the sum  (−1)codim(E) {E}bir E∈S(X )

are integer multiples {Spec k}sb , while at least one term with opposite sign is not a multiple of {Spec k}sb . Thus the whole sum is different from {Spec k}sb .  In order to apply Theorem 4.2.1 and Corollary 4.2.2 to find new classes of non-stably rational varieties, one always needs non-trivial input, namely, a strictly toroidal degeneration such that at least one stratum in the special fiber is not stably rational and such that one can control the potential cancellations in the alternating sum of stable birational types. A convenient method to produce interesting strictly toroidal degenerations is provided by tropical geometry; this method is used in [NO19] to obtain various new stable irrationality results. The upshot of this technique is that one can deduce stable irrationality of a very general member of a family of varieties from the stable irrationality of special varieties in lower dimensions and/or degrees. 4.3. Examples We will discuss a few applications of the obstruction to stable rationality in Theorem 4.2.1. Examples 4.3.1, 4.3.2, 4.3.3 and 4.3.4 have already been obtained by different methods in the literature; here our aim is merely to illustrate the general technique. To the best of our knowledge, Examples 4.3.5 and 4.3.6 are new. More elaborate applications can be found in [NO19], where, among other results, we prove the stable irrationality of very general quartic fivefolds and various new classes of complete intersections (including very general (2, 3) complete intersections in P6 ). Throughout this section, we denote by k an algebraically closed field of characteristic zero. Example 4.3.1. Our first example is taken from Theorem 4.3.1 in [NS19]. We will deduce from Theorem 4.2.1 that a very general quartic double solid over k is not stably rational; this is a special case of a result by Voisin [Vo15]. By Corollary 4.1.2, it suffices to construct one non-stably rational smooth quartic double solid

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over some algebraically closed field of characteristic zero; our base field will be the field K of Puiseux series over k = C. As input we use Artin and Mumford’s famous example of a stably irrational quartic double solid Y0 over C with only isolated ordinary double points as singularities [AM]. Let F0 ∈ C[z0 , . . . , z3 ] be a homogeneous degree 4 polynomial that 3 defines the ramification divisor D0 of the double cover Y0 → PC . Let F be a general 3 homogeneous degree 4 polynomial in C[z0 , . . . , z3 ]. Let D be the divisor in PC[t] defined by F0 − tF = 0 and let Y → P3C[t] be the double cover ramified along D. Then Y is a regular proper C[t]-scheme with special fiber Y0 ; its generic fiber is a smooth quartic double solid. Let Y  → Y be the blow-up at the singular points of Y0 and let X be the normalization of Y  ×C[t] R. By Example 3.2.2, the R-scheme X is strictly semi-stable. Its special fiber has a unique stably irrational stratum, namely, the strict transform of Y0 . Thus, it follows from Theorem 4.2.1 that the smooth quartic double solid XK is stably irrational. Example 4.3.2. The next application concerns stable non-rationality of very general quartic hypersurfaces of dimensions 4 and 5. In the fourfold case, this was first proved by Totaro as a special case of the general bound he established in [To16]; this bound was further improved (and extended to positive characteristic) by Schreieder in [Sch19]. The fivefold case was first proved in [NO19], as an application of the tropical techniques we develop in that paper. Here we will treat both cases in a uniform way, without invoking tropical methods. Let n ∈ {4, 5} and let F ∈ k[z0 , . . . , zn+1 ] be a homogeneous polynomial of degree 4, which we choose to be very general subject to the condition that F is invariant under the transposition of the variables zn and zn+1 . Set X = Proj R[z0 , . . . , zn+1 , y]/(yt − zn zn+1 , y 2 − F ) where the variable y has weight 2. The generic fiber XK is a smooth quartic n+1 hypersurface in PK (we can make the substitution y = zn zn+1 /t because t is invertible in K). The R-scheme X is strictly toroidal: away from the locus Z defined by y = t = zn = zn+1 = 0, the scheme X  = Proj k[[t]][z0 , . . . , zn+1 , y]/(yt − zn zn+1 , y 2 − F ) is regular and its special fiber is a reduced divisor with strict normal crossings, so that X = X  ×k[t] R is strictly semi-stable away from Z. On the other hand, for every i in {0, . . . , n − 1}, the projection morphism : ;  9 zn zn+1 y zn zn+1 y D+ (zi ) → Spec R , , t− zi zi zi2 zi2 zi zi is smooth along Z ∩ D+ (zi ), so that X is also strictly toroidal along Z. The special fiber Xk has three strata. The two irreducible components E1 and E2 , given by zn = 0 and zn+1 = 0, are isomorphic because of the symmetry of F . When n = 4, their intersection is a very general quartic double solid, and thus not stably rational by Example 4.3.1. When n = 5, we conclude similarly using

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the result that very general quartic double fourfolds over k are also not stably rational, by [HPT19]. Thus, in each case, we have {E1 }sb + {E2 }sb − {E1 ∩ E2 }sb = 2{E1 }sb − {E1 ∩ E2 }sb = {Spec k}sb in Z[SBk ], and it follows from Theorem 4.2.1 that XK is not stably rational. Now Corollary 4.1.2 implies that a very general quartic fourfold or fivefold is not stably rational. Example 4.3.3. Next, we prove that very general sextic fivefolds and sixfolds are not stably rational. These cases also fall in the range of results in [To16] and [Sch19]. Our starting points are the stable irrationality of very general complete intersections of three quadrics in Pk6 [HT19, §4.4] and in Pk7 [HPT18b]. Let (Q1 , Q2 , Q3 ) be a very general triple of quadratic forms in k[z0 , . . . , zn ], where n is either 6 or 7. Let F be a general sextic form in k[z0 , . . . , zn ]. Then the R-scheme X = Proj R[z0 , . . . , zn ]/(tF − Q1 Q2 Q3 ) is strictly toroidal, by Example 3.2.4. Since smooth quadrics and smooth intersections of two quadrics in Pnk are rational, the only non-stably rational stratum in Xk is the triple intersection defined by Q1 = Q2 = Q3 = 0. Theorem 4.2.1 now implies that XK is not stably rational, so that a very general sextic fivefold and sixfold are not stably rational by Corollary 4.1.2. Example 4.3.4. In this example, we will prove that for every d ≥ 2, a very general hypersurface in P2k ×k Pk2 of bidegree (2, d) is not stably rational. This was the main result in [BvB18]. As input we use the property that very general hypersurfaces of bidegree (2, 2) in P2k ×k P2k are not stably rational by [HT19, §8.2]. This settles the d = 2 case. We prove the general case by induction on d. Assume that d > 2 and that the result holds for hypersurfaces of bidegree (2, d − 1). Let F, G ∈ k[z0 , z1 , z2 , w0 , w1 , w2 ] be very general bihomogeneous polynomials of bidegree (2, d) and (2, d − 1), respec2 2 tively. Consider the closed subscheme X of PR × R PR defined by tF − w2 G = 0. Then X is strictly toroidal, by the same argument as in Example 3.2.4. The special fiber Xk has three strata: the linear space in Pk2 ×k P2k defined by w2 = 0; the very general bidegree (2, d − 1) hypersurface defined by G = 0, which is not stably rational by the induction hypothesis; and their intersection given by w2 = G = 0, which is a smooth bidegree (2, d− 1) hypersurface in P2k ×k P1k . The latter is a conic bundle over P1k , which is rational by Tsen’s theorem. Now Theorem 4.2.1 implies that XK is not stably rational, so that a very general bidegree (2, d) hypersurface in P2k ×k Pk2 is not stably rational by Corollary 4.1.2. As a further illustration, we will discuss two applications that have not yet appeared in the literature. Example 4.3.5. The first new result is that a very general intersection of a bidegree (1, 2) hypersurface and a bidegree (2, 2) hypersurface in Pk2 ×k Pk4 is not stably

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rational. Such a variety is fibered in quartic del Pezzo surfaces via the projection to P2k ; not much appears to be known about stable rationality of del Pezzo fibrations over Pkn for n ≥ 2. The argument is similar to that in Example 4.3.4. Let F, G, H ∈ k[z0 , z1 , z2 , w0 , . . . , w4 ] be very general bihomogeneous polynomials of bidegree (1, 2), (1, 2) and (2, 2), 4 respectively. Then the closed subscheme of P2R ×R PR defined by F = tH −z2 G = 0 is strictly toroidal, by a similar calculation as in Example 4.3.2. The generic fiber XK is a smooth complete intersection of a bidegree (1, 2) hypersurface and a 4 bidegree (2, 2) hypersurface in P2K ×K PK . The special fiber Xk contains three strata: the two irreducible components E1 and E2 , given by {F = G = 0} and {F = z2 = 0}, respectively, and their intersection E1 ∩ E2 . The component E1 is a very general complete intersection of two bidegree (1, 2) hypersurfaces in Pk2 ×k Pk4 ; this is birational to Pk4 via the projection to the second factor. The component E2 is a smooth bidegree (1, 2) hypersurface in Pk1 ×k P4k , and, therefore, a quadric bundle over Pk1 (which is rational by Tsen’s theorem). The intersection E1 ∩ E2 is a very general intersection of two bidegree (1, 2) hypersurfaces in P1k ×k Pk4 . Such an intersection is not stably rational by Theorem 2 in [HT19] (it is a very general quartic del Pezzo fibration over Pk1 with height invariant h = 20). Now Corollary 4.2.2 implies that XK is not stably rational. Example 4.3.6. Using a similar construction, we can prove that a very general intersection of a bidegree (1, 1) hypersurface and a bidegree (2, 2) hypersurface in P3k ×k Pk3 is not stably rational. This fourfold is a conic bundle over P3k via the second projection. Let F, G ∈ k[z0 , . . . , z3 , w0 , . . . , w3 ] be very general bihomogeneous polynomials of bidegree (2, 2) and (1, 1) respec3 tively. As before, the closed subscheme of PR ×R P3R defined by F = tG − z3 w3 = 0 is strictly toroidal. The generic fiber XK is a smooth complete intersection of a 3 bidegree (1, 1) hypersurface and a bidegree (2, 2) hypersurface in PK ×K P3K . The special fiber Xk has two irreducible components E1 and E2 , given by {F = z3 = 0} and {F = w3 = 0}. The strata E1 and E2 are isomorphic to very general bidegree (2, 2) hypersurfaces in Pk2 ×k P3k and Pk3 ×k P2k , respectively; therefore, they are both non-stably rational by [HPT18a]. Now the result follows from Theorem 4.2.1.

5. The monodromy action If X is a scheme of finite type over k((t)), rather than K, then the motivic volume of X ×k((t)) K defined in [NS19] carries additional structure: an action of the profinite group scheme μ ˆ of roots of unity over k. This structure captures the monodromy action on the cohomology of X and plays an important role in the theory of motivic

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Igusa zeta functions [DL01]. In this final section we will briefly explain how this structure can also be defined for our dimensional refinement of the motivic volume. 5.1. The equivariant Grothendieck ring The equivariant version of the classical Grothendieck ring of varieties was defined by Denef and Loeser; we will follow the construction in Section 2.3 of [NS19], which is the most appropriate for our purposes. Let F be a field of characteristic zero, and let G be a profinite group scheme over F . We say that a quotient group scheme H of G is admissible if the kernel of G(F a ) → H(F a ) is an open subgroup of the profinite group G(F a ), where F a denotes an algebraic closure of F . In particular, H is a finite group scheme over F . Let d be a non-negative integer. The Grothendieck group KG (Vard F ) of F varieties of dimension at most d with G-action is the abelian group with the following presentation: • Generators: isomorphism classes [X]d of F -schemes X of finite type and of dimension at most d endowed with a good G-action. Here “good” means that the action factors through an admissible quotient of G and that we can cover X by G-stable affine open subschemes (the latter condition is always satisfied when X is quasi-projective). Isomorphism classes are taken with respect to G-equivariant isomorphisms. • Relations: we consider two types of relations. 1. Scissor relations: if X is a F -scheme of finite type of dimension at most d with a good G-action and Y is a G-stable closed subscheme of X, then [X]d = [Y ]d + [X \ Y ]d . 2. Trivialization of linear actions: let X be a F -scheme of finite type with a good G-action, and let V be a F -vector scheme of dimension m with a good linear action of G. Assume that dim(X) + m ≤ d. Then [X ×F V ]d = [X ×F Am F ]d where the G-action on X ×F V is the diagonal action and the action on m AF is trivial. We set KG (VarF ) =



KG (Vard F ).

d≥0

This graded abelian group has a unique graded ring structure such that [X]d · [X  ]e = [X ×F X  ]d+e for all F -schemes X, X  of finite type of dimensions at most d and e, respectively, and with good G-action. The G-action on X ×F X  is the diagonal action. We 1 write L for the class [AF ]1 (where A1F carries the trivial G-action) in the ring dim G K (VarF ).

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J. Nicaise and J.C. Ottem If F  is a field extension of F , then we have an obvious base change morphism dim  G KG (Vardim F ) → K (VarF  ) : [X] → [X ×F F ].

If G → G is a continuous morphism of profinite group schemes, then we can also consider the restriction morphism 

dim dim G G ResG G : K (VarF ) → K (VarF ).

Both of these morphisms are ring homomorphisms. 5.2. The monodromy action on the motivic volume Let X be a smooth and proper k((t))-scheme, and let X be a proper flat k[[t]]scheme endowed with an isomorphism of k((t))-schemes X ×k[[t]] k((t)) → X. Assume that X is regular and that the special fiber Xk is a strict normal crossings divisor (not necessarily reduced). Then we call X an snc-model of X. We write  Xk = Ni Ei i∈I

where {Ei | i ∈ I} is the set of irreducible components of Xk , and the coefficients Ni are their multiplicities. For every non-empty subset J of I, we set   7 < o EJ = Ei . Ej , EJ = EJ \ j∈J

i∈I\J

The scheme EJ is a smooth and proper k-scheme of pure codimension |J| − 1 in Xk , and EJo is a dense open subscheme of EJ . The subschemes EJo form a partition of Xk . Let n be the least common multiple of the multiplicities Ni , and let Y be the normalization of X ×k[[t]] k[[t1/n ]]. An easy local calculation shows that Y is strictly toroidal. The k-group scheme μn of nth roots of unity acts on Spec k[[t1/n ]] via the morphism k[[t1/n ]] → k[ζ]/(ζ n − 1) ⊗k k[[t1/n ]] that maps a formal power series φ(t1/n ) to φ(ζ −1 t1/n ). This induces an action of μn on X ×k[[t]] k[[t1/n ]], and also on its normalization Y because Y ×k μn is normal since μn is ´etale over k. For every non-empty subset J of I, the μn -action on Y  o = Y ×X E o . By composition J = Y ×X EJ and E restricts to a μn -action on E J J  o , and it follows J and E with the projection μ ˆ → μn we obtain actions of μ ˆ on E J immediately from the construction that these actions are good. A more explicit J with their μ description of the schemes E ˆ -actions can be found in Section 2.3 of [Ni13]; see also Section 4.1 in [BN16].

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A Refinement of the Motivic Volume Theorem 5.2.1. There exists a unique ring morphism dim μ ˆ Vol : K(Vardim k((t)) ) → K (Vark )

such that, for  every smooth and proper K-scheme X, every snc-model X of X with Xk = i∈I Ni Ei , and every integer e ≥ dim(X), we have  Jo ×k G|J|−1 ]e (5.2.2) (−1)|J|−1 [E Vol([X]e ) = m,k ∅ =J⊂I

=



J ×k P (−1)|J|−1 [E k

|J|−1

]e

(5.2.3)

∅ =J⊂I |J|−1

|J|−1

where μ ˆ acts trivially on the schemes Gm,k and Pk diagram

. It fits into a commutative

Vol

dim ) −−−−→ Kμˆ (Varkdim ) K(Vark((t)) ⏐ ⏐ ⏐ ˆ ⏐  Resμ{1} dim K(VarK ) −−−−→ K(Varkdim ) Vol

where the left vertical arrow is the base change morphism and the right vertical ˆ -action. arrow forgets the μ Moreover, for every integer m > 0, we also have a commutative diagram K(Vardim k((t)) ) ⏐ ⏐ 

Vol

−−−−→

Kμˆ (Vardim k ) ⏐ ⏐Resμˆ  μ(m) ˆ

μ ˆ (m) (Varkdim ) K(Vardim k((t1/m )) ) −−−−→ K Vol

where the left vertical arrow is the base change morphism and the right vertical arrow restricts the μ ˆ(m) = ker(ˆ μ → μm ). ˆ -action to μ Proof. One can simply copy the proof of Theorem A.3.9 in [NS19]; all the arguments remain valid in the dimensional refinement of the Grothendieck ring. The equality of (3.3.3) and (3.3.4) follows from the same calculation as in the proof of Lemma 3.3.1. The arguments in Section A.3 of [NS19] prove the existence and uniqueness of the morphism Vol in the statement, and its compatibility with base change to extensions k((t1/m )). The compatibility with the morphism Vol on K(Vardim K ) follows directly from the fact that Y ×k[[t1/n ]] R is a strictly toroidal proper R-model of X ×k((t)) K.  Acknowledgement It is a pleasure to thank the organizers of the conference Rationality of Algebraic Varieties on the beautiful Schiermonnikoog Island in April 2019 for putting together the event and inviting us to submit a contribution to the proceedings. Our investigation of a common refinement of the Grothendieck ring of varieties and the ring of birational types was motivated by questions from Gerard van der Geer and

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Lenny Taelman, and the very first sketch of this paper was made on the trip back from Schiermonnikoog. The first-named author would also like to thank Evgeny Shinder for the rewarding collaboration that has led to the article [NS19], which is the basis for the present paper. We are grateful to Olivier Benoist, Jean-Louis Colliot-Th´el`ene, Alexander Kuznetsov, Daniel Litt, Sam Payne, Alex Perry and Stefan Schreieder for stimulating discussions during the preparation of this paper and [NO19]. Finally, our thanks go out to the referee for carefully reading the manuscript and making several valuable suggestions. Johannes Nicaise is supported by the EPSRC grant EP/S025839/1, the grant G079218N of the Fund for Scientific Research–Flanders, and a long term structural funding (Methusalem grant) of the Flemish Government. John Christian Ottem is supported by the Research Council of Norway project no. 250104.

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[HPT19] [HT19] [HK06]

[KKMS73] [KT19] [KT21] [LL03] [Lˆe15] [Ma16] [Ni13] [NO19] [NP19] [NS19] [Sch19] [Sh19] [To16] [Vo15]

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Johannes Nicaise Imperial College Department of Mathematics South Kensington Campus London SW72AZ, UK and KU Leuven Department of Mathematics Celestijnenlaan 200B B-3001 Heverlee, Belgium e-mail: [email protected] John Christian Ottem Department of Mathematics University of Oslo Box 1053, Blindern N-0316 Oslo, Norway e-mail: [email protected]

Explicit Rationality of Some Special Fano Fourfolds Francesco Russo and Giovanni Staglian`o Abstract. Recent results of Hassett, Kuznetsov and others pointed out countably many divisors Cd in the open subset of P55 = P(H 0 (OP5 (3))) parametrizing all cubic 4-folds and lead to the conjecture that the cubics corresponding to these divisors should be precisely the rational ones. Rationality has been proved by Fano for the first divisor C14 , in [RS19a] for the divisors C26 and C38 , and in [RS19b] for C42 . In this note we describe explicit birational maps from a general cubic fourfold in C14 , in C26 and in C38 to P4 , providing concrete geometric realizations of the more abstract constructions in [RS19a] and of the theoretical framework developed in [RS19b]. We also exhibit an explicit relationship between the divisor C14 and a certain divisor in the open subset of P39 = P(H 0 (OY (2))) parametrizing smooth quadratic sections of a del Pezzo fivefold Y = G(1, 4) ∩ P8 ⊂ P8 , the so-called Gushel–Mukai fourfolds. Mathematics Subject Classification (2010). Primary 14E08; secondary 14M20, 14N05, 14Q10, 14Q15. Keywords. Rationality of Fano Fourfolds; projective techniques; computational aspects of rationality of fourfolds.

Introduction The rationality of smooth cubic hypersurfaces in P5 is an open problem on which a lot of new and interesting contributions and conjectures appeared in the last decades. The classical work by Fano in [Fan43], correcting some wrong assertions in [Mor40], has been the only known result about the rationality of cubic fourfolds for a long time and, together with a great amount of recent theoretical work on the subject (see for example the survey [Has16]), lead to the expectation that the very general cubic fourfold should be irrational. More precisely, in the moduli space C the locus Rat(C) of rational cubic fourfolds is the union of a countable family of closed subsets Ti ⊆ C, i ∈ N, see [dFF13, Proposition 2.1] and [KT19, Theorem 1]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Farkas et al. (eds.), Rationality of Varieties, Progress in Mathematics 342, https://doi.org/10.1007/978-3-030-75421-1_12

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Hassett defined in [Has99, Has00] (see also [Has16]) via Hodge theory infinitely many irreducible divisors Cd in C and introduced the notion of admissible values d ∈ N, i.e., those even integers d > 6 not divisible by 4, by 9 and nor by any odd prime of the form 2+3m. More recent contributions by Kuznetzsov via derived categories in [Kuz10, Kuz16] (see also [AT14, Has16]) fortified the conjecture that 7 Rat(C) = Cd . d admissible

The first admissible values are d = 14, 26, 38, 42, 62, 74, 78, 86 and Fano showed the rationality of a general cubic fourfold in C14 , see [Fan43, BRS19]. The main results of [RS19a] assure that every cubic fourfold in C26 and C38 is rational. Moreover, very recently in [RS19b] we also showed that every cubic fourfold in C42 is rational. A geometric definition of Cd can be also given as the (closure of the) locus of cubic fourfolds X ⊂ P5 containing an explicit surface Sd ⊂ X. These surfaces are obviously not unique and a standard count of parameters shows that in specific examples the previous locus is a divisor (see [Nue15]). The degree and self-intesection of S = Sd on a cubic fourfold X determine the value d via the formula d = 3 · (S)2X − deg(S)2 .

(0.1)

Moreover, when S has smooth normalization and only a finite number δ of nodes as singularities, the self-intersection (S)2X can be explicitly calculated by the following formula (see [Ful84, Theorem 9.3]): (S)2X = 6h2 + 3h · KS + KS2 − χS + 2δ,

(0.2)

where h ∈ Pic(X) is the hyperplane class. We refer to [Has99, Has00] for more details on the Hodge theoretical definition of Cd , and also to [YY19] for recent interesting contributions on the divisors Cd . Recall that the divisor C14 can be described either as the closure of the locus of cubic fourfolds containing a smooth quintic del Pezzo surface or, equivalently, a smooth quartic rational normal scroll, see [Fan43, BRS19]. Fano proved that the restriction of the linear system of quadrics through a smooth quintic del Pezzo surface, respectively a smooth quartic rational normal scroll, to a general cubic through the surface defines a birational map to P4 , respectively onto a smooth fourdimensional quadric hypersurface. Indeed, a general fiber of the map P5  P4 given by quadrics through a quintic del Pezzo surface is a secant line to it, yielding the birationality of the restriction to X (the other case is similar). The extension of this explicit geometrical approach to rationality for other (admissible) values d appeared to be impossible because the cubic fourfolds containing irreducible surfaces with one apparent double point belong to C14 , see [CR11, BRS19]. In [RS19a] we discovered irreducible surfaces Sd ⊂ P5 admitting a fourdimensional family of 5-secant conics such that through a general point of P5 there passes a unique conic of the family (congruences of 5-secant conics to Sd ) for d=14, 26 and 38. From this we deduced the rationality of a general cubic in

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|H 0 (ISd (3))|, showing that it is a rational section of the universal family of the congruence of 5-secant conics. Here we come back to Fano’s method and we propose an explicit realisation of the previous abstract approach. Some simplifying hypotheses, based on the known examples in [RS19a], suggest that rationality might be related to linear systems of hypersurfaces of degree 3e − 1 having points of multiplicity e ≥ 1 along a right surface Sd contained in the general X ∈ Cd , see Section 1. This expectation is also motivated by the remark, due to J´anos Koll´ ar, that if the surface Sd ⊂ P5 admits a congruence of (3e − 1)-secant curves of degree e ≥ 1 generically transversal to the cubics through Sd (see Section 1 for precise definitions), then the above linear systems contract the curves of the congruence. So, if the general fiber of the map is a curve of the congruence, then a general cubic through Sd is birational to the image of the map. We shall see that this really occurs for the first three admissible values d = 14, 26, 38 and that these linear systems provide by restriction explicit birational maps from a general cubic fourfold in Cd for d = 14, 26, 38 to P4 (and also to a four-dimensional linear section of a G(1, 3), respectively G(1, 4), respectively G(1, 5)), a fact which was not known before at least for d = 26, 38 (see also [Kol19, Section 5, §29]). The theoretical framework explaining the birationality of these maps, the relations with the theory of congruences to the surfaces Sd and to the associated K3 surfaces has been developed recently in [RS19b], where moreover the case d = 42 is considered. Similarly to Hassett’s analysis of cubic fourfolds, in [DIM15, DK18a, DK19, DK18b] the authors studied Gushel–Mukai fourfolds, that is smooth quadratic sections of five-dimensional linear sections of cones over the Grassmannian G(1, 4) ⊂ P9 ; defined their coarse moduli space MGM of dimension 24 and introduced, via 4 Hodge theory and the period map, the analogous definitions of countably many Hodge-special divisors inside M4GM . Our contribution here is an alternative description of a divisor of rational Gushel–Mukai fourfolds inside the moduli space, studied firstly in [DIM15, Subsection 7.3]. We show that a general Gushel–Mukai fourfold W belonging to this divisor is birational to a general cubic fourfold X in the Fano divisor C14 via explicit birational maps given by linear systems with multiplicities along the associated K3 surfaces of X and of W . The maps μ : X  W in this case provide a birational perspective to the association between Pfaffian fourfolds and smooth K3 surfaces of degree 14 and genus 8, developed originally by Beauville and Donagi in [BD85], and which has been the starting point of the modern study of special cubic fourfolds (and of hyperk¨aler manifolds) from different points of view.

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1. Rationality via linear systems of hypersurfaces of degree 3e − 1 having points of multiplicity e along a surface Let us recall the following definitions introduced in [RS19a, Section 1]. Let H be an irreducible proper family of (rational or of fixed arithmetic genus) curves of degree e in P5 whose general element is irreducible. We have a diagram DA AA AAψ π AA  H P5 where π : D → H is the universal family over H and where ψ : D → P5 is the tautological morphism. Suppose moreover that ψ is birational and that a general member [C] ∈ H is (3e − 1)-secant to an irreducible surface S ⊂ P5 , that is C ∩S is a length 3e−1 scheme. We shall call such a family H a congruence of (3e−1)-secant curves of degree e to S. Let us remark that necessarily dim(H) = 4. An irreducible hypersurface X ∈ |H 0 (IS (3))| is said to be transversal to the congruence H if the unique curve of the congruence passing through a general point p ∈ X is not contained in X. A crucial result is the following. Theorem 1.1 ([RS19a]). Let S ⊂ P5 be a surface admitting a congruence consisting of (3e − 1)-secant curves of degree e parametrized by H. If X ∈ |H 0 (IS (3))| is an irreducible hypersurface transversal to H, then X is birational to H. If the map Φ = Φ|H 0 (IS (3))| : P5  P(H 0 (IS (3))) is birational onto its image, then a general hypersurface X ∈ |H 0 (IS (3))| is birational to H. Moreover, under the previous hypothesis on Φ, if a general element in |H 0 (IS (3))| is smooth, then every hypersurface X ∈ |H 0 (IS (3))| with at worst rational singularities is birational to H. Thus, when a surface S ⊂ P5 admits a congruence of (3e − 1)-secant curves of degree e ≥ 1, which is transversal to a general cubic fourfold through it, such a −1 cubic fourfold is birational to H via the map π ◦ ψ|X , see the proof of the above result. Since ψ : D → P5 is birational, we also have a rational map ϕ = π ◦ ψ −1 : P5  H, whose general fiber F = ϕ−1 (ϕ(p)) over the point p ∈ P5 is the unique curve of the congruence passing through p. Conversely, if there exists a map ϕ : P5  Y ⊆ PN , with Y being a four-dimensional variety whose general fiber F is an irreducible curve of degree e which is (3e−1)-secant to a surface S ⊂ P5 , then we have found a congruence for S, a birational realization of the variety H and therefore a concrete representation of the abstract map π ◦ ψ −1 . It is natural to ask what linear systems on P5 can yield maps ϕ : P5  Y as above. Since a linear system in |H 0 (ISe (3e − 1))| contracts the fibers of ϕ, these (complete) linear systems appear as natural potential candidates, as remarked by

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J´anos Koll´ar. A posteriori we shall see that this really occurs with Y = P4 (or with Y a linear section of a Grassmannian of lines) for the first three admissible values d = 14, 26, 38. For e = 1 one should consider linear systems of quadric hypersurfaces through S; for e = 2 quintics having double points along S; for e = 3 hypersurfaces of degree 8 having triple points along S and so on. For e = 1 we have a unique secant line to S passing through a general point of P5 , which is a very strong restriction. Indeed, such a S is a so-called surface with one apparent double point. These surfaces are completely classified in [CR11] and the smooth ones contained in a cubic fourfold are only quintic del Pezzo surfaces and quartic rational normal scrolls. Any case, cubic fourfolds through any OADP irreducible surface belong to the divisor C14 . This was the starting point of Fano’s analysis in [Fan43] (see also [BRS19, Theorem 3.7] for a modern account of Fano’s original arguments using deformations of quartic scrolls). We shall mainly consider the surfaces Sd ⊂ P5 admitting a congruence of 5-secant conics parametrised by a rational variety studied in [RS19a] (but also other new examples) to determine explicitly the rationality of a general X ∈ Cd for d = 14, 26, 38 with e = 2 (or also for other values e ≥ 3). To this aim we shall summarise some well-known facts in the next subsection. 1.1. Linear systems of quintics with double points along a general Sd ∈ Sd Let Sd be an irreducible component of the Hilbert scheme of surfaces in P5 with a fixed Hilbert polynomial p(t) and such that Cd = {[X] ∈ C for which ∃ [Sd ] ∈ Sd : Sd ⊂ X}. One can verify explicitly the previous equality by comparing the Hodge theoretic definition on the left with the geometrical description on the right. A modern count of parameters usually shows that the right side is at least a divisor in C so that equality holds because Cd is an irreducible divisor if not empty. The hard problem is to compute the dimension of the family of Sd ’s contained in a fixed (general) X belonging to the set on the right side above, see [Nue15, RS19a] for some efficient computational arguments based on semicontinuity. For every a, b ∈ N the function h0 (ISb d (a)) is upper semicontinuous on Sd . In particular, there exists an open non empty subsets U ⊆ Sd on which h0 (ISb d (a)) attains a minimum value m = m(a, b). We shall be mainly interested in the case a = 5 and b = 2, when we compute the dimension of the linear system |H 0 (IS2d (5))| for a general Sd ∈ Sd . To this end we consider the exact sequence 0 → IS2d (5) → ISd (5) → NS∗d /P5 (5) → 0.

(1.1)

Suppose that we know h0 (NS∗d /P5 (5)) = y and h0 (ISd (5)) = x for the general Sd ∈ Sd via standard exact sequences (or also computationally) or for some geometrical property of the surfaces. From (1.1) we deduce h0 (IS2d (5)) ≥ x − y for a general

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Sd . By the upper semicontinuity of h0 (IS2d (5)) it will be sufficient to find a surface S ∈ Sd with h0 (IS2 (5)) = x−y to deduce that the same holds for a general Sd ∈ Sd . If π : χd → Sd is the universal family and if a general [Sd ] ∈ Sd is smooth, let V ⊆ Sd be the non empty open set of points [S] ∈ Sd such that S ⊂ P5 is a smooth ∗ surface. Then π −1 (V ) → V is a smooth morphism and the function h0 (NS/P 5 (a)) is upper semicontinuos on V for every a ∈ N. In particular, if there exists [S] ∈ V ∗ such that h0 (NS/P 5 (a)) = z, then for a general [Sd ] ∈ Sd , we have h0 (NS∗d /P5 (a)) ≤ z and hence h0 (IS2d (a)) ≥ m(a, 1) − z. If moreover h0 (IS2  (a)) = m(a, 1) − z for a [S  ] ∈ Sd , then the same holds for a general [Sd ] ∈ Sd . These standard and wellknown remarks will be useful in our analysis of the examples considered in the next sections, where we shall also deal with the case e = 3 and the linear systems |H 0 (IS3d (8))| for d = 14, 38.

2. Birational maps to P4 for cubics in C14 , C26 and C38 Let S14 ⊂ P5 be an isomorphic projection of an octic smooth surface S ⊂ P6 of sectional genus 3 in P6 , obtained as the image of P2 via the linear system of quartic curves with 8 general base points. Let S14 be the irreducible component of the Hilbert scheme parametrizing surfaces S14 ⊂ P5 as above. In [RS19a, Theorem 2] we verified that a general X ∈ C14 contains a surface S14 and that each S14 admits a congruence of 5-secant conics (birationally) parametrized by its symmetric product and transversal to X. Theorem 2.1. For a general surface S14 ⊂ P5 as above we have |H 0 (IS214 (5))| = P4 and this linear system determines a rational map ϕ : P5  P4 whose general fiber is a 5-secant conic to S14 . In particular, the restriction of ϕ to a general cubic through S14 is birational. Proof. A general S14 ∈ S14 is k-normal for every k ≥ 2, see [AR02, Example 3.8], yielding h0 (IS14 (5)) = 141. For a particular smooth S ∈ S14 we have verified that ∗ h0 (NS/P 5 (5)) = 136,

so that for a general S14 ∈ S14 we have h0 (NS∗14 /P5 (5)) ≤ 136. By (1.1) we have h0 (IS214 (5)) ≥ 5 for a general S14 ∈ S14 . Since in the example we studied we have that h0 (IS2 (5)) = 5, the same holds for a general S14 ∈ S14 , see Subsection 1.1. Let ϕ : P5  P4 be the rational map associated to |H 0 (IS214 (5))| = P4 with S14 general. We verified that the closure of a general fiber of ϕ is a 5-secant conic to S14 , concluding the proof.  Let S26 ⊂ P5 be a rational septimic scroll with three nodes recently considered by Farkas and Verra in [FV18], where they also proved that a general X ∈ C26 contains a surface of this kind. Also these surfaces admit a congruence

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of 5-secant conics transversal to X and parametrized by a rational variety, see [RS19a, Remark 6]. Theorem 2.2. For a general surface S26 ⊂ P5 as above we have |H 0 (IS226 (5))| = P4 and this linear system determines a rational map ϕ : P5  P4 whose general fiber is a 5-secant conic to S26 . In particular, the restriction of ϕ to a general cubic through S26 is birational. Proof. For a general S26 ∈ S26 as above, we have h0 (IS26 (5)) = 144,

h0 (NS∗26 /P5 (5)) = 139

and in an explicit example we verified that h0 (IS226 (5)) = 5. Thus |H 0 (IS226 (5))| = P4 for a general S26 , see Subsection 1.1. Let ϕ : P5  P4 be the rational map associated to |H 0 (IS226 (5))| = P4 with S26 general. We verified that a general fiber of the map ϕ is a 5-secant conic to S26 , concluding the proof.  Let S38 ⊂ P5 be a general degree 10 smooth surface of sectional genus 6 obtained as the image of P2 by the linear system of plane curves of degree 10 having 10 fixed triple points. As shown by Nuer in [Nue15], these surfaces are contained in a general [X] ∈ C38 . In [RS19a, Theorem 7] we proved that a general S38 admits a congruence of 5-secant conics transversal to X and parametrised by a rational variety. Theorem 2.3. For a surface S38 ⊂ P5 as above we have |H 0 (IS238 (5))| = P4 and this linear system defines a rational map ϕ : P5  P4 whose general fiber is a 5-secant conic to S38 . In particular, the restriction of ϕ to a general cubic through S38 is birational. Proof. A general S38 ∈ S38 has ideal generated by 10 cubic forms and is thus 5normal by [BEL91, Proposition 1], yielding h0 (IS38 (5)) = 126 for a general S38 (a fact which can also be verified by a direct computation). For a particular smooth ∗ S ∈ S38 we verified that h0 (NS/P 5 (5)) = 121 so that for a general S38 ∈ S38 we 0 ∗ have h (NS38 /P5 (5)) ≤ 121. From (1.1) we deduce h0 (IS238 (5)) ≥ 5 for a general S38 ∈ S38 . Since in the previous explicit example we also have h0 (IS2 (5)) = 5, the same holds for a general S38 ∈ S38 , see Subsection 1.1. We verified that a general fiber of the corresponding rational map ϕ : P5  P4 is a 5-secant conic to S38 , concluding the proof. 

3. Birational maps to linear sections of G(1, 3 + k) for cubics in C14+12k for k ≤ 2 Let S ⊂ P6 be a septimic surface with a node, which is the projection of a smooth del Pezzo surface of degree seven in P7 from a general point on its secant variety. Let  S26 ⊂ P5

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be the projection of S from a general point outside the secant variety Sec(S) ⊂ P6 . These surfaces admit a congruence of 5-secant conics parametrised by a rational variety and a general cubic in C26 contains such a surface, see [RS19a, Theorem 4].  ⊂ P5 as above we have |H 0 (IS2  (5))| = P7 Theorem 3.1. For a general surface S26 26 and this linear system determines a rational map ϕ : P5  Y 4 ⊂ P7 with Y 4 a smooth linear section of G(1, 4) ⊂ P9 . A general fiber of ϕ is a 5-secant conic to   S26 and the restriction of ϕ to a general cubic through S26 is birational.   as above we verified that h0 (IS2  (5)) = 8 and ∈ S26 Proof. For a general S26 26 that the closure of the image of ϕ : P5  P7 is a smooth four-dimensional linear section Y 4 of G(1, 4). Moreover, a general fiber of the corresponding map ϕ is a  5-secant conic to S26 , concluding the proof. 

While studying the rational map ϕ : P5  P4 treated in Theorem 2.3, we realized that its base locus contains an irreducible component of dimension three B ⊂ P5 of degree 6 and sectional genus 3, which has 7 singular points and it is given by the maximal minors of a 3 × 4 matrix of linear forms on P5 . Therefore B ⊂ P5 is a degeneration of the so-called Bordiga scroll. The variety B contains the surface S38 ⊂ P5 and a general cubic through S38 cuts B along S38 and an  octic rational scroll S38 ⊂ P5 with 6 nodes belonging to the singular locus of B.  The scroll S38 is a projection of a smooth octic rational normal scroll S ⊂ P9 from a special P3 cutting the secant variety to S in six points. Let us now describe some  geometrical properties of S38 ⊂ P5 .  ⊂ P5 with 6 Theorem 3.2. A general [X] ∈ C38 contains an octic rational scroll S38  0 5 3 nodes. Moreover, for a general surface S38 ⊂ P we have |H (IS  (8))| = P10 and 38 this linear system determines a rational map ϕ : P5  Y 4 ⊂ P10 with Y 4 a linear  section of G(1, 5) ⊂ P14 . The general fiber of ϕ is an 8-secant twisted cubic to S38  and the restriction of ϕ to a general cubic through S38 is birational. In particular,  an octic rational scroll S38 ⊂ P5 admits a congruence of 8-secant twisted cubics.  ⊂ P5 depends on 47 parameters and it has homoProof. A general octic scroll S38  geneous ideal generated by 10 cubics forms. In an explicit example S ∈ S38 we ver0 ified that h (NS/P5 ) = 47 and that S is contained in smooth cubic hypersurfaces.  Therefore S38 is generically smooth of dimension 47 and the natural incidence cor  respondence in S38 where h0 (IS (3)) = 10 has an × C above the open subset of S38  2  irreducible component C38 of dimension 56. Since (S38 ) = 34 by (0.2), to prove that a general [X] ∈ C38 contains such a surface, we verified in an explicit general  /X ) = 2. example that h0 (NS38   For a general S38 ∈ S38 we have h0 (IS3  (8)) = 11 and the closure of the 38 image of the associated rational map ϕ : P5  P10 is a linear section Y 4 ⊂ P10 of G(1, 5) ⊂ P14 . We verified that a general fiber of ϕ is an 8-secant twisted cubic  to S38 , concluding the proof. 

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We summarize in Tables 1 and 2 some information related to the results of this and of the previous Section. For other examples of this kind, we refer to the tables in [RS19b]. 8-secant 2-secant 5-secant twisted lines conics cubics

d

Surface S ⊂ P5

14

Smooth del Pezzo surface of degree 5

1

0

0

25

35

5

14

Rational normal scroll of degree 4

1

0

0

28

29

2

14

Isomorphic projection of a smooth surface in P6 of degree 8 and sectional genus 3, obtained as the image of P2 via the linear system of quartic curves with 8 general base points

7

1

0

13

49

7

26

Rational scroll of degree 7 with 3 nodes

7

1

0

13

44

2

38

Smooth surface of degree 10 and sectional genus 6, obtained as the image of P2 via the linear system of curves of degree 10 with 10 general triple points

7

1

0

10

47

2

26

Projection of a smooth del Pezzo surface of degree 7 in P7 from a line intersecting the secant variety in one general point

5

1

0

14

42

1

38

Rational scroll of degree 8 with 6 nodes

9

4

1

10

47

2

h0 (IS/P5 (3)) h0 (NS/P5 ) h0 (NS/X )

Table 1. Surfaces S ⊂ P5 contained in a general cubic fourfold [X] ∈ Cd and admitting a congruence of (3e − 1)-secant rational normal curves of degree e ≤ 3.

d 14 14 14

e 1 1 2

Multidegree

Y4

3, 6, 7, 4, 1

P

3, 6, 8, 6, 2

G(1, 3) ⊂ P

3, 15, 19, 9, 1

δ

4 5

9

8

9

8

10

7

P

9

52

256

32

106

4

9

51

246

31

100

9

42

165

18

39

7

5

77

212

43

73

10

5

204

633

94

144

3, 15, 20, 9, 1

P

3, 15, 27, 9, 1

P4

3, 15, 31, 25, 5

G(1, 4) ∩ P ⊂ P

3, 24, 80, 70, 14

4

7

2

3

g(Bred )

10

2

38

deg(Bred )

3

38

2

g(B)

4

26

26

deg(B)

7

G(1, 5) ∩ P

10

⊂P

Table 2. Birational maps from a cubic fourfold [X] ∈ Cd to a fourfold Y 4 defined by the restrictions to X of the linear systems |H 0 (ISe (3e − 1))|, where S ⊂ P5 are the surfaces in Table 1 admitting a congruence of (3e − 1)-secant rational normal curves of degree e ≤ 3. Here, B denotes the base locus of the inverse map, which is a two-dimensional scheme; g stands for the sectional arithmetic genus; δ is the degree of the forms defining the inverse map.

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4. A divisor of rational Gushel–Mukai fourfolds By definition a Gushel–Mukai fourfold Z ⊂ P8 , GM fourfold for short, is a quadratic section of a five-dimensional linear section of a cone over the Grassmannian G(1, 4) ⊂ P9 . Equivalently, we can consider a smooth prime Fano fourfold Z of degree 10 and index 2, that is Pic(Z)  ZH is generated by the class of an ample divisor H such that H 4 = 10 and −KZ ≡ 2H. Then H is very ample and embeds Z in P8 either as a quadratic section of a hyperplane section of G(1, 4) ⊂ P9 (Mukai fourfold, see [Muk89, DIM15]) or as a double cover of G(1, 4) ∩ P7 branched along its intersection with a quadric (Gushel fourfold, see [Gus82, DIM15]). There exists a 24-dimensional coarse moduli space M4GM of GM fourfolds, where the locus of Gushel fourfolds is of codimension 2, see [DIM15, DK19, DK18b]. Moreover, we have a period map p : MGM →D 4 to a 20-dimensional quasi-projective variety D (called the period domain), which is dominant with irreducible four-dimensional fibers (see [DK18b, Corollary 6.3]). For a very general GM fourfold Z ⊂ P8 , the natural inclusion A(G(1, 4)) = H 4 (G(1, 4), Z) ∩ H 2,2 (G(1, 4)) ⊆ A(Z) = H 4 (Z, Z) ∩ H 2,2 (Z) (4.1) is an equality, see [DIM15, Corollary 4.6]. If A(G(1, 4))  A(Z) holds, then Z is said to be a Hodge-special GM fourfold. Inside D there exist countably many arithmetic hypersurfaces Dd , d ∈ N, expressing the previous strict inclusion of lattices, whose union is the so-called Noether-Lefschetz locus. A standard argument yields that Dd = ∅ implies d ≡ 0, 2, 4 (mod 8), see [DIM15, Lemma 6.1]. If d ≡ 2 (mod 8), then Dd = Dd ∪ Dd with Dd and Dd irreducible hypersurfaces interchanged by the natural involution rD : D → D; while if d ≡ 0 (mod 4) then Dd is irreducible. The Hodge-special GM = fourfolds are parametrized by the contable union of hypersurfaces d p−1 (Dd ) ⊂ M4GM , where p−1 (Dd ) = p−1 (Dd ) ∪ p−1 (Dd ) is the union of two irreducible hypersurfaces if d ≡ 2 (mod 8), while p−1 (Dd ) is irreducible if d ≡ 0 (mod 4). Following [DIM15, Section 7], suppose that an ordinary GM fourfold Z ⊂ P8 contains a (smooth) surface S such that [S] ∈ A(Z) \ A(G(1, 4)). Write [S] = aσ3,1 + bσ2,2 in G(1, 4). Then the double points formula for S ⊂ Z yields: (S)2Z = 3a + 4b + 2KS · σ1|S + 2KS2 − 12χ(OS ),

(4.2)

and we have that [Z] ∈ p−1 (Dd ), where d is the determinant (or discriminant) of the intersection matrix in the basis (σ1,1|Z , σ2|Z − σ1,1|Z , [S]), that is d = 4(S)2Z − 2(b2 + (a − b)2 ). When d ≡ 2 (mod 8), we have that [Z] ∈ p if b is even.

−1

(Dd )

if a+b is even, and [Z] ∈ p

(4.3) −1

(Dd )

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Some loci of rational Gushel–Mukai fourfolds of different codimension inside MGM are known since the classical work of Roth, see [Rot49] and also [DIM15, Sec4 tion 7] for recent contributions. As for cubic fourfolds the rationality/irrationality of a (very) general Gushel–Mukai fourfold is unknown and there are striking relations between the two problems. The known results of rationality for GM fourfolds can be summarized by saying that every GM fourfold belonging to one of the two   irreducible hypersurfaces p−1 (D10 ) is rational. In the following, we ) and p−1 (D10  shall give a new geometric description of the hypersurface p−1 (D10 ) from which −1  we derive that a general fourfold in p (D10 ) is birational in a natural way to a general cubic fourfold in the Fano divisor C14 . Recently, after the first version of the present paper, it has been shown in [HS19] that every GM fourfold in p−1 (D20 ) is rational, and in [Sta20b] that there  are rational GM fourfolds in p−1 (D26 ). 4.1. Del Pezzo fivefolds through a K3 surface of degree 14 and genus 8 Let Y ⊂ P8 be a smooth quintic del Pezzo fivefold. As it is well known Y ⊂ P8 is a hyperplane section of G(1, 4) ⊂ P9 , Aut(Y ) has dimension 15 and it coincides with the group of projective transformations of P8 leaving Y fixed. Moreover, two quintic del Pezzo fivefolds are projectively equivalent, so that the Hilbert scheme DP parametrizing these manifolds is irreducible of dimension 65 and generically smooth. Let S be the Hilbert scheme parametrizing K3 surfaces of degree 14 and genus 8 in P8 . Then dim(S) = 99, S is generically smooth and a general [S] ∈ S is a ucker embedding G(1, 5) ⊂ P14 . Let Πp ⊂ G(1, 5) transversal linear section of the Pl¨ 4 be a P representing lines passing through a general p ∈ P5 and parametrized by a fixed hyperplane H ⊂ P5 not passing through p. Let πp : G(1, 5)  G(1, H) ⊂ P9 be the projection which to a line [L] ∈ G(1, 5) \ Πp associates its projection from p into H. Then one verifies that a general linear space M = P8 ⊂ P14 is such that M ∩ Πp = ∅ (see also [Uga02] for general results on smoothness of the projection of surfaces in G(1, 5) via πp ). Thus the K3 surface S = M ∩ G(1, 5) is projected isomorphically from Πp onto a smooth surface S ⊂ G(1, H) ⊂ P9 contained in a hyperplane Hp = πΠp (M ). For p ∈ P5 general, YH = Hp ∩ G(1, H) is a smooth del Pezzo fivefold. Thus, fixing S ⊂ G(1, 4) ⊂ P9 one has an irreducible fivedimensional family of del Pezzo fivefolds containing S parametrised by an open subset of P5 . We now consider the following incidence correspondence I = {([S], [Y ]) : S ⊂ Y } ⊂ S × DP and we introduce p1

S



IB BB BBp2 BB B DP

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the two natural projections. We have seen that [S] ∈ S general is contained in at least a smooth del Pezzo fivefold, i.e., p1 is a surjective morphism, and that p−1 1 ([S])) is irreducible and of dimension 5, yielding that I is irreducible of dimension 104. Hence, on a fixed quintic del Pezzo fivefold Y ⊂ P8 we have a 39-dimensional family of K3 surfaces of degree 14 and genus 8 contained in Y , a fact that can be also verified by an explicit computation in an example. In particular, every quintic del Pezzo fivefold Y ⊂ P8 contains such a family and dim(p2−1 ([Y ])) = 39. Moreover, by standard calculations with Schubert cycles, one also sees that in the Chow ring of G(1, 4) a general such surface [S] can be written as [S] = 9σ3,1 + 5σ2,2 .

(4.4)

Remark 4.1. The five-dimensional family of del Pezzo fivefolds containing a general smooth K3 surface S ⊂ P8 of degree 14 and genus 8 can be considered as the dual of the five-dimensional family of smooth quintic del Pezzo surfaces contained in the associated Pfaffian cubic fourfold. Indeed, following [BD85], let V a vector space of dimension 6 and let G(2, V ) ⊂ Δ ⊂ P(Λ2 V ), G(2, V ∗ ) ⊂ Δ∗ ⊂ P(Λ2 V ∗ ) ucker embeddings of the Grassmann manifolds G(2, V ), respecively be the Pl¨ G(2, V ∗ ). These manifolds are the loci of tensors of rank at most two while the cubic hypersurfaces Δ, respectively Δ∗ , which are the secant varieties of G(2, V ) and of G(2, V ∗ ) respectively, are the loci of tensors of rank at most four. Let L = P8 ⊂ P(Λ2 V ) be general. Then S = L ∩ G(2, V ) ⊂ L is a general smooth K3 surface of degree 14 and genus 8, while X = L⊥ ∩ Δ∗ is a smooth cubic hypersurface in L⊥ = P5 . Let L⊥ = P(U ) with U ⊂ Λ2 V ∗ of dimension 6. To a general subspace W ⊂ U ⊂ Λ2 V ∗ of dimension 5 there corresponds a surjection Λ2 V ∗ → Λ2 W ∗ . Consider the set of [α] ∈ X such that ker(α) ⊂ W . Then α|W is decomposable and since X ∩ G(2, V ∗ ) = ∅, the inclusion U ⊂ Λ2 V ∗ yields an embedding U ⊂ Λ2 W ∗ and hence an embedding L⊥ ⊂ P(Λ2 W ∗ ) such that L⊥ ∩ G(2, W ) is a smooth quintic del Pezzo surface contained in X given exactly by the [α] ∈ X such that ker(α) ⊂ W . In this way one constructs the well-known five-dimensional family of smooth quintic del Pezzo surfaces contained in the Pfaffian cubic X, see [BD85]. 4.2. GM fourfolds through a K3 surface of degree 14 and genus 8 We consider a fixed smooth quintic del Pezzo fivefold Y ⊂ P8 Then the GM fourfolds contained in Y are parametrized by an open subset of P39 = P(H 0 (OY (2))). Let F be the 39-dimensional family of K3 surfaces of degree 14 and genus 8 contained in Y , as described in Subsection 4.1. Consider the incidence correspondence: J = {([S], [Z]) : S ⊂ Z ⊂ Y } ⊂ F × P(H 0 (OY (2)))

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and let π1

F

J LL LLL LLπL2 LLL % P(H 0 (OY (2)))

be the two natural projections. For [S] ∈ F general, the fibre π1−1 (S)  P(IS,Y (2)) is irreducible of dimension h0 (IS,P8 (2))− h0 (IY,P8 (2))− 1 = 9. It follows that J has a unique irreducible component that dominates F which has dimension 39+9 = 48. Using Macaulay2, in a specific example of a pair (S, Z), where [S] ∈ F and Z ⊂ Y is a GM fourfold containing S, we verified that h0 (NS/Z ) = 10, yielding that the dimension of π2 (J) is at least 48 − 10 = 38. Since π2 (J)  P(H 0 (OY (2)))  P39 (a very general GM fourfold cannot contain a K3 surface of degree 14 and genus 8 by [DIM15, Corollary 4.6]), we deduce that π2 (J) ⊂ P(H 0 (OY (2))) is a hypersurface. Moreover, from (4.2), (4.3), and (4.4), it follows that a general GM fourfold in  π2 (J) belongs to p−1 (D10 ). Summarizing we have proved the following: Proposition 4.2. Inside MGM , the closure of the family of fourfolds containing 4 a K3 surface of degree 14 and genus 8, as described in Subsection 4.1, forms an  irreducible divisor which coincides with p−1 (D10 ).  Remark 4.3. The irreducible divisor p−1 (D10 ) ⊂ M4GM was previously described in [DIM15, Proposition 7.4] as the closure in M4GM of the family of fourfolds containing a τ -quadric surface, that is, a linear section of G(1, 3) ⊂ G(1, 4). Note that the class of a τ -quadric surface in G(1, 4) is σ12 · σ1,1 = σ3,1 + σ2,2 .

We point out that a τ -quadric surface S, inside a del pezzo fivefold Y containing it, admits a congruence of 1-secant lines, that is, through a general point of Y there passes exactly one line contained in Y which intersect S. By taking the linear system on Y of hyperplane sections through S, we get a dominant map Y  P4 whose general fibers are the lines of the congruence. In particular, the restriction of this map to a general GM fourfold [X] ∈ P(H 0 (OY (2))) containing S gives a birational map X  P4 . From this it follows that a general fourfold in  p−1 (D10 ) is rational (see also [DIM15, Proposition 7.4]). We will see in Subsection 4.4 that a general K3 surface of degree 14 and genus 8 contained in del Pezzo fivefold Y , as described in Subsection 4.1, admits inside Y a congruence of 5-secant twisted cubics, that is, through a general point of Y , there passes exactly one twisted cubic curve contained in Y that cuts the surface at 5 points. We summarize in Table 3 the description of the family of GM fourfolds obtained in Proposition 4.2 together with all the other known descriptions of families in M4GM . Very recently in [Sta20a] (after the first version of the present paper), this table has been extended with many other examples.

o F. Russo and G. Staglian`

336 Surface S ⊂ Y = G(1, 4) ∩ P8

Class in G(1, 4)

K3 surface of degree 9σ3,1 + 5σ2,2 14 and genus 8

Codim Image in MGM in D 4

1-secant lines 3-secant conics 5-secant cubics to S contained to S contained to S contained in Y and passing in Y and passing in Y and passing though a general though a general though a general point of Y point of Y point of Y

Ref.

1

 D10

9

8

1

Prop. 4.2

τ -quadric surface

σ3,1 + σ2,2

1

 D10

1

0

0

[DIM15]

Quintic del Pezzo surface

3σ3,1 + 2σ2,2

1

 D10

3

0

0

[Rot49]

σ-plane

σ3,1

2

 D10

1

0

0

[Rot49]

Cubic scroll

2σ3,1 + σ2,2

1

D12

2

0

0

[DIM15]

ρ-plane

σ2,2

3

D12

0

0

0

[Rot49]

1

D20

6

1

0

[HS19]

Rational surface of 6σ3,1 + 3σ2,2 degree 9 and genus 2

Table 3. All the known geometric descriptions of families of GM fourfolds.

4.3. Surfaces of degree 10 and sectional genus 6 with a node in P5 obtained as projections of general K3 surfaces of degree 10 and genus 6 Let S  ⊂ P6 be a smooth K3 surface of degree 10 and genus 6. Let S14 ⊂ P5 be the projection of S  from a general point on the secant variety of S  . Then S14 ⊂ P5 is a degree 10 and sectional genus 6 surface with a node, contained in smooth cubic hypersurfaces and whose ideal is generated by 10 cubic forms. On a cubic fourfold X ∈ |H 0 (IS14 (3))| we have (S14 )2 = 38 and d = 3 · 38 − 100 = 14, by (0.1) and (0.2). The surfaces S14 depend on at least 19 + 5 moduli so that the corresponding Hilbert scheme S14 has dimension at least 59. By an explicit computation we find that for a (general) S14 of the previous kind we have h0 (NS14 /P5 ) = 59, proving that S14 is generically smooth of dimension 59 and that the surfaces S14 ’s depend on 24 = 59 − 35 moduli. Moreover, since h0 (IS14 (3)) = 10, and since there exists a particular example of cubic fourfold containing a 14-dimensional family of such surfaces, one deduces the following by semicontinuity (see [Nue15] for details on this kind of arguments). Proposition 4.4. Inside the moduli space C of cubic fourfold, the closure of the family of fourfolds containing a surface S14 of degree 10 and genus 6 with a node as above forms the irreducible divisor C14 . Remark 4.5. The projection from the node q of S14 yields a birational map j : S14  S0 ⊂ P4 onto the projection S0 ⊂ P4 of S  ⊂ P6 from two general points on it, which is a singular surface of degree 8 and sectional genus 6, cut out by one cubic and four quartics. The birational map j can also been described by (a six-dimensional family of) linear systems of quadrics on S14 whose base loci are the union of the point q with a smooth quadratic section of a smooth quintic del Pezzo surface. From this one can find smooth quintic del Pezzo surfaces

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intersecting the given S14 along a smooth curve of degree 10 and genus 6 (we give some computational details in the Macaulay2 package mentioned in Section 5).  4.4. Rationality of the GM fourfolds in p−1 (D10 ) We now illustrate an unknown birational isomorphism between GM fourfolds in  p−1 (D10 ) and Pfaffian cubic fourfolds determined by the associated K3 surfaces.

Theorem 4.6. Let notation be as above. Then: (i) A general surface S14 ⊂ P5 of degree 10 and sectional genus 6 with a node as above admits a congruence of 8-secant twisted cubics and the linear system |H 0 (IS314 (8))| of octic hypersurfaces with triple points along S14 restricted to a general X ∈ |H 0 (IS14 (3))| defines a birational map μ : X  W ⊂ P8 with  W a GM fourfold belonging to p−1 (D10 ). 8 8 (ii) Let Y = G(1, 4) ∩ P ⊂ P be a smooth del Pezzo fivefold, and let S ⊂ Y be a general K3 surface of degree 14 and genus 8 contained in Y as described in Subsection 4.1. Then, inside Y , the surface S admits a congruence of 5-secant 3 twisted cubics and the linear system |H 0 (IS,Y (5))| of quintic hypersurfaces in Y with triple points along S restricted to a general W ∈ |H 0 (IS,Y (2))| defines a birational map μ : W  X ⊂ P5 with X a cubic fourfold belonging to C14 . (iii) The inverse of the map μ : X  W as in (i) is a map μ : W  X as in (ii), and the inverse of the map μ : W  X as in (ii) is a map μ : X  W as in (i). Proof. We keep the notation as above and pick a general element S14 ∈ S14 . The linear system |H 0 (IS14 (3))| satisfies Vermeire’s K3 condition and defines a map φ = φS14 : P5  P9 which is birational onto a hypercubic section Z = φ(P5 ) ⊂ P9 of G(1, 4) ⊂ P9 . Through a general point φ(p) ∈ Z there passes 18 lines contained in Z. Of these 11 are images of the eleven secant lines to S14 passing through p. The remaining 7 come from six 5-secant conics to S14 passing through p and a single 8-secant twisted cubic to S14 passing through p, which is thus transversal to a general X ∈ |H 0 (IS14 (3))| by Theorem 1.1. We deduce that a general S14 admits a congruence of 8-secant twisted cubics. Moreover, by an explicit computation we verified that the linear system |H 0 (IS314 (8))| of octic hypersurfaces with triple points along S14 defines a dominant map ΦS14 : P5  W ⊂ P8 onto a smooth GM fourfold W ⊂ P8 and whose general fibers are the twisted cubics of the congruence of S14 . Now let S ⊂ Y be a general K3 surface of degree 14 and genus 8 contained in a smooth del Pezzo fivefold Y ⊂ P8 as described in Subsection 4.1. The linear system |H 0 (IS,Y (2))| of quadric hypersurfaces in Y containing S defines a map ψ = ψS : Y  P9 which is birational onto a hypercubic section Z  = ψ(P5 ) ⊂ P9 of G(1, 4) ⊂ P9 . Through a general point φ(q) ∈ Z  there passes 18 lines contained in Z  . Of these 9 are images of the nine 1-secant lines to S passing through q and contained in Y . The remaining 9 come from eight 3-secant conics to S passing through q and contained in Y and a single 5-secant twisted cubic to S passing through q and contained in Y . We deduce that a general S admits inside Y a

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congruence of 5-secant twisted cubics. Moreover, by an explicit computation we 3 verified that the linear system |H 0 (IS,Y (5))| of quintic hypersurfaces in Y with triple points along S defines a dominant map ΨS : Y  X ⊂ P5 onto a smooth cubic fourfold X ⊂ P5 and whose general fibers are the twisted cubics of the congruence of S. If we start from a pair (S14 , X), where S14 is a general surface of degree 10 and sectional genus 6 with a node as above and X is a general cubic fourfold containing S14 , then the restriction to X of the map ΦS14 : P5  P9 defined above determines a pair (S, W ), where S is a smooth K3 surface of degree 14 and genus 8 and W ⊂ P8 is a smooth GM fourfold containing S. Conversely, if we start from a pair (S, W ), where S is a general K3 surface of degree 14 and genus 8 and W ⊂ Y = G(1, 4) ∩ P8 is a general GM fourfold containing S, then the restriction to W of the map ΨS : Y  P9 defined above determines a pair (S14 , X), where S14 is a surface of degree 10 and sectional genus 6 with a node and X a smooth cubic fourfold containing S14 . These phenomena are a consequence of the existence of the trisecant flop of the small flop contraction BlS14 X → W ⊂ P8 . We refer to [RS19b] for some details on the theoretical construction of the flop.  Remark 4.7. In the notation of Theorem 4.6, let ΦS14 : P5  W ⊂ P8 denote the rational map defined by the linear system |H 0 (IS314 (8))| of octic hypersurfaces with triple points along a general S14 . If D ⊂ P5 is a general quintic del Pezzo surface intersecting S14 along a smooth curve of degree 10 and genus 6, as constructed in Remark 4.5, we have that ΦS14 (D) ⊂ W is a smooth τ -quadric surface Q which is 5-secant a general K3 surface S ⊂ W of degree 14 and genus 8 (see Remark 4.3 and [DIM15, Proposition 7.4]).

5. Computations via Macaulay2 To study surfaces in P5 admitting congruences of (3e − 1)-secant curves of degree e, the rational maps given by hypersurfaces of degree 3e − 1 having points of multiplicity e along these surfaces and also the lines contained in the images of P5 via the linear system of cubics through these surfaces we mostly used the computer algebra system Macaulay2 [GS19]. Our proofs of various claims exploit the fact that the irreducible components Sd of the Hilbert schemes considered here are explicitly unirational. Therefore, by introducing a finite number of free parameters, one can explicitly construct the generic surface in Sd in function of the specified parameters. Adding more parameters one can also take the generic point of P5 , and then one can for instance compute the generic fiber of the map defined by the cubics through the generic [Sd ] ∈ Sd , which will depend on all these parameters. In principle, there are no theoretical limitations to perform this computation, but in practice this is far beyond what computers can do today. Anyway, the answer we get is equivalent to the one obtained on the original field via a generic specialization of the parameters

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and, above all, the generic specialization commutes with this type of computation. So, using a common computer one can get an experimental proof that a certain property holds or not for the generic [Sd ]. In the affirmative case, one can try to apply some semicontinuity arguments to get a rigorous proof. We now explain how one can ascertain the contents of Theorems 2.1, 2.2, 2.3, and similar results in specific examples. We begin by observing that given the defining homogeneous ideal of a sube variety X ⊂ Pn , the computation of a basis for the linear system |H 0 (IX (d))| of hypersurfaces of degree d with points of multiplicity at least e along X can be perfomed using pure linear algebra. This approach is implemented in the Macaulay2 package Cremona (see [Sta18]), which turns out to be effective for small values of d and e. In practice, in any Macaulay2 session with the Cremona package loaded, if I is a variable containing the ideal of X, we get a rational map defined by a e basis of |H 0 (IX (d))| by the command1 rationalMap(I,d,e). Now we consider a specific example related to Theorem 2.3. In the following code, we produce a pair (f, ϕ) of rational maps: f : P2  P5 is a birational parameterization of a smooth surface S = S38 ⊂ P5 of degree 10 and sectional genus 6 as in Theorem 2.3, and ϕ : P5  P9 is a rational map defined by a basis of cubic hypersurfaces containing S (see also Section 5 of [RS19a]). Here we work over the finite field F10000019 for speed reasons. Macaulay2, version 1.14 with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems, LLLBases, PrimaryDecomposition, ReesAlgebra, TangentCone, Truncations i1 : needsPackage "Cremona"; i2 : f = rationalMap (ZZ/10000019[vars(0..2)],{10,0,0,10}); o2 : RationalMap (rational map from PP^2 to PP^5) i3 : S = image f; i4 : phi = rationalMap S; o4 : RationalMap (cubic rational map from PP^5 to PP^9)

We now compute the rational map ψ defined by the linear system of quintic hypersurfaces of P5 which are singular along S. From the information obtained by its projective degrees we deduce that ψ is a dominant rational map onto P4 with generic fibre of dimension 1 and degree 2 and with base locus of dimension 3 and degree 52 − 19 = 6. i5 : time psi = rationalMap(S,5,2); -- used 9.07309 seconds o5 : RationalMap (rational map from PP^5 to PP^4) i6 : projectiveDegrees psi o6 = {1, 5, 19, 13, 2, 0}

Next we compute a special random fibre F of the map ψ. 1 For all the examples treated in this paper, the linear system of hypersurfaces of degree d with points of multiplicity at least e along X ⊂ Pn coincides with the homogeneous component of degree d of the saturation with respect to the irrelevant ideal of Pn of the epower of the homogeneous ideal of X. So one can also compute it using the code: gens image basis(d,saturate(I^e)).

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i7 : p = point source psi; i8 : F = psi^*(psi(p));

-- a random point on P^5

It easy to verify directly that F is an irreducible 5-secant conic to S passing through p. One can also see that F coincides with the pull-back ϕ−1 (L) of the unique line L ⊂ ϕ(P5 ) ⊂ P9 passing through ϕ(p) that is not the image of a secant line to S passing through p (see [RS19a] for details on this computation). Finally, the following lines of code tell us that the restriction of ψ to a random cubic fourfold containing S is a birational map whose inverse map is defined by forms of degree 9 and has base locus scheme of dimension 2 and degree 92 − 27 = 54. i9 : psi’ = psi|sum(S_*,i->random(ZZ/10000019)*i); o9 : RationalMap (rational map from hypersurface in PP^5 to PP^4) i10 : projectiveDegrees psi’ o10 = {3, 15, 27, 9, 1}

For the convenience of the reader, we have included in a Macaulay2 package (named ExplicitRationality and provided as an ancillary file to our arXiv submission) the examples listed in Table 2 of birational maps between cubic fourfolds and other rational fourfolds. The two examples with d = 38 can be obtained as follows: i11 : needsPackage "ExplicitRationality"; i12 : time example38(); -- used 1.2428 seconds

The above command produces the following 6 rational maps: 1. a parameterization f : P2  P5 as that obtained above of a surface S = S38 ⊂ P5 of degree 10 and sectional genus 6; 2. the rational map ψ : P5  P4 defined by the quintic hypersurfaces with double points along S; 3. the restriction of ψ to a cubic fourfold X containing S, which is a birational map; 4. the linear projection of a smooth scroll surface of degree 8 in P9 from a linear three-dimensional subspace intersecting the secant variety of the scroll in 6 points, so that the image is a scroll surface T ⊂ P5 of degree 8 with 6 nodes; moreover, we have the relation: T ∪ S = X ∩ top(Bs(ψ)), where top(Bs(ψ)) denotes the top component of the base locus of ψ; 5. the rational map η : P5  Z ⊂ P10 defined by the octic hypersurfaces with triple points along T , and where Z ⊂ P10 is a four-dimensional linear section of G(1, 5) ⊂ P14 ; 6. the restriction of η to the cubic fourfold X, which is a birational map onto Z. Now we can quickly get information on the maps, e.g., on the inverse of the last one: i13 : g = last oo; o13 : RationalMap (birational map from hypersurface in PP^5 to four-dimensional subvariety of PP^10) i14 : describe inverse g o14 = rational map defined by forms of degree 5 source variety: four-dimensional variety of degree 14 in PP^10 cut out by 15 hypersurfaces of degree 2 t

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target variety: smooth cubic hypersurface in PP^5 birationality: true projective degrees: {14, 70, 80, 24, 3}

Acknowledgement We wish to thank J´anos Koll´ar for asking about the maps defined by the linear systems of quintics singular along surfaces admitting a congruence of 5-secant conics and for his interest in our subsequent results. The first-named author is very grateful to the organizers of the Schiermonnikoog conference Rationality of Algebraic Varieties for the invitation and for the stimulating atmosphere inspiring the contents of the last section presented here.

References [AR02]

A. Alzati, F. Russo, On the k-normality of projected algebraic varieties, Bull. Braz. Math. Soc. 33 (2002), no. 1, 27–48.

[AT14]

N. Addington and R. Thomas, Hodge theory and derived categories of cubic fourfolds, Duke Math. J. 163 (2014), no. 10, 1886–1927.

[BD85]

A. Beauville and R. Donagi, La vari´et´e des droites d’une hypersurface cubique de dimension 4 (in French), C. R. Math. Acad. Sci. Paris 301 (1985), no. 14, 703–706.

[BEL91] A. Bertram, L. Ein, and R. Lazarsfeld, Vanishing theorems, a theorem of Severi, and the equations defining projective varieties, J. Amer. Math. Soc. 4 (1991), no. 3, 587–602. o, Some loci of rational cubic fourfolds, [BRS19] M. Bolognesi, F. Russo, and G. Staglian` Math. Ann. 373 (2019), no. 1, 165–190. [CR11] C. Ciliberto and F. Russo, On the classification of OADP varieties, Sci. China Math. 54 (2011), 1561–1575, Special Volume on the occasion of the 60 years of Fabrizio Catanese. [DIM15] O. Debarre, A. Iliev, and L. Manivel, Special prime Fano fourfolds of degree 10 and index 2, Recent Advances in Algebraic Geometry: A Volume in Honor of Rob Lazarsfeld’s 60th Birthday (C. Hacon, M. Mustata, and M. Popa, eds.), London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, 2015, pp. 123– 155. [dFF13] T. de Fernex and D. Fusi, Rationality in families of threefolds, Rend. Circ. Mat. Palermo 62 (2013), no. 1, 127–135. [DK18a] O. Debarre and A. Kuznetsov, Gushel–Mukai varieties: Classification and birationalities, Algebr. Geom. 5 (2018), 15–76. , Gushel–Mukai varieties: moduli, Internat. J. Math 31 (2020), no. 2, [DK18b] 2050013, 59 pp. , Gushel–Mukai varieties: Linear spaces and periods, Kyoto J. Math. 59 [DK19] (2019), no. 4, 897–953. [Fan43] G. Fano, Sulle forme cubiche dello spazio a cinque dimensioni contenenti rigate razionali del 4◦ ordine, Comment. Math. Helv. 15 (1943), no. 1, 71–80.

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[Uga02] L. Ugaglia, Subvarieties of the Grassmannian G(1, n) with small secant variety, Comm. Algebra 30 (2002), 4059–4083. [YY19] S. Yang and X. Yu, Rational cubic fourfolds in Hassett divisors, preprint: Comptes Rendus, Math., 358 (2020), 129–137. Francesco Russo e-mail: [email protected] Giovanni Staglian` o e-mail: [email protected] Dipartimento di Matematica e Informatica a degli Studi di Catania Universit` Viale A. Doria 5 I-95125 Catania, Italy

Unramified Cohomology, Algebraic Cycles and Rationality Stefan Schreieder Abstract. This is a survey on unramified cohomology with a view towards its applications to rationality problems. Mathematics Subject Classification (2010). Primary 14E08, 14C25; secondary 14M20. Keywords. Unramified cohomology, algebraic cycles, rationality.

1. Introduction A variety X of dimension n over a field k is said to be rational if it is birational to Pnk , which means that it becomes isomorphic to Pkn after removing proper closed subsets from both sides. More generally, X is said to be stably rational if X × Pkm is rational for some m ≥ 0. Generic projection shows that any variety X of dimension n over a field k is birational to a hypersurface {F = 0} ⊂ Pn+1 . The question whether X is rational k is then equivalent to asking whether over the algebraic closure of k, almost all solutions of the equation F = 0 can be parametrized 1 : 1 by a parameter t ∈ Ank (via rational functions with coefficients in k). Rationality of X thus translates into a very basic question about the solutions of F = 0. Disproving rationality for a given (rationally connected) variety X is in general a subtle problem, which requires the computation of invariants that allow to distinguish irrational varieties from those that are rational. Arguably the most powerful such invariant is unramified cohomology, which has its roots in the work of Bloch–Ogus [BO74] and which was introduced into the subject by Colliot-Th´el`ene– Ojanguren [CTO89]. Unramified cohomology is a generalization of the torsion in the third integral cohomology of smooth complex projective varieties, respectively the (unramified) Brauer group, that has previously been used by Artin–Mumford [AM72] and Saltman [Sal84]. It has recently found many applications in combination with a cycle-theoretic specialization technique that was initiated by Voisin [Voi15] and developed further by Colliot-Th´el`ene–Pirutka [CTP16a] and the author [Sch19a, Sch19b]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Farkas et al. (eds.), Rationality of Varieties, Progress in Mathematics 342, https://doi.org/10.1007/978-3-030-75421-1_13

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The purpose of this paper is to give a detailed survey on unramified cohomlogy with a view towards its applications to rationality problems. We discuss in particular the interactions of unramified cohomology with cycles and correspondences, explain the most important known methods to construct nontrivial unramified cohomology classes in concrete examples and present the aforementioned cycle-theoretic specialization method, which in conjunction with unramified cohomology yields a powerful obstruction to (stable) rationality. In addition, we discuss some the most important functoriality and compatibility results for unramified cohomology. This includes in particular the following two results that do not seem to be recorded well in the literature: functoriality with respect to finite pushforwards (see Proposition 4.7) and compatibility of the residue map with cup products (see Lemmas 2.4 and 3.1). Recent work of Nicaise–Shinder, Kontsevich–Tschinkel, and Nicaise–Ottem [NS19, KT19, NO19] makes the importance of semistable degenerations in the subject apparent. For this reason, we generalize in this survey Merkurjev’s pairing to the case of schemes with normal crossings in Section 6 and show that it has direct consequences for the aforementioned cycle-theoretic degeneration technique, see Theorem 8.5 below. The reader is advised to also consult the excellent survey on unramified cohomology by Colliot-Th´el`ene [CT95]. The present survey complements [CT95] with more recent developments, such as Merkurjev’s pairing (see Sections 5 and 6), decompositions of the diagonal (see Section 7), the specialization method (see Section 8) and some important vanishing results (see Section 10). Moreover, some special emphasize on the computational aspect of the subject is given in Section 9. We end this survey with a few open problems in Section 11 and outline a strategy how unramified cohomology and decompositions of the diagonal may potentially be used to prove the existence of a rationally connected variety that is not unirational, see Proposition 11.8. 1.1. Notation and convention All schemes are separated and locally Noetherian. An algebraic scheme is a scheme of finite type over a field. A variety is an integral algebraic scheme. If k is an uncountable field, a very general point of a k-variety X is a closed point outside a countable union of proper closed (algebraic) subsets. An alteration of a variety X over an algebraically closed field k is a proper generically finite morphism τ : X  → X such that X  is smooth over k. The existence of alterations was shown by de Jong [deJ96].

2. Preliminaries from ´etale cohomology Let k be a field. For a positive integer m that is invertible in k, we denote by μm the sheaf of mth roots of unity. For any scheme X over k, this is a subsheaf of the multiplicative sheaf Gm of invertible functions on X and hence a sheaf in the ´etale

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topology of X. For an integer j ≥ 1, we consider the twists μ⊗j m := μm ⊗· · ·⊗μm (j⊗−j ⊗0 times) and put μm Z/m) := , Hom(μ := Z/m and μ⊗j for j < 0. If k contains m m all mth roots of unity (e.g., if k is algebraically closed), then μ⊗j m  Z/m is a constant sheaf for all j. For a scheme X and a sheaf F in the ´etale topology of X, we denote by H i (X, F ) the ith ´etale cohomology of F . If X = Spec A for some ring A, then we write H i (A, F ) := H i (Spec A, F ). 2.1. Cohomology of fields Since H 1 (K, Gm ) = 0 by Hilbert 90, the long exact sequence associated to the Kummer sequence / μm / Gm / Gm /0 0 (2.1) shows that for any field K in which m is invertible, there is a canonical isomorphism H 1 (K, μm )  K ∗ /(K ∗ )m , where (K ∗ )m ⊂ K ∗ denotes the subgroup of mth powers in K ∗ . Using this, we denote the class in H 1 (K, μm ) that is represented by an element a ∈ K ∗ by (a). Moreover, for a collection of elements a1 , . . . , an ∈ K ∗ , we denote by the symbol (a1 , . . . , an ) the class (a1 , . . . , an ) := (a1 ) ∪ (a2 ) ∪ · · · ∪ (an ) ∈ H n (K, μ⊗n m )

(2.2)

that is given by cup product. We will also need the following well-known result (see, e.g., [Ser97, II.4.2]): for a variety X over an algebraically closed field k in which m is invertible, H i (k(X), μ⊗j m ) = 0 for all i > dim X.

(2.3)

2.2. Commutativity with direct limits If X is a quasi-projective variety over a field k, then for any locally constant torsion ´etale sheaf F and for any point x ∈ X, there are canonical isomorphisms ' ( lim / H i (U, F )  H i

x∈U⊂X

lim U, F ←−

= H i (OX,x , F ).

(2.4)

x∈U⊂X

Here we used that in the above limit, we may restrict to the directed system of affine open subsets U ⊂ X containing x. This guarantees that the transition maps are affine and so the above compatibility between the direct limit of ´etale cohomology groups and the ´etale cohomology of the inverse limit of schemes holds true by [Mil80, p. 88, III.1.16]. Applying (2.4) to the generic point of X, we find in particular lim / H i (U, F )  H i (k(X), F ),

(2.5)

∅ =U ⊂X

where the limit runs through all non-empty open subsets U of X and k(X) denotes the function field of X.

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2.3. Long exact sequence of pairs Proposition 2.1 ([Mil80, III.1.25] or [SGA4.2, V.6.5.4]). Let V be a scheme and let Z ⊂ V be a closed subscheme with complement U . For any ´etale sheaf F on V , the pair (V, Z) gives rise to a long exact sequence / H i (V, F ) / H i (U, F |U ) / H i+1 (V, F ) / H i+1 (V, F ) / · · · , (2.6) ··· Z

where

HZi (V, F )

denotes ´etale cohomology of F with support on Z.

Proof. Let i : Z → V and j : U → V denote the inclusions. Consider the short exact sequence / j! ZU / ZV / i∗ ZZ /0 0 of ´etale sheaves on V , where ZV denotes the constant sheaf on V with stalk Z, j! ZU denotes the sheaf that agrees with ZV on U and is 0 outside of U and i∗ ZZ denotes the pushforward of the constant sheaf with stalk Z on Z to V . For any ´etale sheaf F on V , applying RHom(−, F ) to the above short exact sequence, we get a long exact sequence · · · → Exti (ZV , F ) → Exti (j! ZU , F ) → Exti+1 (i∗ ZZ , F ) → Exti+1 (ZV , F ) → · · · . (2.7) The long exact sequence (2.6) follows from this as there are canonical identifications Exti (j! ZU , F )  H i (U, F ), Exti (ZV , F )  H i (V, F ), Exti (i∗ ZZ , F )  HZi (V, F ), (2.8) see [Mil80, proof of III.1.25] or [SGA4.2, V.6.3].



2.4. Cup products Let V be a scheme and recall that the abelian category Sh´et (V ) of ´etale sheaves on V has enough injectives, see, e.g., [Mil80, Proposition III.1.1]. Let D = D(Sh´et (V )) be the corresponding derived category. For ´etale sheaves F and G on V , we then get canonical isomorphisms Extj (F, G)  HomD (F, G[j]), see, e.g., [Huy06, Proposition 2.56]. Composition in the derived category thus yields for any ´etale sheaves F, G, and H on V a map / Exti+j (H, G). Exti (H, F ) ⊗ Extj (F, G) There is a canonical map Extj (ZV , G) → Extj (F, F ⊗ G) and so for any ´etale sheaves F, G, and H on V , the above construction yields a map / Exti+j (H, F ⊗ G). Exti (H, F ) ⊗ Extj (ZV , G) Via the identifications in (2.8), any class α ∈ H j (V, G) thus yields cup product maps ∪α : H i (V, F ) → H i+j (V, F ⊗ G), ∪α|U : H i (U, F ) → H i+j (U, F ⊗ G), and ∪ α : HZi (V, F ) → HZi+j (V, F ⊗ G).

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We will need the following well-known compatibility of the long exact sequence in Proposition 2.1 with the above cup product maps. Lemma 2.2. Let V be a scheme and let Z ⊂ V be a closed subscheme with U := V \ Z. For any ´etale sheaves F and G on V and for any class α ∈ H j (V, G), cup product with α induces a commutative diagram / H i (U, F )

H i (V, F ) ∪α

 H i+j (V, F ⊗ G)

/ H i+1 (V, F ) Z

∪α|U

 / H i+j (U, F ⊗ G)

/ H i+1 (V, F )

∪α

 / H i+j+1 (V, F ⊗ G) Z

∪α

 / H i+j+1 (V, F ⊗ G)

between the long exact sequences from Proposition 2.1. Proof. Let α ∈ Extj (F, F ⊗ G) be the image of α via the canonical map Extj (ZV , G) → Extj (F, F ⊗ G). As explained above, we may think about α as a morphism α : F → F ⊗ G[j] in the derived category D = D(Sh´et (V )). This morphism yields by [Huy06, Remark 12.57(ii)] a morphism between the following distinguished triangles in D: RHom(i∗ ZZ , F ) α

 RHom(i∗ ZZ , F ⊗ G[j])

/ RHom(ZV , F )

/ RHom(j! ZU , F )

α

 / RHom(ZV , F ⊗ G[j])

/

+1

α

 / RHom(j! ZU , F ⊗ G[j])

+1

/

The above morphism between exact triangles yields a morphism between the associated long exact sequences of cohomology groups, which by (2.8) identifies to the commutative diagram in the lemma, as we want. This concludes the proof.  2.5. Gysin sequence Theorem 2.3 (Gysin sequence). Let V be a regular Noetherian scheme with a regular closed subscheme Z ⊂ V of pure codimension c and complement U := V \Z. Then for any integer m invertible on V , there is a long exact sequence ∗ i ⊗j i+1−2c · · · → H i (V, μ⊗j (Z, μ⊗j−c )→ H i+1 (V, μ⊗j m ) → H (U, μm ) → H m m ) → ··· .

ι

∂

Proof. The theorem is a well-known consequence of the long exact sequence of pairs in Proposition 2.1 together with Gabber’s proof of Grothendieck’s purity conjecture [Fuj02, ILO14]; we give some details for convenience of the reader. Let H iZ (V, μ⊗j etale sheaf on V , associated to the presheaf m ) be the ´ / Hi (U, μ⊗j ), U Z×V U

m

cf. [Mil80, p. 85, III.1.6(e)]. Then there is a local to global spectral sequence p+q ⊗j E2p,q := H p (V, H qZ (V, μ⊗j m )) =⇒ HZ (V, μm ),

(2.9)

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see [SGA4.2, V.6.4]. Since Z and V are regular and Z is of pure codimension c in V , Grothendiecks purity conjecture, proven by Gabber (see [ILO14, XVI.3.1.1] and [Fuj02, 2.1.1]), implies $ if q = 2c; i∗ i∗ μ⊗j−c q m ⊗j H Z (V, μm )  0 else, where i : Z → V denotes the closed immersion. Hence, the spectral sequence (2.9) degenerates at E2 and yields a canonical isomorphism: ) H i−2c (Z, μ⊗j−c m



/ H i (V, μ⊗j ). Z m

(2.10) 

Combining this with (2.6), the theorem follows.

Lemma 2.4. Let V be a regular Noetherian scheme with a regular closed subscheme Z ⊂ V of pure codimension c and complement U := V \Z. Let m be an integer that is invertible on V and let α ∈ H r (V, μ⊗s m ). Then cup product with α is compatible with the Gysin sequence in Theorem 2.3, i.e., there is a commutative diagram: H i (V, μ⊗j m )



∪α

H i+r (V, μ⊗j+s ) m

/ H i (U, μ⊗j m ) 

/ H i+1−2c (Z, μ⊗j−c ) m

∪α|U

/ H i+r (U, μ⊗j+s ) m



/ H i+1 (V, μ⊗j m )

∪α|Z

/ H i+r+1−2c (Z, μ⊗j+s−c ) m



∪α

/ H i+j+1 (V, μ⊗j+s ) m

Proof. It follows from the construction of the isomorphism in (2.10) via the spectral sequence (2.9) that the cup product map i ⊗j ∪α : HZi (V, μ⊗j m )  Ext (i∗ ZZ , μm )

/ Exti+r (i∗ ZZ , μ⊗j+s )  H i+r (V, μ⊗j+s ) m m Z

identifies via (2.10) to the map given by cup product with the restriction of α to Z: ) ∪α|Z : H i−2c (Z, μ⊗j−c m

/ H i+r−2c (Z, μ⊗j+s−c ). m

The commutative diagram in question thus follows directly from Lemma 2.2 and Theorem 2.3. This concludes the proof.  Remark 2.5. In algebraic topology, the cohomology group HZi (V, −) with support corresponds to the relative cohomology group H i (V, U ; −) of the pair (V, U ) and the long exact sequence (2.6) identifies to the long exact sequence of the pair (V, U ). Moreover, the isomorphism in (2.10) corresponds to the Thom isomorphism in differential topology, which asserts that if Z is a closed complex submanifold of a complex manifold V of pure codimension c and with complement U = V \ Z, then H i (V, U ; Z(j))  H i−2c (Z, Z(j − c)), where Z(j) := (2πi)j · Z ⊂ C denotes the jth Tate twist of Z. Combining both results one obtains a topological version of the Gysin sequence in Theorem 2.3. The compatibility result with cup products stated in Lemma 2.2 is in this context proven in [Dol72, VII.8.10].

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3. Residue maps Let A be a discrete valuation ring with fraction field K and residue field κ. Applying Theorem 2.3 to V = Spec A, Z = Spec κ and U = Spec K, we get a long exact sequence of the form ⊗j ⊗j ⊗j · · · → H i (A, μm ) → H i (K, μm ) → H i+1 (A, μm ) → ··· . ) → H i−1 (κ, μ⊗j−1 m (3.1) ∂

One then defines the residue map ⊗j ) ∂A := −∂ : H i (K, μm

/ H i−1 (κ, μ⊗j−1 ), m

(3.2)

as the negative of the above boundary map ∂. Here the minus sign is necessary to make our definition compatible with the definition of the residue map in Galois cohomology or Milnor K-theory, cf. [CT95, §3.3]. By definition, the residue map has the property that its kernel coincides with the image of the natural map / H i (K, μ⊗j H i (A, μ⊗j m ) m ). Lemma 3.1. Let A be a discrete valuation ring with fraction field K and residue i1 ⊗j1 1 field κ. Then for any α ∈ H i1 (A, μ⊗j m ) with image α ∈ H (K, μm ) and any i2 ⊗j2 β ∈ H (K, μm ), we have ∂(β ∪ α) = (∂β) ∪ α,  1 where α ∈ H i1 (κ, μ⊗j m ) denotes the image of α .

Proof. This is a direct consequence of Lemma 2.4.



We also need the following well-known lemma, see, e.g., [CTO89, Proposition 1.3]. Lemma 3.2. Let A be a discrete valuation ring with fraction field K and residue field κ. Let m be an integer invertible in κ and let ν denote the valuation on K that corresponds to A. Then the composition ∂ : H 1 (K, μm )  K ∗ /(K ∗ )m

/ H 0 (κ, Z/m)  Z/m

coincides with the homomorphism that is induced by the valuation ν : K ∗ → Z. Remark 3.3. Since the cup product in ´etale cohomology is graded commutative and bilinear, Lemmas 3.1 and 3.2 completely determine the residue map on symbols (a1 , . . . , an ) as in (2.2), cf. [Sch19b, Lemma 2.1]. 3.1. Compatibility with pullbacks Let K  /K be a field extension and let m be a positive integer that is invertible in K. We can think about this as a morphism f : Spec K  → Spec K of schemes. Since ´etale cohomology is contravariant with respect to arbitrary morphisms of schemes (see, e.g., [Mil80, p. 85, III.1.6(c)]), we thus get pullback maps f ∗ : H i (K, μ⊗j m )

/ H i (K  , μ⊗j m ).

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Let ν  be a discrete valuation on K  with valuation ring A and residue field κA . Assume that the restriction ν = ν  |K of ν  to K is a nontrivial valuation on K with valuation ring A and residue field κA . This is a subfield of κA and so we get a natural morphism g : Spec κA → Spec κA . Let π be a uniformizer of A and let e := ν  (π) be the valuation of π in the valuation ring A , which is a non-negative integer that measures the ramification of the extension A ⊂ A . Then we have a commutative diagram ∂A

) H i (K  , μ⊗j O m

/ H i−1 (κA , μ⊗j−1 ); O m

f∗

(3.3)

e·g∗ ∂A

H i (K, μ⊗j m )

/ H i−1 (κA , μ⊗j−1 ) m

see, e.g., [CTO89, p. 143], where the statement is proven after identifying the above ´etale cohomology groups with the corresponding Galois cohomology groups. 3.2. Compatibility with pushforwards Let now K  /K be a finite field extension and let m be a positive integer that is invertible in K. We fix a discrete valuation ν on K with valuation ring A. Let A ⊂ K  be the integral closure of A. By the Krull–Akizuki theorem (see [Bou72, p. 500, Ch. VII, §2, no. 5, Prop. 5]), A is a Dedekind domain with finitely many maximal ideals m1 , . . . , mr . Each localization Al := Aml is a discrete valuation ring with fraction field K  and we denote its residue field by κAl , which (by the aforementioned Krull Akizuki theorem) is a finite extension of κA . In particular, the natural morphisms f : Spec K  → Spec K and gl : Spec κAl → Spec κA are finite, hence proper, and we get pushforward morphisms i ⊗j / H i (K, μ⊗j / H i (κA , μ⊗j f∗ : H i (K  , μ⊗j m ) m ) and (gl )∗ : H (κA , μm ) m ), l

for all i and all l = 1, . . . , r. Under the additional assumption that A is a finite A-module, we then get the following compatibility result. Lemma 3.4. Let K  /K be a finite field extension and let m be a positive integer that is invertible in K. Let ν be a discrete valuation on K with valuation ring A ⊂ K and let A ⊂ K  be the integral closure of A in K  . Assume that A is a finite A-module. Then in the above notation, the following diagram is commutative:  i

H (K



, μ⊗j m )

l

∂A

/

r l=1

H i−1 (κAl , μ⊗j−1 ). m 

f∗

 H i (K, μ⊗j m )

l

∂A



(3.4)

l (gl )∗

/ H i−1 (κA , μ⊗j−1 ) m

Proof. Let V  := Spec A and U  := Spec K  . Let further Z  ⊂ V  be the union of the finitely many closed points Spec κAl , l = 1, . . . , r. Similarly, we put V := Spec A, U := Spec K and Z := Spec κA . Since A and A are normal of dimension

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one, they are both regular and so the Gysin sequence from Theorem 2.3 holds for (V, Z) and (V  , Z  ). We thus have exact sequences   l ∂A ⊗j / H i (K  , μ⊗j / H i+1 (A , μ⊗j /l H i (A , μm ) ) H i−1 (κAl , μ⊗j−1 ) m m m ), l

(3.5) and H i (A, μ⊗j m )

/ H i (K, μ⊗j m )

∂A

/ H i−1 (κA , μ⊗j−1 ) m

/ H i+1 (A, μ⊗j m ).

(3.6)

Since A is a finite extension of A, there is a natural pushforward map from each of the terms of (3.5) to the respective terms of (3.6). To prove the lemma, we need to see that these morphisms yield a morphism between the (exact) complexes in (3.5) and (3.6). Since the natural morphism f : V  → V is finite, we have Rp f∗ μ⊗j m = 0 for p ≥ 1, see, e.g., [Mil80, VI.2.5]. Applying the same reasoning to the base change of f to U and Z (which we denote by the same letter f ), we obtain natural isomorphisms i ⊗j i  ⊗j i ⊗j H i (U  , μ⊗j m )  H (U, f∗ μm ), H (V , μm )  H (V, f∗ μm )

and

i ⊗j H i (Z  , μ⊗j m )  H (Z, f∗ μm ). The sequence (3.5) is thus by (2.8) isomorphic to the sequence (2.7) with F = ⊗j f∗ μ⊗j m , while (3.6) is isomorphic to (2.7) with F = μm . The norm homomorphism ⊗j induces a natural morphism of ´etale sheaves f∗ μm → μ⊗j m on V . This induces a morphism from the long exact sequence (2.7) with F = f∗ μ⊗j m to that for F = μ⊗j . Using the above identifications, one checks that this morphism of long exact m sequences is nothing but the morphism between (3.5) and (3.6) that is induced by the respective pushforward maps. This concludes the proof of the lemma. 

Remark 3.5. The commutativity in (3.3) can be proven similarly as above, by ⊗j ⊗j ⊗j replacing the norm map f∗ μ⊗j m → μm by the natural map μm → f∗ μm . Note i ⊗j i  ⊗j / however that in this case, the pullback map HZ (V, μm ) HZ  (V , μm ) corresponds via the Gysin isomorphism (2.10) to the morphism / H i−2c (Z  , μ⊗j H i−2c (Z, μ⊗j m ) m ) that is given by e times the pullback morphism, where e denotes the ramification index of the ring extension A /A. 3.3. A consequence of Bloch–Ogus’ theorem The following result, known as injectivity and codimension 1 purity property for ´etale cohomology, is a consequence of Bloch–Ogus’ proof [BO74] of the Gersten conjecture for ´etale cohomology, cf. [CT95, Theorems 3.8.1 and 3.8.2]. Theorem 3.6. Let X be a variety over a field k and let m be a positive integer that is invertible in k. Let x be a point in the smooth locus of X. Then the following hold:

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354 (a) The natural morphism

⊗j H i (OX,x , μm )

/ H i (k(X), μ⊗j ) m

(3.7)

is injective. (b) A class α ∈ H i (k(X), μ⊗j m ) lies in the image of (3.7) if and only if α has trivial residue along each prime divisor on X that passes through x.

4. Unramified cohomology Let K/k be a finitely generated field extension and let ν be a discrete valuation on K. We say that ν is a valuation on K over k, if ν is trivial on k. The valuation ring Aν ⊂ K of ν is a discrete valuation ring with fraction field K and we denote its residue field by κν . By (3.2), we get a residue map / H i−1 (κν , μ⊗j−1 ∂ν := ∂Aν : H i (K, μ⊗j ), m ) m for any positive integer m that is invertible in k. Definition 4.1. Let K/k be a finitely generated field extension. A geometric valuation ν on K over k is a discrete valuation on K over k such that the transcendence degree of κν over k is given by trdegk (κν ) = trdegk (K) − 1. The following lemma goes back to Zariski, see, e.g., [KM08, Lemma 2.45] or [Mer08, Proposition 1.7]. Lemma 4.2. Let K/k be a finitely generated field extension. A discrete valuation ν on K over k is geometric if and only if there is a normal k-variety Y with k(Y )  K such that the valuation ν corresponds to a prime divisor E on Y , i.e., for any φ ∈ K ∗ , ν(φ) = ordE (φ), where we think about φ as a rational function on Y . Definition 4.3 ([CTO89, Mer08]). Let K/k be a finitely generated field extension and let m be a positive integer that is invertible in k. We define the unramified cohomology of K over k with coefficients in μ⊗j m as the subgroup i i ⊗j Hnr (K/k, μ⊗j m ) ⊂ H (K, μm )

that consists of all elements α ∈ H i (K, μ⊗j m ) such that for any geometric valuation ν on K over k, we have ∂ν (α) = 0. Remark 4.4. Unramified cohomology was introduced by Colliot-Th´el`ene and Ojanguren [CTO89, Definition 1.1.1] into the subject. In their original definition (that is also used in the survey [CT95]), they ask ∂ν α = 0 for any discrete valuation ν of K over k. The version in Definition 4.3 where one restricts to geometric valuations has later been introduced by Merkurjev [Mer08, §2.2]. It follows from Proposition 4.10 below that both definitions coincide if there is a smooth projective variety X over k with k(X) = K. In particular, both notions coincide for any

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finitely generated field extension K/k if k is perfect and resolution of singularities is known for varieties over the field k (e.g., char(k) = 0). In general it seems however unclear whether both definitions coincide. We prefer to work with the definition given above, as it has slightly better formal properties (e.g., it admits proper pushforwards, see Proposition 4.7 below). 4.1. Stable invariance One of the most important properties of unramified cohomology is the fact that it is a stable birational invariant (cf. [CTO89, Proposition 1.2] and [Mer08, Example 2.3]), that is, it does not change if one passes from a field K to a purely transcendental extension of K. Lemma 4.5. Let K/k be a finitely generated field extension and let m be a positive integer that is invertible in k. Let n ∈ N and let f : AnK → Spec K be the structure morphism. Then the canonical morphism ⊗j i f ∗ : Hnr (K/k, μm )

∼

i / Hnr (K(An )/k, μ⊗j m )

is an isomorphism. Proof. By induction, it suffices to prove the case n = 1. By the Faddeev sequence (see [GS06, Theorem 6.9.1]), the following is exact:   ∂x/ f∗ / / H i (K, μ⊗j 0 H i (K(A1 ), μ⊗j H i−1 (κ(x), μ⊗j−1 ), m ) m ) m x∈P1K

where x runs through all closed points of P1K and ∂x = ∂OP1 ,x . The zero on the K left implies that f ∗ is injective and so it remains to prove surjectivity. Exactness i in the middle of the above sequence shows that any class α ∈ Hnr (K(A1 )/k, μ⊗j m ) ∗ i ⊗j is of the form f β for some β ∈ H (K, μm ) and we need to show that if f ∗ β is unramified over k, β must have trivial residue at any geometric valuation of K over k. By Lemma 4.2, any such valuation corresponds to a prime divisor E on some k-variety Y with function field K. But then E × P1 is a prime divisor on the k-variety Y ×P1 with function field K(A1 ) and the fact that f ∗ β has trivial residue along this divisor shows by (3.3) that β has trivial residue along E, as we want. (Here we used implicitly the injectivity of the natural map H i−1 (k(E), μ⊗j−1 )→ m H i−1 (k(E)(A1 ), μ⊗j−1 ), which follows from the Faddeev sequence above, applied m to K = k(E).) This concludes the proof of the lemma.  The above lemma has by (2.3) the following immediate consequence. Corollary 4.6. Let X be a variety over an algebraically closed field k and let m be a positive integer that is invertible in k. If X is stably rational, then i Hnr (k(X)/k, μ⊗j m ) =0

for all

i > 0.

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4.2. Functoriality Unramified cohomology has the following functoriality properties. Proposition 4.7. Let K  /K/k be a finitely generated fields extensions, let f : Spec K  → Spec K be the natural morphism and let m be an integer that is invertible in k. ⊗j (a) Then f ∗ : H i (K, μm ) → H i (K  , μ⊗j m ) induces a pullback map i / Hnr (K  /k, μ⊗j m ).

i (K/k, μ⊗j f ∗ : Hnr m )

i ⊗j (b) If f is finite, then f∗ : H i (K  , μ⊗j m ) → H (K, μm ) induces a pushforward map i i / Hnr f∗ : Hnr (K  /k, μ⊗j (K/k, μ⊗j m ) m )

with f∗ ◦ f ∗ = deg(f ) · id. Proof. Item (a) follows directly from (3.3). In item (b) the equality f∗ ◦ f ∗ = deg(f ) · id holds already on the level of ´etale cohomology and so it suffices to show that f∗ is well defined on the subgroup i of unramified classes. To this end, let α ∈ Hnr (K  /k, μ⊗j m ) and let ν be a geometric valuation on K over k. We then need to show that ∂ν f∗ α = 0. Since ν is geometric, we can construct a normal projective variety Y with k(Y )  K such that ν corresponds to the valuation associated to a codimension 1 point y ∈ Y (1) . Since K  /K is a finite field extension, we can further construct a normal projective variety Y  with a surjective morphism f : Y  → Y such that k(Y  )  K  and f ∗ : k(Y ) → k(Y  ) corresponds to the given inclusion K ⊂ K  . Since f is generically finite and surjective, and y ∈ Y is a codimension 1 point, the reduced preimage (f −1 (y))red is given by finitely many codimension 1 points y1 , . . . , yr of Y  . We claim that the integral closure A ⊂ K  of A := OY,y in K  is nothing but the local ring of Y  at the finitely many codimension 1 points y1 , . . . , yr of Y  . Indeed, Spec A → Spec A can be constructed by first taking the fibre product of the proper morphism f : Y  → Y with the inclusion Spec OY,y → Y and precomposing this with the normalization map Spec A → Y  ×Y Spec OY,y . This description also shows that Spec A → Spec A is proper, hence finite as it is clearly quasi-finite. Hence, A is a finite ring extension of A. Moreover, the localizations of A at its finitely many maximal ideals are exactly the local rings OY  ,yl , l = 1, . . . , r. Since i α ∈ Hnr (K  /k, μ⊗j m ) is unramified, we know that ∂OY  ,y α = 0 l

for all l = 1, . . . , r and so the commutative diagram (3.4) shows that ∂ν f∗ α = 0, as we want. This completes the proof.  4.3. Restriction to scheme points and pullbacks for morphisms between smooth projective varieties Theorem 3.6 has the following important consequences, which show that unramified classes on smooth projective varieties can be restricted to any scheme point.

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Proposition 4.8. Let X be a smooth variety over a field k and let m be a positive i integer that is invertible in k. Let α ∈ Hnr (k(X)/k, μ⊗j m ). (a) Then for any x ∈ X, there is a well-defined restriction α|x ∈ H i (κ(x), μ⊗j m ). (b) If X is also proper over k, then α|x ∈ H i (κ(x)/k, μ⊗j m ) is unramified over k. Proof. We begin with the proof of (a). By part (b) in Theorem 3.6, we know that α admits a lift α ˜ ∈ H i (OX,x , μ⊗j m ) and this lift is unique by part (a) of Theorem 3.6. The restriction α|x may thus be defined as image of α ˜ via the natural morphism i ⊗j i / H (κ(x), μ⊗j H (OX,x , μm ) m ). This concludes the proof of (a). To prove (b), let Z be a normal variety over k with k(Z)  κ(x) and let z ∈ Z (1) be a codimension 1 point of Z. We then have to prove that ∂z (α|x ) = 0, or, equivalently, that α|x lies in the image of the natural map / H i (κ(x), μ⊗j H i (OZ,z , μ⊗j (4.1) m ) m ). Since X is proper, we may up to shrinking Z around z assume that the isomorphism k(Z)  κ(x) of fields is induced by a morphism of schemes ι : Z → X such that i the generic point of Z maps to x ∈ X. Since α ∈ Hnr (k(X)/k, μ⊗j m ) is unramified over k, Theorem 3.6 implies that α lies in i ⊗j H i (OX,ι(z) , μ⊗j m ) ⊂ H (k(X), μm ).

(4.2)

By (2.4), this means that there is an open neighbourhood U ⊂ X of ι(z) and a class α ˜ ∈ H i (U, μ⊗j m ) that restricts to α on the generic point Spec k(X). Since ι(z) lies in the closure of the point x ∈ X, it follows that U contains x and so α ˜ has an image in i ⊗j H i (OX,x , μ⊗j m ) ⊂ H (k(X), μm ) which must be α by (4.2). In particular, the restriction α|x ∈ H i (κ(x), μ⊗j m ) that we defined above coincides with the image of α ˜ via the natural map / H i (κ(x), μ⊗j H i (U, μ⊗j m ) m ). Since ι(z) ∈ U , this shows that α|x lies in the image of (4.1), as we want. This concludes the proof of the proposition.  Corollary 4.9. Let f : X → Y be a morphism between smooth proper varieties over a field k and let m be a positive integer that is invertible in k. Then there is a well-defined pullback map i i / Hnr f ∗ : Hnr (k(Y )/k, μ⊗j (k(X)/k, μ⊗j m ) m ) i which is given by restricting a given unramified class α ∈ Hnr (k(Y )/k, μ⊗j m ) to the generic point of the image of f and pulling that back to k(X).

Proof. By Proposition 4.8, the restriction of α to the generic point of the image of f is unramified over k and so is its pullback to k(X) by Proposition 4.7. This proves the corollary. 

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4.4. It is enough to check residues on a smooth proper model The codimension 1 purity property (see item (b) in Theorem 3.6) implies the following, cf. [CT95, Theorem 4.1.1]. Proposition 4.10. Let X be a smooth proper variety over a field k and let m be a ⊗j positive integer that is invertible in k. Then a class α ∈ H i (k(X), μm ) is unramified over k if and only if α has trivial residue along any prime divisor on X. ⊗j Proof. One direction is trivial. For the converse, assume that α ∈ H i (k(X), μm ) has trivial residue along any prime divisor on X. We then need to show that for any normal variety Y over k that is birational to X and for any codimension 1 point y ∈ Y , we have ∂y α = 0. Equivalently, we need to see that α lies in the image of the natural map ⊗j / H i (k(X), μ⊗j H i (OY,y , μm ) m ).

Since Y is normal, the rational map φ : Y  X is defined in codimension 1 and so up to shrinking Y around the codimension 1 point y, we may assume that φ : Y → X is a morphism. Let x = φ(y) ∈ X be the image of y. Then we get a commutative diagram H i (OX,x , μ⊗j ) m QQQ QQQ QQQ QQQ ( H i (k(X), μ⊗j m ) mm6 m m mmm mmm  mmm H i (OY,y , μ⊗j m ) and so our claim follows from part (b) in Theorem 3.6.



4.5. Comparison with usual cohomology Let X be a smooth proper variety over a field k and let m be invertible in k. It follows immediately from Proposition 4.10 that the image of the natural map / H i (k(X), μ⊗j H i (X, μ⊗j m ) m ) lies in the subgroup of unramified classes and so we get a well-defined map i / Hnr H i (X, μ⊗j (k(X)/k, μ⊗j (4.3) m ) m ). It is not hard to show that this is an isomorphism for i = 1 (see [CT95, Proposition 4.2.1]) and surjective for i = 2 and j = 1. However, starting from i = 3, this map is in general neither injective nor surjective. The kernel of (4.3) consists of all cohomology classes α ∈ H i (X, μ⊗j m ) that vanish on some non-empty Zariski open subset of X. For instance, if i = 2j is even and α = cj (E) is the Chern class of a vector bundle E, then α lies in the kernel of (4.3) because any vector bundle is generically trivial. For i = 2 and j = 1, this observation can be used to prove the following, see [CT95, Proposition 4.2.3].

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Proposition 4.11. Let X be a smooth projective variety over a field k and let m be invertible in k. For i = 2 and j = 1, the natural map (4.3) is surjective and its kernel is given by the image of c1 : Pic X → H 2 (X, μm ). In particular, there is a natural isomorphism 2 Hnr (k(X)/k, μm ) 

H 2 (X, μm ) . im(c1 : Pic X → H 2 (X, μm ))

It follows from the Kummer sequence (2.1) that the right-hand side in the above isomorphism is isomorphic to the m-torsion of the Brauer group Br(X) = H 2 (X, Gm ) of X. Remark 4.12. For k = C, Colliot-Th´el`ene–Voisin [CTV12] found a relation between the third unramified cohomology of a smooth projective variety and the failure of the integral Hodge conjecture for codimension two cycles on X. A generalization of this result to cycles of arbitrary codimension was recently given in [Sch21b].

5. Merkurjev’s pairing Let X be a smooth proper variety over a field K and let m be invertible in K. (Here it is important to allow K to be non-algebraically closed, e.g., the function field of a variety over a smaller field k.) Let Z0 (X) denote the group of 0-cycles on X, i.e., the free abelian group generated by the closed points of X. For a closed point z ∈ X, we denote by fz : Spec κ(z) → Spec K the structure morphism. Following Merkurjev [Mer08, §2.4], we then define for any unramified i class α ∈ Hnr (K(X)/K, μ⊗j m ) a class z, α := (fz )∗ (α|z ) ∈ H i (K, μ⊗j m ), where α|z ∈ H i (κ(z), μ⊗j m ) denotes the restriction from Proposition 4.8. We may extend this definition to arbitrary 0-cycles z ∈ Z0 (X) linearly and so we obtain a bilinear pairing  i / H i (K, μ⊗j / z, α. Z0 (X) × Hnr (K(X)/K, μ⊗j (5.1) m ) m ), (z, α) The main result about this pairing is the following proposition due to Merkurjev, which shows that the pairing descends to the level of Chow groups, cf. [Mer08, §2.4]. Proposition 5.1. Let K be a field and let m be an integer that is invertible in K. Let g : C → X be a non-constant morphism between smooth proper K-varieties, where i C is a curve. Then for any α ∈ Hnr (K(X)/K, μ⊗j m ) and any non-zero rational function φ ∈ K(C), we have g∗ div(φ), α = 0, where div(φ) ∈ Div(C) denotes the divisor of zeros and poles of φ. Before we prove Proposition 5.1 in Section 5.2, let us explain some of its applications.

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5.1. Applications of Proposition 5.1 Corollary 5.2. Let X be a smooth proper variety over a field K. Then (5.1) descends to a bilinear pairing  / i / H i (K, μ⊗j CH0 (X) × Hnr (K(X)/K, μ⊗j z, α. (5.2) m ) m ), (z, α) Proof. This is an immediate consequence of Proposition 5.1.



The next result is originally due to Karpenko and Merkurjev, see [KM13, RC-I]. Corollary 5.3. Let X and Y be smooth proper varieties over a field k and let m be an integer that is invertible in k. Then there is a bilinear pairing i / H i (k(X)/k, μ⊗j ), (Γ, α)  / Γ∗ α, CHdim X (X × Y ) × Hnr (k(Y )/k, μ⊗j m ) nr m which via linearity is defined as follows: if Γ ⊂ X × Y is integral and does not dominate the first factor, then Γ∗ α := 0; otherwise, the first projection induces a finite morphism p : Spec κ(γ) → k(X), where γ denotes the generic point of Γ, and we put Γ∗ α := p∗ ((pr∗2 α)|γ ). Proof. Assume that Γ ⊂ X × Y is a subvariety of dimension dim X that dominates X via the first projection and let γ be the generic point of Γ. By Proposition 4.8, i (pr∗2 α)|γ ∈ Hnr (κ(γ)/k, μ⊗j m ) is unramified over k and so i p∗ ((pr∗2 α)|γ ) ∈ Hnr (k(X)/k, μ⊗j m )

is unramified over k by Proposition 4.7. Hence, our definition of Γ∗ α is well defined and we get a bilinear pairing / H i (k(X)/k, μ⊗j ), (Γ, α)  / Γ∗ α. (5.3) Zn (X × Y ) × H i (k(Y )/k, μ⊗j ) nr

m

nr

m

It remains to see that this pairing descends to the level of Chow groups. For this, let K := k(X) denote the function field of X. Then there are natural group homomorphisms i i / Z0 (YK ) and Hnr / Hnr Zdim X (X × Y ) (k(Y )/k, μ⊗j (K(Y )/K, μ⊗j m ) m ). i i ⊗j Since K = k(X) and Hnr (K/k, μ⊗j m ) ⊂ H (K, μm ), this induces a diagram i Zdim X (X × Y ) × Hnr (k(Y )/k, μ⊗j m ) UUUU UUUU UUUU UUUU U*

iii4 iiii i i i iii  iiii i Z0 (YK ) × Hnr (K(Y )/K, μ⊗j m )

H i (K, μ⊗j m )

which is commutative by the definition of the pairings in (5.1) and (5.3). Since the / Z0 (YK ) descends to a map CHdim X (X × Y ) → natural map Zdim X (X × Y )

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CH0 (YK ), we deduce from Corollary 5.2 that the pairing (5.3) satisfies Γ∗ α = 0 whenever Γ is a cycle that is rationally equivalent to 0. This concludes the proof  of the corollary. 5.2. Proof of Proposition 5.1 For the proof of Proposition 5.1, we will need the following compatibility result for the pairing defined in (5.1). Lemma 5.4. Let g : X → Y be a morphism between smooth proper K-varieties. The pairing (5.1) has the following properties. i (i) For any α ∈ Hnr (K(Y )/K, μn⊗j ) and any z ∈ Z0 (X), we have g∗ z, α = z, g ∗ α, i where g ∗ α ∈ Hnr (K(X)/K, μ⊗j m ) is defined by Corollary 4.9. (ii) If X and Y are curves and g is finite and surjective, then for any β ∈ i Hnr (K(X)/K, μ⊗j n ) and any w ∈ Z0 (Y ), we have

g ∗ w, β = w, g∗ β, ⊗j i where g∗ β ∈ Hnr ) is defined by Proposition 4.7 and g ∗ w de(K(Y )/K, μm notes the flat pullback of cycles.

Proof. By linearity, it suffices to prove (i) in the case where z is a closed point of X. Let fz : Spec κ(z) → Spec K be the structure morphism. Then we have z, g ∗ α = (fz )∗ (g ∗ α)|z . If y = g(z) denotes the image of z in Y , with structure morphism fy : Spec κ(y) → Spec K, then g induces a morphism gz : Spec κ(z) → Spec κ(y) with fz = fy ◦ gz and so we find z, g ∗ α = (fz )∗ (g ∗ α)|z = (fy ◦ gz )∗ (gz∗ (α|y )) = deg(gz ) · (fy )∗ (α|y ), where we used (gz )∗ ◦ (gz )∗ = deg(gz ). On the other hand, g∗ z = deg(gz ) · y and so g∗ z, α = deg(gz ) · y, α = deg(gz ) · (fy )∗ (α|y ). This proves item (i) of the lemma. To prove item (ii), it suffices as before to deal with the case where w is a closed point of Y . Since g is finite and surjective and X and Y are both smooth and proper curves, g is flat and so the pullback g ∗ w is defined on the level of cycles. Explicitly, it is given by r  g∗w = ar · z l l=1

where z1 , . . . , zr denote the closed points of X that lie above w and where ar denotes the ramification indices of OY,w ⊂ OX,zi (recall that X and Y are smooth proper curves by the assumption in (ii)). Then we have r r   g ∗ w, β = al · (fzl )∗ (α|zl ), al · zl , α = l=1

l=1

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where fzl : Spec κ(zl ) → Spec K denotes the structure morphism. On the other hand, if fw : Spec κ(w) → Spec K denotes the structure morphism and gzl : Spec κ(zl ) → Spec κ(w) denotes the natural morphism induced by g, then w, g∗ β = (fw )∗ (g∗ β)|w . To simplify this further, let π ∈ OY,w be a parameter. The rational function π yields a class (π) ∈ H 1 (k(Y ), μm )  k(Y )∗ /(k(Y )∗ )m and we have by Lemmas 3.1 and 3.2 the following well-known formula (g∗ β)|w = ∂w (g∗ β ∪ (π)). By the projection formula, g∗ β ∪ (π) = g∗ (β ∪ (g ∗ π)). Using the compatibility (3.4), we thus get ∂w (g∗ β ∪ (π)) = ∂w g∗ (β ∪ (g ∗ π)) =

r  (gzl )∗ ∂zl (β ∪ (g ∗ π)). l=1

∗

Since β is unramified and g π ∈ OX,zl coincides up to a unit with the al th power of a parameter of OX,zl , we deduce from Lemmas 3.1 and 3.2 that ∂zl (β ∪ (g ∗ π)) = al · β|zl . Putting everything together, this yields ' w, g∗ β = (fw )∗ ∂w (g∗ β ∪ (π)) = (fw )∗

r 

( al · (gzl )∗ β|zl

=

l=1

r 

al · (fzl )∗ (α|zl ),

l=1

because fzl = fw ◦ gzl . Hence, g ∗ w, β = w, g∗ β, which concludes the proof of the lemma.  1 Proof of Proposition 5.1. There is a finite morphism ϕ : C → PK with

div(φ) = ϕ∗ (0 − ∞). By Lemma 5.4, we thus find g∗ div(φ), α = g∗ ϕ∗ (0 − ∞), α = ϕ∗ (0 − ∞), g ∗ α = 0 − ∞, ϕ∗ g ∗ α = 0, ϕ∗ g ∗ α − ∞, ϕ∗ g ∗ α. We claim that this last difference vanishes. To see this, note that i (K(P1 )/K, μ⊗j ϕ∗ g ∗ α ∈ Hnr m )

is unramified over K by Proposition 4.7 and Corollary 4.9. Hence, Lemma 4.5 ⊗j implies that there is a class α ∈ H i (K, μm ) with ϕ∗ g ∗ α = f ∗ α ,

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1 where f : PK → Spec K denotes the structure morphism. Using this, we find by the above calculation that

g∗ div(φ), α = 0, f ∗ α  − ∞, f ∗ α  = f∗ 0, α  − f∗ ∞, α  = 0, where we used item (i) of Lemma 5.4 in the second equality and f∗ 0 = f∗ ∞ ∈ Z0 (Spec K) in the last equality. This proves the proposition. 

6. Generalization to schemes with normal crossings Let k be a field. Let X be a pure-dimensional algebraic scheme over k. If Xi with i ∈ I denote the irreducible components of X, then for any non-empty subset J ⊂ I, we define < XJ := Xl . l∈J

Definition 6.1. A pure-dimensional algebraic scheme X over a field k with irreducible components Xi with i ∈ I is called snc (simple normal crossings) scheme if for each non-empty subset J ⊂ I, the subscheme XJ ⊂ X is either empty or smooth of codimension |J|, the cardinality of J. Definition 6.2. Let X be a proper snc scheme over a field k and let m be a positive integer invertible in k. We then define the unramified cohomology of X with values ⊗j in μm as the subgroup  ⊗j i i (X/k, μ⊗j ) Hnr (k(Xl )/k, μm Hnr m )⊂ l∈I

that consists of all collections α = (αl )l∈I of unramified classes αl ∈ H i (k(Xl )/k, μ⊗j m ), such that for all l, l ∈ I αl |Xl ∩Xl = αl |Xl ∩Xl , by which we mean that αl and αl have the same restriction to each component of Xl ∩ Xl via the restriction maps from Proposition 4.8. If X is a smooth proper variety over k, then ⊗j i i Hnr (k(X)/k, μ⊗j (X/k, μm ) = Hnr m ).

For any nonempty subset J ⊂ I, we get a well-defined restriction map / α| / H i (X /k, μ⊗j ), α  H i (X/k, μ⊗j ) nr

m

nr

J

m

XJ

which is defined by picking any index l ∈ J and defining α|XJ on each component of XJ as restriction of αl . This is well defined (i.e., does not depend on the choice of l) by the compatibility condition in Definition 6.2.

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6.1. A pairing on the level of 0-cycles Let X be a proper snc scheme over a field K with irreducible components Xl , l ∈ I. Then there is a bilinear pairing / H i (K, μ⊗j ), Z0 (X) × H i (X/K, μ⊗j ) nr

m

defined by (z, α) 

/ z, α :=



m

(−1)|J|−1 z|XJ , α|XJ ,

(6.1)

∅ =J⊂I

where we note that XJ is smooth and where z|XJ denotes the 0-cycle given by intersecting the 0-cycle z ∈ Z0 (X) with XJ ⊂ X in the naive sense, i.e., we simply keep that part of z that lies on XJ (this is an operation on the level of cycles that does not pass to the level of Chow groups).1 Lemma 6.3. Let X be a proper snc scheme over a field K and let m be a positive integer that is invertible in K. If z is a closed point of X with structure morphism i fz : Spec κ(z) → Spec K and α ∈ Hnr (X/K, μ⊗j m ), then for any component Xl with z ∈ Xl , we have z, α = (fz )∗ (αl |z ). Proof. By definition



z, α =

(−1)|J|−1 (fz )∗ ((α|XJ )|z ).

∅ =J⊂I z∈XJ

Up to replacing X by the union of those components that contain z, we may assume that z ∈ Xl for all l ∈ I. We then find  z, α = (−1)|J|−1 (fz )∗ ((α|XJ )|z ). ∅ =J⊂I

The compatibility condition in Definition 6.2 ensures that (fz )∗ ((α|XJ )|z ) = (fz )∗ ((α|XJ  )|z ). 

for all non-empty J, J ⊂ I. Hence, for any l ∈ I, we have  (−1)|J|−1 (fz )∗ (α|XJ |z ) z, α = ∅ =J⊂I

=−



(−1)|J| (fz )∗ (αl |z )

∅ =J⊂I

 |I|   |I| (−1)r · (fz )∗ (αl |z ) = (fz )∗ (αl |z ), r r=1 |I|  r where we used in the last equality that |I| r=1 r (−1) = −1. This proves the lemma.  =−

1 The formula in (6.1) is motivated by the specialization formula of Nicaise and Shinder in [NS19] which itself is motivated by formulas in motivic integration.

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Proposition 6.4. Let K be a field and let m be an integer that is invertible in K. Let X be a proper snc scheme over K and let g : C → X be a non-constant i morphism from a smooth proper curve C. Then for any α ∈ Hnr (X/K, μ⊗j m ) and any non-zero rational function φ ∈ K(C), we have g∗ div(φ), α = 0, where div(φ) ∈ Div(C) denotes the divisor of zeros and poles of φ. Proof. This is an immediate consequence of Lemma 6.3 and Proposition 5.1.



Corollary 6.5. The pairing (6.1) descends to a bilinear pairing i / H i (K, μ⊗j CH0 (X) × Hnr (X/K, μ⊗j m ) m ) on the level of Chow groups. Proof. This is a direct consequence of Proposition 6.4.



6.2. A pairing on the level of correspondences Let X and Y be proper reduced algebraic schemes over a field k and assume that Y is an snc scheme. Let further m be an integer that is invertible in k. We aim to define a bilinear pairing   / i / Zdim X (X × Y ) × Hnr (Y /k, μ⊗j H i (k(Xl ), μ⊗j ((Γ∗ α)l )l∈I , m ) m ), (Γ, α) l∈I

(6.2) where Xl with l ∈ I denote the irreducible components of X. It suffices to define (Γ∗ α)l for each l ∈ I, i.e., the composition of (6.2) with the natural projection to H i (k(Xl ), μ⊗j m ). To this end, fix l ∈ I and note that flat pullback induces a natural map / Z0 (Yk(X ) ), Zdim X (X × Y ) l which descends to the level of Chow groups, i.e., sends cycles rationally equivalent to 0 on X × Y to cycles rationally equivalent to 0 on Yk(Xl ) . In addition, there is i i ⊗j (Y /k, μ⊗j a natural map Hnr m ) → Hnr (Yk(Xl ) /k(Xl ), μm ). Using this, we define  / (Γ∗ α)l , (Γ, α) by asking that the diagram i Zdim X (X × Y ) × Hnr (Y /k, μ⊗j ) WWWWW m WWWWW WWWWW WWWWW + i ⊗j l H (k(Xl ), μm ) 3 g g g ggggg ggggg g g g g ggg  i , Z0 (Yk(Xl ) ) × l Hnr (Yk(Xl ) /k(Xl ), μ⊗j m )

is commutative, where the lower horizontal arrow is induced by (6.1).

(6.3)

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Corollary 6.6. Let X and Y be proper reduced algebraic schemes over a field k and assume that Y is an snc scheme. Let further m be an integer that is invertible in k. Then the pairing (6.2) descends to a well-defined pairing  i / CHdim X (X × Y ) × Hnr (Y /k, μ⊗j H i (k(Xl ), μ⊗j (6.4) m ) m ). l∈I

Proof. Since (6.3) commutes, Corollary 6.5 implies that Γ∗ α = 0 whenever Γ ∈ Zdim X (X × Y ) is rationally equivalent to 0. This proves the corollary.  Remark 6.7. The above corollary generalizes Corollary 5.3 in two ways: X and Y may be reducible and X may be arbitrarily singular.

7. Decompositions of the diagonal The following notion goes back to Bloch [Blo79] (using an idea of Colliot-Th´el`ene) and Bloch–Srinivas [BS83], and has for instance been studied in [ACTP13] and [Voi15]. Definition 7.1. An algebraic scheme X of pure dimension n over a field k admits a decomposition of the diagonal if [ΔX ] = [X × z] + [ZX ] ∈ CHn (X ×k X),

(7.1)

where ΔX ⊂ X ×k X denotes the diagonal, z ∈ Z0 (X) is a 0-cycle on X and ZX is a cycle on X ×k X which does not dominate any component of the first factor. For instance, X = Pnk admits a decomposition of the diagonal, because CHn (Pn ×k Pn ) is generated by [Pn × {x}] for some k-rational point x ∈ Pn together with cycles that do not dominate the first factor (namely hn−i × hi for i = 1, . . . , n, where hi ⊂ Pn denotes a linear i-dimensional subspace). To give a reducible example with a decomposition of the diagonal, we will now show that the union X = Pnk ∪H Pnk of two copies of Pnk glued along a hypersurface H ⊂ Pnk admits a decomposition of the diagonal as long as H admits a k-rational point. Indeed, if we write X = X1 ∪X2 , where X1 , X2 are the irreducible components of X, then X ×k X has the 4 irreducible components Xi ×k Xj with i, j ∈ {1, 2}. The class of the diagonal ΔX may then be written as [ΔX ] = [ΔX1 ] + [ΔX2 ] ∈ CHn (X × X), where ΔXi ⊂ Xi × Xi denotes the diagonal. Because of the proper pushforward map / CHn (X × X) CHn (Xi × Xi ) for each i = 1, 2, we may use the decomposition of the diagonal of Xi for each i constructed above to get an identity ΔX = [X1 × x1 ] + [X2 × x2 ] + [ZX ] ∈ CHn (X × X), where xi ∈ Xi is a k-rational point of Xi and ZX is a cycle on X ×k X which does not dominate any component of the first factor. This decomposition has not yet

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the form required in (7.1). However, since H contains a k-rational point, any two k-rational points of X are rationally equivalent (they may be joined by a chain of two lines). This implies in particular [X2 × x1 ] = [X2 × x2 ] and so ΔX = [X × x1 ] + [ZX ] ∈ CHn (X × X). This gives a decomposition of the diagonal ΔX as in (7.1), as claimed. Since in the above definition, X is an arbitrary algebraic k-scheme, it may be illustrative to also consider a slightly more exotic example, such as X = Spec C over the ground field k = R. In this case, the choice of a root of −1 in C identifies X ×R X to the disjoint union of two copies of Spec C (the Galois group of X/R permutes the two copies). The diagonal ΔX is isomorphic to Spec C (it corresponds to one of the two components once we fixed a root of −1). Hence, ΔX is a 0-cycle of degree 2 on the proper R-scheme X ×R X. On the other hand, any 0-cycle z on X is just a multiple of X and so the cycle X × z will have degree divisible by 4. This shows that no decomposition as in (7.1) exists (where we note that ZX needs to be empty, as any nonzero cycle on X ×R X dominates both factors). The following lemma generalizes the observation made in this last example, by showing that a proper algebraic scheme X with a decomposition of the diagonal is geometrically connected and of index one, i.e., X contains a 0-cycle of degree one. Lemma 7.2. Let X be a proper algebraic scheme of pure dimension n over a field k. If X admits a decomposition of the diagonal as in (7.1), then the following holds: 1. the 0-cycle z on X in (7.1) has degree one; 2. X is geometrically connected. Proof. Since X is proper, the projection pr1 : X ×k X → X to the first factor gives rise to a proper pushforward map / CHn (X). (pr1 )∗ : CHn (X ×k X) If a decomposition as in (7.1) exists, then (pr1 )∗ [ΔX ] = (pr1 )∗ [X × z] = deg(z) · [X] ∈ CHn (X). Since (pr1 )∗ [ΔX ] = [X] ∈ CHn (X) and because this class is non-torsion (in fact CHn (X) is a free abelian group on the irreducible components of the reduced scheme X red ), the above identity implies deg(z) = 1, as we want. This proves (1). If X admits a decomposition of the diagonal, then for any field extension K/k, the algebraic K-scheme XK = X ×k K admits a decomposition of the diagonal as well, simply by taking flat pullbacks of (7.1). To prove (2), it thus suffices to show that X is connected if it admits a decomposition of the diagonal. For a contradiction, assume that X = #m i=1 Xi is the disjoint union of finitely many algebraic k-schemes X1 , . . . , Xr with r ≥ 2. We may then write the 0-cycle z from r (7.1) in the form z = i=1 zi such that zi is supported on Xi . Pulling back the equality (7.1) to Xi ×k Xj for i = j (this is possible because Xi ×k Xj is an open subscheme of X ×k X and so the inclusion is flat), we find that Xi × zj ∈ CHn (Xi ×k Xj )

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is rationally equivalent to a cycle that does not dominate Xi via the first projection. Pushing this identity forward to the first factor, we find that zj has degree zero. Since r ≥ 2, this holds for all j and so the 0-cycle z has degree zero, contradicting item (1). This concludes the proof.  In the case of varieties, we have the following important result. Lemma 7.3. A variety X over a field k admits a decomposition of the diagonal if and only if there is a 0-cycle z ∈ Z0 (X) on X such that [δX ] = [zK ] ∈ CH0 (XK ),

(7.2)

where K = k(X) and where δX denotes the 0-cycle on XK that is induced by the diagonal ΔX . Proof. Let n := dim X. There is a natural isomorphism lim / CHn (U ×k X)  / CH0 (XK ). ∅ =U ⊂X

Using this, a decomposition of the diagonal (7.1) implies directly an identity as in (7.2) and the converse follows from the localization sequence [Ful98, Proposition 1.8].  The following theorem which in the smooth proper case is due to Merkurjev, relates the notion of decompositions of the diagonal with unramified cohomology, cf. [Mer08, Theorem 2.11]. Theorem 7.4. Let k be a field and let m be a positive integer that is invertible in k. Let X be a proper snc scheme (e.g., a smooth proper variety) over k. If X admits a decomposition of the diagonal, then the natural morphism / H i (X/k, μ⊗j ) ι : H i (k, μ⊗j ) m

is surjective for all i. In particular, algebraically closed.

nr m i ⊗j Hnr (X/k, μm ) =

0 for all i > 0 if k is

Proof. Assume that X admits a decomposition of the diagonal as in (7.1) and let Xl with l ∈ I denote the components of X. By Corollary 6.6, the pairing (6.2) descends to a well-defined pairing  i / CHdim X (X × X) × Hnr (X/k, μ⊗j H i (k(Xl ), μ⊗j (7.3) m ) m ). l∈I

It follows form the definition of this pairing in (6.2) that [ΔX ]∗ α = α for all

i (X/k, μ⊗j α ∈ Hnr m ) ⊂



H i (k(Xl ), μ⊗j m ).

l∈I

On the other hand

[ZX ]∗ α = 0,

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whenever ZX ∈ CHdim X (X × X) does not dominate any component of the first factor. Hence, (7.1) implies 

α = [ΔX ]∗ α = [X × z]∗ α + [ZX ]∗ α = [X × z]∗ α.

If z = s as [xs ] for some integers as and closed points xs ∈ X with structure morphisms fxs : Spec κ(xs ) → Spec k, then ( '  ∗ as (fxs )∗ α|xs , [X × z] α = ι s

where

⊗j i / Hnr ι : H i (k, μm ) (X/k, μ⊗j m ) is the natural morphism. This proves the theorem.



7.1. Connection to rationality and stable birational types For the following result, see [CTP16a, Lemme 1.5] in the smooth case and [Sch19b, Lemma 2.4] in general. Lemma 7.5. A variety X over a field k that is stably rational (or more generally retract rational) admits a decomposition of the diagonal. Proof. Recall that X is retract rational if there are rational maps f : X  PN k and g : PN k  X with g ◦ f = id (here we ask implicitly that the composition g ◦ f is defined). This condition is for instance satisfied if X is stably rational. We N denote by Γf ⊂ X × PN k and Γg ⊂ Pk × X the closure of the graph of f and g, respectively. Replacing X by a projective model, we may assume that X is proper over k. Replacing X by Γf , we may also assume that f is a morphism, which is automatically proper since X is proper. For any field extension K of k, we then get morphisms / CH0 (PN ) and g ∗ : CH0 (PN ) / CH0 (XK ), f∗ : CH0 (XK ) K

∗

K

PN K

where g is defined by pulling back cycles from to (Γf )K (using [Ful98, Definiis smooth) and pushing these cycles forward tion 8.1.2], which works because PN K to XK via the natural proper morphism (Γf )K → XK . Let now K = k(X) be the function field of X. Then [δX ] = g ∗ ◦ f∗ [δX ], because g ◦ f = id implies g ∗ ◦ f∗ = id. On the other hand, CH0 (PN K )  Z is generated by [zK ] for any k-point z of PN . Since f [δ ] has degree one, it follows ∗ X k that f∗ [δX ] = [zK ] and so [δX ] = g ∗ ◦ f∗ [δX ] = g∗ [zK ] = [(g∗ z)K ], which shows that X admits a decomposition of the diagonal by Lemma 7.3.



The above lemma implies that a variety that does not admit a decomposition of the diagonal cannot be stably rational. Motivated by work of Shinder [Shi19], the following generalization is proven in [Sch19c, Theorem 1.1].

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Theorem 7.6. Let k be an uncountable algebraically closed field and let X be a smooth projective k-variety that does not admit a decomposition of the diagonal. Then for any positive integers n and d, X is not stably birational to a very general hypersurface of degree d and dimension n over k. Remark 7.7. Let B := PH 0 (Pn+1 , O(d)) and let Y → B be the universal degree d k n+1 hypersurface in Pk . The conclusion of the above theorem means that there is a countable union (that depends on X) of proper closed subsets of B such that any hypersurface Y , that corresponds to a point outside this countable union, is not stably birational to X. The above result is a consequence of Theorem 7.8 below (see [Sch19c, Theorem 4.1]). To state the result, we need to recall the following terminology. Firstly, for a discrete valuation ring R, a proper flat R-scheme X is called strictly semi-stable, if the special fibre X0 is a geometrically reduced simple normal crossing divisor on X . That is, the components of X0 are smooth Cartier divisors on X and the scheme-theoretic intersection of r different components of X0 is either empty or smooth and equi-dimensional of codimension r in X . Secondly, a proper variety X over a field k has universally trivial Chow group of 0-cycles if for any field extension K/k, the degree map deg : CH0 (XK ) → Z is an isomorphism. If X is smooth and proper over k, then this is equivalent to (7.2) and hence equivalent to the fact that X admits a decomposition of the diagonal. Theorem 7.8. Let R be a discrete valuation ring with algebraically closed residue field k. Let π : X → Spec R and π  : X  → Spec R be flat projective morphisms with geometrically connected fibres such that π is strictly semi-stable and π  is smooth. Assume 1. the special fibre X0 of π has universally trivial Chow group of 0-cycles; 2. the special fibre X0 of π  does not admit a decomposition of the diagonal. Then the geometric generic fibres of π and π  are not stably birational to each other. 7.2. Torsion orders Let X be a rationally chain connected proper variety over a field k (e.g., a smooth Fano variety, see, e.g., [Kol96, Theorem V.2.13]). This means that any two points of the base change XL of X to any algebraically closed field L ⊃ k can be joined by a chain of rational curves (if k is uncountable then it suffices to ask this for L = k). In particular, the degree map deg : CH0 (Xk )

/Z

is an isomorphism and this property remains true if we replace k by a larger algebraically closed field. We thus find that for any 0-cycle z ∈ CH0 (X) of degree one, the 0-cycle δX − zk(X) ∈ CH0 (Xk(X) )

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lies in the kernel of the natural map CH0 (Xk(X) )

/ CH0 (X k(X) ).

But then there must be a finite extension L of k(X), so that the above 0-cycle vanishes already in CH0 (XL ). Since the natural composition / CH0 (XL ) / CH0 (Xk(X) ) CH0 (Xk(X) ) is given by multiplication by deg(L/k), we find that δX − zk(X) ∈ CH0 (Xk(X) ) is a torsion class. This motivates the following definition, that has for instance been studied in [CL17], [Kah17], and [Sch21a]. Definition 7.9. Let X be a proper variety over a field k and let δX be the 0-cycle on Xk(X) that is induced by the diagonal ΔX . Then the torsion order Tor(X) ∈ N ∪ {∞} of X is the smallest positive integer such that the 0-cycle Tor(X) · δX may be written as Tor(X) · δX = zk(X) ∈ CH0 (Xk(X) ), for some zero-cycle z on X, and it is ∞ if no such integer exists. Remark 7.10. By Lemma 7.3, Tor(X) = 1 if and only if X admits a decomposition of the diagonal and the same argument shows that for any positive integer e, e·ΔX admits a decomposition as in (7.1) if and only if Tor(X) < ∞ divides e. If k is algebraically closed and X is smooth and proper, then the torsion order of X bounds the order of any unramified cohomology class α on X of cohomological degree at least 1, as for deg α ≥ 1 and Tor(X) < ∞, 0 = Tor(X) · δX − zk(X) , α = Tor(X) · δX , α = Tor(X) · α, because zk(X) , α = 0 as α restricts to zero on any closed point of X (because k is algebraically closed). Lemma 7.11. Let f : X → Y be a proper dominant morphism between proper k-varieties. If Tor(X) < ∞, then Tor(Y ) | deg(f ) · Tor(X). Proof. The morphism f induces a proper morphism f × f : X × X → Y × Y and hence a proper morphism f  : Xk(X) → Yk(Y ) with 0 = f∗ (Tor(X) · δX − zk(X) ) = deg(f ) · Tor(X) · δY − f∗ zk(X) ∈ CH0 (Yk(Y ) ). This implies Tor(Y ) | deg(f ) · Tor(X), as we want.



Corollary 7.12. Let Y be a proper k-variety of dimension n and let f : Pnk  Y be a dominant rational map. Then Tor(Y ) | deg(f ). Proof. Let X ⊂ Pnk × Y be the closure of the graph of f . Then X is a proper variety over k and the second projection yields a morphism f  : X → Y of degree deg(f  ) = deg(f ). Since X is rational, it has torsion order 1 by Lemma 7.5 and so the corollary follows from Lemma 7.11. 

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8. Specialization method The specialization method that we explain in this section was initiated by Voisin [Voi15], with important improvements by Colliot-Th´el`ene–Pirutka [CTP16a] and later some further improvements by the author [Sch19a, Sch19b]. Notation 8.1. Let R be a discrete valuation ring with fraction field K and algebraically closed residue field k. Let π : X → Spec R be a proper flat R-scheme with connected fibres and denote by X = X × K and Y = X × k the geometric generic fibre and special fibre of π. Fulton showed the following specialization result for Chow groups, see [Ful75, §4.4] and [Ful98, §20.3]. Theorem 8.2. In the notation 8.1, let Xη = X ×R K. Then we have for any integer i a specialization map / CHi (Y ), sp : CHi (Xη ) which is defined by sp(γ) = γ|Y , where γ denotes the closure of γ in the total space X and where γ|Y denotes its restriction to Y , i.e., the pullback of γ to the Cartier divisor Y ⊂ X . Proof. Since R is a discrete valuation ring, the closure γ in X of a cycle γ on the generic fibre Xη will automatically be flat over R. Hence the map in the theorem exists on the level of cycles and we only need to see that it descends to the Chow group modulo rational equivalence. By [Ful98, p. 154, §1.9], the following sequence is exact j∗ / / 0, CHi+1 (Y ) ι∗ / CHi+1 (X ) CHi (Xη ) where ι : Y → X and j : Xη → X denote the natural morphisms. Assume that γ and γ  are two rationally equivalent i-dimensional cycles on Xη with closures γ and γ  in X . Then j ∗ [γ] = j ∗ [γ  ] and so [γ] − [γ  ] = ι∗ ξ for some ξ ∈ CHi+1 (Y ). On the other hand, Y is a principal Cartier divisor on X that contains the support of ι∗ ξ and so ι∗ (ι∗ ξ) = 0 by the definition of ι∗ , see [Ful75, §1.7]. Hence, ι∗ [γ] = ι∗ [γ  ] ∈ CHi (Y ), as we want. This concludes the proof.  Fulton’s theorem has the following consequence. Corollary 8.3 ([Voi15, CTP16a]). In the notation 8.1, if X admits a decomposition of the diagonal and the special fibre Y is pure-dimensional (e.g., this is automatic if X is integral by Krull’s Hauptidealsatz), then Y admits a decomposition of the diagonal as well. Proof. Replacing R by its completion we may assume that in the notation 8.1 R is complete. Let n := dim Y . Since π is flat and Y is pure-dimensional, X is also

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pure-dimensional of dimension n. Assume that there is a decomposition of the diagonal [ΔX ] = [X × z] + [ZX ] ∈ CHn (X ×K X) of X. This implies that there is a finite field extension L/K, so that the above relation holds over L. Since R is complete, the integral closure of R in L is a discrete valuation ring R , see, e.g., [EGAIV, Th´eor`eme 23.1.5 and Corollaire 23.1.6]. Hence, replacing X → Spec R by the finite base change Spec R → Spec R, we may assume that the above relation is defined and holds already over the field K. Applying the specialization map from Theorem 8.2, we then find [ΔY ] = sp([ΔX ]) = [Y × sp(z)] + sp[ZX ] ∈ CHn (Y ×k Y ), where sp(z) ∈ CH0 (Y ) is a 0-cycle on Y . Projecting the closure of ZX to the first factor yields a cycle on X that is automatically flat over R and which has generically dimension at most n − 1. It follows that also the special fibre of this cycle has dimension at most n − 1. We deduce that sp[ZX ] can be represented by a cycle on Y ×k Y whose image via the first projection has dimension at most n − 1 and so it does not dominate any component of Y , as Y is of pure dimension n. Hence, Y admits a decomposition of the diagonal, as we want. This concludes the corollary.  Together with Theorem 7.4, we deduce the following criterion, which has in this form been proven by Colliot-Th´el`ene–Pirutka, thereby generalizing an earlier version of Voisin [Voi15], who initiated the method. Theorem 8.4 ([CTP16a, Th´eor`eme 1.12]). In the notation 8.1, assume that Y is integral and that the following holds: i • Hnr (k(Y )/k, μ⊗j m ) = 0 for some i > 0 and some j; • there is a resolution τ : Y  → Y such that τ∗ : CH0 (YL ) → CH0 (YL ) is an isomorphism for all field extensions L/k.

Then X does not admit a decomposition of the diagonal. Proof. For a contradiction, assume that X admits a decomposition of the diagonal. Then so does Y by Corollary 8.3. By Lemma 7.3, δY = zk(Y ) holds in CH0 (Yk(Y ) ). By assumption, τ∗ : CH0 (YL ) → CH0 (YL ) is an isomorphism for L = k(Y ) and so      we find δY  = zk(Y ) in CH0 (Yk(Y ) ) and for some 0-cycle z on Y . Since k(Y ) =  k(Y ), this shows by Lemma 7.3 that Y admits a decomposition of the diagonal, and so Theorem 7.4 yields i Hnr (k(Y )/k, μ⊗j m ) = 0,

which contradicts our assumptions. This concludes the theorem.



Thanks to our generalization of Merkurjev’s pairing to the case of snc schemes in Section 6, we obtain immediately the following (new) variant of the above theorem.

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Theorem 8.5. In the notation 8.1, assume that Y is a snc scheme over k with i Hnr (Y /k, μ⊗j m ) = 0

for some i > 0 and some integer m that is invertible in k. Then X does not admit a decomposition of the diagonal. Proof. For a contradiction, assume that X admits a decomposition of the diagonal. Then so does Y by Corollary 8.3. Hence, Theorem 7.4 yields i Hnr (Y /k, μ⊗j m ) = 0,

which contradicts our assumptions. This concludes the theorem.



To apply the above theorems to a given projective variety X, one has to construct a resolution of an integral degeneration Y of X, or an R-model X of X whose special fibre is an snc scheme. On the other hand, the special fibre needs to have non-trivial unramified cohomology and practice shows that this usually forces the model X to be highly non-smooth. The presence of the singularities in the special fibre makes it often very hard (and sometimes practically impossible) to compute a model X or a resolution τ : Y  → Y as in the above theorems. This problem was solved in [Sch19a], where it was shown that much more singular models can be used. This method was generalized further in [Sch19b, Proposition 3.1] (allowing alterations instead of resolutions), as follows. Theorem 8.6 ([Sch19a, Sch19b]). In the notation 8.1, assume that X and Y are integral. Let m ≥ 2 be an integer that is invertible in k. Suppose that for some ⊗j i integers i ≥ 1 and j there is a class α ∈ Hnr (k(Y )/k, μm ) of order m such that  for any alteration τ : Y → Y and any (scheme) point x ∈ Y  with τ (x) ∈ Y sing we have ⊗j (τ ∗ α)|x = 0 ∈ H i (κ(x), μm ).

(8.1)

Then X does not admit a decomposition of the diagonal. Remark 8.7. The main point of the above result lies in the fact that in many situations of interest, condition (8.1) is automatically satisfied (cf. Theorem 10.1 below); that is, one often gets the condition (8.1) on the singularities for free, without even computing an alteration (or resolution). Remark 8.8. It is not hard to see that the proof that we give below still works in the case where the special fibre might be reducible and Y in the above theorem is replaced by a reduced component of the special fibre, see [Sch21a, Proposition 6.1] for more details. Remark 8.9. The proof will show more generally that the torsion order of X is either infinite or divisible by m, see Definition 7.9.

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Proof of Theorem 8.6. Let A = OX ,Y be the local ring of X at the generic point of Y . Since Y is reduced, it follows that A is a discrete valuation ring with residue field k(Y ). Since X admits a decomposition of the diagonal, so does Y by Corollary 8.3. We can restrict this identity in the Chow group of Y × Y to Yk(Y ) and get an equality δY = zk(Y ) ∈ CH0 (Yk(Y ) ) where z is a 0-cycle on Y and δY denotes the 0-cycle that is induced by the diagonal. By Gabber’s improvement of de Jong’s alteration theorem [deJ96], there is an alteration τ : Y  → Y whose degree is coprime to m, see [IT14, Theorem 2.1]. We would like to pull back the above relation to the Chow group of 0-cycles of  Yk(Y ) . In general this is impossible if Yk(Y ) is not smooth. Instead we can restrict the above equality to the smooth locus of Yk(Y ) (using flat pullbacks) and pulling sm back this identity to the Chow group of 0-cycles on τ −1 (Yk(Y ) ). By the localization exact sequence (see [Ful98, Proposition 1.8]), we then get an identity  δτ = zk(Y  ) + z  ∈ CH0 (Yk(Y ) ),

(8.2)

  where δτ is the 0-cycle on Yk(Y ) that is induced by the graph of τ inside Y × Y ,  z ∈ CH0 (Y  ) is a 0-cycle on Y  and z  ∈ CH0 (Yk(Y ) ) is a 0-cycle whose support satisfies

supp(z  ) ⊂ τ −1 (Y sing )k(Y ) . The definition of the pairing in (5.1) thus shows by (8.1) that z  , τ ∗ α = 0. Since ⊗j k is algebraically closed and so H i (k, μm ) = 0, we also have zk(Y  ) , τ ∗ α = 0. By linearity, we then get zk(Y  ) + z  , τ ∗ α = 0. Using (8.2), the above relation in CH0 (Yk(Y ) ) thus shows by Corollary 5.2 that δτ , τ ∗ α = 0. On the other hand, the definition of the pairing in (5.1) yields δτ , τ ∗ α = τ∗ δτ , α = deg τ · δY , α = deg τ · α. Hence, deg τ · α = 0 ∈ H i (k(Y ), μ⊗j m ), which contradicts the fact that α is nonzero and deg τ is coprime to m. This concludes the proof of the theorem.  Theorems 8.4, 8.5 and 8.6 yield together with Lemma 7.5 a powerful method to obstruct rationality of varieties that have interesting specializations. Using entirely different methods, Nicaise–Shinder [NS19] and Kontsevich–Tschinkel [KT19] recently showed the following related result.

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Theorem 8.10 ([NS19, KT19]). In the notation 8.1, assume that k has characterLet Yl with l ∈ I denote istic 0, that X is regular and that Y is an snc scheme. > the irreducible components of Y and write YJ := l∈J Yl for any J ⊂ I. If  |J|−1 X ] = [Pdim ] (−1)|J|−1 [YJ × Pk (8.3) k ∅ =J⊂I

holds true in the free abelian group on (stable) birational equivalence classes of smooth k-varieties, then X is not (stably) rational. Remark 8.11. In order to show that (8.3) holds, one has to show in practice that |J|−1 at least one of the terms YJ × Pk is not (stably) rational and that it is not cancelled out in the alternating sum by the remaining terms. This strategy has recently been implemented successfully by Nicaise–Ottem [NO19], who use as an input known stable irrationality results, which in turn are proven by applications of Theorems 8.4 or 8.6.

9. Examples with nontrivial unramified cohomology In this section we collect some of the most important known constructions of rationally connected varieties Y over algebraically closed fields with nontrivial unramified cohomology. All examples have the following approach in common. 1. Start with a smooth projective rational variety S and a nontrivial class α ∈ ⊗j H i (k(S), μm ). 2. Construct another variety Yα (usually of larger dimension) together with a dominant morphism fα : Yα → S such that fα∗ α = 0. This usually requires that the ramification locus of Yα is contained in the ramification locus of α and often both loci will coincide. 3. Construct a rationally connected variety Y with a dominant morphism f : Y → S such that: (i) ´etale locally at any (codimension one) point of S where α has nontrivial residue, the fibration f coincides with the fibration fα up to birational equivalence; (ii) Zariski locally over S, the fibrations f and fα are not birationally equivalent. 4. Show that f ∗ α ∈ H i (k(Y ), μ⊗j m ) is unramified over k by exploiting (3i). The idea is that f ∗ α can possibly only have residues above points on S where α ramifies, but condition (3i) ensures that ´etale locally at such points, f ∗ α is 0 and so it must have trivial residue. 5. Find a reason why f ∗ α is nonzero; this is a tricky point, because f ∗ α will be unramified by item (4) and so it is a priori impossible to check nontriviality by a residue computation. An obviously necessary condition here is condition (3ii).

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Remark 9.1. In [Sch19b] and [Sch21a], it was shown that the following flexible condition ensures the nonvanishing of f ∗ α: f ∗ α is nonzero if there is a degeneration Y0 → S of Y → S, so that there is a k(S)-rational point in the smooth locus of the generic fibre of Y0 → S. Remark 9.2. We emphasize that conditions (3i) and (3ii) do not automatically imply that the aim in items (4) and (5) can be achieved, it should rather be seen as a guideline how to find potential candidates for which one might hope to be able to prove (4) and (5). 9.1. Quadric bundles ´a la Artin–Mumford and Colliot-Th´el`ene–Ojanguren The starting point here is the following result of Arason [Ara75] and Orlov–Vishik– Voevodsky, see [OVV07, Theorem 2.1]. Theorem 9.3. Let K be a field of characteristic 0, let a1 , . . . , an ∈ K ∗ and consider the associated symbol α = (a1 , . . . , an ) ∈ H n (K, μ⊗n m ). Consider the associated / Spec K over K, given by Pfister quadric f : Qα ⎧ ⎫ ⎨  ⎬ n 2 Qα := (−a1 )1 (−a2 )2 · · · (−an )n · zρ() = 0 ⊂ P2K −1 ⎩ ⎭ n ∈{0,1}

where  = (1 , . . . , n ) ∈ {0, 1}n, and where ρ : {0, 1}n → {0, 1, 2, . . . , 2n − 1} n−1 denotes the bijection ρ() = i=0 i · 2i . Then,   / H n (K(Q), μ⊗n ) = {0, α} . ker f ∗ : H n (K, μ⊗n 2 ) 2 In the construction of Artin–Mumford [AM72] and Colliot-Th´el`ene–Ojanguren [CTO89], one considers fibrations f : Y → Pnk whose generic fibre is (stably) birational to a Pfister quadric Qα over K = k(Pn ) with the following special property. Definition 9.4. Let k be a field of characteristic 0 and let K = k(Pn ) for some integer n ≥ 2. Let further a1 , . . . , an−1 , b1 , b2 ∈ K ∗ be nonzero rational functions on Pnk and consider the symbols αj := (a1 , a2 , . . . , an−1 , bj ) ∈ H n (K, μ⊗n 2 ) for j = 1, 2. Then the Pfister quadric Qα1 +α2 associated to α = α1 + α2 is called CTO- type quadric, if the following holds: (1) For any geometric valuation ν on K over k, we have ∂ν αj = 0 for at least one j ∈ {1, 2}. In other words, there is no such valuation ν such that α1 and α2 have both nontrivial residue with respect to ν. (2) αj = 0 for j = 1, 2 (if k = k, this is equivalent to asking that for each j = 1, 2, αj has at least one nontrivial residue at some geometric valuation ν). With this definition and the above theorem, it is easy to prove the following result that goes back to Colliot-Th´el`ene–Ojanguren, see [Sch19a, Proposition 17].

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Proposition 9.5 ([CTO89]). Let k be a field of characteristic 0 and let Y be a kvariety with a dominant morphism f : Y → Pkn , such that the generic fibre of f is stably birational to a CTO-type quadric Qα1 +α2 over the function field k(Pn ). Then n 0 = f ∗ α1 = f ∗ α2 ∈ Hnr (k(Y )/k, μ⊗n 2 ). Proof. By Theorem 9.3, f ∗ (α1 + α2 ) = 0 and so f ∗ α1 = f ∗ α2 (as we work with mod 2 coefficients). Moreover, f ∗ α1 = 0 would by Theorem 9.3 imply α1 = 0 or α1 = α1 +α2 (hence α2 = 0), which contradicts the assumption αj = 0 for j = 1, 2. Hence, 0 = f ∗ α1 = f ∗ α2 ∈ H n (k(Y ), μ⊗n 2 ) and it suffices to show that this class is unramified over k. For this, let ν be a geometric valuation of k(Y ) over k and consider the restriction μ := ν|K of ν to k(Pn ). If μ is trivial, then ∂ν f ∗ αj = 0 is clear. Otherwise, μ is a geometric valuation on k(Pn ) over k. By the definition of CTO-type quadrics, there is some j ∈ {1, 2} with ∂μ αj = 0 and so ∂ν f ∗ αj = 0 follows from the diagram (3.3). This concludes the proof of the proposition.  The main difficulty in this approach is the construction of CTO-type quadrics. The example of Artin–Mumford [AM72] is a conic that is stably birational to a CTO-type quadric surface over k(P2 ). CTO-type quadrics over k(P3 ) have been constructed by Colliot-Th´el`ene–Ojanguren in [CTO89] and the general case of CTO-type quadrics over k(Pn ) for arbitrary n ≥ 2 was established in [Sch19a, Section 6]. An algebraic variant of this construction which leads to fibrations over rational bases whose generic fibres are products of certain Pfister quadrics was established by Peyre [Pey93] and Asok [Aso13]. This approach allows generalizations to μ -coefficients for any prime , but it still relies heavily on Theorem 9.3 (and its analogue for other Norm varieties associated to symbols with μ -coefficients). 9.2. The quadric surface bundle of Hassett–Pirutka–Tschinkel For any smooth quadric surface Q ⊂ P3K over a field K, the kernel of the pullback map / H 2 (K(Q), μ⊗2 ) H 2 (K, μ2⊗2 ) 2 is completely described by Arason in [Ara75], see, e.g., [Pir18, Theorem 3.10]. Moreover, one can use the Hochschild–Serre spectral sequence to show that the 2 image of the above map is given by Hnr (K(Q)/K, μ⊗2 2 ). In [Pir18, Theorem 3.17], Pirutka uses these ingredients to give a general formula for the unramified coho2 mology Hnr (C(Y )/C, μ⊗2 2 ), where Y is a variety over C that admits a fibration 2 f : Y → PC whose generic fibre is a smooth quadric surface over C(P2 ). This general formula has the following beautiful consequence, see [HPT18, Proposition 11]. Proposition 9.6 ([HPT18]). Let g := x20 + x21 + x22 − 2(x0 x1 + x0 x2 + x1 x2 ) be the equation of the conic in P2C that is tangent to the coordinate lines xi = 0 for

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i = 0, 1, 2 and consider the bidegree (2, 2) hypersurface Y ⊂ P2C × P3C , given by the equation 2 Y := {g(x0 , x1 , x2 ) · z02 + x0 x1 · z12 + x0 x2 · z22 + x1 x2 · z32 = 0} ⊂ PC × P3C . 2 If f : Y → PC denotes the natural morphism that is induced by the first projection, then   x1 x2 2 0 = f ∗ ∈ Hnr , (C(Y )/C, μ⊗2 2 ). x0 x0

Even though the above result is formulated over C, the proof remains valid over any algebraically closed field of characteristic different from 2. One of the main differences between the examples in Section 9.1 and 9.2 is that the generic fibre of the quadric surface bundle of Hassett–Pirutka–Tschinkel in Proposition 9.6 is not (stably birational to) a Pfister quadric. On the other hand, a common feature of both results is that they rely on the fact that the kernel of the pullback map / H n (K(Q), μ⊗n ) H n (K, μ⊗n 2 ) 2 is known in both cases: when Q is a Pfister quadric or an arbitrary quadric surface. The kernel of the above map is not known for general quadrics, which yields a nontrivial obstacle when trying to generalize the result of Hassett–Pirutka–Tschinkel from Proposition 9.6 to higher dimensions. 9.3. Generalization The following generalization of Proposition 9.6 was discovered in [Sch19b] and [Sch21a]. Theorem 9.7. Let k be an algebraically closed field and let m be a positive integer that is invertible in k. Assume that there is an element t ∈ k which is transcendental over the prime field of k.2 For d := m · n+1 m , consider the polynomial (d ' n  xi + xd−n x1 x2 · · · xn ∈ k[x0 , x1 , . . . , xn ] g := t · 0 i=0

and the bidegree (d, m) hypersurface Y ⊂ Pn × P2 −1 , given by ⎫ ⎧ g(x0 , x1 , . . . , xn ) · z0m ⎪ ⎪ ⎪ ⎪ ⎬ ⎨   n d− n n m i=1 i 1 2 + x0 x1 x2 · · · xn · zρ() = 0 ⊂ Pnk × P2k −1 , Y := ⎪ ⎪ ⎪ ⎪ ∈{0,1}n ⎭ ⎩ n

 =0

n−1 where ρ : {0, 1} → {0, 1, 2, . . . , 2n − 1} denotes the bijection ρ() = i=0 i · 2i . If f : Y → Pnk denotes the morphism induced by the first projection, then the class   x1 x2 xn ∗ n 0 = f ∈ Hnr , ,..., (k(Y )/k, μ⊗n (9.1) m ) x0 x0 x0 has order m and is unramified over k. n

2 If

k has characteristic 0, then t may also be chosen to be a prime number that is coprime to m

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A formal analogy between the example in the above theorem and that in Proposition 9.6 is as follows: the equation g in Proposition 9.6 defines a conic that is tangent to the three coordinate lines in Pk2 and so g restricts to squares on the coordinate lines. Similarly, the equation g in Theorem 9.7 restricts to a dth power (and hence to an mth power because m | d) on each coordinate hypersurface {xi = 0} ⊂ Pnk . The starting point of Theorem 9.7 is the following result, which generalizes one part of Theorem 9.3 from m = 2 to arbitrary m ≥ 2, see [Sch21a, Corollary 4.2]. Lemma 9.8. Let K be a field and let m be a positive integer that is invertible in K. Let a1 , . . . , an ∈ K ∗ be invertible elements with associated symbol α = (a1 , . . . , an ) ∈ H n (K, μ⊗n m ). Consider further the hypersurface 6  B n m Fα := = 0 ⊂ P2K −1 (−a1 )1 (−a2 )2 · · · (−an )n · zρ() ∈{0,1}n

with structure morphism f : Fα → Spec K. Then ⊗n f ∗ α = 0 ∈ H n (K(Fα ), μm ).

Proof of Theorem 9.7. The case m = 2 follows from [Sch19b, Propositions 5.1 and 6.1], and the general case of arbitrary m ≥ 2 follows from [Sch21a, Proposition 4.1] and [Sch21a, Theorem 5.3]. We give (almost all) the details of the argument in what follows.   x1 x2 xn Let α := ∈ H n (k(Pn ), μ⊗n , ,..., m ). x0 x0 x0 Then we need to show the following two properties: (a) f ∗ α is unramified over k; (b) f ∗ α has order m. Note that these properties are opposing to each other, as (a) amounts to a vanishing result (all residues of the class in (9.1) vanish), while (b) amounts to a non-vanishing result. To prove (b), let F ⊂ k be the algebraic closure of the prime field of k. Then Y is defined over F [t] and so we can consider its degeneration Y0 modulo t, which is n a hypersurface Y0 ⊂ PnF ×P2F −1 with projection f0 : Y0 → PnF . For a contradiction, assume that there is an integer e ∈ {1, 2, . . . , m − 1} with e · f ∗ α = 0. It is not hard to show that this implies the following for the specialization where t = 0: e · f0∗ α = 0 ∈ H n (F (Y0 ), μ⊗n m ). (Here by slight abuse of notation, we use that α ∈ H n (F (Pn ), μ⊗n m ).) However, our construction implies that the generic fibre of f0 : Y0 → PnF has a F -rational point y0 ∈ Y0 in its smooth locus. The class e · f0∗ α can be restricted to this point and so e · f0∗ α|y0 = 0 ∈ H n (κ(y0 ), μ⊗n m ). On the other hand, κ(y0 )  F (Pn ) and the composition n n ⊗n / H n (κ(y0 ), μ⊗n H n (F (Pn ), μ⊗n m ) m )  H (F (P ), μm )

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given by pullback via f0 and restriction to y0 is the identity. Hence, ⊗n e · α = 0 ∈ H n (F (Pn ), μm ),

which is false as one may check by induction on n by taking residues along xn = 0. This proves (b). To prove (a), let x ∈ Y  be a codimension 1 point of a normal birational model of Y . We may assume that there is a birational morphism Y  → Y and so f induces a morphism f  : Y  → Pn . We then need to show that ∂x (f  ∗ α) = 0, where   x1 x2 xn α := ∈ H n (k(Pn ), μ⊗n , ,..., m ). x0 x0 x0 This vanishing is obvious, unless f  (x) is contained in the union of hyperplanes {x0 x1 · · · xn = 0} ⊂ Pnk , which is the ramification locus of α on Pnk . It thus suffices to deal with the case where f  (x) ∈ {x0 x1 · · · xn = 0}. The proof then splits up into two cases, as follows. Case 1. f  (x) ∈ / {g = 0}. In this case, g is a nontrivial mth power in the residue field of the local ring  OY  ,x and so it becomes an mth power in the completion O Y  ,x . The residue n−1 ⊗n−1 ∂x : H n (k(Y ), μ⊗n ) → H (κ(x), μ ) factors through m m ⊗n / H n (Frac O  Y  ,x , μm )

H n (k(Y ), μ⊗n m )

∗

and the image of f ∗ α via the above map vanishes by Lemma 9.8, so that ∂x (f  α) = 0 follows, as we want. Case 2. f  (x) ∈ {g = 0}. The main point about this case is that g = 0 meets each strata of the union of the hyperplanes xi = 0 dimensionally transversely. In particular, f  (x) ∈ {g = 0} implies that the number c of coordinate functions xi that vanish at f  (x) is strictly smaller than the codimension of f  (x) in Pnk : c < codimPnk (f  (x)). It then follows from Lemma 3.1 that the valuation μ on k(Pn ) that is induced by restricting the valuation on k(Y ) that is given by the codimension 1 point x ∈ Y  satisfies ∂μ α = α ∪ β, for some β ∈ H c−1 (κ(μ), μ⊗c−1 ) and α ∈ H n−c (κ(μ), μ⊗n−c ) with m m  n−c  ⊗n−c n−c /H α ∈ im(H (κ(f (x)), μ ) (κ(μ), μ⊗n−c )). m

m



Since k is algebraically closed, κ(f (x)) has cohomological dimension n − codimPnk (f  (x)) < n − c, ) = 0, which implies α = 0. In particular, see (2.3). Hence, H n−c (κ(f  (x)), μ⊗n−c m ∗ ∂μ α = 0 and so ∂x (f α) = 0 follows from (3.3). This completes the proof of Theorem 9.7. 

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10. Vanishing result and applications The examples discussed in Sections 9.1, 9.2 and 9.3 all have the following feature in common: Up to birational equivalence, there is a morphism f : Y → Pnk whose ⊗n generic fibre is smooth and a class α ∈ H n (k(Pn ), μm ) such that f ∗ α is nonzero and unramified. One of the main discoveries in [Sch19a, Sch18, Sch19b, Sch21a] was the observation that all these examples have the following property in common: for any smooth variety Y  with a generically finite dominant morphism τ : Y  → Y , and for any point x ∈ Y  with τ (x) ∈ Y sing , the restriction of τ ∗ f ∗ α to x vanishes: ⊗n (τ ∗ f ∗ α)|x = 0 ∈ H n (κ(x), μm ).

This vanishing result is crucial, as it allows to apply Theorem 8.6 without resolving the singularities of the special fibre (or even the whole family) of the degeneration. The intuition behind this result is as follows: • since the generic fibre of f is smooth, it suffices to show (τ ∗ f ∗ α)|x = 0 whenever f (τ (x)) ∈ Pkn is not the generic point of Pkn ; • if f (τ (x)) is not contained in the ramification locus of α, then the restriction (τ ∗ f ∗ α)|x factors via the restriction of α to the point f (τ (x)) ∈ Pnk . Since the latter is not the generic point, it has codimension at least 1 and so its cohomological dimension is at most n − 1, which shows that α|f (τ (x)) = 0 and so (τ ∗ f ∗ α)|x = 0 as we want. • if f (τ (x)) is contained in the ramification locus of α, then the intuition is as follows: first note that τ ∗ f ∗ α is unramified over k by Proposition 4.7, because f ∗ α is unramified and τ is generically finite. On the other hand, α is by assumption ramified locally around the point f (τ (x)) and so the most natural reason for the fact that τ ∗ f ∗ α is unramified over k would be that this class in fact vanishes ´etale locally around x, so that it can be extended trivially across x. In [Sch19a, Sch18, Sch19b, Sch21a] exactly this phenomenon is observed for all the examples mentioned in Sections 9.1, 9.2 and 9.3. In the case where the generic fibre of f is a smooth quadric, the following general vanishing result, which gives some evidence for the above intuition and which makes it very easy to apply Theorem 8.6 in many situations, is proven in [Sch19b]. Theorem 10.1. Let f : Y → S be a surjective morphism of proper varieties over an algebraically closed field k with char(k) = 2 whose generic fibre is birational to a smooth quadric over k(S). Assume that there is a class α ∈ H n (k(S), μ2⊗n ) with n (k(Y )/k, μ⊗n f ∗ α ∈ Hnr 2 ), where n = dim(S). Then for any dominant generically finite morphism τ : Y  → Y of varieties with Y  smooth over k and for any (scheme) point x ∈ Y  which does not map to the generic point of S via f ◦ τ , we have (τ ∗ f ∗ α)|x = 0 ∈ H n (κ(x), μ2⊗n ). Combining these vanishing results with the examples in Sections 9.1, 9.2 and 9.3, one deduces for instance the following from the specialization result in Theorem 8.6:

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3 • A very general hypersurface X ⊂ P2C × PC of bidegree (2, 2) is not stably 3 2 rational [HPT18]. Since the smooth bidegree (2, 2) hypersurfaces in PC × PC that are known to be rational can be shown to be dense in moduli, this showed in particular that rationality is not an open nor a closed condition in smooth projective families, see [HPT18]. • Generalizations of the aforementioned result to all standard quadric surface bundles over P2 , different from Verra fourfolds and cubic fourfolds containing a plane (see [Sch18, Pau18]), as well as to quadric bundles of arbitrary fibre dimension, see [Sch19a]. +1 • A very general hypersurface X ⊂ PN of dimension N ≥ 3 and degree d ≥ k log2 N + 2 (resp. d ≥ log2 N + 3) is not stably rational if k is an uncountable field of characteristic different from 2 (resp. equal to 2), see [Sch19b, Sch21a]. ar [Kol95] and Totaro [Tot16] that were This improved earlier bounds of Koll´ linear (roughly d ≥ 32 N ). The results in [Sch19b, Sch21a] involve a degeneration of a very general hypersurface to a hypersurface Z that has high multiplicity along a largedimensional linear subspace P . One then needs to pass to the blow-up Y = BlP Z and one exploits that projection from P induces a fibration structure f : Y → Pn where n = dim X − dim P . While one can then use results `a la Theorem 10.1 to get the desired vanishing condition (8.1) from Theorem 8.6 at points that do not dominate Pn via f , this vanishing condition needs to be checked by hand on the exceptional divisor of the blowup BlP Z → Z, which makes the argument somewhat subtle.

Remark 10.2. In dimension N = 5, the logarithmic bound from [Sch19b] shows that very general hypersurfaces of degree at least five in P6k are not stably rational. In the case where k has characteristic 0, this result was improved by Nicaise and Ottem [NO19], who showed that very general quartic fivefolds are stably irrational over fields of characteristic 0. Their result is achieved by an application of Theorem 8.10, and it relies eventually on the stable irrationality of the quadric bundles of Hassett–Pirutka–Tschinkel from Section 9.2, whose discriminant locus has smaller degree than those of the corresponding generalizations in Theorem 9.7 that have been used in (arbitrary dimension) in [Sch19b].

11. Open problems 11.1. Decompositions of the diagonal versus stable rationality Recall from Lemma 7.5 that a variety that is stably rational admits a decomposition of the diagonal. It is natural to wonder whether the converse to this statement holds as well. It is known that a smooth complex projective surface X with CH0 (X)  Z and without torsion in H 2 (X, Z) admits a decomposition of the diagonal, see [BS83, p. 1252, Remark (2)], [ACTP13, Corollary 1.10], [Voi17, Corollary 2.2] or [Kah17]. There are such surfaces that are of general type and so existence of a decomposition of the diagonal is in general not equivalent to stable

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rationality. However, no such counterexample is known if we restrict to the class of rationally connected varieties. Question 11.1. Is there a rationally connected smooth complex projective variety which admits a decomposition of the diagonal, but which is not stably rational? The above question is already open for the Fermat cubic threefold, which admits a decomposition of the diagonal by a result of Colliot–Th´el`ene [CT17], while stable irrationality is unknown. In fact, while any smooth cubic threefold 4 in PC is irrational by a celebrated result of Clemens and Griffiths [CG72], the question whether it is also stably irrational is open for any such cubic; cf. [Voi17] for an interesting connection of this question to the integral Hodge conjecture on abelian varieties. A natural approach to Question 11.1 would be given by the obstruction for stable rationality in [NS19, KT19] (see Theorem 8.10), which might a priori not be sensitive to decompositions of the diagonal. A related classical open question pointed out by one of the referees is as follows. Question 11.2. Is there a variety X over an algebraically closed field k which is not stably rational but such that X ×k Y is rational for some variety Y over k? 11.2. Torsion orders and unirationality Let X be a rationally chain connected projective variety over a field k. Then its torsion order Tor(X) is finite, see Section 7.2. The class of rationally chain connected varieties is closed under several natural operations (e.g., taking products and taking quotients) and so it is natural to investigate how the torsion orders behave when performing these operations. Lemma 11.3. Let X and Y be proper varieties over a field k whose torsion orders are finite. Then Tor(X × Y ) | Tor(X) Tor(Y ). Proof. The diagonal of X × Y corresponds to the product of the diagonals of X and Y . Since Tor(X)ΔX and Tor(Y )ΔY admit decompositions as in (7.1), we conclude that Tor(X) Tor(Y )ΔX×Y admits a similar decomposition and so Tor(X × Y ) | Tor(X) Tor(Y ), as we want.  Lemma 11.4. Let X be a smooth projective variety over a field k whose torsion order is finite. For a positive integer n, we consider the symmetric product S n X = X n / Sym(n) of X. Then Tor(S n X) | n! · Tor(X)n

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Proof. By Lemma 7.11, applied to the quotient map X n → S n X, we have Tor(S n X) | n! · Tor(X n ) and so the claim follows from the previous lemma, which implies Tor(X n ) | Tor(X)n .  Question 11.5. Let X be a smooth complex projective variety with finite torsion order. Is it possible to improve the estimate from Lemma 11.4 for the torsion order of S n X? More precisely, we may ask the following. Question 11.6. Is it true that for any smooth complex projective variety X with finite torsion order and for any positive integer n, we have Tor(S n X) | Tor(X)n (or maybe even Tor(S n X) | Tor(X))? In the opposite direction, it is natural to wonder about the following. Question 11.7. Is there a smooth complex projective variety X that is rationally (chain) connected such that the prime factors of Tor(S n X) are unbounded for n → ∞? A positive answer to Question 11.7 would, by the following proposition, imply the existence of a rationally connected3 smooth complex projective variety that is not unirational, which is a longstanding open problem in the field. Proposition 11.8. Let X be a variety over a field k and assume that there is a dominant rational map f : Pkdim X  X. Then for all n ≥ 1, we have Tor(S n X) | deg(f )n . In particular, the prime factors of Tor(S n X) are bounded for n → ∞. Proof. Taking the nth symmetric power of f , we obtain a dominant rational map X S n f : S n Pdim  S n X. k

The degree of this map may be identified to deg S n f = deg(f )n . Hence, Lemma 7.11 implies Tor(S n X) | deg(f )n · Tor(S n Pdim X ). On the other hand, S n Pdim X is rational by an old result of Mattuck [Mat68]. Hence, Tor(S n Pdim X ) = 1 by Lemma 7.5 and so the proposition follows.  Acknowledgement Thanks to the organizers of the Schiermonnikoog conference on Rationality of Algebraic Varieties in spring 2019 for inviting me to write this survey. Parts of this survey rely on lecture series that I have given in Moscow and Nancy in spring 2019. The comments of two excellent referees significantly improved this text. 3 For smooth complex projective varieties, rationally chain connectedness and rationally connectedness coincide, see [Kol96].

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References [Ara75] J. Kr. Arason, Cohomologische Invarianten quadratischer Formen, J. Algebra 36 (1975), 448–491. [AM72] M. Artin and D. Mumford, Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3) 25 (1972), 75–95. [Aso13] A. Asok, Rationality problems and conjectures of Milnor and Bloch-Kato, Compos. Math. 149 (2013), 1312–1326. [ACTP13] A. Auel, J.-L. Colliot-Th´el`ene and R. Parimala, Universal unramified cohomology of cubic fourfolds containing a plane, in Brauer groups and obstruction problems: moduli spaces and arithmetic (Palo Alto, 2013), Progress in Mathematics, vol. 320, Birkh¨ auser Basel, 2017, 29–56. [Blo79] S. Bloch, On an argument of Mumford in the theory of algebraic cycles, Journ´ees de G´eom´etrie Alg´ebrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, pp. 217–221, Sijthoff & Noordhoff, Alphen aan den Rijn–Germantown, Md., 1980. [BO74] S. Bloch and A. Ogus, Gersten’s conjecture and the homology of schemes, Ann. ´ Norm. Sup´er., 7 (1974), 181–201. Sci. Ec. [BS83] S. Bloch and V. Srinivas, Remarks on correspondences and algebraic cycles, Amer. J. Math. 105 (1983), 1235–1253. [Bou72] N. Bourbaki, Commutative Algebra, Elements of Mathematics, Addison-Wesley Publishing Company, Reading, Massachusetts, 1972. [CG72] C.H. Clemens and P.A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. Math. 95 (1972), 281–356. [CL17] A. Chatzistamatiou and M. Levine, Torsion orders of complete intersections, Algebra & Number Theory 11 (2017), 1779–1835. [CT95] J.-L. Colliot-Th´el`ene, Birational invariants, purity and the Gersten conjecture, K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), 1–64, Proc. Sympos. Pure Math. 58, AMS, Providence, RI, 1995. [CT17] J.-L. Colliot-Th´el`ene, CH0 -trivialit´e universelle d’hypersurfaces cubiques presque diagonales, Algebraic Geometry 4 (5) (2017) 597–602. [CTO89] J.-L. Colliot-Th´el`ene and M. Ojanguren, Vari´et´es unirationnelles non rationnelles: au-del` a de l’exemple d’Artin et Mumford, Invent. Math. 97 (1989), 141–158. [CTP16a] J.-L. Colliot-Th´el`ene and A. Pirutka, Hypersurfaces quartiques de dimension 3: ´ Norm. Sup. 49 (2016), 371–397. non rationalit´e stable, Annales Sc. Ec. [CTV12] J.-L. Colliot-Th´el`ene and C. Voisin, Cohomologie non ramifi´ee et conjecture de Hodge enti`ere, Duke Math. J. 161 (2012), 735–801. ´ 83 [deJ96] A.J. de Jong, Smoothness, semi-stability and alterations, Publ. Math. IHES (1996), 51–93. [Dol72] A. Dold, Lectures on Algebraic Topology, Springer, Berlin, 1972 ´ ´ ements de g´eom´etrie alg´ebrique IV. Etude [EGAIV] J. Dieudonn´e and A. Grothendieck, El´ locale des sch´emas et des morphismes de sch´emas, Premi`ere partie, Publ. Math. ´ 20 (1964), 5–259. IHES

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[Pey93] E. Peyre, Unramified cohomology and rationality problems, Math. Ann. 296 (1993), 247–268. [Pfi65] A. Pfister, Multiplikative quadratische Formen, Arch. Math. 16 (1965), 363–370. [Pir18] A. Pirutka, Varieties that are not stably rational, zero-cycles and unramified cohomology, Algebraic Geometry (Salt Lake City, UT, 2015), Proc. Sympos. Pure Math. 97 459–484. Amer. Math. Soc., Providence, RI, 2018. [Ro96] M. Rost, Chow groups with coefficients, Doc. Math. 1 (1996), 319–393. [Sal84] D. Saltman, Noether’s problem over an algebraically closed field, Invent. Math.77 (1984), 71–84. [Sch18] S. Schreieder, Quadric surface bundles over surfaces and stable rationality, Algebra & Number Theory 12 (2018), 479–490. [Sch19a] S. Schreieder, On the rationality problem for quadric bundles, Duke Math. J. 168 (2019), 187–223. [Sch19b] S. Schreieder, Stably irrational hypersurfaces of small slopes, J. Amer. Math. Soc. 32 (2019), 1171–1199. [Sch19c] S. Schreieder, Variation of stable birational types in positive characteristic, ´ Epijournal de G´eom´etrie Alg´ebrique 3 (2019), Article Nr. 20. [Sch21a] S. Schreieder, Torsion order of Fano hypersurface, Algebra & Number Theory 15 (2021), 241–270. [Sch21b] S. Schreieder, Refined unramified homology of schemes, Preprint 2021, arXiv: 2010.05814v3. [Ser97] J.-P. Serre, Galois cohomology, Springer–Verlag, Berlin, 1997. [SGA4.2] M. Artin, A. Grothendieck, and J.-L. Verdier, Th´eorie des Topos et Cohomologie Etale des Sch´emas, SGA 4, Tome 2, Lecture Notes in Mathematics 270, Springer, Berlin 1972. [Shi19] E. Shinder, Variation of stable birational types of hypersurfaces, with an Appendix by C. Voisin, arXiv:1903.02111. [Tot16] B. Totaro, Hypersurfaces that are not stably rational, J. Amer. Math. Soc. 29 (2016), 883–891. [Voi15] C. Voisin, Unirational threefolds with no universal codimension 2 cycle, Invent. Math. 201 (2015), 207–237. [Voi17] C. Voisin, On the universal CH0 group of cubic hypersurfaces, J. Eur. Math. Soc. 19 (2017), 1619–1653. Stefan Schreieder Institute of Algebraic Geometry Leibniz University Hannover Welfengarten 1 D-30167 Hannover, Germany e-mail: [email protected]

Vanishing Cycles under Base Change and the Integral Hodge Conjecture Mingmin Shen Abstract. In this paper we discuss an obstruction to the integral Hodge conjecture which arises from a certain behavior of vanishing cycles. This allows us to construct new counter-examples to the integral Hodge conjecture. One typical such counter-example is the product of a very general hypersurface of odd dimension and an Enriques surface. Our approach generalizes the degeneration argument of Benoist–Ottem [2]. Mathematics Subject Classification (2010). 14C25. Keywords. Integral Hodge conjecture, vanishing cycles.

1. Introduction In this paper we work over the field C of complex numbers. If X is a smooth projective variety, then the cohomology group of X carries a Hodge structure given by  Hk (X, Z) ⊗ C = Hp,q (X), Hp,q (X) = Hq,p (X). p+q=k

The group of integral Hodge classes, denoted Hdg2p (X, Z), consists of all elements α ∈ H2p (X, Z) such that α ⊗ 1 ∈ H2p (X, Z) ⊗ C is in the summand Hp,p (X). One easily sees that the torsion classes are all integral Hodge classes, i.e., H2p (X, Z)tor ⊆ Hdg2p (X, Z). W. Hodge discovered that the cohomology class [Z] of an algebraic cycle Z on X is always an integral Hodge class. Conjecture 1.1 (Integral Hodge Conjecture). Every integral Hodge class is the cohomology class of an algebraic cycle. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Farkas et al. (eds.), Rationality of Varieties, Progress in Mathematics 342, https://doi.org/10.1007/978-3-030-75421-1_14

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It is known by Atiyah–Hirzebruch [1] that the integral Hodge conjecture is false. Since that paper theory and techniques were developed to construct more counter-examples. Recent developments show that the integral Hodge conjecture is also closely related to some open questions of rationality. For example, Voisin [9] showed that the stable rationality of a cubic threefold is related to the algebraicity of the minimal curve class on its intermediate Jacobian. Similarly, Shen [6] relates the stable rationality of a cubic fourfold to the algebraicity of some surface class that represents the Beauville–Bogomolov form on the variety of lines. In the recent paper [2], Benoist and Ottem used a degeneration argument to show that certain integral Hodge classes are not algebraic. In this paper we generalize their method to produce more counter-examples. Our method is based on the following simple observation. Let Y be a smooth projective variety and U ⊆ Y a dense open subvariety. If ZU is an algebraic cycle on U , then it extends to an algebraic cycle Z on Y by taking the closure. However, a (locally finite) topological cycle zU on U does not necessarily extend to one on Y . The main reason is that the closure of zU might have a nontrivial boundary. Hence a cohomology class being algebraic imposes a strong extension property on the class. We make the following definition to make the discussion easier. Definition 1.2. Let π : X → B be a dominant projective morphism between smooth complete varieties. Let 0 ∈ B be a closed point such that X = X0 := π −1 (0) is a smooth fiber. Let α ∈ Hk (X, R) be a cohomology class with coefficients in a commutative ring R. We say that α is extendable if ˜ a generically finite morphism B ˜→ • there exists a smooth complete variety B, ˜ of 0 ∈ B, B and a preimage 0˜ ∈ B such that ˜ there exists a cohomology class • for all resolutions X˜ of X  := X ×B B, k ˜ α ˜ ∈ H (X , R) such that α = α| ˜ X˜˜ under the identification X = X˜0˜ := π 0) ˜ −1 (˜ 0 ˜ ˜ where π ˜ : X → B is the morphism induced by π. In this paper we require a resolution to be an isomorphism on the smooth locus. Proposition 1.3. Let π : X −→ B be a dominant projective morphism between smooth complete varieties and X = π −1 (0) for some very general point 0 ∈ B. If a cohomology class α ∈ H2p (X, Z) is algebraic, then it is extendable. Proof. Let ηB be the generic point of B and η¯ its geometric generic point. By assumption, 0 ∈ B is a very general point. Let α ∈ H2p (X, Z) be algebraic, namely the class of an algebraic cycle Z. One can identify X = π −1 (0) with the geometric generic fiber Xη¯ . The algebraic cycle Z, viewed as an algebraic cycle on Xη¯ , can then be defined over a finite extension of ηB . Then a standard argument shows ˜ → B and an algebraic cycle that there exists some generically finite morphism B ˜ is a point above 0. The ˜ such that [Z  ] = α, where ˜0 ∈ B Z  on X  := X ×B B ˜ 0  ˜ ˜ ˜ strict transform Z of Z in the resolution X satisfies [Z˜0 ] = α. Thus we can simply ˜ take α ˜ = [Z]. 

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This proposition gives rise to a non-algebraicity criterion as follows. If an integral Hodge class α is not extendable, then this class is not algebraic on a very general fiber. Our first main result is the following non-extendability of vanishing cycles on odd-dimensional smooth hypersurfaces. Theorem 1.4 (Theorem 3.2). Let X −→ B = P1 be a Lefschetz pencil of hypersurfaces in Pn+1 where n is odd. Let X = π −1 (0) be a smooth fiber. Then every non-zero element α ∈ Hn (X, R) is non-extendable, where R is a nonzero commutative ring. Remark 1.5. The global invariant cycle theorem of Deligne [4] implies that, with Qcoefficients, a cohomology class is extendable if and only it has a finite orbit under the monodromy action. Indeed, such a class becomes invariant after a finite base change. Then the invariant cycle theorem provides the required global cohomology class on the total space which restricts to the given class on the fiber. Voisin [8, Theorem 4.1] proved that a rational Hodge class has finite monodromy orbit and hence is always extendable. When the coefficient ring R is a finite field, any cohomology class would have a finite monodromy orbit. The above theorem shows that the global invariant cycle theorem becomes false if we replace the coefficients Q by a finite field. This non-extendability can be used to obstruct algebraicity as follows. For simplicity, we take S to be an Enriques surface. Then H3 (S, Z) = Z/2Z with a generator u. Corollary 1.6 (Corollary 3.3). Let X ⊂ Pn+1 be a very general hypersurface of odd dimension n. For every element α ∈ Hn (X, Z) which is not divisible by 2, the torsion class α ⊗ u ∈ Hn+3 (X × S, Z) is not extendable (in a Lefschetz pencil) and hence not algebraic. Remark 1.7. The proof of the corollary reduces to the non-extendability of the image α ¯ of α in Hn (X, Z/2Z); see Section 3. In [2], Benoist and Ottem considered the case where X = E is a very general elliptic curve. Their method involves an element α ∈ H1 (E, Z). Instead of considering the topological extendability of ¯ ∈ H1 (E, Z/2Z), they consider the degeneration of the double cover E  −→ E α ¯ . The obstruction in the Benoist–Ottem example was given an associated to α interpretation via unramified cohomology by Colliot-Th´el`ene [3]. It is interesting to see if a similar interpretation exists for our generalisation. The counter-examples to the integral Hodge conjecture obtained via the above corollary are all around the range of middle degree cohomology. Our method also works when X is a hyperplane section of a smooth projective variety Y . This more general case is treated in Theorem 3.4.

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2. Vanishing cycles under blow-up 2.1. An induction process Let r ≥ 2 be an integer and let Xr be a complex analytic space with a unique singular point Pr . Assume that Pr has an open neighborhood Ur which is identified as an open subset of {(t, z) ∈ C × Cn+1 : tr = z02 + · · · + zn2 } such that the point Pr corresponds to the point P = (0, 0). Here we take the following convention: we use Pr to denote the singular point when we view Ur as an open subset of Xr ; we use P to denote the singular point when we view Ur as a subset of C × Cn+1 . Let  be a sufficiently small positive real number. Let Dn+1 := {x ∈ Rn+1 : |x| ≤ } be the closed disc of radius . We have continuous maps ϕr,a : Dn+1 −→ Ur ,

ϕr,a (x) = (ξra |x|2/r , x),

(1)

 where ξr = exp( 2πi r ) and a = 0, 1, . . . , r − 1. Let M be the blow-up of M = n+1   at the point P = (0, 0). Let Ur ⊂ M be the strict transform of Ur and C×C ρ : Xr → Xr be the resulting blow-up of Xr at the point Pr . We write

M \{(0, 0)} = V ∪ V0 ∪ · · · ∪ Vn ,

V = {t = 0}, Vi = {zi = 0}.



Then M admits a corresponding open cover M  = V  ∪ V0 ∪ · · · ∪ Vn . Here V  ∼ = C × Cn+1 and the map V  → V ∪ {(0, 0)} is given by Similarly, we have

Vi

(t, w0 , . . . , wn ) → (t, tw0 , . . . , twn ). ∼ = C × Cn+1 and the map Vi → Vi ∪ {(0, 0)} is given by

(t, w0 , . . . , wn ) → (twi , w0 wi , . . . , wi−1 wi , wi , wi+1 wi , . . . , wn wi ). The exceptional divisor E of the blow-up M  → M is isomorphic to Pn+1 and the open cover n 7 (E ∩ Vi ) E = (E ∩ V  ) ∪ i=0

is the standard affine cover associated to the homogeneous coordinates [T : Z0 : · · · : Zn ] of Pn+1 . We have the following commutative diagram Ur

/ M

ρ

 Ur

 /M

Furthermore, Ur ∩ V  is an open subset of the locus in V  defined by the equation tr−2 = w02 + · · · + wn2 .

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Thus Ur ∩ V  is smooth if r = 2, 3; it is singular at the point (t, w) = (0, 0) if r ≥ 4. The intersection Ur ∩ Vi is defined by the equation 2 2 tr wir−2 = w02 + · · · + wi−1 + · · · + wn2 , + 1 + wi+1

which is always smooth. If r = 2, then the exceptional divisor of Ur → Ur is the smooth quadric Q = {T 2 = Z02 + · · · + Zn2 } ⊂ E = Pn+1 . If r ≥ 3, then the exceptional divisor of Ur → Ur is the singular quadric Q = {0 = Z02 + · · · + Zn2 } ⊂ E = Pn+1 . The singular point of Q is Pr := [1 : 0 : · · · : 0] ∈ Pn+1 . The map ϕr,a , restricted to Dn+1 \{0}, lifts to V  , which is given by x = (x0 , . . . , xn ) → (ξra |x|2/r , ξr−a |x|1−2/r θ(x)), where θ(x) =

x |x|

x ∈ Dn+1 \{0},

∈ S n . If r ≥ 3, then the above map extends to ϕr,a : Dn+1 −→ V 

by the same formula and 0 → (0, 0) ∈ V  , which is the singular point of Q . In this case, Xr is locally defined by the equation tr−2 = w02 + · · · + wn2 . The following lemma implies that the same argument can be repeated on (Xr−2 , Pr−2 ) = (Xr , Pr ). Lemma 2.1. (1) If r = 2, then the lifting of ϕ2,a |D n+1 \{0} to Xr can be extended to a continuous map ϕ2,a : [0, ] × S n −→ Xr such that ϕ2,a (ρ, x) = ϕ2,a (ρx) for all (ρ, x) ∈ (0, ] × S n . Furthermore, ϕ2,a (0, x) = [1 : (−1)a x0 : · · · : (−1)a xn ] ∈ Q, which parametrizes an n-sphere in Q ⊂ E ∼ = Pn+1 that vanishes in the n+1  homology of P . In this case, Xr is smooth. (2) If r = 3, then the lifting of ϕr,a |D n+1 \{0} to Xr can be extended to a continuous map ϕr,a : Dn+1 −→ Xr such that ϕr,a (0) = Pr is the singular point of Q . In this case, Xr is smooth. (3) If r ≥ 4, then Xr is singular at the point Pr where Xr is locally defined by an equation r−2 2 2 2 = z0 + z1 + · · · + zn . t where the change of coordinates is given by t = ξr−a t, zi = ξra wi .

394

M. Shen The lifting of ϕr,a |D n+1 \{0} to Xr can be extended to the composition of a homeomorphism x → |x|−2/r x,

Dn+1 −→ Dn+1 ,  and a continuous map

ϕr−2,0 : Dn+1 −→ Xr ,  with ϕr−2,0 (0) = Pr being the singular point of Xr and 2

ϕr−2,0 (x) = (|x| r−2 , x) under the new coordinates, for nonzero x. Proof. For (1), we note that, in this case, the lifting of ϕ2,a restricted to Dn+1 \{0} is given by x = (x0 , . . . , xn ) → ((−1)a |x|, (−1)a θ(x)). Note that (0, ] × S n ∼ = Dn+1 \{0}. It is clear that the above map extends to a  continuous map ϕ2,a : [0, ] × S n −→ Xr as stated. Statement (2) has been explained above. We show the last statement and assume that r ≥ 4. We have already seen that Xr has a unique singular point Pr such that Xr is locally defined by tr−2 = w02 + w12 + · · · + wn2 and that there is a lifting ϕr,a : Dn+1 −→ Xr of ϕr,a given by ϕr,a (x) = (ξra |x|2/r , ξr−a |x|1−2/r θ(x)),

x ∈ Dn+1 \{0}

and ϕr,a (0) = (0, 0) = Pr . With the new coordinates t = ξr−a t, zi = ξra wi , we see that the local defining equation of Xr around Pr becomes t

r−2

= z0 + z1 + · · · + zn . 2

2

2

Furthermore, in terms of the coordinates (t , z ), the map ϕr,a becomes x → (|x|2/r , |x|1−2/r θ(x)) for x = 0 and 0 →  (0, 0). Let  = 1−2/r and define a homeomorphism μ : n+1 n+1 D → D by x →  x = |x|−2/r x for x = 0 and 0 → x = 0. It follows that the composition Dn+1 

μ−1

/ Dn+1

ϕr,a

/ Xr

becomes 2

ϕr−2,0 : x → (t , z ) = (|x | r−2 , x ). In other words, ϕr,a = ϕr−2,0 ◦ μ. This concludes the proof.



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2.2. Applications to vanishing cycles 2.2.1. Local situation. Let Δ ⊂ C be the unit open disc in the complex plane and Δ∗ = Δ\{0}. Let π : X → Δ be a proper map of complex manifolds such that X ∗ → Δ∗ is smooth, where X ∗ = π −1 Δ∗ . We write Xt := π −1 (t), t ∈ Δ. Assume that X0 = π −1 (0) has one ordinary double point P such that we have local coordinates (z0 , . . . , zn ) on an open neighborhood U of P in X and π(z0 , z1 , . . . , zn ) = z02 + z12 + · · · + zn2 . Let ψr : Δ → Δ be the map ψr (t) = tr and Xr := ψr∗ X be the base change of X , where r ≥ 2. Namely, we have the following fiber product square Xr

ψr

πr

 Δ

/X π

ψr

 /Δ

Let Ur = ψr∗ U be the corresponding base change of U . Thus Ur is an open neigh−1 n+1 and it is borhood of the point Pr = ψr (P ) in X2r . Hence we have Ur ⊂ C × C r locally defined by the equation t = zi . In other words, Ur is an open subset of   (t, z0 , . . . , zn ) ∈ C × Cn+1 : tr = z02 + z12 + · · · + zn2 , and Pr is the unique singular point of Ur with coordinates (0, 0, . . . , 0). For any positive real number  ∈ (0, 1), let Sn = {(z0 , . . . , zn ) ∈ U : z02 + · · · + zn2 = 2 , zi ∈ R} ⊂ X2 be a vanishing sphere. Let Dn+1 → X ,

(x0 , . . . , xn ) → (z0 , . . . , zn ) = (x0 , . . . , xn ) ,

where = {(x0 , . . . , xn ) ∈ Rn+1 : x20 + · · · + x2n ≤ 2 } is a small disc whose boundary gives the vanishing sphere Sn . Let ρ : Xr → Xr be the blow-up of Xr at the point Pr . There are r different ways to lift the map Dn+1 \{0} → X ∗ to Xr∗ := πr−1 Δ∗ given by Dn+1

x = (x0 , . . . , xn ) → (ξra |x|2/r , x0 , . . . , xn ),

a = 0, 1, . . . , r − 1.

(2)

Xr

We have seen in Lemma 2.1 that is again singular if r ≥ 4 with a single singular point Pr and the blow-up process can be repeated. Lemma 2.2. The following statements are true. (1) The singularity of Xr can be resolved by successively blowing up the singular points ˜ = X (b) −→ · · · −→ X (2) −→ X (1) = X  −→ Xr X r r r r (b)

where b = [ r2 ] and Xr is smooth. ˜ be the exceptional divisor of the last blow-up Xr(b) −→ Xr(b−1) . (2) Let Q ⊆ X ˜ −→ Δ is the composition ˜ : X Then Q is a component of π ˜ −1 (0), where π

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of all the blow-ups together with πr . If r = 2b is even, then Q is a smooth quadric hypersurface of dimension n; if r = 2b + 1 is odd, then Q is a cone over a smooth quadric hypersurface of dimension n − 1. ˜ (3) If n is odd, then any of the r liftings of the vanishing sphere Sn ⊂ X2 to X ˜ vanishes in Hn (X, Z). (4) If n is even and r is odd, then any of the r liftings of the vanishing sphere ˜ vanishes in Hn (X, ˜ Z). Sn ⊂ X2 to X (5) If n is even and r is also even, then any of the r liftings of the vanishing ˜ is homologous to some sphere S n ⊂ Q. Furthermore, sphere Sn ⊂ X2 to X n the sphere S vanishes in Hn (Pn+1 , Z) under the embedding Q → Pn+1 of Q as a quadric hypersurface. Proof. By Lemma 2.1, the blow-up of a singularity of the form tr = z02 + · · · + zn2 will either resolve the singularity (if r ≤ 3) or turn the singularity into one of the form tr−2 = z02 + · · · + zn2 (if r ≥ 4). As a consequence, the singularity of Xr is resolved by doing the blow-up process b = [ 2r ] times. This proves (1). Statements (1) and (2) of Lemma 2.1 imply that Q ⊆ Pn+1 is a quadric; it is smooth if r is even and it is a cone over a smooth (n − 1)-dimensional quadric if r is odd. This establishes (2). Note that any lifting of the vanishing sphere Sn to Xr extends to a map from the disc of dimension n + 1 and this map is explicitly given by the formula (2). Hence the assumptions of Lemma 2.1 are satisfied. If r is odd, then by (2) of ˜ is the boundary of a disc in X ˜ Lemma 2.1 we know that any lifting of Sn in X and hence it vanishes in homology. This proves (4). Now assume that r is even. In this case, by (1) of Lemma 2.1, we know that ˜ is homologous to an n-sphere S n ⊂ Q. Thus the the lifting of Sn ⊂ X2 to X homology class of the lifting of Sn lands in the image of ˜ Z). Hn (Q, Z) −→ Hn (X, When n is odd, we have Hn (Q, Z) = 0 since a smooth quadric has trivial homology group in odd degree. Thus we obtain the vanishing in (3). If n is even and r is also even, then we have (5) as a consequence of statement (1) of Lemma 2.1.  2.2.2. Global situation. Let X be a smooth algebraic variety of dimension n + 1 and B a smooth curve. Let π : X −→ B be a proper morphism. For a point b ∈ B, we use Xb = π −1 (b) to denote the fiber. Assume that the following conditions hold. • There exists a set T = {b1 , b2 , . . . , bm } ⊂ B of finitely many points such that Xbi = π −1 (bi ) contains exactly one isolated singular point Pi which is an ordinary double point. • The morphism π is smooth over B\T . Let 0 ∈ B be a point not in T and let X = X0 . Thus X is a smooth complete variety over C. Let Δi ⊂ B be a small disc centered at bi . Let Δ∗i := Δi \{bi }.

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Definition 2.3. Let  be a sufficiently small positive real number. For each bi ∈ T , a sphere Sn ⊂ Xti , ti ∈ Δi∗ , is called a vanishing sphere associated to Pi if the following conditions hold: (1) there exist local coordinates z = (z0 , z1 , . . . , zn ) of X at Pi = (0, 0, . . . , 0); (2) there is a local coordinate t on Δi such that π is locally given by t = π(z) = z02 + z12 + · · · + zn2 ; (3) with the above coordinates, we have ti = 2 and Sn is given by all points xi2 = 2 . z = (x0 , x1 , . . . , xn ) with xi ∈ R and Let γ : [0, 1] −→ B\T be a continuous path such that γ(0) = 0 and γ(1) = ti ∈ Δ∗i for some 1 ≤ i ≤ m. The monodromy action along γ is an isomorphism ∼

γ∗ : Hn (X, Z) −→ Hn (Xti , Z). Definition 2.4. We say that a class α ∈ Hn (X, Z) is a primitive vanishing class if there exists a path γ : [0, 1] −→ B\T as above such that γ∗ α ∈ Hn (Xti , Z) is the class of a vanishing sphere associated to Pi for some i. A class α ∈ Hn (X, Z) is a vanishing class (associated to X /B) if it is an integral linear combination of primitive vanishing classes. Proposition 2.5. Let X be a smooth algebraic variety and B a smooth curve. Let ˜ be π : X −→ B be a proper morphism as above. Let 0 ∈ B\T and X = X0 . Let B  ˜ ˜ another smooth curve and let f : B → B be a finite morphism. Let X := X ×B B  ˜ ˜ ˜ be the base change of X and let X be a resolution of X . Let π ˜ : X −→ B be the ˜ such that f (0) ˜ =0 resulting morphism induced from the morphism π. Let 0˜ ∈ B, and hence X ∼ ˜ −1 (˜0). Let j : X → X˜ be the embedding. Let α ∈ Hn (X, Z) be a =π vanishing class associated to X /B. (1) If n is odd, then j∗ α = 0 in Hn (X˜ , Z). (2) If n is even and X˜ is obtained by successively blowing up the singular points, then j∗ α is in the image of N 

Hn (Ql , Z)van −→ Hn (X˜ , Z),

l=1

where Ql runs through all smooth quadric hypersufaces appearing as components of the exceptional set of the morphism X˜ −→ X  and Hn (Ql , Z)van consists of classes β ∈ Hn (Ql , Z) that vanish in Hn (Pn+1 , Z) under the natural embedding Ql ⊂ Pn+1 . ˜ −→ B around Proof. We first look at the local behaviour of the morphism f : B a point bi ∈ T . Assume that ˜ f −1 bi = {b , b , . . . , b } ⊂ B. i,1

i,2

i,mi

 ˜ centered at b such that the For each point bi,l , we can find a small disc Δi,l ⊂ B i,l ˜ morphism f : B −→ B restricts to the analytic map

fi,l : Δi,l −→ Δi ,

z → z ri,l .

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To prove the proposition, we first assume that the resolution X˜ −→ X  is the one obtained by successively blowing up the singular points. Without loss of generality, we may assume that α is a primitive vanishing class. Thus there is a path γ : [0, 1] −→ B\T with γ(0) = 0 ∈ B and γ(1) = ti ∈ Δ∗i such that γ∗ α is the class of a vanishing sphere in Xti . We may choose γ in such a way that its interior avoids all ˜ the branching points of the morphism f . Thus there exists a lifting γ˜ : [0, 1] −→ B ∗ ˜ ˜ such that γ˜ (0) = 0. Then γ˜(1) = ti ∈ Δi,l for some l ∈ {1, 2, . . . , mi }. Furthermore, γ˜∗ α is the class of a lifting S˜n of the vanishing sphere Sn in Xti . If n is odd, then by π −1 Δi,l , Z) Lemma 2.2 (3) we know that the homology class of S˜n vanishes in Hn (˜ ˜ and hence also in Hn (X , Z). Similarly, if n is even, we conclude from Lemma 2.2 (4) and (5). Now assume n is odd. We still need to establish the vanishing on an arbitrary resolution X˜1 of X  . Let X˜ be the resolution of X  obtained by successively blowing up the singular points. Then we can find a third resolution X˜2 which dominates both X˜ and X˜1 , namely we have a diagram

X˜

X˜2  @@@   @@τ τ  @@  @  

X˜1

Let j1 : X → X˜1 , j2 : X → X˜2 and j : X → X˜ be the inclusions of the fiber ˜ in the corresponding models. Set α1 = j1,∗ α, α2 = j2,∗ α and α over 0˜ ∈ B ˜ = j∗ α to be the corresponding homology classes. We have already see that α ˜ = 0. Since these models are isomorphic on an open neighborhood of the fiber X. We have α2 = τ ∗ α  ˜ = 0 and α1 = τ∗ α2 = 0.

3. Applications to the integral Hodge conjecture In this section, we construct a class of new examples of the failure of the integral Hodge conjecture. These generalise the examples of Benoist–Ottem [2]. Let S be an Enriques surface. The cohomology groups of S are described as follows. H0 (S, Z) = Z,

H0 (S, Z/2Z) = Z/2Z,

H1 (S, Z) = 0,

H1 (S, Z/2Z) = Z/2Z,

H2 (S, Z) = Z⊕10 ⊕ Z/2Z,

H2 (S, Z/2Z) = (Z/2Z)⊕12 ,

H3 (S, Z) = Z/2Z,

H3 (S, Z/2Z) = Z/2Z,

H4 (S, Z) = Z,

H4 (S, Z/2Z) = Z/2Z.

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Vanishing Cycles under Base Change

3.1. Special case: hypersurfaces Let X ⊂ Pn+1 be a smooth hypersurface. Assume that the dimension n of X is odd. By the Lefschetz Hyperplane Theorem, we know that Hp (Pn+1 , Z) −→ Hp (X, Z) is an isomorphism for p < n. Thus Hn (X, Z). Then by the universal coefficient theorem for cohomology, we see that ∼ HomZ (Hn (X, Z), Z) ⊕ Hn−1 (X, Z)tor Hn (X, Z) = is also torsion free. Hence we conclude that both H∗ (X, Z) and H∗ (X, Z) are torsion free. Lemma 3.1. The following equality holds n+3

H

(X × S, Z) =

4 

Hn+3−i (X, Z) ⊗ Hi (S, Z).

i=0

unneth formula applied to this case gives Proof. The K¨ ' 4 (  i n+3−i n+3 H (X, Z) ⊗ H (S, Z) (X × S, Z) = H i=0

' ⊕

4 



n+4−i

Tor1 H

(X, Z), H (S, Z)

(

i

i=0

Since the cohomology of X is torsion free, we see that the Tor1 -terms vanish.



Theorem 3.2. Let π : X −→ B = P1 be a Lefschetz pencil of smooth hypersurfaces of odd dimension n. Let X = π −1 (0) be a smooth fiber. Then any non-zero element α ∈ Hn (X, R) is non-extendable, where R is a non-zero commutative ring. Proof. Assume that α is extendable. Then there exists a smooth projective curve ˜ and a finite morphism B ˜ −→ B such that a resolution X˜ of the base change B  ˜ X = X ×B B is obtained by successively blowing up the singular points. Let ˜ be the induced morphism. Furthermore, we have a cohomology class ˜ : X˜ −→ B π n ˜ ˜ of ˜ for some preimage ˜0 ∈ B α ˜ ∈ H (X , R) such that α| ˜ X = α, where X = π ˜ −1 (0) ˜ 0. Let j : X → X be the inclusion. Let β ∈ Hn (X, Z). Since n is odd, we know that β vanishes in Hn (Pn+1 , Z). By Lefschetz theory (see for example [5, §5] or [7, Theorem 2.1]), we know that β is a vanishing class associated to the Lefschetz pencil X −→ B. Then by (1) of Proposition 2.5, we see that j∗ β = 0 in Hn (X˜ , Z). Thus α, β = j ∗ α, ˜ β = ˜ α, j∗ β = 0. This forces that α = 0 since Hn (X, R) = HomZ (Hn (X, Z), R).



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Corollary 3.3. Let i : X → Pn+1 be a smooth hypersurface of odd dimension n and let S be an Enriques surface. Let α ∈ Hn (X, Z) be an element not divisible by 2 and let u ∈ H3 (S, Z) be the unique nonzero element. Then the torsion class α ⊗ u ∈ Hn+3 (X × S, Z) is non-extendable in X × S −→ B for any Lefschetz pencil X −→ B containing X. If X is very general, then α ⊗ u is not algebraic. Proof. Let π : X −→ B = P1 be a Lefschetz pencil of hypersurfaces of dimension n such that for some point 0 ∈ B the corresponding fiber X0 := π −1 (0) ∼ = X. Assume that α ⊗ u is extendable. As in the above proof, there exist a smooth ˜ a finite morphism B ˜ → B, a resolution X˜ of X  := X ×B projective curve B, −1 ˜ an identification X = π ˜ and a cohomology class τ ∈ Hn+3 (X˜ × S, Z) ˜ (0) B, such that τ |X×S = α ⊗ u. Consider the class τ¯ ∈ Hn+3 (X˜ × S, Z/2Z) which is obtained from τ modulo 2. Let u ∈ H1 (S, Z/2Z) be the unique non-zero element which is associated to the K3 covering S˜ −→ S. We view τ¯ as a cohomological correspondence between S and X˜ . Then we get α ˜ := τ¯∗ u ∈ Hn (X˜ , Z/2Z) which satisfies the following condition ¯ τ ∗ u )|X = (¯ τ |X×S )∗ u = (¯ α⊗u ¯)∗ u = α ˜ |X = (¯ α ¯ is the image of u in H3 (S, Z/2Z). where α ¯ is the image of α in Hn (X, Z/2Z) and u The last equality uses the duality relation ¯ u, u  = 1. n It follows that α ¯ ∈ H (X, Z/2Z) is extendable. By the above theorem, we have α ¯ = 0 and hence α is divisible by 2 in Hn (X, Z). This gives a contradiction. It follows from Proposition 1.3 that α ⊗ u is not algebraic for a very general member X in a Lefschetz pencil. In particular, this holds for a very general X.  3.2. General case: hyperplane sections Let Y be a smooth projective variety with a very ample line bundle OY (1) which gives rise to an embedding Y → PN . The same argument as above gives the following. Theorem 3.4. Let π : X → B = P1 be a Lefschetz pencil in |OY (1)|. Let X = π −1 (0) be a smooth fiber and let i : X → Y be the embedding. Assume that dim Y = n + 1 where n is an odd integer. Let R be a nonzero commutative ring. (1) If α ∈ Hn (X, R) is extendable, then we have α, β = 0, for all β ∈ Hn (X, Z)van := ker{i∗ : Hn (X, Z) −→ Hn (Y, Z)}. Furthermore, if Hn (Y, Z) vanishes and Hn−1 (Y, Z) is torsion-free, then every nonzero element α ∈ Hn (X, R) is non-extendable. (2) Let S be an Enriques surface and u ∈ H3 (S, Z) be the unique nonzero element. Let α ∈ Hn (X, Z). If α ⊗ u, viewed as an element in Hn+3 (X × S, Z), is ¯ ∈ Hn (X, Z/2Z) is extendable. extendable (in the family X × S → B), then α

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(3) Assume that Hn (Y, Z) vanishes and that Hn−1 (Y, Z) is torsion-free. If X is very general in |OY (1)|, then for all α ∈ Hn (X, Z) not divisible by 2, the class α ⊗ u ∈ Hn+3 (X × S, Z) is not algebraic. ˜ X  , X˜ , j : X = π ˜ → X˜ as in the previous Proof. We will use the notations B, ˜ −1 (0) proofs. (1) If α is extendable, then there exists α ˜ ∈ Hn (X˜ , R) such that α = j ∗ α. ˜ Then we again have α, β = j ∗ α, ˜ β = α, j∗ β = 0. Lefschetz theory (see [7, Theorem 2.1]) says that Hn (X, Z)van consists of all vanishing classes associated to the Lefschetz pencil X → B. Hence we get j∗ β = 0 for all β ∈ Hn (X, Z)van by (1) of Proposition 2.5. This is the reason why we have the last vanishing in the above equation. Assume that Hn (Y, Z) = 0, then we have Hn (X, Z)van = Hn (X, Z). If Hn−1 (Y, Z) is torsion free, then by Lefschetz hyperplane theorem, we know that Hn−1 (X, Z) is also torsion free. Then the universal coefficient theorem for cohomology becomes Hn (X, R) = HomZ (Hn (X, Z), R). Thus the vanishing of α, β = 0 for all β ∈ Hn (X, Z) implies α = 0 in Hn (X, R). (2) and (3): the proof is the same as that of Corollary 3.3. One only needs to note that under the assumptions of (3), the group Hn+1 (X, Z) is also torsion free by Poincar´e duality. Thus the universal coefficient theorem implies Hn (X, Z/2Z) = Hn (X, Z) ⊗ Z/2Z. Thus α ¯ = 0 in Hn (X, Z/2Z) if and only if α ⊗ u = 0 in Hn+3 (X × S, Z) since Hn (X, Z) ⊗ H3 (S, Z) → Hn+3 (X × S, Z) by K¨ unneth formula and H3 (S, Z) = Z/2Z. Then (3) follows from (1) and (2).



Acknowledgment A large part of the computations in Section 2 were carried out in the summer of 2018 when I was visiting University of Science and Technology of China. I thank Mao Sheng for the invitation. I also thank John Ottem for the interesting discussions related to this paper. I thank the referee for many helpful suggestions for improvement. This research was partially supported by NWO Innovational Research Incentives Scheme 016.Vidi.189.015.

References [1] M. Atiyah and F. Hirzebruch, Analytic cycles on complex manifolds, Topology 1 (1962), p. 25–45. [2] Benoist and J. Ottem, Failure of the integral Hodge conjecture for threefolds of Kodaira dimension zero, Commentarii Mathematici Helvetici, 95 (2020), 27–35.

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[3] J.-L. Colliot-Th´el`ene, Cohomologie non ramifi´ee dans le produit avec une courbe elliptique, Manuscripta Mathematica 160 (2019), 561 – 565. [4] P. Deligne, Th´eorie de Hodge, II, Publ. IHES 40 (1971), 5–57. [5] K. Lamotke, The topology of complex projective varieties after S. Lefschetz, Topology 50 (1981), p. 15–51. [6] M. Shen, Rationality, universal generation and the integral Hodge conjecture, Geometry & Topology, 23 (2019), 2861 – 2898. [7] C. Voisin, Hodge theory and complex algebraic geometry, II, Cambridge University Press, 2003. [8] C. Voisin, Hodge loci, in “Handbook of moduli” (Eds G. Farkas and I. Morrison), Advanced Lectures in Mathematics 25, Volume III, International Press, 507–547 (2013). [9] C. Voisin, On the universal CH0 group of cubic hypersurfaces, JEMS 19, Issue 6 (2017), 1619–1653. Mingmin Shen KdV Institute for Mathematics University of Amsterdam P.O.Box 94248 NL-1090 GE Amsterdam, Netherlands e-mail: [email protected]

The Igusa Quartic and the Prym Map, with Some Rational Moduli Alessandro Verra Abstract. This paper is devoted to the ubiquity of the Igusa quartic B ⊂ P4 in connection to the Prym map p : R6 → A5 . We introduce the moduli space X of those quartic threefolds X cutting twice a quadratic section of B. A general X is 30-nodal and the intermediate Jacobian J(X) of its natural desingularization is a five-dimensional p.p. abelian variety. Let j : X → A5 be the period map sending X to J(X), in the paper we study j and its relation to p. As is well known the degree of p is 27 and its monodromy group endows any smooth fibre F of p with the incidence configuration of 27 lines of a cubic surface. Then the same monodromy defines a map j : D6 → A5 of degree 36, with fibre the configuration of 36 ’double-six’ sets of lines of a cubic surface. We prove that j = j ◦ φ, where φ : X → D6 is birational. This provides a geometric description of j . Finally we describe the relations between the different moduli spaces considered and prove that some, including X , are rational. Mathematics Subject Classification (2010). 14H40, 14H45, 14E08. Keywords. Genus 6 curve, Prym map, Igusa quartic, double six configuration, rational variety.

1. Introduction and preliminaries The Igusa quartic is a well-known quartic threefold in the complex projective space P4 . It originates from classical Algebraic Geometry and Invariant Theory, see [B, Chapter V]. In more recent times, the Igusa quartic has been frequently reconsidered, starting from the work of Igusa and of van der Geer, [I, VdG]; we will denote the Igusa quartic by B. Its dual variety B ∗ ⊂ P4∗ is the equally famous Segre cubic primal, see [S1]. The hypersurface B ∗ is the unique invariant cubic threefold under the action of the symmetric group S6 , determined by its standard irreducible representation of dimension 5, see [FH] 4. Its isomorphic dual acts on B defining a natural isomorphism S6 ∼ = Aut B. Finally, letting P4 ⊂ P5 be the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Farkas et al. (eds.), Rationality of Varieties, Progress in Mathematics 342, https://doi.org/10.1007/978-3-030-75421-1_15

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A. Verra

404

hyperplane {z1 + · · · + z6 = 0}, then the equation of B is obtained putting t = in the pencil z14 + · · · + z64 − t(z12 + · · · + z62 )2 = 0, of all S6 -invariant quartics of P4 , see [B3, CKS, VdG].

1 4

We shall provide a few more remarks about the geometry and history of B and B ∗ . This also serves us to remind of the ubiquity of B in Algebraic Geometry, and illustrates the appearance of B at several spots in our story. Let R6 be the moduli space of genus 6 Prym curves and A5 that of principally polarized abelian varieties of dimension 5. In our paper we recover this ubiquity at a new place, addressing the link between B and the Prym map p : R6 → A5 . We show that the geometry of p, revealed in [DS] and [Do], nicely relates with the geometry of B. We deduce along the way the rationality of most of the moduli spaces involved. Our work frequently relies on the papers [CKS] and [FV]. Theorems A, B, C, D in this section summarize our results. To state these we cannot avoid some preliminaries. We begin from a quintic Del Pezzo surface S and its realization S = P5 ∩ G ⊂ P9 (1) as a smooth linear section of the Pl¨ ucker embedding G of the Grassmannian of planes of P4 . As it is well known, all these surfaces are biregular and projectively equivalent as linear sections of G. From now on P will denote the universal plane over S that is P = {(x, y) ∈ P4 × S | x ∈ Py }, (2) 4 where Py ⊂ P is the plane whose parameter point is y. Consider the natural projections u t (3) P4 ←−−−− P −−−−→ S. 2 Then u is a P -bundle structure on P and t is defined by the tautological sheaf OP (1) := t∗ OP4 (1). We notice that t has degree two and consider its Stein factorization: P t c (4) P

t

P4 .

It is known that the branch divisor of t is B and that c is a small resolution of P, see [CKS, 2.32]. A very important element of our picture is represented by the linear system of divisors |OP (2)|. This is studied in the paper [FV]. Let us recall that a general Q ∈ |OP (2)| is smooth and that the restriction u|Q : Q → S is a conic bundle. Its discriminant is a smooth Prym curve (C, η) of genus 6, that is, C is a smooth, integral genus 6 curve and η ∈ Pic0 C is a non trivial 2-torsion point. Hence (C, η) defines a point

The Igusa Quartic and the Prym Map

405

of the moduli space R6 . Actually C is canonically embedded in S ⊂ P5 and (C, η) has general moduli. In particular we meet here the space of Aut P-isomorphism classes of the conic bundles u|Q : Q → S. We define such a space as a GIT quotient, fixing for it the notation: |OP (2)|// Aut P =: Rcb 6 .

(5)

We fix a general element Q as above. We show in Section 6 that the datum of u|Q is equivalent modulo Aut P to the data of the triple (C, η, s), where (C, η) ∗ is the discriminant of u|Q and s : C → P is the map sending x to Sing u|Q (x). We cb refer to (C, η, s) as the Steiner map of u|Q . Therefore R6 is the space of Aut Pisomorphism classes of Steiner maps as well. Notice that Aut S is isomorphic to Aut P via its action on the fibres of u. Moreover Aut S is the symmetric group S5 and acts linearly on P4 and P5 inducing Aut P and Aut S, [SB]. Let σ : S → P2 be the contraction of four disjoint lines of S and Autσ S ⊂ Aut S the group leaving invariant the exceptional divisor E of σ, then Autσ S is S4 and acts on σ(E) by permutations.  Now let (C, η, s) be a Steiner map. Then σ|C defines2the line bundle ∗ OP2 (1) , that is, a sextic model of the curve C in P . In the paper M := σ|C we will also consider the GIT-quotient of Aut P-isomorphism classes of 4-tuples (C, η, s, M ):  6cb . |OP (2)|// Autσ S =: R (6) At this point, we introduce one more actor in our story, namely the moduli space of 4-nodal Prym plane sextics (C  , η  ). Here C  is a 4-nodal plane sextic and η  ∈ Pic0 C  is a non zero 2-torsion element. We will denote such a moduli space as R6ps . In the paper we prove that: Theorem A. The moduli space Rps 6 of 4-nodal Prym plane sextics is rational.  6cb and the birationality of R  6cb and Rps , see Indeed we prove the rationality of R 6 theorems (5.2) and (5.5). The latter follows from a construction and the methods in [FV, Section 2]. To summarize, we rely on the next commutative diagram, to be explained in detail in Sections 4 and 6 respectively: 

P2 × P2

P

i

P15



t

t

P . 4

Here the map t has degree 2 and is defined by |Ie (1, 1)|, where Ie is the ideal sheaf of e in P2 × P2 and e is a set of 4 general points. The map  is a birational isomorphism and |Ie2 (2, 2)| defines ◦i, where i is the embedding defined by OP (2). Notice also that  defines by pull-back a linear isomorphism  : |OP (2)| → |Ie2 (2, 2)|. Us cb and Rps . Then, focusing our view ing this we will prove the birationality of R 6 6 4 on B and P , we will see very interesting families of threefolds and geometric con-

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figurations. These relate B to the Prym map and in what follows we summarize our results in this respect. At first, starting from a general Q ∈ |OP (2)|, we list some geometric objects related to it and define their moduli spaces. Let ι : P → P be the birational involution induced by t and let Q := ι∗ Q, we notice that Q ∈ |OP (2)| and that the same is true for the ramification divisor of t R := t−1 (B) ∈ |OP (2)|. In particular the set of the fixed points of the involution ι∗ is the union t∗ |OP4 (2)| ∪ {R}. A first object associated to Q is X := t∗ Q = t∗ Q. Then X is a quartic three˜ be X blown up at Sing X. fold with 30 nodes, since Sing X = X ∩ Sing B. Let X Then its intermediate Jacobian is the Prym variety P (C, η) of the discriminant of the conic bundle u : Q → S. Since t∗ X = Q + Q is split and B is the branch locus of t, we have B · X = 2A where A is a quadric. Hence X defines a pencil of quartic threefolds with 30 nodes λa2 + μb = 0, where B = div(b) and A = div(a). We define a pencil P , generated by b and some a2 , as an Igusa pencil. Let X ∈ P we also say that X is an E6 -quartic threefold. As we will see, X is related to the root lattice E6 and its corresponding simple Lie Algebra. Inside the linear system |OP4 (4)| the family of these threefolds is a cone V with vertex B over the family V2 of all double quadrics 2A and V2 parametrizes Igusa pencils. Again, the projective isomorphism classes of V and V2 can be constructed as GIT quotients: X =: V// Aut B and P I =: V2 // Aut B.

(7)

∼ S6 , see Section 7. Here Aut S ⊂ Aut B is the stabilizer of S and Aut B = Clearly one has t∗ X = Q + Q and Q, Q generate the pencil t∗ P . Now consider the plane Py = t∗ u∗ (y), y ∈ S. Restricting X to it we have a union of two conics: X · Py = t∗ (Q · Py ) ∪ t∗ (Q · Py ). Therefore X has two family of conics, induced from the conic bundle structures u|Q and u|Q over S. Notice that Py · B is a double conic. This follows since B is the focal locus of the congruence of planes S, that is the branch divisor of t. See Section 2 and [CS] for facts concerning the classical theory of foci for a family of linear spaces in Pn . Now the intriguing feature of B is that it is actually the focal locus for six congruences of planes, which are smooth linear sections of G. The set of these surfaces is the orbit of S under the action of Aut B on G. Indeed Aut B is the symmetric group S6 and the stabilizer of S under its action is Aut S ∼ = S5 . We will say more about this classical fact in Section 3, [1 ]. Then X is endowed with six pairs of conic bundle structures over the six surfaces: a double six so to say. 1 Passing

to the Segre primal B ∗ ⊂ P4∗ and the dual Grassmannian of lines, these surfaces define the six quintic Del Pezzo components of the well known Fano surface of B ∗ , cf. [D3] 2.

The Igusa Quartic and the Prym Map

407

Let (X, Q) be a pair such that Q ⊂ t∗ X

and Q ∈ |OP (2)|.

We will say that (X, Q) is a marked E6 -quartic threefold. Finally, we set   X := (X, Q) : X is a marked E6 -quartic threefold, X ∈ V // Aut B,

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then the GIT quotient X is endowed with the degree two forgetful map q : X → X , whose associated involution is induced by the exchange of Q and Q. Since Aut S ∼ = S5 , the inclusion Aut S ⊂ Aut B induces a degree six morcb  phism d : R6 → X and the diagram Rcb 6

u:=q◦d d

X q

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X. Fixing the contraction σ : S → P2 , that is Autσ S ⊂ Aut S, we define in the same way  cb := |OP (2)|// Autσ S. R 6 This space parametrizes Aut P-isomorphism classes of triples (X, Q, σ  ), where σ  : S → P2 is one of the five contractions of S to P2 . The relations between the quotient spaces discussed above are described in the forthcoming sections and summed up in the next theorem and diagram. The connections to the Prym map and the 27 lines of the cubic surface are discussed in Section 7. cb cb  cb Let d˜ : R 6 → R6 and d : R6 → X be the forgetful maps. Clearly these are induced by the inclusions Autσ S ⊂ Aut S ⊂ Aut B, so that deg d˜ = 5, deg d = 6. Now let us define the following maps:   First p : Rps 6 → R6 is induced by the assignment (C , η ) → (C, η), where ∗   ∼ η = ν η and ν : C → C is the normalization.  cb → Rps is the birational map from the proof of Theorem (5.5). Then α : R 6 6 Consider also the forgetful map n : Rcb 6 → R6 and finally the period map

j : X → A5 , ˜ Let p : R6 → A5 be Prym map . Then we induced by the assignment of X to J X. have:

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408

Theorem B. The next diagram is commutative. Moreover the period map j : X → A5 has degree 36 and the forgetful map n : Rcb 6 → X has degree 16.  cb R 6

α

R6

n

d˜

R6cb

p

R6ps

u:=q◦d

p j

X

d

A5

(10)

q

X Theorem C. The moduli spaces of E6 -quartic threefolds and of Igusa pencils are rational. See Theorems 6.3 and 6.4. Let F be the configuration of 27 lines of a smooth cubic surface, as we know the fibre of the Prym map p is reflected by F. Now, expecting that the fibre of the period map j : X → A5 reflects the configuration of double sixes of F is natural. The numbers deg j = 36 and deg n = 16 suggest indeed that the following interpretation: First, 36 is the number of double sixes in F and 16 the number of double sixes containing an element of F. A celebrated theorem of Donagi shows that the monodromy group of p is the Weyl group W (E6 ) of permutations preserving the incidence relation of F, see [Do, 4.2]. Let a ∈ A5 be general and p−1 (a) = Fa . Using the monodromy of p we may think of Fa as F. Denoting by DSa (F) its set of double sixes let us consider: 1. D6 := {(s, a)| s ∈ DSa (F), a ∈ A5 } and its degree 36 forgetful map j : D6 → A5 . 2. R := {(s, a, l) | (s, a) ∈ D6 , l ∈ s ⊂ Fa } and its degree 16 and 12 forgetful maps n : R → R6 and u : R → D6 . The irreducibility of D6 , and of R , follows by monodromy considerations. Indeed the monodromy group of p is W (E6 ) and acts transitively on the configuration of double sixes of F, see [D1, 9.1.4]. The above rational maps and Theorem B define the commutative diagrams n

R −−−−→ R6 ⏐ ⏐ ⏐  ⏐p u  j

n

Rcb 6 −−−−→ ⏐ ⏐  u

D6 −−−−→ A5 X so that we can give the following definitions.

R6 ⏐ ⏐p 

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j

−−−−→ A5 ,

Definition 1.1. j : D6 → A5 is the universal family of double sixes of p. Definition 1.2. n : R → R6 is the universal pointed family of double sixes of p.

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409

In Section 8, relying on Donagi’s tetragonal construction, we conclude as follows. Theorem D. The period map j : X → A5 of E6 -quartic threefolds and the universal family j : D6 → A5 , of double sixes of p, are birationally equivalent over A5 . The birationality of R and R6cb also follows. So this is the geometry of E6 quartic threefolds and of their periods, relating their double sixes of conic bundles to the classical ones.

2. Plane geometry in P4 : the congruence S and its focal locus B In this section we introduce more geometry of the quintic Del Pezzo surface S and the Igusa quartic B. On the occasion, before describing our results, we revisit some classical notions related to Grassmannians and their history. About this a warning is due. It is known that an unexpected number of decades passed, after the outstanding work of Grassmann, in order to fix language and theory on Grassmannians, see, e.g., the biographical paper [D] by Dieudonn`e. This implies that the words in use and their meaning have undergone quite contrasting variations. ucker embedding of the Grassmannian of r-spaces in Pn Let G be the Pl¨ with 0 < r < n − 1. Possibly adopting the classical language, we will say that a surface Y ⊂ G is a congruence of r-spaces of Pn . In this language the order of the congruence Y and its class are well defined as follows. For y ∈ G let Py ⊂ Pn be the corresponding r-space, consider the universal r-space I := {(x, y) ∈ Pn × G | x ∈ Py },

(12)

and its projections t

u

(13) Pn ←−−−− I −−−−→ G. Then u is a Pr -bundle structure on I and t is its natural tautological map. In the Chow ring CH ∗ (I) let h be the pull-back by t of the hyperplane class, then the degree of the 0-cycle class u∗ (hr+2 u∗ [Y ]) = (u∗ hr+2 )[Y ] ∈ CH ∗ (G)

(14)

is the order of Y . Counting multiplicities this is just the degree of the scroll union of the r-spaces parametrized by Y .The class of Y is the number of its elements not intersecting properly a general space of codimension r. Let VY be the restriction to Y of the locally free sheaf V = u∗ t∗ OPn (1), in modern terms we can define the order a and the class b of Y via Chern classes: a = deg(c12 (V) − c2 (V)) , b = deg c2 (V). ∗

(15)

Then a is the degree of the cycle t∗ u Y , that is the degree of the second Segre class s2 (V). The degree of Y in the Pl¨ ucker space is a + b = c12 (V). Let α, β ∈ CH2 (G) be the two classes of the families of planes contained in G, we denote by α the class of a plane parametrizing the r-spaces through a fixed codimension 3 space

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of Pn . By β we denote the class of a net of r-spaces in a fixed (r + 1)-space of Pn . Notice that CH2 (G) = Zα ⊕ Zβ and that [Y ] = aα + bβ. Remark 2.1. In spite of the generality of Grassmann’s foundations, most of geometric research was prevalently focusing on the Grassmannian of lines of P3 for long time, [K, XXII 4]. For decades the tripartition of names ruled surface, congruence, complex was largely in use for families of lines in P3 of dimensions 1, 2, 3, [Lo, 7]. Passing to any G, the word complex evolved as synonimous of hypersurface in G. Among many exceptions to this trend, let us mention two papers on the Grassmannians of P4 by Segre and Castelnuovo, [S2, C]. Both are relevant in what follows. For simplicity let Y ⊂ G be a smooth integral variety and let us consider the morphism t|u∗ Y : u∗ Y → Pn , (16) ∗ ∗ r defined by the sheaf t OPn (1). Often we will say that the P -bundle u Y is the universal r-space over Y and that t|u∗ Y is its tautological map. The classical Theory of Foci is concerned with the ramification of this map and the infinitesimal deformations in the family Y . For a point y ∈ Y , following this theory we say that the focal locus of the r-space Py is the restriction to Py of the ramification scheme of t|Y . See [CS] and [Se, 4.6.7] for a modern reconstruction. The general case of interest is clearly when t|u∗ Y is a generically finite morphism. So we assume this since now and we will have in particular dim Y = n − r. Then we fix the notation BY ⊂ P n (17) for the branch scheme of t|u∗ Y : u∗ Y → Pn and say that BY is the focal locus of Y . If deg t|u∗ Y ≥ 2, then BY is a hypersurface, we call it the focal hypersurface of Y . Finally we will say that the fundamental locus of the family Y is F := {x ∈ Pn | dim(t|u∗ Y )−1 (x) ≥ 1}.

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In classical terms F is indeed the locus of points in P which are singular for Y (quite conflicting with modern language). Since now we fix n = 4 and r = 2 so that Y is a congruence of planes in P4 . Moreover we assume that Y is a smooth linear section of G. and keep for this surface the notation S adopted in the introduction. In particular we have as in (1) n

S = P5 ∩ G ⊂ P 9 . S is therefore a quintic Del Pezzo surface, see [D] 8.5 for a detailed description. Let us also recall that the group Aut S is isomorphic to the symmetric group S5 , cf. [SB]. We have Aut S ⊂ Aut P4 (19) 4 and, up to conjugation, its action on P is the permutation of coordinates. As above let V = u∗ t∗ OPn (1) be the rank 3 vector bundle whose projectivization is

The Igusa Quartic and the Prym Map

411

the universal plane over S, then the following properties of V are standard and well known: ∼ OS (1), 1. c1 (V) = 2. c1 (V)2 − c2 (V) = 2, 3. c2 (V) = 3 4. h0 (V) = 5, 5. hi (V) = 0 for i ≥ 1. In particular the congruence of planes S has order 2 and class 3 in CH2 (G), that is, (20) [S] = 2α + 3β. To see that S has order 2 just consider a general point x ∈ P4 and the Schubert variety Gx := {(x, y)|x ∈ Py } = t∗ (x). Then Gx is a smooth quadric of dimension 4 and the linear span P5 of S cuts on it a linear section. Since x is general we can assume that Gx and P5 intersects transversally. Hence the order of S is 2 = deg Gx and its class is 3 = deg c2 (V). V fits in the standard exact sequence 0 → U → H 0 (OS (1)) ⊗ OS → V → 0. Passing to the long exact sequence, it is easy to deduce that h0 (V) = 5 and h (V) = 0 for i ≥ 1. Now let G(5, 9) be the Grassmannian of 5-spaces in the ucker space of G, as is well known the action of Aut P4 on it has a unique open Pl¨ orbit, which is the family of 5-spaces transversal to G. This implies that V is the unique rank 3 vector bundle on S satisfying the above properties and defining an embedding of S in G. In particular S is equivariant with respect to Aut S. i

Remark 2.2. The vector bundle V is also known as the rank 3 Mukai bundle of S, since it is related to Mukai’s higher Brill–Noether theory for a general genus 6 canonical curve C. Indeed C is embedded in S and V ⊗ OC is the kernel of the evaluation map of the unique stable vector bundle on C, of rank 2 and canonical determinant, with 5 linearly independent global sections, see also [M, FV]. Now we want to study the universal plane over S and its tautological morphism. To this purpose we fix the notation P for such a universal plane and consider the diagram I u|P

(21)

tP

P4

t

P

u

S

where t : P → P4 is the tautological morphism, u : P → S is the universal plane over S and the vertical arrow is the inclusion. Since the order of S is two then t is a morphism of degree two.

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Our main interest is in the focal locus of the congruence S, that is the branch hypersurface of t. We will simply denote it by B. The ramification divisor of t will be denoted as (22) R ⊂ P. Proposition 2.1. B is a quartic threefold and R is an element of |OP (2)|. Proof. Let K be a canonical divisor of P and H ∈ |OP (1)|, from the morphism t ∼ OS , the formula for the canonical we have K ∼ −5H + R. Since det V ⊗ OS (KS ) = class of the projective bundle P implies K ∼ −3H. Then R ∼ 2H and B = t∗ R is  a quartic. The quartic threefold B is crucial for our work. One has the following result, for which we refer to [CKS]: Theorem 2.2. The focal locus B is the Igusa quartic threefold and F = Sing B. The Igusa quartic B is a very classical object, see [Ri]. Its geometry is discussed in Chapter V of Baker’s treatise [B]. Its ubiquity in Algebraic Geometry has been mentioned already, see [CKS, D1, D2, Hu, I, VdG]. This implies that B bears more names, like the Castelnuovo–Richmond quartic, and appears in relation to several moduli spaces. Actually, as one can realize reading [C] and [Ri], the variety B was described first by Castelnuovo, followed independently by Richmond. We will use the Stein factorization of t, namely the diagram P P

(23)

t

c t

P4 .

But P is the moduli space of 6 ordered points of P2 considered for instance in [D1, 9.4.17]. On the other hand, we will see various new incarnations of B relating it to Prym curves of genus 6 and their Prym varieties. We recall that B is invariant under the action of the symmetric group S6 so that

S6 ∼ (24) = Aut B ⊂ Aut P4 , cf. [M1, Lemma 5]. Let (z1 : · · · : z6 ) be the projective coordinates in P5 and sk := z1k + · · · + z6k . Putting P4 := {s1 = 0}, the equation of B in P4 becomes s4 − 14 s22 = 0. To continue, we consider a point y ∈ S and the fibre of u|R : R → S at y, say Ry := R · Py . (25) Since the ramification divisor R is an element of |OP (2)|, and no plane is in B, then Ry is always a conic. Consider the plane Py = t(Py ) and By := t∗ Ry . Then it follows Py · B = 2By . (26) Indeed t∗ (Py ) is reducible, since it contains Py , and the branch divisor of t : t∗ (Py ) → Py is Py · B. This implies Py · B = 2By , where By is a conic. According

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413

to the classical theory By is the focal conic of Py . Finally notice also that Aut S ⊂ Aut B. Since Aut S ∼ = S5 and Aut B ∼ = S6 , it follows that the orbit of S is the union of six Del Pezzo surfaces, say SS6 =

6 7

Si .

(27)

i=1

Each surface Si is endowed as in (3) with its universal plane Pi for i = 1, . . . , 6, and the usual diagram t ui (28) P4 ←−−i−− Pi −−−− → Si . The following well-known characterization of B will be also coming into the play. Theorem 2.3. B is the strict dual hypersurface of the Segre cubic primal. The Segre primal is, modulo the action of Aut P4∗ , the unique cubic threefold B in P4∗ whose singular locus consists of 10 double points. Furthermore, 10 is also the maximal number of isolated singular points for a cubic threefold. Moreover the dual map B ∗  B contracts the 15 planes of B ∗ to 15 singular lines of B: Sing B is their union. Sing B ∗ and the 15 planes of B ∗ define a (154 , 106 )-configuration: 4 nodes in each plane and 6 planes through each node, [D3] 2.2. In the Grassmannian G∗ of lines of P4∗ , the Fano variety of lines of B ∗ is the union of 15 planes and 6 Del Pezzo surfaces S1∗ , . . . , S6∗ . These, under the natural isomorphism between G∗ and G, are dual to S1 , . . . , S6 . Finally we fix the convention: S = S1 so that t = t1 and u = u1 . Coming to the fundamental locus F of the congruence S its description is classical, cfr. [B]. We have F = Sing B and the 15 lines of F intersect 3 by 3 along the set of 15 triple points of B. This defines the famous Cremona -Richmond (153 , 153 )-configuration, [C, Ri], cfr. [D1, 9.4.4]. ∗

Remark 2.3. Let us sketch very briefly a direct reconstruction of F . For any x ∈ F observe that u∗ t∗ (x) = Gx · S, where Gx is a smooth four-dimensional quadric. Then, since S is an integral linear section of G and dim t∗ (x) ≥ 1, the fibre t∗ (x) is either a line or a conic in {x} × S. Assume L = u∗ t∗ (x) is a line L, then L is one of the ten lines of S. Hence L is a pencil of planes contained in a given hyperplane of P4 . Let A ⊂ P4 be its base locus, then P contains the surface A × L and t contracts it to A. Now assume that u∗ t∗ (x) is a smooth conic C, then |C| is one of the five pencils of conics of S. Let PC = u∗ C. It turns out that PC is the projectivization of the bundle OP1 (−1)⊕2 ⊕ OP1 and t|PC is its tautological map. Hence QC := t(PC ) is a rank 4 quadric and t|PC is the small contraction of {x} × C ⊂ PC to x = Sing QC . Moving C in the pencil of conics |C|, the point Sing QC moves along a line which is in F . In particular, each irreducible component of t−1 (F ) is one of the 15 surfaces as above, cfr. [CKS].

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3. Conic bundles associated to a genus 6 Prym curve We now study the linear system of divisors |OP (2)|. We have the following result: Proposition 3.1. For m ≥ 1 one has     m+4 m+2 0 , hi (OP (m)) = 0 f or i ≥ 1. + h (OP (m)) = 4 4

(29)

We omit the standard proof and only compute h0 (OP (m)) for the future use. Let P t

c

P

t

P4 .

be the Stein factorization of t, we consider the birational involution induced by t: ι : P → P.

(30)

By Remark (2.3) c is a small contraction, hence the map c∗ : H 0 (OP (m)) → ∗ H 0 (t OP4 (m)) is an isomorphism. Let ι : P → P be the biregular involution induced by t. In particular ∗ 0 0 ι∗ := c−1 ∗ ◦ ι ◦ c∗ : H (OP (m)) → H (OP (m))

(31)

is an involution with eigenspaces + − := t∗ H 0 (OP4 (m)) and Hm := t∗ H 0 (OP4 (m − 2)) ⊗ q− , Hm

(32)

where div(q− ) = R = t−1 (B). This implies the previous equality. Indeed     m+4 m+2 + − + dim Hm = + . h0 (OP (m)) = dim Hm 4 4 We denote the elements of |OP (2)| by Q. Clearly a general fibre of u|Q is a conic. With some abuse we will simply say that u|Q is a conic bundle structure on Q. From [FV, Section 2] we have the following: Proposition 3.2. A general Q is smooth and each fibre of u|Q is a conic of rank ≥ 2. Let Q be general as in the above statement and let Qy := (u|Q )∗ (y), then C := {y ∈ S | rank Qy = 2}

(33)

is a smooth curve endowed with the embedding sending y to Sing Qy , we denote it as s : C → P.

(34)

The Igusa Quartic and the Prym Map

415

Consider the surface QC = u∗ (C) then Sing QC = s(C). Moreover it is easy to see ˜ C → QC fits in the Cartesian square that the normalization map n : Q n ˜ C −−− −→ QC Q ⏐ ⏐ ⏐ ⏐ uC  ˜C  u

(35)

π C˜ −−−−→ C,

˜ C → C and u ˜C is the Stein factorization of uC . Then π is where uC is u ◦ n : Q an ´etale 2 : 1 cover, defined by a 2-torsion element η ∈ Pic C. The next theorem summarizes our situation. Theorem 3.3. Let Q be general as above. The following hold: 1. 2. 3. 4. 5.

C ∈ | − 2KS |, η is non trivial, C is a general genus 6 curve. u : C → S ⊂ P5 is the canonical embedding, t ◦ s : C → P4 is defined by ωC ⊗ η.

Proof. Some basic results on conic bundles and genus 6 curves imply the statement. (1) is given by the formula for the discriminant of a conic bundle, see [Sa]. Statements (2), (3), (4) are known properties one retrieves in [FV, Section 2]. To establih statement (5), we argue as follows: Consider the section ˜ Then, by adjunction formula ˜ C → C˜ and let n ˜ := n|C. C˜ := n∗ s(C) of u ˜C : Q C ∗ ˜ on QC , the line bundle nC˜ OP (1) is ωC˜ . Moreover one can show that n∗C˜ H 0 (OP (1)) is an eigenspace of the involution defined by π. Since its dimension is 5 then ∗ 0 ∗ 0 ∼ nC  ˜ H (OP (1)) = π H (ωC ⊗ η). This implies ωC ⊗ η = OP (1). We assume that ωC ⊗ η is very ample, since it is true for a general C of genus 6, [DS]. We will also say that (C, η) is the discriminant of u : Q → S. Now (C, η) defines a point in the moduli space R6 of genus 6 Prym curves. Starting from such a point we are interested in the set of possible realizations of (C, η) as the discriminant of a conic bundle Q ∈ |OP (2)|. We say that Q, Q ∈ |OP (2)| are isomorphic if there exists a biregular map f : Q → Q such that u|Q = u |Q ◦ f . We are interested to study the isomorphism classes of |OP (2)|. Let Ie be the ideal sheaf of a set e of four general points in P2 × P2 , to this purpose we will use the main diagram of rational maps in [FV] p. 529 and the induced linear isomorphism |OP (2) ∼ = |I 2 (2, 2)|. e

Let us give a more precise explanation. This isomorphism depends on the choice of a contraction σ : S → P2 (36) of four disjoint exceptional lines. Let ei = σ(Ei ), i = 1, . . . , 4, Ei being an exceptional line. Then, setting e := {e1 , . . . , e4 }, we have a set of points in general linear

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position and the same is true for its diagonal embedding e ⊂ P2 × P2 . Notice also that dim |Ie (1, 1)| = 4. Definition 3.1. te : P2 × P2  P4 is the rational map associated to |Ie (1, 1)|. It is easy to see that deg te = 2. The same is true replacing te by tE := te ◦ (σ × idP2 ) : S × P2  P4 .

(37)

The strict relation between the tautological morphism t and te is described in the following diagram that is mentioned in [FV, p. 529]. This can be written as follows: 1

 P



2

P2 × P2

σ−1 ×idP2

P

S

t

tE

S × P2

u

P4

(38) Here 1 , 2 are birational morphisms resolving the indeterminacy of the birational map  := 1 ◦ −1 : S × P2  P. (39) 2 ∗ Let L ∈ |σ OP2 (1)|, as in [FV] 2 (3) the birational map  can be defined via the exact sequence j

r

0 → V → H 0 (OS (L)) ⊗ OS (L) →

4 

OEi (2L) → 0

i=1

where Ei = σ −1 (ei ) is the exceptional line of σ over ei , r is the natural multiplication of sections and V = Ker r is the previously considered rank 3 Mukai bundle of S. Dualizing and projectivizing the morphism of locally free sheaves j one obtains  : S × P2  P. Remark 3.1. It is useful to add some remarks on 1 and 2 . It turns out that PEi := u−1 (Ei ) is biregular to the blowing up of P3 along a line. Moreover let Fi ⊂ PEi be the exceptional divisor of such a blowing up, then Fi = P1 × Ei and u|Fi is the projection onto Ei . We have:  → S × P2 is the blowing up of the disjoint union F1 ∪ · · · ∪ F4 . (a) 2 : P (Fi ) then each fibre of u ◦ 1 : F˜i → Ei is P2 (b) Consider the threefold F˜i := −1 2 blown up in a point. (c) The contraction 1 (F˜i ) of F˜i is Ei × P2 and 1 : F˜i → Ei × P2 is the blowing up of Ei × {ei }. (d) Let PEi be the strict transform of PEi by 2 , then 1 contracts PEi to Ei ×{ei }. In what follows we let φ := (σ × idP2 ) ◦ , and consider the birational map φ : P  P2 × P2 . 

(40)

Let Q ∈ |OP (2)| be general and let Q ⊂ P × P be its strict transform by φ. To understand Q consider QEi := Q · PEi . It turns out that QEi is P1 × P1 blown 2

2

The Igusa Quartic and the Prym Map

417

up in two distinct points o1 , o2 and that u : QEi → Ei is the conic bundle defined by the pencil |Io1 ,o2 (1, 1)|, where Io1 ,o2 is the ideal sheaf of {o1 , o2 } in QEi . Now let E1 , E2 be the exceptional lines of QEi over o1 , o2 , then φ blows E1 , E2 up and flops down the resulting surfaces to two lines. These surfaces intersect at a double  point of Q= which is the contraction of QEi . Moreover φ : Q → Q is biregular on U = Q \ ( i=1,...,4 QEi ). The resulting Q is singular at e and has bidegree (2, 2). Let Ie be the ideal sheaf of e in P2 × P2 , then Q ∈ |Ie (2, 2)|. The map φ is studied in [FV] 2. We conclude by pointing out the obvious consequence of our discussion, which will be a useful tool. Let h := (σ × idP2 )∗ ◦ 1∗ ◦ ∗1 ,

(41)

Theorem 3.4. The map h : H 0 (OP (2)) → H 0 (Ie2 (2, 2)) is an isomorphism.

4. Igusa pencils and E6 -quartic threefolds Now we introduce the useful notion of an Igusa pencil of quartic threefolds. As in (30) consider the birational involution ι : P → P induced by the morphism t : P → P4 . As in above let R = t−1 (B), we know that ι∗ acts linearly on |OP (2)| so that its set of fixed points is t∗ |OP4 (2)| ∪ {R}.

(42) ∗ ∗ ˜ Let P ⊂ |OP (2)| be a ι -invariant pencil on which ι is not the identity, then ι∗ acts on P˜ with exactly two fixed elements. These are R and t∗ A, where A ⊂ P4 is a quadric. Equivalently P˜ is the pull-back by t of the pencil of quartic threefolds P , generated by B and the double quadric 2A. Let A = div(a) and div(b) = B, the equation of P is P = λa2 + μb = 0. (43) Definition 4.1. We say that P is an Igusa pencil of quartic threefolds. Definition 4.2. We say that a threefold X ∈ P is an E6 -quartic threefold. The family of these pencils is parametrized by |OP4 (2)|. Assuming P general and X general in P , let us see some properties of X and of P . For Q ∈ |OP (2)| we will set Q := ι∗ Q. At first we emphasize that for a general X the threefold t∗ X is split: t∗ X = Q + Q, where Q, Q are smooth and Q = Q. The base scheme of P is the contact surface {a2 = b = 0} = 2A · X ⊂ P4 , where A = div(a) is a smooth quadric transversal to B. Let y ∈ S and Py ⊂ P4 the corresponding plane. We know that Py · B = 2By , where By is a conic. Let by be the equation of By in Py . Restricting λa2 + μb to Py we obtain the sum of squares λa2 + μb2y .

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This defines the union of two conics of Py , namely t∗ Qy and t∗ Qy . Furthermore the covering t : t−1 (Py ) → Py splits, since it is branched on 2By . Let Py := ι∗ (Py ), then t∗ (Py ) = Py + Py . In particular [Py +Py ] = [H 2 ] in CH 2 (P) and Py is a birational section of u : P → S. Theorem 4.1. Assuming X and P are general as above, then the following hold: 1. Sing X is A ∩ Sing B and consists of 30 ordinary double points. 2. The discriminants of u|Q and u|Q are smooth genus 6 Prym curves. 3. t|Q : Q → X and t|Q :→ X are small contractions to X. Proof. By Bertini’s Theorem X ∈ P is smooth outside the surface Ba := {a = b = 0}. Moreover Sing Ba = Sing B ∩ A and this set consists of 30 ordinary double points of X. Let x ∈ Ba − Sing Ba then B is smooth at x and hence {a2 = 0} is the unique element of P = {λa2 + μb = 0} which is singular at x. This implies (1). Statement (2) follows from Theorem 3.3 since Q, Q are general in |OP (2)|. Then they are smooth with no fibre of rank ≤ 1. For (3) it suffices to recall that the fibre of t at a general x ∈ Sing B is P1 .  Let Q and Q be as above, then consider the Cartesian square of birational morphisms ˜ −−−−→ Q X ⏐ ⏐ ⏐ ⏐ t|Q  (44)  t|Q

Q −−−−→ X. Since X is general the set Sing X = X ∩ Sing B is general in Sing B and the fibre ˜ → X be the birational morphism of t|Q and t|Q at o ∈ Sing X is P1 . Let β : X ∗ 1 induced by the diagram, then β (o) = P × P1 and β is the blowing up of Sing X. ˜ is smooth and, denoting by T the intermediate Jacobian of T , one has Then X the following statement, which follows from [CG, Lemma 3.11]: ˜ ∼ JQ ∼ = JX = JQ.

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Finally let (C, η) and (C, η) be the discriminants curves of u|Q and u|Q respectively. Consider the corresponding Prym varieties P (C, η) and P (C, η) as in [B1]. Then JQ ∼ (46) = P (C, η) , JQ ∼ = P (C, η). Now let us recall that t : P → P4 is just one of the six double coverings ti : Pi → P4 , i = 1, . . . , 6, considered in (27). Hence t∗ X = Q + Q is just one of the splittings t∗i X = Qi + Qi ⊂ Pi , i = 1, . . . , 6.

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Hence birationally X is naturally endowed with six pairs of conic bundle structures as follows: {ui|Qi : Qi → Si , and ui|Qi : Qi → Si }. (48) To simplify our notation we set: ui := ui|Qi , and ui := ui|Qi .

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Definition 4.3. We refer to the set {ui , ui i = 1, . . . , 6} as the double six of conic bundle structures of X. The set of corresponding discriminants is the double six of Prym curves of X: {(Ci , ηi ) , (C i , η i ),

i = 1, . . . , 6}.

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Remark 4.1. One recognizes here the shadow of the set of 27 lines on the cubic surface, via its 36 Schl¨afli double sixes. Let p : R6 → A5 be the Prym map, sending (C, η) to its Prym P (C, η), [DS, D1]. The map p has degree 27. The Weyl group of E6 , preserving the incidence of the 27 lines on a cubic surface is its monodromy group. So far we see that the double six of X defines a subset of the fibre of p at ˜ in A5 . the moduli point of J X

5. Moduli of Prym sextics of genus 6 It is time to address the rationality results for some moduli spaces related to R6 and their relations. This involves the moduli of 4-nodal Prym plane sextics of genus six. With this in mind we start from a general genus 6 Prym curve (C, η). We know from Theorem 3.3 and [FV] that (C, η) is biregular to the discriminant of a smooth Q ∈ |OP (2)|. According to the same theorem we know more: consider u|Q : Q → S ⊂ P5 then C is canonical in P5 . Moreover let s:C→P

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be the map sending y ∈ C to Sing Qy , then s ◦ t : C → P4 is the Prym canonical map of (C, η). Consider Cs := s(C) and its ideal sheaf ICs (2) in P, we have the standard exact sequence 0 → ICs (2) → OP (2) → OCs (2) → 0. ⊗2 . Let (C, η) be general then: Notice also that OCs (1) ∼ = ωC ⊗ η and OCs (2) ∼ = ωC

Proposition 5.1. One has h0 (ICs (2)) = 1 and hi (ICs (2)) = 0, i ≥ 1. Proof. We have h0 (OCs (2)) = 15 and h0 (OP (2)) = 16 from 3.1. Hence the previous exact sequence implies h0 (ICs (2)) ≥ 1. Assume the latter inequality is strict, then it follows that dim t∗ H 0 (OP (2)) ∩ H 0 (ICs (2)) ≥ 1 in H 0 (OP (2)). Equivalently a quadric contains the Prym canonical model t(Cs ). But then the moduli point of (C, η) in R6 is in the ramification of the Prym map p, [B1] 7.5, and (C, η) is not general: a contradiction. 

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Let PC ⊂ P be the universal plane over C. Then PC is defined by the embedding of C in G via VC := V ⊗ OC , where V is u∗ t∗ OP4 (1). Then VC is uniquely defined as the kernel of the evaluation of global sections of the unique rank two Mukai bundle over C, see [M, Section 5] for its definition and uniqueness. Moreover the map s is uniquely defined by an exact sequence v

0 → NC → VC →s ωC ⊗ η → 0 for a given one-dimensional space < vs > ⊂ Hom(VC , ωC ⊗ η). Conversely, starting from a general v ∈ Hom(VC , ωC ⊗η), one obtains an exact sequence as above and a section sv : C → PC ⊂ P. For a general v, sv (C) is certainly contained in a unique Qv ∈ |OP (2)|, as prescribed in (5.1). However C does not necessarily coincide with the discriminant curve of the conic bundle u|Qv nor sv : C → Qv is the map sending y ∈ C to the singular point of the fibre of u|Qv at y. This prompts us for the next definition. Let (C, η, s) be any triple such that: 1. C ∈ |OS (2)| is smooth and (C, η) is a Prym curve, 2. s : C → PC is a section of the P2 -bundle u|P : PC → C, 3. s is defined by an epimorphism v ∈ Hom(VC , ωC ⊗ η), 4. a unique Q ∈ |OP (2)| contains s(C). Definition 5.1. (C, η, s) is a Steiner map of the pair (C, η) if the discriminant of u|Q is the pair (C, η) and s : C → Q is the map sending y ∈ C to the singular ∗ point of u|Q (y). The name is motivated by some relation to the classical notion of Steinerian, the locus of singular points of the quadratic polars of a hypersurface, [D], 1.1.6. If (C, η, s) is a Steiner map, Q is smooth and each fibre of u|Q has rank ≥ 2, cfr. [FV] 2. Now a general Q ∈ |OP (2)| defines a triple (C, η, s) satisfying the above conditions and such that s is the Steiner map of u|Q. Therefore, giving a Steiner map (C, η, s), is equivalent to give a general Q with discriminant (C, η) and Steiner map equal to s. Let t := (C, η, s), t := (C  , η  , s ) be two Steiner maps and u|Q, u|Q their respectively associated conic bundles. We will say that t and t are isomorphic if there exists a biregular map α : Q → Q such that u ◦ α = u|Q. It is easy to see that α∗ OQ (1) ∼ = OQ (1), so that α = a|Q for some a ∈ Aut P. To construct the moduli of Steiner maps, somehow abusing of the word moduli, we rely on Geometric Invariant Theory and construct the GIT quotient |OP (2)|// Aut P. Since P is the universal plane of P4 over S, then Aut P is the subgroup of Aut P4 leaving invariant the congruence of planes {Px , x ∈ S}. This is actually the symmetric ∼ Aut S. group S5 = Definition 5.2. R6cb := |OP (2)|// Aut P is the moduli space of Steiner maps. cb By [MF] 0.2 (2) the GIT quotient Rcb 6 is reduced, irreducible and normal. R6 has dimension 15 and dominates R6 via the forgetful map, [FV]. In [SB] Shepherd Barron proves that the quotient P(V )//S5 of any finite representation V of S5 is rational. This implies the next theorem.

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Theorem 5.2. R6cb is rational. We recall from (5.1) that giving a general marked E6 -quartic (X, Q) is equivalent to give its Steiner map (C, η, s). Hence R6cb is also the GIT quotient parametrizing Aut P classes of marked E6 -quartics. Continuing in the same vein let σ : S → P2 be a fixed contraction of four lines and Autσ ⊂ Aut S the stabilizer of its exceptional divisor:  cb Definition 5.3. R 6 = |OP (2)// Autσ S.  cb Theorem 5.3. R 6 is rational. Proof. We give a standard proof related to the geometry of |OP (2)| and similar to Theorem 6 in [SB]. S4 is isomorphic to Autσ S and acts on the exceptional divisor E := E1 + E2 + E3 + E4 of σ as the permutation group of its irreducible components. Ei is a line in the Grassmannian G that is a pencil of planes in P4 . Let u : Pi → Ei be the universal plane and Ti := t(Pi ). Then Ti is a hyperplane and t|Ti : Pi → Ti is the contraction of Ei × Fi to Fi , where Fi is the base line of the pencil Ei . Clearly S4 acts on the set {T1 , . . . , T4 }. More precisely S4 = Autσ S is the stabilizer of the point oσ = T1 ∩ · · · ∩ T4 under the action of Aut S on P4 . S4 acts in the same way on the set {2P1 , . . . , 2P4 } ⊂ |OP (2)| and this spans P(W ), where W ⊂ V := H 0 (OP (2)| is an isomorphic representation of dimension 4. Passing to the dual spaces and projectivizing we obtain a linear projection π : P(V ∗ ) → P(W ∗ ). After the blowing up γ of its center, we obtain a P12 -bundle p : Pσ → P(W ∗ ) and the S4 -linearized line bundle γ ∗ OP(V ∗ ) (1). By the same proof of [SB] Theorem 6, this descends to a P12 -bundle on P3 //S4 , which is rational by  the symmetric functions theorem. This implies the statement.  6cb → Rcb Now the inclusion Aut Sσ ⊂ Aut S induces a morphism d˜ : R 6 of ∗ degree 5. Let (C, η, s) be a Steiner map and let M := σ OP2 (1) ⊗ OC . Then M defines the plane sextic model σ(C) and it is one of the five elements of the Brill–Noether locus W62 (C) := {M ∈ Pic6 C | h0 (M ) ≥ 3}.

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Consider the set of 4-tuples {(C, η, s, M ), M ∈ and its orbit F under the action of Aut S. It is easy to see that the action of Autσ S on F leaves each M invariant. In particular this establishes a natural bijection between W62 (C) and the fibre of d˜ at the isomorphism class of (C, η, s). Then the next theorem follows. W62 (C)}

 cb is the parameter space for the Autσ S-isomorphism classes of Theorem 5.4. R 6  cb → Rcb is the forgetful map. the fourtuples (C, η, s, M ) and d˜ : R 6 6 Definition 5.4. A Prym plane sextic of genus 6 is a pair (C  , η  ) such that: 1. C  ⊂ P2 is a 4-nodal plane sextic, 2. Sing C  is a set of 4 points in general position, 3. η  ∈ Pic C  is a non zero 2-torsion element 4. η  is endowed with an isomorphism η ⊗2 ∼ = OC  .

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Definition 5.5. R6ps is the moduli space of genus 6 Prym sextics (C  , η  ). C  is integral and stable of arithmetic genus 10. Let Rg be the compactification of Rg as in [FL], then (C  , η  ) defines a point in the boundary R10 \ R10 . ps This assignment defines a generically injective rational map Rps 6  R10 . R6 is ps cb  irreducible, [FV] 1. In the sequel we prove that R6 and R6 are birational. We will see that a 4-tuple (C, η, s, M ) defines uniquely a pair (C  , η  ) up to isomorphisms and conversely. Let ν : C → C  be the normalization map, ν ∗ defines the exact sequence of 2-torsion groups ν∗

0 → Z42 → Pic2 C  → Pic2 C → 0.

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It is indeed well known that Ker ν ∗ is determined by the 4-nodes of C  as C∗ 4 . Definition 5.6. We fix the notation ν ∗ η  =: η. As in above, let us denote by e ⊂ P2 × P2 the diagonal embedding of Sing C  and by Ie its ideal sheaf. Our goal is now to prove the following theorem.  cb and Rps are birational. Theorem 5.5. R 6 6 The proof relies on [FV] and [B2]. We use diagram (38) and the isomorphism h : H 0 (OP (2)) → H 0 (Ie2 (2, 2)), defined in (41). We assume that Q ∈ |Ie2 (2, 2)| is general so that Sing Q consists of 4 nodes at e. Let (z1 : z2 : z3 ) be coordinates in P2 then the equation of Q is ⎛ ⎞ z1   z1 z2 z3 A ⎝z2 ⎠ = 0, (54) z3 where A is a symmetric matrix of quadratic forms. Let u : P2 × P2 → P2 be the first projection then u |Q : Q → P2 is a conic bundle. Since Q is general every fibre of u |Q is a conic of rank ≥ 2 and the curve C  := {det A = 0} is a 4-nodal sextic, singular at u (e). For such a general Q , the matrix A defines an exact sequence of vector bundles 0 → OP2 (−4)⊕3 → OP2 (−2)⊕3 → η  → 0 A







(55)



where η is a non zero 2-torsion element of Pic C . (C , η ) is a genus 6 Prym plane sextic. Conversely a general Prym plane sextic (C  , η  ) uniquely defines a resolution of η  as above up to isomorphisms. See [B2] thm. B and [FV] 1. The resolution of η  defines a conic bundle Q ∈ |Ie2 (2, 2)| as in (54), up to the action of PGL(3) on A. Remark 5.1. Let s : C  → P2 ×P2 be the Steiner map of u |Q , s is an embedding, [FV] 2. Let M := s∗ OP2 ×P2 (1, 0), the sequence (55) implies OP2 ×P2 (0, 2) ∼ = η  ⊗ M ⊗2 .

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Proof of Theorem 5.5. From diagram (38) we have the commutative diagram φ

P2 × P2

P



u

u

σ−1

P2 .

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S



In it u is the projection onto the first factor. Moreover we have φ = ◦(σ −1 ×idP2 ) as in (40) and  = 2 ◦ −1 1 as in (38). In particular the strict transform by φ of a general element Q ∈ |OP (2)| is obtained applying to Q the projective isomorphism φ : |OP (2)| → |Ie2 (2, 2)|,

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associated to the isomorphism h−1 : H 0 (OP (2)) → H 0 (Ie2 (2, 2)) of Theorem 3.4. Let Q = φ (Q) be the strict transform of Q, then φ|Q : Q → Q is birational. Moreover the conic bundle u |Q is birationally equivalent to u|Q and fits in the diagram diagram Q

φ|Q

u

P2

Q u

σ

−1

S

Now, following a typical method under these circumstances, we use the above ps ps  cb  cb discussion to define rational maps α : R 6  R6 and β : R6  R6 which ps  cb and R are irreducible, it follows that these are inverse to each other. Since R 6 6 spaces are birational. Let (C, η, s) be general and Q ∈ |OP (2)| such that the discriminant of u|Q is (C, η) and s is the Steiner map. Consider Q = φ (Q) then the discriminant of u |Q is a genus 6 Prym sextic (C  , η  ) such that the following diagram is commutative: φ−1

s (C) ←−−−− s(C) C C ⏐ ⏐ s⏐ s ⏐ σ

C  ←−−−− C. So we have C  = σ∗ C and σ ∗ η  ∼ = η. The construction of (C  , η  ) from (C, η, s) defines a rational map ps  cb α:R (58) 6 → R6 . The inverse construction is clear: from (C  , η  ) one retrieves the sequence (55) and hence Q . The strict transform of Q by φ is Q and Q defines (C, η, s). This defines β = α−1 .  The next rationality result follows immediately from the latter theorem and (5.2). Theorem 5.6 (Theorem A). The moduli space of 4-nodal Prym sextics is rational.

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6. Moduli of Igusa pencils and of E6 -quartics In this section we study two new moduli spaces related to the previous ones, the moduli of Igusa pencils and the moduli of E6 -quartic threefolds. Both are defined as GIT quotients and we prove that both are rational. Then we discuss some rational maps between these four spaces. As we know an Igusa pencil is generated by the Igusa quartic B, and by a double quadric. Therefore we can assume that its equation is λa2 + μb = 0, where b is the equation of B and a is a non zero quadratic form. Let V ⊂ P69 := |OP4 (4)|

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be the union of the lines which are Igusa pencils as above. Then V is a cone, of vertex the element B, over the 2-Veronese embedding of P14 := |OP4 (2)|. The latter is just V2 := {2A, A ∈ P14 } ⊂ V. (60) Notice also that each Igusa pencil P contains a unique double quadric 2A, otherwise each element of P would be union of two quadrics. Let P = {λa2 + μb = 0} be general and X general in P . Then Sing X is the transversal intersection A ∩ B and consists of 30 ordinary double points, distributed in pairs on the 15 lines whose union is Sing B. Proposition 6.1. No element of P \ {B} is projectively equivalent to B. Proof. Let t : P → P4 be our usual tautological morphism and t : P → P4 its ∗ −1 Stein factorization as in (9). Then the pencil t P is generated by B := t (B) and −1 A = t (A). For a general P the finite double cover t : A → A is branched on a surface A ∩ B which is singular at the finite set Sing B ∩ A. Hence A has isolated ∗ singularities. Let u : P  P1 be the rational map defined by t P , we consider on P the following open condition: ∗ 1. every X ∈ t P \ {B} has isolated singularities, 2. the ramification scheme of u is reduced at Sing B. We claim that this condition is not empty. This easily implies the statement. To prove the claim a well-known pencil is available. This is the unique pencil of S6 invariant quartic forms: {s1 = λs22 +μs4 = 0}. This satisfes (1), see [VdG] Theorem 4.1 and [CKS] tTheorem 3.3. The proof of (2) is a straightforward computation.  Proposition 6.2. Two general Igusa pencils P1 , P2 are projectively equivalent iff ∃ ψ ∈ Aut B | P2 = ψ(P1 ). In the same way two general X1 , X2 ∈ V are projectively equivalent iff ∃ ψ ∈ Aut B | X2 = ψ(X1 ). Proof. Let P1 , P2 be projectively equivalent then P2 = ψ(P1 ) for some ψ ∈ Aut P4 . By the previous proposition no element of P2 \ {B} is projectively equivalent to B. Hence it follows ψ(B) = B and ψ ∈ Aut B. The converse is obvious. In the same way let X1 , X2 ∈ V be general. Then Xi , (i = 1, 2), belongs to a unique

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Igusa pencil contained in the cone V, say Pi = {λa2i + μb = 0}. Moreover Sing Xi is the transversal intersection Sing B ∩ Ai , where Ai = {ai = 0}. We claim that h0 (ISing Xi (2)) = 1. Under this claim assume that ψ is a projective isomorphism such that ψ(X1 ) = X2 . Then it follows ψ(Sing X1 ) = Sing X2 and ψ(A1 ) = A2 . In particular this also implies that ψ(P1 ) = P2 . Hence, by the former proof, ψ ∈ Aut B. This implies the statement up to proving the claim. For this consider the standard exact sequence of ideal sheaves 0 → ISing B (2) → ISing Xi (2) → ISing X| Sing B (2) → 0. (61) ∼ Then observe that ISing X| Sing B (2) = OSing B because Sing X is a quadratic section of Sing B, namely by the quadric A. Moreover notice that no quadric contains Sing B. Indeed a general tangent hyperplane section Y = P3 ∩ B is a general Kummer surface, [VdG]. If Sing B is in a quadric then Z = Y ∩ Sing B is in a quadric Q of P3 and Sing Y = Z ∪ {o}. By the 166 configuration of Sing Y , o belongs to a trope T that is a conic through 6 nodes. Then T ⊂ Q and Y ∩ Q contains the 16 tropes: a contradiction. Then the claim follows passing to the associated long exact sequence of (61). From this, since h0 (ISing B (2)) = 0, it follows h0 (OSing X (2)) = h0 (OSing B ) = 1.  ∼ Aut B ⊂ Aut P4 and consider the induced action of it Now recall that S6 = on V2 . The latter proposition motivates the next definitions. Definition 6.1. The moduli space of Igusa pencils is the GIT quotient P I := V2 // S6 .

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Definition 6.2. The moduli space of E6 -quartic threefolds is the GIT quotient X := V//S6 .

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One can deduce the rationality of P I from the geometry of the Segre cubic primal B ∗ ⊂ P4∗ , the dual of B appearing in Theorem 2.3. A proof can be given as follows. Theorem 6.3. The moduli space P I of Igusa pencils is rational.

 Proof. In P5 with coordinates (z1 : . . . z6 ) let P4 be the hyperplane { i=1,...,6 zi = 4 0}, we fix the standard representation of S6 on P4 , acting of  on P 4 by1permutation  4 5 the coordinates of P . The equation of B in P is b = i=1,...,6 zi − 4 ( i=1,...,6 zi2 )2 and S6 acts as Aut B on B, see the Introduction. As is well known the unique S6 -invariant cubic threefold in P4 is defined by z13 + · · · + z63 = 0

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and it is the Segre cubic primal B ∗ , up to projective equivalence. We have to prove the rationality of |OP4 (2)|//S6 . To this purpose we consider B ∗ and the standard exact sequence 0 → ISing B ∗ (2) → OP4 (2) → OSing B ∗ (2) → 0,

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ISing B ∗ being the ideal sheaf of Sing B ∗ . As is well known its long exact sequence is 0 → H 0 (ISing B ∗ (2)) → H 0 (OP4 (2)) → H 0 (OSing B ∗ (2)) → 0. Let I := H 0 (ISing B ∗ (2)), since dim I = 5 it follows that P(I) is the linear system of polar quadrics of B ∗ . This is the family of quadrics Qt such that t = (t1 : · · · : t6 ) ∈ P4 and Qt is the restriction to P4 of the polar quadric at t to the cubic { i=1,...,6 zi3 = 0} of P5 . In other words Qt = {z1 + · · · + z6 = 0} ∩ {t1 z12 + · · · + t6 z62 = 0}. Clearly S6 acts on P(I), moreover let σ ∈ S6 then σ(Qt ) = Qσ(t) . This implies that these actions of S6 on P4 and P(I) are isomorphic. Hence P(I)//S6 is rational, since it is isomorphic to P4 //S6 . Indeed the latter is the weighted projective space P(2, 3, 4, 5, 6), [D2] p. 281 and 9.1. Let us prove the rationality of P I . Of course we can fix a direct sum decomposition W := I ⊕ J of the previous representation W := H 0 (OP4 (2)) of the group S6 , [FH] 1.5. We consider the projection map p : P(W ) → P(I) and the blowing up j : P(W ) → P(W ) of P(J). Then p lifts to a projective bundle p : P(W ) → P(I). Since the action of S6 on P(W ) is linear and equivariant, p descends to a projective bundle over a non empty open set U ⊂ P(I ∗ )//S6 , say p : P(W ) //S6 → U ⊂ P(I ∗ )//S6 . See [MF] 7.1 and [SB] 7. Since U is rational we have |OP4 (2)|//S6 ∼ = P(W ) //S6 ∼ = P10 × U.



To conclude this section let us define the dominant rational map f : X  P I .

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Let X ∈ V be general and P the unique Igusa pencil containing it. Denoting by [X] and [P ] their moduli, we set by definition f ([X]) = [P ]. Then the fibre of f at [P ] is the image of P in X via the moduli map. Therefore f is a fibration in rational curves. Theorem 6.4 (Theorem C). The moduli space X is rational. Proof. Let f  := f ◦ σ, where σ : D6 → X is a birational morphism and D6 is smooth. Then the general fibre of f  is P1 . Therefore it suffices to show that f  admits a rational section s : P I  D6 . This implies that X is birational to P I × P1 and the statement follows because P I is rational. To construct s we use  → V be the exceptional divisor of the cone V blown up in its vertex v. Let σ ˜:V ˜ the exceptional divisor. Then the projection of V, from v such a blow up and E  → V2 such that E˜ is the image of the obvious onto V2 , lifts to a P1 -bundle p˜ : V   and coincides on E ˜ biregular section s˜ : V2 → V. The action of S6 on V lifts to V with the standard action of S6 on V2 . Passing to the corresponding GIT quotients, it is clear that s˜ is the pull-back of a section s of f  . 

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7. Revisiting the Prym map In the next section we prove our conclusive result, stated in Section 1 as Theorem D, relating the Prym map p : R6 → A5 and the period map j : X → A5 for the family of E6 -quartic threefolds. To this purpose we review in this section the beautiful picture of the Prym map p and Donagi’s tetragonal construction, [Do] and [DS]. The map p is in fact behind most results and geometric constructions we have seen so far. Let us recall that p : R6 → A5 associates to the moduli point of a smooth Prym curve (C, η) of genus 6 the moduli point of its Prym variety P (C, η). This is a principally polarized abelian variety of dimension 5. It is constructed from (C, η) via the ´etale double covering π : C˜ → C, defined by the nontrivial 2-torsion line bundle η ∈ Pic0 C. Actually P (C, η) is the connected component of zero of the so-called norm map N m : Pic0 C˜ → Pic0 C, sending OC˜ (d) to OC (π∗ d). It admits a natural principal polarization. The exceptional beauty of p is due to its relation to the exceptional Lie algebra E6 and to cubic surfaces. As shown by Donagi and Smith P , has degree 27, [DS]. Of course this is also the degree of the forgetful map f : C˜ → C, where C is the universal Fano variety of lines over the family C of smooth cubic surfaces in P3 . The mentioned theorem of Donagi shows that the monodromy groups of p and of f are isomorphic, [Do] Theorem 4.2. The monodromy group is isomorphic to the Weyl group W (E6 ) of E6 and acts on the fibre of f as the group preserving the incidence relation of the 27 lines of a cubic surface. Therefore we can view a general fibre F of p as the configuration of these lines. We proceed keeping this in mind. From Donagi’s construction it follows that two distinct elements of F are incident lines iff they are directly associated by such a construction as follows, see [Do] 2.5, 4.1. Let (C, η) be a general genus 6 Prym curve and l ∈ F ⊂ R6 its moduli point. Then C has exactly 5 line bundles L of degree 4 and with h0 (L) = 2, forming the Brill–Noether locus W41 (C) := {L ∈ Pic4 (C) | h0 (L) ≥ 2}. From a triple (C, η, L) one constructs as in [Do] 2.5 two new triples (C  , η  , L ) and (C  , η  , L ). They define two points l , l ∈ F so that l, l , l are coplanar lines of F. One says that l , l , l form a triality. and also that l , l are directly associated to l. After labeling by the subscript i = 1, . . . , 5, we obtain, from the five triples (C, η, Li ), exactly eleven elements of F: l, li , li . These are the lines of F incident to l. Let us fix the notation F+ = {li , li i = 1, . . . 5, } , F− = {nj , j = 1, . . . , 16}

(67)

for the set of lines respectively intersecting or non intersecting l. Then F decomposes as F = {l} ∪ F+ ∪ F− . (68)

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Now let us also recall the bijection induced by Serre duality, namely W41 (C) ←→ W62 (C). This is sending L ∈ W41 (C) to M := ωC ⊗ L−1 , where M belongs to W62 (C) and W62 (C) := {M ∈ Pic6 C | h0 (M ) ≥ 3}. The latter set consists of five line bundles, defining five sextic models C  ⊂ P2 of C modulo Aut P2 . This prompts us to give a new look at the fibre F of P in terms of Prym plane sextics. Therefore let (C  , η  ) be a Prym plane sextic of genus 6 so ∼ ν ∗ η  . As already remarked in (53) that ν : C → C  is the normalization and η = we have the exact sequence ν∗

0 → Z42 → Pic0 C2 → Pic02 C → 0 of 2-torsion groups and η  ∈ ν ∗ −1 (η). Moreover the embedding C  ⊂ P2 is determined by the line bundle M ∼ = ν ∗ OP2 (1) ∈ W62 (C) such that M ∼ = ωC ⊗ L−1 . Finally let f : Rps → R6 , (69)   be the rational map induced by the assignment s (C , η ) → (C, η, M ) → (C, η). Since |W62 (C)| = 5 and |ν ∗ −1 (η)| = 16 one can deduce as in [FV] p. 524 the next property. Proposition 7.1. The natural map f : Rps → R6 has degree 80. To continue let us fix the moduli point l ∈ F of (C, η) and recall from [Do] that the set F+ , of ten incident lines to l distinct from l, is recovered applying just once the tetragonal constructions to the five line bundles of W41 (C). In particular, fixing an element [C, η] in the fibre F of the Prym map P , one has the decomposition + + F = {[C, η]} ∪ {[Cik ], i = 1, . . . , 5 , k = 1, 2} , ηik

∪ {[Cj− , ηj− ], j = 1, . . . , 16}

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−

= {l} ∪ F ∪ F . +

Here the elements labeled by + are obtained from (C, η) applying once the tetragonal construction to each Li ∈ W41 (C), i = 1, . . . , 5. The label − corresponds to the elements of the set F− of lines disjoint from l. Starting from (C, η) these are obtained after a sequence of two tetragonal constructions applied to the elements of F. Notice that the number 16 does not appear by chance when counting the afli double sixs containg as an element the line l. Indeed there exist number of Schl¨ 36 double sixs and 36 × 12 = 27 × 16. Actually a natural bijection exists between F− and the set of double sixes containing l and n as elements not in the same six. The bijection is constructed as follows. Let n ∈ F− then a standard exercise on the configuration F of 27 lines shows that exactly five lines of n1 , . . . , n5 ∈ F− satisfy the following conditions: |ni ∩ nj | = 0 , |l ∩ nk | = 0 , |n ∩ nk | = 1,

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for 1 ≤ i = j ≤ 5 and 1 ≤ k ≤ 5. Since |l ∩ n| = 0, then n defines the six of disjoint lines {l n1 , . . . , n5 } and hence a unique double six {l n1 , . . . , n5 } , {n m1 , . . . , m5 }.

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We do not expand further this matter but for reconsidering diagram (10):  cb R 6

n

d˜

R6cb

p

Rps 6

α

u:=q◦d d

R6 p

j

X

A5

q

X Let us recall what are the maps in the diagram: α is birational and was defined in Theorem 5.5. The inclusions Autσ ⊂ Aut S ⊂ Aut B correspond to the inclusions d˜ d  cb → S4 ⊂ S5 ⊂ S6 . This defines a sequence of maps of GIT quotients R Rcb 6 6 → X. We have deg d˜ = 5, deg d = 6 and deg u = 12, since q is the degree two map defined in (9). The assignment (C, η, s) → (C, η) defines the diagonal map n and p is defined by (C  , η  ) → (C, η), where η ∼ = ν ∗ η  and ν : C → C  is the normalization. By [FV] one has deg p = 80. j is the period map as in (44) and deg p = 27. The commutativity of the diagram easily follows from the given descriptions of its maps, whose known degrees are sufficient to determine the other ones. In particular we obtain the following theorem, mentioned as B in the Introduction. Theorem 7.2 (Theorem B). The period map j has degree 36 and deg n = 16.

8. The period map j and the universal set of double sixes Since Theorem C is proven we pass to Theorem D. Let us recall from our construction in (10) that the Prym map p defines the variety D6 of pairs (s, a) such that s ⊂ p−1 (a) is a double six and a ∈ A5 is general. Also, we have the variety R of triples (l, s, a) such that (s, a) ∈ D6 and l ∈ s ⊂ p−1 (a). This gives the commutative diagram in (11): n

R −−−−→ ⏐ ⏐  d j

R6 ⏐ ⏐p 

D6 −−−−→ A5 .

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We defined j : D6 → A5 as the universal set of double sixes of p and n : R → R6 as the universal set of pointed doubles sixes of p. On the other hand the diagram n

Rcb 6 −−−−→ ⏐ ⏐  u X

R6 ⏐ ⏐p 

j

−−−−→ A5 ,

appears in our main commutative diagram (10) as a subdiagram. Theorem 8.1 (Theorem D). The period map j : X → A5 for E6 -quartic threefolds and the universal set j : D6 → A5 of double sixes of p are birational over A5 . Proof of the theorem. We use the main diagram and Donagi’s tetragonal construction for the Prym map P . Let x ∈ X be the moduli point of a general X. Applying the diagram we consider the fibre of u : R6cb → X at x. Then u−1 (x) consists of twelve Steiner maps modulo Aut S. We can put these in the following matrix: 9 : (C1 , η1 , s1 ) . . . . . . (C6 , η6 , s6 ) . (72) (C 1 , η 1 s1 ) . . . . . . (C 6 , η 6 , s1 ) Since u = q ◦ d we have put in the columns the two elements of a same fibre of q. We fix the convention that Qi and Qi are the conic bundles defined by the Steiner maps (Ci , ηi , si ) and (C i , η i , si ), where (Ci , ηi ) and (C i , η i ) are their discriminants. Let a = j(x), the moduli points in R6 of these discriminants are denoted as li and li . These are points in P −1 (a), therefore we can think of these as points of the configuration F = P −1 (a) of 27 lines of a smooth cubic surface. By generic smoothness we can assume that J ◦ u is smooth over a. Then, since j ◦ u = p ◦ n, it follows that n is an embedding along u∗ (x), in particular n|u∗ (x) is injective. Claim: Let us consider s := {l1 , l1 , . . . , l6 , l6 } = n(u−1 (x))

(73)

then s is a double six of lines in F. The claim implies that there exists a rational map φ : X  D6 , of varieties over A5 , defined as follows. Let [s] ∈ D6 be the moduli point of s, we set by definition φ(x) = [s]. Since P (s) = a it follows that the period map j factors through φ so that j = j ◦ φ. But then φ is birational because deg j = deg j .  Proof of the claim. At first we prove that li , li are skew lines of the configuration F. Assume not, then (C i , η i ) is obtained from (Ci , ηi ) after one step of the tetragonal construction as in [Do] 2.5, see Section 8. This is equivalent to say that exactly one Li in W41 (Ci ) exists so that the set realized from (Ci , ηi , Li ) after the tetragonal construction is {(Ci , ηi , Li ), (C i , η i , Li ), (C i , η i , Li )}, where (Ci , ηi ), (C i , η i ), (C i , η i ) are the coplanar lines li , li , li in the plane spanned by li , li . But this implies that the assignment to (Ci , ηi , si ) of (Ci , ηi , si , Li ) defines a

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 cb → Rcb , whose degree is 5: against the irreducibility of rational section of u : R 6 6 cb  R6 . To continue let us consider the following sets: Bi := {m ∈ s | m ∩ li = ∅ , m ∩ li = ∅}

(74)

and, more interestingly for this proof, the two sets Pi := {m ∈ s | m ∩ li = ∅ , m ∩ li = ∅} , P i := {m ∈ s | m ∩ li = ∅ , m ∩ li = ∅}. (75) We denote their cardinalities by bi and pi , pi . Now the standard monodromy ˜ E6 implies pi = pi and that pi does not depend  cb argument applied to d : R 6 → Q on i. So we denote it by p. For the same reason we denote bi by b. We claim b = 0 and at first prove the statement. b = 0 implies p ≥ 1. If not each element of u−1 (x) would correspond to a line m ∈ s which is disjoint from any other one but m. Hence F would contain 12 disjoint lines: a contradiction. Notice also that p ≤ 5 because 5 is the number of lines of a smooth cubic surface intersecting li in exactly one point and not intersecting li . We show that p = 5, which implies that s is a double six. Indeed it is a standard property of F that Pi ∪ P i ∪ {li , li } is a double six if p = 5: precisely it is the unique double six containing {li , li } as a subset intersecting both its sixes. This implies s = Pi ∪ P i ∪ {li , li } i = . . . , 6. To prove p = 5 we use monodromy again. Let Pi = {ni,1 , . . . , ni,p }, then these elements of s are lines intersecting li and not li . Working as above, the property that li and ni,k are incident uniquely reconstructs a line bundle Li,k ∈ W41 (Ci ) such that the plane spanned by li and ni,k is determined by the tetragonal construc cb → tion applied to (Ci , ηi , Li,k ). Now recall that the degree 5 morphism d˜ : R 6 cb R6 coincides with the forgetful map. Therefore the assignment to (Ci , ηi , si ) of ˜ Since R  cb is irreducible and deg d˜ = 5 (Ci , ηi , si , Lik ) defines a multisection of d. 6 it follows p = 5. Finally we show our claim that b = 0. Notice that exactly 5 lines intersect both li and li , hence b ≤ 5. Assume b ≥ 1 and, for t = 1, . . . , 6, consider the set Bt of 5 elements of s incident to lt and lt . Let I be the set of 30 pairs (Bt , nt,k ) such that nt,k ∈ Bt , k = 1, . . . , 5. Consider its projection f : I → s. Since |s| = 12, the number (m) of sets containing a given m ∈ s is not constant. But then distinct values of (m) define distinct components of Rcb 6 : against its irreducibility. This completes the proof.  Acknowledgment We are indebted to Gabi Farkas for several conversations during the preparation of the related joint paper [FV]. We also thank Ivan Cheltsov, Igor Dolgachev and Alexander Kuznetsov for a very useful correspondence on the subject of this paper. This work was partially supported by INdAM-GNSAGA and by the projects PRIN-2015 ’Geometry of Algebraic Varieties’ and PRIN-2017 ’Moduli Theory and Birational Classification’.

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