Brauer Groups and Obstruction Problems: Moduli Spaces and Arithmetic [1st ed.] 3319468510, 978-3-319-46851-8, 978-3-319-46852-5, 3319468529

Table of contents :
Front Matter....Pages i-ix
The Brauer Group Is Not a Derived Invariant....Pages 1-5
Twisted Derived Equivalences for Affine Schemes....Pages 7-12
Rational Points on Twisted K3 Surfaces and Derived Equivalences....Pages 13-28
Universal Unramified Cohomology of Cubic Fourfolds Containing a Plane....Pages 29-55
Universal Spaces for Unramified Galois Cohomology....Pages 57-86
Rational Points on K3 Surfaces and Derived Equivalence....Pages 87-113
Unramified Brauer Classes on Cyclic Covers of the Projective Plane....Pages 115-153
Arithmetically Cohen–Macaulay Bundles on Cubic Fourfolds Containing a Plane....Pages 155-175
Brauer Groups on K3 Surfaces and Arithmetic Applications....Pages 177-218
On a Local-Global Principle for H 3 of Function Fields of Surfaces over a Finite Field....Pages 219-230
Cohomology and the Brauer Group of Double Covers....Pages 231-247

Citation preview

Progress in Mathematics 320

Asher Auel Brendan Hassett Anthony Várilly-Alvarado Bianca Viray Editors

Brauer Groups and Obstruction Problems Moduli Spaces and Arithmetic

Progress in Mathematics Volume 320

Series Editors Antoine Chambert-Loir, Université Paris-Diderot, Paris, France Jiang-Hua Lu, The University of Hong Kong, Hong Kong SAR, China Yuri Tschinkel, Courant Institute of Mathematical Sciences, New York, USA

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

Asher Auel • Brendan Hassett Anthony Várilly-Alvarado • Bianca Viray Editors

Brauer Groups and Obstruction Problems Moduli Spaces and Arithmetic

Editors Asher Auel Department of Mathematics Yale University New Haven, Connecticut, USA

Brendan Hassett Department of Mathematics Brown University Providence, Rhode Island, USA

Anthony Várilly-Alvarado Department of Mathematics MS-136 Rice University Houston, Texas, USA

Bianca Viray Department of Mathematics University of Washington Seattle, Washington, USA

ISSN 0743-1643 ISSN 2296-505X (electronic) Progress in Mathematics ISBN 978-3-319-46851-8 ISBN 978-3-319-46852-5 (eBook) DOI 10.1007/978-3-319-46852-5 Library of Congress Control Number: 2017930680 Mathematics Subject Classification (2010): 14F05, 14F22, 14E08, 14G05, 14J28, 14J35, 14J60 © Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved 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, express 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.

Printed on acid-free paper This book is published under the trade name Birkhäuser, www.birkhauser-science.com The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents Nicolas Addington The Brauer Group Is Not a Derived Invariant . . . . . . . . . . . . . . . . . . . . . . .1 Benjamin Antieau Twisted Derived Equivalences for Affine Schemes . . . . . . . . . . . . . . . . . . . 7 Kenneth Ascher, Krishna Dasaratha, Alexander Perry, and Rong Zhou Rational Points on Twisted K3 Surfaces and Derived Equivalences . 13 Asher Auel, Jean-Louis Colliot-Th´el`ene, and Raman Parimala Universal Unramified Cohomology of Cubic Fourfolds Containing a Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Fedor Bogomolov and Yuri Tschinkel Universal Spaces for Unramified Galois Cohomology . . . . . . . . . . . . . . . 57 Brendan Hassett and Yuri Tschinkel Rational Points on K3 Surfaces and Derived Equivalence . . . . . . . . . . 87 Colin Ingalls, Andrew Obus, Ekin Ozman, and Bianca Viray Unramified Brauer Classes on Cyclic Covers of the Projective Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Mart´ı Lahoz, Emanuele Macr`ı, and Paolo Stellari Arithmetically Cohen–Macaulay Bundles on Cubic Fourfolds Containing a Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Kelly McKinnie, Justin Sawon, Sho Tanimoto, and Anthony V´ arilly-Alvarado Brauer Groups on K3 Surfaces and Arithmetic Applications . . . . . . 177 Alena Pirutka On a Local-Global Principle for H 3 of Function Fields of Surfaces over a Finite Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Alexei N. Skorobogatov Cohomology and the Brauer Group of Double Covers . . . . . . . . . . . . . 231

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Preface This volume grew out of a workshop that we organized at the American Institute of Mathematics from February 25 through March 1, 2013. The meeting brought together experts from two different fields: number theorists interested in rational points, and complex algebraic geometers working on derived categories of coherent sheaves. We were motivated by fresh developments in the arithmetic of K3 surfaces, which suggest that cohomological obstructions to the existence and distribution of rational points on K3 surfaces can be fruitfully studied via moduli spaces of twisted sheaves. Our aim was to encourage cross-pollination between the two fields and to explore concrete instances when the derived category of coherent sheaves on a variety over a number field determines some of its arithmetic. Within this framework, we aim to extend to K3 surfaces a number of powerful tools for analyzing rational points on elliptic curves: isogenies among curves, torsion points, modular curves, and the resulting descent techniques. Let S1 and S2 denote complex K3 surfaces. An isomorphism ∼

ι : H 2 (S1 , Z) −→ H 2 (S2 , Z) that is compatible with the cup product and Hodge structures causes S1 and S2 to be isomorphic by the Torelli theorem. We might weaken this by asking only that these lattices be stably isomorphic, or equivalently, that there is an isomorphism between the lattices of transcendental classes ∼

T (S1 ) −→ T (S2 ).

(1)

Orlov has shown that this relation coincides with the a priori algebraic notion of derived equivalence, i.e., an equivalence of the bounded derived categories of coherent sheaves ∼ Db (S2 ) −→ Db (S1 ) as triangulated categories over C. Such equivalences manifest themselves as interpretations of S2 as a moduli space of sheaves over S1 and vice versa. The manuscript by Hassett and Tschinkel explores the implications for K3 surfaces over finite and local fields. Antieau’s contribution takes a broader perspective, contrasting the local and global features of derived equivalence by looking at twisted derived equivalences among affine varieties. On a K3 surface S, we arrive at the notion of cyclic isogeny by weakening (1) further to an inclusion T (X) ,→ T (S)

(2)

with cokernel isomorphic to Z/nZ, for some projective variety X. Such an isogeny arises from, and gives rise to, an element of the Brauer group Br(S) of S, which can be lifted to an element of the second cohomology of S with Z/nZ coefficients. These cohomology classes on K3 surfaces are the analog of torsion points for elliptic curves. In 2005, van Geemen exploited the interplay between Brauer elements and cyclic isogenies to give a complete description of the order 2 Brauer vii

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classes on a degree 2 K3 surface with Picard rank 1. The paper of McKinnie, Sawon, Tanimoto, and V´ arilly-Alvarado generalizes van Geemen’s work to higher order Brauer classes of higher degree K3 surfaces. Their paper uses lattice-theoretic techniques to study the components of moduli spaces of K3 surfaces with level structure coming from the Brauer group. This is an essential step toward a comprehensive geometric interpretation for the second cohomology of a K3 surface, with applications to the structure of rational points. Van Geemen’s work is generalized in different directions in the contributions by Skorobogatov and by Ingalls, Obus, Ozman, and Viray. Skorobogatov undertakes a systematic study of 2-torsion Brauer elements on double covers of rational surfaces; Ingalls, Obus, Ozman, and Viray look at higher-degree cyclic covers of P2 . These cases should play a central role in the study of explicit examples in the future. Both contributions handle the case of arbitrary Picard rank. The isogeny (2) suggests that X provides a geometric representative of an element α ∈ Br(S)[n]. Similarly, one expects an equivalence between the derived category of twisted sheaves Db (S, α) and an admissible subcategory of Db (X). An important motivating example is when X is a smooth cubic fourfold containing a plane, in which case X is birational to a quadric surface bundle over P2 whose relative Lagrangian Grassmannian has the structure of an ´etale locally trivial P1 -bundle over a K3 surface S of degree 2, giving rise to a Brauer class α ∈ Br(S)[2]. In this case, there is an inclusion T (X) ,→ T (S), which has index 2 when α ∈ Br(S) is nontrivial, and Kuznetsov has shown that Db (S, α) is equivalent to an admissible subcategory of Db (X). The cubic fourfold X is rational as soon as α ∈ Br(S)[2] becomes trivial. Using derived category techniques, Lahoz, Macr`ı, and Stellari add to this geometric picture by providing an elegant geometric reconstruction of the K3 surface S as a moduli space of vector bundles on the cubic fourfold X. Given the tight connection between Brauer groups and derived equivalences for K3 surfaces, one could ask whether the Brauer group is invariant under derived equivalences. Addington’s contribution answers this in the negative by showing that the Brauer group fails to be invariant under derived equivalences of Calabi–Yau threefolds. The paper of Ascher, Dasaratha, Perry, and Zong also demonstrates the limitations of derived equivalence by showing that naive guesses on the relation between the structure of rational points and twisted derived equivalence for K3 surfaces are incorrect. The key role of the Brauer group naturally leads to more systematic analysis of higher degree unramified cohomology. Motivated by the rationality problem for cubic fourfolds, Auel, Colliot-Th´el`ene, and Parimala obtain the universal triviality of the unramified cohomology of degree 3 for cubic fourfolds containing a plane. More generally, inspired by a result of Merkurjev, they develop the notion of the universal triviality of the Chow group of 0-cycles as a possible obstruction to rationality, which has since been instrumental to recent breakthroughs in the (stable) rationality problem. Pirutka

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establishes a local-global criterion for the decomposability of elements in degree 3 cohomology – decompositions involving Brauer elements – for surfaces over finite fields. Finally, Bogomolov and Tschinkel offer a broad classifying space framework for unramified cohomology in higher degrees for varieties over the algebraic closure of a finite field. Acknowledgments. We are grateful to the American Institute of Mathematics for their support of this workshop and the collaborative environment that made this work possible. The first editor was partially supported by NSF grant DMS-0903039 and by NSA Young Investigator grant H98230-13-1-0291. The second editor was partially supported by NSF grant DMS-1551514. The third editor was partially supported by NSF CAREER grant DMS-1352291. And the fourth editor was partially supported by NSA Young Investigator grant H98230-15-1-0054.

The Brauer Group Is Not a Derived Invariant Nicolas Addington Abstract. In this short note we observe that the recent examples of derived-equivalent Calabi–Yau 3-folds with different fundamental groups also have different Brauer groups, using a little topological K-theory. Mathematics Subject Classification (2010). 14F05, 14F22, 14J32. Keywords. Brauer groups, derived equivalence, Calabi–Yau threefolds.

Some years ago Gross and Popescu [12] studied a simply-connected Calabi–Yau 3-fold X fibered in non-principally polarized abelian surfaces. They expected that its derived category would be equivalent to that of the dual abelian fibration Y , which is again a Calabi–Yau 3-fold but with π1 (Y ) = (Z8 )2 , the largest known fundamental group of any Calabi–Yau 3fold. This derived equivalence was later proved by Bak [2] and Schnell [23]. Ignoring the singular fibers it is just a family version of Mukai’s classic derived equivalence between an abelian variety and its dual [19], but of course the singular fibers require much more work. As Schnell pointed out, it is a bit surprising to have derived-equivalent Calabi–Yau 3-folds with different fundamental groups, since for example the Hodge numbers of a 3-fold are derived invariants [22, Cor. C]. Gross and Pavanelli [11] showed that Br(X) = (Z8 )2 , the largest known Brauer group of any Calabi–Yau 3-fold. In this note we will show that the finite abelian group H1 (X, Z) ⊕ Br(X) is a derived invariant of Calabi–Yau 3-folds; thus in this example we must have Br(Y ) = 0, and in particular the Brauer group alone is not a derived invariant. This too is a bit surprising, since the Brauer group is a derived invariant of K3 surfaces: if X is a K3 surface then Br(X) ∼ = Hom(T (X), Q/Z) [7, Lem. 5.4.1], where T (X) = N S(X)⊥ ⊂ 2 H (X, Z) is the transcendental lattice, which is a derived invariant by work of Orlov [20]. Since an earlier version of this note first circulated, Hosono and Takagi [13] have found a second example of derived-equivalent Calabi–Yau 3-folds with different fundamental groups. Their X and Y are constructed from spaces of 5×5 symmetric matrices in what is likely an instance of homological © Springer International Publishing AG 2017 A. Auel (eds.) et al., Brauer Groups and Obstruction Problems, Progress in Mathematics 320, DOI 10.1007/978-3-319-46852-5_1

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projective duality [14], and one has π1 (X) = Z2 and π1 (Y ) = 0. While Br(X) and Br(Y ) are not known, from our result we see that Br(Y ) ∼ = Z2 ⊕ Br(X), so they are different. An explicit order-2 element of Br(Y ) arises naturally in Hosono and Takagi’s construction [13, Prop. 3.2.1]. It is worth mentioning that both π1 and Br are birational invariants, so while birational Calabi–Yau 3-folds are derived equivalent [5], the converse is not true. In addition to the two examples just mentioned, there is the Pfaffian–Grassmannian derived equivalence of Borisov and C˘ald˘araru [4]. In that example X is a complete intersection in a Grassmannian, so H1 (X, Z) = Br(X) = 0, so from our result we see that H1 (Y, Z) = Br(Y ) = 0 as well; to show that X and Y are not birational Borisov and C˘ald˘araru use a more sophisticated minimal model program argument. Before proving our result we fix terminology. Definition. A Calabi–Yau 3-fold is a smooth complex projective 3-fold X with ωX ∼ = OX and b1 (X) = 0. In particular H1 (X, Z) may be torsion. This is in contrast to the case of surfaces, where ωX ∼ = OX and b1 (X) = 0 force π1 (X) = 0 [17, Thm. 13]. There are several reasons not to require π1 (X) = 0 for Calabi–Yau 3-folds. As we have just seen, a simply-connected Calabi–Yau 3-fold may be derived equivalent to a non-simply-connected one; it may also be mirror to a non-simply-connected one. Perhaps the best reason is that families of simply-connected and non-simply-connected Calabi–Yau 3folds can be connected by “extremal transitions,” that is, by performing a birational contraction and then smoothing; most known families of Calabi– Yau 3-folds can be connected by extremal transitions [10, 18]. Definition. The Brauer group of a smooth complex projective variety X is 2 ∗ Br(X) = tors(Han (X, OX )),

where tors denotes the torsion subgroup. This used to be called the cohomological Brauer group until it was shown to coincide with the honest Brauer group [8]. From the exact sequence ∗ H 2 (X, OX ) → H 2 (X, OX ) → H 3 (X, Z) → H 3 (X, OX )

we see that if X is a Calabi–Yau 3-fold then Br(X) = tors(H 3 (X, Z)). That is, the Brauer group of a Calabi–Yau 3-fold is entirely topological, in contrast to that of a K3 surface which is entirely analytic. Proposition. Let X and Y be Calabi–Yau 3-folds with Db (X) ∼ = Db (Y ). Then ∼ H1 (X, Z) ⊕ Br(X) = H1 (Y, Z) ⊕ Br(Y ). Proof. Brunner and Distler [6, §2.5] analyzed the boundary maps in the Atiyah–Hirzebruch spectral sequence and saw that for a Calabi–Yau 3-fold X, or indeed any closed oriented 6-manifold with b1 (X) = 0, it degenerates at the E2 page. Thus there is a short exact sequence 1 0 → H 5 (X, Z) → Ktop (X) → H 3 (X, Z) → 0,

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∗ where Ktop (X) is topological K-theory. Since H 5 (X, Z) = H1 (X, Z) is torsion, this gives an exact sequence 1 0 → H1 (X, Z) → tors(Ktop (X)) → Br(X) → 0.

(1)

While it is not strictly necessary for our purposes, they also got an exact sequence 0 0 → Br(X)∗ → tors(Ktop (X)) → H1 (X, Z)∗ → 0; (2) ∗ here if A is a finite abelian group then the dual group A := Hom(A, Q/Z), which is non-canonically isomorphic to A. ∗ (X) more carefully using the Doran and Morgan [9, §4] analyzed Ktop fact that c1 (X) = 0 and showed that the sequences (1) and (2) are in fact split. 0 1 Now the proposition follows from the fact that Ktop and Ktop are deb rived invariants [1, §2.1]. In a bit more detail, if Φ : D (X) → Db (Y ) and Ψ : Db (Y ) → Db (X) are inverse equivalences, then by [20, Thm. 2.2] there are objects E, F ∈ Db (X × Y ) such that ∗ Φ(−) = πY ∗ (E ⊗ πX (−)),

Ψ(−) = πX∗ (F ⊗ πY∗ (−)),

and arguing as in [16, Lem. 5.32] we find that the same formulas define inverse isomorphisms K ∗ (X) → K ∗ (Y ) and K ∗ (Y ) → K ∗ (X): use the fact that the pushforward on K ∗ satisfies a projection formula and is compatible with the pushforward on Db .  We conclude with a remark on H1 and Br in mirror symmetry. Batyrev and Kreuzer [3] predicted that mirror symmetry exchanges H1 and Br, having calculated both groups for all Calabi–Yau hypersurfaces in 4-dimensional toric varieties. In all their examples the groups are quite small: either H1 = 0 and Br = Z2 , Z3 , or Z5 , or vice versa. This prediction does not seem to be right in general. On the one hand it is contradicted by a prediction of Gross and Pavanelli [11, Rem. 1.5], based on calculations in Pavanelli’s thesis [21], that if X is the abelian fibration above, with H1 (X) = 0 and Br(X) = (Z8 )2 , ˇ has π1 (X) ˇ = Br(X) ˇ = Z8 . Even more seriously, Hosono then its mirror X and Takagi’s X and Y have the same mirror according to [15], but different H1 and Br as we have discussed. Mirror symmetry is expected to exchange 0 1 and Ktop , however, so mirror Calabi–Yau 3-folds should have the same Ktop H1 ⊕ Br. Acknowledgments. I thank P. Aspinwall and A. C˘ald˘araru for helpful discussions, and S. Hosono and H. Takagi for encouraging me to publish this note.

References [1] N. Addington and R. Thomas. Hodge theory and derived categories of cubic fourfolds. Duke Math. J., 163(10):1885–1927, 2014. Also arXiv:1211.3758. [2] A. Bak. The spectral construction for a (1,8)-polarized family of abelian varieties. Preprint, arXiv:0903.5488.

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[3] V. Batyrev and M. Kreuzer. Integral cohomology and mirror symmetry for Calabi-Yau 3-folds. In Mirror symmetry. V, volume 38 of AMS/IP Stud. Adv. Math., pages 255–270. Amer. Math. Soc., Providence, RI, 2006. Also math/0505432. [4] L. Borisov and A. C˘ ald˘ araru. The Pfaffian-Grassmannian derived equivalence. J. Algebraic Geom., 18(2):201–222, 2009. Also math/0608404. [5] T. Bridgeland. Flops and derived categories. Invent. Math., 147(3):613–632, 2002. Also math/0009053. [6] I. Brunner and J. Distler. Torsion D-branes in nongeometrical phases. Adv. Theor. Math. Phys., 5(2):265–309, 2001. Also hep-th/0102018. [7] A. C˘ ald˘ araru. Derived categories of twisted sheaves on Calabi–Yau manifolds. PhD thesis, Cornell University, 2000. Available at math.wisc.edu/~andreic/ publications/ThesisSingleSpaced.pdf. [8] A. J. de Jong. A result of Gabber. Available at math.columbia.edu/~dejong/ papers/2-gabber.pdf. [9] C. Doran and J. Morgan. Algebraic topology of Calabi-Yau threefolds in toric varieties. Geom. Topol., 11:597–642, 2007. Also math.AG/0605074. [10] P. Green and T. H¨ ubsch. Connecting moduli spaces of Calabi–Yau threefolds. Comm. Math. Phys., 119(3):431–441, 1988. [11] M. Gross and S. Pavanelli. A Calabi-Yau threefold with Brauer group (Z/8Z)2 . Proc. Amer. Math. Soc., 136(1):1–9, 2008. Also math/0512182. [12] M. Gross and S. Popescu. Calabi-Yau threefolds and moduli of abelian surfaces. I. Compositio Math., 127(2):169–228, 2001. Also math/0001089. [13] S. Hosono and H. Takagi. Double quintic symmetroids, Reye congruences, and their derived equivalence. Preprint, arXiv:1302.5883. [14] S. Hosono and H. Takagi. Duality between S 2 P4 and the double quintic symmetroid. Preprint, arXiv:1302.5881. [15] S. Hosono and H. Takagi. Determinantal quintics and mirror symmetry of Reye congruences. Comm. Math. Phys., 329(3):1171–1218, 2014. Also arXiv:1208.1813. [16] D. Huybrechts. Fourier-Mukai transforms in algebraic geometry. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford, 2006. [17] K. Kodaira. On the structure of compact complex analytic surfaces. I. Amer. J. Math., 86:751–798, 1964. [18] M. Kreuzer and H. Skarke. Complete classification of reflexive polyhedra in four dimensions. Adv. Theor. Math. Phys., 4(6):1209–1230, 2000. Also hepth/0002240. ˆ with its application to Picard [19] S. Mukai. Duality between D(X) and D(X) sheaves. Nagoya Math. J., 81:153–175, 1981. [20] D. Orlov. Equivalences of derived categories and K3 surfaces. J. Math. Sci. (New York), 84(5):1361–1381, 1997. Also alg-geom/9606006. [21] S. Pavanelli. Mirror symmetry for a two parameter family of Calabi-Yau threefolds with Euler characteristic 0. PhD thesis, University of Warwick, 2003.

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[22] M. Popa and C. Schnell. Derived invariance of the number of holomorphic 1´ Norm. Sup´er. (4), 44(3):527–536, 2011. forms and vector fields. Ann. Sci. Ec. Also arXiv:0912.4040. [23] C. Schnell. The fundamental group is not a derived invariant. In Derived categories in algebraic geometry, EMS Ser. Congr. Rep., pages 279–285. Eur. Math. Soc., Z¨ urich, 2012. Also arXiv:1112.3586. Nicolas Addington Department of Mathematics University of Oregon Eugene, OR 97403-1222 USA e-mail: [email protected]

Twisted Derived Equivalences for Affine Schemes Benjamin Antieau Abstract. We show how work of Rickard and To¨en completely resolves the question of when two twisted affine schemes are derived equivalent. Mathematics Subject Classification (2010). 14F22, 14F05, 16D90. Keywords. Brauer groups and twisted derived equivalences.

1. Introduction The question of when Db (X) is equivalent as a k-linear triangulated category to Db (Y ) for two varieties X and Y over a field k has been extensively studied ˆ ' Db (A) for an abelian variety A and its dual since Mukai proved that Db (A) Aˆ [8]. Since in general A and Aˆ are not isomorphic, derived equivalence is a weaker condition than isomorphism. However, derived equivalence nevertheless does preserve a great deal of information: derived equivalent varieties have the same dimension, the same algebraic K-theory, and the same Hochschild homology. The cohomological Brauer group of a scheme X is Br0 (X) = H´e2t (X, Gm )tors . When X is quasi-compact, there is an inclusion Br(X) ⊆ Br0 (X), where Br(X) denotes the Brauer group of X, which classifies Brauer equivalence classes of Azumaya algebras on X. In many cases of interest, Br(X) = Br0 (X). Examples include all quasi-projective schemes over affine schemes [3]. The Brauer group comes into play because in many problems on moduli of vector bundles, there is an obstruction, living in the Brauer group of the coarse moduli space, to the existence of a universal vector bundle. Another way to say this is that this class in the Brauer group is the obstruction to the coarse moduli space being fine. See [1, Introduction]. At times one then obtains an equivalence Db (X) ' Db (Y, β), where Db (Y, β) is the derived category of β-twisted coherent sheaves. Particular cases of this arise in the study of K3 © Springer International Publishing AG 2017 A. Auel (eds.) et al., Brauer Groups and Obstruction Problems, Progress in Mathematics 320, DOI 10.1007/978-3-319-46852-5_2

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surfaces for example, and especially in the work of Huybrechts, Macr`ı, and Stellari [4–6]. The systematic study of when Db (X, α) ' Db (Y, β) began with C˘ ald˘araru’s thesis [1]. In this short note, we are interested in the following two problems. Problem 1.1. Let R be a commutative ring, let X and Y be two locally noetherian R-schemes, and fix α ∈ Br0 (X) and β ∈ Br0 (Y ). Determine when there exists an R-linear equivalence of triangulated categories Db (X, α) ' Db (Y, β). Problem 1.2. Let R be a commutative ring, let X and Y be two quasicompact and quasi-separated R-schemes, and fix α ∈ Br0 (X) and β ∈ Br0 (Y ). Determine when there exists an R-linear equivalence of triangulated categories Dperf (X, α) ' Dperf (Y, β). Here, Db (X, α) denotes the bounded derived category of α-twisted coherent sheaves, while Dperf (X, α) is the triangulated category of perfect complexes of α-twisted OX -modules. When X is regular and noetherian, the existence locally of finite-length finitely generated locally free resolutions implies that the natural map Dperf (X, α) → Db (X, α) is an equivalence of R-linear triangulated categories. Question 1.3. Are Problems 1.1 and 1.2 equivalent for X and Y locally noetherian, quasi-compact, and quasi-separated? The contents of our paper are as follows. In Section 2, we give some background on twisted derived categories and equivalences between them. Then, in Section 3, the affine case of Problems 1.1 and 1.2 is solved completely, and we explain how work of Rickard shows that these two problems are equivalent for affine schemes. We do not claim that this result is new, but rather that it is not as well-known as it should be. Acknowledgments. We would like to thank the organizers of the AIM workshop “Brauer groups and obstruction problems” for facilitating a stimulating week in February 2013.

2. Twisted derived categories Let X be a scheme, and take α ∈ H´e2t (X, Gm ). Then, α is represented by a Gm -gerbe X → X. There is a good notion of quasi-coherent sheaf on X, or of coherent sheaf when X is locally noetherian. An OX -module F comes naturally with a left action of the sheaf Gm,X . Moreover, there is a second, inertial action, which can be described as saying that a section u ∈ Gm,X (U ) over U → X acts on F(U ) via the isomorphism u∗ FU → FU , which induces an isomorphism u∗ : F(U ) → F(U ). There is an associated left action of the inertial action. An α-twisted OX -module is by definition an OX -module F for which these two left actions agree. It is shown in [7, Proposition 2.1.3.3] that this agrees with the definition of α-twisted sheaf given by C˘ald˘araru.

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If X is a scheme and α ∈ H´e2t (X, Gm ), write Dperf (X, α) for the triangulated category of complexes of α-twisted sheaves that are ´etale locally quasi-isomorphic to finite-length complexes of vector bundles. This is naturally a full subcategory of Dqc (X, α) of complexes of α-twisted sheaves with (α-twisted) quasi-coherent cohomology sheaves. If X is regular and noetherian, then Dperf (X, α) ' Db (X, α), the bounded derived category of α-twisted coherent OX -modules. Let A be an Azumaya algebra on X with class α (so that α ∈ Br(X) in this case). A complex of right A-modules P (in the abelian category ModOX ) is perfect if there is an open affine cover {Ui }i∈I of X such that PUi is quasi-isomorphic to a bounded complex of finitely generated projective right Γ(Ui , A)-modules. The derived category of perfect complexes of right Amodules will be denoted Dperf (X, A). Then, as explained in [1], Dperf (X, α) ' Dperf (X, A). In the same way, there is a big derived category of all complexes of right A-modules with quasi-coherent cohomology sheaves Dqc (X, A) and an equivalence Dqc (X, α) ' Dqc (X, A). In the next section, we will need dg enhancements of these categories. Write Perf(X, α) and QC(X, α) for dg enhancements of Dperf (X, α) and Dqc (X, α), respectively. These are pretriangulated dg categories. The big dg category QC(X, α) is constructed for example in To¨en [12]. The small dg category Perf(X, α) can then be taken to be the dg category of compact objects in QC(X, α).

3. Twisted derived equivalences over affine schemes Many of us first learned of twisted derived categories from C˘ald˘araru’s thesis [1] and the paper [2]. In that paper, C˘ald˘araru cites a private communication from Yekutieli giving the following theorem [1, Theorem 6.2]. Theorem 3.1. Suppose that R is a commutative local ring and that A and B are Azumaya R-algebras with classes α, β ∈ Br(R). Then, the following are equivalent: 1. α = β in Br(R); 2. A and B are derived Morita equivalent over R – that is, there is an R-linear equivalence of triangulated categories Db (A) ' Db (B). It is the purpose of our paper to advertise the fact that the condition that R be local is unnecessary. Theorem 3.2. Suppose that R is a commutative ring and that α, β ∈ Br(R). Then, the following are equivalent: 1. α = β in Br(R); 2. there is an R-linear equivalence of triangulated categories Dperf (R, α) ' Dperf (R, β). Moreover, if R is noetherian, these are equivalent to:

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3. there is an R-linear equivalence of triangulated categories Db (R, α) ' Db (R, β). Proof. Let X = Spec R. Since α, β ∈ Br(R), we can assume that α is represented by an Azumaya R-algebra A, and that β is represented by an Azumaya R-algebra B. In this case, Dperf (X, α) ' Dperf (X, A) ' Db (projA ), where Db (projA ) is the bounded derived category of finitely generated projective A-modules. The second equivalence follows because on an affine scheme, every perfect complex is quasi-isomorphic to a bounded complex of finitely generated projectives (see [11, Proposition 2.3.1(d)]), and this generalizes in a straightforward way to Azumaya algebras on an affine scheme. When α = β, the Azumaya algebras A and B are Brauer equivalent. This means that there exist finitely generated projective R-modules M and N and an R-algebra isomorphism A ⊗R EndR (M ) ∼ = B ⊗R EndR (N ). It follows from classical Morita theory that there is an equivalence ModA ' ModB of abelian categories of right A and B-modules. From this it follows immediately that Db (projA ) ' Db (projB ). This proves that (1) implies (2). So, suppose that Dperf (R, α) ' Dperf (R, β), or in other words that Db (projA ) ' Db (projB ). Rickard’s theorem [9, Theorem 6.4] as refined in [10] implies that there is a tilting complex inducing an R-linear equivalence Db (projA ) ' Db (projB ). (Rickard’s theorem does not imply that this is the equivalence we began with, but it is still R-linear.) The existence of the tilting complex implies that there is an equivalence of R-linear dg category enhancements Perf(R, α) ' Perf(R, β), which is then a derived Morita equivalence. Recall that QC(R, α) is equivalent to the derived category of dg modules over Perf(R, α), and similarly for QC(R, β). Hence, the equivalence of small dg categories Perf(R, α) ' Perf(R, β) induces an equivalence of the “big” R-linear dg categories QC(R, α) ' QC(R, β). These are locally presentable dg categories with descent in the language of [12]. Now, the derived Brauer group of R, denoted dBr(R), classifies locally presentable dg categories with descent over R that are ´etale locally equivalent to QC(R). Since Spec R is affine, the R-linear equivalence QC(R, α) ' QC(R, β) means that α and β define the same element of dBr(R) (see [12, Section 3]). But, dBr(R) ∼ = H´e1t (Spec R, Z) × H´e2t (Spec R, Gm ) by [12, Theorem 1.1]. Since Br(R) ⊆ dBr(R), it follows that α = β, and so (2) implies (1). Finally, the fact that (2) and (3) are equivalent follows from [9, Propositions 8.1, 8.2]. This completes the proof.  Remark 3.3. By [3], Br(R) = Br0 (R) = H´e2t (Spec R, Gm )tors . We expand briefly on the philosophy of the proof. Write QC for the ´etale stack of locally presentable dg categories with dg category of sections over f : Y → X the locally presentable dg category QC(Y ), which is a dg category enhancement of the derived category Dqc (Y ) of complexes of OY -modules with quasi-coherent cohomology sheaves. The derived Brauer group dBr(X)

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of a scheme classifies stacks of locally presentable dg categories that are ´etale locally equivalent to QC up to equivalence of stacks. Motto 3.4. The Brauer group classifies Azumaya algebras A up to derived Morita equivalence of stacks of dg categories of complexes of A-modules. For α ∈ dBr(X), write QC(α) for the associated stack. For instance, if α is the Brauer class of an Azumaya algebra A over X then the dg category of sections over f : Y → X of QC(α) is QC(Y, f ∗ A), which is a dg category enhancement of Dqc (Y, A) ' Dqc (Y, α). The key point in the proof of the theorem was that over an affine scheme Spec R, giving an equivalence of stacks QC(α) ' QC(β) is equivalent to giving an R-linear equivalence of the global sections QC(R, α) ' QC(R, β). On non-affine schemes, giving an equivalence of global sections is, not surprisingly, insufficient. The following example is due to C˘ald˘araru [1, Example 1.3.16]. Let X be a smooth projective K3 surface over the complex numbers given as a double cover of P2 branched along a smooth sextic curve. The involution φ of X given by interchanging the sheets of the cover has the property that φ∗ α = −α for α ∈ Br(X). Clearly φ induces an equivalence Db (X, α) ' Db (X, −α). But, since Br(X) contains non-zero p-torsion for every prime p, there is a class α ∈ Br(X) such that α 6= −α. Thus, the theorem fails in the non-affine case. The problem is that the equivalence does not respect restriction to open subsets of X. We now prove the conjecture suggested by C˘ald˘araru after [2, Theorem 6.2]. Corollary 3.5. Suppose that R and S are commutative rings and that there is an equivalence of triangulated categories Dperf (R, α) ' Dperf (S, β) for α ∈ Br(R) and β ∈ Br(S). Then, there exists a ring isomorphism φ : R → S such that φ∗ (α) = β in Br(S). Proof. Let A be an Azumaya algebra with class α over R, and let B be an Azumaya algebra over S with class β. Then, our hypotheses say that Db (projA ) ' Db (projB ). By Rickard [9, Proposition 9.2], the centers of A and B are isomorphic. As before, by Rickard [9, Proposition 9.2], there is an isomorphism φ : R → S, and there are equivalences Dperf (S, β) ' Dperf (R, α) ' Dperf (S, φ∗ (α)). The composition induces a ring automorphism σ : S → S. So, by composing on the φ on the right with σ −1 , we can assume that Dperf (S, β) ' Dperf (S, φ∗ (α)) is S-linear. The corollary follows now from Theorem 3.2.  We end by observing that the condition of R-linearity in Theorem 3.2 is necessary. Remark 3.6. Consider the field k = C(w, x, y, z) and the quaternion division algebras (w, x) and (y, z) over k. Then, these algebras are evidently derived Morita equivalent over C (they are even isomorphic over C). However, [(w, x)] 6= [(y, z)] in Br(k).

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References ald˘ araru, Derived categories of twisted sheaves on Calabi-Yau manifolds. Ph.D. [1] A. C˘ thesis, Cornell University (2000), available at http://www.math.wisc.edu/~andreic/. [2] , Derived categories of twisted sheaves on elliptic threefolds, J. Reine Angew. Math. 544 (2002), 161–179. [3] A. J. de Jong, A result of Gabber, available at http://www.math.columbia.edu/ ~dejong/. [4] D. Huybrechts and P. Stellari, Equivalences of twisted K3 surfaces, Math. Ann. 332 (2005), no. 4, 901–936. , Proof of C˘ ald˘ araru’s conjecture. Appendix: “Moduli spaces of twisted sheaves [5] on a projective variety” by K. Yoshioka, Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math., vol. 45, Math. Soc. Japan, Tokyo, 2006, pp. 31–42. [6] D. Huybrechts, E. Macr`ı, and P. Stellari, Derived equivalences of K3 surfaces and orientation, Duke Math. J. 149 (2009), no. 3, 461–507. [7] M. Lieblich, Moduli of twisted sheaves, Duke Math. J. 138 (2007), no. 1, 23–118. ˆ with its application to Picard sheaves, [8] S. Mukai, Duality between D(X) and D(X) Nagoya Math. J. 81 (1981), 153–175. [9] J. Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456. , Derived equivalences as derived functors, J. London Math. Soc. (2) 43 (1991), [10] no. 1, 37–48. [11] R. W. Thomason and T. Trobaugh, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkh¨ auser Boston, Boston, MA, 1990, pp. 247–435. [12] B. To¨ en, Derived Azumaya algebras and generators for twisted derived categories, Invent. Math. 189 (2012), no. 3, 581–652.

Benjamin Antieau Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago 851 S. Morgan St. Chicago, IL 60607 USA e-mail: [email protected]

Rational Points on Twisted K3 Surfaces and Derived Equivalences Kenneth Ascher, Krishna Dasaratha, Alexander Perry and Rong Zhou Abstract. Using a construction of Hassett and V´ arilly-Alvarado, we produce derived equivalent twisted K3 surfaces over Q, Q2 , and R, where one has a rational point and the other does not. This answers negatively a question recently raised by Hassett and Tschinkel. Mathematics Subject Classification (2010). 14G05, 14J28, 18E30. Keywords. Rational points, twisted K3 surfaces, derived categories.

1. Introduction A twisted K3 surface is a pair (X, α), where X is a K3 surface and α ∈ Br(X) is a Brauer class. In a recent survey paper [5], Hassett and Tschinkel asked whether the existence of a rational point on a twisted K3 surface is invariant under derived equivalence. More precisely, they asked: Question. Let (X1 , α1 ) and (X2 , α2 ) be twisted K3 surfaces over a field k. Suppose there is a k-linear equivalence Db (X1 , α1 ) ' Db (X2 , α2 ) of twisted derived categories. Then is the existence of a k-point of (X1 , α1 ) equivalent to the existence of a k-point of (X2 , α2 )? By definition, a k-point of a twisted K3 surface (X, α) is a point x ∈ X(k) such that the evaluation α(x) = 0 ∈ Br(k). Equivalently, it is a k-point of the Gm -gerbe over X associated to α. In [5], it is shown that for the untwisted case of the question where α1 , α2 vanish, the answer is positive over certain fields k, e.g., R, finite fields, and p-adic fields (provided the Xi have good reduction, or p ≥ 7 and the Xi This project was supported by the 2015 AWS and NSF grant DMS-1161523. K.A. was also partially supported by NSF grant DMS-1162367. A.P. was also partially supported by NSF GRFP grant DGE1144152.

© Springer International Publishing AG 2017 A. Auel (eds.) et al., Brauer Groups and Obstruction Problems, Progress in Mathematics 320, DOI 10.1007/978-3-319-46852-5_3

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have ADE reduction). The purpose of this paper is to show that if α1 , α2 are allowed to be nontrivial, the answer to the question is negative for k = Q, Q2 , and R. We work over a field k of characteristic not equal to 2, and consider a double cover Y → P2 × P2 ramified over a divisor of bidegree (2, 2). The projection πi : Y → P2 onto the i-th P2 factor, i = 1, 2, realizes Y as a quadric fibration. Provided that the discriminant divisor of πi is smooth, the Stein factorization of the relative Fano variety of lines of πi is a K3 surface Xi , which comes with a natural Brauer class αi . In this setup, we prove the following result. Theorem 1.1. There is a k-linear equivalence Db (X1 , α1 ) ' Db (X2 , α2 ). We note that this result seems to be known to the experts (at least for k = C), but we could not find a proof in the literature. Hassett and V´ arilly-Alvarado studied the above construction of twisted K3s in relation to rational points [6]. They show that over k = Q, if certain conditions are imposed on the branch divisor Z ⊂ P2 × P2 of Y , the class α1 gives a (transcendental) Brauer–Manin obstruction to the Hasse principle on X1 . A priori, α2 need not obstruct the existence of rational points on X2 . In fact, it is possible that X2 has rational points, but the conditions imposed on Z result in very large coefficients of the defining equation of X2 , making a computer search for points infeasible. In this paper, we observe that the 2-adic condition imposed by Hassett and V´arilly-Alvarado can be relaxed, while still guaranteeing α1 gives a Brauer–Manin obstruction (see Lemma 4.5). The upshot is that the defining coefficients of X2 are much smaller, making it easy to find rational points with a computer. Up to modifying the αi by a Brauer class pulled back from k = Q, we obtain the desired example over Q. We also check the example “localizes” over Q2 and R. More precisely, we prove: Theorem 1.2. For k = Q, Q2 , or R, the divisor Z ⊂ P2 × P2 can be chosen so that there are Brauer classes αi0 ∈ Br(Xi ), congruent to αi modulo Im(Br(k) → Br(Xi )), such that: 1. There is a k-linear equivalence Db (X1 , α10 ) ' Db (X2 , α20 ), 2. (X1 , α10 ) has no k-point, 3. (X2 , α20 ) has a k-point. Acknowledgements. The authors thank the organizers of the 2015 Arizona Winter School, where this work began. We thank Asher Auel, Brendan Hassett, and Sho Tanimoto for useful discussions. Above all, we thank Tony V´arilly-Alvarado for suggesting this project, and for his help and encouragement along the way.

2. Construction of the twisted K3 surfaces In this section, k denotes a base field of characteristic not equal to 2.

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2.1. Quadric fibrations We start by reviewing some terminology on quadric fibrations. Let S be a variety over k, i.e., an integral, separated scheme of finite type over k. Let E be a rank n ≥ 2 vector bundle on S, i.e., a locally free OS -module of rank n. Our convention is that the projective bundle of E is the morphism p : P(E) = ProjS (Sym• (E∗ )) → S. A quadric fibration is determined by a line bundle L on S and a nonzero section s ∈ Γ(P(E), OP(E) (2) ⊗ p∗ L) = Γ(S, Sym2 (E∗ ) ⊗ L). Namely, the zero locus of s on P(E) defines a subvariety Q, and the restriction π : Q → S of p : P(E) → S is the associated quadric fibration, which if flat is of relative dimension n − 2. Below we will be specifically interested in flat quadric fibrations of relative dimension 2, which we refer to as quadric surface fibrations. Note that the section of Sym2 (E∗ ) ⊗ L defining a quadric fibration corresponds to a morphism q : E → E∗ ⊗ L. Taking the determinant gives rise to a section of det(E∗ )2 ⊗ Ln whose vanishing defines the discriminant locus D ⊂ S, which is a divisor provided π : Q → S is generically smooth. The fibration π : Q → S is said to have simple degeneration if the fiber over every closed point of S is a quadric of corank ≤ 1. We note that if π : Q → S is flat and generically smooth and S is smooth over k, then the discriminant divisor D is smooth over k if and only if Q is smooth over k and π has simple degeneration [1, Proposition 1.6]. 2.2. Twisted K3 surfaces Let V1 and V2 be 3-dimensional vector spaces over k. We denote by Hi the hyperplane class on P(Vi ); by abuse of notation, we denote by the same letter the pullback of Hi to any variety mapping to P(Vi ). Let π : Y → P(V1 ) × P(V2 ) be the double cover of P(V1 ) × P(V2 ) ramified over a smooth divisor Z in the linear system |2H1 + 2H2 |. Let pri : P(V1 ) × P(V2 ) → P(Vi ) be the i-th projection, and let πi = pri ◦ π : Y → P(Vi ). Lemma 2.1. Let E1 = (V2 ⊗ O) ⊕ O(H1 ) on P(V1 ) and E2 = (V1 ⊗ O) ⊕ O(H2 ) on P(V2 ). Then for i = 1, 2 there is a commutative diagram Y πi

 { P(Vi )

ji

/ P(Ei )

pi

where ji is a closed immersion, j1∗ OP(E1 ) (1) = OY (H2 ), j2∗ OP(E2 ) (1) = OY (H1 ). Moreover, Y is cut out on P(Ei ) by a section of OP(Ei ) (2) ⊗ O(2Hi ), so that πi is a quadric surface fibration.

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Proof. Consider the case i = 1. The morphism j1 : Y → P(E1 ) is given by the π1 -very ample line bundle OY (H2 ). More precisely, using that π∗ (OY ) = O ⊕ O(−H1 − H2 ), we find π1∗ (OY (H2 )) = pr1∗ (O(H2 ) ⊕ O(−H1 )) = (V2∗ ⊗ O) ⊕ O(−H1 ) = E∗1 . Working locally on P(V1 ), we see the canonical map π1∗ E∗1 = π1∗ π1∗ (OY (H2 )) → OY (H2 ) is surjective and the corresponding morphism j1 : Y → P(E1 ) is an immersion. By construction j1∗ OP(E1 ) (1) = OY (H2 ). Moreover, if ζ denotes the class of OP(E1 ) (1) in Pic(P(E1 )), then it is easy to compute [Y ] = 2ζ + 2H1 ∈ Pic(P(E1 )) by using the intersection numbers H12 H22 = 2 and H1 H23 = 0 on Y . So Y is indeed a quadric surface fibration, cut out by a section of OP(E1 ) (2) ⊗ O(2H1 ) on P(E1 ).  Let Di denote the discriminant divisor of πi : Y → P(Vi ). It follows from the lemma that Di is defined by a section of det(E∗i )2 ⊗ O(8Hi ) = O(6Hi ), i.e., Di ⊂ P(Vi ) is a sextic curve. Let fi : Xi → P(Vi ) be the double cover of P(Vi ) ramified over Di . If Di is smooth (equivalently, if πi has simple degeneration), then Xi is a smooth K3 surface. Moreover, Xi comes equipped with an Azumaya algebra Ai , as follows. In general, suppose given a generically smooth quadric surface fibration π : Q → S over a smooth k-variety S, with smooth discriminant divisor and simple degeneration. Let F → S be the relative Fano variety of lines of π. It follows from [7, Proposition 3.3] that Stein factorization gives morphisms g

f

F −→ X −→ S, where g is an ´etale locally trivial P1 -bundle over X and f is the double cover of S branched along the discriminant divisor D. The morphism g corresponds to an Azumaya algebra A on X. Applying this discussion to πi : Y → P(Vi ), we see that if Di is smooth, then Xi is equipped with an Azumaya algebra Ai . Of course Ai represents a Brauer class αi ∈ Br(Xi ), so we can regard the pair (Xi , Ai ) as a twisted K3 surface.

3. Derived equivalence of the twisted K3 surfaces In this section, we prove the twisted K3 surfaces (Xi , Ai ) of the previous section are derived equivalent. Our proof works over any field k of characteristic not equal to 2, and gives an explicit functor inducing the equivalence. The key tool is Kuznetsov’s semiorthogonal decomposition of the derived category of a quadric fibration [9].

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3.1. Conventions All triangulated categories appearing below will be k-linear, and functors between them will be k-linear and exact. For a variety X, we denote by Db (X) the bounded derived category of coherent sheaves on X, regarded as a triangulated category. More generally, for any sheaf of OX -algebras A which is coherent as an OX -module, we denote by Db (X, A) the bounded derived category of coherent sheaves of right Amodules on X. We note that if A is an Azumaya algebra corresponding to a Brauer class α ∈ Br(X), then the bounded derived category of α-twisted sheaves Db (X, α) is equivalent to Db (X, A). As a rule, all functors we consider are derived. More precisely, for a morphism of varieties f : X → Y , we simply write f∗ : Db (X) → Db (Y ) for the derived pushforward (provided f is proper) and f ∗ : Db (Y ) → Db (X) for the derived pullback (provided f has finite Tor-dimension). Similarly, for F, G ∈ Db (X), we write F ⊗ G ∈ Db (X) for the derived tensor product. 3.2. Semiorthogonal decompositions One way to understand the derived category of a variety (or more generally a triangulated category) is by “decomposing” it into simpler pieces. This is formalized by the notion of a semiorthogonal decomposition, which plays a central role in the rest of this section. We summarize the rudiments of this theory; see, e.g., [3] and [4] for a more detailed exposition. Definition 3.1. Let T be a triangulated category. A semiorthogonal decomposition T = hA1 , . . . , An i is a sequence of full triangulated subcategories A1 , . . . , An of T – called the components of the decomposition – such that: 1. Hom(F, G) = 0 for all F ∈ Ai , G ∈ Aj if i > j. 2. For any F ∈ T, there is a sequence of morphisms 0 = Fn → Fn−1 → · · · → F1 → F0 = F, such that Cone(Fi → Fi−1 ) ∈ Ai . Semiorthogonal decompositions are closely related to the notion of an admissible subcategory of a triangulated category. Such a subcategory A ⊂ T is by definition a full triangulated subcategory whose inclusion i : A ,→ T admits right and left adjoints i! : T → A and i∗ : T → A. For X a smooth proper variety over k, the components of any semiorthogonal decomposition of Db (X) are in fact admissible subcategories. The simplest examples of admissible subcategories come from exceptional objects. An object F ∈ T of a triangulated category is called exceptional if  k if p = 0, Hom(F, F[p]) = 0 if p 6= 0. If X is a proper variety and F ∈ Db (X) is exceptional, then the full triangulated subcategory hFi ⊂ Db (X) generated by F is admissible and equivalent

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to the derived category of a point via Db (Spec(k)) → Db (X) : V 7→ V ⊗F. To simplify notation, we write F in place of hFi when hFi appears as a component in a semiorthogonal decomposition, i.e., instead of Db (X) = h. . . , hFi, . . . i we write Db (X) = h. . . , F, . . . i. Example 3.2. It is easy to see any line bundle on projective space Pn is exceptional as an object of Db (Pn ). In fact, Beilinson [2] showed Db (Pn ) has a semiorthogonal decomposition into n + 1 line bundles, namely Db (Pn ) = hO, O(1), . . . , O(n)i. Given one semiorthogonal decomposition of a triangulated category T, others can be obtained via mutation functors. If i : A ,→ T is the inclusion of an admissible subcategory, the left and right mutation functors LA : T → T and RA : T → T are defined by the formulas LA (F) = Cone(ii! F → F)

and

RA (F) = Cone(F → ii∗ F)[−1],

where ii! F → F and F → ii∗ F are the counit and unit morphisms of the adjunctions. These functors satisfy the following basic properties. Lemma 3.3. The mutation functors LA and RA annihilate A. Moreover, they restrict to mutually inverse equivalences ∼

LA |⊥ A : ⊥ A − → A⊥

and

∼

RA |A⊥ : A⊥ − → ⊥ A,

where A⊥ and ⊥ A are the right and left orthogonal categories to A, i.e., the full subcategories of T defined by A⊥ = {F ∈ T | Hom(G, F) = 0 for all G ∈ A} , ⊥

A = {F ∈ T | Hom(F, G) = 0 for all G ∈ A} .

The following lemma describes the action of mutation functors on a semiorthogonal decomposition. Lemma 3.4. Let T = hA1 , . . . , An i be a semiorthogonal decomposition with admissible components. Then for 1 ≤ i ≤ n − 1 there is a semiorthogonal decomposition T = hA1 , . . . , Ai−1 , LAi (Ai+1 ), Ai , Ai+2 , . . . , An i, and for 2 ≤ i ≤ n there is a semiorthogonal decomposition T = hA1 , . . . , Ai−2 , Ai , RAi (Ai−1 ), Ai+1 , . . . , An i. We will also need the following lemma, which allows us to compute the effect of a mutation functor in a special case. It follows easily from Serre duality. Lemma 3.5. Let X be a smooth projective variety over k. Assume that we are given a semiorthogonal decomposition Db (X) = hA1 , . . . , An i. Then we have LhA1 ,...,An−1 i (An ) = An ⊗ ωX , where An ⊗ ωX denotes the image of An under the autoequivalence F 7→ F ⊗ ωX of Db (X).

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3.3. Derived categories of quadric fibrations Let π : Q → S be a quadric fibration associated to a rank n vector bundle E and a section of Sym2 (E∗ ) ⊗ L, as in Section 2.1. Then there is an associated even Clifford algebra C`0 , which is a sheaf of algebras on S given as a certain quotient of the tensor algebra T• (E ⊗ E ⊗ L∗ ). For the precise definition, see [1, Section 1.5] (cf. [9, Section 3.3]). We note that C`0 admits an OS module filtration of length b n2 c with associated graded pieces ∧2i E ⊗ (L∗ )i . In case the fibration π : Q → S is flat and S is smooth over k, Kuznestov [9] established a semiorthogonal decomposition of Db (Q) into a copy of Db (S, C`0 ) and a number of copies of Db (S). In fact, Kuznetsov stated his result under the assumption that k is algebraically closed of characteristic 0, but as explained in [1, Theorem 2.11], the proof works without this hypothesis. Theorem 3.6 ([9, Theorem 4.2]). Let π : Q → S be a flat quadric fibration of relative dimension n − 2 over a smooth k-variety S. Let OQ (1) denote the restriction of OP(E) (1) to Q. Then the functor π ∗ : Db (S) → Db (Q) is fully faithful, and there is a fully faithful functor Φ : Db (S, C`0 ) → Db (Q) such that there is a semiorthogonal decomposition Db (Q) = hΦ(Db (S, C`0 )), π ∗ Db (S) ⊗ OQ (1), . . . , π ∗ Db (S) ⊗ OQ (n − 2)i. Remark 3.7. The functor Φ : Db (S, C`0 ) → Db (Q) is given by an explicit Fourier–Mukai kernel, see [9, Section 4]. Now assume π : Q → S is a generically smooth quadric surface fibration over a smooth k-variety S, with smooth discriminant divisor and simple degeneration. As in the discussion at the end of Section 2.2, the double cover f : X → S ramified over D is equipped with an Azumaya algebra A. In terms of this data, we have the following alternative description of Db (S, C`0 ), see [1, Proposition B.3] or [10, Lemma 4.2]. Lemma 3.8. In the above situation, there is an isomorphism f∗ A ∼ = C`0 . In ∼ particular, there is an equivalence f∗ : Db (X, A) − → Db (S, C`0 ). 3.4. Derived equivalence Let π : Y → P(V1 ) × P(V2 ) be as in Section 2.2. Assume the discriminant divisors Di of the quadric fibrations πi : Y → P(Vi ) are smooth, so that we get associated twisted K3 surfaces (Xi , Ai ). Let C`0,i denote the even Clifford algebra of the quadric fibration πi : Y → P(Vi ). Then Lemma 3.8 ∼ gives an equivalence fi∗ : Db (Xi , Ai ) − → Db (P(Vi ), C`0,i ). Finally, denote by b b Φi : D (P(Vi ), C`0,i ) → D (Y ) the fully faithful functor from Theorem 3.6. In this setup, we prove the following result. Theorem 3.9. Assume D1 and D2 are smooth. Then there is an equivalence Db (X1 , A1 ) ' Db (X2 , A2 ) given by the composition −1 f2∗ ◦ Φ∗2 ◦ ROY (H2 ) ◦ LOY (H1 ) ◦ Φ1 ◦ f1∗ : Db (X1 , A1 ) → Db (X2 , A2 ),

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where • • • •

LOY (H1 ) is the left mutation functor through hOY (H1 )i ⊂ Db (Y ), ROY (H2 ) is the right mutation functor through hOY (H2 )i ⊂ Db (Y ), Φ∗2 is the left adjoint of Φ2 , −1 f2∗ is the inverse of the equivalence ∼

f2∗ : Db (X2 , A2 ) − → Db (P(V2 ), C`0,2 ). The theorem is an immediate consequence of the following proposition. We note that the proposition holds without assuming smoothness of the discriminant divisors Di . Proposition 3.10. There is an equivalence Db (P(V1 ), C`0,1 ) ' Db (P(V2 ), C`0,2 ) given by the composition Φ∗2 ◦ ROY (H2 ) ◦ LOY (H1 ) ◦ Φ1 : Db (P(V1 ), C`0,1 ) → Db (P(V2 ), C`0,2 ). Proof. Set Ci = Φi (Db (P(Vi ), C`0,i )) ⊂ Db (Y ). Theorem 3.6 gives semiorthogonal decompositions Db (Y ) = hC1 , π1∗ Db (P(V1 )) ⊗ O(H2 ), π1∗ Db (P(V1 )) ⊗ O(2H2 )i, Db (Y ) = hC2 , π2∗ Db (P(V2 )) ⊗ O(H1 ), π2∗ Db (P(V2 )) ⊗ O(2H1 )i. Recall Beilinson’s decomposition Db (P(Vi )) = hO, O(Hi ), O(2Hi )i (see Example 3.2). In each of the above decompositions of Db (Y ), we replace the first copy of Db (P(Vi )) by Beilinson’s decomposition and the second copy by the same decomposition twisted by O(Hi ): Db (Y ) = hC1 , O(H2 ), O(H1 + H2 ), O(2H1 + H2 ), O(H1 + 2H2 ), O(2H1 + 2H2 ), O(3H1 + 2H2 )i, Db (Y ) = hC2 , O(H1 ), O(H1 + H2 ), O(H1 + 2H2 ), O(2H1 + H2 ), O(2H1 + 2H2 ), O(2H1 + 3H2 )i.

(3.1) (3.2)

We perform a sequence of mutations that identifies the categories generated by the exceptional objects in (3.1) and (3.2). First consider (3.1). Mutate O(3H1 + 2H2 ) to the far left of the decomposition. Note that Y is smooth with canonical class KY = −2H1 − 2H2 , so by Lemma 3.5 the result of the mutation is Db (Y ) = hO(H1 ), C1 , O(H2 ), O(H1 + H2 ), O(2H1 + H2 ), O(H1 + 2H2 ), O(2H1 + 2H2 )i. Left mutating C1 through O(H1 ) then gives a decomposition Db (Y ) = hLO(H1 ) C1 , O(H1 ), O(H2 ), O(H1 + H2 ), O(2H1 + H2 ), O(H1 + 2H2 ), O(2H1 + 2H2 )i.

(3.3)

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By the same argument, we obtain from (3.2) a similar decomposition Db (Y ) = hLO(H2 ) C2 , O(H2 ), O(H1 ), O(H1 + H2 ), O(H1 + 2H2 ), O(2H1 + H2 ), O(2H1 + 2H2 )i.

(3.4)

Up to permutation, the exceptional objects in the decompositions (3.3) and (3.4) agree, hence they generate the same subcategory of Db (Y ). It follows that LO(H1 ) C1 and LO(H2 ) C2 coincide, as both are the right orthogonal to the same subcategory. Now the proposition follows since RO(H2 ) ◦ LO(H2 ) ∼ = id on ⊥ hO(H2 )i by Lemma 3.3.  Remark 3.11. The equivalence Db (X1 , α1 ) ' Db (X2 , α2 ) of Theorem 3.9 implies other relations between X1 and X2 . For instance, over k = C it implies the Picard numbers of X1 and X2 agree. Indeed, it suffices to note that the equivalence induces an isomorphism of twisted transcendental lattices T(X1 , α1 ) ∼ = T(X2 , α2 ), whose ranks are the same as the usual transcendental lattices (see [8]).

4. Equations for the twisted K3 surfaces and local invariants Let k be a number field. Then for any place v of k, class field theory provides an embedding invv : Br(kv ) → Q/Z (which is an isomorphism for nonarchimedian v). Now let X be a smooth, projective, geometrically integral variety over k. Any subset S ⊂ Br(X) cuts out a subset X(Ak )S ⊂ X(Ak ) of the adelic points of X, given by ( ) X S X(Ak ) = (xv ) ∈ X(Ak ) | invv α(xv ) = 0 for all α ∈ S . v

For fixed (xv ) ∈ X(Ak ) and α ∈ Br(X), the evaluation α(xv ) = 0 for all but finitely many v, so the above sum is well-defined. Moreover, class field theory gives inclusions X(k) ⊂ X(Ak )S ⊂ X(Ak ). Hence, if X(Ak )S is empty for some S, then X has no k-points. We note that if X(Ak )S is empty but X(Ak ) is not, then S is said to give a Brauer–Manin obstruction to the Hasse principle. See [12, 5.2] for more details. In this section, we describe conditions on the divisor Z ⊂ P(V1 )×P(V2 ) from Section 2.2, which allow us to control the local invariants invv α1 (xv ) for any v-adic point xv ∈ X1 (kv ). In the end, we will see that if k = Q and enough conditions are met, then ( 0 if v is finite, invv α1 (xv ) = 1 if v is real, 2 for all xv ∈ X1 (kv ). Hence X1 (Ak )α1 is empty and X1 has no k-points. Our discussion follows [6] very closely, and differs only in the treatment of the 2-adic place (see Lemma 4.5).

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4.1. Equations for the twisted K3 surfaces Let the notation be as in Section 2.2 (in particular k may be any field of characteristic not equal to 2). Choose coordinates x0 , x1 , x2 on P(V1 ) and y0 , y1 , y2 on P(V2 ). The equation defining Z can be written as A(x0 , x1 , x2 )y02 + B(x0 , x1 , x2 )y0 y1 + C(x0 , x1 , x2 )y0 y2 + D(x0 , x1 , x2 )y12 + E(x0 , x1 , x2 )y1 y2 + F (x0 , x1 , x2 )y22 ,

(4.1)

where A, . . . , F are degree 2 homogeneous polynomials in the xi , or as A0 (y0 , y1 , y2 )x20 + B 0 (y0 , y1 , y2 )x0 x1 + C 0 (y0 , y1 , y2 )x0 x2 + D0 (y0 , y1 , y2 )x21 + E 0 (y0 , y1 , y2 )x1 x2 + F 0 (y0 , y1 , y2 )x22 .

(4.2)

where A0 , . . . , F 0 are degree 2 homogeneous polynomials in the yi . The first or second expression is useful depending on whether we regard Y as a quadric fibration over P(V1 ) or P(V2 ). The following lemma summarizes the computations of [6, Section 3]. Lemma 4.1. (1) Let  2A B M =  B 2D C E

 C E . 2F

Then the discriminant curve D1 ⊂ P(V1 ) is defined by det(M ) = 0, and X1 is defined in the weighted projective space P(1, 1, 1, 3) with coordinates x0 , x1 , x2 , w by 1 w2 = − det(M ). 2 The analogous statements hold for D2 ⊂ P(V2 ) and X2 with M replaced by  0  2A B0 C0 M 0 =  B 0 2D0 E 0  . C0 E 0 2F 0 (2) Define MA = 4DF − E 2 ,

MD = 4AF − C 2 ,

MF = 4AD − B 2 .

Assume D1 ⊂ P(V1 ) is smooth, so that we have a twisted K3 surface (X1 , α1 ). Then the image of α1 under the injection Br(X1 ) → Br(k(X1 )) (where k(X1 ) is the function field of X1 ) can be represented by any of the following Hilbert symbols: (−MF , A), (−MD , A), (−MF , D), (−MA , D), (−MD , F ), (−MA , F ). 0 0 For MA0 0 , MD 0 , MF 0 defined similarly, the analogous statement holds for α2 ∈ Br(X2 ).

From now on, we assume D1 ⊂ P(V1 ) is smooth, so that (X1 , α1 ) is defined.

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4.2. Conditions controlling the local invariants The following result holds by Proposition 4.1 and Lemma 4.2 of [6]. It allows us to control local invariants at finite places of bad reduction, assuming the place is not 2-adic and the singularities are mild. Proposition 4.2. Let F be a finite extension of Qp for p 6= 2, and denote by OF the ring of integers of F . Let X be a K3 surface over F . Let X → Spec(OF ) be a flat, proper morphism from a regular scheme X, with generic fiber Xη ∼ = X. Assume the singular locus of the geometric special fiber Xs consists of less than 8 points, each of which is an ordinary double point. If X(F ) 6= ∅, then for any 2-power torsion Brauer class α ∈ Br(X)[2∞ ], the map X(F ) → Br(F ) given by evaluation of α is constant. In particular, α(x) = 0 for all x ∈ X(F ) if this holds for a single x. The next result is [6, Lemma 4.4]. It guarantees that the local invariants of α1 ∈ Br(X1 ) vanish at finite places of good reduction, away from the prime 2. Lemma 4.3. Let k be a number field. Let v be a finite place of good reduction for X1 which is not 2-adic. Then invv α1 (x) = 0 for all x ∈ X1 (kv ). We are left to control the real and 2-adic invariants of α1 ∈ Br(X1 ). The following result, which is [6, Corollary 4.6], gives conditions which guarantee α1 is nontrivial at any real point of X1 . Lemma 4.4. Let k = Q. Assume the polynomials A, . . . , F from (4.1), when regarded as quadratic forms, satisfy: 1. A, D, and F are negative definite, 2. B, C, and E are positive definite. If ∞ denotes the real place, then inv∞ α1 (x) = 1/2 for all x ∈ X1 (R). The following lemma improves [6, Lemma 4.7], giving conditions such that α1 is trivial at every 2-adic point of X1 . Lemma 4.5. Let k = Q. Write the polynomials A, . . . , F ∈ Q[x0 , x1 , x2 ] from (4.1) as A = A1 x20 + A2 x0 x1 + A3 x0 x2 + A4 x21 + A5 x1 x2 + A6 x22 , B = B1 x20 + B2 x0 x1 + B3 x0 x2 + B4 x21 + B5 x1 x2 + B6 x22 , C = C1 x20 + C2 x0 x1 + C3 x0 x2 + C4 x21 + C5 x1 x2 + C6 x22 , D = D1 x20 + D2 x0 x1 + D3 x0 x2 + D4 x21 + D5 x1 x2 + D6 x22 , E = E1 x20 + E2 x0 x1 + E3 x0 x2 + E4 x21 + E5 x1 x2 + E6 x22 , F = F1 x20 + F2 x0 x1 + F3 x0 x2 + F4 x21 + F5 x1 x2 + F6 x22 . Suppose the coefficients of A, . . . , F satisfy: 1. The 2-adic valuation of A1 , B1 , C6 , D4 , E4 , and F6 is 0. 2. The 2-adic valuation of all other coefficients is > 0. Then inv2 α1 (x) = 0 for all x ∈ X1 (Q2 ).

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Proof. Let x = [x0 , x1 , x2 , w] be a point of X1 (Q2 ) ⊂ P(1, 1, 1, 3)(Q2 ). By scaling the coordinates, we may assume x0 , x1 , x2 ∈ Z2 and at least one of the xi is a unit. By Lemma 4.1, the Hilbert symbols (B 2 − 4AD, A),

(E 2 − 4DF, D),

(C 2 − 4AF, F )

all represent the image of α1 in Br(k(X)). According to whether x0 , x1 , or x2 is a 2-adic unit, the first, second, or third of these representatives can be used to see inv2 α1 (x) = 0. For instance, suppose x0 is a 2-adic unit. Then by our assumptions on coefficients, A(x) and B(x)2 − 4A(x)D(x) are also 2-adic units. In particular, they are nonzero, so (B(x)2 − 4A(x)D(x), A(x))2 represents α1 (x) ∈ Br(Q2 ). Recall (see for example [11, p. 20, Theorem 1]) that if s, t ∈ Z× 2 , then s−1 t−1 (s, t)2 = (−1) 2 2 . But by our assumptions B(x)2 − 4A(x)D(x) ≡ 1 (mod 4), so the formula gives (B(x)2 − 4A(x)D(x), A(x))2 = 1. Thus α1 (x) = 0 ∈ Br(Q2 ). The same argument works when x1 or x2 is a 2-adic unit, using the other representatives for α1 from above. 

5. Proof of Theorem 1.2 Consider the following quadrics in Z[x0 , x1 , x2 ]: A = −5x20 + 4x0 x2 − 4x21 + 2x1 x2 − 4x22 , B = 5x20 + 2x0 x1 − 2x0 x2 + 2x21 + 2x1 x2 + 4x22 , C = 4x20 + 2x0 x1 − 4x0 x2 + 2x21 − 2x1 x2 + 5x22 , D = −4x20 − 2x0 x1 − x21 − 2x1 x2 − 4x22 , E = 4x20 + 3x21 + 4x22 , F = −4x20 + 4x0 x1 + 2x0 x2 − 2x21 − 4x1 x2 − 5x22 . Inserting these polynomials in (4.1) gives the equation of a (2,2) divisor Z ⊂ P(V1 ) × P(V2 ), which we regard as a variety over Q. As in Section 2.2, Z gives rise to a branched double cover π : Y → P(V1 ) × P(V2 ), which is a quadric fibration via projection to each factor. Lemma 4.1 gives explicit equations for the discriminant curves Di ⊂ P(Vi ), and the Jacobian criterion can be used to

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25

check the Di are smooth. Hence, by Section 2.2, we get associated twisted K3 surfaces (Xi , αi ), which we will use to prove Theorem 1.2. Remark 5.1. The quadrics A, . . . , F above were found using the algorithm described in [6, Section 6], modified in two ways. First, we omitted the steps related to checking the geometric Picard number of X1 is 1, since it was not our goal to produce an example with this property. Second, instead of using [6, Lemma 4.7] to constrain the quadrics, we used our Lemma 4.5, which results in much smaller coefficients. Indeed, the equations for X1 and X2 are: w2 = − 4x60 − 308x50 x1 − 190x40 x21 − 278x30 x31 − 203x20 x41 − 40x0 x51 − 28x61 + 18x50 x2 + 460x40 x1 x2 + 276x30 x21 x2 + 474x20 x31 x2 + 40x0 x41 x2 + 98x51 x2 − 25x40 x22 − 820x30 x1 x22 − 247x20 x21 x22 − 374x0 x31 x22 − 2x41 x22 + 20x30 x32 + 652x20 x1 x32 + 14x0 x21 x32 + 270x31 x32 − 20x20 x42 − 562x0 x1 x42 − 105x21 x42 − 8x0 x52 + 166x1 x52 − 4x62 , w2 = 236y06 − 740y05 y + 1268y04 y12 − 1092y03 y13 + 624y02 y14 − 164y0 y15 + 32y16 − 616y05 y2 + 416y04 y1 y2 − 96y03 y12 y2 − 976y02 y13 y2 + 548y0 y14 y2 − 288y15 y2 + 1236y04 y22 − 456y03 y1 y22 + 1484y02 y12 y22 − 356y0 y13 y22 + 676y14 y22 − 1332y03 y23 − 804y02 y1 y23 − 372y0 y12 y23 − 1024y13 y23 + 1036y02 y24 + 768y0 y1 y24 + 812y12 y24 − 472y0 y25 − 388y1 y25 + 40y26 . In contrast, most of the coefficients appearing in the corresponding equations in [6] have 5 or 6 digits. Smaller coefficients are crucial in making a computer search for points of X2 feasible. The following proposition is reduced to a series of computations by the results in Section 4.2. We postpone its proof to the end of the section. Proposition 5.2. (1) X1 (Qv ) 6= ∅ for all places v, or equivalently X1 (AQ ) 6= ∅. (2) The local invariants of the class α1 ∈ Br(X1 ) satisfy ( 0 if v is finite, invv α1 (xv ) = 1 if v is real, 2 for all xv ∈ X1 (Qv ). In particular, α1 obstructs the existence of Q-points on X1 , and hence gives a Brauer–Manin obstruction to the Hasse principle. Using the proposition, we construct the examples which prove Theorem 1.2. The necessary computations that appear below were carried out using Magma1 . 1 Code

available at http://www.math.brown.edu/~kenascher/magma/magma.html.

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5.1. Example over Q Using the equation for X2 given by Lemma 4.1, it can be checked that x = [1, 1, 1, 0] ∈ P(1, 1, 1, 3) is a Q-point of X2 . Let β = α2 (x) ∈ Br(Q), let βi be the constant class given by the image of β under Br(Q) → Br(Xi ), and set αi0 = βi−1 αi . Then there is a Q-linear equivalence Db (X1 , α10 ) ' Db (X2 , α20 ) induced by the equivalence Db (X1 , α1 ) ' Db (X2 , α2 ) of Theorem 3.9. By Proposition 5.2, X1 has no Q-point, so a fortiori the pair (X1 , α10 ) has no Q-point. On the other hand, by construction x is a Q-point of (X2 , α20 ). 5.2. Example over Q2 Replace the pairs (Xi , αi ) defined above √over Q by their base changes to Q2 . It can be checked that x = [−3, −1, 1, 357008] ∈ P(1, 1, 1, 3) is a Q2 -point of X2 (note that Hensel’s lemma can be used to see 357008 is a 2-adic square). One then checks that β = α2 (x) = (B(x)2 − 4A(x)D(x), A(x))2 is nontrivial. Let βi be the constant class given by the image of β under Br(Q) → Br(Xi ), and set αi0 = βi−1 αi . Then there is a Q2 -linear equivalence Db (X1 , α10 ) ' Db (X2 , α20 ) induced by the equivalence of Theorem 3.9. By Proposition 5.2, α1 (y) is trivial for any y ∈ X1 (Q2 ), and hence α10 (y) = α2−1 (x)α1 (y) is nontrivial (since α2 (x) is). Thus (X1 , α10 ) has no Q2 -points. On the other hand, by design x is a point of (X2 , α20 ). 5.3. Example over R Replace the pairs (Xi , αi ) by their base changes to R. Then Theorem 3.9 still gives an R-linear equivalence Db (X1 , α1 ) ' Db (X2 , α2 ). Moreover, Proposition 5.2 shows α1 (x) is nontrivial for any x ∈ X1 (R), so (X1 , α1 ) has no R-points. On the other hand, using Lemma 4.1, it can be checked that the point √ x = [4, 3, 3, 5204] ∈ P(1, 1, 1, 3) lies on X2 and that α2 (x) = (B(x)2 − 4A(x)D(x), A(x))∞ is trivial. Hence x is an R-point of (X2 , α2 ). 5.4. Proof of Proposition 5.2 5.4.1. Local points. We first check that X1 (Qv ) 6= ∅ for all v. This is obvious when v = ∞. Let v = p be a finite prime of good reduction with p > 22. Then if (X1 )p is a smooth reduction of X1 at p, there is an Fp -point of (X1 )p by the Weil conjectures. This lifts to a Qp -point of X1 by Hensel’s lemma. It therefore suffices to check that X1 (Qp ) 6= ∅ for primes p of bad reduction for X1 and for all primes p < 22. A Gr¨obner basis calculation as in [6, Section 5.1] can be used to show the primes of bad reduction for X1 are: 2, 5, 7, 307, 4591, 27077, 371857, 6902849, 104388233, 541264119547919951, 6097863609641310921149279, 2616678388926286398002864469014842817095009312844790479.

Rational Points on Twisted K3 Surfaces and Derived Equivalences

27

In the table below, we list for each prime p of bad reduction and each p < 22 the (x0 , x1 , x2 ) coordinates of a Qp -point of X1 . (By Lemma 4.1, (x0 , x1 , x2 ) gives a Qp -point if − 12 det(M )(x0 , x1 , x2 ) is a square in Qp , which can be checked using Hensel’s lemma). p 2 3 5 7 11 13 17 19 307 4591 27077 371857 6902849 104388233 541264119547919951 6097863609641310921149279 2616678388926286398002864469014842817095009312844790479

(x0 , x1 , x2 ) (-1,0,-1) (-1,-1,1) (-1,-1,0) (-1,-1,1) (-1,-1,0) (-1,-1,1) (-1,-1,-1) (-1,-1,-1) (-1,-1,-1) (-1,-1,0) (-1,-1,-1) (-1,-1,-1) (-1,0,0) (-1,-1,-1) (-1,-1,1) (-1,1,-1) (-1,-1,0)

5.4.2. Local invariants. One computes that for each prime p 6= 2 of bad reduction, X1,Qp satisfies the assumptions of Proposition 4.2. Moreover, using the representatives for α1 given in Lemma 4.1, it can be computed that α1 is trivial when evaluated at the Qp -points specified in the table above. We conclude by Proposition 4.2 that invv α1 (xv ) = 0 at the non-2-adic finite places v of bad reduction. On the other hand, at the non-2-adic finite places of good reduction, we also have invv α1 (xv ) = 0 by Lemma 4.3. Finally, it is straightforward to check the quadrics A, . . . , F satisfy the hypotheses of Lemmas 4.4 and 4.5. The conclusions of these lemmas give Proposition 5.2(2) at the real and 2-adic places. 

References [1] Asher Auel, Marcello Bernardara, and Michele Bolognesi. Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems. J. Math. Pures Appl. (9), 102(1):249–291, 2014. [2] Alexander Beilinson. Coherent sheaves on Pn and problems in linear algebra. Funktsional. Anal. i Prilozhen., 12(3):68–69, 1978. [3] Alexei Bondal. Representations of associative algebras and coherent sheaves. Izv. Akad. Nauk SSSR Ser. Mat., 53(1):25–44, 1989. [4] Alexei Bondal and Mikhail Kapranov. Representable functors, Serre functors, and mutations. Mathematics of the USSR-Izvestiya, 35(3):519–541, 1990.

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[5] Brendan Hassett and Yuri Tschinkel. Rational points on K3 surfaces and derived equivalence. In A. Auel, B. Hassett, A. V´ arilly-Alvarado, and B. Viray, editors, Brauer Groups and Obstruction Problems: Moduli Spaces and Arithmetic, volume 320 of Progress in Mathematics, pages 87–113, 2017. [6] Brendan Hassett and Anthony V´ arilly-Alvarado. Failure of the Hasse principle on general K3 surfaces. J. Inst. Math. Jussieu, 12(4):853–877, 2013. [7] Brendan Hassett, Anthony V´ arilly-Alvarado, and Patrick Varilly. Transcendental obstructions to weak approximation on general K3 surfaces. Adv. Math., 228(3):1377–1404, 2011. [8] Daniel Huybrechts and Paolo Stellari. Equivalences of twisted K3 surfaces. Math. Ann., 332(4):901–936, 2005. [9] Alexander Kuznetsov. Derived categories of quadric fibrations and intersections of quadrics. Adv. Math., 218(5):1340–1369, 2008. [10] Alexander Kuznetsov. Derived categories of cubic fourfolds. In Cohomological and geometric approaches to rationality problems, volume 282 of Progress in Mathematics, pages 219–243. Birkh¨ auser Boston, Inc., Boston, MA, 2010. [11] J.-P. Serre. A Course in Arithmetic. Springer-Verlag, New York-Heidelberg, 1973. [12] Alexei Skorobogatov. Torsors and Rational Points, volume 144 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2001. Kenneth Ascher Mathematics Department, Brown University Providence, RI USA e-mail: [email protected] Krishna Dasaratha Department of Economics Harvard University Cambridge, MA USA e-mail: [email protected] Alexander Perry Department of Mathematics Columbia University New York, NY USA e-mail: [email protected] Rong Zhou Department of Mathematics Harvard University Cambridge, MA USA e-mail: [email protected]

Universal Unramified Cohomology of Cubic Fourfolds Containing a Plane Asher Auel, Jean-Louis Colliot-Th´el`ene and Raman Parimala Abstract. We prove the universal triviality of the third unramified cohomology group of a very general complex cubic fourfold containing a plane. The proof uses results on the unramified cohomology of quadrics due to Kahn, Rost, and Sujatha. Mathematics Subject Classification (2010). 11E20, 11E88, 14C25, 14D06, 14E08, 14E08, 14F22, 14J28. Keywords. Unramified cohomology, 0-cycle, universal triviality of CH0 , complex surfaces, cubic fourfold, quadric fibration, Clifford algebra.

Introduction Let X be a smooth cubic fourfold, i.e., a smooth cubic hypersurface in P5 . A well-known problem in algebraic geometry concerns the rationality of X over C. Expectation. The very general cubic fourfold over C is irrational. Here, “very general” is usually taken to mean “in the complement of a countable union of Zariski closed subsets” in the moduli space of cubic fourfolds. At present, however, not a single cubic fourfold is provably irrational, though many families of rational cubic fourfolds are known. If X contains a plane P (i.e., a linear two-dimensional subvariety of P5 ), e → P2 then X is birational to the total space of a quadric surface bundle X 2 by projecting from P . Its discriminant divisor D ⊂ P is a sextic curve. The rationality of X in this case is also a well-known problem. Expectation. The very general cubic fourfold containing a plane over C is irrational. Assuming that the discriminant divisor D is smooth, the discriminant double cover S → P2 branched along D is then a K3 surface of degree 2 e → P2 gives rise to a and the even Clifford algebra of the quadric fibration X © Springer International Publishing AG 2017 A. Auel (eds.) et al., Brauer Groups and Obstruction Problems, Progress in Mathematics 320, DOI 10.1007/978-3-319-46852-5_4

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Brauer class β ∈ Br(S), called the Clifford invariant of X. This invariant does not depend on the choice of P (if β is not zero, the plane P is actually the unique plane contained in X). By classical results in the theory of quadratic forms (e.g., [46, Thm. 6.3]), β is trivial if and only if the quadric surface e → P2 has a rational section. Thus if β is trivial, then X is rational bundle X (see [38, Thm. 3.1]), though it may happen that X is rational even when β is not trivial (see [4, Thm. 11]). Some families of rational cubic fourfolds have been described by Fano [32], Tregub [58], [59], and Beauville–Donagi [8]. In particular, pfaffian cubic fourfolds, defined by pfaffians of skew-symmetric 6×6 matrices of linear forms, are rational. Hassett [37] describes, via lattice theory, Noether–Lefschetz divisors Cd in the moduli space C of cubic fourfolds. In particular, C14 is the closure of the locus of pfaffian cubic fourfolds and C8 is the locus of cubic fourfolds containing a plane. Hassett [38] identifies countably many divisors of C8 – the union of which is Zariski dense in C8 – consisting of rational cubic fourfolds with trivial Clifford invariant. A natural class of birational invariants arises from unramified cohomology groups. The unramified cohomology of a rational variety is trivial (i.e., reduces to the cohomology of the ground field), cf. [18, Thm. 1.5] and [19, §2 and Thm. 4.1.5]. Such invariants have been known to provide useful obstructions to rationality. For a smooth cubic fourfold X over C, the unramified cohomology groups i Hnr (X/C, Q/Z(i − 1)) vanish for 0 ≤ i ≤ 3, see Theorem 2.4. The case i = 1 follows from the Kummer sequence (see Theorem 2.4). For i = 2, one appeals to the Leray spectral sequence and a version of the Lefschetz hyperplane theorem due to M. Noether, as in [52, Thm. A.1]. For i = 3, the proof relies on the integral Hodge conjecture for cycles of codimension 2 on smooth cubic fourfolds, a result proved by Voisin [63, Thm. 18] building on [51] and [67]. Something stronger is known when i ≤ 2, namely that for any field i extension F/C, the natural map H i (F, Q/Z(i−1)) → Hnr (XF /F, Q/Z(i−1)) is an isomorphism, see Theorem 2.4. In this case, we say that the unramified cohomology is universally trivial. The universal behavior of an unramified cohomology group can lead to a finer obstruction to rationality. While we do not know if the unramified cohomology of an arbitrary smooth cubic fourfold over C is universally trivial in degree 3, our main result is the following.1 Theorem 1. Let X ⊂ P5 be a very general cubic fourfold containing a plane 3 over C. Then Hnr (X/C, Q/Z(2)) is universally trivial, i.e., the natural map 3 3 H (F, Q/Z(2)) → Hnr (XF /F, Q/Z(2)) is an isomorphism for every field extension F/C. 1 After

the first version of this article appeared, C. Voisin [64] was able to prove, using different techniques, that any smooth cubic fourfold has universally trivial unramified cohomology in degree 3.

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Here, “very general” can be interpreted as the condition that the quadric e → P2 attached to the plane P ⊂ X has simple degeneration surface bundle X (see Proposition 4.1) and that the K3 surface S → P2 of degree 2, defined by the rulings of the quadric bundle, has Picard rank 1. A very general cubic fourfold containing a plane has nontrivial Clifford invariant. Theorem 1 implies the corresponding statement with µ⊗2 2 coefficients as well (see Corollary 5.6). This article is organized as follows. In §1, we expand on the notion of universal triviality and, more generally, universal torsion by a positive integer, of the Chow group A0 (X) of zero-cycles of degree zero on a smooth proper variety X. We state an extension of a theorem of Merkurjev [49, Thm. 2.11] on the relationship between universal torsion properties of A0 (X) and analogous properties for unramified cohomology groups and, more generally, unramified classes in cycle modules. We also recall that rationally connected varieties, and among them Fano varieties, satisfy such a universal torsion property for A0 (X). In §2, we discuss the specific case of cubic hypersurfaces X ⊂ Pn+1 for n ≥ 2. Already in this case, the universal triviality is an open question. We register a folklore proof that A0 (X) is killed by 18, and by 2 as soon as X contains a k-line. In particular, for any smooth cubic hypersurface X ⊂ Pn+1 over C, with n ≥ 2, and any field extension F/C, the group A0 (XF ) is 2-torsion, which also follows from the existence of a unirational parameterization of X of degree 2. The only possible interesting unramified cohomology groups are thus those with coefficients Z/2Z. We recall the known results on these groups. Our main result, Theorem 1, then appears as one step beyond what was known, but still leaving open the question of the universal triviality of A0 (X). In §3, we recall some results on the unramified cohomology groups of quadrics in degrees at most 3. The most important is due to Kahn, Rost, and Sujatha (see Theorem 3.3), who build upon earlier work of Merkurjev, Suslin, and Rost. In §4, we discuss the fibration into 2-dimensional quadrics over the projective plane, as well as its corresponding even Clifford algebra, associated to a smooth cubic fourfold containing a given plane. In §5, we use the results of the previous two sections to prove Theorem 1. The “very general” hypothesis allows us to construct well-behaved parameters at the local rings of curves on P2 that split in S (see Lemma 5.3). We thank the American Institute of Mathematics for sponsoring the workshop “Brauer groups and obstruction problems: moduli spaces and arithmetic” held February 25 to March 1, 2013, in Palo Alto, California. This work emerges from a problem session group formed during the workshop, and involved the participation of Marcello Bernardara, Jodi Black, Evangelia Gazaki, Andrew Kresch, Eric Riedl, Olivier Wittenberg, and Matthew Woolf. We thank Andrew Kresch for further discussions after the AIM meeting.

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The first author is partially supported by National Science Foundation grant MSPRF DMS-0903039 and by NSA Young Investigator grant H9823013-1-0291. The second author is partially supported by the Agence Nationale de la Recherche grant number ANR-12-BL01-0005. The third author is partially supported by National Science Foundation grant DMS-1001872, DMS1401319, and DMS-1463882.

1. Unramified cohomology and Chow group of 0-cycles 1.1. Unramified elements A general framework for the notion of “unramified element” is established in [19, §2]. Let k be a field and denote by Localk the category of local k-algebras together with local k-algebra homomorphisms. Let Ab be the category of abelian groups and let M : Localk → Ab be a functor. For any field K/k the group of unramified elements of M in K/k is the intersection \
Mnr (K/k) = im M (O) → M (K) k⊂O⊂K

over all rank 1 discrete valuations rings k ⊂ O ⊂ K with Frac(O) = K. There is a natural map M (k) → Mnr (K/k) and we say that the group of unramified elements Mnr (K/k) is trivial if this map is surjective. For X an integral scheme of finite type over a field k, in this paper we write Mnr (X/k) := Mnr (k(X)/k). By this definition, the group Mnr (X/k) is a k-birational invariant of integral schemes of finite type over k. We will be mostly concerned with the functor M = H´eit (−, µ) with coefficients µ either ⊗(i−1) µ2 (under the assumption char(k) 6= 2) or Q/Z(i − 1) := lim µ⊗(i−1) , −→ m the direct limit being taken over all integers m coprime to the characteristic of k.2 In this case, Mnr (X/k) is called the unramified cohomology group i Hnr (X, µ) of X with coefficients in µ. Theorem 1.1. Let F be a field and n a nonnegative integer prime to the characteristic. Then the natural map H i (F, µ⊗(i−1) ) → H i (F, Q/Z(i − 1)) n ⊗(i−1)

is injective and the natural map lim H i (F, µn ) → H i (F, Q/Z(i−1)) is an −→ isomorphism, where the limit is taken over all n prime to the characteristic. This is a well-known consequence of the norm residue isomorphism theorem (previously known as the Bloch–Kato conjecture) in degree i−1. In fact, we use it only for i = 3, in which case it is a consequence of the Merkurjev– Suslin theorem; for i = 3 and p = 2, this is a consequence of a theorem of Merkurjev. The case of i ≤ 3 is proved by Kato [42, Lemma 2.3], whose argument can be adapted once one knows the norm residue isomorphism. For 2 For

a description of the torsion divisible by the characteristic of k, see [34, App. A].

Universal Unramified Cohomology of Cubic Fourfolds

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lack of a standard reference, and at the request of the referee, we give this argument. Proof of Theorem 1.1. Given integers n > 0, m > 0 prime to the characteristic of F , and given an integer i, we have an exact sequence of Galois modules ⊗i ⊗i 1 → µ⊗i n → µnm → µm → 1

inducing a long exact sequence of Galois cohomology groups i ⊗i i+1 i+1 · · · → H i (F, µ⊗i (F, µ⊗i (F, µ⊗i nm ) → H (F, µm ) → H n )→H nm ) → · · ·

The map from Milnor K-theory to Galois cohomology induced by taking a symbol to the cup product of its entries gives rise to a commutative diagram KiM (F )/nm

/ H i (F, µ⊗i nm )

 KiM (F )/m

 / H i (F, µ⊗i m)

where the left vertical map is the quotient map (which is surjective) and ⊗i the right vertical map is induced by µ⊗i nm → µm . The fact that the diagram commutes reduces to the case of i = 1, hence to the commutativity of the diagram / H 1 (F, µnm ) F × /F ×nm  / H 1 (F, µm )

 F × /F ×m

which itself follows from applying the long exact sequence in Galois cohomology to the commutative diagram of Kummer sequences 1

/ µnm

/ Gm

1

 / µm

 / Gm

/ Gm

/1

/ Gm

/1

n

The norm residue isomorphism theorem, proved by Voevodsky, Rost, and Weibel, asserts that the map KiM (F )/n → H i (F, µ⊗i n ) is surjective. Coni ⊗i sidering the above commutative squares, the map H i (F, µ⊗i nm ) → H (F, µm ) is thus surjective. It then follows from the long exact sequence that the map i+1 H i+1 (F, µ⊗i (F, µ⊗i n ) → H nm ) is injective. Since Galois cohomology commutes with direct limits, we arrive at the desired conclusion.  If F/k is a field extension, we write XF = X ×k F . If X is geometrically integral over k, we say that Mnr (X/k) is universally trivial if Mnr (XF /F ) is trivial for every field extension F/k.

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Proposition 1.2 ([19, §2 and Thm. 4.1.5]). Let M : Localk → Ab be a functor satisfying the following conditions: • If O is a discrete valuation ring containing
k, with fraction field
K and residue field κ, then ker M (O) → M (K) ⊂ ker M (O) → M (κ) . • If A is a regular local ring of dimension
T 2 containing k, with fraction
field K, then im M (A) → M (K) = ht(p)=1 im M (Ap ) → M (K) . • The group Mnr (A1k /k) is universally trivial. Then Mnr (Pnk /k) is universally trivial. The functor H´eit (−, µ) satisfies the conditions of Proposition 1.2 (cf. [19, i (X/k, µ) is universally Thm. 4.1.5]), hence if X is a k-rational variety, then Hnr trivial. Moreover, the same conclusion holds if X is retract k-rational, in particular, stably k-rational, which can be proved using [43, Cor. RC.12–13], see [49, Prop. 2.15]. Denote by Ab• the category of graded abelian groups. An important class of functors M : Localk → Ab• arise from the cycle-modules of Rost [55, Rem. 5.2]. In particular, unramified cohomology arises from a cycle-module. A cycle module M comes equipped with residue maps of graded degree −1 M ∂ M i (k(X)) − → M i−1 (k(x)) x∈X (1)

for any integral k-variety X. If X is smooth and proper, then the kernel is i (X/k). Mnr 1.2. Chow groups of 0-cycles Denote by CHd (X), resp. CHd (X), the Chow group of d-dimensional cycles, resp. codimension-d cycles, on a smooth variety X over a field k, up to rational equivalence. If X is proper over k, then there is a well-defined degree map CH0 (X) → Z, and we denote by A0 (X) its kernel, called the Chow group of 0-cycles of degree 0. The group A0 (X) is a k-birational invariant of smooth, proper, integral varieties over a field k, see [23, Prop. 6.3] requiring resolution of singularities and [33, Ex. 16.1.11] in general. The computation of the Chow groups of projective space goes back to Severi. For 0-cycles, one easily sees that A0 (Pnk ) = 0. We say that a smooth proper variety X of dimension n over a field k has a decomposition of the diagonal if we can write ∆X = P ×k X + Z n

(1)

in CH (X ×k X), where P is a 0-cycle of degree 1 and Z is a cycle with support in X ×k V for some proper closed subvariety V ⊂ X. The weaker notion of a rational decomposition of the diagonal N ∆X = N (P ×k X) + Z for some N ≥ 1, was studied by Bloch and Srinivas [12]. For X a proper k-variety, we say that A0 (X) is universally trivial if A0 (XF ) = 0 for every field extension F/k. We remark that if X is k-rational, then A0 (X) is universally trivial. In fact, the same conclusion holds if X is retract k-rational, in particular, stably k-rational, which can be proved using

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[43, Cor. RC.12]. To check triviality of A0 (XF ) over every field extension F/k seems like quite a burden. However, usually it suffices to check it over the function field by the following lemma, inspired by Merkurjev [49, Thm. 2.11], which shows the interest in decompositions of the diagonal. Lemma 1.3. Let X be a geometrically irreducible smooth proper variety over a field k. Assume that X has a 0-cycle of degree 1. The group A0 (X) is universally trivial if and only if X has a decomposition of the diagonal if and only if A0 (Xk(X) ) = 0. Proof. If A0 (X) is universally trivial, then A0 (Xk(X) ) = 0 by definition. Let us prove that if A0 (Xk(X) ) = 0, then X has a decomposition of the diagonal. Write n = dim(X). Let ξ ∈ Xk(X) be the k(X)-rational point which is the image of the “diagonal morphism” Spec k(X) → X ×k Spec k(X). Let P be a fixed 0-cycle of degree 1 on X. By hypothesis, we have ξ = Pk(X) in CH0 (Xk(X) ). The closures of Pk(X) and ξ in X ×k X are P ×k X and the diagonal ∆X , respectively. By the closure in X ×k X of a 0-cycle on Xk(X) , we mean the sum, taken with multiplicity, of the closures of each closed point in the support of the 0-cycle on Xk(X) . Hence the class of ∆X − P ×k X is in the kernel of the contravariant map CHn (X ×k X) → CHn (Xk(X) ). Since CHn (Xk(X) ) is the inductive limit of CHn (X ×k U ) over all dense open subvarieties U of X, we have that ∆X −P ×k X vanishes in some CHn (X×k U ). We thus have a decomposition of the diagonal ∆X = P ×k X + Z

(2)

n

in CH (X ×k X), where Z is a cycle with support in X × V for some proper closed subvariety V ⊂ X. Now we prove that if X has a decomposition of the diagonal, then A0 (X) is universally trivial. Each n-cycle T on X ×k X induces a homomorphism T∗ : CH0 (X) → CH0 (X) defined by T∗ (z) = (p1 )∗ (T.p∗2 z), where pi : X ×k X → X are the two projections. The map T 7→ T∗ is itself a homomorphism CHn (X ×k X) = CHn (X ×k X) → HomZ (CH0 (X), CH0 (X)) by [33, Cor. 16.1.2]. We note that (∆X )∗ is the identity map and (P ×k X)∗ (z) = deg(z)P. By the easy moving lemma for 0-cycles on a smooth variety (cf. [20, p. 599]), for a proper closed subvariety V ⊂ X, every 0-cycle on X is rationally equivalent to one with support away from V . This implies that Z∗ = 0 for any d-cycle with support on X ×k V for a proper closed subvariety V ⊂ X. Thus a decomposition of the diagonal ∆X = P ×k X + Z as in (1) implies that the identity map restricted to A0 (X) is zero. For any field extension F/k, we have the base-change ∆XF = PF ×F XF + ZF of the decomposition of the diagonal (1), hence the same argument as above shows that A0 (XF ) = 0. We conclude that A0 (X) is universally trivial. 

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A. Auel, J.-L. Colliot-Th´el`ene and R. Parimala

Let M be a cycle module and let X be a smooth, proper, geometrically connected variety over the field k. Let N be a positive integer. We say that Mnr (X/k) is universally N -torsion if the cokernel of the natural homomorphism M (F ) → Mnr (XF /F ) is killed by N for every field extension F/k. We say that A0 (X) is universally N -torsion if A0 (XF ) is killed by N for every field extension F/k. The index i(X) of a variety X is the smallest positive degree of a 0-cycle. Theorem 1.4. Let X be a smooth proper geometrically connected variety over a field k. Let N > 0 be an integer. If the Chow group A0 (X) of 0-cycles of degree 0 is universally N -torsion, then for every cycle module M over k, the group Mnr (X/k) is universally i(X)N -torsion. Proof. This is a direct consequence of Lemma 1.3 and [43, Prop. RC.9(2)].



The case N = 1 is a generalization of a theorem of Merkurjev [49, Thm. 2.11], who also establishes a converse statement. For any N > 0, one may extend Merkurjev’s method to prove a weak converse: if Mnr (X/k) is universally N -torsion for every cycle module M , then A0 (X) is also universally N -torsion. We point out that the appearance of the index of X, in the statement of Theorem 1.4, is necessary. For any quadric X over any field k of characteristic 6= 2, we have that A0 (X) is universally trivial (see [57]). However, if X is the 4-dimensional quadric associated to an anisotropic Albert form X over a field
3 k of characteristic 6= 2, then coker H 3 (F, Q/Z(2)) → Hnr (X/k, Q/Z(2)) ∼ = Z/2Z by [41, Thm. 5]. Hence A0 (X) is universally trivial, while there are nontrivial unramified elements of some cycle module over X. Note that the index of an anisotropic quadric is 2. We remark that if X is a smooth proper variety of dimension n over a field k and X has a unirational parameterization Pnk 99K X of degree N , then A0 (X) is universally N -torsion. This follows, when k has characteristic 0, from a standard restriction-corestriction argument using resolution of singularities, see [23, Prop. 6.4]. In general, a different argument is needed. In particular, if N is the greatest common divisor of the degrees of possible unirational parameterizations of X, then A0 (X) is universally N -torsion. The following lemmas will be used in the next section. Lemma 1.5. Let X be a proper variety over a field k with X(k) 6= ∅. Let N > 0 be an integer. If for every finite extension K/k and any two K-points P, Q ∈ X(K) the class of the 0-cycle P − Q is N -torsion in A0 (XK ), then A0 (X) is N -torsion. Proof. Fixing P ∈ X(k), any element of A0 (X) can be written as a linear combination of 0-cycles of degree 0 of the form Z − deg(Z)P , where Z is a closed point of X. Let K be the residue field of Z. Consider the morphism f : XK → X. Since K ⊗k K has K as a direct factor, there is a corresponding K-rational point ζ of XK lying over Z, such that f∗ ζ = Z. By hypothesis ζ − PK is N -torsion, hence f∗ (ζ − PK ) = Z − deg(Z)P is N -torsion. 

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The hypothesis that X has a k-point is essential. Indeed, there exist projective homogeneous varieties X under a connected linear algebraic group over a field k such that A0 (X) 6= 0 (see [16, §18]), whereas any such variety satisfies A0 (X) = 0 as soon as it has a k-point [16, Cor. 4.3]. Lemma 1.6. Let k be an algebraically closed field and K/k a field extension. Let X be a smooth projective connected variety over k. Then the natural map CH0 (X) → CH0 (XK ) is injective. Proof. Let z be a 0-cycle on X that becomes rationally equivalent to zero on XK . Then there exists a subextension L of K/k that is finitely generated over k, such that z becomes rationally equivalent to zero on XL . In fact, we can spread out the rational equivalence to a finitely generated k-algebra A. Since k is algebraically closed, there are many k-points on Spec A, at which we can specialize the rational equivalence.  Lemma 1.7. Let X be a smooth proper connected variety over an algebraically closed field k of infinite transcendence degree over its prime field. If A0 (X) = 0, then there exists an integer N > 0 such that A0 (X) is universally N -torsion. Proof. The variety X is defined over an algebraically closed subfield L ⊂ k, with L algebraic over a field finitely generated over its prime field. That is, there exists a variety X0 over L with X ∼ = X0 ×L k. Let η be the generic point of X0 . Let P be an L-point of X0 . One may embed the function field F = L(X0 ) into k, by the transcendence degree hypothesis on k. Let K be the algebraic closure of F inside k. By Lemma 1.6 and the hypothesis that A0 (X) = 0, we have that A0 (X0 ×L F ) = 0. This implies that there is a finite extension E/F of fields such that ηE − PE = 0 in A0 (X0 ⊗L E). Taking the corestriction to F , one finds that N (ηF − PF ) = 0 in A0 (X0 ×L F ), hence in A0 (X) as well. Arguing as in the proof of Lemma 1.3, we conclude that A0 (X) is universally N -torsion.  1.3. Connections with complex geometry The universal torsion of A0 (X) puts strong restrictions on the variety X. For example, the following result is well known. Proposition 1.8. Let X be a smooth proper geometrically irreducible variety over a field k of characteristic zero. If A0 (X) is universally N -torsion for some positive integer N , then H 0 (X, ΩiX) = 0 and H i (X, OX) = 0 for all i ≥ 1. Proof. Over a complex surface, the result goes back to Bloch’s proof [9, App. Lec. 1] of Mumford’s [50] result on 2-forms on surfaces, exploiting a decomposition of the diagonal and the action of cycles on various cohomology theories. Aspects of this argument were developed in [12]. A proof over the complex numbers can be found in [62, Cor. 10.18, §10.2.2]. Over a general field of characteristic zero, the argument is sketched in [31, p. 187].  ⊗n Over C, the universal triviality of A0 (X) does not imply H 0 (X, ωX )=0 for all n > 1. Otherwise, a surface over C with A0 (X) = 0 would additionally

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A. Auel, J.-L. Colliot-Th´el`ene and R. Parimala

⊗2 satisfy P2 (X) = h0 (X, ωX ) = 0, hence would be rational by Castelnovo’s criterion. It is however well known that there exist nonrational complex surfaces X, with pg (X) = q(X) = 0 and for which A0 (X) = 0. Enriques surfaces were the first examples, extensively studied in [29], [30, p. 294] with some examples considered earlier in [54], see also [15]. Such examples emerge in the context of Bloch’s conjecture on 0-cycles on surfaces, see [10]. We remark that for an 1 Enriques surface X, we have that Hnr (X/C, Z/2Z) = H´e1t (X, Z/2Z) = Z/2Z. Hence by Theorem 1.4, we see that A0 (X) is not universally trivial.

The following result was stated without detailed proof as the last remark of [12]. As we show, it is an immediate consequence of a result in [25]. A more geometric proof was recently shown to us by C. Voisin.3 Proposition 1.9. Let X be a smooth proper connected surface over C. Suppose i (X(C), Z) are torsionfree and that A0 (X) = 0. Then that all groups HBetti A0 (X) is universally trivial. Proof. By Lemma 1.7, we have that A0 (X) is universally N -torsion. Hence by Lemma 1.8, we have that H i (X, OX ) = 0 for all i ≥ 1. Hence pg (X) = q(X) = 0, and thus b3 (X) = b1 (X) = 2q(X) = 0. The torsion-free hypothesis on cohomology finally allows us to conclude, from [25, Thm. 3.10(d)], that A0 (XF ) = 0 for any field extension F/C.  Corollary 1.10. Let X be a smooth proper connected surface over C. Suppose that the N´eron-Severi group NS(X) of X is torsionfree and that A0 (X) = 0. Then A0 (X) is universally trivial. 1 (X(C), Z) is clearly torsionfree. For any smooth proProof. The group HBetti per connected variety X over C of dimension d there is an isomorphism 2 2 N S(X)tors ' HBetti (X(C), Z)tors , and the finite groups HBetti (X(C), Z)tors 2d−1 and HBetti (X(C), Z)tors are dual to each other. The hypotheses thus imply i that for the surface X, all groups HBetti (X(C), Z) are torsionfree, and we apply Proposition 1.9. 

The first surfaces X of general type with pg (X) = q(X) = 0 were constructed in [14] and [35]. Simply connected surfaces X of general type for which pg (X) = 0 were constructed by Barlow [6], who also proved that A0 (X) = 0 for some of them. See also the recent paper [60]. For such surfaces, Pic(X) = NS(X) has no torsion, thus Corollary 1.10 applies. The group A0 (X) is universally trivial, but the surfaces are far from being rational, since they are of general type. The interested reader can find how to adapt Proposition 1.9 and Corollary 1.10 over an algebraically closed field of infinite transcendence degree over its prime field. 3 See

[65, Cor. 2.2].

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A smooth projective variety X over a field k is called rationally chain connected if for every algebraically closed field extension K/k, any two Kpoints of X can be connected by a chain of rational curves. Smooth, geometrically unirational varieties are rationally chain connected. It is a theorem of Campana [13] and Koll´ ar–Miyaoka–Mori [47] that any smooth projective Fano variety is rationally chain connected. If X is rationally chain connected, then A0 (XK ) = 0 for any algebraically closed field extension K/k. While a standard argument then proves that A0 (XF ) is torsion for every field extension F/k, the following more precise result, in the spirit of Lemma 1.7 above, is known. Proposition 1.11 ([20, Prop. 11]). Let X be a smooth, projective, rationally chain connected variety over a field k. Then there exists an integer N > 0 such that A0 (X) is universally N -torsion. There exist rationally connected varieties X over an algebraically closed field of characteristic zero with A0 (X) not universally trivial. Indeed, let X 2 (X, Q/Z(1)) ∼ be a unirational threefold with Hnr = Br(X) 6= 0, see, e.g., [2]. Then by Theorem 1.4, A0 (X) is not universally trivial. Question. Does there exists a smooth Fano variety X over an algebraically closed field of characteristic 0 with A0 (X) not universally trivial?4

2. Chow groups of 0-cycles on cubic hypersurfaces Now we will discuss the situation for cubic hypersurfaces X ⊂ Pn+1 with n ≥ 2. Quite a few years ago, one of us learned the following argument from the Dean of Trinity College, Cambridge. Proposition 2.1. Let X ⊂ Pn+1 , with n ≥ 2, be an arbitrary cubic hypersurk face over an arbitrary field k. Then A0 (X) is 18-torsion. If X(k) 6= ∅, then A0 (X) is 6-torsion. If X contains a k-line, then A0 (X) is 2-torsion. Proof. Assuming the assertion in the case X(k) 6= ∅, we can deduce the general case from a norm argument, noting that X acquires a rational point after an extension of degree 3. So we assume that X(k) 6= ∅. By Lemma 1.5, to prove the general statement over an arbitrary field k, it suffices to prove that for any distinct points P, Q ∈ X(k), the class of the 0-cycle P − Q is 6-torsion, and that it is 2-torsion if X contains a k-line. Choose a P3k ⊂ Pn+1 k containing P and Q. If X contains a k-line L, take such a P3k which contains the line L. Then X ∩ P3k is either P3k or is a cubic surface in P3k . In the first case, P is rationally equivalent to Q on P3k , hence on X. Thus we can assume, for the rest of the proof, that X ⊂ P3k is a cubic surface, possibly singular. 4 Partly

motivated by our work, C. Voisin [64] has proven the existence of such examples, namely, (resolutions of) certain special quartic double solids. A variation of her method was successfully applied, in joint work of the second author with A. Pirutka [24], to certain smooth quartic threefolds.

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We will sketch a route to prove the result, leaving the minute details to the reader. We first show that if the cubic surface X contains a k-line L, then for any two distinct points P, Q ∈ X(k), we have 2(P − Q) = 0 ∈ A0 (X). It is enough to prove this when Q is a k-point on L. If P also lies on L, then P − Q = 0. If P does not lie on L, let Π be the plane spanned by P and L. If it is contained in X, then clearly P − Q = 0. If not, then it cuts out on X a cubic curve C, one component of which is the line L, the other component is a conic. At this juncture, we leave it to the reader to consider the various possible cases and show that 2(P − Q) is rationally equivalent to zero on C, hence on X. The coefficient 2 is the degree of intersection of the conic with the line. We may now assume that the cubic surface X does not contain a kline. If P and Q were both singular, then the line through P and Q would intersect the surface with multiplicity at least 4, hence would be contained in X, which is excluded. We may thus assume that P is a regular k-point. First assume that Q is singular. Let Π be a plane which contains P and Q. It is not contained in X. Its trace on X is a cubic curve C in the plane Π, such that C is singular at Q. A discussion of cases then shows that 6(Q − P ) = 0 on C, hence on X (the occurrence of 6 rather than 2, comes from allowing nonperfect fields in characteristic 3). Now suppose that both P and Q are regular, hence smooth, k-points. Let TP ⊂ P3k be the tangent plane to X at P , and TQ ⊂ P3k the tangent plane to X at Q. Since X contains no k-line, the tangent planes are distinct. Let L = TP ∩ TQ be their intersection. This line is not contained in X, which it intersects in a zero-cycle z of degree 3 over k. The trace of X on the plane TP is a cubic curve CP which is singular at P and contains the 0-cycle z. We leave it to the reader to check that 6P − 2z = 0 ∈ A0 (CP ), hence in A0 (X). Similarly, 6Q − 2z = 0 ∈ A0 (CQ ), hence in A0 (X). Thus 6(P − Q) = 0 ∈ A0 (X) in all cases.  For smooth cubic hypersurfaces, the last part of Proposition 2.1 is also a consequence of the fact that a cubic hypersurface containing a line has unirational parameterizations of degree 2. This fact that was likely known to M. Noether (cf. [17, App. B]). Theorem 2.2. Let X ⊂ Pn+1 , with n ≥ 2, be a smooth cubic hypersurface k containing a k-line L. Then X has a unirational parameterization of degree 2. Proof. Denote by W the variety of pairs (p, l) where p ∈ L and l is a line in Pn+1 tangent to X at p. Then the projection W → L is a Zariski locally k trivial Pn−1 -bundle. A general such line l intersects X in one further point, defining a rational map g : W → X. This map is two-to-one. Indeed, as before, for a general point p ∈ X, the plane through p and L meets X in the union of L and a smooth conic. Generally, that conic meets L in two points. The lines through p, and tangent to X, are exactly those connecting p to these two points of intersection.

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We can give another argument, following [36, §2.1]. Projecting from L , where displays X as birational to the total space of a conic bundle Y → Pn−1 k Y is the blow-up of X in L. Each point P in the base Pn−1 corresponds to a k plane containing L, and intersecting X with this plane is the union of L and a conic CP ; this conic is the fiber above P in the conic bundle. Let M be the incidence variety of pairs (P, p) where P ∈ Pn−1 and p ∈ L, such that p ∈ CP . k Then the projection M → Pn−1 has degree 2, the fiber over each P is exactly k the two points of intersection of L and CP . The other projection M → L displays M as the total space of a Zariski locally trivial Pn−2 -bundle, hence M is rational. Then consider the base change of Y → Pn−1 by M → Pn−1 . k k The resulting conic bundle M ×Pn−1 Y → M has a tautological rational k section. Thus M ×Pn−1 Y is rational (being a conic bundle over M with a k rational section) and the projection to Y has degree 2.  Over an algebraically closed field, any cubic hypersurface X ⊂ Pn+1 , with n ≥ 2, contains a line. Indeed, by taking hyperplane sections, one is reduced to the well-known fact that any cubic surface X ⊂ P3 contains a line over an algebraically closed field, see [53, Prop. 7.2] for instance. Proposition 2.1 and Theorem 1.4 then yield the following corollary. , with n ≥ 2, be a smooth cubic hypersurface Corollary 2.3. Let X ⊂ Pn+1 k over a field k. If X contains a k-line (e.g., if k is an algebraically closed field), then A0 (X) is universally 2-torsion and thus for every cycle module M over k, the group Mnr (X/k) is universally 2-torsion. Hassett and Tschinkel [40, §7.5] provide explicit conditions under which cubic fourfolds over C admit unirational parameterizations of odd degree. These conditions would imply that such cubic fourfolds are Zariski dense in the moduli space, in particular, are dense in the Noether–Lefschetz divisors Cd , with d = 2(m2 + m + 1) for m ≥ 1 (in which case the cubic fourfolds are expected to admit unirational parameterizations of degree m2 − m + 1). Together with Corollary 2.3, this would imply that such cubic fourfolds X have universally trivial A0 (X), and thus for every cycle module M over C, the group Mnr (X/k) is universally trivial.5 Corollary 2.3 leaves the following questions open. Question. In any of the following cases, does there exist a smooth cubic with A0 (X) not universally trivial (for instance with hypersurface X ⊂ Pn+1 k ⊗(i−1) i Hnr (X/C, μ2 ) not universally trivial for some i ≥ 1)? a) When k = C and n ≥ 3? b) When k = C((t)) or k = C(t) and n ≥ 3? 5 In

recent work, C. Voisin [65] was able to prove that A0 (X) is universally trivial for smooth cubic fourfolds X in Cd for any d not divisible by 4 without constructing unirational parameterizations of odd degree. More recently, the conditions of Hassett and Tschinkel have been clarified and generalized [39, §4] to also apply to cubic fourfolds X in Cd for any d not divisible by 4. Some cases of small d (e.g., d ≤ 38), where the conditions are met, are also explained. None of these results cover the case of cubic fourfolds containing a plane.

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c) When k is a finite field and n ≥ 3? d) When k = C((x))((y)) and n ≥ 4? e) When k = Qp and n ≥ 4? The following theorem gathers previously known results. Theorem 2.4. Let X ⊂ Pn+1 be a smooth cubic hypersurface over a field k k i of characteristic zero. Then Hnr (X/k, Q/Z(1)) is universally trivial for all 3 (X/k, Q/Z(2)) is trivial for all n ≥ 3 and 0 ≤ i ≤ 2. If k = C, then Hnr n ≥ 3. Proof. Let F/k be a field extension. For any complete intersection Y ⊂ Pn+1 F of dimension ≥ 3 over F , the restriction map on Picard groups Pic(Pn+1 F ) → Pic(Y ) is an isomorphism and the natural map on Brauer groups Br(F ) → Br(Y ) is an isomorphism, see [52, Thm. A.1] for details. By purity, for any smooth variety Y over F , we have that 1 Hnr (Y /F, Q/Z(1)) = H´e1t (Y, Q/Z(1))

2 and Hnr (Y /F, Q/Z(1)) = Br(Y ),

see [26, Cor. 3.2, Prop. 4.1] and [11]. From the Kummer sequence for a projective and geometrically connected variety X over F , we get an exact sequence 1 → F × /F ×n → H´e1t (X, µn ) → Pic(X)[n] → 0 which yields an exact sequence 0 → H 1 (F, Q/Z(1)) → H´e1t (X, Q/Z(1)) → Pic(X)tors → 0. upon taking direct limits. When X ⊂ Pn+1 is a smooth cubic hypersurface, the above considerk i (X/k, Q/Z(1)) is universally trivial for i = 1, 2. Also ations imply that Hnr 0 Hnr (X/k, Q/Z(1)) is universally trivial since X is geometrically irreducible. Now assume k = C. If n = 3, then X contains a line hence is birational to a conic fibration over P2 , as in the proof of Theorem 2.2. For a conic 3 bundle Y over a complex surface, one has Hnr (Y /C, Q/Z(2)) = 0 (cf. [21, Cor. 3.1(a)]). If n = 4 and X contains a plane, then X is birational to a fibration Y → P2 in 2-dimensional quadrics, and once again [21, Cor. 3.1(a)] 3 yields Hnr (X/C, Q/Z(2)) = 0. For n ≥ 2 arbitrary, X is unirational hence rationally chain connected. Then [28, Thm. 1.1] implies that the integral Hodge conjecture for codimen3 sion 2 cycles on X is equivalent to the vanishing of Hnr (X/C, Q/Z(2)). For smooth cubic threefolds, the integral Hodge conjecture holds for codimension 2 cycles, as H 4 (X, Z) is generated by a line. This yields another 3 proof of Hnr (X/C, Q/Z(2)) = 0 in the case n = 3. For smooth cubic fourfolds, the integral Hodge conjecture for codimension 2 cycles is a result of Voisin [63, Thm. 18], building on [51] and [67]. We 3 thus get Hnr (X/C, Q/Z(2)) = 0 for a smooth cubic hypersurface X ⊂ P5 by [28, Thm. 1.1].

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For n ≥ 5, the Lefschetz theorems show that codimension 2 cycles on the hypersurface X ⊂ Pn+1 are complete intersections, hence the integral Hodge conjecture for such cycles holds. Again by [28, Thm. 1.1], this implies 3 Hnr (X/C, Q/Z(2)) = 0.  Our main result, Theorem 1, is a first step beyond the mentioned results.

3. Unramified cohomology of quadrics Let Q be a smooth quadric over a field k of characteristic 6= 2 defined by the vanishing of a quadratic form q. We note that the dimension of Q (as a k-variety) is 2 less than the dimension of q (as a quadratic form). When Q has even dimension, one defines the discriminant d(Q) ∈ H 1 (k, µ2 ) of Q to be the (signed) discriminant of q. If Q has even dimension and trivial discriminant or has odd dimension, then define the Clifford invariant c(Q) ∈ Br(k) of Q to be the Clifford invariant of q, i.e., the Brauer class of the Clifford algebra C(q) or the even Clifford algebra C0 (q), respectively. We point out that when q has even rank and trivial discriminant, then a choice of splitting of the center induces a decomposition C0 (q) ∼ = C0+ (q) × C0− (q), with C(q), C0+ (q), − and C0 (q) all Brauer equivalent central simple k-algebras. Under the given constraints on dimension and discriminant, these invariants only depend on the similarity class of q, and thus yield well-defined invariants of Q. The following two results are well known (cf. [1, §5, p. 485]; see also the proof of [27, Th´eor`eme 2.5]), though we could not find the second one stated in the literature. Theorem 3.1. Let k be a field of characteristic 6= 2. Let Q be a smooth quadric surface over k. Then (
0 if d(Q) is nontrivial, ker Br(k) → Br(k(Q)) = Z/2Z · c(Q) if d(Q) is trivial, where c(Q) is the Clifford invariant of Q. Proposition 3.2. Let k be a field of characteristic 6= 2. Let Q be a quadric surface cone over k, the base of which is a smooth conic Q0 . Then
ker Br(k) → Br(k(Q)) = Z/2Z · c(Q0 ), where c(Q0 ) is the Clifford invariant of Q0 .
1 Proof. In this case k(Q) ∼ = k(Q
0 ×k Pk ), hence ker Br(k) → Br(k(Q)) equals ker Br(k) → Br(k(Q0 )) . Thus the proposition follows from the case of smooth conics, a result going back to Witt [66, Satz p. 465].  Finally, the deepest result we will need is the following one concerning the degree three unramified cohomology of a quadric.

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Theorem 3.3 (Kahn–Rost–Sujatha [41, Thm. 5]). Let k be a field of characteristic 6= 2. Let Q be a smooth quadric surface over k. Then the natural map 3 H 3 (k, Q/Z(2)) → Hnr (Q/k, Q/Z(2)) is surjective. The following is an amplification of one the main results of Arason’s thesis. Theorem 3.4. Let k be a field of characteristic 6= 2. Let Q be a smooth quadric surface over k defined by a nondegenerate quadratic form q of rank 4. Then the kernel of the map H 3 (k, Q/Z(2)) → H 3 (k(Q), Q/Z(2)) coincides with the ⊗2 3 kernel of the map H 3 (k, µ⊗2 2 ) → H (k(Q), µ2 ), and it is equal to the set of symbols {(a, b, c) : q is similar to a subform of  −a, −b, −c }. Proof. By a standard norm argument, the kernel of H 3 (k, Q/Z(2)) → H 3 (k(Q), Q/Z(2)) is 2-torsion. By Merkurjev’s theorem (see Theorem 1.1), the two kernels in the Theorem thus coincide. The precise description of the kernel with coefficients Z/2Z is Arason’s [1, Satz 5.6].  However, for our purposes, we will only need to know that certain special symbols are contained in this kernel. We can give a direct proof of this fact. Lemma 3.5. Let k be a field of characteristic √ 6= 2. If < 1, −a, −b, abd > is isotropic over k, then for w any norm from k( d)/k, the symbol (a, b, w) in H 3 (k, µ⊗2 2 ) is trivial. √ Proof. Put l = k( d). As < 1, −a, −b, abd > is isotropic, there exist x, y, u, v ∈ k such that x2 − ay 2 = b(u2 − adv 2 ) 6= 0. The class (a, x2 − ay 2 ) is trivial, hence (a, b, w) = (a, (x2 − ay 2 )/(u2 − adv 2 ), w) = (a, u2 − adv 2 , w). Let w = Nl/k (w0 ). By the projection formula, we have that (a, b, w) = coresk(√d)/k (a, u2 − adv 2 , w0 ) √ which is trivial since (a, u2 − adv 2 ) = (a, u2 − a( dv)2 ) ∈ H 2 (l, µ⊗2 2 ) is trivial. 

4. Cubic fourfolds containing a plane and Clifford algebras Let X be a smooth cubic fourfold over a field k. Suppose X ⊂ P5k = P(V ) contains a plane P = P(W ), where W ⊂ V is a dimension 3 linear subspace of e be the blow-up of X along P and π : X e → P(V /W ) the projection V . Let X from P . We will write P2k = P(V /W ). Then the blow-up of P5k along P is isomorphic to the total space of the projective bundle p : P(E ) → P2k , where

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e → P2 embeds as a quadric E = (W ⊗ OP2k ) ⊕ OP2k (−1), and in which π : X k surface bundle. Now choose homogeneous coordinates (x0 : x1 : x2 : y0 : y1 : y2 ) on P5k . Since Autk (P5 ) acts transitively on the set of planes in P5k , without loss of generality, we can assume that P = {x0 = x1 = x2 = 0}. Write the equation of X as X X amn ym yn + bp yp + c = 0 0≤m≤n≤2

0≤p≤2

for homogeneous linear polynomials amn , quadratic polynomials bp , and a cubic polynomial c in k[x0 , x1 , x2 ]. Then we define a line bundle-valued quadratic form q : E → OP2k (1) over P2k by X X q(y0 , y1 , y2 , z) = amn ym yn + bp yp z + cz 2 (3) 0≤m≤n≤2

0≤p≤2

on local sections yi of OP2k and z of OP2k (−1). Of course, given X, the quadratic form q is only well-defined up to scalar multiplication by an element of Γ(X, Gm ) = k × . The quadric fibration associated to (E , q, OP2k (1)) is pree → P2 . The associated bilinear form bq : E → E ∨ ⊗ OP2 (1) has cisely π : X k k Gram matrix   2a00 a01 a02 b0  a01 2a11 a12 b1    (4)  a02 a12 2a22 b2  b0 b1 b2 2c whose determinant ∆ is a homogeneous sextic polynomial defining the discriminant divisor D ⊂ P2k . Proposition 4.1. Let X be a smooth cubic fourfold containing a plane P over e → P2 the associated quadric a field k of characteristic 6= 2. Denote by π : X k 2 surface bundle, D ⊂ Pk the discriminant divisor, and U = P2k r D. Then the following are equivalent: a) The divisor D is smooth over k. b) The fibers of q are nondegenerate over points of U and have a radical of dimension 1 over points of D. c) The fibers of π are smooth quadric surfaces over points of U and are quadric surface cones with isolated singularity over points of D. d) There is no other plane in X ×k k meeting P ×k k. In this case, we say that π has simple degeneration. Proof. The equivalence between a) and b) is proved in [7, I Prop. 1.2(iii)] over an algebraically closed field and [3, Prop. 1.6] in general. The equivalence of b) and c) follows from the fact that the singular locus of a quadric is the projectivization of its radical. The statement that d) implies a) appears without proof in [61, §1 Lemme 2], and holds over a general field. Finally, another plane intersecting P nontrivially will give rise, in the projection, to a singular line or plane in a fiber of the quadric fibration, contradicting c). 

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Let C0 be the even Clifford algebra associated to (E , q, OP2k (1)), cf. [3, §1.5]. It is a locally free OP2k -algebra of rank 8 whose center Z is a locally free quadratic OP2k -algebra (cf. [44, IV Prop. 4.8.3]). The discriminant cover r : S = Spec Z → P2 is a finite flat double cover branched along the sextic D ⊂ P2k . Assuming simple degeneration, then S is a smooth K3 surface of degree 2 over k. We say that a cubic fourfold X containing a given plane P is very general if the quadric surface bundle associated to P has simple degeneration and the associated K3 surface S has geometric Picard number 1 (equivalently, r∗ : Pic(P2k¯ ) → Pic(Sk¯ ) is an isomorphism, since a hyperplane section will map to a polarization of degree 2). If, furthermore, the ground field k is algebraically closed, then by rigidity of the N´eron–Severi group, r∗ : Pic(P2F ) → Pic(SF ) is an isomorphism for any extension F of k. We note that in the moduli space C8 of smooth cubic fourfolds containing a plane over C, the locus of very general ones (in our definition) is the complement of countably many closed subvarieties (see [37, Thm. 1.0.1]). Thus this notion of very general agrees with the usual notion in algebraic geometry. A cubic fourfold X containing a plane P over C is very general if and only if the Chow group CH2 (X) of cycles of codimension 2 is spanned by the classes of P and of a fiber Z of the quadric fibration (see [61, §1 Prop. 2] and its proof). Still assuming simple degeneration, the even Clifford algebra C0 , considered over its center, defines an Azumaya quaternion algebra B0 over S (cf. [48, Prop. 3.13] and [5, Prop. 1.13]). We refer to the Brauer class βX,P ∈ Br(S) of B0 as the Clifford invariant of the pair (X, P ). Lemma 4.2. Let X be a smooth cubic fourfold containing a plane P over a field k of characteristic 6= 2. Assume that the quadric surface bundle associated to P has simple degeneration. Let S be the associated K3 surface of degree 2. If X contains another plane P 0 (necessarily skew to P ), then the fibration e → P2 has a rational section. In this case, the generic fiber of π is an π:X k isotropic quadric over k(P2 ), hence is a k(P2 )-rational variety. In particular, the k-variety X is k-rational. Moreover, βX,P = 0 ∈ Br(S). e → P2 is constructed. One fixes an Proof. Recall how the fibration π : X k 5 arbitrary plane Q ⊂ Pk which does not meet P . The morphism π is induced by the morphism $ : X \ P → Q sending a point x of X not on P to the unique point of intersection of the linear space spanned by P and x with the linear space Q. Let P 0 be another plane in X. By Proposition 4.1, P 0 must be skew to P . Take the plane Q to be P 0 . On points of P 0 ⊂ X the map $ : X \ P 0 → P 0 is the identity. Thus π has a rational section, the generic fiber is an isotropic quadric, hence rational over k(P2 ), and the even Clifford invariant of this quadric is trivial.  In view of this lemma, when given a smooth cubic fourfold X containing a plane P whose associated quadric fibration has simple degeneration (e.g.,

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X is very general), we shall abuse notation and write βX ∈ Br(S) instead of βX,P ∈ Br(S). We point out that a very general cubic fourfold X has nontrivial Clifford invariant βX . Indeed, suppose that βX is trivial. By a result in the theory of quadratic forms (cf. [46, Thm. 6.3] or [56, 2 Thm. 14.1, Lemma 14.2]) β is e → P2 has a rational section. The Zariski closure trivial if and only if π : X e (in X) of a rational section of π would provide a cycle of dimension 2 in X having one point of intersection with a general fiber Z. Such a cycle cannot be rationally equivalent to any linear combination of the plane P and Z, since both P and Z have even intersection with Z. Hence CH2 (X) would have rank at least 3, and hence X cannot be very general (see [38, Thm. 3.1]). The generic fiber of (E , q, OP2k (1)) is a quadratic form of rank 4 over k(P2 ) with values in a k(P2 )-vector space of dimension 1. Choosing a generator l of OP2k (1) over k(P2 ), we arrive at a usual quadratic form (E, q) with discriminant extension k(S)/k(P2 ). The generic fiber of βX is then in the image of the restriction map Br(k(P2 )) → Br(k(S)) by the fundamental relations for the even Clifford algebra (cf. [45, Thm. 9.12]). Explicitly, the full Clifford algebra C(E, q) is central simple over k(P2 ) and its restriction to k(S) is Brauer equivalent to the even Clifford algebra C0 (E, q), hence with the generic fiber of βX . We note that C(E, q) depends on the choice of l, while C0 (E, q) does not. We now construct a particular Brauer class on k(P2 ) restricting to βX on k(S). This will play a crucial rˆole in the proof of Theorem 1. Proposition 4.3. Let X be a smooth cubic fourfold containing a plane P over a field k of characteristic 6= 2. Assume that the associated quadric surface bundle has simple degeneration along a divisor D ⊂ P2k and let S be the associated K3 surface of degree 2. Given a choice of homogeneous coordinates on P5k , there exists a line L ⊂ P2k and a quaternion algebra α over k(P2 ) whose restriction to k(S) is isomorphic to the generic fiber of the Clifford invariant βX , and such that α has ramification only at the generic points of D and L. Proof. For a choice of homogeneous coordinates on P5k , let L ⊂ P2k be the line whose equation is a00 from (3). The smoothness of X implies that a00 is nonzero. Then on A2k = P2k r L, the choice of a00 determines a trivialization ψ : O(1)|A2k → OA2k , with respect to which the quadratic form q 0 = ψ ◦ q|A2k : E |A2k → OA2k (given by the dehomogenization of equation (3) associated to ψ) represents 1. Letting V = P2k r (D ∪ L) ⊂ A2k , we have that (E |V , q 0 |V , OV ) is a regular quadratic form of rank 4 on V . In this case, we have the full Clifford algebra C = C (E |V , q 0 |V , OV ) at our disposal, which is an Azumaya algebra of degree 4 on V . Let us prove that C |k(P2 ) is Brauer equivalent to a symbol. Since q 0 |k(P2 ) represents 1, we have a diagonalization q 0 |k(P2 ) =< 1, a, b, abd >

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where d ∈ k(P2 )× /k(P2 )×2 is the discriminant. Then in Br(k(P2 )) we have [C |k(P2 ) ] = (a, b) + (a, abd) + (b, abd) + (−1, −d) = (−ab, −ad) by the formula [56, Ch. 2, Def. 12.7,Ch. 9, Rem. 2.12] relating the Clifford invariant to the 2nd Hasse–Witt invariant. Letting α = (−ab, −ad) ∈ Br(k(P2 )), we see that α coincides with [C |k(P2 ) ] in Br(k(P2 )), hence is unramified at all codimension 1 points of V , i.e., α is ramified at most at the generic points of D and L. The even Clifford algebra is a similarity class invariant, hence we have an an isomorphism C0 (E |V , q 0 |V , OV ) ∼ = C0 (E |V , q|V , O(1)|V ) over V , hence over the inverse image of V in S.
Finally, we have C |k(S) ∼ = M2 B0 |k(S) , hence α restricted to k(S) is Brauer equivalent to the generic fiber of the Clifford invariant βX ∈ Br(S). 

5. Proof of the main result Let us recall the statement. Theorem 5.1. Let X ⊂ P5 be a very general cubic fourfold containing a plane 3 (X/C, Q/Z(2)) is universally trivial. P ⊂ P5 over C. Then Hnr Let k be a field of characteristic 6= 2. Let X ⊂ P5k be a smooth cubic e → P2 be the associated quadric fourfold containing a plane over k. Let π : X k surface bundle. We assume that π has simple degeneration along a smooth divisor D ⊂ P2k , see Proposition 4.1. Denote by Q the generic fiber of π; it is a smooth quadric surface over k(P2 ). For any field extension F/k, we will need to refer to the following commutative diagram of Bloch–Ogus complexes 0

0

/ H 3 (F, Q/Z(2))

/ H 3 (F (P2 ), Q/Z(2))





3 / Hnr (X/F, Q/Z(2))

{∂γ }

3 / Hnr (QF /F (P2 ), Q/Z(2))

/

/

M

M

H 2 (F (γ), Q/Z(1))



(5)

H 2 (F (Qγ ), Q/Z(1))

where the sums are taken over all points γ of codimension 1 of P2F . The top row is the exact sequence defining the unramified cohomology of P2F (which is constant, by Proposition 1.2) via the residue maps ∂γ . The bottom row is the complex arising from taking residues on F (Q) at points of e whose image in P2 is of codimension 1 on P2 (recall codimension 1 of X F F 3 2 that Hnr (QF /F (P ), Q/Z(2)) consists of classes over F (Q) which have trivial residues at all rank 1 discrete valuations trivial on F (P2 )). Here, Qγ denotes eC → C, where the generic fiber of the restricted quadric fibration π|C : X C ⊂ P2F is a projective integral curve with generic point γ. Under the simple degeneration hypothesis, each Qγ is an integral quadric surface over the residue field F (γ). The bottom complex is also exact. This is obvious except 3 at the term Hnr (QF /F (P2 ), Q/Z(2)), though we shall not use the exactness at that point. The vertical maps in the diagram are the natural ones.

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49

3 3 Let ψ ∈ Hnr (XF /F, Q/Z(2)) ⊂ Hnr (QF /F (P2 ), Q/Z(2)). By Theorem 3.3, this is the image of an element ξ ∈ H 3 (F (P2 ), Q/Z(2)). By the commutativity of the right-hand square, and since the bottom row of diagram (5) is a complex, for each γ of codimension 1 in P2F we know that
∂γ (ξ) ∈ ker H 2 (F (γ), Q/Z(1)) → H 2 (F (Qγ ), Q/Z(1)) . (6)

We recall that if F is any field, then H 2 (F, Q/Z(1)) is isomorphic to the prime-to-p part of the Brauer group Br(F ), where p is the characteristic of F . With the notation of §4, where the line L is defined by a00 = 0, let d = ∆/a600 . Then div(d) = D − 6L, and the class of d in k(P2 )/k(P2 )×2 is the discriminant of the quadric Q. Definition 5.2. Fix L as above. Let F/k be a field extension and ξ a class in H 3 (F (P2 ), Q/Z(2)). We call an integral curve C ⊂ P2F with generic point γ a bad curve (for ξ) if C is different from DF and LF and if ∂γ (ξ) 6= 0 in H 2 (F (γ), Q/Z(1)). There are finitely many bad curves for each given ξ ∈ H 3 (F (P2 ), Q/Z(2)). Let α ∈ H 2 (k(P2 ), µ2 ) be the class of a quaternion algebra attached to π : e → P2 and the choice of a line L, as in Proposition 4.3. Theorem 3.1 X k and Proposition 3.2 imply that the following statements hold concerning bad curves: a) The class d|γ ∈ H 1 (F (γ), µ2 ) is trivial. b) The class α|γ ∈ H 2 (F (γ), µ2 ) is nontrivial and coincides with the class c(Q)|γ ∈ Br(F (γ)). c) The class ∂γ (ξ) ∈ H 2 (F (γ), Q/Z(1)) also coincides with the class c(Q)|γ ∈ Br(F (γ)). For curves C split by the discriminant extension (e.g., for bad curves), we will construct special rational functions that are parameters along C and are norms from the discriminant extension. This is where the very general hypothesis will be used. Lemma 5.3. Let k be a field of characteristic 6= 2. Let r : S → P2k be a finite flat morphism of degree 2 branched over a smooth curve D of even degree 2m. Assume that r∗ : Pic(P2k ) → Pic(S) is an isomorphism. Choose 2 a line L ⊂ P2k and √ a function d2 ∈ k(P ) with divisor D − 2mL that satisfies 2 k(S) = k(P )( d). If C ⊂ Pk is an integral curve (with generic point γ) different from D and L such that d|γ ∈ H 1 (k(γ), µ2 ) is trivial, then there exists a function fγ ∈ k(P2 ) whose divisor is C − 2nL for some positive integer n, and which is a norm from k(S)/k(P2 ). Proof. Under our hypothesis, Pic(S) = CH1 (S) = ZH, where H = r∗ L by flat pull-back. As a consequence, the action of Aut(S/P2 ) on Pic(S) is trivial. If the class of d is a square in k(γ), then f −1 C splits as C1 ∪ C2 , with C1 and C2 both having class nH for some positive integer n. Hence we have that C1 − nH = div(gγ ) for some gγ ∈ k(S). However, by proper pushforward, r∗ (C1 − nH) = C − 2nL, hence C − 2nL = div(Nk(S)/k(P2 ) (gγ )). Set fγ = Nk(S)/k(P2 ) (gγ ). 

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Remark 5.4. From the proof of Lemma 5.3, one sees that every bad curve C has even degree. When D has degree 6, using some further intersection theory, one can even prove that any such bad curve has degree 6, though we shall not need this. Now assume that k = C. Let α ∈ Br(C(P2 )) be the class of the full Clifford algebra (E , q, O(1)) over P2 r (D ∪ L), as in Proposition 4.3. Then α is unramified, except possibly at the generic points of D and L. Now assume that X is very general. Fix a field F/C. For each bad curve C over F , we will choose a parameter fγ ∈ F (P2 ) at the generic point γ of C as in Lemma 5.3, which we may apply since r∗ : Pic(P2F ) → Pic(SF ) is an isomorphism by the rigidity of the N´eron–Severi group. Denote by Y f= fγ , where the product is taken over all bad curves. Let us continue with the proof of Theorem 1. Recall that we have lifted 3 ψ ∈ Hnr (XF /F, Q/Z(2)) to an element ξ ∈ H 3 (F (P2 ), Q/Z(2)). We now compute the residues of ξ along codimension one points of P2F , where we need only worry about the generic points of D, any bad curves C, and L. Let us first consider the generic point η of D. The hypothesis that X is very general implies simple degeneration, hence that the F (η)-quadric Qη is a quadric cone over a smooth F (η)-conic. The conic is split by Tsen’s theorem since it is the base change of a smooth C(η)-conic. Hence Proposition 3.2, and the commutativity of the right hand square of diagram (5), implies that ξ is unramified at η. Now, we will compare ξ with the class α ∪ (f ), whose ramification we control. Relying on Theorem 1.1, we are implicitly considering an inclusion 3 2 H 3 (F (P2 ), µ⊗2 By construction, the function f is a 2 ) ⊂ H (F (P ), Q/Z(2)). √ 2 norm from the extension F (P )( d)/F (P2 ), i.e., is of the form f = g 2 − dh2 for some g, h ∈ F (P2 ). Also, f has its zeros and poles only along the bad curves and L, hence in particular, f is a unit at the generic point η of D. √ The extension F (P2 )( d)/F (P2 ) is totally ramified at√η. In particular, any unit in the local ring at η which is a norm from F (P2 )( d) reduces to a square in the residue field F (η). Thus f lifts to a square in the completion F[ (P2 )η . Now we consider the residues of α ∪ (f ) ∈ H 3 (F (P2 ), µ⊗2 2 ) along codimension one points of P2F . Both α ∈ H 2 (F (P2 ), µ2 ) and (f ) ∈ H 1 (F (P2 ), µ2 ) are unramified away from the generic points of D, L, and the bad curves C. At the generic point η of D, the function f is a square in the completion F[ (P2 )η . Thus, the residue of α ∪ (f ) at η is zero. At a bad curve, α is regular and the valuation of f is one. Thus the residue at such a curve is α|γ = c(Q)|γ . Thus, we have that the difference ξ − α ∪ (f ) ∈ H 3 (F (P2 ), Q/Z(2)) has trivial residues away from L, hence it comes from a constant class ξ0 in the image of H 3 (F, Q/Z(2)) → H 3 (F (P2 ), Q/Z(2)). Now we show that α∪(f ) vanishes when restricted to H 3 (F (Q), µ⊗2 2 ). In the notation of the proof of Proposition 4.3, q|F (P2 ) =< 1, a, b, abd > becomes

Universal Unramified Cohomology of Cubic Fourfolds

51

√ isotropic over F (Q) and α = (−ab, −ad). Since f is a norm from F (P2 )( d), Lemma 3.5 implies that√(a, b, f ) is trivial in H 3 (F (Q), µ⊗2 2 ). Further, since f is a norm from F (P2 )( d), we have (d, f ) = 0. But then α ∪ (f ) = (ab, ad, f ) = (a, b, f ) + (a, a, f ) + (ab, d, f ) 3 is trivial in H 3 (F (Q), µ⊗2 2 ) as well. Thus ψ ∈ Hnr (XF /F, Q/Z(2)) is the 3 image of the constant class ξ0 ∈ H (F, Q/Z(2)).

Remark 5.5. We can give a different argument, using Arason’s result (see Theorem 3.4), for the vanishing of α ∪ (f ) in H 3 (F (Q), µ⊗2 2 ). As in the notation of the proof of Proposition 4.3, we write qk(P2 ) = < 1, a, b, abd > and α = (−ab, −ad). The 3-Pfister form associated to α ∪ (f ) decomposes as  −ab, −ad, f  = < 1, ab, ad, bd >⊥ −f < 1, ab, ad, bd > . √ √ from F (P2 )( f ). Since f is a norm from F (P2 )( d) we have that d is a norm √ Thus d is a similarity factor of the norm form of F (P2 )( f )/F (P2 ), i.e., we have an isometry < 1, −f > ∼ = < d, −df >. Hence the 3-Pfister form  −ab, −ad, f  = < 1, ab, ad, bd >⊥ −f < 1, ab, ad, bd > = < ab, ad, bd >⊥< 1, −f >⊥ −f < ab, bd, ad > = < ab, ad, bd >⊥< d, −df >⊥ −f < ab, bd, ad > = < d, ab, ad, bd >⊥ −f < d, ab, bd, ad > contains the form < d, ad, bd, ab > = d < 1, a, b, abd >. Thus, by Theorem 3.3, 3 we have that α ∪ (f ) is trivial in Hnr (QF /F (P2 ), Q/Z(2)). Corollary 5.6. Let X ⊂ P5 be a very general cubic fourfold containing a plane 3 P ⊂ P5 over C. Then Hnr (X/C, µ⊗2 2 ) is universally trivial. Proof. In the following commutative diagram H 3 (F, µ⊗2 2 )

3 / Hnr (XF /F, µ⊗2 2 )

 H 3 (F, Q/Z(2))

 3 / Hnr (XF /F, Q/Z(2))

the horizontal maps are injective since we may specialize to an F -rational point. The vertical maps are injective by Theorem 1.1. By Theorem 5.1, any 3 3 ψ ∈ Hnr (XF /F, µ⊗2 2 ) ⊂ Hnr (XF /F, Q/Z(2))

is the image of a constant class ξ0 ∈ H 3 (F, Q/Z(2)), which by the diagram is 2-torsion, hence comes from an element in H 3 (F, µ⊗2 2 ). Since the right hand side vertical map is injective, the proof is complete. 

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, Two remarks on four-dimensional cubics, Uspekhi Mat. Nauk 48 (1993), no. 2(290), 201–202. Translation in Russian Math. Surveys 48 (1993), no. 2, 206–208. 30 C. Voisin, Bloch’s conjecture for Catanese and Barlow surfaces, J. Differential Geom. 97 (2014), no. 1, 149–175. 38 , Th´eor`eme de Torelli pour les cubiques de P5 , Invent. math. 86 (1986), no. 3, 577–601. 45, 46 , Hodge theory and complex algebraic geometry. I, II. Translated from the French by Leila Schneps. Cambridge Studies in Advanced Mathematics 76, 77, Cambridge University Press, Cambridge, 2007. 37 , Some aspects of the Hodge conjecture, Jpn. J. Math. 2 (2007), no. 2, 261–296. 30, 42 , Unirational threefolds with no universal codimension 2 cycle, Invent. math. 201 (2015), no. 1, 207–237. 30, 39 , On the universal CH0 group of cubic hypersurfaces, preprint arxiv.org/abs/1407.7261, to appear in J. Eur. Math. Soc. 38, 41 ¨ E. Witt, Uber ein Gegenbeispiel zum Normensatz, Math. Z. 39 (1935), no. 1, 462–467. 43 S. Zucker, The Hodge conjecture for cubic fourfolds, Compos. Math. 34 (1977), no. 2, 199–209. 30, 42

Asher Auel Department of Mathematics Yale University 10 Hillhouse Avenue, New Haven, CT 06511 USA e-mail: [email protected] Jean-Louis Colliot-Th´el`ene CNRS Universit´e Paris-Sud Math´ematiques, Bˆ atiment 425, 91405 Orsay Cedex France e-mail: [email protected] R. Parimala Department of Mathematics & CS Emory University 400 Dowman Drive, Atlanta, GA 30322 USA e-mail: [email protected]

Universal Spaces for Unramified Galois Cohomology Fedor Bogomolov and Yuri Tschinkel Abstract. We construct and study universal spaces for birational invariants of algebraic varieties over algebraic closures of finite fields. Mathematics Subject Classification (2010). 12G05, 20J06, 11R58. Keywords. Galois groups of function fields, unramified cohomology, universal spaces, anabelian geometry.

Introduction Let ` be a prime. Recall that in topology, there exist unique (up to homotopy) topological spaces K(Z/`n , m) such that • K(Z/`n , m) is homotopically trivial up to dimension m−1, in particular, Hi (K(Z/`n , m), Z/`n ) = 0, m

n

for 0 < i < m;

n

• H (K(Z/` , m), Z/` ) is cyclic, with a distinguished generator κm ; • for every topological space X and every α ∈ Hm (X, Z/`n ) there is a unique, up to homotopy, continuous map µX,α : X → K(Z/`n , m) such that µ∗X,α (κm ) = α. This reduces many questions about singular cohomology to the study of these universal spaces (see, e.g., [1, Chapter 2]). Analogous theories exist for other contravariant functors, for example, topological K-theory, or the theory of cobordisms. The study of moduli spaces in algebraic geometry can be viewed, broadly speaking, as an incarnation of the same idea of universal spaces. Here we propose a similar theory for unramified cohomology, developed in connection with the study of birational properties of algebraic varieties [4], [15]. The Bloch–Kato conjecture proved by Rost and Voevodsky [30], with a patch by Weibel, combined with techniques and results from birational anabelian geometry in [9], implies that an unramified class in the cohomology © Springer International Publishing AG 2017 A. Auel (eds.) et al., Brauer Groups and Obstruction Problems, Progress in Mathematics 320, DOI 10.1007/978-3-319-46852-5_5

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of the function field K = k(X) of an algebraic variety X over an algebraic ¯ p , with finite constant coefficients, is induced closure of a finite field k = F from the cohomology of a finite abelian group Ga . This, together with our prior work on centralizers of elements of Galois groups of function fields [8], implies our main result: Theorem. Let ` and p be distinct primes, K = k(X) the function field of an ¯ p , GK its absolute Galois algebraic variety X of dimension ≥ 2 over k = F n i group, and αK ∈ Hnr (GK , Z/` ), i ≥ 2, an unramified class. Then there exists a finite set J of finite-dimensional k-vector spaces Vj , j ∈ J, depending on αK , such that αK is induced, via a rational map, from an unramified class in the cohomology of an explicit open subset of the quotient of Y P := P(Vj ) j∈J a

by a finite abelian `-group G , acting projectively on each factor. Thus, the spaces P/Ga serve as universal spaces for all finite birational ¯ p . The theorem fails for H1 beinvariants of algebraic varieties over k = F nr cause all such elements are induced from abelian varieties and H1nr vanishes for every smooth proper separably rationally connected variety over an algebraically closed field (see, e.g., [16, Corollary 3.6]). Actions of finite abelian groups Ga on products of projective are V2 spaces described by central extensions of Ga , i.e., by subspaces in (Ga ). This allows us to present unramified classes of X in terms of configurations of subspaces of skew-symmetric matrices. For example, if the unramified Brauer group of X is trivial, then all finite birational invariants of X are encoded already in the combinatorics of configurations of liftable subgroups in finite abelian quotients of the absolute Galois group GK (see Section 1 for the definition). The program towards the construction of universal spaces for unramified cohomology was outlined in [4] and [5]. The recent proof of the Bloch–Kato conjecture allows us to complete this program, in a more precise and constructive form. This approach to birational invariants leads to many new questions: • Is there a smaller class of configurations with this universal property? • How does this structure interact with Sylow subgroups of GK ? • Is there an extension to cohomology with Z` -coefficients? An equally simple description of models for `-adic invariants would provide insights into higher-dimensional Langlands correspondence. ¯ • What are the analogs of universal spaces for varieties over k = Q? Counterexamples to our main result arise from bad reduction places, already for abelian varieties [4]. Here is the roadmap of the paper: In Section 1 we recall basic facts about stable and unramified cohomology. In Section 3 we provide some background on valuation theory. In Section 5 we investigate Galois cohomology groups of ¯ p and their function fields of higher-dimensional algebraic varieties over k = F

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images in cohomology of finite groups. In Section 6 we introduce and study unramified cohomology of algebraic varieties. Section 7 contains the proof of our main theorem, modulo geometric considerations presented in Sections 8 and 9. Acknowledgments. We are indebted to the referee for many detailed suggestions and remarks. We are grateful to A. Pirutka for her interest and insightful comments. The first author was supported by NSF grant DMS-1001662 and by AG Laboratory GU-HSE grant RF government ag. 11 11.G34.31.0023. The second author was partially supported by NSF grants 0901777 and 1160859.

1. Stable cohomology Let G be a pro-finite group. We will write Ga = G/[G, G]

Gc = G/[[G, G], G]

and

for the abelianization, respectively, the second lower central series quotient of G; throughout the paper, we write [G, G] and [[G, G], G] for topological closures of algebraic subgroups generated by the corresponding commutators. We have a canonical central extension π

a 1 → Z → Gc −→ Ga → 1.

(1.1)

i

Let M be a topological G-module and H (G, M ) its (continuous) icohomology group. These groups are contravariant with respect to G and covariant with respect to M . In this paper, G is either a finite group or a Galois group (see [1] for background on group cohomology and [28] for background on Galois cohomology). We will sometimes omit the coefficient module M from the notation. Our goal is to investigate incarnations of Galois cohomology of function fields in cohomology of finite groups. For example, let K = k(X) be the function field of an algebraic variety X over an algebraically closed field k; varieties birational to X are called models of K. We do not assume a model to be proper over k. Let GK be the absolute Galois group of K and π ˆ1 (X) the ´etale fundamental group of X, with respect to some basepoint. The choice of a base point will not affect our considerations and we omit it from our notation. When we work with GK , we take M to be either Q/Z or Z/`n , for some prime ` invertible in k, with trivial G-action. We have natural homomorphisms κ∗

η˜∗

X X H∗ (ˆ π1 (X)) −→ H∗et (X) −→ H∗ (GK ),

where the right arrow arises from the embedding of the generic point Xη → X. We will write ηX : GK → π ˆ1 (X) and ∗ ∗ ηX = η˜X ◦ κ∗X : H∗ (ˆ π1 (X)) → H∗ (GK )

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for the corresponding map in cohomology. We say that a class αK ∈ H∗ (GK ) is defined (or represented) on a model X of K if there exists a class αX ∈ H∗et (X) such that ∗ αK = η˜X (αX ).

Let G be a finite group. A continuous homomorphism χ:π ˆ1 (X) → G gives rise to homomorphisms in cohomology η∗

χ∗

X H∗ (G) −→ H∗ (ˆ π1 (X)) −→ H∗ (GK ).

Conversely, every αK ∈ H∗ (GK ) arises in this way: there exist • a model X of K, • a continuous homomorphism χ as above, • and a class αG ∈ H∗ (G) such that ∗ αK = ηX (χ∗ (αG )). This follows from the description of ´etale cohomology of points, see [21]. In such situations we say that αK is defined on X and is induced from χ. A version of this construction arises as follows: assume that the characteristic of k does not divide the order of G. Let V be a faithful representation of G over k, and X an algebraic variety over k with function field K = k(X) ' k(V )G , the field of invariants; we will write X = V /G and call it a quotient. Even more generally, let Y be a quasi-projective algebraic variety over k with a generically free action of G, and X = Y /G the quotient. This situation gives rise to a natural surjective continuous homomorphism GK → G and induced homomorphisms on cohomology siK : Hi (G) → Hi (GK ). The following lemma shows that we have many choices in realizing a class αK ∈ Hi (GK ): Lemma 1.1. [4] Assume that αK ∈ Hi (GK ) is represented by a class αX ∈ Hiet (X) on some affine irreducible model X of K and is induced from a surjective continuous homomorphism χ : π ˆ1 (X) → G and a class αG ∈ Hi (G). Let V be a faithful representation of G over k and V ◦ ⊂ V the locus where the action is free. Then, for every x ∈ X and v ∈ V ◦ there exists a map f = fx : X → V /G such that • f (x) = v and • the restriction of αX to X ◦ = f −1 (V ◦ /G) ⊂ X is equal to f ∗ (αG ).

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Proof. We follow the proof in [4]. The homomorphism χ : π ˆ1 (X) → G defines ˜ → X, by an affine variety X. ˜ The ring k[X] ˜ is a a finite ´etale covering π : X ∗ ˜ defines k[G] -module. Every finite-dimensional k[G]-submodule W ⊂ k[X] ˜ → W. a G-equivariant map X Let e ∈ k[G] be the unit element of G. For any G-orbit G · y ∈ V there is a G-linear homomorphism ly : k[G] → V, ˜ such which maps the orbit G · e to G · y. Let x ˜ ∈ π −1 (x). Choose h ∈ k[X] that h(˜ x) = 1, h(g · x ˜) = 0, g 6= e. ˜ and defines a regular G-map Then h generates a k[G]-submodule W ⊂ k[X] ˜ → W = k[G], with h(˜ h:X x) = e ∈ k[G]. The map f := ly ◦ h is a regular G-map satisfying the first property. Let X ◦ = f −1 (V ◦ /G) ⊂ X. It is a nonempty affine subvariety. We have a compatible diagram of G-maps ˜◦ ˜ X ⊂ X V◦ o πG

 V ◦ /G o

π0

 X◦

f

π

⊂

 X

and the maps πG and π induce the same cover π0 . This implies the second claim.  We can achieve even more flexibility for G-maps, under a projectivity condition on V : we say that a G-module V is projective if for every finitedimensional representation W of G with a G-surjection µ:W →V there exists a G-section θ:V →W

with

µ ◦ θ = id.

This condition holds, for example, for regular representations over arbitrary fields or when the order of G is invertible in k. Now let {Sj }j∈J be a finite set of G-orbits in a generically free G-variety Y with stabilizers Hj so that Sj ' G/Hj . Consider a faithful representation V of G and a subset {Tj }j∈J of G-orbits in V with stabilizers Qj , with Hj ⊂ Qj , Tj = G/Qj . Consider regular G-maps fj : Sj → Tj , for j ∈ J. Applying the argument above to finite sets of orbits, we obtain: Lemma 1.2. Assume that V is a projective G-module. Then there is a regular G-map f : Y → V such that f = fj , for all j ∈ J. We return to our setup: X = Y /G, K = k(X), and χ : GK → G, inducing siK : H∗ (G) → H∗ (GK ).

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The groups His,K (G) := Hi (G)/Ker(siK ) are called stable cohomology groups with respect to K = k(X). Let \ Ker(si ) := Ker(siK ), K

over all function fields K = k(X) as above. In fact, Ker(si ) = Ker(sik(V /G) ), for some faithful representation V of G over k, in particular, this is independent of the choice of V (see [7, Proposition 4.3]). The groups His (G) := Hi (G)/Ker(si ) are called stable cohomology groups of G (with coefficients in M = Z/`n or Q/Z); they depend on the ground field k. These define contravariant functors in G. For example, for a subgroup H ⊂ G we have a restriction homomorphism resG/H : H∗s (G) → H∗s (H). Furthermore: • While usual group cohomology Hi (G) can be nontrivial for infinitely many i (even for cyclic groups), stable cohomology groups His (G) vanish for i > dim(V ), where V is a faithful representation. • We have His (G) ⊆ His (Syl` (G))NG (Syl` (G)) , where the coefficient group M is Z/`n or Q` /Z` , NG (H) is the normalizer of H in G, and Syl` (G) an `-Sylow subgroup of G. The determination of the stable cohomology ring H∗s (G, Z/`n ) := ⊕i His (G, Z/`n ) is a nontrivial problem, see, e.g., [6] for a computation of stable cohomology of alternating groups. For finite abelian groups G, we have ∗ ^ H∗s (G, Z/`n ) ⊂ (H1 (Zm , Z/`n )), (1.2) induced by a surjection Zm → G. For central extensions of finite groups as in (1.1), the kernel of πa∗ : H∗s (Ga ) → H∗s (Gc ) contains the ideal I = I(Gc ) generated by
R2 = R2 (Gc ) := Ker H2s (Ga ) → H2s (Gc ) . (see, for example, [11, Section 8]). An important role in the computation of this subring of H∗s (Gc ) is played by the fan Σ = Σ(Gc ) = {σ}, the set of noncyclic liftable subgroups σ of Ga , and the complete fan ¯ = Σ(G ¯ c ) = {σ}, Σ

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consisting of all nontrivial liftable subgroups σ ⊂ Ga : a subgroup σ is liftable if and only if the full preimage σ ˜ of σ in Gc is abelian. The fan Σ defines 2 2 a a subgroup R (Σ) ⊆ Hs (G ) as the set of all elements which vanish upon restriction to every σ ∈ Σ. Note that for any σ ˜ and σ as above, the natural homomorphism of cohomology groups His (σ) → Hi (˜ σ) is injective; indeed, stable cohomology of any finite abelian group Ga with any finite coefficients coindices with the image of the group cohomology of Ga in the group cohomology of any finite rank free abelian group surjecting onto Ga . Using this fact, we have R2 ⊆ R2 (Σ). Lemma 1.3. For every α ∈ I(Gc ) ⊆ H∗s (Ga ) and every σ ∈ Σ(Gc ) the restriction of α to σ is trivial. Definition 1.4. Let 1 → Z → Gc → Ga → 1 be a central extension of finite groups, with Ga abelian. A ∆-pair (I, D) of Ga is a set of subgroups I ⊆ D ⊆ Ga such • • •

that ¯ c ), I ∈ Σ(G D is noncyclic, ¯ c ). for every δ ∈ D, the subgroup hI, δi ∈ Σ(G

This definition depends on Gc . Assume we have a commutative diagram of central extensions ˜c ˜a /G /G /1 / Z˜ 1

1

 / Gc

 /Z

γ

 / Ga

/ 1.

˜ D) ˜ of G ˜ a surjects onto a ∆-pair (I, D) of Ga if Definition 1.5. A ∆-pair (I, ˜ = I and γ(D) ˜ = D. γ(I) Definition 1.6. A class α ∈ His (Ga ) is unramified with respect to a ∆-pair if its restriction to D is induced from D/I, i.e., there exists a β ∈ His (D/I) such that φ(α) = ψ(β), for the natural homomorphisms in the diagram: H∗s (Ga )

φ

/ H∗s (D) o

ψ

H∗s (D/I)

Recall that a cohomology class β ∈ H∗ (GK ) is unramified if for every divisorial valuation ν of K the restriction of β to GKν , the Galois group of the completion of K with respect to ν, is induced from the quotient GKν /Iν , where Iν ⊂ GK is the corresponding inertia subgroup.

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When G is a finite group and K = k(V /G), the field of functions of V /G for some faithful representation V of G, the stable cohomology groups of G are naturally subgroups of the corresponding cohomology groups of GK , and unramified stable cohomology classes are those which are unramified when considered as classes in H∗ (GK ). In particular, the images of unramified classes in Ga with respect to ∆-pairs are mapped to unramified classes in H∗ (Gc ), as the following lemma shows. Lemma 1.7. Consider a homomorphism ˜ a → Ga γ:G ˜ a ) be the induced and a class α ∈ Hi (Ga ), for i ≥ 2. Let α ˜ := γ ∗ (α) ∈ Hi (G a ˜ ˜ ˜ class. Let (I, D) be a ∆-pair in G . Assume that one of the following holds: ˜ = 0, • γ(I) ˜ is cyclic, • γ(D) • γ induces a surjection of ∆-pairs ˜ D) ˜ → (I, D) (I, and α ∈ His (Ga ) is unramified with respect to (I, D). ˜ a ) is unramified with respect to (I, ˜ D). ˜ Then α ˜ ∈ His (G Proof. The first case is evident. In the second case, the stable cohomology ˜ is trivial. Consider the third condition. By assumption, γ induces a of D ˜ I˜ → D/I. Passing to cohomology we get a commutative homomorphism D/ diagram / H∗s (D) H∗s (D/I)  ˜ / H∗s (D),

 ˜ I) ˜ H∗s (D/ and thus the claim.



2. Central extensions and isoclinism Let Ga and Z be finite abelian `-groups. Central extensions of Ga by Z are parametrized by H2 (Ga , Z); for α ∈ H2 (Ga , Z) we let Gcα be the corresponding central extension: π

a 1 → Z → Gcα −→ Ga → 1.

Fix an embedding Z ,→ (Q/Z)r , consider the exact sequence 1 → Z → (Q/Z)r → (Q/Z)r → 1, and the induced long exact sequence in cohomology δ

H1 (Ga , (Q/Z)r ) −→ H2 (Ga , Z) → H2 (Ga , (Q/Z)r ).

(2.1)

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We say that α, α ˜ ∈ H2 (Ga , Z) and the corresponding extensions are isoclinic if α−α ˜ ∈ δ(H1 (Ga , (Q/Z)r )). This notion does not depend on the chosen embedding Z ,→ (Q/Z)r and is equivalent to the standard definition of isoclinic in the theory of `-groups (as in [17]). Lemma 2.1. If α, α ˜ ∈ H2 (Ga , Z) are isoclinic, then the corresponding extena sions of G define the same set of ∆-pairs in Ga . Proof. A pair of subgroups (I, D) is a ∆-pair in Ga , with respect to a central extension Gc , if their preimages commute in Gc , i.e., [πa−1 (I), πa−1 (D)] = 0

in Z.

Consider the homomorphism πa∗ : H2 (Ga , Q/Z) → H2 (Gc , Q/Z), and note that Ker(πa∗ ) only depends on the isoclinism class of the extension. V2 a V2 a Furthermore, H2 (Ga , Q/Z) is dual to (G ). Let R ⊂ (G ) be the subgroup which is dual to Ker(πa∗ ). It remains to observe that (I, D) is a ∆-pair for Gc if and only if πa−1 (I) ∧ πa−1 (D) intersects R trivially; thus the notion of a ∆-pair is an invariant of the isoclinism class of the extension.  Lemma 2.2. If α, α ˜ ∈ H2 (Ga , Z) are isoclinic, then there exist faithful repre˜ sentations V, V of Gcα and Gcα˜ over k such that V /Gcα and V˜ /Gcα˜ are birational. Proof. Explicit construction: Let χ1 , . . . , χr be a basis of Hom(Z, k × ) and put V := ⊕rj=1 Vj and V˜ = ⊕rj=1 V˜j , where

Gc

c

G V˜j = IndZ α˜ (χj ). Note that the projectivizations P(Vj ) := (Vj \0)/k × and P(V˜j ) are canonically isomorphic as Ga -representations. The group (k × )r acts on V and V˜ , and both V /Gcα and V /Gcα˜ are birational to     r r Y Y  P(Vj ) /Ga ×  k × /χj (Z) . 

Vj = IndZ α (χj )

and

j=1

j=1

Lemma 2.3. Consider a central extension of finite groups π

a 1 → Z → Gc −→ Ga → 1

and let V = ⊕j Vj be a faithful representation of Gc as in Lemma 2.2, i.e., Gc × each Q Vj = IndZ (χj ), where {χj }j∈J is a basis of Hom(Z, k ). Let P := j∈J P(Vj ). Then: 1. Ga acts faithfully on P.

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2. For any subgroup σ ⊂ Ga the subset of σ-fixed points Pσ ⊂ P is nonempty ¯ c ). if and only if σ ∈ Σ(G 3. Each irreducible component of Pσ is a product of projective subspaces of P(Vj ), corresponding to different eigenspaces of σ in Vj , and distinct irreducible components are disjoint. 4. Each irreducible component of Pσ is stable under the action of Hσ ⊂ Gc , the maximal subgroup such that [Hσ , πa−1 (σ)] = 1 in Gc ; the action of σ Gc /Hσ on the set of components S of P isσ free. a ◦ 5. The action of G on P := P \ σ∈Σ\0 P is free. ¯ Proof. Since the order of Gc is coprime to the characteristic of k, every g ⊂ Gc is semi-simple and we can decompose M Vj = Vj (λi (g)), i

as Q a`sum of eigenspaces. The subset of g-fixed points splits as a product j i P(Vj (λi (g))), where the product runs over different eigenvalues in different Vj . It follows that the subset of g-fixed points Pg ⊂ P is a union of products of projective subspaces of P(Vj ). ¯ then its elements can be simultaneously diagonalized. Hence If σ ∈ Σ, Q the subset of fixed points in P = j P(Vj ) is a union of products of projective subspaces, and there is a Zariski open subvariety of P on which the action of σ is free. ¯ Then the same Let σ := hg, hi ⊂ Ga be a subgroup such that σ ∈ / Σ. holds for the images of g, h in GL(Vj ), for at least one j ∈ J. Thus the commutator [g, h] ∈ GL(Vj ) is a nontrivial scalar matrix, hence they have no common eigenvectors, i.e., no common fixed points in P(Vj ). Thus if σ ∈ / ¯ then σ has no fixed points in Q P(Vj ). Note that projective subspaces Σ, j corresponding to different eigenvalues of g do not intersect in P(Vj ) and hence Pσ splits into a disjoint union of products of projective subspaces of different P(Vj ). ˜ i has a fixed Assume that [h, σ ˜ ] = 1 in Gc for some h ∈ Gc . Then hh, σ point in each component of Pσ and h maps every component of Pσ into itself. Thus a subgroup H ⊂ Gc , with [H, σ ˜ ] = 1 maps every component of Pσ into itself. ¯ for some γ ∈ σ. Then for some j, the images Assume that hh, γi ∈ / Σ, of h, γ in GL(Vj ) have nonintersecting invariant subvarieties in P(Vj ). In particular, h does not preserve any component of Pσ .  Lemma 2.4. Let K = k(X) be a function field with Galois group GK . Given c a surjection G QK → G , onto some finite central extension of an abelian group a G , let P = j P(Vj ) be the space constructed in Lemma 2.3. Then there is a rational map % : X 99K P/Ga such that • % maps the generic point of X into P◦ /Ga ;

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• the homomorphism sK : H∗s (Ga , Z/`n ) → H∗ (GK , Z/`n ) factors through the cohomology of P◦ /Ga . Proof. Let X ◦ ⊂ X be an open affine subvariety such that π1 (X ◦ ) surjects ˜ ◦ → X ◦ be the induced unramified Gc covering. Then k[X ˜ ◦] onto Gc . Let X c decomposes into an infinite direct sum of G -representations. Fix a point ˜ ◦ and consider its orbit. The restriction of k[X ˜ ◦ ] to this orbit defines a X c regular quotient G -representation isomorphic to k[G]. Since the order of G ˜ ◦ ] projecting isomorphically is coprime to p, we have a direct summand of k[X to k[G] under the above homomorphism. This subspace of regular functions ˜ ◦ defines a G-equivariant map to V and hence a map X 99K V /Gc with on X desired properties. 

3. Basic valuation theory ¯ p , K = k(X) its function field, and GK the Let X be a variety over k = F absolute Galois group of K. A valuation of K is a homomorphism ν : K × → Γν onto a totally ordered abelian group Γν such that its extension to K, via ν(0) = ∞, satisfies the nonarchimedean triangle inequality. A divisorial valuation measures the order of a rational function along a divisor on some model X of K. Let oν denote the valuation ring and mν the corresponding maximal ideal. The residue field will be denoted by K ν ; in general, it need not be finitely generated over k, see Example 9 in [29]. We write VK for the set of (equivalence classes of) valuations of K and DVK for the subset of divisorial valuations. Let Z ⊂ X be an affine subset, Z = Spec(oZ ), and ν ∈ VK . A valuation ν is said to have a center on Z, cX (ν)◦ ⊆ Z if and only if ν(f ) ≥ 0, for all f ∈ oZ ; the center is the closed subvariety of Z corresponding to the prime ideal defined by ν(f ) > 0. For ν ∈ VK , let Dν ⊂ GK denote a decomposition group of ν and Iν ⊂ Dν the inertia subgroup; we have GK ν = Dν /Iν (see, e.g., [19, Section 5] for the description of the inertia subgroup in terms of the value group and the description of the Galois group of the residue field). The pro-`-quotients of these groups will be denoted by GK , Dν , and Iν , respectively. We will always assume that p 6= `. The corresponding abelianizations will be denoted a c c c by GK , Dνa , and Iνa ; their canonical central extensions by GK , DK , and IK . a Under our assumptions, GK is a free abelian pro-`-group.

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Lemma 3.1. For ν ∈ VK consider the commutative diagram / GK Dν πν

 Dνa



δνa

π

/ Ga , K

where πv and π are the canonical projections and δνa is the induced homomorphism. Then δνa is injective with primitive (i.e., nondivisible) image. In particular, δνa embeds Iνa as a primitive subgroup of Dνa . a Proof. We have GK = Hom(K × , Z` ) and Dνa = Hom(Kν× , Z` ). We have exact sequences × 1 → o× ν → K → Γν → 1 and × 1 → (1 + mν )× → o× ν → K nu → 1. ¯ p (X) with Q-independent values of ν(x) are alNote that the elements of F gebraically independent. Thus the Q-rank of Γν is ≤ n and the Z` rank of Hom(Γν , Z` ) is also ≤ n; it is a free Zl -module of finite rank. Taking a finitely generated subgroup S ⊂ K × of the same rank, with an isomorphism Hom(Γν , Z` ) = Hom(S, Z` ), we obtain a direct splitting (depending on S):

Hom(K × , Z` ) = Hom(Γν , Z` ) ⊕ Hom(o× ν , Z` ). The right summand contains Hom(K ∗ν , Z` ) as a primitive subgroup. This implies that a Dνa = Iνa ⊕ GK , ν  where Hom(Γν , Z` ) = Iνa . Remark 3.2. If the residue field K ν is finitely generated over k, then there is a model X of K such that the center of ν is realized by a subvariety Xν ⊂ X. Indeed, in this case there is a finite subset of elements fi ∈ oν which generate K and reduce to a generating subset of K ν . The subring k[f1 , . . . , fn ] defines an affine model X of K and its image B in K ν a finitely generated subring of K ν ; hence we have an inverse embedding of affine varieties XB ⊂ X with desired properties. c Let Σ(GK ) be the set of primitive topologically noncyclic subgroups of a c GK whose preimage in GK is abelian. By [8, Section 6], we have:

Theorem 3.3. Assume that dim(X) ≥ 2. Then rkZ` (σ) ≤ dim(X),

c for all σ ∈ Σ(GK ).

The following key result gives a valuation-theoretic interpretation of a liftable subgroups in GK ; it is crucial for the reconstruction of function fields in [9] and [10]. Theorem 3.4. [8, Corollary 6.4.4] Assume that dim(X) ≥ 2 and let σ ∈ c Σ(GK ). Then there exists a valuation ν ∈ VK such that Iνa is a subgroup of σ of Z` -corank at most 1 and σ ⊆ Dνa .

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4. Liftable subgroups and their configurations ¯ p . In Let K = k(X) be the function field of an algebraic variety over k = F c this section, we compare the structure of the fan Σ(GK ) with fans in its finite quotients. Consider the canonical central extension c a 1 → ZK → GK → GK → 1.

(4.1)

Lemma 4.1. We have c c ZK = [GK , GK ].

Proof. This holds for function fields of curves since the corresponding pro`-quotients of their absolute Galois Q a groups are free. In higher dimensions, a embedds into the product E GE , where E ranges over function fields of GK c a c maps to , the center of GK → GE curves E ⊂ K. Under the projection to GK zero, hence the claim.  Lemma 4.2. Consider commutative diagrams of continuous homomorphisms / ZK / Gc / Ga /1 1 K K c γK

γK

   /Z / Gc / Ga / 1, 1 c c . Assume that where G is finite, with fixed surjective γK and surjective γK c Z ⊂ G is a quotient of ZK such that

Ker H2 (Ga ) → H2 (Gc ) = Ker H2 (Ga ) → H2 (GcK ) , with Q/Z-coefficients. Then Gc is unique modulo isoclinism. Proof. Assume that Gc1 , Gc2 are two such extensions of Ga with Z1 , Z2 , respectively, and put G := Gc1 ×Ga Gc2 . We have a natural surjection G → Ga and an inclusion Z1 × Z2 ,→ G. Moreover, [G, G] ⊆ Z1 × Z2 . By Lemma 4.1, c ZK is generated by commutators in GK . There is natural diagonal projection c c GK → G which maps ZK onto [G, G]. The image of GK in G is a subgroup c a a c ˜c ˜ ˜ ˜ G with G  G and [G , G ] = [G, G]. By the maximality assumption, we obtain that both projections of [G, G] into Z1 and Z2 are are isomorphisms; this implies isoclinism.  We proceed to investigate the properties of fans under such factorizations. Let a γK : GK → Ga be a continuous surjective homomorphism onto a finite group. We choose a maximal finite central extension Gc of Ga as in Lemma 4.2. Corollary 4.3. Given continuous surjective homomorphisms ˜K γ a γ ˜ a −→ GK −→ G Ga ,

˜a

(4.2)

with G a finite group, there is a unique (modulo isoclinism of lower rows) diagram of central extensions

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F. Bogomolov and Y. Tschinkel 1

/ ZK

/ Gc

1

 / Z˜

 ˜c /G

/ Ga K

K c γ ˜K

/1

γ ˜K

 ˜a /G

γc

/1

γ

  / Gc /Z 1 c ˜ Z. , γ c and maximal Z, with surjective γ˜K

 / Ga

/1

Proof. Evident.



We will use the following observation: Lemma 4.4. Let G a be a profinite abelian group and γj ˜a, G a −→ G j

j = 1, . . . , n,

a collection of continuous surjective homomorphisms onto finite groups. Then there exists a continuous surjection ˜a γ : Ga → G onto a finite group such that each γj factors through γ: γ

˜a → G ˜a. γj : G a −→ G j ˜ a to be the image of G a in the direct product Proof. We can choose G ˜1 × · · · × G ˜n. G



˜ a , preserving We are interested in factorizations (4.2), with finite G liftable subgroups and their configurations. Throughout we will be working with the canonical, modulo isoclinism, diagram as in Corollary 4.3, i.e., a ˜ c ) and the factorization as in Equation (4.2) will canonically determine Σ(G a ˜ set of ∆-pairs in G , by Lemma 2.1. Let ΣE (Gc ) := {σ ∈ Σ(Gc ) | σ = γK (σK ),

c for some σK ∈ Σ(GK )}

be the subset of extendable subgroups. Lemma 4.5. Given a continuous surjective homomorphism a γK : GK → Ga

onto a finite abelian group there exists a factorization ˜K γ a γ ˜ a −→ GK −→ G Ga ,

γK = γ ◦ γ˜K ,

˜ a , such that for all σ ∈ Σ(Gc ) we have: if σ is nonextendable, with finite G ˜ c ) with γ(˜ then there is no σ ˜ ∈ Σ(G σ ) = σ.

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Proof. First we prove the statement for one nonextendable σ. Write a GK = proj lim Gaι , ι∈I

γιι0 : Gaι −→ Gaι0 ,

ι0  ι,

(4.3)

a where the limit is over finite continuous quotients of GK . Assume that for all ι, there is some σι ∈ Σ(Gcι ) surjecting onto σ; this implies that there exist such σι0 , for all ι0  ι, with γιι0 (σι ) = σι0 . a a By compactness of GK , there exists a closed liftable σK ⊂ GK surjecting onto σ. This contradicts our assumption that σ is nonextendable. Thus there is a required factorization γ a ˜ a −→ GK →G Ga .

Let {σ1 , . . . , σn } = Σ(Gc ) \ ΣE (Gc ). For each j, let γj a ˜ a −→ GK −→ G Ga j be the factorization constructed above. Now we apply Lemma 4.4, combined with Corollary 4.3, and obtain factorizations of γj : a ˜a → G ˜ j → Ga , γ : G ˜ a → Ga , GK →G

˜ c ) surjecting Assume that there is some j for which there exists a σ ˜ ∈ Σ(G ˜ onto σj . Then image σ ˜ in Gj must be liftable, contradicting the construction in the first part.  Let I ⊆ D ⊆ Ga . be a ∆-pair (see Definition 1.4). Throughout, we assume that Ga arises as a finite quotient of the Galois group GK of some function field K, in particular, the corresponding Gc is determined as in Lemma 4.2, up to isoclinism. We say that (I, D) is extendable if there exists a valuation ν ∈ VK and subgroups I a ⊆ Iνa ,

Da ⊆ Dνa ,

I a ⊂ Da ,

such that γK (I a ) = I, γK (Da ) = D. ˜ D) ˜ is said to surject onto (I, D) if Recall that a ∆-pair (I, ˜ = I, γ(D) ˜ = D. γ(I) We will need the following strengthening of Lemma 4.5 Proposition 4.6. Given a continuous surjective homomorphism a GK → Ga

onto a finite abelian group there exists a factorization a ˜ a → Ga , GK →G ˜ a , such that for all ∆-pairs (I, D) in Ga we have: if (I, D) with finite G ˜ D) ˜ in G ˜a nonextendable and D is not liftable, then there is no ∆-pair (I, surjecting onto (I, D).

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Proof. As in the proof of Lemma 4.5, it suffices to establish the statement for one nonextendable ∆-pair; indeed, there are only finitely many ∆-pairs in Ga and the same application of Lemma 4.4 will then establish it for all. a Assume that there is no finite quotient of GK with the desired property. We start with a factorization γ a ˜ a −→ GK →G Ga

˜ a satisfies the conclusions of Lemma 4.5, i.e., no σ ∈ Σ(Gc ) \ such that G c ˜ c ). ΣE (G ) is the image of a σ ˜ ∈ Σ(G Let (I, D) be a nonextendable ∆-pair. If D/I is cyclic or trivial, then, ˜ c it is not liftable to Σ(G ˜ c ). Thus in fact, D ∈ Σ(Gc ) and by assumption on G c ˜ D) ˜ surjects onto (I, D), then hI, ˜ gi ⊂ Σ(G ˜ ) surjects onto D, if a ∆-pair (I, ˜ and hence D lifts to Σ(G ˜ c ), contradicting our assumption. for some g ∈ D, Thus it suffices to consider ∆-pairs (I, D) with D/I noncyclic. ˜ D) ˜ in G ˜ a surjecting onto By our assumption, there exists a ∆-pair (I, (I, D). Choose representatives g1 , . . . , gn ∈ D \ I for nontrivial elements of ˜ surjecting onto gj under γ. Note that for each j, D/I and g˜j ∈ D ¯ c ), σj := hgj , Ii ∈ Σ(G

˜ ∈ Σ( ¯ G ˜ c ). σ ˜j := h˜ gj , Ii

˜ a , all σj are and that σ ˜j surject onto σj . By Lemma 4.5 and our choice of G extendable. Then n \ σ ˜j , I˜ ⊆ j=1

˜ by and if we replace and rename the original D ˜ := D

n X

σ ˜j ,

j=1

˜ D) ˜ is a ∆-pair in G ˜ a surjecting onto (I, D). then (I, Now we consider a projective system of finite continuous quotients a ˜ aι → G ˜ a → Ga , GK →G

γιι0 : Gaι −→ Gaι0 ,

ι0  ι.

˜ ι ) in G ˜ a surjecting onto Assume that for each ι there exists a ∆-pair (I˜ι , D ι (I, D). Iterating the construction above, we construct, for each ι, a collection of liftable subgroups σ ˜ι,1 , . . . , σ ˜ι,n ˜ ι ) of the form and a ∆-pair (I˜ι , D I˜ι ⊆

n \ j=1

σ ˜ι,j ,

˜ ι := D

n X

σ ˜ι,j ,

j=1

such that ˜ ι0 ), for each ι0  ι, ˜ ι ) surjects onto (I˜ι0 , D • (I˜ι , D

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a and in particular onto (I, D). By compactness of GK (see Lemma 4.5), there exist closed subgroups c σK,1 , . . . , σK,n ∈ Σ(GK );

the closed subgroups I a :=

\

σK,j ,

Da := hσK,j ij

a surject onto I, resp. D. of GK Note that if a pair (I, D0 ) ⊂ (I, D) is non extendable, then (I, D) is also not extendable. Thus we can assume that proper sub-pairs (I, D0 ) ⊂ (I, D) are extendable. In particular, any liftable subgroup in Da is equal to hI a , gi, g ∈ / I a , for some g ∈ Da \ I a . By [8, Lemma 6.4.3] and [8, Corollary 6.4.4] any liftable subgroup Lg contains a subgroup Ig of corank ≤ 1 which consists of flag elements and the a a . If Ig = Lg , for some g, then I a consists and Dν,g group Ig is contained in Iν,g of flag elements and hence by [8, Lemma 6.4.3] and [8, Corollary 6.4.4] there is a ν such that Ig ⊂ Iνa and Da ⊂ Dνa and hence (I, D) is liftable, contradicting our assumption. Thus we can assume that for any Lg the subgroup Ig has corank exactly one. Let us show that Ig = I a . Assume that h ∈ I a is not a flag map. By results mentioned above, h ⊂ Dµah , for some valuation µh with the property that any commuting pair hh, xi is contained in the image of hh, Iµ i. The image of hh, Iµ i = H a ⊂ Ga is a liftable subgroup, but then Da ⊂ H a and hence Da is also liftable, contradicting our assumption on Da . Our assumption on (I, D) implies that any closed subgroup containing hI, δi, with δ ∈ D \ I, lifts to an abelian group. The theory developed in [8] a which lift describes all pairs (g, h) of topologically independent elements in GK c to commuting pairs in GK : they are realized as Z` -valued maps on K × /k × = P(K), a projective space over k, with the property that g(xy) = g(x) + g(y), for all x, y. The so-called flag maps f are maps such that every finitedimensional subspace Pn ⊂ P(K) admits a flag of projective subspaces P1 ⊂ . . . ⊂ Pr = Pn so that f is constant on Pi \ Pi−1 , for all i = 2, . . . , r. A flag map defines

1. a natural scale on K: a sequence of linear subspaces Lγ ⊂ K over k parametrized by an ordered abelian group Γ with the property that Lγ ⊂ Lβ if γ > β in Γ, 2. a map ν : K × → Γ, where ν(x) = β if x ∈ Lβ and is not contained in Lγ ⊂ Lβ . Moreover, x · Lγ = Lγ+ν(x) , i.e., the scale is invariant under multiplication in K × . Thus to any multiplicative flag map f on P(K) we can associate a nonarchimedean valuation ν of K with value group Γ. We have f (x) = f∗ (ν(x)), where f∗ is a homomorphism Γ → Z` . Note that a flag map f defines a unique order, and hence the value group of the valuation. Of course, similar homomorphisms exist for refinements of this valuation, but the latter is defined intrinsically by the flag map f .

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The main result of [8] states that for any pair of (g, h) as above there is a basis (f, δ) of hg, hi such that f is a flag map defining (canonically!) a valuation νf and δ belongs to the decomposition group of νf . This holds for ¯ p ; a slightly more complicated version is valid for funcfunction fields over F tion fields over arbitrary algebraically closed fields k. The property of δ to be in the decomposition group of ν is also described in terms of projective geometry of the level sets in P(K). In particular, for any such δ there is a maximal c valuation ν such that δ ∈ Dνa and every σ ∈ Σ(GK ) containing δ is contained a in hIν , δi. The above general description of commuting pairs provides also a a description of pairs (I a , Da ) in GK . Since by assumption I 6= D, the same a a a holds for I 6= D in GK and hence Da /I a has topologically independent a . elements g1 , g2 , since we assumed that Da is not a liftable subgroup GK a a a Therefore, all elements in I are flag and hence I ⊆ Iν , for some ν, and Da ⊆ Dνa . In particular, the initial pair (I, D) was extendable which completes the proof of the proposition. 

5. Galois cohomology of function fields ¯ p , with p 6= `, and X is an algebraic In [9], [10] we proved that if k = F variety over k of dimension ≥ 2, then K = k(X) is encoded, up to purely c , the second lower series quotient of GK . Related inseparable extensions, by GK reconstruction results have been obtained in [24], [22], [25]. The proof of the Bloch–Kato conjecture by Voevodsky, Rost, and Weibel, substantially advanced our understanding of the relations between fields and their Galois groups, in particular, their Galois cohomology. Indeed, consider the diagram GK? ??  ?? π  πc  ??   ??   ?   c / Ga . GK K πa The following theorem relates the Bloch–Kato conjecture to statements in Galois-cohomology, with coefficients in Z/`n (see also [12], [13], [26]). ¯ p , p 6= `, and K = k(X) be Theorem 5.1 ([3], [11, Theorem 11]). Let k = F the function field of an algebraic variety of dimension ≥ 2. The Bloch–Kato conjecture for K is equivalent to: 1. The map a π ∗ : H∗ (GK , Z/`n ) → H∗ (G K , Z/`n )

is surjective, and 2. Ker(πa∗ ) = Ker(π ∗ ). This implies that the Galois cohomology of the pro-`- quotient GK of the absolute Galois group GK encodes important birational information of X.

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c For example, in the case above, GK , and hence K, modulo purely-inseparable extensions, can be recovered from the cup-products

H1 (GK , Z/`n ) ∪ H1 (GK , Z/`n ) → H2 (GK , Z/`n ),

n ∈ N.

From now on, we will frequently omit the coefficient ring Z/`n from notation. The first part of the Bloch-Kato theorem says that every αK ∈ Hi (GK ) is induced from a cohomology class αa ∈ Hi (Ga ) of some finite abelian quotient GK → Ga . An immediate application of this is the following proposition: Proposition 5.2. Let αK ∈ Hi (GK ) be defined on a model X of K and induced from a continuous surjective homomorphism χ : π ˆ1 (X) → G onto a finite be the class representing αK on X. Then there group. Let α = αX ∈ Hiet (X) S exists a finite cover X = j Xj by Zariski open subvarieties such that, for each j, the restriction αj := α|Xj is induced from a continuous surjective homomorphism χj : π ˆ (X ) → Ga onto a finite abelian group and a class Vi 1 1 a j a a i α ∈ Hs (G ) = (H (G )). Proof. We first apply the Bloch-Kato theorem to V ◦ /G and find a Zariski open subset U := Uα◦ of V ◦ /G such that the restriction αU is as claimed, i.e., Vi 1 a induced from a class αa ∈ His (Ga ) = (H (G )), for some homomorphism χa : Gk(V ◦ /G) → Ga to a finite abelian group. Note that this homomorphism is unramified on U . By Lemma 1.1, for every x ∈ X there exists a map f : X → V /G such that f (x) ⊂ U and the restriction of α to f −1 (U ) ⊂ X equals f ∗ (χ∗a (αa )). The claim follows by choosing a finite cover by open subvarieties with these properties.  The second part implies the following: Corollary 5.3. Let αK ∈ Hi (GK ). Assume that we are given finitely many quotients χj : GK → Gaj onto finite abelian groups and classes αja ∈ Hi (Gaj ) with χ∗j (αja ) = αK , for all j. Then there exists a continuous finite quotient GK → Gc onto a finite central extension of an abelian group Ga such that • χj factor through Gc , i.e., there exist surjective homomorphisms ψj : Gc → Gaj , for all j; • there exists a class αc ∈ Hi (Gc ) with αc = ψj∗ (αja ),

for all j.

Lemma 5.4. Let X be a normal variety with function field K. Assume that αK ∈ Hi (GK ) is defined on X and induced from a homomorphism χ : π ˆ1 (X) → G to a finite group G. Consider the sequence χ

χK : GK → π ˆ1 (X) −→ G. Then χK (Iν ) = 0, for every ν such that cX (ν)◦ ⊂ X.

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Proof. An ´etale cover of X induces an ´etale cover of the generic point of cX (ν), thus the cover is unramified in ν, i.e., χK (Iν ) = 0.  Corollary 5.5. Let αK ∈ Hi (GK ). Let X be a normal projective model of K S and j Xj a finite cover by open subvarieties such that αK is defined on Xj , for each j, and is induced from a class αja ∈ Hi (Gaj ), via a homomorphism χj : π ˆ1 (Xj ) → Gaj to some finite abelian group. Then there exist a diagram GK  ???  ?? π πc  ??   ??    / Ga Gc πa where Gc is a finite `-group which is a central extension of Ga , and a class α ∈ Hi (Ga )/I(Gc ) such that α induces αK and for any extendable ∆-pair (I, D) ⊂ Ga α has a representative in Hi (Ga ) which is unramified with respect to (I, D). Proof. Each αja is unramified on all ν such that the generic point cX (ν)◦ ⊂ Xj , by Lemma 5.4. Since αj are induced from a finite number of finite abelian quotients Gaj of GK there exists an abelian quotient Ga of GK with surjections GK → Ga → Gai ; it follows that all classes αj are simultaneously induced from Ga . Note that αj define the same class already on GcK and hence on ˜ c of Gc with a abelian quotient G ˜ a which surjects onto some finite quotient G K a G . For each ν such that the center of ν is in Xj , the image of Iν in Gaj is ˜ a is induced from trivial, and the restriction of αj to the image of Dν in G the image of Dν /Iν . For any extendable ∆-pair (I, D) ⊂ Ga there exists a j and a projection a ˜ → Ga which maps I to a trivial group. Since on the corresponding central G j ˜ c all αj define the same class α, we obtain that the image of αj extension G j ˜a ˜ c ) is induced from D/I, for all extendable ∆-pairs in G ˜a.  in H (G )/I(G

6. Unramified cohomology An important class of birational invariants of algebraic varieties are unramified cohomology groups, with finite constant coefficients (see [4], [15]). These are defined as follows: Let ν be a divisorial valuation of K. We have a natural homomorphism ∂ν : Hi (GK ) → Hi−1 (GK ν ). Classes in ker(∂ν ) are called unramified with respect to ν. The unramified cohomology is \ Hinr (GK ) := Ker(∂ν ) ⊂ Hi (GK ). ν∈DVK

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For i = 2 this is the unramified Brauer group which was used to provide counterexamples to Noether’s problem, i.e., nonrational varieties of type V /G, where V is a faithful representation of a finite group G (see [27], [2]). Generally, for ν ∈ VK and α ∈ Hi (GK ) let αν ∈ Hi (Dν ) be the restriction of α to the decomposition subgroup Dν ⊂ GK of ν. Lemma 6.1. A class α is in Ker(∂ν ) ⊆ Hi (GK ), for ν ∈ DVK , if and only if αν is induced from the quotient GK ν = Dν /Iν . In particular, αν is welldefined as an element in Hi (GK ν ). Proof. Since ν is divisorial, the exact sequence 1 → Iν → Dν → GK ν → 1 where Iν and Dν are quotients of the inertia, respectively decomposition, subgroups, by wild inertia, admits a noncanonical splitting, i.e., Dν is noncanonical direct product of GK ν = Dν /Iν with the corresponding inertia group, which is a torsion-free central procyclic subgroup of Dν . This follows a from Lemma 3.1, using that Iν is abelian and GK is a free abelian pro-` ν group. Thus V∗ 1 H∗ (Dν ) = H∗ (GK ν ) ⊗ H (Iν ). We have H1 (Iν , Z/`n ) = H0 (Iν , Z/`n ) = Z/`n and V∗ 1 (H (Iν , Z/`n )) = H1 (Iν , Z/`n ) ⊕ H0 (Iν , Z/`n ). Thus Hi (Dν ) = Hi−1 (GK ν ) ⊗ H1 (Iν ) ⊕ Hi (GK ν ) and the differential ∂ν coincides with the projection onto the first summand. Hence ∂ν (α) = 0 is equivalent to αν being induced from GK ν = Dν /Iν .  Combining the considerations above we obtain the notion of unramified stable cohomology H∗s,nr (G) of a finite group G: a stable cohomology class α ∈ His (G) is unramified if and only if it is contained in the kernel of the composition ∂

ν His (G) → Hi (GK ) −→ Hi−1 (GK ν ),

for every valuation ν ∈ DVK , where K = k(V /G) for some faithful representation of G. This does not depend on the choice of V , provided ` 6= char(k). These groups are contravariant in G and form a subring H∗s,nr (G) ⊂ H∗s (G). Furthermore: • If V /G is stably rational, then His,nr (G) = 0, for all i ≥ 2.

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F. Bogomolov and Y. Tschinkel • We have His,nr (G) ⊆ His,nr (Syl` (G))NG (Syl` (G)) .

Remark 6.2. In [7], we proved that His,nr (G) = 0, i ≥ 1, for most quasi-simple groups of Lie type. A complete result for quasi-simple groups and i = 2 was obtained in [20]. Note that the `-Sylow-subgroups of finite simple groups often have stably-rational fields of invariants; this provides an alternative approach to our vanishing theorem. Lemma 6.3. Let a γK : GK → Ga a ∗ a be a continuous surjective homomorphism and αK = γK (αa ) ∈ Hi (GK ), for a a a i some α ∈ Hs (G ). If α is unramified with respect to every extendable ∆a a ). ∈ Hinr (GK pair in Ga , then αK

Proof. For ν ∈ DVK , let D := γK (Dνa ) and I := γK (Iνa ). Then either D is cyclic or (D, I) is an extendable ∆-pair in Ga . We have a commutative diagram His (Ga ) ∗ γK

 a ) Hi (GK

φ

/ His (D) o  / Hi (Dνa ) o

ψ

His (D/I)  Hi (Dνa /Iνa ).

a is unramified with respect to ν, by Lemma 1.7. In either case, αK



7. Main theorem ¯ p of tr deg (K) ≥ 2 Theorem 7.1. Let K = k(X) be a function field over k = F k i and αK ∈ Hnr (GK ), with ` 6= p and i > 1. Then there exist a continuous ˜ a onto a finite abelian `-group, fitting into a diagram homomorphism GK → G GK  ˜c / Z˜ /G 1 ˜ a ) such that and a class α ˜ a ∈ Hi (G 1. αK is induced from α ˜a, c ∗ a ∗ ˜ c ). 2. α ˜ := πa (˜ α ) ∈ Hs,nr (G

˜a /G

Conversely, every αK ∈ Hi (GK ) induced from ˜ c as above is in Hinr (GK ). on some G

V∗

/1

˜ a )) and unramified (H1 (G

In this section we begin the proof of Theorem 7.1, reducing it to geometric statements addressed in Sections 8 and 9. Fix an unramified class αK ∈ Hinr (GK ) ⊂ Hi (GK ).

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By Theorem 5.1, we have a surjection a π ∗ : Hi (GK ) → Hi (GK ), a a a let αK ∈ Hi (GK ) be a class such that π ∗ (αK ) = αK . Let a γK : GK → Ga a be a continuous quotient onto a finite abelian `-group such that αK is induced Vi 1 a a i a from a class α ∈ Hs (G ) = (H (G )). We have a diagram of central extensions: / Ga /1 / Gc / ZK 1 K K

 /Z

1

 / Gc

γK

πa

 / Ga

/1

where the lower row is uniquely defined, up to isoclinism, as in Lemma 4.2. The group Ga might be too small, i.e., it may happen that πa∗ (αa ) ∈ / His,nr (Gc ). a ˜ a fitting into a →G Our goal is to pass to an intermediate finite quotient GK commutative diagram below, / ZK / Gc / Ga /1 1 K K γ ˜K

1

 / Z˜

 ˜c /G

 ˜a /G

1

 /Z

 / Gc

 / Ga

/1

γ

/1

where the vertical arrows are surjections onto finite `-groups, and such that a ˜ a ) with αK is induced from a class α ˜ a ∈ H∗ (G ˜ c ). π ˜a∗ (˜ αa ) ∈ His,nr (G There are two possibilities: 1. There exists a finite quotient GK → Ga such that αK is induced from αa ∈ His (Ga ) which is unramified with respect to every extendable ∆pair (I, D) in Ga . This case is treated in Lemma 7.2. 2. On every finite quotient, the class αa inducing αK is ramified on some extendable ∆-pair (I, D). This possibility is eliminated by Lemma 7.3. Lemma 7.2. Assume that αa ∈ His (Ga ) is unramified with respect to every extendable ∆-pair (I, D) in Ga . Then there is a factorization ˜K γ a γ ˜ a −→ GK −→ G Ga ,

γK = γ ◦ γ˜K ,

˜ a , such that with finite G ˜ c ), πa∗ (˜ αa ) ∈ His,nr (G

α ˜ a := γ ∗ (αa ).

(7.1)

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˜ a be the quotient constructed in Proposition 4.6, i.e., if (I, D) is Proof. Let G ˜ D) ˜ surjects onto (I, D). not an extendable ∆-pair in Ga , then no ∆-pair (I, a ˜ D) ˜ in G ˜ one of the following holds: Thus, for each ∆-pair (I, ˜ ˜ • either γ(I) = 0 or γ(D) cyclic, ˜ γ(D)) ˜ is an extendable ∆-pair in Ga . • or (γ(I), ˜ c , we obtain that α Applying Lemma 1.7 and Lemma 6.3 to V /G ˜ a := γ ∗ (αa ) ˜ ˜ is unramified with respect to (I, D).  Lemma 7.3. There exists a finite quotient GK → Ga such that αa ∈ Hi (Ga ) induces αK and is unramified on every extendable ∆-pair (I, D) in Ga . The proof of this lemma is presented in Section 8, in the case when K admits a smooth projective model; a reduction to the smooth case is postponed until Section 9.

8. The smooth case Let X be a smooth projective irreducible variety over an algebraically closed field k with function field K = k(X). By the Bloch-Ogus theorem, there is an isomorphism i Hinr (GK ) = H0Zar (X, Het (X)), i where Het is an ´etale cohomology sheaf (see also Theorem 4.1.1 in [14]). In particular, a class αK ∈ Hinr (GK ) can be represented by a finite collection of classesS{αn }n∈N , with αn defined on some Zariski open affine Xn ⊂ X, with X = n Xn , such T that the restrictions of αn to some common open affine subvariety X ◦ ⊂ n Xn coincide. We will need the following strengthening: Lemma 8.1. Let X be a smooth variety with function field K and S = {x1 , . . . , xr } ⊂ X a finite set of points. Given a class αK ∈ Hinr (GK ) there exist • a Zariski open subset US ⊂ X, containing S, and • a class αS ∈ Hiet (US ) such that αK and αS coincide on some dense Zariski open subset US◦ ⊂ US , i.e., for every representation of αK by {αn }n∈N as above there exists some dense Zariski open US◦ ⊂ US , containing S, such that the restrictions of αS and all αn to US◦ coincide. Proof. By the argument in [14, Theorem 4.1.1], αK has a representative α = αX ∈ H0Zar (X, Hi (Z/`n )). Hence, there is a covering of X by Zariski open subsets Xn with αn representing α in Xn . Moreover, using the refinement of the Bloch-Ogus exact sequence for semi-local rings as in [23, Theorem 1.1], we can assume that one of the subsets contains the finite set S.  Fix a representation of αK ∈ Hinr (GK ) by {αn }n∈N as above. Each class αn ∈ Hiet (XS n ) is represented by a finite collection {Xnm } of affine charts Xnm , with Xnm = Xn and finite ´etale covers ˜ nm → Xnm , ψnm : X

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such that the restrictions αnm := αn |Xnm are induced from homomorphisms χnm : π ˆ1 (Xnm ) → Gnm onto finite groups. Proposition 5.2 implies that there is further refinement of the cover by affine subcovers [ X= Xj , j

such that for each j there exist • a finite abelian `-group Gaj • a surjection χK,j : GK → Gaj , unramified over Xj , and • a class αj ∈ Hi (Gaj ) inducing α via χK,j . Corollary 5.3 implies that there exists a finite quotient πc : GK → Gc onto a central extension of an abelian group Ga such that the projections χK,j factor through Gc and the images of αj in Hi (Gc ) coincide. In particular, αK is induced from αc ∈ Hi (Gc ). We claim that αc is unramified on every pair (πc (Iν ), πc (Dν )), for ν ∈ VK . Indeed, for each ν, χK,j (Iν ) = 0 on at least one of the charts Xj , thus the restriction of αj to χK,j (Dν ) is induced from χK,j (Dν )/χK,j (Iν ); since αc ∈ Hi (Gc ) is induced from αj , we have the same property for αc , with respect to the pair (πc (Iν ), πc (Dν )). The description of the action in Lemma 2.3 identifies subgroups of Ga acting on products of projective spaces with fixed points with images of inertia subgroups of some valuations, and images of their decomposition subgroups with subgroups preserving the corresponding components, see Lemmas 2.2 and 2.3. The rest of the argument is similar to the proof of Lemma 7.2. Let GK → Ga be an intermediate quotient surjecting onto each Gaj and ˜ a → Ga GK → G the intermediate finite quotient constructed in Proposition 4.6. In particular, ˜ a to Ga is either of the form the projection of every ∆-pair in G j (πa,j (Iν ), πa,j (Dν )), ˜ c be a central extension as in Corollary 4.3, for some ν ∈ VK , or cyclic. Let G ˜ a ), induced by αa constructed surjecting onto Gc . We have classes α ˜ ja ∈ Hi (G j c ˜ c ). above, and mapping to the same class α ˜ ∈ Hi (G ˜ c ). Indeed, for every ∆-pair (I, ˜ D) ˜ in G ˜a We claim that α ˜ c ∈ His,nr (G ˜ projects to a cyclic group in Ga or it is extendable, i.e., image of either D ˜ D). ˜ In some (Iν , Dν ). In the first case, all elements α ˜ j are unramified on (I, the second case, at least one of the α˜j is unramified on it.

9. Reduction to the smooth case In absense of resolution of singularities in positive characteristic, we reduce to the smooth case via the de Jong-Gabber alterations theorem (see [18]): The Galois group GK contains a closed subgroup GK˜ of finite index, coprime to `, such that:

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F. Bogomolov and Y. Tschinkel ˜ corresponding to G ˜ admits a smooth proper • The function field K K model, i.e., there exists a generically finite morphism of proper varieties ˜ → X, ρ:X

˜ smooth and K ˜ = k(X). ˜ of degree |GK /GK˜ | with X n Let αK ∈ Hnr (GK ) be an unramified class. Its restriction αK˜ to a class in Hn (GK˜ ) is also unramified. By results in Section 8, there exists a surjection ˜c GK˜ → G

(9.1)

onto a finite abelian `-group such that αK˜ is unduced from a class in α ˜c ∈ c n ˜ Hs,nr (G ). Lemma 9.1. There exists a diagram GK˜

˜ /G

 GK

 /G

π ˜c

˜c //G

˜ where the vertical arrows are injections, with image of index coprime to `, G n and G are finite groups, and αK is induced from an element αG ∈ Hnr (G). ˜ In particular, Syl` (G) ' Syl` (G). Proof. Fix a finite continuous quotient GK → G0 such that αK is induced from some αG0 ∈ Hi (G0 ). Note that for every intermediate quotient GK → G → G0 there exists an αG ∈ Hi (G) inducing αK . It suffices to find a sufficiently large G such that the sujection (9.1) factors through a subgroup of G. This is a standard fact in Galois theory. ˜ is also unramified. Since the ˜ ∈ Hi (G) Since α ˜ c ∈ His,nr (Gc ) its image α ˜ index (G : G) is coprime to `, and since the unramified α ˜ is induced from an element αG ∈ Hi (G), αG is also unramified, as claimed.  At this stage, we cannot yet guarantee that G is a central extension of an abelian group, nor that it is an `-group. However, we know that tr(resGK /GK˜ (αK )) = αK ∈ Hinr (GK ), modulo multiplication by an element in (Z/`n )× . We need the following version of resolution of singularities in positive characteristic: Theorem 9.2. [18] Let G be a finite `-group and Y a smooth variety over a perfect field with a generically free action of G. Then there exists a G-variety Y˜ with a proper G-equivariant birational map Y˜ → Y such that Y˜ /G is smooth. We are very grateful to D. Abramovich for providing the reference and indicating the main steps of the proof in [18]:

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• Theorem VIII.1.1 gives an equivariant modification Y 0 of Y with a regular log structure on Y 0 such that the action is very tame, i.e., the stabilizers in G of points in Y 0 are abelian and act as subgroups of tori in toroidal charts of the log structure Y 0 . • By Theorem VI.3.2, the quotient Y 0 /G of a log regular variety by a very tame action is log regular. • By Theorem VIII.3.4.9, which is a step in Theorem VIII.1.1, a log regular variety has a resolution of singularities. We return to the proof of Theorem 7.1. Start with a suitable faith˜ and construct a ful representation V of G, and thus of Syl` (G) = Syl` (G), diagram πY / Y = Y˜ /Syl` (G) V¯ πG

 V¯ /G

where Y = Y˜ /Syl` (G) is the smooth projective variety from Theorem 9.2, and πY is a Syl` (G)-equivariant map from a G-equivariant projective closure of V . Let L = k(Y ) be the S function field of Y . Given a class αL ∈ Hinr (GL ) we have a covering of Y = n Yn by affine Zariski open subsets and a finite set of classes {αn } representing αL , as considered in Section 8. Pick a point v ∈ V¯ . The image S := πY (G · b) of its G-orbit is a finite set of points. By Lemma 8.1, there exist a dense Zariski open subset US and a class αS ∈ Hiet (US ) coinciding with αL on some dense Zariski open subset US◦ ⊂ US . Its preimage ¯ ◦ := π −1 (U ◦ ) ⊂ V¯ U S

Y

S

is a dense Zariski open subset containing G · v. Put \ ¯v := ¯S◦ ) ⊂ V¯ , g(U U g∈G

¯v ) ⊂ it is a G-stable dense Zariski open subvariety containing v. Its image πG (U ¯ ¯ V /G is a Zariski open subset containing πG (G · v). Note that πv : Uv /Syl` (G) → US is a birational morphism to an open subset and πv∗ (αS ) is well-defined ¯v /Syl` (G). It follows that the trace in ´etale cohomology of U ¯v /G) trπG (πv∗ (αS )) ∈ Hiet (U is well-defined and coincides with (G : Syl` (G)) · αS at the generic point of V¯ /G. Thus we have a covering of V¯ /G by Zariski open subsets of the form ¯v /G, v ∈ V¯ , with cohomology classes representing αK on each chart. There U ¯v /G with extensions of αS to each U ¯v /G. We exists a finite subcovering by U can now apply Proposition 5.2 to produce a finite subcover such that on each chart, the class is induced from homomorphisms onto finite abelian groups, and proceed as in Section 8. 

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References [1] A. Adem and R. J. Milgram. Cohomology of finite groups, volume 309 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 2004. [2] F. Bogomolov. The Brauer group of quotient spaces of linear representations. Izv. Akad. Nauk SSSR Ser. Mat., 51(3):485–516, 688, 1987. [3] F. Bogomolov. On two conjectures in birational algebraic geometry. In Algebraic geometry and analytic geometry (Tokyo, 1990), ICM-90 Satell. Conf. Proc., pages 26–52. Springer, Tokyo, 1991. [4] F. Bogomolov. Stable cohomology of groups and algebraic varieties. Mat. Sb., 183(5):3–28, 1992. [5] F. Bogomolov. Stable cohomology of finite and profinite groups. In Algebraic groups, pages 19–49. Universit¨ atsverlag G¨ ottingen, G¨ ottingen, 2007. [6] F. Bogomolov and Chr. B¨ ohning. Stable cohomology of alternating groups. Cent. Eur. J. Math., 12(2):212–228, 2014. [7] F. Bogomolov, T. Petrov, and Y. Tschinkel. Unramified cohomology of finite groups of Lie type. In Cohomological and geometric approaches to rationality problems, volume 282 of Progr. Math., pages 55–73. Birkh¨ auser Boston Inc., Boston, MA, 2010. [8] F. Bogomolov and Y. Tschinkel. Commuting elements of Galois groups of function fields. In Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), volume 3 of Int. Press Lect. Ser., pages 75–120. Int. Press, Somerville, MA, 2002. [9] F. Bogomolov and Y. Tschinkel. Reconstruction of function fields. Geom. Funct. Anal., 18(2):400–462, 2008. [10] F. Bogomolov and Y. Tschinkel. Reconstruction of higher-dimensional function fields. Mosc. Math. J., 11(2):185–204, 406, 2011. [11] F. Bogomolov and Y. Tschinkel. Introduction to birational anabelian geometry. In Current developments in algebraic geometry, volume 59 of Math. Sci. Res. Inst. Publ., pages 17–63. Cambridge Univ. Press, Cambridge, 2012. [12] S. K. Chebolu, I. Efrat, and J. Min´ aˇc. Quotients of absolute Galois groups which determine the entire Galois cohomology. Math. Ann., 352(1):205–221, 2012. [13] S. K. Chebolu and J. Min´ aˇc. Absolute Galois groups viewed from small quotients and the Bloch-Kato conjecture. In New topological contexts for Galois theory and algebraic geometry (BIRS 2008), volume 16 of Geom. Topol. Monogr., pages 31–47. Geom. Topol. Publ., Coventry, 2009. [14] J.-L. Colliot-Th´el`ene. Birational invariants, purity and the Gersten conjecture. In K-Theory and Algebraic Geometry : Connections with Quadratic Forms and Division Algebras, AMS Summer Research Institute, Santa Barbara 1992, volume 58, Part I of Proceedings of Symposia in Pure Mathematics, pages 1–64. 1995. [15] 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(1):141– 158, 1989.

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[16] Olivier Debarre. Vari´et´es rationnellement connexes (d’apr`es T. Graber, J. Harris, J. Starr et A. J. de Jong). Ast´erisque, (290):Exp. No. 905, ix, 243–266, 2003. S´eminaire Bourbaki. Vol. 2001/2002. [17] P. Hall. The classification of prime-power groups. J. Reine Angew. Math., 182:130–141, 1940. [18] L. Illusie, Y. Laszlo, and F. Orgogozo. Travaux de Gabber sur l’uniformisation locale et la cohomologie etale des schemas quasi-excellents. Seminaire a l’Ecole polytechnique 2006–2008, 2012. arXiv:1207.3648. [19] Franz-Viktor Kuhlmann. Valuation theoretic and model theoretic aspects of local uniformization. In Resolution of singularities (Obergurgl, 1997), volume 181 of Progr. Math., pages 381–456. Birkh¨ auser, Basel, 2000. [20] B. Kunyavski˘ı. The Bogomolov multiplier of finite simple groups. In Cohomological and geometric approaches to rationality problems, volume 282 of Progr. Math., pages 209–217. Birkh¨ auser Boston Inc., Boston, MA, 2010. ´ [21] J. Milne. Etale cohomology. Princeton Mathematical Series 33. Princeton University Press, 1980. [22] Sh. Mochizuki. The local pro-p anabelian geometry of curves. Invent. Math., 138(2):319–423, 1999. [23] Ivan Panin and Kirill Zainoulline. Variations on the Bloch-Ogus theorem. Doc. Math., 8:51–67 (electronic), 2003. [24] F. Pop. On Grothendieck’s conjecture of birational anabelian geometry. Ann. of Math. (2), 139(1):145–182, 1994. [25] F. Pop. On the birational anabelian program initiated by Bogomolov I. Invent. Math., 187(3):511–533, 2012. [26] L. Positselski. Koszul property and Bogomolov’s conjecture. Int. Math. Res. Not., (31):1901–1936, 2005. [27] D. J. Saltman. Noether’s problem over an algebraically closed field. Invent. Math., 77(1):71–84, 1984. [28] J.-P. Serre. Galois cohomology. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2002. [29] M. Vaqui´e. Valuations and local uniformization. In Singularity Theory and its Applications, pages 477–527. Math. Soc. Japan, 2006. [30] Vladimir Voevodsky. On motivic cohomology with Z/l-coefficients. Ann. of Math. (2), 174(1):401–438, 2011. Fedor Bogomolov Courant Institute of Mathematical Sciences New York University 251 Mercer Str., New York, NY 10012 USA and Laboratory of Algebraic Geometry National Research University Higher School of Economics 7 Vavilova Str., Moscow, 117312 Russia e-mail: [email protected]

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Yuri Tschinkel Courant Institute of Mathematical Sciences New York University 251 Mercer Str., New York, NY 10012 USA and Simons Foundation 160 Fifth Avenue, New York, NY 10010 USA e-mail: [email protected]

Rational Points on K3 Surfaces and Derived Equivalence Brendan Hassett and Yuri Tschinkel Abstract. We study K3 surfaces over non-closed fields and relate the notion of derived equivalence to arithmetic problems. Mathematics Subject Classification (2010). 14J28, 14G20, 14F05, 11G25. Keywords. K3 surface, derived category of coherent sheaves, rational point.

The geometry of vector bundles and derived categories on complex K3 surfaces has developed rapidly since Mukai’s seminal work [45]. Many foundational questions have been answered: • the existence of vector bundles and twisted sheaves with prescribed invariants; • geometric interpretations of isogenies between K3 surfaces [50, 15]; • the global Torelli theorem for holomorphic symplectic manifolds [58, 28]; • the analysis of stability conditions and its implications for birational geometry of moduli spaces of vector bundles and more general objects in the derived category [9, 8, 12]. Given the precision and power of these results, it is natural to seek arithmetic applications of this circle of ideas. Questions about zero cycles on K3 surfaces have attracted the attention of Beauville-Voisin [10], Huybrechts [27], and other authors. Our focus in this note is on rational points over non-closed fields of arithmetic interest. We seek to relate the notion of derived equivalence to arithmetic problems over various fields. Our guiding questions are: Question 1. Let X and Y be K3 surfaces, derived equivalent over a field F . Does the existence/density of rational points of X imply the same for Y ? Given α ∈ Br(X), let (X, α) denote the twisted K3 surface associated with α, i.e., if P → X is an étale projective bundle representing α, of relative dimension r − 1, then (X, α) = [P/ SLr ]. © Springer International Publishing AG 2017 A. Auel (eds.) et al., Brauer Groups and Obstruction Problems, Progress in Mathematics 320, DOI 10.1007/978-3-319-46852-5_6

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Question 2. Suppose that (X, α) and (Y, β) are derived equivalent over F . Does the existence of a rational point on the former imply the same for the latter? Note that an F -rational point of (X, α) corresponds to an x ∈ X(F ) such that α|x = 0 ∈ Br(F ). After this paper was released, Ascher, Dasaratha, Perry, and Zhou [6] found that Question 2 has a negative answer, even over local fields. We shall consider these questions for F finite, p-adic, real, and local with algebraically closed residue field. These will serve as a foundation for studying how the geometry of K3 surfaces interacts with Diophantine questions over local and global fields. For instance, is the Hasse/Brauer-Manin formalism over global fields compatible with (twisted) derived equivalence? See [21, 20, 42] for concrete applications to rational points problems. In this paper, we first review general properties of derived equivalence over arbitrary base fields. We then offer examples which illuminate some of the challenges in applying derived category techniques. The case of finite and real fields is presented first – here the picture is well developed. Local fields of equicharacteristic zero are also fairly well understood, at least for K3 surfaces with semistable or other mild reduction. The analogous questions in mixed characteristic remain largely open, but comparison with the geometric case suggests a number of avenues for future investigation. Acknowledgments. We are grateful to Jean-Louis Colliot-Thélène, Daniel Huybrechts, Christian Liedtke, Johannes Nicaise, Sho Tanimoto, Anthony Várilly-Alvarado, Olivier Wittenberg, and Letao Zhang for helpful conversations. We thank Vivek Shende for pointing out the application of A’Campo’s Theorem. We have benefited enormously from the thoughtful feedback of the referees. The first author was supported by NSF grants 0901645, 0968349, and 1148609; the second author was supported by NSF grants 0968318 and 1160859. We are grateful to the American Institute of Mathematics for sponsoring workshops where these ideas were explored.

1. Generalities on derived equivalence for K3 surfaces 1.1. Definitions Let X and Y denote K3 surfaces over a field F . Let p and q be the projections from X × Y to X and Y respectively. Let E ∈ Db (X × Y ) be an object of the bounded derived category of coherent sheaves, which may be represented by a perfect complex of locally free sheaves. The Fourier-Mukai transform is defined ΦE : Db (X) → F 7→

Db (Y ) q∗ (E ⊗ p∗ F),

where push forward and tensor product are the derived operations.

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We say X and Y are derived equivalent if there exists an equivalence of triangulated categories over F ∼

Φ : Db (X) → Db (Y ). A fundamental theorem of Orlov [50] implies that Φ arises as the FourierMukai transform ΦE for some perfect complex E ∈ Db (X ×Y ). When we refer to Fourier-Mukai transforms below they will induce derived equivalences. 1.2. Mukai lattices Suppose X is a K3 surface defined over C and consider its Mukai lattice ˜ ˜ H(X, Z) = H(X, Z) := H 0 (X, Z)(−1) ⊕ H 2 (X, Z) ⊕ H 4 (X, Z)(1), where we apply Tate twists to get a Hodge structure of weight two. Mukai ˜ vectors refer to type (1, 1) vectors in H(X, Z). Let (, ) denote the natural ˜ nondegenerate pairing on H(X, Z); this coincides with the intersection pairing on H 2 (X, Z) and the negative of the intersection pairing on the other summands. There is an induced homomorphism on the level of cohomology: ˜ ˜ φE : H(X, Z) → H(Y, p Z) η 7→ q∗ ( TdX×Y · ch(E) ∪ p∗ η). Note that φE ch(F) = ch(ΦE (F)). For X defined over a non-closed field F , there are analogous constructions in `-adic and other flavors of cohomology [38, § 2]. When working ˜ `-adically, we interpret H(X, Z` ) as a Galois representation rather than a Hodge structure. Observe that φE is defined on Hodge structures, de Rham cohomologies, and `-adic cohomologies – and these are all compatible. 1.3. Characterizations over the complex numbers Theorem 3. [50, §3] Let X and Y be K3 surfaces over C, with transcendental cohomology groups T (X) := Pic(X)⊥ ⊂ H 2 (X, Z),

T (Y ) := Pic(Y )⊥ ⊂ H 2 (Y, Z).

The following are equivalent: ˜ ˜ • there exists an isometry of Hodge structures H(X, Z) ' H(Y, Z); • there exists an isometry of Hodge structures T (X) ' T (Y ); • X and Y are derived equivalent; • Y is isomorphic to a moduli space of stable vector bundles over X, admitting a universal family E → X × Y , i.e., Y = Mv (X) for a Mukai vector v such that there exists a Mukai vector w with (v, w) = 1. See [50, §3.8] and [25, §3] for discussion of the fourth condition; for our purposes, we need not distinguish among various notions of stability. This has been extended to arbitrary fields as follows: Theorem 4. [38, Th. 1.1] Let X and Y be K3 surfaces over an algebraically closed field F of characteristic 6= 2. Then the third and the fourth statements are equivalent.

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Morover, a derived equivalence between X and Y induces Galois-compatible isomorphisms between their `-adic Mukai lattices. See [29, §16.4] for more discussion. 1.4. Descending derived equivalence Let F be a field of characteristic zero with algebraic closure F¯ . Given Y smooth and projective over F , let Y¯ denote the corresponding variety over F¯ . We say Y is of K3 type if Y¯ is deformation equivalent to the Hilbert scheme of a K3 surface. Lemma 5. Suppose Y1 and Y2 are of K3 type over F and there exists an ∼ isomorphism ι : Y¯1 → Y¯2 defined over F¯ inducing ∼

ι∗ : H 2 (Y¯2 , Z` ) → H 2 (Y¯1 , Z` ) compatible with the action of Gal(F¯ /F ). Then Y1 ' Y2 over F . Proof. In characteristic zero, Galois fixed points of Isom(Y¯1 , Y¯2 ) correspond to isomorphisms between the varieties defined over F . Automorphisms of Y¯1 (resp. Y¯2 ) act transitively and faithfully on this set by pre-composition (resp. post-composition). The Torelli theorem implies that the automorphism group of a K3 surface has a faithful representation in its second cohomology. This holds true for manifolds of K3 type as well – see [39, Prop. 1.9] as well as previous work of Beauville and Kaledin-Verbitsky. The Galois invariance of ι∗ implies that ι is Galois-fixed, hence defined over F .  Suppose that X and Y are K3 surfaces over F . Given a derived equivalence ∼ Φ : Db (X) → Db (Y ) over F the induced ∼ ˜ ¯ ˜ X, ¯ Z` ) → φ : H( H(Y , Z` )

is compatible Gal(F¯ /F ) actions. We consider whether the converse holds. We use the notation Mv (X) for the moduli stack of vector bundles/ complexes over X; this is typically a Gm -gerbe over Mv (X) due to homotheties. Proposition 6. Suppose that X and Y are K3 surfaces over F . Let Φ : ∼ ¯ → D(X) D(Y¯ ) be a derived equivalence over F¯ such that the induced ∼

˜ X, ¯ Z` ) → H( ˜ Y¯ , Z` ) φ : H( is compatible with Galois actions. Write v = φ−1 (1, 0, 0) and w = φ(1, 0, 0). Then we have isomorphisms Y ' Mv (X) and X ' Mw (Y ) inducing the derived equivalence over F¯ . Moreover, there exist Gm -gerbes X → X and Y → Y and isomorphisms Y ' Mv (X) and X ' Mw (Y ) such that X → X and Y → Y admit trivializations over F¯ .

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¯ Proof. Choosing an appropriate polarization on X, the moduli space Mv (X) is a K3 surface. This follows from [45, Prop. 4.2]; Mukai’s argument shows that the requisite polarization can be found over a non-closed field, as we just have to avoid walls orthogonal to certain Mukai vectors. ˜ X, ¯ Z` ) that is Galois invariant and There exists a Mukai vector v 0 ∈ H( 0 0 −1 satisfies (v, v ) = 1, e.g., v = φ (0, 0, −1). Thus the universal sheaf E → X × Mv (X) has the following property: on basechange to F¯ it may be obtained as the pull-back of a universal sheaf on the coarse space ¯ × Mv (X). ¯ E¯ → X Indeed, the universal sheaf exists wherever Mv0 (X) admits a rational point (see [38, Th. 3.16]). We apply Lemma 5 to Y and Mv (X). The Torelli Theorem implies Y¯ ' ¯ with the induced isomorphism compatible with the Galois actions on Mv (X) `-adic cohomology. Thus it descends to give Y ' Mv (X) defined over F .  Remark 7. Can X and Y fail to be derived equivalent over F ? How do the pull-back homomorphisms Br(F ) → Br(X),

Br(F ) → Br(Y )

compare? They must have the same kernel if X and Y are derived equivalent over F . Remark 8. Sosna [56] has given examples of complex K3 surfaces that are derived equivalent but not isomorphic to their complex conjugates. 1.5. Cycle-theoretic invariants of derived equivalence Proposition 9. Let X and Y be derived equivalent K3 surfaces over a field F of characteristic 6= 2. Then Pic(X) and Pic(Y ) are stably isomorphic as Gal(F¯ /F )-modules, and Br(X)[n] ' Br(Y )[n] provided n is not divisible by the characteristic. Even over C, this result does not extend to higher dimensional varieties [2]. Proof. The statement on the Picard groups follows from the Chow realization of the Fourier-Mukai transform – see [38, §2.7] for discussion. We write ˜ ét (X, µn ) = H 0 (X, Z/nZ) ⊕ H 2 (X, µn ) ⊕ H 4 (X, µ⊗2 ). H ét ét ét n The Chow-theoretic interpretation of the Fourier-Mukai kernel gives the realization ˜ ét (X, µn ) → H ˜ ét (Y, µn ), φ:H compatible with cycle class maps [38, Prop. 2.10]. Modding out by the images of the cycle class maps we get the desired equality of Brauer groups. 

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Recall that the index ind(X) of a smooth projective variety X over a field F is the greatest common divisor of the degrees of field extensions F 0 /F over which X(F 0 ) 6= ∅, or equivalently, the lengths of zero-dimensional subschemes Z ⊂ X. Given a bounded complex of locally free sheaves on X E = {E−M → E−M +1 → · · · → EN } we may define the Chern character X ch(E) = (−1)j ch(Ej ) j

in Chow groups with Q coefficients. The degree zero and one pieces yield the rank and first Chern class of E, expressed as alternating sums of the ranks and determinants of the terms, respectively. Similarly, we may define c2 (E) = − ch2 (E) + ch1 (E)2 /2, a quadratic expression in the Chern classes of the Ej with integer coefficients. Modulo the Z algebra generated by the first Chern classes of the Ej , we may write X c2 (E) ≡ (−1)j c2 (Ej ). j

Lemma 10. If (S, h) is a smooth projective surface over F , then ind(S) = gcd{c2 (E) : E vector bundle on S} = gcd{c2 (E) : E ∈ Db (S)}. Proof. Consider the ‘decomposible index’ inddec(S) := gcd{D1 · D2 : D1 , D2 very ample divisors on S} which is equal to gcd{D1 · D2 : D1 , D2 divisors on S}, because for any divisor D the divisor D + N h is very ample for N  0. All three quantities in the assertion divide inddec(S), so we work modulo this quantity. By the analysis of Chern classes above, the second and third quantities agree. Given a reduced zero-dimensional subscheme Z ⊂ S we have a resolution 0 → E−2 → E−1 → OS → OZ → 0 with E−2 and E−1 vector bundles. This implies that gcd{c2 (E) : E vector bundle on S}| ind(S). Conversely, given a vector bundle E there exists a twist E ⊗ OS (N h) that is globally generated and c2 (E ⊗ OS (N h)) ≡ c2 (E)

(mod inddec(S)).

Thus there exists a zero-cycle Z with degree c2 (E ⊗ OS (N h)) and ind(S)| gcd{c2 (E) : E vector bundle on S}.



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Proposition 11. If X is a K3 surface over a field F , then ind(X)| gcd{24, D1 · D2 where D1 , D2 are divisors on X}. This follows from Lemma 10 and the fact that c2 (TX ) = 24. BeauvilleVoisin [10] and Huybrechts [26] have studied the corresponding subgroup of CH0 (XF¯ ). Proposition 12. Let X and Y be derived equivalent K3 surfaces over a field F . Then ind(X) = ind(Y ). The proof will require the notion of a spherical object on a K3 surface. This is an object S ∈ Db (X) with Ext0 (S, S) = Ext2 (S, S) = F,

Exti (S, S) = 0,

i 6= 0, 2.

These satisfy the following • (v(S), v(S)) = −2; • rigid simple vector bundles are spherical; • each spherical object S has the associated spherical twist [29, 16.2.4]: TS : Db (X) → Db (X), an autoequivalence such that the induced homomorphism on the Mukai lattice is the reflection associated with v(S); • each spherical object S¯ on XF¯ is defined over a finite extension F 0 /F [26, 5.4]; ˜ • over C, each v = (r, D, s) ∈ H(X, Z) ∩ H 1,1 with (v, v) = −2 arises from a spherical object, which may be taken to be a rigid vector bundle E if r > 0 [34]; • under the same assumptions, for each polarization h on X there is a unique h-slope stable vector bundle E with v(E) = v [27, 5.1.iii]. Proof. Proposition 6 allows us to express Y = Mv (X) for v = (r, ah, s) where h is a polarization on X and a2 h2 = 2rs. It also yields a Mukai vector ˜ w = (r0 , bg, s0 ) ∈ H(X, Z` ) with (v, w) = abg · h − rs0 − sr0 = 1. Thus we have hr, si = h1i

(mod g · h).

(1.1)

Consider a Fourier-Mukai transform realizing the equivalence Φ : Db (X) → Db (Y ) and the induced homomorphism φ on the Mukai lattice. Note that φ(v) = (0, 0, 1) reflecting the fact that a point on Y corresponds to a sheaf on X with Mukai vector v. Suppose that Y has a rational point over a field of degree n over F ; let Z ⊂ Y denote the corresponding subscheme of length n. Applying Φ−1 to

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OZ gives an element of the derived category with Mukai vector (nr, nah, ns) and c2 (Φ−1 (OZ )) =

c1 (Φ−1 (OZ ))2 − χ(Φ−1 (OZ )) + 2 rank(Φ−1 (OZ )) 2

which equals n(nrs + r − s). Following the proof of Lemma 10, we compute c2 (Φ−1 (OZ ))

(mod inddec(X)).

First suppose that r and s have different parity, so that gcd(nrs + r − s, 2rs) = gcd(nrs + r − s, rs). Then using (1.1) we obtain hnrs + r − s, rsi = hr − s, rsi = hr − s, ri hr − s, si 2

= h−s, ri hr, si = hr, si = h1i

(mod g · h).

If r and s are both even, then g · h must be odd and hnrs + r − s, 2rsi = hnrs + r − s, rsi

(mod g · h)

and repeating the argument above gives the desired conclusion. Now suppose that r and s are both odd. It follows that h2 ≡ 2 (mod 4) and we write h2 = 2γ − 2 for some even integer γ. Let S denote the spherical object associated with h so that v(S) = (1, h, γ). Applying TS to the Mukai vector (r, ah, s) 7→ (r, ah, s) + ((r, ah, s), (1, h, γ)) (1, h, γ), we obtain a new vector with rank r + (ah2 − s − rγ), which is even. This reduces us to the previous situation. In each case, we find c2 (Φ−1 (OZ )) ≡ n

(mod inddec(X)),

whence ind(X)|n. Varying over all degrees n, we find ind(X)| ind(Y ) and the proposition follows.



The last result raises the question of whether spherical objects are defined over the ground field: Question 13. Let X be a K3 surface over a field F . Suppose that S¯ is a ¯ ∈ Pic(XF¯ ) is a divisor defined over spherical object on XF¯ such that c1 (S) X. When does S¯ come from an object S on X? Kuleshov [34, 35] gives a partial description of how to generate all exceptional bundles on K3 surfaces of Picard rank one through ‘restructuring’ operations and ‘dragons’. It would be worthwhile to analyze which of these operations could be defined over the ground field.

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Example 14. We give an example of a K3 surface X over a field F with Pic(X) = Pic(XF¯ ) = Zh and a rigid sheaf E over XF¯ that fails to descend to F . Choose (X, h) to be a degree fourteen K3 surface defined over R with X(R) = ∅. This may be constructed as follows: Fix a smooth conic C and quadric threefold Q with C ⊂ Q ⊂ P4 ,

Q(R) = ∅.

0

Let X denote a complete intersection of Q with a cubic containing C; we have X 0 (R) = ∅ and X 0 admits a lattice polarization g C

g 6 2

C 2 . −2

Write h = 2g − C so that (X 0 , h) is a degree 14 K3 surface containing a conic. Let X be a small deformation of X 0 with Pic(XC ) = Zh. The K3 surface X is Pfaffian if and only if it admits a vector bundle E with v(E) = (2, h, 4) corresponding to the classifying morphism X → Gr(2, 6). However, note that c2 (E) = 5 which would mean that ind(X) = 1. On the other hand, if X(R) = ∅, then ind(X) = 2.

2. Examples of derived equivalence 2.1. Elliptic fibrations The paper [3] has a detailed discussion of derived equivalences among genus one curves over function fields. In this section we work over a field F of characteristic zero. A K3 surface X is elliptic if it admits a morphism X → C to a curve of genus zero with fibers of genus one. We allow C = P1 or a non-split conic over F . Lemma 15. A K3 surface X is elliptic if and only if it admits a non-trivial divisor D with D2 = 0. Proof. If X is elliptic, then the pull back of a non-trivial divisor from C gives a square-zero class; we focus on the converse. This is well-known if F is algebraically closed [52, §6, Th. 1]. Indeed, let C+ ⊂ H 2 (X, R) denote the component of the positive cone {η : (η, η) > 0} containing an ample divisor and Γ ⊂ Aut(H 2 (X, Z)) the group generated by Picard-Lefschetz reflections ρR associated with (−2)-curves R ∈ Pic(X). Then the Kähler cone of X is a fundamental domain for the action of Γ on C+ , in the sense that no two elements of the cone are in the same orbit and each orbit in C+ meets the closure of the Kähler cone. Cohomology classes in the

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interior of the cone have trivial Γ-stabilizer; classes on walls of the boundary associated with (−2)-curves are stabilized by the associated reflections. Thus if Pic(X) represents zero, then there exists a non-zero divisor D in the closure of the Kähler cone with D2 = 0, which induces an elliptic fibration X → P1 . Given a divisor D ∈ C+ , we can be a bit more precise about the γ ∈ Γ required to take D to a nef divisor. We can write γ = ρR1 · · · ρRm where each Rj is the class of an irreducible rational curve contained in the fixed part of the linear series |D| (cf. proof in [52, §6], [54, §2]). Now suppose F is not algebraically closed and D is defined over F . ¯ and thus on Γ and the The group Gal(F¯ /F ) acts on the (−2)-curves on X orbit Γ · D. The Galois action on R1 , . . . , Rm may be nontrivial. However, the fundamental domain description guarantees a unique f ∈ Γ · D in the nef cone of X, which is necessarily Galois invariant; some multiple of this divisor is defined over F . This semiample divisor induces our elliptic fibration.  Proposition 16. Let X be an elliptic K3 surface over F and Y another K3 surface derived equivalent to X over F . Then Y is elliptic over F . Proof. By Proposition 9, the Picard groups of X and Y are stably isomorphic as lattices   0 1 Pic(X) ⊕ U ' Pic(Y ) ⊕ U, U = . 1 0 In particular, Pic(X) and Pic(Y ) share the same discriminant and p-adic invariants. A rank-two indefinite lattice represents zero if and only if its discriminant is a square. In higher ranks, an indefinite lattice represents zero if and only if it represents zero p-adically for each p. Thus Pic(Y ) admits a square-zero class and is elliptic by Lemma 15.  An elliptic K3 surface J → C is Jacobian if it admits a section C → J. It has geometric Picard group containing f Σ

f 0 1

Σ 1 −2

(2.1)

where f is a fiber and Σ is the section. Jacobian elliptic surfaces admit numerous autoequivalences [11, §5]. Let a, b ∈ Z with a > 0 and (a, b) = 1. The moduli space of rank a degree b indecomposable vector bundles on fibers of J → C is an elliptic fibration with section, isomorphic to J over C. This reflects Atiyah’s classification of vector bundles over elliptic curves [7, Th. 7]. The associated Fourier-Mukai transform induces an autoequivalence of J act˜ ing on H(J, Z) via an element of   c a ∈ SL2 (Z) d b

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obtained as in [11, Th. 3.2,5.3]. This acts on the Mukai vectors (or their `-adic analogues) (1, 0, 0), (0, 0, 1) ∈ H 0 (J, Z)(−1) ⊕ H 2 (J, Z) ⊕ H 4 (J, Z)(1) by the formula (1, 0, 0) 7→ (0, af, c),

(0, 0, 1) 7→ (b, d(f + Σ), 0).

The action is transitive on the (−2)-vectors in the lattice of algebraic classes     0 −1 0 1 H 0 (J, Z)(−1) ⊕ H 4 (J, Z)(1) ⊕ hf, Σi ' ⊕ . (2.2) −1 0 1 −2 Thus we have established the following: Proposition 17. Let J → C be a Jacobian elliptic K3 surface over F . Then autoequivalences of J defined over F induce a representation of SL2 (Z) on (2.2) as above. The point is that here autoequivalences are defined over the ground field. Over C, the availability of autoequivalences has strong consequences for Jacobian elliptic K3 surfaces. For instance, orientation preserving automorphisms of the Mukai lattice of a complex K3 surface arise from autoequivalences [23, Th. 1.6]. This implies that derived equivalent Jacobian elliptic K3 surfaces are isomorphic [24, Cor. 2.7.3]. The autoequivalences are generated via spherical twists associated with rigid objects. As we have seen (e.g., in Example 14), rigid objects over non-closed fields need not descend. It is natural to ask the following question: Question 18. Let J1 and J2 denote Jacobian elliptic K3 surfaces over a field F of characteristic zero. If J1 and J2 are derived equivalent does it follow that J1 ' J2 ? If the geometric Picard group has rank two, then the isomorphism follows from Proposition 17. We recall the classical Ogg-Shafarevich theory for elliptic fibrations, following [15, 4.4.1,5.4.5]: Let F be algebraically closed of characteristic zero and J → P1 a Jacobian elliptic K3 surface. We may interpret X(J/P1 ) ' Br(J) and each α in this group may be realized by an elliptically fibered K3 surface X → P1 with Jacobian fibration J → P1 . If α has order n, then we have natural exact sequences 0 → Z/nZ → Br(J) → Br(X) → 0 and 0 → T (X) → T (J) → Z/nZ → 0. Note that if Y = Pic (X/P1 ), then [Y ] = c[X] ∈ X(J/P1 ). If c is relatively prime to n, so there exists an integer b with bc ≡ 1 (mod n), then we have c

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Picb (Y ) = X as well. It follows that X and Y are derived equivalent with the universal sheaves inducing the Fourier-Mukai transform [11, Th. 1.2]. Proposition 19. Let F be algebraically closed of characteristic zero. Let φ : X → P1 be an elliptic K3 surface with Jacobian fibration J(X) → P1 . Let α ∈ Br(J(X)) denote the Brauer class associated with [X] in the Tate-Shafarevich group of J(X) → P1 . Then X is derived equivalent to the pair (J(X), α). This follows from the proof of Căldăraru’s conjecture; see [30, 1.vi] as well as [15, 4.4.1] for the fundamental identification between the twisting data and the Tate-Shafarevich group. Question 20. Let X and Y be K3 surfaces derived equivalent over a field F . Suppose we have an elliptic fibration X → C over F . Does Y admit a fibration Y → C such that ∼ X → Picb (Y /C) & . C for some integer b? By symmetry we have X ' Picb (Y /C) and Y ' Picc (X/C) for b, c ∈ Z. As before, if n = ord([X]) = ord([Y ]) in the Tate-Shafarevich group, then bc ≡ 1 (mod n). A positive answer to Question 20 would imply: • X dominates Y over F and vice versa. • If X and Y are elliptic K3 surfaces derived equivalent over a field F of characteristic zero, then X(F ) 6= ∅ if and only if Y (F ) 6= ∅. 2.2. Rank one K3 surfaces We recall the general picture: Proposition 21 ([49, Prop. 1.10], [57]). Let X/C be a K3 surface with Pic(X) = Zh, where h2 = 2n. Then the number m of isomorphism classes of K3 surfaces Y derived equivalent to X is given by m = 2τ (n)−1 ,

where τ (n) = number of prime factors of n.

Example 22. The first case where there are multiple isomorphism classes is degree twelve. Let (X, h) be such a K3 surface and Y = M(2,h,3) (X) the moduli space of stable vector bundles E → X with rk(E) = 2,

c1 (E) = h,

χ(E) = 2 + 3 = 5,

whence c2 (E) = 5. Note that if Y (F ) 6= ∅, then X admits an effective zerocycle of degree five and therefore a zero-cycle of degree one. Indeed, if E → X is a vector bundle corresponding to [E] ∈ Y (F ), then a generic σ ∈ Γ(X, E) vanishes at five points on X. As we vary σ, we get a four-parameter family of such cycles. Moreover, the cycle h2 has degree twelve, relatively prime to five. Is X(F ) 6= ∅ when Y (F ) 6= ∅?

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2.3. Rank two K3 surfaces Exhibiting pairs of non-isomorphic derived equivalent complex K3 surfaces of rank two is a problem on quadratic forms [24, §3]. Suppose that Pic(XC ) = ΠX and Pic(YC ) = ΠY and X and Y are derived equivalent. Orlov’s Theorem implies T (X) ' T (Y ) which means that ΠX and ΠY have isomorphic discriminant groups/p-adic invariants. Thus we have to exhibit p-adically equivalent rank-two even indefinite lattices that are not equivalent over Z. (Proposition 9 asserts this is a necessary condition over general fields of characteristic zero.) Example 23. We are grateful to Sho Tanimoto and Letao Zhang for assistance with this example. Consider the lattices ΠX = C f

C 2 13

f 13 , 12

ΠY = D g

D 8 15

g 15 10

which both have discriminant 145. Note that ΠX represents −2, (2f − C)2 = (25C − 2f )2 = −2, but that ΠY fails to represent −2. Let X be a K3 surface over F with split Picard group ΠX over a field F . We assume that C and f are ample. The moduli space Y = M(2,C+f,10) (X) has Picard group 2C (C + f )/2

2C 8 15

(C + f )/2 ' ΠY , 15 10

while M(2,D,2) (Y ) has Picard group D/2 2g

D/2 2 15

2g 15 ' ΠX 40

and is isomorphic to X. These surfaces have the following properties: • X and Y admit decomposable zero cycles of degree 1 over F ; • X(F ) 6= ∅: the rational points arise from the smooth rational curves with classes 2f − C and 25C − 2f , both of which admit zero-cycles of odd degree and thus are ' P1 over F ; • Y (F 0 ) is dense for some finite extension F 0 /F , due to the fact that | Aut(YC )| = ∞. We do not know whether • X(F 0 ) is dense for any finite extension F 0 /F ; • Y (F ) 6= ∅.

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3. Finite and real fields The `-adic interpretation of the Fourier-Mukai transform yields Theorem 24 ([38], [29, 16.4.3]). Let X and Y be K3 surfaces derived equivalent over a finite field F . Then for each finite extension F 0 /F we have |X(F 0 )| = |(Y (F 0 )|. For the case of general surfaces see [22]. We have a similarly complete picture over the real numbers. We review results of Nikulin [47, §3] [48, §2] on real K3 surfaces. Let X be a K3 surface over R, XC the corresponding complex K3 surface, and ϕ the action of the anti-holomorphic involution (complex conjugation) of XC on H 2 (XC , Z). Let Λ± ⊂ H 2 (XC , Z) denote the eigenlattices where ϕ acts via ±1. If D is a divisor on X defined over R, then ϕ([D]) = −D; the sign reflects the fact that complex conjugation reverses the sign of (1, 1) forms. In Galois-theoretic terms, the cycle class of a divisor lives naturally ˜ ± denote H 2 (XC , Z(1)) and twisting by −1 accounts for the sign change. Let Λ ˜ − contains the degree zero the eigenlattices of the Mukai lattice; note that Λ and four summands. Again, the sign change reflects the fact that these are twisted in the Mukai lattice. We introduce the key invariants: Let r denote the rank of Λ− . The discriminant groups of Λ± are two-elementary groups of order 2a where a is ˜ ± have discriminant groups of the same a non-negative integer. Note that Λ order. Finally, we set ( 0, if (λ, ϕ(λ)) ≡ 0(mod 2) for each λ ∈ Λ, δϕ = 1, otherwise. Note that δϕ can be computed via the Mukai lattice ˜ δϕ = 0 iff (λ, ϕ(λ)) ≡ 0(mod 2) for each λ ∈ Λ, as the degree zero and four summands always give even intersections. We observe the following: Proposition 25. Let X and Y be K3 surfaces over R, derived equivalent over R. Then (r(X), a(X), δϕ,X ) = (r(Y ), a(Y ), δϕ,Y ). Proof. The derived equivalence induces an isomorphism ˜ C , Z) ' H(Y ˜ C , Z) H(X compatible with the conjugation actions. Since (r, a, δϕ ) can be read off from the Mukai lattice, the equality follows.  The topological type of a real K3 surface is governed by these invariants. Let Tg denote a compact orientable surface of genus g.

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Proposition 26 ([47, Th. 3.10.6], [48, 2.2]). Let X be a real K3 surface with invariants (r, a, δϕ ). Then the manifold X(R) is orientable and  ∅, if (r, a, δϕ ) = (10, 10, 0),    T t T , if (r, a, δϕ ) = (10, 8, 0), 1 1 X(R) = k  Tg t (T0 ) , otherwise, where    g = (22 − r − a)/2, k = (r − a)/2. Corollary 27. Let X and Y be K3 surfaces defined and derived equivalent over R. Then X(R) and Y (R) are diffeomorphic. In particular, X(R) 6= ∅ if and only if Y (R) 6= ∅. The last statement also follows from Proposition 12: A variety over R has a real point if and only if its index is one. (This was pointed out to us by Colliot-Thélène.) Example 28. Let X and Y be derived equivalent K3 surfaces, defined over R; assume they have Picard rank one. Then Y = Mv (X) for some isotropic ˜ Mukai vector v = (r, f, s) ∈ H(X(C), Z) with (r, s) = 1. For a vector bundle E of this type note that c2 (E) = c1 (E)2 /2 + rχ(OX ) − χ(E) = rs + r − s, which is odd as r and s are not both even. Then a global section of E gives an odd-degree cycle on X over R, hence an R-point.

4. Geometric case: local fields with complex residue field We start with a general definition: Let R be a discrete valuation ring with quotient field F . Given a K3 surface X over F , a model is a flat proper algebraic space X → ∆ = Spec(R) with generic fiber X. We say X has good reduction if there exists a smooth model and ADE reduction if there is a model with (at worst) rational double points in the central fiber X0 . Let F = C((t)) with algebraic closure F¯ , obtained by adjoining all the ¯ for its roots of t. Let X be a projective K3 surface over F = C((t)); write X ¯ base-change to F . Consider the monodromy action ¯ Z) → H 2 (X, ¯ Z) T : H 2 (X, associated with a counterclockwise loop about t = 0. This is quasi-unipotent, i.e., there exist e, f ∈ N such that (T e − I)f = 0 [37]; we choose e, f minimal with this property. If X and Y are derived equivalent over F , then Orlov’s Theorem and the discussion preceding Proposition 6 implies that their Mukai lattices admit a monodromy equivariant isomorphism. Write ∆ = Spec C[[t]] and fix a projective model X →∆ and a resolution $ : X0 → ∆

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such that the central fiber $−1 (0) is a normal crossings divisor, perhaps with multiplicities along some components. Let X◦0 ⊂ $−1 (0) denote the smooth locus of the central fiber, i.e., the points of mulitiplicity one. A’Campo [1, Th. 1] proved that χ(X◦0 ) = 2 + trace(T ). This may be interpreted as the alternating sum of the traces of the monodromy matrices on all the cohomology groups of X. An application of Hensel’s Lemma yields Proposition 29. If trace(T ) 6= −2, then X → ∆ admits a section, i.e., X(F ) 6= ∅. This result was previously obtained by Nicaise [46, Cor. 6.6]; his techniques are also applicable in mixed characteristic under appropriate tameness assumptions. Proposition 29 applies when T is unipotent (e = 1). This is the case when there exists a resolution X 0 → ∆ with central fiber reduced normal crossings. (See the Appendix and Theorem 38 for further analysis.) If X and Y are derived equivalent over F , then Orlov’s Theorem and the discussion preceding Proposition 6 implies that their Mukai lattices admit a monodromy equivariant isomorphism. Thus the characteristic polynomials of their monodromy matrices are equal. Corollary 30. Suppose that X and Y are derived equivalent K3 surfaces with monodromy satisfying trace(T ) 6= −2. Then both X(F ) and Y (F ) are nonempty. Let X → ∆ denote a nonsingular model with central fiber X0 . Inclusion ¯ ,→ X and retraction of X to X0 induce X ¯ Z) → H2 (X , Z) → H2 (X0 , Z); % : H2 (X, elements of the kernel are called vanishing cycles. The local invariant cycle theorem [16, 5.3.4] gives an exact sequence %∗

T −I

¯ Q) −→ H 2 (X, ¯ Q). H 2 (X0 , Q) −→ H 2 (X, The kernel of % is dual to the cokernel of %∗ ; we may identify this with the image of (T − I). Thus we define the lattice of vanishing classes of X as ¯ Z). This carries the structure saturation of the image of (T − I) in H 2 (X, of a representation of a cyclic group, i.e., the action of T . Since (T − I) acts trivially on H 0 and H 4 , the lattice of vanishing cycles may also be in interpreted as the saturation of the image of (T − I) on the full Mukai lattice ˜ H(X, Z). Again, Orlov’s Theorem implies Proposition 31. For K3 surfaces over F = C((t)), lattices of vanishing classes are a derived invariant. We have results in some cases where T is semisimple, i.e., when T e = I for some e ∈ N.

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Definition 32. Let X be a K3 surface over F = C((t)) with monodromy T . It is of ADE type if • the lattice of vanishing classes admits a finite-index sublattice isomorphic to a direct sum of negative definite lattices associated with root systems of types An , Dn , E6 , E7 , or E8 ; • the action of T on the vanishing classes may be expressed as a product of reflections in roots in these lattices. Roughly, the monodromy is in the product of Weyl groups associated with the vanishing classes. Since the lattice of vanishing classes is a derived invariant, being of ADE type is as well. Proposition 33. X is of ADE type if and only if it has ADE reduction. An application of Hensel’s Lemma gives Corollary 34. If X is of ADE type, then X(F ) is non-empty. Proof. Suppose X admits a model X → ∆ with ADE singularities in the central fiber. Replacing X with a small modification, we may assume that it has Q-factorial terminal singularities [33, Th. 6.25]. Such a model is maximal among small partial resolutions of the original family; see [53, §1,8] and [31] for interpretations in the context of families of surfaces with ADE singularities. The point of passing to this model is to ensure that the cycles collapsed ¯ by the specialization X X0 are in fact vanishing cycles. After a suitable basechange ∆1 → ∆ t = te1 , our family admits a simultaneous resolution [4, 14] Xe → X ×∆ ∆1 , i.e., ρ0 : Xe0 → X0 is the minimal resolution. As Xe0 is a smooth degeneration of X it is also a K3 surface. Moreover, the vanishing cycles of X are spanned (over Q) by ρ0 -exceptional curves and the monodromy of X is a product of reflections in the product of Weyl groups indexed by the singularities of X0 . Thus X is of ADE type. Conversely, assume that X satisfies the requisite monodromy condition. After a basechange ∆1 → ∆ as above, the Torelli Theorem gives a smooth (Type I Kulikov) model Xe → ∆1 . We analyze the central fiber Xe0 . The lattice of vanishing classes specializes to the saturation of a direct sum of ADE lattices in the Picard group of Xe0 . Moreover, we may interpret T as acting birationally on Xe, resolved after a flop of vanishing smooth rational curves in Xe0 [53, Thm. 8.2]. This action

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is over the action t1 → ζt1 with ζ = exp(2πi/e). Contracting the vanishing (−2)-curves, we obtain a model Xe → Y & . ∆1 with ADE singularities in the central fiber Y0 , over which T is regular. The quotient X := Y/T → ∆1 /T = ∆ is the desired model of X. 

5. Semistable models over p-adic fields Let F be a p-adic field with ring of integers R. We start with the case of good reduction, which follows from Theorem 24 and Hensel’s Lemma: Corollary 35. Let X and Y be K3 surfaces over F , which are derived equivalent and have good reduction over F . Then X(F ) 6= ∅ if and only if Y (F ) 6= ∅. We can extend this as follows: Proposition 36. Assume that the residue characteristic p ≥ 7. Let X and Y be K3 surfaces defined and derived equivalent over F . Assume X and Y admit models X , Y → Spec(R) that are regular and have ADE singularities in the central fiber. Then X(F ) 6= ∅ if and only if Y (F ) 6= ∅. Proof. Let k be the finite residue field, X0 and Y0 the resulting reductions, ¯ and X˜0 and Y˜0 their minimal resolutions over k. Our assumptions on p guarantee that the classification and deformations of rational double points over k coincides with the classification in characteristic 0 [5]. Applying Artin’s version of Brieskorn simultaneous resolution [4, Th. 2], there exists a finite extension Spec(R1 ) → Spec(R) and proper models Xe → X ×Spec(R) Spec(R0 ) → Spec(R0 ), Ye → Y ×Spec(R) Spec(R0 ) → Spec(R0 ), in the category of algebraic spaces, with central fibers Xe0 and Ye0 . The Fourier-Mukai transform induces isomorphisms ˜ ét (X, ¯ Z` ) → H ˜ ét (Y¯ , Z` ) H compatible with monodromy and the action of Frobenius. Specializing yields an isomorphism 2 e 2 e ψ : Hét (X0 , Z` ) → Hét (Y0 , Z` ).

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Note that since Xe, Ye are not projective over Spec(R0 ), there is not an evident interpretation of this as a derived equivalence over Spec(R0 ). (See [13] for such interpretations for K3 fibrations over complex curves.) Furthermore ψ is far from unique, as we may compose with reflections arising from exceptional curves in either Xe0 → X0 or Ye0 → Y0 associated with vanishing cycles of X or Y. ¯ Z` ) Let L (resp. M ) denote the lattice of vanishing classes in H 2 (X, 2 ¯ (resp. H (Y , Z` )). As in the proof of Proposition 31, L ' M , compatibly with the monodromy and Frobenius actions. (Since X and Y have ADE reduction, the local invariant cycle theorem still applies to their models.) Their orthogonal complements in the Mukai lattice L⊥ and M ⊥ are isomorphic as well. The central fibers X0 and Y0 are obtained from X˜0 and Y˜0 by blowing down the (−2)-curves classes associated with L and M respectively. Let X◦ ⊂ X0 and Y◦ ⊂ Y0 denote the smooth loci, i.e., the complements of the rational curves associated with L and M respectively. We can relate compactly supported cohomology to our lattices: 2 2 L⊥ ⊗ Q` ⊃ Hc,ét (X¯◦ , Q` ), M ⊥ ⊗ Q` ⊃ Hc,ét (Y¯◦ , Q` ), with the difference reflecting contributions from H 0 and H 4 . Thus ψ induces an isomorphism 2 2 Hc,ét (X¯◦ , Q` ) ' Hc,ét (Y¯◦ , Q` ), compatible with Galois actions. The Weil conjectures yield then that |X◦ (k)| = |Y◦ (k)|. Rational points of X and Y over F correspond to sections of X → Spec(R) and Y → Spec(R). Since X and Y are regular, these reduce to points of X◦ (k) and Y◦ (k). Hensel’s Lemma then implies our result.  Question 37. Is admitting a model with good or ADE reduction a derived invariant? Y. Matsumoto and C. Liedtke have recently addressed this. Having po2 ¯ tentially good reduction is governed by whether Hét (X, Q` ) is unramified, under technical assumptions [40, Th. 1.1]. These are satisified if there exists a Kulikov model after some basechange [41, p. 2]. The ramification condition depends on the `-adic cohomology and thus only on the derived equivalence class. Proposition 33 suggests a monodromy characterization of ADE reduction in mixed characteristic; under technical assumptions, there exists such a model if the cohomology is unramified [41, Th. 5.1].

Appendix: Semistability and derived equivalence To understand the implications of derived equivalence for rational points over local fields, we must first describe how monodromy governs the existence of models with good properties. For K3 surfaces, it is widely expected that unipotent monodromy should suffice to guarantee the existence of a Kulikov

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model. Over C this is widely known to experts; the referee advised us that this was addressed in correspondence among R. Friedman, D. Morrison, and F. Scattone in 1983. As we are unaware of a published account, we offer an argument: Theorem 38. Let X be a K3 surface over F = C((t)). Then X admits a Kulikov model if and only if its monodromy is unipotent. Corollary 39. Let X and Y be derived equivalent K3 surfaces over F . Then X admits a Kulikov model if and only if Y admits a Kulikov model. As we have seen, if X and Y are derived equivalent over F , then their Mukai lattices admit a monodromy-equivariant isomorphism; thus the characteristic polynomials of their monodromy matrices are equal. The remainder of this section is devoted the proof of Theorem 38. We start with a review of basic results on Kulikov models. Let R = C[[t]], ∆ = Spec(R), and ∆◦ = Spec(F ). The monodromy T of X over C((t)) satisfies (T e − I)f = 0 for some e, f ∈ N. We take e and f minimal with this property. The semistable reduction theorem [32] implies there exists an integer n ≥ 1 such that after basechange to R2 = C[[t2 ]], F2 = C((t2 )),

tn2 = t,

there exists a flat proper π2 : X2 → ∆2 = Spec(R2 ) such that • the generic fiber is the basechange of X to F2 ; • the central fiber π2−1 (0) is a reduced normal crossings divisor. We call this a semistable model for X. It is well-known that semistable reductions have unipotent monodromy so e|n. By work of Kulikov and Persson-Pinkham [36, 51], there exists a semistable modification of X2 $ : Xe → ∆2 with trivial canonical class, i.e., there exists a birational map X2 99K Xe that is an isomorphism away from the central fibers. We call this a Kulikov model for X. Furthermore, the structure of the central fiber Xe0 can be described in more detail: Type I. Xe0 is a K3 surface and f = 1. Type II. Xe0 is a chain of surfaces glued along elliptic curves, with rational surfaces at the end points and elliptic ruled surfaces in between; here f = 2. Type III. Xe0 is a union of rational surfaces and f = 3.

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We will say more about the Type III case: It determines a combinatorial triangulation of the sphere with vertices indexed by irreducible components, edges indexed by double curves, and ‘triangles’ indexed by triple points [44]. We analyze this combinatorial structure of Xe0 in terms of the integer m. Let Xe0 = ∪ni=1 Vi denote the irreducible components, V˜i their normalizations, and Dij ⊂ V˜i the double curves over Vi ∩ Vj . Definition 40. Xe0 is in minus-one form if for each double curve Dij we have 0 2 0 2 (Dij )Vi = −1 if Dij is a smooth component of Dij and (Dij )Vi = 1 if Dij is nodal. Miranda-Morrison [43] have shown that after elementary transformation of Xe, we may assume that Xe is in minus-one form. The following are equivalent [17, §3],[19, 0.5,7.1]: • the logarithm of the monodromy is m times a primitive matrix; • Xe0 admits a ‘special µm action’, i.e., acting trivially on the sets of components, double/triple points, and Picard groups of the irreducible components; • Xe0 admits ‘special m-bands of hexagons’, i.e., the triangulation coming from the components of Xe0 arises as a degree m refinement of another triangulation. In other words, Xe0 ‘looks like’ it is obtained from applying semistable reduction to the degree m basechange of a Kulikov model. Its central fiber Xe00 can readily be described [17, 4.1] – its triangulation is the one with refinement equal to the triangulation of Xe0 , and its components are contractions of the corresponding components of Xe0 . (When we refer to Xe00 below in the Type III case, we mean the surface defined by this process.) For Type II we can do something similar [18, §2]. After elementary modifications, we may assume the elliptic surfaces are minimal. Then following are equivalent: • the logarithm of the monodromy is m times a primitive matrix; • Xe0 = V0 ∪E . . . ∪E Vm is a chain of m + 1 surfaces glued along copies of an elliptic curve E, where V0 and Vm are rational and V1 , . . . , Vm−1 are minimal surfaces ruled over E. Again Xe0 ‘looks like’ it is obtained from applying semistable reduction to another Kulikov model with central fiber Xe00 = V0 ∪E Vm . Moreover (E 2 )V0 + (E 2 )Vm = 0 and Xe00 is d-semistable in the sense of Friedman [18, 2.1]. (When we refer to Xe00 below in the Type II case, we mean the surface defined by this process.) There are refined Kulikov models taking into account polarizations: Let (X, g) be a polarized K3 surface over F of degree 2d. Shepherd-Barron [55] has shown there exists a Kulikov model $ : Xe → ∆2 with the following properties:

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• there exists a specialization of g to a nef Cartier divisor on the central fiber Xe0 ; • g is semi-ample relative to ∆2 , inducing |g|

Xe

→ &

Z .

∆2 where Xe0 → Z0 is birational and Z0 has rational double points, normal crossings, or singularities with local equations xy = zt = 0. These will be called quasi-polarized Kulikov models and their central fibers admissible degenerations of degree 2d. Recall the construction in sections five and six of [19]: Let D denote the period domain for degree 2d K3 surfaces and Γ the corresponding arithmetic group – the orientation-preserving automorphisms of the cohomology lattice H 2 (X, Z) fixing g. Fix an admissible degeneration (Y0 , g) of degree 2d and its image (Z0 , h), with deformation spaces Def(Y0 , g) → Def(Z0 , h); the morphism arises because g is semiample over the deformation space. Let Γ\DNY ⊃ Γ\D 0

denote the partial toroidal compactification parametrizing limiting mixed Hodge structures with monodromy weight filtration given by a nilpotent NY0 associated with Y0 (see [19, p.27]). We do keep track of the stack structure. Given a holomorphic mapping f : {t : 0 < |t| < 1} → Γ\D, that is locally liftable (lifting locally to D), with unipotent monodromy Γconjugate to NY0 , then f extends to f : {t : |t| < 1} → Γ\D. The period map extends to an étale morphism [19, 5.3.5,6.2] Def(Y0 , g) → Γ\D. Thus the partial compactification admits a (local) universal family. Theorem 38 thus boils down to The smallest positive integer n for which we have a Kulikov model equals the smallest positive integer e such that T e is unipotent. Proof. We show that a Kulikov model exists provided the monodromy is unipotent. Suppose we have unipotent monodromy over R1 = C[[t1 ]], te1 = t, and semistable reduction X2 → ∆2 = Spec(R2 ),

R2 = Spec(C[[t2 ]]), tme 2 = t.

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Let Xe → ∆2 denote a Kulikov model, obtained after applying elementary transformations as specified above. Write 1 mN = log(T e ) = (T e − I) − (T e − I)2 2 where m ∈ N and N is primitive (cf.[19, 1.2] for the Type III case). Let Xe00 be the candidate for the ‘replacement’ Kulikov model, i.e., the central fiber of the Kulikov model we expect to find Xe0 → ∆1 . In the Type I case Xe00 = Xe0 by Torelli, so we focus on the Type II and III cases. Lemma 41. Suppose that Xe0 admits a degree 2d semiample divisor g. Then Xe0 admits one as well, denoted by g 0 . 0

Proof. In the Type II case, this follows from [18, Th. 2.3]. The discussion there shows how (after elementary modification) the divisor can be chosen to induce the morphism Xe0 → Xe00 collapsing V1 ∪E . . .∪E Vm−1 to E, interpreted as the normal crossings locus of Xe00 . For Type III, we rely on Proposition 4.2 of [17], which gives an analogous process for modifying the coefficients of h so that it is trivial or a sum of fibers on the special bands of hexagons. However, Friedman’s result does not indicate whether the resulting line bundle is nef. This can be achieved after birational modifications of the total space [55, Th. 1].  We can apply the Friedman-Scattone compactification construction to both (Xe0 , g) and (Xe00 , g 0 ), with N = NXe0 and mN = NXe0 . Thus we obtain 0 two compactifications Γ\DmN → Γ\DN ⊃ Γ\D, both with universal families of degree 2d K3 surfaces and admissible degenerations. To construct Xe0 → ∆1 = Spec(R1 ) we use the diagram ∆2 ↓ ∆1

→ Γ\DmN ↓ Γ\DN .

The liftability criterion for mappings to the toriodal compactifications gives an arrow ∆1 → Γ\DN making the diagram commute. The induced universal family on this space induces a family Xe0 → ∆1 , agreeing with our original family for t1 6= 0 by the Torelli Theorem. More precisely, the monodromies of the two families over ∆◦1 = Spec(R1 ) \ {0} are identified and automorphisms of K3 surfaces act faithfully on cohomology, so they are isomorphic over ∆◦1 . Thus Xe0 → ∆1 is the desired model. 

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Unramified Brauer Classes on Cyclic Covers of the Projective Plane Colin Ingalls, Andrew Obus, Ekin Ozman and Bianca Viray With an Appendix by Hugh Thomas Abstract. Let X → P2 be a p-cyclic cover branched over a smooth, connected curve C of degree divisible by p, defined over a separably closed field of characteristic different from p. We show that all (unramified) p-torsion Brauer classes on X that are fixed by Aut(X/P2 ) arise as pullbacks of certain Brauer classes on k(P2 ) that are unramified away from C and a fixed line L. We completely characterize these Brauer classes on k(P2 ) and relate the kernel of the pullback map to the Picard group of X. If p = 2, we give a second construction, which works over any base field of characteristic not 2, that uses Clifford algebras arising from symmetric resolutions of line bundles on C to yield Azumaya represen√ tatives for the 2-torsion Brauer classes on X. We show that when −1 is in our base field, both constructions give the same result. Mathematics Subject Classification (2010). Primary: 14F22; Secondary: 12G05, 14J28, 14J50, 15A66, 16K50. Keywords. Brauer group, Clifford algebra, K3 surface, p-cyclic cover, symbol algebra, quadratic form.

1. Introduction The Brauer group Br X of a smooth projective variety X is a fundamental object of study and has a variety of applications. For instance, Artin and Mumford used the birational invariance of the Brauer group to exhibit non-rational unirational three-folds [2]. The Brauer group also encodes arithmetic information. For a variety X over a global field k, the Brauer group C.I. was partially supported by an NSERC Discovery Grant. A.O. was partially supported by NSF Grant DMS-1265290. B.V. was partially supported by NSF grant DMS-1002933. H.T. was partially supported by an NSERC Discovery grant.

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of X can obstruct the existence of k-rational points, even when there are no local obstructions [25]. In a different direction, Brauer groups are related to derived categories. More precisely, derived categories of varieties are sometimes equivalent to derived categories of the category of sheaves of modules over an Azumaya algebra, or twisted sheaves; when this occurs these algebras and twistings are classified by the Brauer group. For each of these applications, it is often necessary to work with elements of the Brauer group quite explicitly, say as Azumaya algebras over X or as central simple algebras over the function field of X (We refer the reader to Section 2 for background on the Brauer group and Brauer elements). Unfortunately, there are few general methods for obtaining such representatives. In 2005, van Geemen gave a construction of all 2-torsion Brauer classes on a degree 2 K3 surface of Picard rank 1. Precisely he proves: Theorem ([31, §9]). Let π : X → P2C be a double cover branched over a smooth sextic curve C; then X is a degree 2 K3 surface. Assume that X has Picard rank 1. Then there is a natural isomorphism   Pic C ∼ [2]. (1.1) Br X[2] → KC Moreover, for every α ∈ Br X[2] there is a geometric construction of an Azumaya algebra on X representing α; the construction involves exactly one of 1. a double cover of P2 × P2 branched over a (2, 2) curve, 2. a degree 8 K3 surface, i.e., an intersection of 3 quadrics in P5 , or 3. a cubic 4-fold containing a plane. The main result of this paper is an extension of van Geemen’s result, including the geometric constructions of Azumaya algebras, to degree 2 K3 surfaces of any Picard rank and, more generally, to any smooth double cover of P2 . Theorem 1.1. Let k be a separably closed field of characteristic different from 2 and let π : X → P2k be a double cover branched over a smooth curve C of degree 2d. Then there is an exact sequence   Pic X Pic C AX 0→ → [2] −→ Br X[2] → 0. ZH + 2 Pic X KC Moreover, for every D ∈ (Pic C/KC ) [2] there is a geometric description of an Azumaya algebra representing AX (D); for C in an open dense set of the moduli space of degree 2d plane curves, the construction involves exactly one of 1. a (2, 2) hypersurface of Pd−1 × P2 , 2. a (2, 1) hypersurface of P2d−1 × P2 , or 3. a cubic (2d − 2)-fold containing a (2d − 4)-dimensional linear space. Note that if X is a degree 2 K3 surface, then d = 3 and we exactly recover van Geemen’s result. (In Section 7, we show that the open dense set contains all degree 2 K3 surfaces of Picard rank 1.)

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Remark 1.2. If one is interested only in extending (1.1), then van Geemen states a similar result (he attributes the proof to the referee) that holds for more general surfaces, although still retaining an assumption on the Picard rank [31, Thm. 6.2]. More generally, recent work of Creutz and the last author extend (1.1) and [31, Thm. 6.2] to any smooth surface that is birational to a double cover of a ruled surface [11, Thm. II]. This result also gives a central simple k(X)-algebra representative of every 2-torsion Brauer class; however, this k(X)-algebra may not extend to an Azumaya algebra over all of X. A result of Catanese ([6]) on symmetric resolutions of line bundles allows us to extend van Geemen’s geometric construction and obtain a set-theoretic map   Pic C AX [2] → Br X[2]. KC Yet, it is difficult to discern from this construction whether AX is even a group homomorphism, let alone prove surjectivity or determine the kernel. Van Geemen’s proof of the isomorphism (1.1) proceeds via the isomorphism between the Brauer group and the dual of the transcendental lattice of X. The argument relies on a classification of the index 2 sublattices of the transcendental lattice, which does not seem feasible for an arbitrary double plane X. We avoid this classification by using ramified Brauer classes on P2 . In doing so, we prove a result for p-cyclic covers of P2 . Let X be a smooth projective geometrically integral p-cyclic cover of P2 over a separably closed field k of characteristic different from p, branched over a smooth curve C of degree pd, for some d. Let L be a general line in P2 that intersects C in pd distinct points, and let U := P2 \ (C ∪ L) Theorem 1.3. Let ζ denote a primitive pth root of unity. There is an action of Z[ζ] on Br X and we have a commutative diagram where the top row is exact. 0

/

Pic X ZH+(1−ζ) Pic X

/

Pic C ZL 


[p] _

ψ

/ Br X[1 − ζ] _

/0

φ

 Br U [p]

π∗

 / Br k(X)[p].

To complete the proof of   Theorem 1.1, we show that the map AX factors C through a map AU : Pic [2] → Br U [2], and then show that AU and φ KC agree by comparing ramification data. We expect Theorem 1.3 to be of independent interest. Even though we do not describe φ or ψ by constructing central simple algebras or Azumaya algebras, the theorem should be helpful in constructing such representatives. Indeed, this result shows that every (1 − ζ)-torsion Brauer class on X can be obtained by pullback of a ramified Brauer class on P2 , which are much simpler objects to understand.

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1.1. Related work and open questions Some of the questions and ideas in this paper have been considered previously by a number of authors. Although we do not need their work in the proofs, we would be remiss if we didn’t mention them. We do so briefly here. 1.1.1. Pulling back Brauer classes. The kernel of the map π ∗ : (Br P2 \C)[2] → Br X was studied previously by Ford, who determined bounds for its dimension as an F2 vector space [13]. 1.1.2. Symmetric resolutions of line bundles. Our proof relies on work of Catanese about the existence of symmetric resolutions of line bundles. These symmetric resolutions have been used by many authors, sometimes to explicitly construct Brauer classes. Examples include [4, 5, 8]. 1.2. Notation Throughout, p will be a fixed prime and k will denote a separably closed field of characteristic q 6= p (q may be 0). Let C = V (f ) ⊂ P2k be a smooth and irreducible curve of degree e and let L = V (`) ⊂ P2k be a line that intersects C in e distinct points. Define ˜ 6= L. U := P2 \(C ∪L). We will use `˜ to denote any linear form such that V (`) Unless otherwise stated, we will assume that e = pd for some integer d. In this case, we can define π : X → P2 to be the p-cyclic cover of P2
branched over C. The morphism π gives an isomorphism from C˜ := π −1 (C) red to C. We set b1 (X) and b2 (X) equal to the first and second Betti numbers of X, respectively. That is, bi (X) = dim Hi (X, Qr ) for any prime r 6= q, when i = 1, 2. These invariants are well defined since X is a smooth surface p p (see, e.g., [17, §3.5]). Let hp,q (X) := hq (X, ωX ), that is, dimk Hq (X, ωX ). The Picard rank of X is denoted ρ(X). For any field F , we let Fs denote the separable closure. Note that all sheaves and cohomology are in the ´etale topology unless otherwise stated; we refer the reader to [28] for the background on ´etale cohomology. 1.3. Outline We begin by recalling some basic definitions and facts about Brauer groups and Clifford algebras in Section 2. In Section 3, we determine the numerical invariants of X and show that there exists a natural ring action Z[ζ] on Br X, where ζ is a pth root of unity. Section 4 contains the proof of Theorem 1.3. Starting with Section 5 we specialize to the case p = 2; there we construct the maps AX and AU . In Section 6, we prove that φ and AU agree, thereby completing the proof of Theorem 1.1. We specialize to degree 2 K3 surfaces in Section 7 and show that the open dense subset in Theorem 1.1 contains all Picard rank 1 surfaces. Lastly, we have an appendix by Hugh Thomas which proves a combinatorial proposition needed in Section 5.

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Acknowledgements. We thank Asher Auel and Brendan Hassett for expert advice and the anonymous referee for his or her comments. This project began at the American Institute of Mathematics during a workshop on Brauer groups and obstruction problems. We thank AIM for providing us with excellent working conditions. We also thank the other members of our working group at the conference, Evangelia Gazaki, Alexei Skorobogatov, and Olivier Wittenberg, for helpful discussions.

2. Background on the Brauer group and Clifford algebras Let F be a field. Definition 2.1. Let S be the monoid of isomorphism classes of central simple algebras over F with finite rank, with operation given by tensor product. We say that two elements A, B of S are Morita equivalent if and only if the matrix algebras Mn (A) and Mk (B) are isomorphic for some positive integers n, k. Then the Brauer group Br F of F is defined to be the quotient of S by this equivalence relation. Note that Br F is an abelian group under tensor product since tensor product commutes with taking matrix algebras. The inverse of an element [A] ∈ Br F is [Aop ] where Aop is the opposite algebra obtained by reversing the order of multiplication in A. The following is a classical fact from class field theory that relates Brauer groups to cohomology. Theorem 2.2 ([27, Corollary 3.16]). For any separable closure Fs of F there exists a natural isomorphism between Br F and H2 (Gal(Fs /F ), Fs× ). The definition of the Brauer group can be extended to a scheme Y as in the series [16–18]. An OY -algebra A is an Azumaya algebra over Y if it is a coherent locally free OY -module and the specialization of A at each point y ∈ Y is a central simple algebra over the residue field at y. Two Azumaya algebras A, A0 over Y are Morita equivalent if there exist locally free OY -modules E, E 0 of finite rank such that A ⊗OY End(E) is isomorphic to A0 ⊗OY End(E 0 ). This relation induces an equivalence relation and the tensor product operation carries over the quotient as in the case of central simple algebras over fields. The Azumaya Brauer group BrAz Y of Y is the abelian group given by the equivalence classes of Azumaya algebras. The cohomological Brauer group of Y , denoted by Brcohom Y , is the torsion part of the second cohomology group with values in Gm , the sheaf of units. If Y is smooth, this cohomology group is torsion, so Brcohom Y = H2 (Y, Gm ). Furthermore, if Y is a normal surface, if Y is the separated union of two affine schemes, or if Y is quasi-projective, then by Gabber’s theorem the cohomological Brauer group is the same as the Azumaya Brauer group [12].

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Henceforth, we will only consider smooth quasiprojective varieties in which case all the above notions agree, thus we will simply refer to the Brauer group of Y or Br Y . Remark 2.3. We have Br(Spec F ) = Br F since H2 (Spec F, Gm ) ∼ = H2 (Gal(Fs /F ), F × ). s

It is possible to relate Br Y and the Brauer group of its function field Br k(Y ) using the functoriality of ´etale cohomology. If Y is an integral scheme, then using the inclusion Spec k(Y ) → Y , we obtain a map Br Y → Br k(Y ). Theorem 2.4 ([28, Chap. IV, Cor. 2.6]). Let Y be a regular integral scheme with function field k(Y ). Then the induced map Br Y → Br k(Y ) is an injection. Using Theorem 2.4, we will regard elements of Br Y as elements of the Brauer group of a field, Br k(Y ). 2.1. Cyclic algebras It is a deep theorem of Merkurjev and Suslin that the Brauer group is generated by a special type of central simple algebra, namely the cyclic algebras. In addition, cyclic algebras are the central simple algebras that are easiest to construct and study. We recall their basic properties. Definition 2.5. Let F 0 /F be a cyclic Galois extension of degree m, fix an isomorphism χ : Gal(F 0 /F ) → Z/mZ, and let τ := χ−1 (1). Let b ∈ F × . Then the cyclic algebra associated to χ and b is F 0 [x; τ ] , hχ, bim := m (x − b) where F 0 [x; τ ] denotes the twisted polynomial ring, i.e., αx = xτ (α) for any α ∈ F 0. Remark 2.6. By abuse of notation, we sometimes use hF 0 /F, bim to denote hχ, bim , even though the cyclic algebra does depend on the choice of χ (equivalently τ ). If F 0 /F is a cyclic extension of degree m, we will sometimes denote the cyclic algebra hk(YF 0 )/k(Y ), f im as hF 0 /F, f im since Gal(k(YF 0 )/k(Y )) is isomorphic to Gal(F 0 /F ). Example 2.7. Every quaternion algebra is cyclic if the characteristic of the field F is not 2. The following presentation of a cyclic algebra is useful for computations. We refer to [15, Sections 2.5, 4.7] for its proof. Proposition 2.8. Assume that the characteristic of F is relatively prime to m and fix b ∈ F × , F 0 /F as above. We consider b as representing an element of H1 (F, µm ) via the Kummer isomorphism F × /F ×m ' H1 (F, µm ). Let χ be an element in Hom(Gal(Fs /F ), Z/mZ), which is isomorphic to H1 (F, Z/mZ).

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Consider the cup product H1 (F, Z/mZ)×H1 (F, µm ) → H2 (k, µm ) ' Br F [m]. The image of the tuple (χ, b) under this cup product is the cyclic algebra hχ, bim . If F contains a primitive mth root of unity (in particular, this means char(F ) - m), then we may define m-symbol algebras. Definition 2.9. Assume that F contains a primitive mth root of unity and let a, b, ∈ F × . Then the m-symbol algebra ha, bim is the F -algebra generated by x and y such that xm = a, y m = b and xy = ζm yx where ζm is a primitive mth root of unity. Theorem 2.10. Assume that F contains a primitive mth root of unity and let a, b, ∈ F × . Then any cyclic algebra hχ, bim is isomorphic to the symbol algebra ha, bim for some a ∈ F × . Proof. We use the cohomological interpretation of hχ, bim given in Proposition 2.8. Since F contains a root of unity ζm , we have an isomorphism Z/mZ → µm as Gal(Fs /F )-modules. This induces the following isomorphisms: H1 (F, Z/mZ) ' H1 (F, µm ) ' F × /(F × )m . Say a, b ∈ F × are fixed. The elements a, b can be seen in the corresponding cohomology groups using the mentioned isomorphisms. Then using the cup product H1 (F, Z/mZ) × H1 (F, µm ) → H2 (F, µm ) we get an element of H2 (F, µm ) ' Br F [m].  Now we will list some basic properties of cyclic and symbol algebras which will be useful in later sections. For proofs of these facts we refer to [15], [29], and [30]. Definition 2.11. A central simple algebra A over F is called split if A is isomorphic to the matrix algebra Mn (F ). If F 0 is a field such that the F 0 algebra A ⊗F F 0 is isomorphic to Mr (F 0 ) for some integer r ≥ 1, then we say that F 0 splits A. Proposition 2.12. Let A be central simple algebra over a field F . i) A is a cyclic algebra if and only if there exists a cyclic extension of F splitting A. More precisely, A is split by F 0 , an m-cyclic Galois extension of F, if and only if A is isomorphic to hχ, bim for some b ∈ F × and some isomorphism χ : Gal(F 0 /F ) → Z/m. ii) If A is a cyclic algebra over F and F 0 is a field extension of F, then A ⊗F F 0 is a cyclic algebra over F 0 . iii) In the Brauer group of F , [hχ, aim ] + [hχ, bim ] = [hχ, abim ]

and

[hχ, aim ] + [hψ, aim ] = [hχ + ψ, aim ] 1

for all a, b ∈ F × and χ, ψ ∈ H (G, Z/mZ), where G = Gal(F 0 /F ). In particular, if F contains a primitive mth root of unity, then for all a, b, c ∈ F × [ha, bim ] + [ha, cim ] = [ha, bcim ]

and

[ha, cim ] + [hb, cim ] = [hab, cim ].

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iv) Two cyclic algebras (χ, a) and (χ, b) are equivalent if and only if a/b is the norm of an element in F 0 , where F 0 is an m-cyclic Galois extension of F and χ : Gal(F 0 /F ) → Z/mZ is a fixed isomorphism. v) If F contains a primitive mth root of unity, then ha, (−a)n im is trivial in Br F for all a ∈ F × and n ∈ Z. For completeness we restate the theorem of Merkurjev and Suslin which was mentioned in the beginning of the section. Theorem 2.13 (Merkurjev-Suslin, see, e.g., [15, Thm. 4.6.6]). Assume that char(F ) - m. Then any representative of a class of order dividing m in Br F is Morita equivalent (but not necessarily isomorphic) to the tensor product of cyclic algebras. 2.2. Residues of Brauer elements An important map that we need in the next sections is the residue (or ramification) map. We will give the explicit description of this map for cyclic algebras. Using Theorem 2.13 it is possible to extend this map to any element of Br k(Y ) whose order is not divisible by the characteristic. We begin with defining this map for a complete discrete valuation ring R with field of fractions K. Let v denote the discrete valuation associated to R and π be a uniformizer such that vR (π) = 1. The residue field R/πR is denoted by F and has characteristic q. Theorem 2.14. Let Kun denote the maximal unramified extension of K. Let [A] ∈ Br K be an element of prime order q. Then [A] is split by Kun . Proof. This is Theorem 10.1 in [29].



For any abelian torsion group G, let G0 denote the prime-to-q component. Since K is complete, the valuation v extends uniquely to Kun and gives a Gal(Kun /K)-equivariant map between Kun and Z where the action on Z is the trivial action. Therefore, there exists an induced map × 0 f1 : H2 (Gal(Kun /K), Kun ) → H2 (Gal(Kun /K), Z)0 .

Now consider the short exact sequence 0 → Z → Q → Q/Z → 0. The associated long exact sequence gives a coboundary map, which is an isomorphism f2

H1 (Gal(Kun /K), Q/Z) −→ H2 (Gal(Kun /K), Z). Using these arguments, we can define the residue map as follows: Definition 2.15. The residue (ramification) map ∂v : (Br K)0 → Hom(Gal(Kun /K), Q/Z)0

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is defined as the composition: × 0 Br(K)0 ∼ ) = H2 (Gal(Kun /K), Kun

f1

−→ H2 (Gal(Kun /K), Z)0 ∼ = H1 (Gal(Kun /K), Q/Z)0 = Hom(Gal(Kun /K), Q/Z)0

The first isomorphism follows from Theorem 2.14, and the second isomorphism is given by f2−1 as described above. The last equality follows from the definition of group cohomology. Note that Hom denotes continuous homomorphisms. As seen from the definition, computing the residue map is not easy for a general element of the Brauer group. However, for cyclic algebras it can be done as follows. Let R, K, v, F, π be as above and let L be a cyclic Galois extension of K of degree m where m is relatively prime to q. Assume that the residue field extension is also cyclic of degree m. Fix an isomorphism χ : Gal(L/K) → Z/mZ and let τ = χ−1 (1). Proposition 2.16. The map ∂v (hχ, πim ) ∈ Hom(Gal(Kun /K), Q/Z) factors through Gal(L/K) and it sends τ ∈ Gal(L/K) to 1/m + Z. Proof. We will sketch the proof, for more details see [29]. Note that by assumption L/K is unramified of degree m. Let G denote the Galois group Gal(L/K). The image of [A] under the residue map is the image of [A] under the isomorphisms H2 (G, L× ) → H2 (G, Z) ∼ = H1 (G, Q/Z) using the naturality of the inflation map. Note that H2 (G, L× ) = (L× )G /NG (L× ) = K × /NG (L× ) and H2 (G, Z) = ZG /NG (Z) = Z/nZ, where NG denotes the norm map. Then [A] ∈ Br(K) corresponds to π ∈ K × , which maps to 1 ∈ Z. The rest follows from the definition of the coboundary map f2 : H1 (Gal(Kun /K), Q/Z) → H2 (Gal(Kun /K), Z).  In fact the residue map is onto and its kernel is the prime-to-q part of the Brauer group of R. Theorem 2.17 (Theorem 10.3, [29]). The following sequence is exact: 0 → (Br R)0 → (Br K)0 → Hom(Gal(Kun /K), Q/Z)0 → 0. Now we will define the residue map for cyclic Brauer classes over the function field of a surface. By the Merkurjev-Suslin theorem (Theorem 2.13), this defines it on all Brauer classes of order prime to the characteristic. Let Y be a complete, smooth algebraic surface over an separably closed field k. Let Y (1) denote the set of codimension one points, i.e., the set of all irreducible curves on Y . Then the following sequence is exact by [16–18]: M ⊕ ∂C 0 → Br Y → Br k(Y ) −−C−−→ H1 (k(C), Q/Z). C∈Y (1)

The map Br Y → Br k(Y ) is the injection map of Theorem 2.4. The sum is taken over all irreducible curves on Y , and k(C) denotes the field of rational

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functions on C. Let χ ∈ H1 (k(C), Q/Z) = Hom(Gal(k(C)s /k(C)), Q/Z). Then χ has kernel Gal(k(C)s /L) where L/k(C) is cyclic Galois of degree m. Therefore χ is determined by choosing a generator τ of Gal(L/k(C)) such 1 that χ(τ ) = m + Z, similarly to the case in Proposition 2.16. By Theorem 2.10, every L cyclic algebra of order m is a symbol algebra. The image of ha, bim under C ∂C is the tuple of residues of ha, bim along C for every irreducible curve C ⊂ X. Theorem 2.18. The residue of ha, bim along C is the cyclic Galois extension L of k(C) obtained by adjoining the mth root of avC (b) b−vC (a) . A rigorous proof of this theorem relies on Milnor K-theory and can be found on Section 7.5.1 of [15]. For a similar computation see [9, p. 10]. Definition 2.19. The combination of curves C in Y1 such that ∂C (A) 6= 0 is called the ramification divisor of A on Y . Remark 2.20. If ha, bim has nontrivial ramification along C, then C is a prime divisor of a or b. 2.3. Symbols and residues of Clifford algebras In this section, we will recall and prove some preliminary results about symbols and residues of Clifford algebras, which will be helpful for studying 2torsion Brauer classes. Throughout, we work over √ a field F of characteristic −1 ∈ F ; a similar analysis different from 2. For simplicity, we assume that √ can be carried out in the case that −1 ∈ / F , but is more complicated to state. By an abuse of notation, we will conflate diagonal quadratic forms in n variables and diagonal n × n matrices; they will both be denoted by [α1 , α2 , . . . , αn ],

with αi ∈ F.

First we recall a well-known proposition which follows from GramSchmidt. Proposition 2.21. Let M = (mij ) be a symmetric n × n matrix over a field F with corresponding quadratic form Q and let Mi denote the upper left i × i principal minor. If Mi 6= 0 for all i, then Q is equivalent to the diagonal quadratic form [M1 , M2 /M1 , . . . , Mn /Mn−1 ]. Given a non-degenerate quadratic form Q, we may consider the Clifford algebra Cl(Q), a finite dimensional simple algebra over F . The multiplication in Cl(Q) encodes the geometry determined by Q on V . More precisely, let Q : V → F be a non-degenerate quadratic form on a finite dimensional F vector space V . L∞ Np Definition 2.22. Let (V, Q) be a quadratic space and T (V ) := p=0 V be its tensor algebra. The Clifford algebra Cl(Q) is T /IQ , where IQ is two sided ideal in T (V ) generated by {v ⊗ v − Q(v)|v ∈ V }. Clifford algebras can be characterized as follows:

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Proposition 2.23. Let (V, Q) be a quadratic space. Let C be an F -algebra with unit 1C . Then C is a Clifford algebra for Q if it has the following properties: 1. There exists an F -module map fC : V → C such that fC (v)2 = Q(v)1C for any v in V . 2. C is ‘minimal’ with respect to the above property, i.e., for any F -algebra A that satisfies Property (1), there exists a unique F -algebra homomorphism φ : C → A, such that fA (v) = φ(fC (v)) for any v ∈ V . Proof. See pages 103–104 of [23].



As a corollary we see that equivalent quadratic forms give isomorphic Clifford algebras. Corollary 2.24. Let ι be an isometry of (V, Q) into (W, Q0 ). Then there exists an algebra isomorphism ιCl : Cl(Q) → Cl(Q0 ) extending ι such that the following diagram commutes: Cl(Q) O V

ιCl

/ Cl(Q0 ) O

ι

/ W.

Proof. Follows from Proposition 2.23.



Remark 2.25. Let BQ be the associated symmetric bilinear form defined by BQ (v1 , v2 ) = Q(v1 +v2 )−Q(v1 )−Q(v2 ). By Property (1) of Proposition 2.23, v1 v2 = −v2 v1 if and only if v1 and v2 are orthogonal with respect to BQ . Therefore, we can say that the multiplication on the Clifford algebra encodes the geometry determined by Q on V . The tensor algebra T (V ) has a Z/2Z-grading given by T0 (V ) =

i M O

V

and T1 (V ) =

i≥0,even

i M O

V.

i≥1,odd

There is a canonical map from T (V ) onto Cl(Q). Using this map and the grading on T , one can give a Z/2Z-grading on Cl(Q) such that Cl(Q) = Cl0 (Q) ⊕ Cl1 (Q). The part Cl0 (Q) is a subalgebra; we refer to is as the even Clifford algebra of Q. Remark 2.26. Another way to see the grading on the Clifford algebra is using the isometry v 7→ −v. This isometry extends to an involution of Cl(Q) using Corollary 2.24. The Clifford algebra and even Clifford algebra of Q have the following properties. Proposition 2.27. Let Q be the diagonal quadratic form [m1 , . . . , mn ]. 1. Cl0 (Q) = Cl([−m1 m2 , . . . , −m1 mn ]).

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2. If n is odd, then Cl0 (Q) is a central simple algebra over F and in Br F we have X Cl0 (Q) = hmi , mj i2 . 1≤i − 12 . Now rk(F ) = 1, 2, 3 and the three cases need to be analysed separately. Case A: rk(F ) = 1. Then c1 (F ) > 0 and F is a line bundle. Moreover, we have a commutative diagram F _  q φ

 Mx

" / / Kx .

Since F is torsion free, the composition φ can only vanish or be an injection. If φ is trivial, then it factors through OY (−H)⊕2 which is semistable with

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µ = −1 (recall that OY (H) generates Pic(Y )). Thus we get a contradiction. So assume that φ is injective and consider the following commutative diagram F _ ϕ

φ

IP ∪Q,Y 



 / Kx

$ / / IP ∪l0 ,Y .

Again, F cannot inject neither in IP ∪Q,Y nor in IP ∪l0 ,Y , and we get a contradiction. Case B: rk(F ) = 2. In this case c1 (F ) > −1 and we have a commutative diagram  / F / / F2 F1 _  (3.3) _ _   OY (−H)⊕2 

 / Mx

 / / Kx .

At this point we have to analyse some additional cases. Case B.1: rk(F1 ) = 2. Since Kx is torsion free, F2 = 0. On the other hand, OY (−H)⊕2 is semistable, so c1 (F1 ) 6 −2. Then c1 (F ) 6 −2, so µ(F ) 6 −1 and F does not destabilize Mx . Case B.2: rk(F1 ) = rk(F2 ) = 1. In that situation, c1 (F1 ) = −1, so F1 ∼ = OY (−H) and c1 (F2 ) = 0. On the other hand, since F2 ,→ Kx , by (2.8), we have F2 ,→ IP ∪Q . Thus (3.3) can be rewritten as  / F / / F2 OY (−H) _ _ _   OY (−H)⊕2

 / Mx

 / / Kx

  OY (−H) 

 /G

 / / G2 .

In that case, ch2 (F ) = ch2 (OY (−H))+ch2 (F2 ) 6 ch2 (Mx )·H 2 rk(Mx )

2

4

4

2

H2 2

2

−(P +Q) = − H2 . Since

ch2 (F )·H H = − P ·H = −H 4 12 > − 4 > rk(F ) , F does not destabilize Mx . Case B.3: rk(F2 ) = 2. In this case, F ∼ = F2 . If c1 (F ) > 0, then F ∼ = Kx . Thus (2.7) splits, which gives a contradiction. If c1 (F ) = −1, then F ,→ Kx and, by (2.8), we have F ,→ IP⊕2∪Q . Hence F is the extension of two ideal sheaves. Since we have assumed that F is semistable, we have that

0 → IZ1 (−H) → F → IZ2 → 0, where codim Z1 and codim Z2 are greater or equal than 2. Moreover, Z1 is possibly empty and P ∪ Q ⊆ Z2 . Thus, ch2 (F ) = ch2 (IZ1 (−H)) + ch2 (IZ2 ) =

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− Z1 − Z2 6 H2 − P ∪ Q. Hence, the same computation as at the end of Case B.2 shows that F does not destabilize Mx . Case C: rk(F ) = 3. Now c1 (F ) > −1 and we can consider again a diagram as (3.3). We can distinguish two possibilities. Case C.1: rk(F2 ) = 2. Since F is semistable, − 13 6 µ(F ) 6 µ(F2 ) 6 µ(Kx ) = 0. As 2µ(F2 ) is an integer, µ(F2 ) = c1 (F2 ) = 0, F1 ∼ = OY (−H) and c1 (F ) = −1. Hence we rewrite again (3.3) as  / / F2 / F OY (−H) _ _ _   OY (−H)⊕2 

 / Mx

 / / Kx

  OY (−H)

 /G

 / / T.

Since G is a torsion-free sheaf of rank 1 and c1 (G) = −1, if OY (−H) ∼ 6 G, = then G ∼ = IZ (−H) with Z 6= ∅ with codim Z > 2. Hence we get a contradiction since Hom(OY (−H), IZ (−H)) = 0. Thus, G ∼ = OY (−H) contradicting the fact that Mx is non-split. Case C.2: rk(F2 ) = 1. Since OY (−H)⊕2 is semistable, c1 (F1 ) 6 −2. On the other hand, also Kx is semistable, so c1 (F2 ) 6 0. Then c1 (F ) 6 −2 and µ(F ) 6 − 32 . Therefore, F does not destabilize Mx . This completes the proof of Proposition 3.1. In particular, we proved that S is the closure of a component of the moduli space of stable ACM bundles with Chern character (4, −2H, −P, l, 41 ). 3.1. Universal family In this section we assume that S is smooth. Then the above discussion can be summarized by saying that there exists a twisted universal family M ∈ Coh(S × Y, p∗1 α) such that the Fourier–Mukai functor ΦM : Db (S, α) → Db (Y ) is fully faithful and it factors through TY . Recall that ΦM (−) := (pY )∗ (M ⊗ p∗S (−)). As the Kuznetsov’s functor providing the full embedding of Db (S, α) into b D (Y ) is a composition of a Fourier–Mukai functor and mutations, finding M amounts to finding the Fourier–Mukai kernel of their composition. For this, consider   S := (σ × id)∗ p∗Ye E 0 ⊗p∗e π∗ B0 (π × id)∗ (f × id)∗ (p∗1 Eα∨ ⊗ O∆S ) Y

where pYe : Ye × S → Ye is the natural projection, p1 : S × S → S is the projection on the first factor, ∆S ⊂ S × S is the diagonal, and E 0 is defined

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in (1.6). From (2.2) we have   S|Y ×{x} ∼ = σ∗ p∗Ye E 0 ⊗p∗Ye π∗ B0 (π × id)∗ (f × id)∗ (p∗1 Eα∨ ⊗ O∆S ) e Y ×{x}   0 ∗ ∗ ∨ ∼ = σ∗ E ⊗π∗ B0 π ((f × id)∗ (p1 Eα ⊗ O∆S ))|P2 ×{x}    ∼ = σ∗ E 0 ⊗π∗ B0 π ∗ f∗ Eα∨ ⊗ O∆S |S×{x} ∼ = σ∗ (E 0 ⊗π∗ B0 π ∗ f∗ (Eα∨ ⊗ C(x))) ∼ = σ∗ Φ(Lx ) ∼ = Il ,Q . x

f (x)

Then, the universal family M over Y × S such that M|Y ×{x} ∼ = Mx can be described as M := (σ × id)∗ ◦ Lp∗e OYe (h−H) ◦ Rp∗e OYe (−h) ◦ (σ × id)∗ S[−1], Y

Y

where Lp∗e OYe (h−H) and Rp∗e OYe (−h) denote the corresponding left and right Y Y mutations. The fact that M ∈ Db (S × Y, p∗1 α) is actually a locally free sheaf follows from the fact that M|x×Y ∼ = Mx is locally free, for all x ∈ S. This was observed above. Acknowledgements. Parts of this paper were written while the three authors were visiting the University of Bonn and the University of Barcelona whose warm hospitality is gratefully acknowledged. We also thank the American Institute of Mathematics for sponsoring the workshop “Brauer groups and obstruction problems: moduli spaces and arithmetic” held February 25 to March 1, 2013, in Palo Alto, California, where parts of this paper were discussed. It is a pleasure to thank Nick Addington, Asher Auel, Marcello Bernardara, Robin Hartshorne, Daniel Huybrechts, Nathan Ilten, Sukhendu Mehrotra, Nicolas Perrin, Antonio Rapagnetta, and Pawel Sosna for very useful conversations and comments.

References [1] N. Addington, The Derived Category of the Intersection of Four Quadrics, PhD Thesis, Madison. [2] N. Addington and R. Thomas, Hodge theory and derived categories of cubic fourfolds, Duke Math. J. 163 (2014), 1885–1927. [3] M. Bernardara, E. Macr`ı, S. Mehrotra, P. Stellari, A categorical invariant for cubic threefolds, Adv. Math. 229 (2012), 770–803. [4] A. C˘ ald˘ araru, Derived categories of twisted sheaves on Calabi-Yau manifolds, PhD-Thesis, Cornell University (2000). [5] M. Casanellas, R. Hartshorne, ACM bundles on cubic surfaces, J. Eur. Math. Soc. 13 (2011), 709–731. [6] M. Casanellas, R. Hartshorne, F. Gleiss, F.-O. Schreyer,Stable Ulrich bundles, Internat. J. Math. 23 (2012), 1250083.

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[7] D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), 35–64. [8] D. Huybrechts, Fourier–Mukai transforms in algebraic geometry, Oxford Mathematical Monographs (2006). [9] D. Huybrechts, M. Lehn, The geometry of moduli spaces of sheaves, Cambridge Mathematical Library, Cambridge University Press, Cambridge, second edition, (2010). [10] A. Kuznetsov, Derived categories of cubic fourfolds, in: Cohomological and geometric approaches to rationality problems, 219–243, Progr. Math. 282, Birkh¨ auser Boston, Boston (2010). [11] A. Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics, Adv. Math. 218 (2008), 1340–1369. ´ [12] A. Kuznetsov, Homological projective duality, Publ. Math. Inst. Hautes Etudes Sci. 105 (2007), 157–220. [13] A. Kuznetsov, Instanton bundles on Fano threefolds, Cent. Eur. J. Math. 10 (2012), 1198–1231. [14] A. Kuznetsov, D. Markushevich, Symplectic structures on moduli spaces of sheaves via the Atiyah class, J. Geom. Phys. 59 (2009), 843–860. [15] M. Lahoz, E. Macr`ı, P. Stellari, Arithmetically Cohen-Macaulay bundles on cubic threefolds, Algebr. Geom. 2 (2015), 231–269. [16] E. Macr`ı, P. Stellari, Fano varieties of cubic fourfolds containing a plane, Math. Ann. 354 (2012), 1147–1176. ´ [17] J.S. Milne, Etale Cohomology, Princeton Mathematical Series 33, Princeton University Press (1980). [18] D. Orlov, Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Russian Acad. Sci. Izv. Math. 41 (1993), 133–141. [19] Y. Yoshino, Cohen–Macaulay modules over Cohen–Macaulay rings, London Math. Soc. Lecture Note Ser. 146, Cambridge Univ. Press, Cambridge (1990).

Mart´ı Lahoz Institut de Math´ematiques de Jussieu, Paris Rive Gauche (UMR 7586) Universit´e Paris Diderot / Universit´e Pierre et Marie Curie Bˆ atiment Sophie Germain, Case 7012, 75205 Paris Cedex 13 France e-mail: [email protected] Emanuele Macr`ı Department of Mathematics The Ohio State University 231 W 18th Avenue, Columbus, OH 43210 USA Current address: Department of Mathematics Northeastern University 360 Huntington Avenue, Boston, MA 02115 USA e-mail: [email protected]

ACM Bundles on Cubic Fourfolds Containing a Plane Paolo Stellari Dipartimento di Matematica “F. Enriques” Universit` a degli Studi di Milano Via Cesare Saldini 50, 20133 Milano Italy e-mail: [email protected]

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Brauer Groups on K3 Surfaces and Arithmetic Applications Kelly McKinnie, Justin Sawon, Sho Tanimoto and Anthony V´arilly-Alvarado Abstract. For a prime p, we study subgroups of order p of the Brauer group Br(S) of a general complex polarized K3 surface of degree 2d, generalizing earlier work of van Geemen. These groups correspond to sublattices of index p of the transcendental lattice TS of S; we classify these lattices up to isomorphism using Nikulin’s discriminant form technique. We then study geometric realizations of p-torsion Brauer elements as Brauer-Severi varieties in a few cases via projective duality. We use one of these constructions for an arithmetic application, giving new kinds of counter-examples to weak approximation on K3 surfaces of degree two, accounted for by transcendental Brauer-Manin obstructions. Mathematics Subject Classification (2010). 14J28, 14G05, 14F22. Keywords. K3 surfaces, Brauer groups, projective duality, special cubic fourfolds, weak approximation.

1. Introduction Let S be a smooth projective geometrically integral variety over a number field k; write A for the ring of adeles of k. Assume that S(A) 6= ∅. It is wellknown that elements of the Brauer group Br(S) := H2et (S, Gm ) can obstruct the existence of k-rational points of S [31]; in such cases we say there is a Brauer-Manin obstruction to the Hasse principle. Brauer elements can also explain why sometimes the image of the diagonal map S(k) ,→ S(A) fails to be dense; in such cases we say there is a Brauer-Manin obstruction to weak approximation (see Section 5.2 for more details). In order to show that a particular Brauer element obstructs the Hasse principle or weak approximation, one often needs a geometric realization of the Brauer element, especially if the element remains non-trivial after extension of scalars to an algebraic closure (such elements are known as transcendental elements). In [46] van Geemen studied transcendental 2-torsion © Springer International Publishing AG 2017 A. Auel (eds.) et al., Brauer Groups and Obstruction Problems, Progress in Mathematics 320, DOI 10.1007/978-3-319-46852-5_9

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Brauer elements on general complex K3 surfaces S of degree two. Using lattice-theoretic methods, he gave a classification into three cases, and described geometric realizations of the Brauer elements. In the first case, S is the double cover of P2 branched over the sextic discriminant curve of the quadric surface fibration on a cubic fourfold containing a plane; Hassett, V´arilly-Alvarado, and Varilly [12] showed that the Brauer element in this example can obstruct weak approximation. The second case involves a double cover of P2 × P2 branched over a hypersurface of bidegree (2, 2); Hassett and V´arilly-Alvarado [13] showed that the Brauer element in this example can obstruct the Hasse principle. In the third case, S is the double cover of P2 branched over the sextic discriminant curve of the net of quadrics defining a K3 surface of degree eight in P5 ; transcendental Brauer elements of this type have not yet been used for arithmetic applications. The goal of this paper is to extend this earlier work in several directions: Classification of order p subgroups of general K3 surfaces. Let S be a general complex polarized K3 surface of degree 2d, and write TS = NS(S)⊥ ⊂ H2 (S, Z) for its transcendental lattice. Let p be an odd prime. To classify subgroups of order p in Br(S) we use the correspondence {subgroups of order p in Br(S)} ←→ {sublattices of index p in TS } furnished by the exponential sequence and elementary lattice-theoretic properties of H2 (S, Z) (see Section 2 of [46]). We apply Nikulin’s discriminant form technique [38] to classify sublattices of index p in TS up to isomorphism, and we count the number of lattices in each isomorphism class. This is the content of Sections 2.1–2.5. Our main result in this direction is Theorem 9, showing there are three or four classes of p-torsion subgroups of Br(S), according to whether p - d or not, respectively. We expect that each class of subgroups is associated to a geometric construction for p-torsion Brauer elements, like in the case of 2-torsion. Indeed, the lattice theory already suggests a strong connection between certain p-torsion classes on K3 surfaces of degree two and higher degree K3 surfaces or special cubic fourfolds. We explore these connections in Sections 2.6 and 2.7 following Mukai [35] and building on Hassett [10], respectively. Geometric realization of Mukai dualities. Having classified p-torsion elements of the Brauer group, we next look for geometric realizations as Brauer-Severi varieties. In the third case of van Geemen’s analysis of 2-torsion Brauer elements on degree two K3 surfaces, the Brauer element on S comes from the Fano variety of maximal isotropic subspaces inside the quadrics defining the associated degree eight K3 surface X. We describe how S can also be interpreted as a Mukai moduli space of stable sheaves on X (Lemma 12) and how the Brauer element is the obstruction to the fineness of this moduli space (Lemma 15). This example then admits a vast generalization: given a K3 surface X of degree 2dp2 , there exists a ‘Mukai dual’ K3 surface S given by a moduli space of stable sheaves on X, and a p-torsion Brauer element on S obstructing the existence of a universal sheaf. In some low degree cases,

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including the case d = 1 and p = 2 above, this Mukai duality can be realized as projective duality, and the Brauer element can be realized geometrically as a Brauer-Severi variety. In Section 3.3 we describe the case d = 1 and p = 3, showing that the 3-torsion Brauer element on S comes from the Fano variety of cubic surface scrolls inside a certain net of Fano fourfolds containing X. In Section 3.4 we describe the case d = 2 and p = 2 as an instance of projective duality, though we leave as an open question the geometric realization of the resulting 2-torsion Brauer element. The Mukai dualities we discuss have been studied before [35, 22, 23, 20, 21], although we hope that our exposition will be useful to arithmetic geometers. Explicit obstructions to weak approximation. Returning to the third case in van Geemen’s classification of 2-torsion Brauer elements on K3 surfaces of degree two, we construct an explicit K3 surface S of degree two, together with a transcendental 2-torsion element α ∈ Br(S) that obstructs weak approximation; see Theorem 27. We are able to compute an obstruction by interpreting α as the even Clifford algebra of the discriminant cover S → P2 of a given net of quadrics in P5 , following Auel, Bernardara and Bolognesi [2]. We use elementary properties of Clifford algebras to represent α as a product of two quaternion algebras over the function field k(S). Along the way we prove a curious result (Corollary 25), which explains why it may be difficult to use elements of the form α to obstruct the Hasse principle (see Section 5.5 as well). It would be interesting to use the construction of Section 3.3 to build a K3 surface S (of degree 2) with an obstruction to the Hasse principle arising from a 3-torsion in Br(S). At present, no examples like this are known to exists, and recent work of Ieronymou and Skorobogatov naturally raises this problem; see the discussion after Corollary 1.3 of [17]. Acknowledgements. The authors thank Asher Auel, Brendan Hassett, Danny Krashen, Alexander Kuznetsov, Sukhendu Mehrotra, and Ronald van Luijk for several discussions on this work. We are grateful to the referees for their careful review of the manuscript; their suggestions greatly improved the exposition of the paper. We also thank the American Institute for Mathematics, Palo Alto, for hosting the workshop “Brauer groups and obstruction problems: moduli spaces and arithmetic” and for travel funding. The second and fourth authors were supported by the NSF under grant numbers DMS-1206309 and DMS-1103659/DMS-1352291, respectively.

2. Lattice gymnastics Let S be a complex, projective K3 surface with N´eron-Severi group NS(S) isomorphic to Zh. The intersection form makes the singular cohomology group H2 (S, Z) into a lattice. Write TS = NS(S)⊥ for the transcendental lattice of S, and let p be a prime number. By §2.1 of [46], a nontrivial element α ∈ Br(S)[p] gives rise to a surjective homomorphism α : TS → Z/pZ. The

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kernel of this homomorphism is a sublattice of TS of index p. Conversely, a sublattice Γ ⊂ TS of index p determines a subgroup hαi ⊆ Br(S) of order p. Accordingly, we write Γhαi = Γ for such a sublattice. In §9 of [46], van Geemen classifies the isomorphism types of sublattices of index 2 in TS . He shows that there are three isomorphism types, and for each type he offers an auxiliary variety, together with a geometric construction that takes the auxiliary variety and recovers the original K3 surface S together with a Brauer-Severi bundle over S corresponding to the unique nontrivial element of hαi ⊂ Br(S)[2]. One might hope that for odd p, each isomorphism type of Γhαi has an associated geometric construction that could be used for arithmetic applications. Thus, it is of interest to classify sublattices Γhαi ⊂ TS of odd prime index p up to isomorphism. Our strategy is to generalize the proofs of Propositions 3.3 and 9.2 in [46]. 2.1. Set-up Let (Γ, (· , ·)) be a lattice, i.e., a free Z-module of finite rank, together with a nondegenerate integral symmetric bilinear form (· , ·). We write O(Γ) for the group of orthogonal transformations of Γ. Denote by Γ∗ the dual lattice Hom(Γ, Z); the bilinear form on Γ can be extended Q-bilinearly to Γ∗ . We embed Γ ⊆ Γ∗ via the map γ 7→ [φγ : Γ → Z, δ 7→ (δ, γ)]. The discriminant group d(Γ) of Γ is Γ∗ /Γ; it is a finite abelian group whose order is the discriminant of Γ. A lattice is unimodular if its discriminant group is trivial. If Γ is an even lattice, i.e., (γ, γ) ∈ 2Z for all γ ∈ Γ, then there is a quadratic form q : d(Γ) → Q/2Z

x + Γ 7→ (x, x) mod 2Z,

called the discriminant form of Γ. One also obtains a symmetric bilinear form b : d(Γ) × d(Γ) → Q/Z, which is characterized by the identity q(x + y) − q(x) − q(y) ≡ 2b(x, y) mod 2Z. Nikulin showed in Corollary 1.13.3 of [38] that an even indefinite lattice whose rank exceeds (by at least 2) the minimal number of generators of d(Γ) is determined by its rank, signature and discriminant quadratic form. We will use this fact in what follows, without explicitly mentioning it every time. We write d(Γ)p for the p-Sylow subgroup of d(Γ), and qp for the restriction of q to this subgroup. By Proposition 1.2.2 of [38] there is an orthogonal L decomposition q = p qp as p runs over prime numbers dividing the order of d(Γ). Let S be a complex projective K3 surface. By §1 of [30], we can write H 2 (S, Z) ∼ = U1 ⊕ U2 ⊕ U3 ⊕ E8 (−1)2 =: ΛK3 ,

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where the Ui are hyperbolic planes (i.e., even unimodular lattices of signature (1, 1)), and E8 (−1) is the unique negative definite even unimodular lattice of rank eight. In particular, ΛK3 is even, unimodular, and has signature (3, 19). If NS(S) = Zh, with h2 = 2d > 0, then by Theorem 1.1.2 of [38], the inclusion NS(S) = Zh ,→ ΛK3 is unique up to isometry, and therefore we may assume that NS(S) = Zh ∼ = Z(1, d) ,→ U1 ,→ U1 ⊕ Λ0 = ΛK3 , where Λ0 = U2 ⊕ U3 ⊕ E8 (−1)2 . Let v = (1, −d) ∈ U1 , so that v 2 = −2d. Then TS ∼ = Zv ⊕ Λ0 ∼ = h−2di ⊕ Λ0 . 2.2. Discriminant groups of p-torsion Brauer classes We begin by analyzing the homomorphism α : TS → Z/pZ associated to a nonzero element α ∈ Br(S)[p]. Since the lattice Λ0 is unimodular, and hence self-dual, there is a λα ∈ Λ0 , whose class in Λ0 /pΛ0 is uniquely determined, such that the homomorphism α can be expressed as α : TS zv + λ0

→ Z/pZ, 7 → ziα + hλα , λ0 i mod p,

for some integer iα , which we may assume is in the range 0 ≤ iα ≤ p − 1. If −1 iα 6= 0, we write i−1 α for the inverse of iα modulo p in the range 1 ≤ iα ≤ p−1. −1 Since α and iα α have the same kernel Γhαi , and since the kernel determines the subgroup hαi ⊆ Br(S)[p], replacing α with i−1 α α, we may assume that iα = 1. Define cα ∈ Z by λ2α = −2cα ; the class of cα modulo p is uniquely determined by α. The lattice Λ0 is even, unimodular and has signature (2, 18). Applying Theorem 1.1.2 of [38], we conclude that any embedding λα Z ,→ Λ0 is unique up to isometry. Therefore, without loss of generality, we assume that λα = (1, −cα ) ∈ U2 ,→ U2 ⊕ U3 ⊕ E8 (−1)2 = Λ0 . Let Λ00 = U3 ⊕ E8 (−1)2 ⊂ Λ0 , and let Γhαi = ker(α). We compute Γhαi = {zv + (a, b) + λ00 | z ∈ Z, (a, b) ∈ U2 , λ00 ∈ Λ00 , and ziα − acα + b ≡ 0 mod p} = {zv + (a, kp − ziα + acα ) + λ00 | z, a, k ∈ Z, λ00 ∈ Λ00 } = {z(v + (0, −iα )) + a(1, cα ) + k(0, p) + λ00 | z, a, k ∈ Z, λ00 ∈ Λ00 }. Hence, for fixed d and p, the lattice Γhαi (and by extension, its discriminant form), is determined by the values iα and cα modulo p. Let Mα be the rank three lattice with Gram matrix   −2d −iα 0  −iα 2cα p  . 0 p 0

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Our computation shows that Γhαi ∼ = Mα ⊕ Λ00 . Unimodularity of Λ00 implies that Γhαi and Mα have isomorphic discriminant groups and isometric discriminant forms. When we need to, we will set Γhαi = Γiα ,cα . Theorem 1. Let S be a complex projective K3 surface of degree 2d such that NS(S) ∼ = Zh, and write TS := NS(S)⊥ ⊂ H2 (S, Z) for its transcendental lattice. Let p be a prime number, and let α ∈ Br(S)[p] be nontrivial, with associated index p sublattice Γhαi = Γiα ,cα ⊂ TS . 1. If iα = 0, then 
Z/2dZ ⊕ Z/p2 Z if p - cα and p is odd, ∼ d Γhαi = Z/2dZ ⊕ Z/pZ ⊕ Z/pZ if p | cα or p = 2. 2. If iα = 1, then Z/2dp2 Z if p - (1 + 4cα d), Z/2dpZ ⊕ Z/pZ if p | (1 + 4cα d).
Proof. Each element in d Γhαi ∼ = d(Mα ) = Mα∗ /Mα is represented by a γ ∈ γ0 0 Mα∗ satisfying 2dp2 γ ∈ Mα . In other words, we can write γ = 2dp 2 , with γ ∈ 0 Mα . Therefore, we represent elements of d(Mα ) as triples γ = (A, B, C) ∈ Mα such that ((A, B, C), (x, y, z)) ∈ 2dp2 Z for all x, y, z ∈ Z. Taking (x, y, z) to be (1, 0, 0), (0, 1, 0) and (0, 0, 1), in turn, we see this happens if and only if there exist some k0 , k1 , k2 ∈ Z such that d Γhαi




∼ =



A = −p(iα k2 +pk0 ), B = 2dpk2 , C = 2dpk1 − i2α k2 − piα k0 − 4cα dk2 . Case 1: iα = 0. In this case the equations for A, B, C reduce to A = −p2 k0 , B = 2dpk2 , C = 2dpk1 − 4cα dk2 , for some k0 , k1 , k2 ∈ Z. In particular, the triples v1 :=

(p2 , 0, 0) (0, 2dp, −4cα d) (0, 0, 2dp) , v2 := and v3 := 2dp2 2dp2 2dp2

represent non-trivial elements of d(Mα ). If p is odd, then the elements v1 and v2 generate subgroups of respective orders 2d and p2 / gcd(cα , p), and these subgroups intersect trivially. If p - cα , this shows that d(Mα ) ∼ = Z/2dZ ⊕ Z/p2 Z. If p | cα , then v1 , v2 and v3 are independent elements that generate subgroups of respective orders 2d, p and p, showing that d(Mα )
∼ = Z/2dZ ⊕ Z/pZ ⊕ Z/pZ. If p = 2, then v2 has order two, and thus d Γhαi ∼ = Z/2dZ ⊕ Z/2Z ⊕ Z/2Z. Case 2: iα = 1. Let k2 = −1, and k0 = k1 = 0. Then v4 :=

(p, −2dp, 1 + 4cα d) 2dp2

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is an element of d(Mα ). Because of its first component, v4 generates a subgroup of order divisible by 2dp, hence d(Mα ) is isomorphic to either Z/2dp2 Z or Z/2dpZ ⊕ Z/pZ. Therefore, if p - (1 + 4cα d), then (2dp)v4 is not trivial in d(Mα ), and hence d(Mα ) ∼ = Z/2dp2 Z. On the other hand, if p | (1 + 4cα d), then v4 and 1 (0, 0, 2dp) v5 := 2dp2 generate subgroups, of order 2dp and p respectively, which intersect trivially. Therefore, d(Mα ) ∼  = Z/2dpZ ⊕ Z/pZ. 2.3. Isomorphism classes of lattices Our next task is to determine when the lattices appearing in Theorem 1 are isomorphic. We do this by comparing their discriminant forms. We begin by comparing lattices with cylic discriminant groups. Throughout this section, we retain the notation of Theorem 1. Proposition 2. Let
p be an odd prime. Consider the lattices Γhαi in Theorem 1 for which d Γhαi
∼ = Z/2dp2 Z is a cyclic group, generated by an element v. Write q : d Γhαi → Q/2Z for the discriminant form of Γhαi . (i) If p | d, then all such lattices are isomorphic. (ii) If p - d, then there are two isomorphism classes of lattices. The isomorphism type of Γhαi depends only on whether −2dp2 q(v) is a square modulo p or not. Remark. The analogous proposition when p = 2 is handled by van Geemen in Proposition 9.2 of [46]. Proof. Suppose first that p | d. It follows from Theorem 1 and its proof that 1 iα = 1, p - (1 + 4cα d), and v4 = 2dp 2 (p, −2dp, 1 + 4cα d) is a generator for
∼ d(Mα ) = d Γhαi . Then 1 (1 + 4cα d), 2dp2 and Γhα0 i = Γ1,cα0 are isomorphic if there

q(v4 ) = − and so two lattices Γhαi = Γ1,cα
× exists x ∈ Z/2dp2 Z such that

(1 + 4cα d) ≡ x2 (1 + 4cα0 d) mod m, 2

(1)

where m = 4dp . Such an x exists if and only if (1) has a solution when m = p. Indeed, if the latter congruence has a solution, then so does (1) for all m = pn with n > 1, by Hensel’s lemma, and if q 6= p is an odd prime dividing d, then (1) has a solution for all m = q n with n > 0, again by Hensel’s lemma (the case n = 1 being trivial since q | d). Finally, if 2 - d, then (1) clearly has a solution when m = 4, and if on the other hand 2 | d, then (1) has a solution when m = 8, and thus for any m = 2n with n > 2, by another application of Hensel’s lemma. Putting all this together using the Chinese remainder theorem, we obtain a solution for (1) for m = 4dp2 , as claimed. Since p | d, it is clear that (1) has a solution when m = p. This proves (i).

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Next, assume that p - d, and let Γhαi = Γiα ,cα and Γhα0 i = Γiα0 ,cα0 be two lattices with cyclic discriminant group. By Theorem 1 and its proof, we may assume that the discriminant group of Γhαi is generated by either v4 or v1 + v2 =

(p2 , 2dp, −4cα d) , 2dp2

and likewise for Γhα0 i . We computed q(v4 ) above; now note that q(v1 + v2 ) = −

1 (p2 + 4cα d). 2dp2

Thus, we see that −2dp2 q(v1 + v2 ) and −2dp2 q(v
4 ) are integers not divisible by p. Write v and v 0 for the generators of d Γhαi and d(Γhα0 i ), respectively. Then Γhαi and Γhα0 i are isomorphic if and only if the congruence − 2dp2 q(v) ≡ −2dp2 q(v 0 )x2 mod m

(2)

2

has a solution when m = 4dp . Arguing as in (i), this is equivalent to (2) having a solution when m = p.  Suppose next that p - d and that the discriminant group of a lattice Γhαi is not cyclic. By Theorem 1, we have d(Γhαi ) ∼ = Z/2dZ ⊕ Z/pZ ⊕ Z/pZ, and there are two possible lattices with this discriminant group, characterized by the value of iα and cα . These two lattices are in fact isomorphic, as the following lemma shows. Lemma 3. Let p be an odd prime such that p - d. There is a
unique lattice Γhαi , up to isomorphism, whose discriminant group d Γhαi is not cyclic.
∼ Moreover, in this case we have d Γhαi = Z/2dZ ⊕ Z/pZ ⊕ Z/pZ.
Proof. We have already shown that d Γhαi ∼ = Z/2dZ ⊕ Z/pZ ⊕ Z/pZ. Let Γ be the lattice Γhαi = Γ0,0 determined by iα = 0 and cα = 0, and let Γ0 be the lattice Γhα0 i = Γiα0 ,cα0 determined by iα0 = 1 and cα0 with p | (1 + 4cα0 d). Write q and q 0 for their respective discriminant quadratic forms. We show that q and q 0 are isometric. Using the notation of the proof of Theorem 1, we may assume that d(Γ) = hv1 i ⊕ hv2 , v3 i and d(Γ0 ) = hpv4 i ⊕ h2dv4 , v5 i. Recall that v1 and pv4 have order 2d in their respective discriminant groups, while v2 , v3 , 2dv4 and v5 each have order p. By Proposition 1.2.1 of [38], we know that q = q ⊕ q and q 0 = q 0 ⊕ q0 0 Z/2dZ

d(Γ)p

Z/2dZ

d(Γ )p

0

Thus, to prove that q and q are isometric, it suffices to exhibit an x ∈ (Z/2dZ)× such that q(xv1 ) ≡ q 0 (pv4 )

(mod 2Z),

∼

and an isomorphism φ : d(Γ0 )p − → d(Γ)p of Z/pZ-vector spaces such that q(φ(v)) ≡ q 0 (v)

(mod 2Z)

for all v ∈ d (Γ0 )p .

(3)

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To prove x exists, argue using Hensel’s lemma and the Chinese remainder theorem, as in Proposition 2. Using {v2 , v3 } and {2dv4 , v5 } as bases for d(Γ)p and d(Γ0 )p , one can check that the transformation ! 1 0 φ= α0 d) − d(1+4c −2d p is a witness to (3), where cα0 ∈ {0, . . . , p − 1} is the constant for the lattice Γ0 such that p | (1 + 4cα0 d).  Finally, we treat the case when p | d. It cannot also be the case that p | (1 + 4cα d), so the lattices in Theorem 1 with iα = 1 and p | (1 + 4cα d) cannot occur. This leaves three possible distinct discriminant groups for lattices Γhαi . First, isomorphism classes of lattices with cyclic discriminant group are handled in Proposition 2. Second, there is only one lattice, up to isomorphism, with discriminant group Z/2dZ ⊕ Z/pZ ⊕ Z/pZ, characterized by iα = 0 and cα = 0. Thus, it remains to understand when two lattices with discriminant group Z/2dZ ⊕ Z/p2 Z are isomorphic to each other. Lemma 4. Let p be an odd prime such that p | d. Then there are two lattices
2 Γhαi , up to isomorphism, with discriminant group d Γhαi ∼ Z/2dZ⊕Z/p Z. = Proof. We show that two lattices Γhαi and Γhα0 i with discriminant group Z/2dZ ⊕ Z/p2 Z are isomorphic if and only if cα /cα0 is a quadratic residue modulo p. Write d = pe · d0 , where p - d0 , so that Z/2dZ ⊕ Z/p2 Z ∼ = Z/2d0 Z ⊕ Z/pe Z ⊕ Z/p2 Z. Using the notation of the proof of Theorem 1, we may assume that and d(Γhα0 i ) = hpe v1 i ⊕ h2d0 v1 , v20 i, | {z }

d(Γhαi ) = hpe v1 i ⊕ h2d0 v1 , v2 i | {z } d(Γhαi )p

d(Γhα0 i )p

where 2dp2 v20 = (0, 2dp, −4cα0 d). Recall that pe v1 and 2d0 v1 have orders 2d0 and pe , respectively, while v2 and v20 , each have order p2 . As in the proof of Lemma 3, the quadratic forms qα and qα0 associated to our lattices are isomorphic if and only if there is an φ ∈ Aut(Z/pe Z ⊕ Z/p2 Z) such that (qα )p (φ(v)) ≡ (qα0 )p (v)

(mod 2Z)

for all v ∈ d(Γhα0 i )p .

The symmetric matrices associated to (qα )p and (qα0 )p are respectively equal to ! ! 0 0 − 2d 0 − 2d 0 pe pe and . α 0 − 2c 0 − 2cpα2 0 p2 First, suppose that cα /cα0 is a quadratic residue modulo p. Then we can take  φ=

 1 0 , 0 y

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where y ∈ Z/p2 Z satisfies −

2cα 2cα0 y 2 ≡ − p2 p2

(mod 2Z).

Such a y exists by Hensel’s lemma and because cα /cα0 is a quadratic residue modulo p. Now suppose that (qα )p and (qα0 )p are isometric. Then their associated bilinear forms must be isomorphic, and there is an A ∈ Aut(Z/pe Z ⊕ Z/p2 Z), which can be represented by a 2 × 2 matrix, such that ! ! −2d0 −2d0 0 0 e e p p (mod Z). (4) ·A≡ At · −2cα0 −2cα 0 0 p2 p2   a 0 If e ≥ 3, then we may assume that A has the form , in which case b c the (2, 2) entry in the congruence (4) reads −

2cα 2cα0 c2 ≡ − p2 p2

(mod Z),

(5)

and we conclude that cα /cα0 is a quadratic residue modulo p. If e = 2, then A ∈ GL2 (Z/p2 Z). Multiplying (4) by p2 and taking determinants we arrive at thesame  conclusion. Finally, if e = 1, then we may assume that A has the a b form , in which case the (2, 2) entry in the congruence (4) is again 0 c given by (5), and cα /cα0 is a quadratic residue modulo p.  2.4. Counting lattices We continue using the notation of Theorem 1; in particular, S denotes a complex projective K3 surface with NS(S) ∼ = Zh. The purpose of this section is to count, for each nontrivial hαi ⊂ Br(S)[p], the number of lattices in each isomorphism class of Γhαi ⊂ TS . Since Γhαi ⊆ TS has index p, we know that pTS ⊆ Γhαi ⊆ TS and thus Hα := Γhαi /pTS ⊆ TS /pTS ∼ = F21 p . We may consider Hα as a hyperplane in F21 p . Conversely, to every hyperplane in TS /pTS , we may associate an index p sublattice Γhαi of TS . Thus, the projective space P ((TS /pTS )∗ ) parametrizes index p-sublattices of TS . Using the identification TS ∼ = Zv ⊕ Λ0 , and setting v ∗ = −v/2d, the intersection form on TS allows us to identify P ((TS /pTS )∗ ) ∼ = P (Fp · v ∗ ⊕ Λ0 /pΛ0 ) . The set of index p lattices Γhαi that have iα = 0 are then identified with P19 (Fp ) = P (Λ0 /pΛ0 ) ⊆ P (Fp · v ∗ ⊕ Λ0 /pΛ0 )

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while the set of lattices with iα = 1 can be identified with the distinguished open affine A20 (Fp ) = Λ0 /pΛ0 ⊆ P (Fp · v ∗ ⊕ Λ0 /pΛ0 ) λα 7→ [v ∗ + λα ]. Define the quadratic form Q : Λ0 /pΛ0 → Fp hλ, λi mod p. 2 A lattice Γhαi is determined by the quantities iα and cα . Recall that −2cα = hλα , λα i, so for example, lattices with iα = 1 and a prescribed cα correspond to Fp -points on the affine quadric {Q(x) = cα } ⊂ A20 . The following wellknown lemma will help us count the required points on quadrics. We include a proof here for completeness. λ 7→ −

Lemma 5. Let p be an odd prime, and let Q be a nondegenerate, homogeneous quadratic form in 2n variables over Fp . Assume that (−1)n disc(Q) ∈ F×2 p . Write f (n) for the number of zeroes of Q (including the trivial zero). For 0 6= i ∈ Fp , let gi (n) denote the number of solutions to the equation Q = i. Then f (n) = pn−1 (pn + p − 1) and gi (n) = pn−1 (pn − 1). In particular, g(n) := gi (n) is independent of i. Proof. The hypothesis on p and Q imply that Q is isometric to the form Q∼ = x1 x2 + · · · + x2n−1 x2n . We then note that f (n) satisfies the recurrence relation f (n) = f (n − 1)(2p − 1) + (p2n−2 − f (n − 1))(p − 1), because, informally, f (n) = #(zeroes of x1 x2 + · · · + x2n−3 x2n−2 ) · #(zeroes of x2n−1 x2n ) + #(nonzero values of x1 x2 + · · · x2n−3 x2n−2 ) · #(zeroes of x2n−1 x2n − i)), F× p.

where i ∈ The initial condition f (1) = 2p − 1 then allows us to determine f (n), and we obtain the claimed quantity. The function gi (n) obeys the same recurrence relation, but with initial condition gi (1) = p − 1.  We begin by counting lattices in isomorphism classes with cyclic discriminant group. The following proposition is a complement to Proposition 2. Proposition 6. Let
p be an odd prime. Consider the lattices Γhαi in Theorem 1 for which d Γhαi
∼ = Z/2dp2 Z is a cyclic group, generated by an element v. Write q : d Γhαi → Q/2Z for the discriminant form of Γhαi . (i) If p | d, then there are p20 such lattices, all isomorphic to each other.

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(ii) If p  d, then there are two isomorphism classes of lattices. The iso2 is a square morphism class corresponding  to the case where −2dp q(v)  10 1 10 modulo p has 2 p p + 1 lattices. The other class has 12 p10 p10 − 1 lattices. 2, Proof. We have discussed the isomorphism classes of Γα in Proposition  so we focus on the lattice counts. If p | d, then p  (1 + 4cα d). Hence d Γα is cyclic if and only if iα = 1. Lattices with iα = 1 are in one-to-one correspondence with points in the distinguished open affine A20 (Fp ) = Λ /pΛ ⊆ P (Fp · v ∗ ⊕ Λ /pΛ ). There are thus p20 such lattices. Next, suppose that p  d. Let us call the two isomorphism classes in part (ii) of the proposition Cs and Cn 1 . Let Cs,iα = {Γα ∈ Cs | Γα = Γiα ,cα for some cα } and similarly  for Cn,iα . If iα = 0, then d Γα is cyclic only if p  cα . In this case d Γα is generated by v1 + v2 and −2dp2 q(v1 + v2 ) = p2 + 4cα d. This is a square modulo p if and only if cα d is a square modulo p. Note that ×2 #{x ∈ F× p | xd ∈ Fp } = (p−1)/2

and

#{x ∈ F× / F×2 p | xd ∈ p } = (p−1)/2.

In particular, of the p − 1 non-zero cα ’s mod p, there are (p − 1)/2 such that cα d is a square (mod p). Therefore, using the notation of Lemma 5, we have #{λ ∈ (A20 \ {0})(Fp ) | Q(λ) ∈ F×2 p }=

p−1 · g(10). 2

Since iα = 0, the λ are in P19 (Fp ) and we must divide our count by p − 1 to obtain p9 (p10 − 1) p − 1 g(10) · = . #Cs,0 = 2 p−1 2 p9 (p10 − 1) The same calculation shows #Cn,0 = . 2     If iα = 1, then d Γα is cyclic only if p  (1+4cα d). In this case d Γα is generated by v4 and −2dp2 q(v4 ) = 1 + 4cα d. Since 1 + 4cα d ≡ 1 + 4cα d (mod p) if and only if cα ≡ cα (mod p), we see that as sets Fp = {1 + 4cα d|cα ∈ Fp }. Therefore #{cα | 0 = 1 + 4cα d = x2 for all x ∈ F× p}=

p−1 . 2

Since cα = 0 makes 1 + 4cα d a square modulo p, we see that   p−1 1 − 1 g(10) = p9 (p11 − p10 + p + 1), #Cs,1 = f (10) + 2 2 1 p−1 g(10) = p9 (p10 − 1)(p − 1). #Cn,1 = 2 2 1 Here C stands for cyclic, s stands for −2dp2 q(v) is a square modulo p and n stand for −2dp2 q(v) is a non-square modulo p.

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Finally,
#Cs = #Cs,0 + #Cs,1 = 12 p10 p10 + 1 ,
#Cn = #Cn,0 + #Cn,1 = 12 p10 p10 − 1 . Proposition 7. Suppose that p is an odd prime with p - d. There are

 p20 − 1 p−1

lattices Γhαi in Theorem 1 with noncyclic discriminant group. 21

−1 Proof. This is clear, since there are a total of pp−1 lattices Γhαi and the ones 20 with cyclic group account for a total of p lattices. 

Proposition 8. Suppose that p is an odd prime with p | d. Consider lattices Γhαi as in Theorem 1. There are •

1 9 10 2 p (p

− 1) lattices with iα = 0, p - cα and cα is a quadratic residue modulo p, • 12 p9 (p10 − 1) lattices with iα = 0, p - cα and cα is a quadratic nonresidue modulo p, 9 10 −1) lattices with iα = 0 and p | cα , • (p +1)(p p−1 • p20 lattices with iα = 1. Proof. The first two types of lattices can be counted the same way we counted Cs,0 and Cn,0 in Proposition 6. The third type corresponds to Fp -points of a smooth quadric in P19 , of which there are f (10)−1 p−1 . The fourth type were counted in Proposition 6.  2.5. Summary Theorem 9. Let S be a complex projective K3 surface of degree 2d with N´eronSeveri group NS(S) isomorphic to Zh, and write TS := hhi⊥ ⊂ H2 (S, Z) for its transcendental lattice. Let p be an odd prime, and let α ∈ Br(S)[p] be nontrivial, with associated index p sublattice Γhαi = Γiα ,cα ⊂ TS . Write
q : d Γhαi → Q/2Z for the discriminant form of Γhαi . 1. If p - d, then there are three isomorphism classes of lattices Γα . They are classified in Table 1. 2. If p | d, then there are four isomorphism classes of lattices Γα . They are classified in Table 2. 2.6. Lattice theory for Mukai dual K3 surfaces Let S be a complex K3 surface of degree two with NS(S) ∼ = Zh. In §9 of [46], van Geemen showed that sublattices of index two Γhαi of TS in a particular isomorphism class naturally give rise to K3 surfaces of degree eight via a primitive embedding Γhαi ,→ ΛK3 , using the surjectivity of the period map for K3 surfaces. In this section, we explain a well-known generalization of this framework due to Mukai [35]. Using the notation of Section 2.1, we let S denote a complex projective K3 surface of degree 2d with NS(S) ∼ = Zh, and we fix a primitive vector v ∈ H2 (S, Z) such that v 2 = −2d and TS ∼ = h−2di ⊕ Λ0 . Then Γ := h−2dp2 i ⊕ Λ0

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d Γhαi




Distinguishing Feature Number of Γhαi

Z/2dp2 Z = hvi

−2dp2 q(v) ≡ 2 mod p

1 10 10 p (p + 1) 2

Z/2dp2 Z = hvi

−2dp2 q(v) 6≡ 2 mod p

1 10 10 p (p − 1) 2

Z/2dZ ⊕ Z/pZ ⊕ Z/pZ Table 1. Sublattices Γhαi = Γiα ,cα symbol 2 stands for “a square”.

d Γhαi




p20 − 1 p−1 ⊆ TS of index p - d. The

iα

cα mod p

Number of Γhαi

Z/2dZ ⊕ Z/p2 Z

0

quadratic residue

1 9 10 p (p − 1) 2

Z/2dZ ⊕ Z/p2 Z

0

quadratic nonresidue

1 9 10 p (p − 1) 2

Z/2dZ ⊕ Z/pZ ⊕ Z/pZ

0

0

(p9 + 1)(p10 − 1) p−1

Z/2dp2 Z

1

no restriction

p20

Table 2. Sublattices Γhαi = Γiα ,cα ⊆ TS of index p | d.

is a sublattice of index p of TS . One checks that d (Γ) ∼ = Z/2dp2 Z, generated v by 2dp , from which it follows that −2dp2 q(v) = 1, which is a square modulo p. By Theorem 9, if p - d, then there are 12 p10 (p10 + 1) sublattices of TS of index p isomorphic to Γ, and if p | d, there are p20 such lattices. Up to isomorphism there is a unique primitive embedding Γ ,→ ΛK3 . By the surjectivity of the period map for K3 surfaces, there is a complex projective K3 surface Y of degree 2dp2 such that TY ∼ = Γ. Moreover, we obtain an isomorphism of rational Hodge structures H2 (Y, Q) → H2 (S, Q). The Hodge isogenies above appear naturally in Mukai’s work on moduli spaces of stable sheaves on K3 surfaces (see Section 3.1 below for precise definitions). In this context, one starts with a polarized K3 surface X of degree 2dp2 such that NS(X) = Zh0 , and one defines S := MX (p, h0 , dp) to be the moduli space of sheaves on X with Mukai vector v 0 := (p, h0 , dp). Then S is a K3 surface, and by Theorem 1.5 of [35], there is an isomorphism of Hodge structures between v 0⊥ /Zv 0 and H2 (S, Z), which is compatible with the Mukai pairing on v 0⊥ /Zv 0 and cup product on H2 (S, Z). Moreover, there

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is a rational Hodge isometry H2 (X, Q) → v 0⊥ /Zv 0 ⊗ Q x 7→

  h0 · x 0, x, p

mapping h0 /p to h := (0, h0 /p, 2d), which is integral in v 0⊥ /Zv 0 ⊗ Q because (0, h0 /p, 2d) − v 0 /p = (−1, 0, d). Composing the two isometries, we obtain a rational Hodge isometry φ : H2 (X, Q) → H2 (S, Q) mapping h0 /p to an integral class h ∈ H 2 (S, Q) such that h2 = 2d, giving S a polarization of degree 2d. The isometry φ induces an injection TX ,→ TS whose image has index p. As in Section 2.1, since NS(X) ∼ = Zh0 , we have 2 0 2 2 ∼ ∼ TX = h−2dp i ⊕ Λ , so d (TX ) = Z/2dp Z, and −2dp q(u) is a square modulo p, where q is the discriminant form on d (TX ) = hui. Mukai’s moduli spaces of stable sheaves therefore give geometric manifestations to the lattice theory discussed in this section. In Section 3, we explain some geometric constructions realizing the Mukai duality between the surfaces X and S via projective dualities, and in Section 5, we use one of these constructions (the case d = 1 and p = 2) for an arithmetic application. 2.7. Special Cubic fourfolds Continuing the theme of the previous section, we explore the connection between certain sublattices of TS of index p on general K3 surfaces and special cubic fourfolds. Geometric correspondences explaining these latticetheoretic connections have arithmetic applications: such correspondences can yield Brauer-Severi bundles representing a generator for an order p subgroup of Br(S). This idea was exploited in [12] to obtain counterexamples to weak approximation on a K3 surface, starting from a cubic fourfold containing a plane. The results of this section are easily derived from general work of Hassett [10] on special cubic fourfolds. We include them here to alert the arithmetically inclined audience about a source of constructions of transcendental Brauer classes on K3 surfaces. For example, the results of this section suggest that cubic fourfolds containing a del Pezzo surface of degree 6 form a source of transcendental 3-torsion elements on a K3 surface of degree 2. It would be very interesting to have a geometric correspondence capable of producing such a 3-torsion element, as a Brauer-Severi bundle, starting from the special cubic fourfold. Recall that a special cubic fourfold Y ⊆ P5 is a smooth cubic fourfold that contains a surface T not homologous to a complete intersection. Let h denote the hyperplane class of P5 ; assume that the lattice K := hh2 , T i ⊂ H 4 (Y, Z) is saturated. The discriminant of (Y, K) is the determinant of the Gram matrix of K. The nonspecial cohomology of (Y, K) is the orthogonal complement K ⊥ of K with respect to the intersection form. By Theorem 1.0.1 of [10], special cubic fourfolds (Y, K) of discriminant D form an irreducible

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divisor CD of the moduli space of cubic fourfolds. In what follows, we write KD for the special cohomology lattice of a special cubic fourfold in CD . Proposition 10. Let S be a K3 surface of degree 2d with NS(S) = Zh and p a prime number. For a nonzero α ∈ Br(S)[p], denote by Γhαi its corresponding sublattice of index p in TS . 1. Suppose that d = 1 and p > 3. There is precisely one isomorphism class of lattices Γhαi ⊂ TS such that there exist a special cubic fourfold (Y, K) of discriminant 2p2 , and an isomorphism of lattices Γhαi ∼ = (K ⊥ )(−1). 2. Suppose that p = 3. There is an isomorphism class of lattices Γhαi ⊂ TS such that there exist a special cubic fourfold (Y, K) of discriminant 18d, and an isomorphism of lattices Γhαi ∼ = (K ⊥ )(−1) if and only if (6, d) = 1 and if q is a prime dividing d, then q ≡ 1 mod 3. Remark. The divisor CD is nonempty if and only if D > 6 and D ≡ 0 or 2 mod 6 (see Theorem 1.0.1 of [10]). This implies that if p > 3 and d ≡ 2 mod 6, then no special cubic fourfolds have nonspecial cohomology isomorphic to a twist of an index p sublattice Γhαi ⊂ TS for a K3 surface S of degree 2d. Proof of Proposition 10. (1.) In Proposition 3.2.5 of [10], Hassett shows that the discriminant group ⊥ d(K2p 2 (−1)) is cyclic, and a generator u can be chosen so that its value for 2

the discriminant quadratic form is − 4p6p−1 2 . Then   4p2 − 1 1 2 2 −2p q(u) = −2p · − ≡ − mod p, 6p2 3 so −2p2 q(u) is a quadratic residue modulo p if p ≡ 1 mod 3 and otherwise it is a quadratic nonresidue. Let Γhαi be an index p sublattice of TS with cyclic discriminant group d(Γhαi ) generated by v, and such that the quadratic characters of −2p2 q(v) and −2p2 q(u) coincide (such a lattice exists, and is ⊥ unique up to isomorphism, by Theorem 9). We claim that Γhαi ∼ = K2p 2 (−1). Since d = 1, using Theorem 1, either iα = 0 and p - cα , or iα = 1 and p - 1 + 4cα . On the other hand, Proposition 2 shows that the isomorphism class of Γhαi depends only on whether −2p2 q(v) is a square modulo p or not. Thus, each lattice Γhαi with iα = 1 and p - 1 + 4cα is isomorphic to a lattice Γhα0 i with iα0 = 0 and p - cα0 . Consequently, we may assume that iα = 0 and p - cα . As in the proof of Proposition 2, there is a generator whose value for the discriminant form is − 2p12 (p2 + 4cα ). Therefore, there is an isomorphism ⊥ 2 × Γhαi ∼ such that = K2p 2 (−1)if and only if there exists an x ∈ (Z/2p Z) −

4p2 − 1 p2 + 4cα 2 ≡− x mod 2Z. 2 6p 2p2

Multiplying by 2p2 this becomes 4p2 − 1 ≡ (p2 + 4cα )x2 mod 4p2 Z. 3

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Modulo 4 there is always such an x. Modulo p, we need 1 ≡ −3cα x2 mod p. If p ≡ 1 mod 3, then −3 and cα are squares modulo p. If p ≡ 2 mod 3, then both are not squares modulo p. Thus −3cα is always a square modulo p. ⊥ (2.) We know from Proposition 3.2.5 in [10] that d(K18d (−1)) ∼ = Z/6dZ ⊕ Z/3Z. If 3 divides d, then no discriminant group in Theorem 9 is isomorphic ⊥ ⊥ to d(K18d (−1)). Assume that 3 - d; then d(K18d (−1)) ∼ = Z/2dZ ⊕ Z/3Z ⊕ Z/3Z. Let u = (3, 0) ∈ Z/6dZ ⊕ Z/3Z be a generator for the subgroup 3 ⊥ . By Z/2dZ of d(K18d (−1)). Using Proposition 3.2.5 of [10], we have q(u) = 2d Theorem 9, there is unique isomorphism class Γhαi with d(Γhαi ) = Z/2dZ ⊕ Z/3Z ⊕ Z/3Z; without loss of generality, we may assume that iα = 0 and cα = 0. In this case, the vector v1 in the proof of Theorem 1 is a generator 1 for the subgroup Z/2dZ of d(Γhαi ), and its value for the quadratic form is − 2d . ⊥ Thus, to have an isomorphism d(Γhαi ) ∼ = d(K18d (−1)), we need x ∈ (Z/2dZ)× such that 1 3 ≡ − x2 mod 2Z. 2d 2d Multiplying by 2d we have 3 ≡ −x2 mod 4d. Such an x exists if and only if 2 - d and if for any prime q | d, we have q ≡ 1 mod 3. This shows that the conditions on d in the statement of the proposition are necessary. To see they are sufficient, we need only show that ⊥ (−1)) is isometric to the 3-Sylow part of d(Γhαi ). the 3-Sylow part of d(K18d ⊥ The intersection forms of d(K18d (−1))3 and d(Γhαi )3 are given by 2    0 23 0 3d and , 2 0 − 23 0 3 respectively. Note that under the necessary conditions, we have d ≡ 1 mod 6. This implies that 32 d ≡ 23 mod 2Z. It follows from this that the two discriminant forms are isometric.  Remark. The proof of Proposition 10 shows that, when d = 1 and p ≡ 1 mod 3, the twisted nonspecial cohomology of a special cubic fourfold of discriminant 2p2 and the transcendental lattice of a general K3 surface of degree 2p2 are both isomorphic to the same sublattice of index p in TS . While seemingly surprising, this phenomenon reflects the existence of associated K3 surfaces, in the sense of Hassett, for cubic fourfolds in C2p2 ; see Theorem 5.1.3 of [10]. Next, we elaborate on the geometric connection between special cubic fourfolds and K3 surfaces when d = 1 and p > 3. Let D0 be the local period domain of marked special cubic fourfolds (Y, K2p2 ) of discriminant 2p2 . The ⊥ domain D0 is an open subset of a quadratic hypersurface in P(K2p 2 ⊗ C) and is a connected component of ⊥ {[ω] ∈ P(K2p 2 ⊗ C) | (ω, ω) = 0,

(ω, ω ¯ ) < 0}.

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Let Γ2p2 be the arithmetic group such that D := Γ2p2 \D0 is the global period domain of special cubic fourfolds of discriminant 2p2 (see Section 2.2 of [10] for the definition of this group). The Torelli theorem for cubic fourfolds implies that the period map Ce2p2 → D from the normalization Ce2p2 of C2p2 is an open immersion [49]. Let Λ2 be the primitive cohomology lattice of a degree two polarized K3 surface. Write N 0 for the local period domain of K3 surfaces of degree two, which is an open domain of {[ω] ∈ P(Λ2 ⊗ C) | (ω, ω) = 0,

(ω, ω ¯ ) > 0}.

Let Γ2 be the arithmetic group such that N := Γ2 \N 0 is the global period domain for K3 surfaces of degree two. Surjectivity of the period map for K3 surfaces identifies the global period domain N with the coarse moduli space K2 of degree two polarized K3 surfaces. ⊥ Proposition 11. An embedding j : K2p 2 ,→ −Λ2 induces a dominant morphism C2p2 → K2 of quasi-projective varieties.

Proof. The divisor C2p2 is algebraic by Theorem 3.1.2 of [10]. Next, we describe a holomorphic map C2p2 → K2 , by mirroring the argument of Lemma 3.2 of [26]. Proposition 10 allows us to identify the local period domain D0 ⊥ ⊥ with N 0 . We show that O(K2p 2 ) ⊂ O(Λ2 ). Identifying K2p2 with a index ⊥ p sublattice of −Λ2 , we may consider the subgroup M = (−Λ2 )/K2p 2 of ⊥ d(K2p2 ), which is isotropic. By Proposition 1.4.1 of [38], we have ⊥ ∗ ⊥ Λ2 = {x ∈ (K2p 2 ) | x mod K2p2 ∈ M }. ⊥ ∗ ⊥ Any map ϕ ∈ O(K2p 2 ) naturally extends to (K2p2 ) . Hence ϕ induces an ⊥ ⊥ isomorphism on d(K2p2 ). The group d(K2p2 ) being cyclic, M is preserved by ϕ. This shows that ϕ induces an isomorphism on Λ2 , i.e., ϕ ∈ O(Λ2 ). To see that the holomorphic map C2p2 → K2 obtained thus far is algebraic, one argues as in the proof of Proposition 2.2.2 of [10]. The morphism is dominant because the map C2p2 → D is an open immersion. 

Remark. There is an analogous morphism of coarse moduli spaces K2p2 → K2 encoding the Mukai duality explained in Section 2.6. Kond¯o has studied this morphism in detail [26]; it has degree p10 (p10 + 1).

3. Mukai dual K3 surfaces Moduli spaces of sheaves on K3 surfaces were first studied by Mukai [34, 36]. The theory was further developed by G¨ ottsche and Huybrechts [9], O’Grady [39], Yoshioka [50], and others. We will mainly be interested in two-dimensional moduli spaces; see Mukai [35]. A general reference for these moduli spaces is the book of Huybrechts and Lehn [15]. The modern approach also relies heavily on Fourier-Mukai transforms [33], and their twisted version due to C˘ald˘araru [5, 6]. A general reference for these derived equivalences is the book of Huybrechts [14].

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Our goal in this section is to elaborate on Section 2.6 and describe certain K3 surfaces and their Mukai dual surfaces, which are (twisted) derived equivalent. The Mukai dual surfaces can be described in several ways: they are moduli spaces of stable sheaves on the original K3 surface, and they also arise via projective duality. The former approach leads us to natural elements of the Brauer group, whereas the latter gives explicit equations for the Mukai dual surface, and is therefore indispensible for arithmetic applications. 3.1. Set-up Let X be a general K3 surface of degree 2k. By general, we mean that NS(X) ∼ = Zh where h is the primitive ample divisor with h2 = 2k that polarizes X. Fix a Mukai vector v = (a, bh, c) ∈ H0 (X, Z) ⊕ H2 (X, Z) ⊕ H4 (X, Z), and define MX (v) to be the moduli space of stable sheaves E on X with Mukai vector   1 v(E) := ch(E)Td1/2 = r, c1 (E), r + c1 (E)2 − c2 (E) = v. 2 Here stable means µ-stable with respect to the polarization h of X, i.e., any proper subsheaf F of E must have slope µ(F) :=

c1 (F) · h r(F)

µ(E) :=

c1 (E) · h r(E)

strictly less than the slope

of E. In the examples that interest us, v will be primitive (i.e., gcd(a, b, c) = 1), in which case µ-stability coincides with the related notion of Gieseker stability, and it also coincides with µ-semistability and Gieseker semistability. Mukai [34] proved that MX (v) is smooth of dimension v 2 + 2 := b2 h2 − 2ac + 2, and it admits a holomorphic symplectic structure. In particular, v 2 must be at least −2 if there exists a stable sheaf E with v(E) = v. When v is primitive, MX (v) is compact; it is an irreducible symplectic variety. When v is also isotropic (i.e., v 2 := b2 h2 − 2ac = 0), S := MX (v) is a K3 surface [35]. The degree of S will be 2ac/gcd(a, c)2 . If n := gcd(a, bh2 , c) = 1, then S is a fine moduli space, the universal sheaf on X × S induces an equivalence between the derived categories of coherent sheaves on X and S, and we say they are Mukai dual [33]. If n 6= 1, then there is an n-torsion Brauer element α on S obstructing the existence of a universal sheaf. Instead, there is a twisted universal sheaf and the derived category of X is equivalent to the derived category of α-twisted sheaves on S (see C˘ ald˘ araru [5, 6]). Some particular cases are when h2 = 2k = 2dn2 and v = (n, h, nd). These cases were studied by Hassett and Tschinkel [11] (d = 1), Iliev and

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Ranestad [21] (d = 2 and n = 2), the second author [42], and Markushevich [32]. By the general principles outlined above, S is a K3 surface of degree 2d that comes equipped with an n-torsion Brauer element α. In some low degree cases the Mukai duality can be realized by projective duality. The starting point is to describe X as a linear section of a homogeneous variety. In the next three sections we will consider the cases • d = 1, Brauer • d = 1, Brauer • d = 2, Brauer

n = 2, producing a degree two K3 surface S with a 2-torsion element, n = 3, producing a degree two K3 surface S with a 3-torsion element, n = 2, producing a degree four K3 surface S with a 2-torsion element.

The first of these is precisely the remaining example of van Geemen. In the first two cases, we represent the Brauer elements by Brauer-Severi varieties with fibres isomorphic to P3 and P2 , respectively. For general K3 surfaces over C, Huybrechts and Schr¨oer [16] proved that the Brauer group equals the cohomological Brauer group, i.e., the group of sheaves of Azumaya algebras up to equivalence is isomorphic to the torsion part of the analytic cohomology group H2 (S, O∗ ). Their proof involves showing that any n-torsion element in H2 (S, O∗ ) can be represented by a Brauer-Severi variety with fibres Pn−1 , which we call a minimal BrauerSeveri variety. For a K3 surface arising as a non-fine moduli space of sheaves, with an associated n-torsion Brauer element obstructing the existence of a universal sheaf, there are natural ways to represent the Brauer element as a Brauer-Severi variety, but in general they do not produce a minimal BrauerSeveri variety. In terms of sheaves of Azumaya algebras, the representatives are all Morita equivalent (naturally) but we do not necessarily obtain a sheaf of Azumaya algebras of minimal rank in this way. For example, in cases one and two above, the moduli space approach produces non-minimal BrauerSeveri varieties with fibres isomorphic to P3 and P5 , respectively. However, there is also a geometric approach that produces a minimal Brauer-Severi variety in case two. We leave as an open question the explicit construction of Brauer-Severi varieties in the third case. 3.2. The degree eight/degree two duality A general degree eight K3 surface X is a complete intersection of three quadrics in P5 . To describe it as a linear section, we embed Y := P5 = P(V ) in P20 = P(Sym2 V ) using the Veronese embedding. The K3 surface X will be the intersection of Y ⊂ P20 with a codimension three linear subspace P(U ) = P17 . Now we projectively dualize. The dual variety Yˇ is a sextic hypersurface 20 ˇ in P = P(Sym2 V ∗ ), the determinantal variety, and P(U ⊥ ) = P2 intersects this hypersurface in a plane sextic curve C. Here U ⊥ ⊂ Sym2 V ∗ denotes the annihilator of U . We therefore obtain a degree two K3 surface S as the double

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cover of P2 branched over the sextic C. We say that X and S are projectively dual varieties. Lemma 12. The K3 surface S can be naturally identified with the moduli space MX (2, h, 2). Proof. This is Example 0.9 of Mukai [34], and Example 2.2 in [36]; a very detailed discussion is given by Ingalls and Khalid [22]. The basic idea is as ˇ 20 corresponds to a hyperplane Hp in P20 , which follows. A point p in P2 ⊂ P 5 20 intersects Y = P ⊂ P in a quadric four-fold Zp . The complete picture is: dually Y = P5 = P(V ) ,→ P(Sym2 V ) = P20 ∪ ∪ Zp ,→ Hp = P19 ∪ ∪ X ,→ P(U ) = P17

ˇ 20 Yˇ ,→ P(Sym2 V ∗ ) = P ∪ ∪ C ,→ P(U ⊥ ) = P2 ∪ p.

If p ∈ P2 \C, then Zp is a smooth quadric. Now every smooth quadric in P5 can be identified with the Grassmannian Gr(2, 4). The Grassmannian comes with two natural vector bundles of rank two: the universal bundle E and the universal quotient bundle F , which fit in the exact sequence 0 → E → C4 ⊗ O → F → 0. Dualizing gives 0 → F ∗ → (C4 )∗ ⊗ O → E ∗ → 0, so E can also be regarded as a quotient bundle on Gr(2, 4). Restricting E ∗ and F to X via the embedding X ⊂ Zp ∼ = Gr(2, 4) yields two stable vector bundles on X with Mukai vectors v = (2, h, 2) (see [22] for details, particularly page 450 and Corollary 3.5). Note that we had to choose an identification Zp ∼ = Gr(2, 4), but a different identification yields the same pair of bundles E ∗ |X and F |X , up to interchanging them (the automorphism group of Gr(2, 4) has two connected components, and as homogeneous bundles, E ∗ and F are invariant under pullbacks by automorphisms in the connected component of the identity, and interchanged by pullbacks by automorphisms in the other component). Alternatively, the identification Zp ∼ = Gr(2, 4) could be made canonical in the following way: Recall that there are two P3 -families of maximal isotropic planes contained in a smooth quadric. The subfamily of maximal isotropic planes from, say, the first family that pass through a fixed point of the quadric will be parametrized by a line P1 in P3 . Equivalently, each point of the quadric gives a plane C2 in C4 , and this leads to an isomorphism of the quadric with Gr(2, 4). The second family will yield a second isomorphism of the quadric with Gr(2, 4), and of course the automorphism of Gr(2, 4) given by composing these isomorphisms will interchange E ∗ and F. If p ∈ C, then Zp is a singular quadric. Assuming the K3 surface X is general, this singular quadric will always be of rank five, so Zp will be a cone over a smooth quadric threefold. This quadric in P4 can be identified with ∗

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the lagrangian Grassmannian LGr(2, 4). The blow-up Z˜p of Zp at the apex of the cone is therefore a P1 -bundle over LGr(2, 4). The embedding X ⊂ Zp lifts to an embedding X ⊂ Z˜p because X does not contain the apex. Now the universal bundle E and universal quotient bundle F over LGr(2, 4) are dual, yielding a self-dual sequence 0 → F ∗ → C4 ⊗ O → F → 0. Thus the pair of bundles on X degenerate to isomorphic bundles in this case, as pulling back E ∗ ∼ = F to Z˜p , and then restricting to X ⊂ Z˜p , yields isomorphic bundles. We conclude that the double cover S of P2 branched over C naturally parametrizes a family of stable bundles on X with Mukai vectors v = (2, h, 2). Now let us show that every bundle in MX (2, h, 2) arises in this way. Claim. A stable bundle E with Mukai vector v(E) = (2, h, 2) satisfies  4, if i = 0, i i h (E) = dimH (X, E) = 0, otherwise. Proof. This follows from standard arguments involving stable sheaves. Firstly, H2 (X, E) ∼ = H0 (X, E ∨ )∨ vanishes because E ∨ is also a stable bundle, with ∨ slope µ(E ) = −h2 /2 = −4. Next, suppose that H1 (X, E) is non-vanishing. Then Ext1 (O, E ∨ ) = H1 (X, E ∨ ) ∼ = H1 (X, E)∨ is also non-vanishing, so there is a non-trivial extension 0 → E ∨ → F → O → 0. Now F has Mukai vector (3, −h, 3) and (3, −h, 3)2 = (−h)2 − 2.3.3 = −10 < −2, so F cannot be stable. Let G ⊂ F be a destabilizing sheaf; then G has slope µ(G) =

−h2 c1 (G) · h ≥ µ(F) = . r(G) 3

Moreover, G is necessarily of rank 1 or 2, so writing c1 (G) = dh with d ∈ Z, we find that d ≥ −r(G)/3 > −1. Therefore d ≥ 0 and µ(G) ≥ 0. Let g be the composition G → F → O. The kernel of g is then a subsheaf of E ∨ with slope µ(kerg) ≥ 0; by the stability of E ∨ , kerg must vanish. So G ∼ = O and G → F gives a splitting of the exact sequence defining F, contradicting the fact that the extension class is non-trivial. We conclude that H1 (X, E) must vanish. Finally, RiemannRoch gives Z χ(E) = (2, h, 2)(1, 0, 1) = 4, X

so h0 (E) = 4. This completes the proof of the claim.



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It follows from the claim that E has precisely four independent sections. Moreover, we can show that E is generated by its sections, i.e., the evaluation map H0 (X, E) ⊗ O −→ E is surjective. Roughly, if the evaluation map were not surjective, it would factor through H0 (X, E) ⊗ O −→ E 0 −→ E. One can then argue that E 0 has Mukai vector (2, h, 2 − k) where k ≥ 1, and hence the kernel F in F −→ H0 (X, E) ⊗ O −→ E 0 will have Mukai vector v(F) = (2, −h, 2 + k). Since v(F)2 = −4k < −2, F will be unstable. Looking at the slope, we see that F must have a section. The composition O −→ F −→ H0 (X, E) ⊗ O will identify O isomorphically with its image in H0 (X, E) ⊗ O, which will look like hsi ⊗ O for some non-zero section s ∈ H0 (X, E). But then hsi ⊗ O will lie in the kernel of the evaluation map H0 (X, E) ⊗ O → E. This is only possible if s = 0, a contradiction. Thus every stable sheaf E with Mukai vector (2, h, 2) is naturally a quotient of the trivial rank four bundle, implying that there is a classifying map X → Gr(2, 4) such that E is the pullback of the universal quotient bundle on Gr(2, 4). Generically, the classifying map will be an embedding and compatible with the embeddings into P5 , and thus Gr(2, 4) can be identified with a smooth quadric four-fold containing X, i.e., we have X ⊂ Gr(2, 4) ∼ = Zp ⊂ P5 for some p ∈ P2 \C. But then E belongs to the family of bundles on X parametrized by S, as described above. Note that the covering involution is given by mapping E to the cokernel of the adjoint map E ∗ −→ H0 (X, E)∗ ⊗ O. In the non-generic case, the stable bundle E obtained in this way fits into a self-dual sequence ∼ H0 (X, E) ⊗ O −→ E. E ∗ −→ H0 (X, E)∗ ⊗ O = In particular, there is a skew two-form on H0 (X, E), and the classifying map factors through the lagrangian Grassmannian X −→ LGr(2, 4) ⊂ Gr(2, 4); it is no longer an embedding. The lagrangian Grassmannian is a hyperplane section of the usual Grassmannian, LGr(2, 4) ⊂ P4 , and the required singular quadric Zp containing X is a cone over LGr(2, 4).  Remark. We could instead observe that S parametrizes a complete family of stable sheaves on X, which therefore must be all of MX (v), since the latter is two-dimensional.

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Lemma 13. The K3 surface S comes with a Brauer-Severi variety W → S whose fibres are isomorphic to P3 . Proof. We have a family Z → P2 of quadric fourfolds over P2 . Let W → P2 be the Fano variety of maximal isotropic planes (∼ = P2 ) contained in the 2 2 fibres of Z → P . If p ∈ P \C, then Zp is a smooth quadric fourfold, which therefore contains two P3 -families of maximal isotropic planes. Therefore the fibre Wp will consist of two copies of P3 . If p ∈ C, then Zp is a singular quadric fourfold, and the above familes degenerate to a single P3 -family of maximal isotropic planes. Therefore the fibre Wp will be a single P3 . The morphism W → P2 therefore factors through the double cover of 2 P branched over C, i.e., it factors through W → S → P2 (this is the Stein factorization). Then W → S is the required Brauer-Severi variety, with fibres isomorphic to P3 ; see Proposition 3.3 of [12].  Lemma 14. The Brauer-Severi variety W → S gives a class α ∈ Br(S) in the Brauer group of S of order two. Proof. A priori, the order of the class α must divide four, as locally W is the projectivization of a rank four bundle. By Proposition B.3 of Auel, Bernardara, and Bolognesi [2] α is also the class arising from the even Clifford algebra on the discriminant cover S → P2 . Specifically, a P2 -family of quadrics in P5 is equivalent to a quadratic form q in six variables over the field C(P2 ). Because the rank of q is even, the corresponding Clifford algebra C(q) is a central simple algebra over C(P2 ), whereas the centre of the even Clifford algebra C0 (q) is the quadratic extension C(S) of C(P2 ) given by adjoining the square root of the sextic discriminant; C0 (q) is then a central simple algebra over C(S) (see Lam [29]). The result of Auel et al. identifies α ∈ Br(S) ⊂ Br(C(S)) with the Brauer class of C0 (q). Now the Clifford algebra C(q) admits a canonical involution σ sending x1 ⊗· · ·⊗xk to xk ⊗· · ·⊗x1 . This anti-automorphism induces an automorphism of Azumaya algebras C(q) ⊗ C(q) −→ x ⊗ y 7−→

End(C(q)) (z 7→ xzσ(y)),

implying that the Brauer class of C(q) in Br(C(P2 )) has order two. Finally, the Brauer class of C0 (q) is the pull-back of the Brauer class of C(q) to C(S). To see this, let V be the underlying six-dimensional vector space of the quadratic form q. Then the map V ⊗C(P2 ) C(S) −→ v

7−→

EndC0 (q) (C0 (q) ⊕ C1 (q))   0 v v 0

induces the required isomorphism of Azumaya algebras ∼ = C(q) ⊗C(P2 ) C(S) −→ EndC0 (q) (C0 (q) ⊕ C1 (q)) ∼ = C0 (q) ⊗C(P2 ) M2×2 (C(P2 ))

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by the universal property of the Clifford algebra. It follows that the Brauer class of C0 (q), and hence α ∈ Br(S), also has order two.  Remark. It is not obvious from the Clifford algebra description that α ∈ Br(S) is non-trivial, but this follows from the following lemma, since we saw in Section 3.1 that S is a non-fine moduli space when n := gcd(2, h2 , 2) = 2 is greater than 1. Lemma 15. The class α ∈ Br(S) of the Brauer-Severi variety W → S is the obstruction to the existence of a universal sheaf for the moduli space S = MX (2, h, 2). Proof. Universal sheaves exist locally, so let S = ∪i Si be a cover such that there exists a local universal sheaf Ui on each X × Si . Denote by p and q the projections from X × S to X and S, respectively. On the overlap (X × Si ) ∩ (X × Sj ) Ui and Uj will differ by tensoring with q ∗ Lij , where Lij is a line bundle on Sij := Si ∩ Sj . The collection of line bundles Lij defines a holomorphic gerbe on S, whose Brauer class is the obstruction to the existence of a universal sheaf on X × S. By the claim in the proof above, H0 (X, Ui |X×{s} ) is four-dimensional for all s ∈ Si . Therefore q∗ Ui is a locally free sheaf of rank four on Si . Moreover, q∗ Uj = q∗ (q ∗ Lij ⊗ Ui ) = Lij ⊗ q∗ Ui . Therefore the local P3 -bundles P(q∗ Ui ) patch together to give a globally defined P3 -bundle on S. Claim. This P3 -bundle can be identified with the Brauer-Severi variety W → S. Proof. Let E := Ui |X×{s} . Recall that E is realized as a quotient H0 (X, E) ⊗ O −→ E, which is the pullback of C4 ⊗ O −→ F by the classifying map X → Gr(2, 4). Therefore a line in H0 (X, E) corresponds to a line ` in C4 . But each line ` in C4 determines a maximal isotropic plane in Gr(2, 4), namely, the set of planes in C4 containing ` is isomorphic to P(C4 /`) ∼ = P2 . Thus the family of lines in H0 (X, E) gives one half of the Fano variety of maximal isotropic planes in Gr(2, 4), parametrized by P(C4 ) = P3 . To get the other half, recall that the covering involution of S → P2 takes s to the point representing the cokernel E 0 of E ∗ −→ H0 (X, E)∗ ⊗ O. For this sheaf E 0 , a line in H0 (X, E 0 ) = H0 (X, E)∗ will correspond to a line ` in (C4 )∗ , or equivalently, a hyperplane `⊥ in C4 . Each hyperplane `⊥ in C4 determines a maximal isotropic plane in Gr(2, 4), namely, the set of planes in C4 contained in `⊥ is isomorphic to P(`⊥ ) ∼ = P2 . This gives the other half

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of the Fano variety of maximal isotropic planes in Gr(2, 4), parametrized by P(C4 )∗ = (P3 )∗ . In summary, if the points s and s0 ∈ S sitting above p ∈ P2 represent sheaves E and E 0 on X, then the family of lines in H0 (X, E) and H0 (X, E 0 ) can be identified with Wp , the Fano variety of maximal isotropic planes contained in the quadric Zp ∼ = Gr(2, 4). But this implies that the P3 -bundle on S given locally by P(q∗ Ui ) is precisely the Brauer-Severi variety W → S, proving the claim.  It follows from the claim that if there exists a universal sheaf U on X × S, then the Brauer-Severi variety W → S is the projectivization of the rank four bundle q∗ U, and hence the Brauer class α is trivial. Conversely, if α is trivial, then the P3 -bundle is the projectivization of a rank four bundle V on S. Moreover, V must be locally isomorphic to q∗ Ui , i.e., it must be equal to Mi ⊗ q∗ Ui for some line bundle Mi on Si . Then the local universal sheaves q ∗ Mi ⊗ Ui on X × Si will patch together to give a global universal sheaf U on X × S.  3.3. The degree eighteen/degree two duality The fact that general K3 surfaces of degrees four, six, and eight are complete intersections is classical. Mukai [37] extended this analysis by showing that K3 surfaces of degrees ten to eighteen are linear sections of homogeneous varieties. In particular, a general degree eighteen K3 surface X is a linear section of a certain homogeneous variety Y := G2 /P . This homogenous variety Y is five-dimensional and embeds in P(V ) = P13 (here V is the adjoint representation of G2 , and Y is the orbit of the maximal weight vector). The K3 surface X will be the intersection of Y ⊂ P13 with a codimension three linear subspace P(U ) = P10 . As before, we projectively dualize. The dual variety Yˇ is again a sextic ˇ 13 = P(V ∗ ) and P(U ⊥ ) = P2 intersects this hypersurface in hypersurface in P a plane sextic curve C. So once again the projective dual of X is a degree two K3 surface S, the double cover of P2 branched over C. The geometry of this projective duality was studied extensively by Kapustka and Ranestad [23], and we shall use their results below. Lemma 16. The K3 surface S can be naturally identified with the moduli space MX (3, h, 3). ˇ 13 corresponds to Proof. This is Theorem 1.2 of [23]. A point p in P2 ⊂ P 13 13 a hyperplane Hp in P , which intersects Y ⊂ P in a Fano fourfold Zp of genus ten and index two. The picture is: dually Y = G2 /P ∪ Zp ∪ X

,→ P(V ) = P13 ∪ ,→ Hp = P12 ∪ ,→ P(U ) = P10

Yˇ ∪ C

ˇ 13 ,→ P(V ∗ ) = P ∪ ,→ P(U ⊥ ) = P2 ∪ p.

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If p ∈ P2 \C, then Zp is smooth. Kuznetsov [27] showed that Zp admits a pair of vector bundles of rank three, each with six independent sections. This result was clarified by Kapustka and Ranestad, who showed that Zp admits a unique embedding as a linear section of the Grassmannian Gr(3, 6), up to automorphisms of Gr(3, 6) of course. Denoting the universal bundle and universal quotient bundle on the Grassmannian by E and F respectively, the required rank three bundles on Zp are precisely the restrictions of E ∗ and F . Further restricting the bundles to X ⊂ Zp yields two stable vector bundles on X with Mukai vectors v = (3, h, 3). If p ∈ C, then Zp is singular and the pair of bundles on X degenerate to isomorphic bundles in this case. Thus the double cover S of P2 branched over C naturally parametrizes a family of stable bundles on X with Mukai vectors v = (3, h, 3). Since this is a complete family, and the moduli space MX (v) is two-dimensional, we conclude that S ∼  = MX (3, h, 3). Lemma 17. The K3 surface S comes with a Brauer-Severi variety W → S whose fibres are isomorphic to P2 . Proof. We have a family Z → P2 of Fano fourfolds of genus ten and index two over P2 . Let W → P2 be the Fano variety of cubic surface scrolls contained in the fibres of Z → P2 . If p ∈ P2 \C, then Zp is smooth and Proposition 1.5 of [23] states that there are two disjoint P2 -families of cubic surface scrolls on Zp . Therefore the fibre Wp will consist of two copies of P2 . If p ∈ C, then Zp is singular, the above families degenerate to a single P2 -family of cubic surface scrolls, and the fibre Wp is a single P2 . The morphism W → P2 therefore factors through the double cover of 2 P branched over C, i.e., it factors through W → S → P2 (this is the Stein factorization). Then W → S is the required Brauer-Severi variety, with fibres isomorphic to P2 .  Remark. The Brauer-Severi variety W → S gives a class α in the Brauer group of S whose order divides three. If α is non-trivial, it will therefore be 3-torsion. Non-triviality will follow from the next lemma. Lemma 18. The class α ∈ Br(S) of the Brauer-Severi variety W → S is the obstruction to the existence of a universal sheaf for the moduli space S = MX (3, h, 3). Proof. As in Lemma 15, we let S = ∪i Si be a cover such that there exists a local universal sheaf Ui on each X × Si . These local universal sheaves will differ by tensoring with q ∗ Lij , where q : X × S → S is projection to the second factor, and the collection of line bundles Lij on Si ∩ Sj will define a holomorphic gerbe on S, whose Brauer class obstructs the existence of a universal sheaf on X × S. Applying the same argument as earlier, one can show that the space H0 (X, Ui |X×{s} ) is six-dimensional for all s ∈ Si . Therefore q∗ Ui is a locally

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free sheaf of rank six on Si . Moreover, q∗ Uj = Lij ⊗ q∗ Ui , so the local P5 bundles P(q∗ Ui ) patch together to give a global P5 -bundle on S. Claim: This P5 -bundle can be identified with the second symmetric power Sym2 W → S of the Brauer-Severi variety W → S. Remark. We follow the convention that applying Sym2 to a projective space means taking Sym2 of the underlying vector space, then projectivizing. Thus Sym2 W → S denotes the Brauer-Severi variety that is locally given by P(Sym2 Ei ) → Si , where Ei → Si are rank three bundles such that W |Si ∼ = P(Ei ). In fact, this operation can be applied directly to the corresponding Azumaya algebra. In Section 5 of [45], Suslin described how to construct exterior powers λi A of an Azumaya algebra A, an idea that was further developed by Parimala and Sridharan [40]. The symmetric powers si A of an Azumaya algebra can be constructed in a similar way; for example, see Section 3.A of Knus et al.’s book [25]. Proof. The bundle E := Ui |X×{s} is realized as a quotient H0 (X, E) ⊗ O −→ E, which is the pullback of C6 ⊗ O −→ F by the classifying map X ,→ Zp ,→ Gr(3, 6). Therefore a line in H0 (X, E) corresponds to a line ` in C6 . The set of 3-planes in C6 containing ` then determines a subvariety T` ⊂ Gr(3, 6) of codimension three, isomorphic to Gr(2, 5). For a general line `, T` ∩ Zp will be a curve. However, for some choices of `, the intersection T` ∩ Zp is not transversal; instead, T` ∩ Zp is a cubic surface scroll (two-dimensional). Moreover, the set Wp(1) := {` ⊂ C6 |T` ∩ Zp is a cubic surface scroll} is isomorphic to P2 , embedded as a Veronese surface in the space P5 of all lines in C6 (see Section 3 of [23], particularly Proposition 3.13). Since the Veronese embedding is given by the second symmetric power, P(C3 ) ,→ P(Sym2 C3 ), we see that the family of lines in H0 (X, E) can be canonically identified with (1) (1) Sym2 Wp , where Wp ∼ = P2 is one half of the Fano variety Wp of cubic surface scrolls in Zp . To recover the other half of the Fano variety Wp , we consider instead hyperplanes in H0 (X, E). These correspond to hyperplanes `⊥ in C6 , which determine subvarieties T`⊥ ⊂ Gr(3, 6), again isomorphic to Gr(3, 5) ∼ = Gr(2, 5), parametrizing 3-planes in `⊥ . The set Wp(2) := {`⊥ ⊂ C6 |T`⊥ ∩ Zp is a cubic surface scroll} is isomorphic to P2 , again embedded as a Veronese surface in the space (P5 )∗ of all hyperplanes in C6 . Thus the family of hyperplanes in H0 (X, E) can be

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canonically identified with Sym2 Wp , where Wp is the other half of the Fano variety Wp of cubic surface scrolls in Zp . Finally, we observe that the covering involution of S → P2 takes the point s representing E to the point representing the cokernel E 0 of E ∗ −→ H0 (X, E)∗ ⊗ O. Then a hyperplane in H0 (X, E) will correspond to a line in H0 (X, E 0 ) ∼ = H0 (X, E)∗ . Altogether, we have shown that the P5 -bundle on S given locally by P(q∗ Ui ) is precisely the Brauer-Severi variety Sym2 W → S, proving the claim.  The symmetric power s2 A of an Azumaya algebra is Brauer equivalent to A⊗k A (see page 33 of [25]). Equivalently, the Brauer class of Sym2 W → S is the same as the Brauer class of the tensor product ⊗2 W → S, which is given by α2 . If there exists a universal sheaf U on X×S, then the claim implies that the Brauer-Severi variety Sym2 W → S will be the projectivization of the rank six bundle q∗ U, and hence its Brauer class α2 will be trivial. Since the order of α divides three, we conclude that α is trivial. Conversely, if α is trivial, then the P5 -bundle is the projectivization of a rank six bundle V on S. Moreover, V is locally isomorphic to q∗ Ui , i.e., equal to Mi ⊗ q∗ Ui for some line bundle Mi on Si . The local universal sheaves q ∗ Mi ⊗ Ui will then patch together to give a global universal sheaf U on X × S.  3.4. The degree sixteen/degree four duality By Mukai’s results [37] every degree sixteen K3 surface X is a linear section of the Lagrangian Grassmannian Y := LGr(3, 6). The homogeneous variety Y is six-dimensional and embeds in P(V ) = P13 . The K3 surface X will be the intersection of Y ⊂ P13 with a codimension four linear subspace P(U ) = P9 . As before, we projectively dualize. The dual variety Yˇ is a quartic hyˇ 13 = P(V ∗ ) and P(U ⊥ ) = P3 intersects this hypersurface in persurface in P a quartic K3 surface S. This is the projective dual of X. It was studied by Iliev and Ranestad [20, 21]. Lemma 19. The K3 surface S can be naturally identified with the moduli space MX (2, h, 4). Proof. This is Theorem 3.4.8 of [20]. A point p in Yˇ corresponds to a hyperplane Hp in P13 that is tangent to Y ⊂ P13 . In particular, if p ∈ S ⊂ Yˇ , then Hp intersects Y in a singular fivefold Zp . The picture is: dually Y = LGr(3, 6) ,→ P(V ) = P13 ∪ ∪ Zp ,→ Hp = P12 ∪ ∪ X ,→ P(U ) = P9

Yˇ ∪ S ∪ p.

ˇ 13 ,→ P(V ∗ ) = P ∪ ,→ P(U ⊥ ) = P3

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The five-fold Zp will have a single node at the point of tangency of Hp and Y . Projecting from this node yields an embedding of the blow-up Z˜p in P11 . Note that Zp is degree sixteen in P12 , whereas Z˜p will be degree fourteen in P11 . In fact, by Theorem 3.3.4 of [20], Z˜p embeds as a linear section of the Grassmannian Gr(2, 6), which itself embeds as a degree fourteen subvariety of P14 . The picture is: Gr(2, 6) ,→ P14 ∪ ∪ Z˜p ,→ P11 . The node of Zp does not lie on the K3 surface X, so that the embedding X ⊂ Zp lifts to an embedding X ⊂ Z˜p . Composing this with the embedding Z˜p ⊂ Gr(2, 6) yields an embedding of X in the Grassmannian. We then obtain a rank two bundle on X by restricting the dual E ∗ of the universal bundle of the Grassmannian. Iliev and Ranestad prove that this vector bundle on X is stable with Mukai vector v = (2, h, 4). Thus the quartic K3 surface S naturally parametrizes a family of stable bundles on X with Mukai vectors v = (2, h, 4). Since this is a complete family, and the moduli space MX (v) is two-dimensional, we conclude that S∼  = MX (2, h, 4). Question. The moduli space MX (2, h, 4) is not fine. Rather, there is a 2torsion Brauer class on the K3 surface S = MX (2, h, 4) obstructing the existence of a universal sheaf. Can a Brauer-Severi variety W → S representing this Brauer class be described in a natural way? Does it have fibres isomorphic to P1 or to P3 ? Remark. Kuznetsov also studied projective duality for the Lagrangian Grassmannian LGr(3, 6), in Section 7 of [27]. He constructed a conic bundle over the smooth part of the quartic hypersurface Yˇ (Lemma 7.8 [27]). This P1 bundle is a Brauer-Severi variety representing the Brauer class of a certain Azumaya algebra on (the smooth part of) Yˇ (Proposition 7.9 [27]). When restricted to S ⊂ Yˇ , presumably this Brauer class gives the obstruction to a universal sheaf for the moduli space S = MX (2, h, 4), and the conic bundle gives W → S. Assuming this is true, we would still like to find an interpretation of W → S in terms of Fano varieties of the hyperplane sections Zp , as in the previous examples.

4. Other dualities Other examples of projective dualities do not appear to lead to Brauer elements on K3 surfaces. A K3 surface X of degree fourteen embeds as a linear ˇ 14 is section of the Grassmannian Y := Gr(2, 6) ⊂ P14 . The dual variety Yˇ ⊂ P a cubic hypersurface, and the projective dual of X is a Pfaffian cubic fourfold F . The picture is:

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dually ˇ 14 Y = Gr(2, 6) ,→ P Yˇ ,→ P ∪ ∪ ∪ ∪ X ,→ P8 F ,→ P5 . Beauville and Donagi [4] showed that the Fano variety of lines on a (general) cubic fourfold is a four-dimensional holomorphic symplectic manifold, and in particular, the Fano variety of lines on the Pfaffian cubic F is isomorphic to the Hilbert scheme Hilb[2] X of two points on X. We can write Hilb[2] X as a Mukai moduli space MX (1, 0, −1). Unfortunately it is a fine moduli space, so it does not come with a Brauer element. Iliev and Ranestad [19] associated a second cubic fourfold to a degree fourteen K3 surface X, which they called the apolar cubic. The Fano variety of lines on the apolar cubic parametrizes presentations of the Pfaffian cubic as a sum of ten cubes. This Fano variety is also isomorphic to Hilb[2] X, though with a different polarization. So again there is no Brauer element; in any case, the apolar cubic does not arise from projective duality. Another duality, studied by Iliev and Markushevich [18], is between pairs of degree twelve K3 surfaces. A K3 surface X of degree twelve is a linear section of a ten-dimensional spinor variety in P15 . The projective dual is another K3 surface S of degree twelve, which can be identified with MX (2, h, 3). However, since this is a fine moduli space, it does not come with a Brauer element. There are also interesting derived equivalences between these dual varieties. The appropriate machinery is Kuznetsov’s Homological Projective Duality, and some of these examples are studied from that point of view in [27, 28]. 14

5. Application: Failure of weak approximation Let X be a K3 surface of degree eight over a number field k, given as a complete intersection of three quadrics in P5 X = V (Q1 , Q2 , Q3 ) ⊆ P5 = Proj k[x0 , . . . , x5 ]. In this section we give an explicit description, in terms of quaternion algebras over function fields, for the class α ∈ Br(S) that obstructs the existence of a universal sheaf on the Mukai moduli space S = MX (2, h, 2) described in §3.2. We then use the description of α to exhibit K3 surfaces S of degree two that fail to satisfy weak approximation on account of α, via a Brauer-Manin obstruction. The incidence correspondence Z := {xQ1 + yQ2 + zQ3 = 0} ⊂ P2 × P5 , has X as it base locus, and projection map Z → P2 is a family of quadric fourfolds. Let W → P2 be the Fano variety of maximal isotropic planes contained in the fibers of Z → P2 . We saw in Lemmas 13 and 14 (and their proofs) that the Stein factorization W → S → P2 consists of the discriminant

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cover S → P2 and a Brauer-Severi variety W → S that is ´etale-locally a P3 bundle over S. The image of the corresponding Brauer class α ∈ Br(S) in Br (k(S)) is thus an algebra of degree 4 and exponent 2; a result of Albert ensures that it is Brauer equivalent to a bi-quaternion algebra; see [1]. To compute this biquaternion algebra, we use the interpretation of α ∈ Br(S) as the class associated to the even Clifford algebra on the discriminant cover S → P2 (see the proof of Lemma 14). For i = 1, 2 and 3, let Mi denote the Gram symmetric matrix associated to the quadric Qi . The Gram symmetric matrix of the quadratic form q in six variables associated to S → P2 is M (x, y, z) := xM0 + yM1 + zM2 , and the signed discriminant of q is ∆ := − det(xM0 + yM1 + zM2 ). Thus, we may write S as the surface in P(1, 1, 1, 3) = Proj k[x, y, z, w] given by w2 = − det (M (x, y, z)) . (6) √ The discriminant algebra k(P2 )( ∆) is the function field k(S). To compute α as the class in im (Br(S) → Br (k(S))) of the even Clifford algebra C0 (q), we recall some facts about quadratic forms. Notation. Given nonzero elements a and b of a field K, write (a, b) for the quaternion algebra which, as a four-dimensional K-vector space, is spanned by 1, i, j, and ij, with multiplication determined by the relations i2 = a, j 2 = b and ij = −ji. Abusing notation, we sometimes also denote by (a, b) the class of the quaternion algebra in Br(K). 5.1. Quadratic forms of rank 6 and the even Clifford algebras Let q be a nondegenerate quadratic form of even rank over a field K of charac√  teristic not two. Let ∆ be the signed discriminant of q, and let L = K ∆ be the discriminant extension (we assume that ∆ is not a square in K). Write c(q) ∈ Br(K) (resp. c0 (q) ∈ Br(L)) for the class of the Clifford algebra C(q) (resp. the even Clifford algebra C0 (q)). A straightforward generalization of the last part of the proof of Lemma 14 establishes the following lemma. Lemma 20. We have c0 (q) = c(q) ⊗K L as classes in Br(L).



Lemma 21. Let a ∈ K ∗ , and write hai for the rank one quadratic form aX 2 . Let q, q1 , and q2 be nondegenerate quadratic forms of even rank, with respective signed discriminants ∆, ∆1 , and ∆2 . (i) c(hai ⊗ q) = c(q) ⊗ (a, ∆), (ii) c(q1 ⊥ q2 ) = c(q1 ) ⊗ c(q2 ) ⊗ (∆1 , ∆2 ), (iii) c(q ⊥ ha, −ai) = c(q). Proof. Items (i) and (ii) follow from Proposition IV.8.1.1 of [24]. For (iii), recall that c(ha, bi) = (a, b), so that by (ii) we have c(q ⊥ ha, −ai) = c(q) ⊗ (a, −a) ⊗ (1, ∆) and both (a, −a) and (1, ∆) are trivial in Br(K). 

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We specialize to quadratic forms q of rank six; diagonalizing, we may assume that q = ha1 , . . . , a6 i for some ai ∈ k, i = 1, . . . , 6. Lemma 21(iii) allows us to add hyperbolic planes to q without so changing the class of c(q). Consider the quadratic form ha1 , a2 , a3 , a4 , a5 , a6 i ⊥ ha1 a2 a3 , −a1 a2 a3 i = ha1 , a2 , a3 , a1 a2 a3 i ⊥ ha4 , a5 , a6 , −a1 a2 a3 i, which is equivalent to the sum ha1 a2 a3 i ⊗ h1, a1 a2 , a2 a3 , a1 a3 i ⊥ h−a1 a2 a3 i ⊗ h1, −a1 a2 a3 a4 , −a1 a2 a3 a5 , −a1 a2 a3 a6 i.

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The quadratic form h1, a1 a2 , a2 a3 , a1 a3 i is the norm form of the quaternion algebra (−a1 a2 , −a1 a3 ). Applying Lemma 21 to the forms q1 = h1, a1 a2 i and q2 = ha2 a3 , a1 a3 i we obtain c(h1, a1 a2 , a2 a3 , a1 a3 i) = (−a1 a2 , −a1 a3 ). Applying Lemma 21 to (7) we compute the class of c(q) and obtain (−a1 a2 , −a1 a3 ) ⊗ c(h1, −a1 a2 a3 a4 , −a1 a2 a3 a5 , −a1 a2 a3 a6 i) ⊗ (−a1 a2 a3 , ∆(q)) ∈ Br(k). √  ∆ , the quaternion algebra Over the discriminant extension L = K (−a1 a2 a3 , ∆) splits, and we have an equivalence of quadratic forms h1, −a1 a2 a3 a4 , −a1 a2 a3 a5 , −a1 a2 a3 a6 i ∼ = h1, −a1 a2 a3 a4 , −a1 a2 a3 a5 , a4 a5 i, the latter of which is the norm form of the quaternion algebra (a1 a2 a3 a4 , a1 a2 a3 a5 ). Putting this all together, we obtain c(q) ⊗K L = (−a1 a2 , −a1 a3 ) ⊗ (a1 a2 a3 a4 , a1 a2 a3 a5 ) ∈ BrL. Lemma 20 then allows us to conclude the following. Proposition 22. Let q = ha1 , . . . , a6 i be a nondegenerate diagonal quadratic form of rank six over a field K of characteristic not two, with nontrivial discriminant extension L. Then c0 (q) = (−a1 a2 , −a1 a3 ) ⊗ (a1 a2 a3 a4 , a1 a2 a3 a5 ) ∈ Br(L).



Corollary 23. Let q be a nondegenerate quadratic form of rank 6 over a field K of characteristic different from 2, with nontrivial discriminant extension L. Write mi for the determinant of the leading principal i × i minor of the Gram symmetric matrix of q. Then c0 (q) = (−m2 , −m1 m3 ) ⊗ (m4 , −m3 m5 ) ∈ Br(L).

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Proof. Symmetric Gaussian elimination of M allows us to diagonalize M to the matrix diag(m1 , m2 /m1 , . . . , m6 /m5 ); See the proof of Lemma 12 in [3] for details on this operation. Proposition 22 implies that c0 (q) = (−m2 , −m1 m3 /m2 ) ⊗ (m4 , m3 m5 /m4 ). Finally, we have the equalities of classes in Br(L) (−m2 , −m1 m3 /m2 ) = (−m2 , −m1 m2 m3 ) = (−m2 , m2 ) ⊗ (−m2 , −m1 m3 ) = (−m2 , −m1 m3 ) and similarly (m4 , m3 m5 /m4 ) = (m4 , −m3 m5 ).



5.2. Brauer-Manin obstructions Let S be a smooth projective geometrically integral variety over a number field k. Write k for a fixed algebraic closure of k, and let S denote the fibered product S ×k k. Write A for the ring of adeles ofQk, and Ω for the set of places of k. Since S is projective, the sets S(A) and v∈Ω S(kv ) coincide; here kv denotes the completion of k at v ∈ Ω. A class C of varieties as above is said to satisfy the H asse principle if S(A) 6= ∅ =⇒ S(k) 6= ∅ for every S ∈ C. We say Q that S satisfies weak approximation if the diagonal embedding of S(k) in v∈Ω S(kv ) = S(A) is dense for the product topology of the v-adic topologies. Manin used class field theory to observe that any subset S of the Brauer group Br(S) = H2et (S, Gm ) gives rise to an intermediate set S(k) ⊆ S(A)S ⊆ S(A),

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where S(k) denotes the closure of S(k) in S(A); see [31]. These intermediate sets can thus obstruct the Hasse principle (if S(A) 6= ∅ yet S(A)S = ∅), and weak approximation (if S(A) 6= S(A)S ). This kind of obstruction is known as a Brauer-Manin obstruction. For each xv ∈ S(kv ), there is an evaluation map Br(S) → Br(kv ), α 7→ α(xv ) obtained by applying the functor H2et (−, Gm ) to the morphism Spec kv → S corresponding to xv . The set S(A)S is the intersection over α ∈ S of the sets   X α S(A) := (xv ) ∈ S(A) : invv (α(xv )) = 0 ; v∈Ω

here invv : Br(kv ) → Q/Z is the local invariant map at v from local class field theory. There is a filtration on the Brauer group Br0 (S) ⊆ Br1 (S) ⊆ Br(S)

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where Br0 (S) := im (Br(Spec k) → Br(S)) is the
subgroup of constant Brauer elements, and Br1 (S) := ker Br(S) → Br(S) is the subgroup of algebraic Brauer elements. Classes α ∈ Br(S) \ Br1 (S) are called transcendental. K3 surfaces are some of the simplest varieties on which transcendental classes exist: curves and surfaces of negative Kodaira dimension have trivial geometric Brauer groups. For example, if S is a K3 surface with S(A) 6= ∅ and Pic(S) = NS(S) ∼ = Z, then any nonconstant class in Br(S) is transcendental, because there is an isomorphism ∼

Br1 (S)/Br0 (S) − → H1 (Gal(k/k), Pic(S)), coming from the Hochschild-Serre spectral sequence, and the group H1 (Gal(k/k), Pic(S)) is trivial because Pic(S) is free with trivial Galois action in this case. Details for the material in this subsection can be found in the surveys [41, 48] and Chapter 5 of Skorobogatov’s book [44]. 5.3. Local invariants at the real place We return to the situation at the beginning of §5, specializing to the case where k = Q. So let X be a K3 surface of degree eight over Q, and let S ⊆ P(1, 1, 1, 3) be the associated degree two K3 surface, together with the class α ∈ Br(S). Write, as before, M (x, y, z) for the Gram symmetric matrix of the quadratic form q in six variables associated to S → P2 . Let P0 = [x0 , y0 , z0 , w0 ] ∈ S(R) be a real point of S. From (6) it follows that det (M (x0 , y0 , z0 )) < 0, so the signature of the symmetric matrix M (x0 , y0 , z0 ) is (1, 5), (5, 1) or (3, 3). Lemma 24. Write ∞ for the real place of Q. We have ( 0 if Sign(M (x0 , y0 , z0 )) = (3, 3), inv∞ (α(P0 )) = 1 if Sign(M (x0 , y0 , z0 )) = (1, 5) or (5, 1). 2 Proof. This is an application of Proposition 22, noting that q has coefficients in K := Q(P2 ), and that the discriminant extension L is Q(S). The proof of Lemma 14 shows that α = c0 (q) in Br(Q(S)). We deduce from Proposition 22 that α = (−a1 a2 , −a1 a3 ) ⊗ (a1 a2 a3 a4 , a1 a2 a3 a5 ) ∈ Br(Q(S)). (9) We may now compute invariants for the specialization α(P0 ). For example, suppose that Sign(M (x0 , y0 , z0 )) = (3, 3). Without loss of generality, we may assume that ai (P0 ) is positive for i = 1, 2 and 3, and negative for i = 4, 5, and 6. Then all the entries of the quaternion algebras in (9) are negative, and hence α(P0 ) = 0 as an element of Br(R). Consequently, inv∞ (α(P0 )) = 0. The other possible signatures for M (x0 , y0 , z0 ) are handled similarly.  Corollary 25. Suppose that S(R) 6= ∅. Then there exists a point P ∈ S(R) such that inv∞ (α(P )) = 0.

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Proof. The set S(R) of real points of S is a 2-dimensional real manifold since it is not empty; see [43], p. 106. Hence there is a point P = [x0 , y0 , z0 , w0 ] ∈ S(R) such that det(M (x0 , y0 , z0 )) 6= 0, because the set of real points of the discriminant curve in S is either empty or has real dimension 1. Consider the signed projective plane S = R3 \ {(0, 0, 0)}/R>0 , which is topologically a sphere. Write Q = [x0 , y0 , z0 ] ∈ S and −Q = [−x0 , −y0 , −z0 ] ∈ S for the two points in S corresponding to P . If the signature of M (Q) is (3, 3), then by Lemma 24 we are done. Since the signature of M (−Q) is negative that of M (Q), we may assume that M (Q) has signature (1, 5) and M (−Q) has signature (5, 1). Let γ denote the discriminant curve in S, i.e., γ = {[x, y, z] ∈ S | det(M (x, y, z)) = 0}, which is a disjoint union of smooth closed curves. We claim there is a line ` ⊂ S that contains Q and −Q, and that meets γ transversally. This is an application of Bertini’s theorem: Consider Q as a point in P2 , and note that the set of lines passing through Q is a P1 . Bertini assures us that (over C) the set of lines through Q meeting the curve det(M (x, y, z)) = 0 in P2 transversally forms a nonempty open subset U ⊆ P1 . Hence U (R) = U ∩ P1 (R) 6= ∅, and any element of this set gives a line ` ⊂ S, as desired. Let f be the restriction of det(M (x, y, z)) to `. Then f has simple roots by transversality of γ ∩ `. This implies that as we travel from Q to −Q along ` and cross γ, the signature of M (x, y, z) will change from (a, b) to either (a + 1, b − 1) or (a − 1, b + 1), and starting from signature (1, 5), we must reach signature (5, 1). Hence, along `, there must be a point R ∈ S such that the signature of M (R) is (3, 3), and consequently det(M (R)) < 0. Lifting R to a point in S(R), and applying Lemma 24, we obtain the desired result.  5.4. An explicit example Let X be the K3 surface of degree eight over Q given as the smooth complete intersection of three quadrics in P5 with Gram matrices  −6 1  −3 M1 :=  3  −1 1

1 26 3 2 2 3

−3 3 2 1 2 −3

3 2 1 28 0 0

−1 2 2 0 12 1 

and

  1 0 −1 3     −3 , M2 :=  0 −3 0    1 1  8 3

8 2  −1 M3 :=  −2  0 0

2 32 0 0 −3 −2

−1 0 8 −1 3 0

−2 0 −1 24 −3 −1

−1 8 1 −2 −2 3 0 −3 3 −3 28 3

0 1 24 2 −3 −3  0 −2  0  . −1  3  32

−3 −2 2 −2 −1 −2

1 −2 −3 −1 28 3

 3 3  −3 , −2  3  16

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Proposition 26. Let S be the K3 surface of degree two in P(1, 1, 1, 3) = Proj Q[x, y, z, w] given by w2 = − det (xM1 + yM2 + zM3 ) . Then Pic(S) = NS(S) ∼ = Z. Proof. We follow the strategy of §5.3 of [13], which can be summarized as follows:
1. Prove that Pic S × F3 is isomorphic to Z2 , generated by the two components of the pullback from P2 of a tritangent line to the sextic branch curve. 2. Find a prime p > 3 of good reduction of S for which the reduced sextic branch curve has no tritangent line. 3. Apply Proposition 5.3 of [13] to conclude that Pic(S) ∼ = Z: otherwise the tritangent line over F3 would lift to Q, giving rise to a tritangent line over Fp for any other prime p of good reduction for S. The surface S has good reduction at 3. An equation for S × F3 is given by w2 = (x + 2z)(x4 y + 2x3 y 2 + 2x3 z 2 + 2x2 y 3 + x2 z 3 + 2xy 4 + xy 3 z + 2xy 2 z 2 + xyz 3 + 2y 4 z + y 3 z 2 + 2z 5 ) + (x2 y + y 3 )2 , from which it is clear that the line x+2z = 0 is tritangent to the branch sextic on P2 . The pullback of
this tritangent line to S ×F3 generates a rank two sublattice of Pic S × F3 . Let f be the characteristic polynomial for the action
of Frobenius on H´e2t S × F 3 , Q` , where ` 6= 3 is a prime number. Normalize
this polynomial by setting f3 (t) = 3−22 f (3t). Then the rank of Pic S × F3 is bounded above by the number of roots of f3 (t) that are roots of unity; see Corollary 2.3 of [47]. The computation of f3 (t) is standard: it suffices to determine #S(F3n ) for n = 1, . . . , 10; the Lefschetz trace formula and the functional equation for f then allows one to
determine enough traces of powers of Frobenius acting on H´e2t S × F 3 , Q` to reconstruct f by elementary linear algebra. See [47] for details. We obtain f3 (t) =

1 (t − 1)2 (3t20 + t19 + 2t18 + t17 + 3t16 + t15 + 2t14 − t13 − t12 3 − t11 − t9 − t8 − t7 + 2t6 + t5 + 3t4 + t3 + 2t2 + t + 3).

The roots of the degree 20 factor of
f3 (t) are not roots of unity, because they are not integral. Hence Pic S × F3 ∼ = Z2 . A Gr¨obner basis computation, using [8, Algorithm 8], shows that the reduction of S at 5 (which is smooth) has no line tritangent to the branch curve. This concludes the proof of the proposition.  By Corollary 23, the Brauer class α ∈ Br(S) arising from X is represented over Br (Q(S)) by tensor product of quaternion algebras (−m2 , −m1 m3 ) ⊗ (m4 , −m3 m5 ),

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where m1 = −6x + 8z, m2 = −157x2 − 46xy + 12xz − y 2 + 68yz + 252z 2 , m3 = −512x3 − 3884x2 y − 1790x2 z − 1094xy 2 − 48xyz + 370xz 2 − 24y 3 + 1618y 2 z + 6580yz 2 + 1984z 3 , m4 = −14896x4 − 112256x3 y − 64196x3 z − 13639x2 y 2 − 88686x2 yz − 31415x2 z 2 + 1230xy 3 + 28380xy 2 z + 190454xyz 2 + 66580xz 3 − 1967y 4 − 14274y 3 z + 12573y 2 z 2 + 148652yz 3 + 46212z 4 , m5 = −154622x5 − 1832494x4 y − 1088428x4 z − 3261270x3 y 2 − 6264622x3 yz − 2086758x3 z 2 − 353890x2 y 3 − 2306720x2 y 2 z − 992652x2 yz 2 − 124086x2 z 3 + 2698xy 4 + 587200xy 3 z + 6271452xy 2 z 2 + 9184426xyz 3 + 2279020xz 4 − 51948y 5 − 439790y 4 z − 82534y 3 z 2 + 4374124y 2 z 3 + 5413502yz 4 + 1214952z 5 . (10) Consider the real points on S given by P1 := [1, 2, −1, 924]

√ and P2 := [0, −1, 1, 1863673].

Using Lemma 24, we compute 1 . (11) 2 The point P1 , embedded diagonally in S(A), lies in the set S(A)α ; see (8). Let (Pv ) ∈ S(A) be the adelic point given by ( P1 , if v 6= ∞, Pv = P2 , otherwise. inv∞ (α(P1 )) = 0,

and

inv∞ (α(P2 )) =

The containment P1 ∈ S(A)α and (11) together imply that X 1 invv α(Pv ) = ∈ Q/Z. 2 v Hence (Pv ) ∈ S(A) \ S(A)α , which shows that S is does not satisfy weak approximation on account of α. As explained in §5.2, Proposition 26 implies that
H1 Gal(Q/Q), Pic(S) = 0, so there is no algebraic Brauer-Manin obstruction to weak approximation on S. We summarize our results in the following theorem. Theorem 27. Let M1 , M2 and M3 be the three symmetric matrices defined above. Let S be the K3 surface of degree two in P(1, 1, 1, 3) = Proj Q[x, y, z, w] given by w2 = − det (xM1 + yM2 + zM3 )

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Let α ∈ Br (Q(S)) be the tensor product of quaternion algebras (−m2 , −m1 m3 ) ⊗ (m4 , −m3 m5 ), with m1 , . . . , m5 as in (10). Then α extends to an element of Br(S) that gives rise to a transcendental Brauer-Manin obstruction to weak approximation on S. 5.5. What about the Hasse principle? It is natural to ask if elements α ∈ Br(S)[2] as above can obstruct the existence of rational points on S. This does not happen for the surface of Theorem 27: the point P1 is rational. A Brauer-Manin obstruction to the Hasse principle arising from a 2torsion Brauer element α requires the image of the evaluation maps evα,p : S(Qv ) → 12 Z/Z,

P 7→ invv (α(P ))

be constant for all places v, including the infinite place. Otherwise, an adelic point (Pv ) ∈ S(A) P can be modified at a place where evα,v is nonαconstant to arrange that v evα,v (Pv ) = 0, which means that (PvP ) ∈ S(A) , so α does not obstruct the Hasse principle. We must also have v evα,v (Pv ) = 21 for every (Pv ) ∈ S(A). The evaluation map evα,p can only take nonzero values at a finite number of places: the places of bad reduction for S, the places where α ramifies, and the infinite place. To obtain an obstruction to the Hasse principle from α, we must have evα,∞ (P ) = 0 for all points P ∈ S(R), by Corollary 25. We expect that an argument similar to that of Lemma 4.4 of [13], shows that, for any prime p 6= 2 of good reduction for α, we have evp,α (P ) = 0 for all P ∈ S(Qp ). For primes p 6= 2 of bad reduction, Proposition 4.1 and Lemma 4.2 of [13] show that evα,p is constant, provided the singular locus of the reduction of S consists of at most 7 ordinary double points. We thus expect that the a reasonable way to construct a counterexample to the Hasse principle using elements of the form α is to pick matrices M1 , M2 , and M3 in such a way that evα,2 (P ) = 21 for all points P ∈ S(Q2 ); an analysis similar to that in Section 4.3 of [13] may prove sufficient for this purpose.

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[24] M.-A. Knus, Quadratic and Hermitian Forms over Rings, Grundleren der mathematischen Wissenschaften 294, Springer, Berlin, 1991. 5.1 [25] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions, American Mathematical Society Colloquium Publications 44, American Mathematical Society, Providence, RI, 1998. 3.3 [26] S. Kond¯ o, On the Kodaira dimension of the moduli space of K3 surfaces, Compositio Math. 89, no. 3, 251–299 2.7 [27] A.G. Kuznetsov, Hyperplane sections and derived categories, Izv. Math. 70 (2006), no. 3, 447–547. 3.3, 3.4, 4 ´ [28] A. Kuznetsov, Homological projective duality, Publ. Math. Inst. Hautes Etudes Sci. 105 (2007), 157–220. 4 [29] T.Y. Lam, Introduction to quadratic forms over fields, Graduate Studies in Mathematics 67, American Mathematical Society, Providence, RI, 2005. 3.2 [30] E. Looijenga and C. Peters, Torelli theorems for K¨ ahler K3 surfaces, Compositio Math. 42 (1980/81), 145186. 2.1 [31] Y.I. Manin, Le groupe de Brauer-Grothendieck en g´eom´etrie diophantienne, Actes du Congr`es International des Math´ematiciens (Nice, 1970), 401–411, Gauthier-Villars, Paris, 1971. 1, 5.2 [32] D. Markushevich, Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces, Manuscripta Math. 120 (2006), no. 2, 131–150. 3.1 ˆ with its application to Picard [33] S. Mukai, Duality between D(X) and D(X) sheaves, Nagoya Math. J. 81 (1981), 153–175. 3, 3.1 [34] S. Mukai, Symplectic structure of the moduli space of simple sheaves on an abelian or K3 surface, Invent. Math. 77 (1984), 101–116. 3, 3.1, 3.2 [35] S. Mukai, On the moduli space of bundles on K3 surfaces I , Vector bundles on algebraic varieties (Bombay, 1984), 341–413, Tata Inst. Fund. Res. Stud. Math. 11, Tata Inst. Fund. Res., Bombay, 1987. 1, 1, 2.6, 3, 3.1 [36] S. Mukai, Moduli of vector bundles on K3 surfaces, and symplectic manifolds, Sugaku Expositions 1 (1988), no. 2, 139–174. 3, 3.2 [37] S. Mukai, Curves, K3 surfaces and Fano 3-folds of genus ≤ 10, Algebraic geometry and commutative algebra, Vol. I, 357–377, Kinokuniya, Tokyo, 1988. 3.3, 3.4 [38] V. V. Nikulin, Integral Symmetric Bilinear Forms and some of their Applications, Math. USSR Izvestija 14 (1980), 103–167. 1, 2.1, 2.2, 2.3, 2.7 [39] K. O’Grady, The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface, J. Algebraic Geom. 6 (1997), no. 4, 599–644. 3 [40] R. Parimala and R. Sridharan, Reduced norms and Pfaffians via Brauer-Severi schemes, Recent advances in real algebraic geometry and quadratic forms (Berkeley, CA, 1990/1991; San Francisco, CA, 1991), 351–363, Contemp. Math. 155, Amer. Math. Soc., Providence, RI, 1994. 3.3 [41] E. Peyre, Obstructions au principe de Hasse et a ` l’approximation faible, S´eminaire Bourbaki 56-`eme ann´ee, expos 931, Ast´erisque 299, Soci´et´e Math´ematique de France, 2005. 5.2 [42] J. Sawon, Lagrangian fibrations on Hilbert schemes of points on K3 surfaces, J. Algebraic Geom. 16 (2007), no. 3, 477–497. 3.1

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[43] I. R. Shafarevich Basic algebraic geometry 1: Varieties in projective space, Springer-Verlag, Berlin, 1994. 5.3 [44] A. Skorobogatov, Torsors and Rational Points, Cambridge Tracts in Mathematics 144, Cambridge University Press, Cambridge, UK, 2001. 5.2 [45] A.A. Suslin, K-theory and K-cohomology of certain group varieties, Algebraic K-theory, 53–74, Adv. Soviet Math. 4, Amer. Math. Soc., Providence, RI, 1991. 3.3 [46] B. van Geemen, Some remarks on Brauer groups of K3 surfaces, Adv. Math. 197 (2005), no. 1, 222–247. 1, 1, 2, 2.3, 2.6 [47] R. van Luijk, K3 surfaces with Picard number one and infinitely many rational points, Algebra Number Theory 1 (2007), no. 1, 1–15. 5.4 [48] A. V´ arilly-Alvarado, Arithmetic of del Pezzo surfaces, Birational Geometry, Rational Curves, and Arithmetic, 293–319 (F. Bogomolov, B. Hassett and Y. Tschinkel eds.) Simons Symposia 1, Springer, New York, 2013. 5.2 [49] C. Voisin, Th´eor`eme de Torelli pour les cubiques de P5 , Invent. Math. 86 (1986), no. 3, 577–601. 2.7 [50] K. Yoshioka, Moduli spaces of stable sheaves on abelian surfaces, Math. Ann. 321 (2001), no. 4, 817–884. 3 Kelly McKinnie Department of Mathematical Sciences University of Montana Missoula MT 59812-0864 USA e-mail: [email protected] Justin Sawon Department of Mathematics University of North Carolina Chapel Hill NC 27599-3250 USA e-mail: [email protected] Sho Tanimoto Department of Mathematics MS 136 Rice University 6100 S. Main St., Houston TX 77005-1982 USA e-mail: [email protected] Anthony V´ arilly-Alvarado Department of Mathematics MS 136 Rice University 6100 S. Main St., Houston TX 77005-1982 USA e-mail: [email protected]

On a Local-Global Principle for H 3 of Function Fields of Surfaces over a Finite Field Alena Pirutka Abstract. Let K be the function field of a smooth projective surface S over a finite field F. In this article, following the work of Parimala and Suresh, we establish a local-global principle for the divisibility of elements in H 3 (K, Z/`) by elements in H 2 (K, Z/`), l 6= car.K. Mathematics Subject Classification (2010). 12G05, 11G25, 14J20. Keywords. Unramified cohomology, Galois cohomology, ramification, surfaces over finite fields, local global principles.

1. Introduction Let K be a field of one of two following types: (i) K is the function field of a smooth projective surface S over a finite field F of characteristic p; (ii) K is the function field of a regular (relative) curve, proper over a ring of integers of a p-adic field. Recall that for a field k, the u-invariant u(k) is defined as a maximal dimension of an anisotropic quadratic form over k [7]. Another arithmetical invariant which one can associate to k is the period-index exponent: an integer d (if it exists), such that for any element α of the Brauer group Br k we have indα|(perα)d . Recall that indα and perα have the same prime factors, so that for a fixed α ∈ Br k one can find an integer d = d(α) as above. For k = K of one of the types above, these invariants are now understood. More precisely, if K is of type (ii), Saltman [14, 15] showed that indα|(perα)2 for (perα, p) = 1. For K of type (i), using the techniques of twisted sheaves and also some Saltman’s results on the ramification of α, Lieblich [11] established that indα|(perα)2 as well. For the u-invariant, Parimala and Suresh [12] established that u(K) = 8 for K of type (ii) and p 6= 2 (in the case (i) one easily sees that u(K) = 8 as well). This result has been also obtained by different methods by Harbater, Hartmann and Krashen [4] and Heath-Brown and Leep [5, 10] (which also contains the case p = 2). One © Springer International Publishing AG 2017 A. Auel (eds.) et al., Brauer Groups and Obstruction Problems, Progress in Mathematics 320, DOI 10.1007/978-3-319-46852-5_10

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of crucial steps in the proof of Parimala and Suresh is to establish a localglobal principle for the divisibility of elements in H 3 (K, Z/`) by symbols α ∈ H 2 (K, Z/`), ` 6= p, for K of type (i) and (ii), which also uses Saltman’s results [14, 16] on the classification of the ramification points of α. In [13], they also apply such a local-global principle to establish the vanishing of the third unramified cohomology group of conic fibrations over S. In this note, we follow the arguments of Parimala and Suresh and we establish the local-global principle in the general case for K of type (i), where we consider α ∈ H 2 (K, Z/`) not only a symbol. The main technical difficulty is that for α a symbol there is no so-called «hot» points in the classification of Saltman, which was also the case considered in [16, 17]. Here we extend the techniques of Saltman, Parimala and Suresh to the local study of these points. Our main result is the following: Theorem 1.1. Let K be the function field of a smooth projective surface S over a finite field and let ` be a prime, ` 6= char(K). Assume that K contains a primitive `th root of unity. Let ξ ∈ H 3 (K, Z/`) and α ∈ H 2 (K, Z/`) be such that the union ramS (ξ) ∪ ramS (α) is a simple normal crossings divisor. Assume that for any point x ∈ S (1) there exists fx ∈ Kx∗ in the field of fractions of the completion of the local ring of S at x, such that ξ = α ∪ fx in H 3 (Kx , Z/`). Then there exists a function f ∈ K ∗ such that ξ = α ∪ f in H 3 (K, Z/`). In section 2, we first fix the notations and recall some elements of Saltman’s approach on the ramification of α ∈ Br k, then we give some additional properties of so-called «hot» points. This allows us to deduce the local-global principle 1.1 in section 3. Acknowledgements. I would like to thank Parimala and Suresh for their comments and corrections on the first versions of this article. The work was partially supported by the Swiss National Science Foundation (grant 137928), I am grateful to Andrew Kresch for the invitation to the University of Zürich.

2. Classification of the ramification points, complements on ’hot’ points 2.1. Notations and first properties 2.1.1. Residues and unramified cohomology. Let A be a discrete valuation ring of rank one with fraction field K and the residue field κ, let π be a uniformizing parameter of A. For any i ≥ 1, j ∈ Z and n an integer invertible in κ we have the residue maps in Galois cohomology ∂

A i−1 H i (K, µ⊗j (κ, µ⊗j−1 ). n )→H n

If ∂A (x) = 0 for x ∈ H i (K, µ⊗j n ) we say that x is unramified. In this case we define the specialisation x ¯ of x as x ¯ = ∂A (x ∪ π).

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If A is a regular ring with fraction field K, an element x ∈ H i (K, µ⊗j n ) is called unramified on A if it is unramified with respect to all discrete valuations corresponding to height one prime ideals in A. Let k be a field. For L a function field over k, n an integer invertible in k, i ≥ 1 and j ∈ Z we denote \ ∂A i i−1 Hnr (L/k, µ⊗j Ker[H i (L, µ⊗j (kA , µ⊗j−1 )], n )= n )→H n A

where A runs through all discrete valuation rings of rank one with k ⊂ A and fraction field L. Here we denote by kA the residue field of A and by ∂A the residue map. If X is an integral variety over k, we denote def

i i ⊗j Hnr (X, µ⊗j n ) = Hnr (k(X)/k, µn ),

where k(X) is the function field of X. If X is a smooth and projective variety, then we also have \ i Hnr (X, µ⊗j Ker∂x , (1) n )= x∈X (1) def

where we denote ∂x = ∂OX,x (see [1]). We will use the following vanishing results for varieties over finite fields. Proposition 2.1. Let F be a finite field and let ` be a prime different from the characteristic of F. 2 (C, µ` ) = 0. (i) If C/F is a smooth projective curve, then Hnr 3 (S, µ⊗2 (ii) If S/F is a smooth projective surface, then Hnr ` ) = 0. 2 Proof. For (i) note that Hnr (C, µ` ) is the `-torsion subgroup of Br C (see [1]), the group which is zero as C is a smooth projective curve over a finite field [3]. The statement (ii) is established in [2] p.790.  d+1 Remark 2.2. A part of the Kato conjecture states that Hnr (X, µ⊗d ` ) = 0 for X a smooth projective variety of dimension d, defined over a finite field F. This conjecture has been recently established by Kerz and Saito [9], using also the arguments of Jannsen [6].

2.1.2. Function fields of surfaces over a finite field. In the rest of this section we use the following notations : F is a finite field of characteristic p, ` is a prime different from p, S is a smooth projective surface over F and K is the function field of S. If P is a point of S, AˆP denotes the completion of the def local ring AP = OS,P at P at its maximal ideal, KP is the field of fractions of AˆP . We assume that K contains `th roots of unity. We denote by α a (fixed for what follows) element of H 2 (K, Z/`). For any x ∈ S (1) a codimension one point of S we have the residue maps in Galois cohomology, as in the previous section: ∂x : H i (K, Z/`) → H i−1 (κ(x), Z/`), i ≥ 1

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where κ(x) is the residue field of x; the set of points x ∈ S (1) such that ∂x (α) is non zero is finite. We define the ramification divisor of α as X ramS α = x. x∈S (1) ,∂x (α)6=0

We write ramS α =

m X

Ci

i=1

where Ci ⊂ S, i = 1 . . . m are integral curves and we denote ui = ∂Ci (α), ui ∈ H 1 (κ(Ci ), Z/`) = κ(Ci )∗ /κ(Ci )∗` . If P ∈ S is a closed point, we will say that α is unramified at P if P ∈ / ramS α. Lemma 2.3. Let P ∈ S be a closed point. If α is unramified at P , then α is trivial in H 2 (KP , Z/`). Proof. As α is unramified at P , we have that α comes from He´2t (AP , Z/`) (cf. [1, §3.6, §3.8]). It is then trivial in H 2 (KP , Z/`), as the group He´2t (AˆP , Z/`) = H 2 (κ(P ), Z/`) is zero as κ(P ) is a finite field.  In the rest of this section we assume that ramS α is a simple normal crossings divisor. 2.2. Classification of points Let P ∈ S be a closed point. We recall the classification of [16] with respect to the divisor ramS α : 1. If P ∈ / ramS α, then it is called a neutral point. 2. If P is on only one irreducible curve in the ramification divisor, then it is called a curve point. 3. As the divisor ramS α is a simple normal crossings divisor, the remaining case is when P is only on two curves of the ramification divisor, in this case it is called a nodal point. Assume that P is on the curves Ci and Cj for some i 6= j. Recall that we denote ui = ∂Ci (α). By the reciprocity law (see [8]), ∂P (ui ) = −∂P (uj ). Then the following cases may occur: (a) ui and uj are ramified at P . Then P is called a cold point; (b) ui and uj are unramified at P . Denote the specialisations of ui (resp. uj ) at P by ui (P ), uj (P ) ∈ H 1 (κ(P ), Z/`). (i) If ui (P ) and uj (P ) are trivial, P is called a cool point. (ii) If ui (P ) and uj (P ) are non trivial and generate the same subgroup of H 1 (κ(P ), Z/`), then P is called a chilly point. (iii) If ui (P ) and uj (P ) do not generate the same subgroup of H 1 (κ(P ), Z/`), then P is called a hot point. Since κ(P ) is a finite field , we have H 1 (κ(P ), Z/`) ' Z/`. We then get that for a hot point one of the specialisations ui (P ) and uj (P ), say uj (P ), should be trivial. Then we will call P a hot nonneutral point on Ci and hot neutral point on Cj .

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Consider a graph whose vertices are curves in the ramification locus and whose edges correspond to chilly or hot points. We will call a hot chilly circuit a loop in this graph or a connected component with more than one hot point on curves in this component. 2.3. Local description Lemma 2.4. Let P ∈ S be a closed point. (i) If P ∈ Ci is a curve point with πi a local parameter of Ci in K, then in H 2 (KP , Z/`) we have α = u ∪ πi for some unit u ∈ AP . (ii) If P ∈ Ci ∩ Cj is a nodal point with πi , πj local parameters of Ci and Cj in K, then in H 2 (KP , Z/`) we have 1. α = 0 if P is a cool point; 2. α = uπj ∪ vπis for some units u, v ∈ AP , 1 ≤ s ≤ ` − 1, if P is a cold point; 3. α = u ∪ πi πjs for some unit u ∈ AP , 1 ≤ s ≤ ` − 1, if P is a chilly point; 4. α = u ∪ πi , for some unit u ∈ AP , if P is a hot point, non-neutral on Ci . Proof. For (i) we write (see [14, Proposition 2.1]) α = α0 + (u, πi ) where α0 is unramified at P and u is a unit in AP . By lemma 2.3, α0 is zero in H 2 (KP , Z/`), so that we get (i). For (ii), all the statements but the last one are in [13, Lemma 1.3]. To see the last one, we write (see [14, Proposition 2.1]) α = α0 + (u, πi ) + (v, πj ) where α0 is unramified at P , u, v are units in AP and the image of v in κ(Cj ) is ∂Cj (α). As above, α0 is zero by lemma 2.3. The element v is also zero in AˆP because v(P ) is zero in H 1 (κ(P ), Z/`) by the definition of a hot point, so that v ∈ H 1 (AˆP , Z/`) = H 1 (κ(P ), Z/`) is zero as well.  Let Ci be a curve in the ramification divisor ramS α and let πi be a prime defining Ci in K. We define a residual class βi of α at Ci as βi = ∂Ci (α ∪ πi ) (see [13, Remark 2.6], [16, p.820]). Proposition 2.5. If Ci ∈ ram √ S α contains no hot points, then the element βi is trivial over Li = κ(Ci )( ` ui ). Proof. Using proposition 2.1, it is sufficient to see that βi is unramified over Li . Let v be a discrete valuation on Li , let v 0 be the induced valuation on κ(Ci ) and let e be the valuation of a uniformizing parameter of κ(Ci ) in Li . We have the following commutative diagram (cf. [1, Proposition 3.3.1]): H 2 (Li , Z/`) O

∂v

res

H 2 (κ(Ci ), Z/`)

/ H 1 (κ(v), Z/`) O e·res

∂v 0

/ H 1 (κ(v 0 ), Z/`).

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We may assume that the valuation v 0 corresponds to a point P ∈ Ci . We have the following cases: 1. if P is a curve point, in AˆP , we have α = u ∪ πi for some unit u by the local description 2.4, and so ∂v0 (βi ) = ∂P ∂Ci (u ∪ πi ∪ πi ) = 0. 2. if P is a cool point, α is trivial in AˆP by the local description, hence ∂v0 (βi ) = 0. 3. if P is a cold point, the extension Li /κ(Ci ) is ramified at P , hence l|e. As κ(v 0 ) is a finite field, ∂v0 (βi ) becomes trivial in κ(v). 4. if P is a chilly point, from the local description we deduce that ∂v0 (βi ) = uj (P ) which equals to ui (P )s for some 1 ≤ s ≤ l − 1 by the definition of a chilly point, hence trivial over κ(v). Thus βi is unramified over Li and hence trivial. 

3. The local-global principle In this section we prove the local-global principle 1.1. 3.1. Additional notations Let ξ ∈ H 3 (K, Z/`) be as in theorem 1.1. As the divisor ramS (ξ) ∪ ramS (α) is a simple normal crossings divisor, we can write n X Ci ramS (ξ) ∪ ramS (α) = i=1

where n ≥ m and Ci are integral curves. Here ξ could be ramified at some Ci with i ≤ m and we have Cj ∈ ramS (ξ) \ ramS (α) for m < j ≤ n. We denote C = {C1 , . . . Cm } and T = {C1 , . . . Cm , Cm+1 , . . . Cn }. Let P a finite set of closed points on S, consisting of all the points of intersections Ci ∩ Cj and at least one point from each component Ci , let B be the semilocal ring of S at P and let πi be a (fixed from the beginning) prime defining Ci in B. Recall that we denote ui = ∂Ci (α) ∈ κ(Ci )∗ /κ(Ci )∗` . We put, for 1 ≤ i ≤ n, similarly as before: βi = ∂Ci (α ∪ πi ) ∈ H 2 (κ(Ci ), Z/`)

(2)

fi = fCi = πiti fi0 ∈ KCi ,

(3)

and where vCi (fi0 ) = 0. 3.2. Plan of the proof Vanishing of the third unramified cohomology of S from proposition 2.1 implies that to establish theorem 1.1 it is sufficient to prove that there exists f ∈ K such that for any x ∈ S (1) , ∂x (ξ) = ∂x (α ∪ f ) in the residue field κ(x). This is done essentially in two steps. 1. First we get the condition above for x = Ci ; to do this we first work locally around the double points P .

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2. Next we adjust the element f from the previous step to get the condition for other curves in the support of div(f ). 3.3. Analyzing curves and points in the ramification divisor 3.3.1. Local description at nodal points. Proposition 3.1. Let P ∈ Ci ∩ Cj be a nodal point. Then 1. If P is a chilly point, then for any 0 ≤ ri ≤ ` − 1 there exists r 0 ≤ rj ≤ ` − 1 such that ξ − α ∪ πiri πj j is unramified on AˆP ; t 2. If P is a hot point, non-neutral on Ci , then ξ − α ∪ πiri πjj is unramified on AˆP for any 0 ≤ ri ≤ ` − 11 , where tj is defined in (3). Proof. The first part is in [13, Lemma 3.1]. To see the second, let us first write t0j = vπj (fi ) = vπj (fi0 ). By the reciprocity law (cf. [8]), we have ∂P (∂Ci (ξ)) = −∂P (∂Cj (ξ)), and so ∂P (∂Ci (α ∪ fi )) = −∂P (∂Cj (α ∪ fj )). ˆ In AP we have α = u ∪ πi by the local description. This gives t

∂P (∂Ci (u ∪ πi ∪ πiti fi0 )) = −∂P (∂Cj (u ∪ πi ∪ πjj fj0 )). The left-hand side is 0

∂P (u(Ci ) ∪ fi0 (Ci )−1 ) = u(P )−(tj +) , where we denote u(Ci ), fi0 (Ci ) the images of u, fi0 in κ(Ci ) and t0

 = vP (fi0 (Ci )/πjj (Ci )). The right-hand side is −tj ∂P (u(Cj ) ∪ πi (Cj )) = −tj u(P ), where we denote u(Cj ), πi (Cj ) the images of u, πi in κ(Cj ). We then get tj = t − t0j + (mod `) as u(P ) is not an lth power. We then can write fi = πiti πjj fi00 , 00 where vP (fi (Ci )) =  (mod `). t

Claim. ∂Ci (ξ − α ∪ πiri πjj ) = 0 in the completion κ(Ci )P , the residue field of the extension of Ci to the completion KP . In fact, in κ(Ci )P , we have t − 00 fi ) t πiti πjj ) + ∂Ci (u

∂Ci (ξ) = ∂Ci (u ∪ πi ∪ πiti πjj = ∂Ci (u ∪ πi ∪

∪ πi ∪ πj− fi00 )

t

= ∂Ci (α ∪ πiri πjj ) − u(Ci ) ∪ πj (Ci )− fi00 (Ci ) (for the last equality we observe that ∂Ci (u ∪ πi ∪ πi ) is zero in κ(Ci )P ) and u(Ci ) ∪ πj (Ci )− fi00 (Ci ) is unramified at P and, hence, zero in κ(Ci )P . r r ˆP . Also note one can see that for rj 6= tj mod `, ξ − α ∪ πi i πj j is ramified on A that tj depends only on Cj , but not on a point P and not on ri . 1 Moreover,

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t From the claim we deduce that ξ − α ∪ πiri πjj is ramified on AˆP at most at πj ; using reciprocity (cf.[13, Lemma 1.2]), it is then unramified. 

Using the proposition above, we would like to associate an integer ri to each curve Ci globally, and not only locally in each double point. To do this, we will need to avoid the hot chilly circuits. Lemma 3.2. Let P ∈ S be a chilly or a hot nodal point. Let S 0 → S be the blowing-up of S at P . Then for any point x ∈ S 0(1) there exists fx ∈ Kx∗ such that ξ = α ∪ fx in H 3 (Kx , Z/`). Proof. We need only to consider the case when x = E is an exceptional divisor of the blowing-up. If P is a chilly point, then the statement is established in [13, Theorem 3.4]. Assume that P is a hot point, which is non-neutral t on Ci . Then, taking fE = πiri πjj we see that ξ − α ∪ fE is unramified on KE (for any ri ). In fact, we have an injection of local rings AP → OS 0 ,E , so that AˆP is contained on the completion of OS 0 ,E at its maximal ideal. Since ξ −α∪fE is unramifed on AˆP by proposition 3.1, it is unramified at E. Hence ξ − α ∪ fE comes from He´3t (AˆE , Z/`) = H 3 (κ(E), Z/`) = 0 as κ(E) is the field of functions of a curve over a finite field.  Lemma 3.3. There exists a smooth and projective surface S 0 → S, obtained as a blowing-up of S in some hot or chilly points, such that on S 0 there are no hot chilly circuits for α and such that we still have that for any point x ∈ S 0(1) there exists fx ∈ Kx∗ with ξ = α ∪ fx in H 3 (Kx , Z/`). Proof. In fact, suppose that we have a hot chilly circuit and let P ∈ Ci ∩ Cj be a hot or a chilly point on a curve in this circuit. If P is a hot point, which is non-neutral on Ci , let us consider the blowing-up of P . Let E be the exceptional divisor of the blowing up at P , and denote again Ci , Cj the strict transforms of Ci and Cj , let Pi = Ci ∩ E and Pj = Cj ∩ E. Using the local description, we easily check that Pj is a hot point, non-neutral on E and that Pi is a chilly point. We may then suppose that there is a chilly point on curves in the circuit. By a construction of [16, 2.9] we may break the circuit by successive blowing-ups of this point, and (some of) the double points above. Thus by successive blowing-ups of hot and chilly points we can obtain a surface S 0 with no hot chilly circuits. We may find a corresponding function fx for each exceptional divisor by the previous lemma.  Up to replacing S by S 0 as in the previous lemma, from now on, we assume that there are no hot chilly circuits. Now we can choose constants ri globally. Lemma 3.4. Assume that there are no hot chilly circuits. Then we may assor ciate to each curve Ci ∈ C an integer 0 ≤ ri ≤ ` − 1 such that ξ − α ∪ πiri πj j is unramified on AˆP for any nodal point P . If there is a hot point on Ci , then ri = ti .

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Proof. Since there are no hot chilly circuits, we may successively choose the integers ri as in the statement using lemma 3.1.  3.3.2. First choice of f . For 1 ≤ i ≤ m, let ri be as in lemma 3.4. For i ≥ m + 1, we put ri = vCi (fCi ). Proposition 3.5. Assume that there are no hot chilly circuits. Then there exists a function f ∈ K ∗ such that n X div(f ) = ri Ci + F, i=1

where F does not pass through any point in P, and such that ξ − α ∪ f is unramified on AˆP for all P ∈ P. Proof. By [13, Lemma 3.2], for any non hot point P ∈ Ci ∩ Cj for Ci , Cj ∈ T , r one can associate wP ∈ AP , such that ξ − α ∪ wP πiri πj j is unramified at AˆP , where the ri ’s are as above. We put wP = 1 if P is a hot point. Then r ξ−α∪wP πiri πj j is unramified at AˆP by lemma 3.4. For any other point P ∈ P S (i.e., for P ∈ P \ (Ci ∩ Cj )) we put wP = 1. Next we choose the function f satisfying the conditions of the proposition by the same construction as in [13, r Q Lemma 3.3]. We recall shortly this construction. First consider g = πiri . i=1

Then we can choose an element uP ∈ AP for any P ∈ P as follows. If P ∈ / Ci for all i, we put uP = 1. If P is on only one Ci we put uP = gπi−ri −r and if P ∈ Ci ∩ Cj we put uP = gπi−ri πj j . Now let w ∈ B be such that w(P ) = wP /uP for all P ∈ P. Then one checks that f = wg satisfies the conditions of the proposition.  3.3.3. A better choice of f . To choose an adjusting term we will need to make a more precise choice of the function f : Proposition 3.6. After possibly blowing up the surface S, there exists an element f ∈ K ∗ such that Pn (i) div(f ) = i=1 ri Ci + E; (ii) ξ − α ∪ f is unramified at Ci for any Ci ∈ T ; (iii) E does not passe through any point of P; (iv) (E · Ci )P is a multiple of ` at any intersection point of E and Ci , i = 1, . . . n. Proof. Let us consider f as in proposition 3.5. First we claim that for i ≤ m the residue γi := ∂Ci (ξ − α ∪ f ) may be written as ui ∪ bi for√some constant bi . In fact, it is sufficient to prove that γi is trivial over κ(Ci )( ` ui ). We have ∂Ci (ξ) = ∂Ci (α∪fi ) = ti βi +ui ∪gi for some gi and ∂Ci (α ∪ f ) = ri βi + ui ∪ hi for some hi , where βi are defined as in (2). If Ci has hot points, then ri = ti by construction √ and the claim is clear. If Ci has no hot points, then βi is trivial over κ(Ci )( ` ui ) by proposition 2.5, hence so is γi .

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Note that for i > m the residue γi := ∂Ci (ξ − α ∪ f ) is trivial by the choice of f . We proceed next exactly as in [13, Theorem 3.4]. We have 1. ξ − α ∪ f is unramified at AˆP for all P ∈ P by the choice of f , so γi is trivial in the completion κ(Ci )P at all points P ∈ P ∩ Ci . Hence bi is a √ norm from κ(Ci )P ( ` ui ). 2. By weak approximation, we then can find ai ∈ κ(Ci ) which is a norm √ from κ(Ci )( ` ui ) and such that ai bi (P ) = 1 for all P ∈ P ∩ Ci . As ui ∪ bi = ui ∪ ai bi we may assume that bi (P ) = 1 for all P ∈ P ∩ Ci . 3. We take b ∈ B a unit such that bi is the image of b in B/πi . 4. Changing f by bf , the conditions (i)–(iii) are now satisfied. By [13, Theorem 3.4. p.15] (E · Ci )P is a multiple of `, after possibly some blowing ups of the points in the intersection of E and Ci .  3.4. Adjusting term for other curves Let f and E be as in proposition 3.6. Let P 0 be a finite set of closed points on S, consisting of P and all intersection points of Ci and E, and at least one point from each component of E and at least one non hot point from each Ci . √ Proposition 3.7. There exists u ∈ K ∗ and x ∈ K ∗ a norm from K( ` u), such that (i) div(u)S= −E + E 0 + `U , where E 0 does not pass through any point in P 0 \ ( Ci ∩ Cj ); i6=j

(ii) the image of u in κ(Ci )∗ /κ(Ci )∗` equals to ui ; (iii) div(x) = −E + E 00 + `U 0 where E 00 does not pass through any point in P 0 and any intersection point Ci ∩ E 0 ; (iv) if D is an irreducible curve in Supp(E 00 ), then the specialization of α at D is unramified at every discrete valuation of κ(D) centered on a closed point of D. S Proof. Let P 00 = P 0 \ ( Ci ∩ Cj ) and let B 0 be the semilocal ring of S at i6=j

P 00 . By [17, Proposition 0.3], one can find v ∈ B 0 such that v = ui mod πi . By [13, Lemma 2.1], under the assumption that (E · Ci )P is a multiple of ` at any intersection point of E and Ci , i = 1, . . . n, one can find an element z ∈ K ∗ such that: 1. z is a unit at Ci and maps to an `th power in κ(Ci ) for all i; 2. div(z) = −E + Z1 + `Z2 where the support of Z1 does not contain any point in P. Then the element u = vz satisfies the conditions (i)–(ii) of the proposition. By [13, Lemma 2.3] we can construct x satisfying (iii). One then verifies (iv) as in [13, Lemma 2.4]. 

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3.4.1. End of the proof of theorem 1.1. Let x be as in proposition 3.7. We will now show that ξ − α ∪ (f x) is unramified at any codimension one point D ∈ S. By proposition 2.1(ii) we will then get ξ = α ∪ (f x). We have div(f x) =

n X

ri Ci + E 00 + `U 0 .

i=1

1. If D ∈ T , then ξ − α ∪ f is unramified at D by the previous section. If D = Ci with i ≤ m, then α = α0 + (u, πi ), where α0 is unramified on Ci , by the choice √ of u. Hence ∂D (α ∪ x) = ∂Ci (u ∪ πi ∪ x) = 0 as x is a norm from K( ` u). If D = Ci with i > m, then α and x are unramified at D, then so is α ∪ x. 2. If D ∈ / T ∪ Supp(f x) or D ∈ Supp(U 0 ), then ξ − α ∪ (f x) is unramified at D. 3. Assume now that D ∈ Supp(f x) \ (T ∪ U 0 ), i.e., D ∈ Supp(E 00 ). We have that ξ and α ∪ f are unramified at D. So it is sufficient to show that ∂D (α ∪ x) = 0 which follows from proposition 3.7(iv). 

References [1] J.-L. Colliot-Thélène, 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, Part 1, Amer. Math. Soc., Providence, RI, 1995. [2] J.-L. Colliot-Thélène, J.-J. Sansuc et C. Soulé, Torsion dans le groupe de Chow de codimension deux, Duke Math. J. 50 (1983), no. 3, 763–801. [3] A. Grothendieck, Le groupe de Brauer, I, II, III. Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam; Masson, Paris 1968. [4] D. Harbater, J. Hartmann, D. Krashen, Applications of patching to quadratic forms and central simple algebras, Invent. Math. 178 (2009), no. 2, 231–263. [5] D. R. Heath-Brown, Zeros of p-adic forms, Proc. Lond. Math. Soc. (3) 100 (2010), no. 2, 560–584. [6] U. Jannsen and S. Saito, Kato conjecture and motivic cohomology over finite fields, arXiv:0910.2815. [7] I. Kaplansky, Quadratic forms, J. Math. Soc. Japan 5, (1953). pp. 200–207. [8] K. Kato, A Hasse principle for two-dimensional global fields, J. reine angew. Math. 366 (1986), 142–183. [9] M. Kerz and Sh. Saito, Cohomological Hasse principle and motivic cohomology for arithmetic schemes, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 123– 183. [10] D. Leep, The u-invariant of p-adic function fields, J. reine angew. Math. 679 (2013), 65–73. [11] M. Lieblich, The period-index problem for fields of transcendence degree 2, Ann. of Math. (2) 182 (2015), no. 2, 391–427. [12] R. Parimala and V. Suresh, The u-invariant of the function fields of p-adic curves, Ann. of Math. (2) 172 (2010), no. 2, 1391–1405.

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[13] R. Parimala and V. Suresh, Degree 3 Cohomology of Function Fields of Surfaces, Int. Math. Res. Not. 2016 (2016), no. 14, 4341–4374. [14] D.J. Saltman, Division algebras over p-adic curves, J. Ramanujan Math. Soc. 12 (1997), no. 1, 25–47. [15] D.J. Saltman, Correction to: "Division algebras over p-adic curves", J. Ramanujan Math. Soc. 13 (1998), no. 2, 125–129. [16] D.J. Saltman, Cyclic algebras over p-adic curves, J. Algebra 314 (2007), no. 2, 817–843. [17] D.J. Saltman, Division algebras over surfaces, J. Algebra 320 (2008), no. 4, 1543–1585. Alena Pirutka Centre de Mathématiques Laurent Schwartz CNRS – Ecole Polytechnique 91128 Palaiseau France e-mail: [email protected]

Cohomology and the Brauer Group of Double Covers Alexei N. Skorobogatov Abstract. We calculate the 2-torsion subgroup of the Brauer group of a ramified double covering of a rational surface over an algebraically closed field. This and more general results are obtained by working with cohomology with mod 2 coefficients. Applications to the Brauer groups of K3 surfaces are discussed. Mathematics Subject Classification (2010). 14F22, 14J28. Keywords. Double covers, Brauer group, K3 surfaces.

1. Introduction Let k be a field and let d be a positive integer not divisible by the characteristic of k. Let π : X → S be a cyclic covering of degree d, where X and S are smooth, projective and geometrically integral varieties over k. Let i : C → S and j : C → X be the natural closed embeddings of the branch locus of π, so that i = πj. Assume that C is non-empty and smooth, but not necessarily connected. The covering π induces a map of d-torsion subgroups of the Brauer groups of S and X π ∗ : Br(S)[d] −→ Br(X)[d], and it would be interesting to understand its cokernel. Our main tool for investigating this problem is a canonical map Φ : H1 (C, Z/d)[θ]/i∗ H1 (S, Z/d) −→ Br(X)[d]/π ∗ (Br(S)[d]) which is defined as follows. Consider the Gysin sequence in ´etale cohomology (or Betti cohomology if k = C) attached to the smooth pair (S, C): θ

θ

H0 (C, Z/d) −→ H2 (S, µd ) −→ H2 (S \ C, µd ) −→ H1 (C, Z/d) −→ H3 (S, µd ), see §2. Let us write H1 (C, Z/d)[θ] for the kernel of the Gysin map θ. The Gysin sequence for (S, C) is linked with a similar sequence for (X, C) by the maps induced by π. Since π ∗ induces the zero map on H1 (C, Z/d), we obtain a map from H1 (C, Z/d)[θ] to the quotient of H2 (X, µd ) by the sum of subgroups © Springer International Publishing AG 2017 A. Auel (eds.) et al., Brauer Groups and Obstruction Problems, Progress in Mathematics 320, DOI 10.1007/978-3-319-46852-5_11

231

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π ∗ H2 (S, µd ) and θH0 (C, Z/d), and show that i∗ H1 (S, Z/d) is contained in its kernel (see Lemma 2.1 below). The subgroup θH0 (C, Z/d) ⊂ H2 (X, µd ) consists of algebraic classes, so the Kummer sequence 0 −→ Pic(X)/d −→ H2 (X, µd ) −→ Br(X)[d] −→ 0 and a similar sequence for S give us our map Φ. In the case S = P2k the Gysin map θ : H1 (C, Z/d) → H3 (P2k , µd ) can be explicitly computed in terms of corestrictions [1, Prop. 5.3]. In this paper we obtain results about the kernel and the cokernel of Φ for d = 2. A description of Φ in the general case can be read off from the exact sequence (16) in §3, however it may be too complicated for practical purposes. We give a more transparent description in Theorem 1.1 below for a separably closed ground field and under additional assumptions on S. We need to introduce some notation. Let us denote by Pic(X)[π∗ ] the kernel of the natural map π∗ : Pic(X) → Pic(S). Using diagram (4) below, it is not hard to see that j ∗ maps Pic(X)[π∗ ] to the 2-torsion subgroup Pic(C)[2]. ¯ Let k¯ be a separable closure of k, and let Γ = Gal(k/k). Write S = ¯ In Proposition 3.5 we show that if C is geometrically connected and S ×k k. H1 (S, Z/2) = 0 (for example, S is geometrically simply connected), then Φ gives rise to an exact sequence 0 −→ j ∗ (Pic(X)[π∗ ]) −→ Pic(C)[2][θ] −→ Br(X)[2]/π ∗ (Br(S)[2]). We also give a description of the cokernel of the last map in this sequence. Note that Pic(X)[π∗ ] = 0 when Pic(X) ∼ = Z, so in this case Φ is injective (cf. Example 1 in Section 4.3). See Example 3 in Section 4.3 for a case when Φ is surjective. When S is a surface, we obtain an explicit description of the kernel and the cokernel of Φ over k¯ under certain simplifying assumptions. Over k¯ the Kummer sequence gives an isomorphism H1 (C, Z/2) = Pic(C)[2]. Theorem 1.1. Let k be a field of characteristic different from 2 with a separa¯ Let S be a smooth, projective and geometrically integral surface ble closure k. over k with Pic(S)[2] = Br(S)[2] = 0, for example, a geometrically rational surface. For any finite surjective morphism π : X → S of degree 2 ramified in a non-empty smooth curve j : C ,→ X, the map Φ : Pic(C)[2] → Br(X)[2] gives rise to an exact sequence of Γ-modules 0 → Pic(C)[2]/j ∗ (Pic(X)[π∗ ]) → Br(X)[2] → Pic(S)even /π∗ Pic(X) → 0, (1) where Pic(S)even is the subgroup of Pic(S) consisting of the classes that have even intersection with each connected component of C. Remarks. 1. The class of C in Pic(S) is divisible by 2. Thus if C is geometrically connected we have Pic(S)even = Pic(S). 2. When k has characteristic zero, the condition Br(S)[2] = 0 follows from Pic(S)[2] = 0 and H2 (S, OS ) = 0. Indeed, by Hodge theory the latter

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233

condition implies the triviality of the divisible subgroup of Br(S), see formula (8.7) in [8, §III.8]. Then Br(S)[2] = 0 follows from Pic(S)[2] = 0 by formula (8.12) of loc. cit. When k = k¯ the 2-torsion subgroup Br(X)[2] was studied by T.J. Ford in [5]. He used the results of Knus, Parimala and Srinivas [13], in particular, the observation that for an unramified double cover π : V → U the cokernel of the canonical adjunction map (Z/2)U → π∗ (Z/2)V is isomorphic to (Z/2)U . Although the methods used by Ford appear to be rather general, the results of [5] are proved under the assumption that S has Picard rank 1, so they apply to double covers of the projective plane but not to double covers of more general rational surfaces. In the last few years there was a renewed interest in constructing elements of Br(X) motivated in part by the desire to compute the Brauer–Manin obstruction on X. Van Geemen’s explicit geometric construction [20] of elements of Br(X)[2] as Azumaya algebras was recently extended in [12] to arbitrary double covers X → P2 ramified in a smooth curve C. In particular, the exact sequence of [12, Thm. 1.1] represents Br(X)[2] as the quotient of the 2-torsion subgroup of Pic(C)/KC by Pic(X)/(Zπ ∗ O(1) + 2Pic(X)), where KC is the canonical class of C. See [6] for another recent work on the subject. When S is a rational surface, [20, Thm. 6.2] says that under some additional assumptions there is an injective map Pic(C)[2] → Br(X)[2] and calculates the size of its cokernel. Finally, in their recent paper [2] B. Creutz and B. Viray give a presentation of the elements of Br(X)[2] by central simple algebras when S is a ruled surface. The main point of this article is to show that if one is only interested in Φ as a homomorphism of Galois modules, and not in a geometric construction of resulting Azumaya algebras on X, then fairly general results can be obtained using only standard cohomological methods without recourse to the geometry of underlying varieties. Our approach was influenced by the calculations of Colliot-Th´el`ene and Wittenberg in [1, §5.1], which seems to be the only place in the literature where the ground field is not assumed to be separably closed. After some preliminaries on cyclic covers in §2 we set up in §3 a cohomological machinery for working with a double cover of a smooth, projective and geometrically integral variety S over an arbitrary field k, char(k) 6= 2, ramified in a smooth subvariety C of codimension 1. In §4 we assume that S is a surface such that H1 (S, Z/2) = 0. In §4.1 we spell out some useful exact sequences for the cohomology groups of double covers that are most likely well known but do not seem to be readily available in the literature. In Proposition 4.1 we obtain the exact sequence π

∗ 0 −→ H1 (C, Z/2) −→ H2 (X, Z/2)/π ∗ H2 (S, Z/2) −→ H2 (S, Z/2)⊥C −→ 0,

where H2 (S, Z/2)⊥C is the subgroup consisting of the elements orthogonal to the connected components of C with the respect to the cup-product pairing. In Corollary 4.2 we establish the following amusing fact: if C is geometrically

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connected, then for any i ≥ 1 we have a canonical isomorphism of Γ-modules Hi (Z/2, H2 (X, Z/2)) = H1 (C, Z/2), where Z/2 is the automorphism group of the covering π : X → S. Theorem 1.1 is proved in §4.2. For concrete calculations of the Galois module structure of Br(X)[2] using (1) we need to have enough information about Pic(S), Pic(X), Pic(C)[2] and the natural maps between these Galois modules. We sketch a few applications in §4.3. Corollary 4.5 concerns arbitrary K3 surfaces with a non-symplectic involution having a non-empty set of fixed points. We also show that the geometric Brauer group of a diagonal quartic surface defined over any field of characteristic not equal to 2 contains a Galois-invariant element of order 2, see Proposition 4.6. This paper originates from the workshop “Brauer groups and obstruction problems” at the American Institute of Mathematics, whose hospitality is gratefully acknowledged. I would like to thank Bianca Viray for introducing me to the subject of this note and helpful discussions, Martin Bright for the calculation mentioned at the end of §4.3, Jean-Louis Colliot-Th´el`ene and Yuri Zarhin for their interest in this paper. The work on the paper was carried out at the Institute for the Information Transmission Problems of the Russian Academy of Sciences at the expense of the Russian Foundation for Sciences (project number 14-50-00150).

2. Cyclic covers In this section we do not assume the field k to be algebraically closed. Let π : X → S, i : C ,→ S and j : C ,→ X be as in the first paragraph of the introduction. Since π and i are finite morphisms, π∗ and i∗ are exact functors between respective categories of ´etale sheaves [14, Cor. II.3.6]. In particular, for any ´etale sheaf E on X we have Rn π∗ E = 0 for all n > 0, and similarly for i∗ . Thus the spectral sequence Hp (S, Rq π∗ E) ⇒ Hp+q (X, E) degenerates, so that for n ≥ 1 we have canonical isomorphisms Hn (S, π∗ E) = Hn (X, E),

Hn (S, i∗ F) = Hn (C, F),

for any ´etale sheaf E on X and any ´etale sheaf F on C. We shall use these isomorphisms without further comment. Let OX and OS be the structure sheaves. There is a natural map of coherent sheaves π∗ OX → OS that induces norm on the function fields, see [15, Lecture 10]. The composition of the canonical map OS → π∗ OX with π∗ OX → OS sends a function f to f d . This gives natural morphisms of ´etale sheaves Gm,S −→ π∗ Gm,X , π∗ Gm,X −→ Gm,S , whose composition is [d] : Gm,S → Gm,S . The first of these morphisms is injective and the second one is surjective. The canonical morphism Gm,X → j∗ Gm,C gives rise to the morphism π∗ Gm,X −→ π∗ j∗ Gm,C = i∗ Gm,C ,

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where the last equality follows from i = πj. A verification on stalks shows the commutativity of the following diagram of ´etale sheaves on S: −→

π∗ Gm,X ↓

Gm,S ↓

[d]

−→

i∗ Gm,C

(2)

i∗ Gm,C

Let T be the kernel of π∗ Gm,X → Gm,S . We can extend (2) to a commutative diagram with exact rows 0 −→ 0 −→

−→

T ↓ i∗ (µd )C

−→

π∗ Gm,X ↓

−→

[d]

−→

i∗ Gm,C

Gm,S ↓

−→

i∗ Gm,C

−→

0 (3) 0

∗

The map T → i∗ (µd )C is adjoint to the isomorphism i T −→(µ ˜ d )C . The map of sheaves π∗ Gm,X → Gm,S induces a homomorphism of abelian groups π∗ : Pic(X) → Pic(S), whose kernel we denoted by Pic(X)[π∗ ]. We deduce from (2) a commutative diagram with exact rows, which shows that j∗ maps Pic(X)[π∗ ] to Pic(C)[d]: 0

/ Pic(X)[π∗ ]

0

 / Pic(C)[d]

/ Pic(X)

j∗

π∗

/ Pic(S)

[d]

 / Pic(C)

j∗

 / Pic(C)

(4)

i∗

Let U be the complement to i(C) in S, and V be the complement to j(C) in X. We write ρ : U → S and σ : V → X for the corresponding open embeddings, so that there is a commutative diagram V

σ

/X

ρ

 /S

π

 U

π

The cohomological (relative) purity theorem for the smooth pair (S, i(C)) gives the following canonical isomorphisms: ρ∗ (µd )U = (µd )S , R1 ρ∗ (µd )U = i∗ (Z/d)C , Rn ρ∗ (µd )U = 0 for n > 1, (5) and similarly for σ : V → X, see [3, Arcata, Thm. V.3.4, pp. 63–64]. The restriction of π turns V into a U -torsor with the structure group µd . We denote the class of this torsor by [V /U ] ∈ H1 (U, µd ). The spectral sequence Hp (S, Rq ρ∗ (µd )U ) ⇒ Hp+q (U, µd ) combined with the purity theorem (5) gives rise to the Gysin exact sequence: ρ∗

∂

ρ∗

θ

∂

0 → H1 (S, µd ) → H1 (U, µd ) → H0 (C, Z/d) → H2 (S, µd ) → H2 (U, µd ) → θ

ρ∗

∂

θ

H1 (C, Z/d) → H3 (S, µd ) → H3 (U, µd ) → H2 (C, Z/d) → H4 (S, µd ) The maps marked with ∂ are defined by this sequence. The maps marked with θ are called the Gysin maps; we denote by Hn (C, Z/2)[θ] the kernel of the corresponding Gysin map.

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Lemma 2.1. The map i∗ : H1 (S, Z/d) → H1 (C, Z/d) coincides with the composition H1 (S, Z/d)

ρ∗

/ H1 (U, Z/d)

/ H2 (U, µd )

∂

/ H1 (C, Z/d),

where the middle arrow is the cup-product with [V /U ]. In particular, i∗ H1 (S, Z/d) is contained in H1 (C, Z/d)[θ]. Moreover, ∗ 1 i H (S, Z/d) is contained in the kernel of the map from H1 (C, Z/d)[θ] to Br(X)[d]/π ∗ (Br(S)[d]) defined by the Gysin sequences for (S, C) and (X, C), so that the map Φ from the Introduction is well definied. Proof. Let us first consider the map 1

∂

0

H (U, µd ) −→ H (C, Z/d) =

n M

H0 (Ci , Z/d) = (Z/d)n ,

i=1

where C1 , . . . , Cn are the irreducible components of C, which are the same as the connected components. For each i = 1, . . . , n we have ∂i ([V /U ]) = mi ∈ H0 (Ci , Z/d) = Z/d, where mi is a positive integer such that for a local equation ti of Ci in an i open affine subset A ⊂ S, the restriction of X to A is given by xd = utm for i ∗ some u ∈ k[A] . Since X is smooth, we must have mi = 1. To show that for any x ∈ H1 (S, Z/d) we have ∂([V /U ] ∪ ρ∗ (x)) = ∗ i (x) we can replace C by each Ci . In view of the inclusion H1 (Ci , Z/d) ⊂ H1 (k(Ci ), Z/d) we can replace S by Spec(OCi ), where OCi is the local ring at the generic point of Ci , replace U by Spec(k(S)) and replace Ci by Spec(k(Ci )). By the conclusion of the previous paragraph, [V /U ] ∈ H1 (k(S), µd ) = k(S)∗ /k(S)∗d goes to 1 ∈ Z/d under the map defined by the discrete valuation of OCi . Since ∗ ρ∗ (x) ∈ OC , the usual formula for the residue of [V /U ]∪ρ∗ (x), understood as i an element of Br(k(S))[d], now gives that ∂i ([V /U ] ∪ ρ∗ (x)) is the restriction of x to H1 (Ci , Z/d). The second statement is now immediate from the Gysin sequence. The pullback of the torsor V → U to V is trivial, hence [V /U ] is in the kernel of π ∗ : H1 (U, µd ) → H1 (V, µd ). This implies that [V /U ]∪ρ∗ (x) is in the kernel of π ∗ : H2 (U, µd ) → H2 (V, µd ), and the last statement of the lemma follows. 

3. Cohomology of double covers From now on we let d = 2 and assume that the characteristic of k is not 2. We write Z/2 in place of µ2 . By restricting the map of ´etale sheaves π∗ Gm,X → Gm,S to the kernels of [2] we obtain a map f : π∗ (Z/2)X → (Z/2)S . The following lemma is a variation on the Smith exact sequence in topology.

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Lemma 3.1. In the bounded derived category D(S) of ´etale sheaves on S the cone of f : π∗ (Z/2)X → (Z/2)S is canonically quasi-isomorphic to Rρ∗ (Z/2)U [1]. Thus in D(S) there is a canonical exact triangle Rρ∗ (Z/2)U −→ π∗ (Z/2)X −→ (Z/2)S .

(6)

The associated exact sequence of cohomology is f

0 −→ (Z/2)S −→ π∗ (Z/2)X −→ (Z/2)S −→ i∗ (Z/2)C −→ 0.

(7)

Proof. We have an exact sequence of Knus, Parimala and Srinivas 0 −→ (Z/2)U −→ π∗ (Z/2)V −→ (Z/2)U −→ 0.

(8)

Indeed, for any geometric point of U the sequence of stalks of (8) is 0 −→ Z/2 −→ (Z/2)2 −→ Z/2 −→ 0, where the second arrow is the diagonal embedding and the third arrow is the sum. This sequence is visibly exact, so (8) is exact, too. For any ´etale sheaf F on S we have a canonical truncation morphism in the derived category D(S) ρ∗ F = τ≤0 Rρ∗ F −→ Rρ∗ F. Since ρ∗ π∗ (Z/2)V = π∗ σ∗ (Z/2)V = π∗ (Z/2)X , we get a commutative diagram of exact triangles in D(S) π∗ (Z/2)X  Rρ∗ (π∗ (Z/2)V )

f

/ (Z/2)S

/ Cone(f )

 / Rρ∗ (Z/2)U

 / Rρ∗ (Z/2)U [1]

(9)

where the bottom triangle is obtained by applying the derived functor Rρ∗ to (8) and shifting by 1. We claim that the morphism Cone(f ) → Rρ∗ (Z/2)U [1] defined by diagram (9), is a quasi-isomorphism. For this we note that the long exact sequence of cohomology attached to the exact triangle Rρ∗ (Z/2)U → Rρ∗ (π∗ (Z/2)V ) → Rρ∗ (Z/2)U gives the exact sequence (7), provided we can justify the surjectivity of the fourth arrow in (7). This can be checked on stalks. The stalk of i∗ (Z/2)C at x∈ / i(C) is zero, so we can assume x ∈ i(C). Then the sequence of stalks of (7) is 0 −→ Z/2−→Z/2 ˜ −→ Z/2−→Z/2 ˜ −→ 0, which is exact, because the middle arrow here is the zero map. Hence (7) is exact. By the purity theorem (5) we now conclude from diagram (9) that the morphism Cone(f ) → Rρ∗ (Z/2)U [1] in D(S) induces an isomorphism on each cohomology group, so is indeed a quasi-isomorphism. 

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The long exact sequence of cohomology associated to (6) has the following form: δ

π

α

∗ −→ Hn−1 (S, Z/2) −→ Hn (U, Z/2) −→ Hn (X, Z/2) −→ Hn (S, Z/2) −→ (10) which is the definition of the maps α and δ.

Corollary 3.2. For any n ≥ 0 the following diagram commutes: Hn (X, Z/2) O g

σ∗

/ Hn (V, Z/2) O

α

π∗

Hn (S, Z/2)

ρ

∗

(11)

π∗

/ Hn (U, Z/2)

Proof. By Lemma 3.1, after extending (9) to the left we obtain a commutative diagram / π∗ (Z/2)X Rρ∗ (Z/2)U ∼ =

 Rρ∗ (Z/2)U

 / Rρ∗ (π∗ (Z/2)V )

which implies the commutativity of the upper triangle of (11). For the lower triangle it is enough to note that the truncation map (Z/2)S = τ≤0 Rρ∗ (Z/2)U −→ Rρ∗ (Z/2)U composed with the first arrow in (6) is the natural map (Z/2)S → π∗ (Z/2)X .  Lemma 3.1 implies that the obvious commutative diagram of exact triangles / π∗ Gm,X / Gm,S T  T

[2]

[2]

 / π∗ Gm,X



[2]

/ Gm,S

gives rise to a commutative diagram in D(S) with exact rows and columns: Rρ∗ (Z/2)U

/ π∗ (Z/2)X

/ (Z/2)S

 T

 / π∗ Gm,X

 / Gm,S

 T

[2]

[2]

 / π∗ Gm,X



(12)

[2]

/ Gm,S

Lemma 3.3. The map T → i∗ (Z/2)C from (3) coincides with the composition of the differential T → Rρ∗ (Z/2)U [1] attached to the left hand column of (12)
and the truncation map Rρ∗ (Z/2)U [1] → τ≥1 Rρ∗ (Z/2)U [1] = i∗ (Z/2)C .

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Proof. This can be checked by a direct calculation on stalks. For this we note that the composed map T → Rρ∗ (Z/2)U [1] → i∗ (Z/2)C can be viewed as provided by the snake lemma applied to the middle and right hand columns of (12). If a geometric point x ∈ S is not in i(C), then the stalk of i∗ (Z/2)C at x is zero, so there is nothing to check. The stalk of T at x ∈ i(C) is an invertible element √ of the strictly local ring Ox,X with norm 1. We can write Ox,X = Ox,S [ t], where t ∈ Ox,S is a local √ equation of C. Extracting a square root we write our element as (a + b t)2 for some a, b ∈ Ox,S such that (a2 − b√2 t)2 = 1, that is, a2 − b2 t = s where s = ±1. We need √ to check that (a + b t)2 is congruent to s modulo the maximal ideal ( t), which is immediate.  From diagram (12) we obtain a commutative diagram of abelian groups with exact rows and columns

[2]

[2]

δ

 / H2 (U, Z/2)

π∗

/ Pic(S)

π∗

 / Pic(S)

 / H2 (X, Z/2)

π∗

 / H2 (S, Z/2)

 Br(X)[2]

π∗

 / Br(S)[2]

[2]

 / H1 (S, T )

 k∗  H1 (S, Z/2)

/ Pic(X)

/ H1 (S, T )

k∗

 / Pic(X)

[2]

cl

α

cl

 0

 0

(13) where cl is the cycle class map. From (13) we cut out the following commutative diagram with exact rows: 0

/ Pic(X)[π∗ ] 

0

β

/ H2 (U, Z/2)/δH1 (S, Z/2)

/ Pic(X) 

π∗

cl

/ H2 (X, Z/2)

π∗

/ π∗ Pic(X) 

/0

cl

/ H2 (S, Z/2)[δ]

/0 (14)

where H2 (S, Z/2)[δ] is the kernel of δ, and the map β is defined by the diagram. Lemma 3.4. The composition of δ : Hn−1 (S, Z/2) → Hn (U, Z/2) with the map ∂ : Hn (U, Z/2) → Hn−1 (C, Z/2) from the Gysin sequence is the restriction map i∗ .

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Proof. The composition of the differential (Z/2)S → Rρ∗ (Z/2)U [1] defined by (6) with the truncation map Rρ∗ (Z/2)U [1] → i∗ (Z/2)C is the map (Z/2)S → i∗ (Z/2)C given by the restriction to C. (One could also give an alternative argument based on Lemma 2.1.)  We would like to trim (14) a bit more. By Corollary 3.2 the map π ∗ : H2 (S, Z/2) → H2 (X, Z/2) factors through H2 (U, Z/2). Since π∗ π ∗ = [2], the subgroup π ∗ H2 (S, Z/2) of H2 (X, Z/2) is in the kernel of the map π∗ to H2 (S, Z/2). The Gysin sequence shows that the quotient of H2 (U, Z/2) by ρ∗ H2 (S, Z/2) is H1 (C, Z/2)[θ]. Taking quotients by the images of H2 (S, Z/2) we obtain from (14) the following commutative diagram with exact rows, where we have also used Lemma 3.4: / Pic(X)

/ Pic(X)[π∗ ]

0

/

0



β

H1 (C,Z/2)[θ] i∗ H1 (S,Z/2)

/



π∗

/ π∗ Pic(X)

π∗

 / H2 (S, Z/2)[δ]

cl

H2 (X,Z/2) π ∗ H2 (S,Z/2)

/0

(15)

cl

/0

The composition π∗ π ∗ : Pic(S) → Pic(S) is [2], hence 2Pic(S) ⊂ π∗ Pic(X). In other words, π∗ Pic(X) contains 2Pic(S), which is the kernel of the cycle class map Pic(S) → H2 (S, Z/2). Since 2Pic(S) is the surjective image of π ∗ Pic(S) ⊂ Pic(X), which is in the kernel of the middle vertical map in (15), an application of the snake lemma to (15) gives rise to the following exact sequence H1 (C, Z/2)[θ]/i∗ H1 (S, Z/2) Br(X)[2] H2 (S, Z/2)[δ] → ∗ → →0 β(Pic(X)[π∗ ]) π (Br(S)[2]) cl(π∗ Pic(X)) (16) From construction it is clear that the second arrow in (16) is given by the map Φ defined in the introduction. To say more about Φ we need to make some simplifying assumptions. The case when the ground field k is algebraically closed will be considered in the next section. 0→

Proposition 3.5. If C is geometrically connected and H1 (S, Z/2) = 0, then the Gysin map θ : H1 (C, Z/2) → H3 (S, Z/2) factors through Pic(C)[2]. Let us denote by Pic(C)[2][θ] the kernel of the resulting map Pic(C)[2] → H3 (S, Z/2). Then Φ gives rise to the exact sequence 0 −→

Pic(C)[2][θ] Br(X)[2] H2 (S, Z/2)[δ] −→ −→ −→ 0. j ∗ (Pic(X)[π∗ ]) π ∗ (Br(S)[2]) cl(π∗ Pic(X))

Proof. The spectral sequence Hp (k, Hq (S, Z/2)) ⇒ Hp+q (S, Z/2) shows that in our assumptions the structure morphism S → Spec(k) induces an isomorphism H1 (k, Z/2)−→H ˜ 1 (S, Z/2). Since C is geometrically connected, we have 0 ∗ H (C, Gm ) = k . The Kummer sequence then gives an exact sequence 0 −→ k ∗ /k ∗2 −→ H1 (C, Z/2) −→ Pic(C)[2] −→ 0 The second arrow here is the map H1 (k, Z/2) → H1 (C, Z/2) induced by the structure morphism C → Spec(k), hence its image is i∗ H1 (S, Z/2).

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The composed map H1 (S, T ) → H2 (U, Z/2) → H1 (C, Z/2), after passing to the quotients by the image of H1 (k, Z/2), becomes β

Pic(X)[π∗ ] −→ H2 (U, Z/2)/δH1 (S, Z/2) −→ Pic(C)[2]. By Lemma 3.3 this composition is the restriction map j ∗ .



Remarks. 1. If S = P2k , then C is a geometrically irreducible curve. We have Pic(S) = Z and Br(S) = Br(k). 2. If Pic(X) = Z, then Pic(S) = Z and Pic(X)[π∗ ] = 0. 3. If we assume that S = P2k and Pic(X) = Z, then Φ is an injective map from Pic(C)[2][θ] to the cokernel of the natural map Br(k)[2] → Br(X)[2].

4. Surfaces 4.1. Cohomology of double covers of surfaces In this section we describe H2 (X, Z/2), where X → S is a double covering of a geometrically simply connected surface. This material is related to the classical Smith theory and is probably well known to the experts. We spell out these descriptions here as they do not seem to be readily available in this form in the literature. Proposition 4.1. Let S be a smooth, projective and geometrically integral surface over k with H1 (S, Z/2) = 0, for example, a geometrically simply connected surface. For any finite surjective morphism π : X → S of degree 2 ramified in a non-empty smooth curve C we have an exact sequence of Γmodules π

∗ 0 −→ H1 (C, Z/2) −→ H2 (X, Z/2)/π ∗ H2 (S, Z/2) −→ H2 (S, Z/2)⊥C −→ 0, (17) where H2 (S, Z/2)⊥C is the subgroup consisting of the elements orthogonal to the connected components of C with the respect to the cup-product pairing.

Proof. We obtain (17) from the bottom exact sequence of (15) considered ¯ By the Poincar´e duality we have H3 (S, Z/2) = 0. This implies that over k. H1 (C, Z/2)[θ] is all of H1 (C, Z/2). It remains to identify H2 (S, Z/2)[δ] with H2 (S, Z/2)⊥C . As H3 (S, Z/2) = 0 the Gysin sequence gives the injectivity of the map H3 (U , Z/2) → H2 (C, Z/2). By Lemma 3.4 we now obtain that H2 (S, Z/2)[δ] is the kernel of the restriction map i∗ : H2 (S, Z/2) −→ H2 (C, Z/2) =

n M

H2 (C i , Z/2) = (Z/2)n ,

i=1

where C 1 , . . . , C n are the connected components of C. Each restriction map H2 (S, Z/2) −→ H2 (C i , Z/2) = Z/2 coincides with the cup-product with the class of C i in H2 (S, Z/2), and the proposition follows. 

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Corollary 4.2. In the assumptions of Proposition 4.1 assume further that C is geometrically connected. Then we have the following properties: (i) the map π ∗ : H2 (S, Z/2) → H2 (X, Z/2) is injective; (ii) the map π∗ : H2 (X, Z/2) → H2 (S, Z/2) is surjective; (iii) there is an exact sequence of Γ-modules π

∗ H2 (S, Z/2) −→ 0; 0 −→ H1 (C, Z/2) −→ H2 (X, Z/2)/π ∗ H2 (S, Z/2) −→ (18) (iv) The automorphism group of the double covering π : X → S defines an action of Z/2 on H2 (X, Z/2). For each i ≥ 1 there is a canonical isomorphism of Γ-modules

Hi (Z/2, H2 (X, Z/2)) = H1 (C, Z/2). Proof. (i) In the commutative diagram (11) the map α : H2 (U , Z/2) → H2 (X, Z/2) is injective because its kernel is the image of H1 (S, Z/2) = 0. Since C is connected we have H0 (C, Z/2) = Z/2. The class of the connected unramified double covering π : V → U is a non-zero element of H1 (U , Z/2). As H1 (S, Z/2) = 0, the Gysin sequence shows that ρ∗ : H2 (S, Z/2) → H2 (U , Z/2) is injective. By the commutativity of the diagram (11) we conclude that π ∗ : H2 (S, Z/2) → H2 (X, Z/2) is injective. (ii) By Proposition 4.1 we need to show that the class of C in H2 (S, Z/2) is zero. This class comes from the class [C] ∈ Pic(S) under the cycle map. But C is the ramification divisor of π : X → S, so [C] is divisible by 2 in Pic(S). (iii) This follows from (ii) and the exact sequence (17). (iv) Let ι : X → X be the involution defined by the double covering π : X → S. It is well known that for even i ≥ 2 the group Hi (Z/2, H2 (X, Z/2)) is canonically isomorphic to Ker(Id − ι∗ )/Im(Id + ι∗ ). For odd i ≥ 1 the same is true once we replace ι∗ by −ι∗ . Since the coefficient group is Z/2, the sign plays no role and hence the answer is the same for even and odd i. The map Id + ι∗ can be written as the composition π

π∗

∗ H2 (S, Z/2) −→ H2 (X, Z/2). H2 (X, Z/2) −→

By (i) the second map here is injective, hence Ker(Id + ι∗ ) = H2 (X, Z/2)[π∗ ]. By (ii) the first map here is surjective, hence Im(Id+ι∗ ) = π ∗ H2 (S, Z/2).  The case when C is a curve of degree 4 in S = P2k , so that X is a del Pezzo surface of degree 2, was discussed in [4, Section IX.1], see also [21, Lemma 1.1]. The exact sequence (18) is the analogue of a similar sequence for double coverings of curves. Let π : C → D be a double covering of smooth, projective and geometrically integral curves ramified in a non-empty 0-dimensional closed subscheme B ⊂ D. Let us denote by (Z/2)B the permutation Γ¯ module whose generators bijectively correspond to the k-points of B. Let

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B (Z/2)B be the submodule of vectors with the zero sum of coor0 ⊂ (Z/2) ¯ is even, the diagonal Z/2 ⊂ (Z/2)B is contained in dinates. Since |B(k)| (Z/2)B 0 . Using the same methods as above, that is, the Gysin sequence and (10), one obtains the well known exact sequence of Γ-modules π

∗ ∗ 1 1 1 0 −→ (Z/2)B 0 /Z/2 −→ H (C, Z/2)/π H (D, Z/2) −→ H (D, Z/2) −→ 0 (19) and the injectivity of π ∗ : H1 (D, Z/2) → H1 (C, Z/2). The proof is left to the reader.

Remarks. 1. Let us return to the situation of Proposition 4.1. In this case the proof of Corollary 4.2 shows that the kernel of the map π ∗ : H2 (S, Z/2) → H2 (X, Z/2) is the subgroup LC generated by the classes of the connected curves C i , for i = 1, . . . , n. Thus the non-degenerate cup-product pairing H2 (S, Z/2) × H2 (S, Z/2) −→ Z/2 induces an isomorphism of Γ-modules π ∗ H2 (S, Z/2) = H2 (S, Z/2)/LC = Hom(H2 (S, Z/2)⊥C , Z/2). 2. The cup-product pairing satisfies the property (π ∗ (x), π ∗ (y)) = 2(x, y) for any x, y ∈ H2 (S, Z/2). Therefore, π ∗ H2 (S, Z/2) is a hyperbolic subspace of H2 (X, Z/2) with respect to the intersection pairing. In the particular case when C is a disjoint union of projective lines, so that H1 (C, Z/2) = 0, we obtain that π ∗ H2 (S, Z/2) is a maximal hyperbolic subspace of H2 (X, Z/2). This situation arises when X is the blowing-up of the eight fixed points of a symplectic involution on a K3 surface, and S is the K3 surface which is the quotient of X by this involution. Likewise, it arises when X is the blowing-up of the sixteen fixed points of the antipodal involution of an abelian surface, and S is the associated Kummer surface. 4.2. Proof of Theorem 1.1 Let k be an algebraically closed field of characteristic not equal to 2. We now assume that S is a surface such that Pic(S)[2] = Br(S)[2] = 0. Since S is projective, the Kummer sequence gives H1 (S, Z/2) = 0. The vanishing of Br(S)[2] implies the surjectivity of the class map Pic(S)/2−→H ˜ 2 (S, Z/2). 1 The Kummer sequence for C gives an isomorphism H (C, Z/2) = Pic(C)[2]. By Poincar´e duality H1 (S, Z/2) = 0 implies H3 (S, Z/2) = 0. β

Lemma 4.3. The composition Pic(X)[π∗ ] −→ H2 (U, Z/2) → Pic(C)[2] is the restriction map j ∗ . Proof. Since k is algebraically closed, and both S and X are connected varieties, from the definition of T we obtain H1 (S, T ) = Pic(X)[π∗ ]. By Lemma 3.3 the map j ∗ : H1 (S, T ) = Pic(X)[π∗ ] → Pic(C)[2] factors through H2 (U, Z/2), as required. 

244

A.N. Skorobogatov Therefore, in our situation, (16) takes the form 0 −→

H2 (S, Z/2)[δ] Pic(C)[2] −→ Br(X)[2] −→ −→ 0. j ∗ Pic(X)[π∗ ] cl(π∗ Pic(X))

(20)

To complete the proof of Theorem 1.1 we use the following lemma. Lemma 4.4. The kernel of δ : H2 (S, Z/2) −→ H3 (U, Z/2) is Pic(S)even /2Pic(S). Proof. By Lemma 3.4 the composition of δ : H2 (S, Z/2) → H3 (U, Z/2) with the map H3 (U, Z/2) → H2 (C, Z/2) from the Gysin sequence is the restriction map i∗ . Since H3 (S, Z/2) = 0, we obtain from the Gysin sequence that H2 (S, Z/2)[δ] is the kernel of the restriction map H2 (S, Z/2) −→ H2 (C, Z/2) = (Z/2)π0 (C) . If C 0 is a connected component of C, the corresponding restriction map Pic(S)/2 → Z/2 is the cup-product modulo 2 with the class of C 0 in Pic(S).  Remark. If U is affine, we have H3 (U, Z/2) = 0 by the affine Lefschetz theorem. 4.3. Surfaces to which the theorem can be applied A K3 surface X that is a double cover of a rational surface S has an involution σ such that S = X/σ. This involution is non-symplectic, that is, it acts on H0 (X, Ω2 ) by −1. Furthermore, the set of fixed points X σ is a non-empty smooth curve which is not necessarily connected [17, Section 4]. Corollary 4.5. Let X be a smooth K3 surface over C with a non-symplectic involution σ such that X σ 6= ∅. Let π : X → X/σ be the quotient map, and let j : X σ → X be the natural closed embedding. Then there is an exact sequence 0 → Pic(X σ )[2]/j ∗ (Pic(X)[π∗ ]) → Br(X)[2] → Pic(X/σ)even /π∗ Pic(X) → 0. Proof. By [17, Section 4] S = X/σ is a smooth rational surface and C = X σ is a non-empty smooth curve in S. Thus the corollary follows from Theorem 1.1.  Below we list examples of del Pezzo surfaces S doubly covered by K3 (or more general) surfaces. Here k is a field of characteristic different from 2. (1) Let S = P2k and let π : X → P2k be a double covering ramified in a smooth curve of even degree C. If Pic(X) is generated by π ∗ O(1), the exact sequence (1) takes the form Φ

0 −→ Pic(C)[2] −→ Br(X)[2] −→ Z/2 −→ 0. This is the exact sequence of [5, Thm. 2.4] or [20, Thm. 6.2]. Without assuming that the Picard rank of X is 1, with some extra work one can deduce from (1) the exact sequence of [12, Thm. 1.1] mentioned in the introduction.

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(2) See [20, Examples 5.6, 6.3] for K3 surfaces that are double covers of the quadric P1k × P1k or the ruled surface F1 . (3) A del Pezzo surface S of degree 4 is a smooth complete intersection of two quadrics in P4k . A smooth complete intersection of the projective cone over S and a quadric in P5k that does not pass through its vertex is a K3 surface doubly covering S. According to [19, Cor. 3.3] for any del Pezzo surface S of degree 4 over k there exists a curve D of genus 2 and a 2-covering Y of the Jacobian J of D with the following property. Let X be the (smooth) Kummer surface associated to Y . Then J[2] acts on X and on S and there is a J[2]-equivariant finite morphism π : X → S of degree 2 ramified in a canonical curve C of genus 5. This morphism maps the sixteen rational curves of X (corresponding to the J[2]-torsor in Y which is the inverse image of 0 under the canonical map Y → J = Y /J[2]) to the sixteen lines in S. See [19, §3.2] for details; see also [16] and [7]. Since the classes of lines on S generate Pic(S), we see that in this case we have π∗ Pic(X) = Pic(S). Thus (1) gives rise to an exact sequence j∗

Φ

Pic(X)[π∗ ] −→ Pic(C)[2] −→ Br(X)[2] −→ 0, so in this case Φ is surjective. (4) A del Pezzo surface S of degree 3 is a smooth cubic surface in P3k . A smooth complete intersection of the projective cone over S and a quadric in P4k not passing through the vertex of the cone is a K3 surface that is a double cover of S. (5) Finally, a del Pezzo surface S of degree 2 is a double cover of P2k ramified in a smooth quartic curve C. The class of C in Pic(S) is divisible by 2, so there is a double cover X → S ramified exactly in C. If C is given by a quartic form f (x, y, z) = 0, then X is a quartic K3 surface with equation t4 = cf (x, y, z) for some c ∈ k ∗ . In the particular case when f (x, y, z) is a diagonal quartic form we obtain the following statement. Proposition 4.6. Let k be a field of characteristic different from 2, and let X ⊂ P3k be the surface given by ax4 + by 4 + cz 4 + dw4 = 0, where a, b, c, d ∈ k ∗ . The group Br(X)Γ contains an element of order 2. Proof. The branch curve C is isomorphic to the Fermat quartic curve, so we may assume f (x, y, z) = x4 + y 4 + z 4 . It is well known that Pic(X) ' Z20 , so that Br(X)[2] ' (Z/2)2 . The group Pic(S) is generated by the (−1)-curves in S that map to the 28 bitangents to C in P2k¯ . On the other hand, Pic(X) is generated by the obvious 48 lines on X, see [18, Lemma 1]. The morphism π : X → S maps these 48 lines to those (−1)-curves in S that lie above the lines in P2k¯ which meet C in exactly one point with multiplicity 4. Hence π∗ Pic(X) is the subgroup of Pic(S) generated by the (−1)-curves above such lines in P2k¯ . A calculation performed by Martin Bright shows that π∗ Pic(X) is a subgroup of Pic(S) of index 2. Now (1) shows that j ∗ (Pic(X)[π∗ ]) has index

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2 in Pic(C)[2]. We conclude from Theorem 1.1 that Φ : Pic(C)[2] → Br(X)[2] factors through an injective map of Galois modules Z/2 → Br(X)[2].  Questions. 1. When is the non-zero element of Br(X)[2]Γ from Proposition 4.6 contained in the image of the natural map Br(X) → Br(X)? 2. How does the Galois group Γ act on Br(X)[2]? In other terms, what ¯ ¯ is the smallest √ subfield K ⊂ k such that Gal(k/K) acts trivially on Br(X)[2]? Is K = k( abcd)? For k = Q a theorem of Evis Ieronymou [9, Thm. 5.2] says that if 2 is not in the subgroup of Q∗ generated by −1, 4, b/a, c/a, d/a and Q∗4 , then the map Br(X){2} → Br(X) is zero, where Br(X){2} is the 2-primary subgroup of Br(X). Thus in this case no non-zero element of Br(X)[2]Γ comes from Br(X). I do not know what happens if this condition is not fulfilled. See [9, 10, 11] and references in these papers for known facts about the Brauer group of diagonal quartic surfaces.

References [1] J.-L. Colliot-Th´el`ene et O. Wittenberg. Groupe de Brauer et points entiers de deux familles de surfaces cubiques affines. Amer. J. Math. 134 (2012) 1303– 1327. [2] B. Creutz and B. Viray. On Brauer groups of double covers of ruled surfaces. Math. Ann. 362 (2015) 1169–1200. [3] SGA 4 12 Cohomologie ´etale. P. Deligne, avec la collaboration de J.F. Boutot, A. Grothendieck, L. Illusie et J.L. Verdier. Springer-Verlag, 1977. [4] I. Dolgachev and D. Ortland. Point sets in projective spaces and theta functions. Ast´erisque 165, Soc. Math. France, 1988. [5] T.J. Ford. The Brauer group and ramified double covers of surfaces. Comm. in Algebra 20 (1992) 3793–3803. [6] T.J. Ford. The Brauer group of an affine double plane associated to a hyperelliptic curve. arXiv:1303.5690 [7] B. Gross and J. Harris. On some geometric constructions related to theta characteristics. Contributions to automrphic forms, geometry, and number theory, Johns Hopkins Press, 2004, 279–311. [8] A. Grothendieck. Le groupe de Brauer I, II, III. Dix expos´es sur la cohomologie des sch´emas. North-Holland, 1968, 46–188. [9] E. Ieronymou. Diagonal quartic surfaces and transcendental elements of the Brauer group. J. Inst. Math. Jussieu 9 (2010) 769–798. [10] E. Ieronymou and A.N. Skorobogatov. Odd order Brauer–Manin obstruction on diagonal quartic surfaces. Adv. Math. 270 (2015) 181–205. Corrigendum available at http://wwwf.imperial.ac.uk/~anskor/publ.htm [11] E. Ieronymou, A.N. Skorobogatov and Yu.G. Zarhin. On the Brauer group of diagonal quartic surfaces. (With an appendix by Peter Swinnerton-Dyer.) J. London Math. Soc. 83 (2011) 659–672.

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[12] C. Ingalls, A. Obus, E. Ozman and B. Viray. Unramified Brauer classes of cyclic covers of the projective plane. Brauer Groups and Obstruction Problems: Moduli Spaces and Arithmetic, volume 320 of Progr. Math., pages 115–153. Birkh¨ auser/Springer, 2017. [13] M.A. Knus, R. Parimala and V. Srinivas. Azumaya algebras with involutions. J. Algebra 130 (1990) 65–82. ´ [14] J.S. Milne. Etale cohomology. Princeton Mathematical Series 33, Princeton University Press, 1980. [15] D. Mumford. Lectures on curves on an algebraic surface. Princeton University Press, 1966. [16] V.V. Nikulin. An analogue of the Torelli theorem for Kummer surfaces of Jacobians. Izvestia Akad. Nauk SSSR Ser. Mat. 38 (1974) 22–41 (Russian). English translation: Math. USSR-Izv. 8 (1974) 21-41. [17] V.V. Nikulin. Quotient-groups of groups of automorphisms of hyperbolic forms by subgroups generated by 2-reflections. Algebro-geometric applications. Current problems in mathematics 18, VINITI, Moscow, 1981, 3–114 (Russian). English translation: J. Soviet Math. 22 (1983) 1401–1475. [18] R.G.E. Pinch and H.P.F. Swinnerton-Dyer. Arithmetic of diagonal quartic surfaces. I. In: L-functions and arithmetic (Durham, 1989) London Math. Soc. Lecture Notes Ser. 153 Cambridge University Press, 1991, pp. 317–338. [19] A. Skorobogatov. Del Pezzo surfaces of degree 4 and their relation to Kummer surfaces. L’Enseignement math. 56 (2010) 73–85 [20] B. van Geemen. Some remarks on the Brauer groups of K3 surfaces. Adv. Math. 197 (2005) 222–247. [21] Yu.G. Zarhin. Del Pezzo surfaces of degree 2 and Jacobians without complex multiplication. Proc. St. Petersburg Math. Soc. 11 (2005) 81–91 (Russian). English translation: AMS Translations 218 (2006) 67–75. Alexei N. Skorobogatov Department of Mathematics, South Kensington Campus Imperial College London SW7 2BZ England U.K. and Institute for the Information Transmission Problems Russian Academy of Sciences 19 Bolshoi Karetnyi, Moscow, 127994 Russia e-mail: [email protected]