A First Course in Algebraic Geometry and Algebraic Varieties 9781800612655, 9781800612747, 9781800612662, 9781800612679

Table of contents :
Contents
Preface
About the Author
Acknowledgments
1. Basics on Commutative Algebra
1.1 Ideals and Operations on Ideals
1.2 UFDs and PIDs
1.3 Polynomial Rings
1.3.1 Polynomials in D[x], where D a UFD
1.3.2 The case D = K a field
1.3.3 Resultant of two polynomials in D[x]
1.3.4 Resultant in D[x1, . . . ,xn] and elimination
1.4 Noetherian Rings and the Hilbert Basis Theorem
1.5 R-Modules, R-Algebras and Finiteness Conditions
1.6 Integrality
1.7 Zariski’s Lemma
1.8 Transcendence Degree
1.9 Tensor Products of R-Modules and of R-Algebras
1.9.1 Restriction and extension of scalars
1.9.2 Tensor product of algebras
1.10 Graded Rings and Modules, Homogeneous Ideals
1.10.1 Homogeneous polynomials
1.10.2 Graded modules and graded morphisms
1.11 Localization
1.11.1 Local rings and localization
1.12 Krull-Dimension of a Ring
Exercises
2. Algebraic Affine Sets
2.1 Algebraic Affine Sets and Ideals
2.2 Hilbert “Nullstellensatz”
2.3 Some Consequences of Hilbert “Nullstellensatz” and of Elimination Theory
2.3.1 Study’s principle
2.3.2 Intersections of affine plane curves
Exercises
3. Algebraic Projective Sets
3.1 Algebraic Projective Sets
3.2 Homogeneous “Hilbert Nullstellensatz”
3.3 Fundamental Examples and Remarks
3.3.1 Points
3.3.2 Coordinate linear subspaces
3.3.3 Hyperplanes and the dual projective space
3.3.4 Fundamental affine open sets (or affine charts) of Pn
3.3.5 Projective closure of affine sets
3.3.6 Projective subspaces and their ideals
3.3.7 Projective and affine subspaces
3.3.8 Homographies, projectivities and affinities
3.3.9 Projective cones
3.3.10 Projective hypersurfaces and projective closure of affine hypersurfaces
3.3.11 Proper closed subsets of P2
3.3.12 Affine and projective twisted cubics
Exercises
4. Topological Properties and Algebraic Varieties
4.1 Irreducible Topological Spaces
4.1.1 Coordinate rings, ideals and irreducibility
4.1.2 Algebraic varieties
4.2 Noetherian Spaces: Irreducible Components
4.3 Combinatorial Dimension
Exercises
5. Regular and Rational Functions on Algebraic Varieties
5.1 Basics on Sheaves
5.2 Regular Functions
5.3 Rational Functions
5.3.1 Consequences of the fundamental theorem on regular and rational functions
5.3.2 Examples
Exercises
6. Morphisms of Algebraic Varieties
6.1 Morphisms
6.2 Morphisms with (Quasi) Affine Target
6.3 Morphisms with (Quasi) Projective Target
6.4 Local Properties of Morphisms: Affine Open Coverings of an Algebraic Variety
6.5 Veronese Morphism: Divisors and Linear Systems
6.5.1 Veronese morphism and consequences
6.5.2 Divisors and linear systems
Exercises
7. Products of Algebraic Varieties
7.1 Products of Affine Varieties
7.2 Products of Projective Varieties
7.2.1 Segre morphism and the product of projective spaces
7.2.2 Products of projective varieties
7.3 Products of Algebraic Varieties
7.4 Products of Morphisms
7.5 Diagonals, Graph of a Morphism and Fiber-Products
Exercises
8. Rational Maps of Algebraic Varieties
8.1 Rational and Birational Maps
8.1.1 Some properties and some examples of (bi)rational maps
8.2 Unirational and Rational Varieties
8.2.1 Stereographic projection of a rank-four quadric surface
8.2.2 Monoids
8.2.3 Blow-up of Pn at a point
8.2.4 Blow-ups and resolution of singularities
Exercises
9. Completeness of Projective Varieties
9.1 Complete Algebraic Var
9.2 The Main Theorem of Elimination Theory
9.2.1 Consequences of the main theorem of elimination theory
Exercises
10. Dimension of Algebraic Varieties
10.1 Dimension of an Algebraic Variety
10.2 Comparison on Various Definitions of “Dimension”
10.3 Dimension and Intersections
10.4 Complete Intersections
Exercises
11. Fiber-Dimension: Semicontinuity
11.1 Fibers of a Dominant Morphism
11.2 Semicontinuity
Exercises
12. Tangent Spaces: Smoothness of Algebraic Varieties
12.1 Tangent Space at a Point of an Affine Variety: Smoothness
12.2 Tangent Space at a Point of a Projective Variety: Smoothness
12.3 Zariski Tangent Space of an Algebraic Variety: Intrinsic Definition of Smoothness
Exercises
Solutions to Exercises
Bibliography
Index

Citation preview

Essential Textbooks in Mathematics ISSN: 2059-7657

The Essential Textbooks in Mathematics series explores the most important topics that undergraduate students in Pure and Applied Mathematics are expected to be familiar with. Written by senior academics as well lecturers recognised for their teaching skills, they offer, in around 200 to 400 pages, a precise, introductory approach to advanced mathematical theories and concepts in pure and applied subjects (e.g. Probability Theory, Statistics, Computational Methods, etc.). Their lively style, focused scope, and pedagogical material make them ideal learning tools at a very affordable price. Published: A First Course in Algebraic Geometry and Algebraic Varieties by Flaminio Flamini (University of Rome “Tor Vergata”, Italy) Analysis in Euclidean Space by Joaquim Bruna (Universitat Autònoma de Barcelona, Spain & Barcelona Graduate School of Mathematics, Spain) Introduction to Number Theory by Richard Michael Hill (University College London, UK) A Friendly Approach to Functional Analysis by Amol Sasane (London School of Economics, UK) A Sequential Introduction to Real Analysis by J M Speight (University of Leeds, UK)



Published by World Scientific Publishing Europe Ltd. 57 Shelton Street, Covent Garden, London WC2H 9HE Head office: 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

Library of Congress Cataloging-in-Publication Data Names: Flamini, Flaminio, author. Title: A first course in algebraic geometry and algebraic varieties / Flaminio Flamini, University of Rome “Tor Vergata”, Italy. Description: New Jersey : World Scientific, [2023] | Series: Essential textbooks in mathematics, 2059-7657 | Includes bibliographical references and index. Identifiers: LCCN 2022028876 | ISBN 9781800612655 (hardcover) | ISBN 9781800612747 (paperback) | ISBN 9781800612662 (ebook) | ISBN 9781800612679 (ebook other) Subjects: LCSH: Geometry, Algebraic--Textbooks. Classification: LCC QA564 .F48 2023 | DDC 516.3/5--dc23/eng/20220708 LC record available at https://lccn.loc.gov/2022028876 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2023 by World Scientific Publishing Europe Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/Q0373#t=suppl Desk Editors: Soundararajan Raghuraman/Adam Binnie/Shi Ying Koe Typeset by Stallion Press Email: [email protected] Printed in Singapore

To my family Arnaldo, Anna Maria, Ornella, Francesca, Lucia. Their love and support are constant motivation and always will be.

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Preface

Geometry is the study of shapes and spaces. Nonetheless these two notions vary according to the different types of geometry, of which Mathematics is rich, and how their study develops over the time. Algebraic Geometry is a branch of Mathematics whose fundamental objects to study are basically Algebraic Varieties which, in simple words, are geometric manifestations of solutions of systems of polynomial equations in several indeterminates. Algebraic Geometry enters into the game when equation solving leaves off, and it becomes even more important to understand the intrinsic and global properties of the solutions instead of explicitly finding solutions. This leads into one of the deepest areas in all of Mathematics, both conceptually and in terms of techniques. Roots of Algebraic Geometry date back to Hellenistic Greeks and Medieval mathematicians, who interpreted some polynomial equations via a geometric point of view. The approach of using geometrical constructions to attack algebraic problems was also adopted by some Renaissance Italian mathematicians, such as G. Cardano and N. Fontana “Tartaglia”, and then favored by most French Mathematicians in the 16th and 17th centuries, including R. Descartes and P. de Fermat, who introduced the well-known coordinate geometry. In the same period, B. Pascal and G. Desargues developed the synthetic notion of projective geometry. It took simultaneous 19th century developments of Non-Euclidean Geometry and of Abelian Integrals in order to bring old algebraic ideas back into the geometrical realm. On the one hand, A. Cayley introduced homogeneous polynomials to study properties of projective spaces and F. Klein, in his Erlangen Program (1872), introduced the method of characterizing all vii

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A First Course in Algebraic Geometry and Algebraic Varieties

geometries with the use of group theory and of projective geometry. Klein’s approach led M. Noether and members of the 20th century Italian school of Algebraic Geometry (e.g. L. Cremona, C. Segre, G. Castelnuovo, F. Enriques, O. Chisini, P. del Pezzo, G. Fano, B. Segre, F. Severi just to mention a few) to the classification of some algebraic varieties up to birational equivalence. On the other hand, ideas of N. H. Abel about Abelian Integrals would have led B. Riemann to the development of Riemann surfaces and to the contribution of H. Poincar´e. In the same period algebrization of Algebraic Geometry by D. Hilbert and E. Noether entered into the game and Hilbert’s Nullstellensatz and Noether’s factorization theorem can be viewed as first steps to connect Algebraic Geometry to Commutative Algebra. In the 20th century, B. L. van der Waerden, O. Zariski and A. Weil developed foundations for Algebraic Geometry based on ring valuation theory, ideals and modules. In 1950–1960s, J. P. Serre and A. Grothendieck completely revised the foundations making use of sheaf theory. Later, from about 1960, and largely led by A. Grothendieck, the idea of schemes was worked out with the support of advanced techniques in homological algebra. From that moment a period of rapid developments of the theory started, which steadily continues nowadays, giving life to several new applications, such as Topology, Number Theory, Complex Analysis, Singularities, Moduli, Tropical Geometry. For a more detailed historical description of developments of Algebraic Geometry through the centuries, we refer the reader to the wonderful first chapter in Ueno (1997). Names mentioned above, and more precisely the importance these names have had and still have, reveal why Algebraic Geometry occupies a central place in Mathematics. At the same time, constant developments of the discipline clarify its deep relationship with several branches of Natural Sciences and Engineering, for example Computer Vision, Coding Theory and Cryptography, Theoretical Physics, Statistical and Machine Learning, just to mention a few. The aim of this book is to provide a “gentle” introduction to the study of Algebraic Varieties. The content of the book arose from courses in Algebraic Geometry that I taught at the University of Rome “Tor Vergata” beginning in 2014. Topics in the book are based on what I have learnt from Sernesi (1992), from handwritten notes from when I was a young Master’s student, now published in Ciliberto (2021), as well as from Hartshorne (1977), Mumford (1988, 1995), and from several research courses taught by mathematicians, mentioned in the following Acknowledgments section, a few years later during my PhD.

Preface

ix

The approach in this book is purely algebraic. Chapter 1 contains notation and main results in Commutative Algebra that will be used in the following chapters; the book is thereby self-contained, making it more convenient for the reader. Chapter 2 focuses on algebraic affine sets and Hilbert Nullstellensatz, analyzing in detail and with several examples the important correspondence between radical ideals and algebraic sets in an affine space and Zariski topology. The same philosophy is then used in Chapter 3 for algebraic sets in a projective space, using the notion of homogeneous elements and ideals, projective closure of an affine algebraic set, and so on. Chapter 4 focuses on topological properties, such as irreducibility and Noetherianity, which allow us to define Algebraic Varieties. After a brief introduction on sheaves, Chapter 5 takes into account regular and rational functions defined on an algebraic variety and the definition of its structural sheaf; many consequences and examples are then discussed. In Chapter 6, we introduce the concepts of morphisms and of isomorphism classes of algebraic varieties; we also examine important examples of morphisms such as the Veronese morphism and morphisms induced by linear systems of divisors. Chapter 7 deals with products of algebraic varieties and the Segre morphism, the graph of a morphism and fiberproducts of morphisms of algebraic varieties. This machinery is then used in Chapter 8 to introduce the definition and basic properties of rational and birational maps among algebraic varieties, which lead to the important notion of birational classification of algebraic varieties, a milestone in Algebraic Geometry. Several examples of rational and unirational varieties are considered, as well as fundamental birational maps, like stereographic projections and blow-ups. In particular we discuss how the blow-up process can be used for desingularization of some wellknown singular plane curves. Chapter 9 contains the fundamental result concerning completeness of projective varieties and its numerous important consequences. In Chapter 10, we introduce various notions of dimension of an algebraic variety, proving that the given notions suitably coincide. We also study the behavior of the dimension when one intersects an algebraic variety with a certain finite number of hypersurfaces, naturally arriving at the notion of complete intersection and set-theoretically complete intersection. A similar analysis occurs in Chapter 11, where semicontinuity for the dimension of the fibers of a dominant morphism between two algebraic varieties is proved. Chapter 12 contains the notion of tangent space, both by making use of coordinates

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A First Course in Algebraic Geometry and Algebraic Varieties

for varieties embedded in either an affine or a projective space, and in the intrinsic definition of Zariski tangent space and derivations. This, together with the definition of dimension, allows to introduce the notion of smooth, respectively singular, point/locus of an algebraic variety. At the end of each chapter the reader can find some suggested summary exercises; detailed solutions are contained at the end of the text in Chapter Solutions to Exercises. The book ends with a useful bibliography, for further information on the topics treated in the book, and with a detailed index of all the symbols and terminology used throughout the text.

About the Author

Flaminio Flamini is full professor in Geometry at the University of Rome “Tor Vergata”. He graduated in Mathematics at the University of Rome “La Sapienza” and got his PhD in Mathematics from the Consortium of Universities of Rome “La Sapienza” and “Roma Tre”. After completing his PhD, he spent a short period at the University of Illinois at Chicago as visiting scholar and then returned to Italy as an Assistant Professor at the University of L’Aquila. His research interests focus on Algebraic Geometry, and he is the author of more than 40 published papers. He is also co-author of some books concerning Commutative Algebra and its applications, and textbooks on Linear Algebra for undergraduate students. Since 2009 he has been member of the Scientific Board of the PhD School in Mathematics of the Department of Mathematics at the University of Rome “Tor Vergata”. He has been Principal Investigator or a member of numerous Committee Boards which award research grants.

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Acknowledgments

There are many people I should thank, perhaps first and foremost, many of my colleagues who make me daily appreciate the pleasure of collaboration, discussing Mathematics, and doing research together. On the other hand, if I take into account how my professional path began, I cannot forget those people who have turned out to be central to my academic journey from my early days as a young Master’s and PhD student, those who have always believed in me and have encouraged me to do better and better. These are certainly (in alphabetical order) Ciro Ciliberto, Edoardo Sernesi ed Alessandro Verra. Besides them, I had also the privilege as a student to attend advanced courses in Algebraic Geometry taught by Enrico Arbarello, Lucia Caporaso, Fabrizio Catanese, Luca Chiantini, Igor Dolgachev, Lawrence Ein, Robert Lazarsfeld, Angelo F. Lopez, Rick Miranda and Rita Pardini. From all these people I have learnt not only concepts and techniques in Algebraic Geometry, but mainly the passion to face this amazing, elegant, and inspiring research area. I will always be grateful to them. It is a pleasure to deeply thank my colleagues and friends Gilberto Bini, Seonja Kim, Andreas L. Knutsen, and Alessandra Sarti, not only for valuable suggestions on either the content of the text or how to produce figures with TikzPgf, but also for their words of encouragement and appreciation regarding this book.

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A First Course in Algebraic Geometry and Algebraic Varieties

Fianlly, I wish to warmly thank my former students Marco Carfagnini, Stefano Cipolla, Fabrizio Clementi, Daniele Di Tullio, Francesco Recupero and Emiliano Torti, who attended my Algebraic Geometry courses, for all the questions they asked me, for having carefully read the first draft of this book, and for pointing out some typos and inaccuracies.

Contents

Preface

vii

About the Author

xi

Acknowledgments

xiii

1.

Basics on Commutative Algebra 1.1 1.2 1.3

Ideals and Operations on Ideals . . . . . . . . . . . UFDs and PIDs . . . . . . . . . . . . . . . . . . . . Polynomial Rings . . . . . . . . . . . . . . . . . . . 1.3.1 Polynomials in D[x], where D a UFD . . . 1.3.2 The case D = K a field . . . . . . . . . . . 1.3.3 Resultant of two polynomials in D[x] . . . 1.3.4 Resultant in D[x1 , . . . , xn ] and elimination 1.4 Noetherian Rings and the Hilbert Basis Theorem . 1.5 R-Modules, R-Algebras and Finiteness Conditions 1.6 Integrality . . . . . . . . . . . . . . . . . . . . . . . 1.7 Zariski’s Lemma . . . . . . . . . . . . . . . . . . . . 1.8 Transcendence Degree . . . . . . . . . . . . . . . . 1.9 Tensor Products of R-Modules and of R-Algebras . 1.9.1 Restriction and extension of scalars . . . . 1.9.2 Tensor product of algebras . . . . . . . . . 1.10 Graded Rings and Modules, Homogeneous Ideals . 1.10.1 Homogeneous polynomials . . . . . . . . . 1.10.2 Graded modules and graded morphisms . .

xv

1 . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

2 4 5 5 8 9 12 13 16 19 21 23 27 29 30 31 35 41

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A First Course in Algebraic Geometry and Algebraic Varieties

1.11 Localization . . . . . . . . . . . . . 1.11.1 Local rings and localization 1.12 Krull-Dimension of a Ring . . . . . Exercises . . . . . . . . . . . . . . . . . . 2.

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

Algebraic Affine Sets and Ideals . . . . . . . . . Hilbert “Nullstellensatz” . . . . . . . . . . . . . Some Consequences of Hilbert “Nullstellensatz” and of Elimination Theory . . . . . . . . . . . . 2.3.1 Study’s principle . . . . . . . . . . . . . 2.3.2 Intersections of affine plane curves . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

55 70

. . . .

74 74 75 76

. . . .

. . . .

. . . .

Algebraic Projective Sets . . . . . . . . . . . . . . . . . Homogeneous “Hilbert Nullstellensatz” . . . . . . . . . Fundamental Examples and Remarks . . . . . . . . . . 3.3.1 Points . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Coordinate linear subspaces . . . . . . . . . . . 3.3.3 Hyperplanes and the dual projective space . . 3.3.4 Fundamental affine open sets (or affine charts) of Pn . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Projective closure of affine sets . . . . . . . . . 3.3.6 Projective subspaces and their ideals . . . . . . 3.3.7 Projective and affine subspaces . . . . . . . . . 3.3.8 Homographies, projectivities and affinities . . . 3.3.9 Projective cones . . . . . . . . . . . . . . . . . 3.3.10 Projective hypersurfaces and projective closure of affine hypersurfaces . . . . . . . . . . . . . . 3.3.11 Proper closed subsets of P2 . . . . . . . . . . . 3.3.12 Affine and projective twisted cubics . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Topological Properties and Algebraic Varieties 4.1

42 46 48 52 55

. . . .

Algebraic Projective Sets 3.1 3.2 3.3

4.

. . . .

Algebraic Affine Sets 2.1 2.2 2.3

3.

. . . .

79 . . . . . .

80 83 86 86 87 87

. . . . . .

87 89 91 94 95 99

. 99 . 100 . 101 . 106 109

Irreducible Topological Spaces . . . . . . . . . . . . . . . 109 4.1.1 Coordinate rings, ideals and irreducibility . . . . 112

xvii

Contents

4.1.2 Algebraic varieties . . 4.2 Noetherian Spaces: Irreducible 4.3 Combinatorial Dimension . . Exercises . . . . . . . . . . . . . . . 5.

. . . . . . . . Components . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . . . . on . . . . . .

Morphisms of Algebraic Varieties Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . Morphisms with (Quasi) Affine Target . . . . . . . . . Morphisms with (Quasi) Projective Target . . . . . . . Local Properties of Morphisms: Affine Open Coverings of an Algebraic Variety . . . . . . . . . . . . . . . . . . 6.5 Veronese Morphism: Divisors and Linear Systems . . . 6.5.1 Veronese morphism and consequences . . . . . 6.5.2 Divisors and linear systems . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141 143 147

. 149 . 151 . 160 . . . . .

Products of Algebraic Varieties Products of Affine Varieties . . . . . . . Products of Projective Varieties . . . . . 7.2.1 Segre morphism and the product spaces . . . . . . . . . . . . . . . 7.2.2 Products of projective varieties . 7.3 Products of Algebraic Varieties . . . . . 7.4 Products of Morphisms . . . . . . . . . . 7.5 Diagonals, Graph of a Morphism and Fiber-Products . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . .

125 127 130

149

6.1 6.2 6.3 6.4

7.1 7.2

114 115 118 123

125

Basics on Sheaves . . . . . . . . . . . . . . . . . . . . Regular Functions . . . . . . . . . . . . . . . . . . . . Rational Functions . . . . . . . . . . . . . . . . . . . 5.3.1 Consequences of the fundamental theorem regular and rational functions . . . . . . . . . 5.3.2 Examples . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.

. . . .

Regular and Rational Functions on Algebraic Varieties 5.1 5.2 5.3

6.

. . . .

164 166 171 174 177 179

. . . . . . . . . . . . . . . . of projective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 179 . 182 . . . .

184 186 188 189

. . . . . . . . . 191 . . . . . . . . . 193

xviii

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A First Course in Algebraic Geometry and Algebraic Varieties

Rational Maps of Algebraic Varieties

195

8.1

Rational and Birational Maps . . . . . . . . . . 8.1.1 Some properties and some examples of (bi)rational maps . . . . . . . . . . . 8.2 Unirational and Rational Varieties . . . . . . . 8.2.1 Stereographic projection of a rank-four quadric surface . . . . . . . . . . . . . . 8.2.2 Monoids . . . . . . . . . . . . . . . . . . 8.2.3 Blow-up of Pn at a point . . . . . . . . 8.2.4 Blow-ups and resolution of singularities Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 9.

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . of . . . .

. . . . . . 217 . . . . . . 219 . . . . . . 221 . . . . . . 222

Dimension of Algebraic Varieties 10.1 Dimension of an Algebraic Variety . 10.2 Comparison on Various Definitions of “Dimension” . . . . . . . . . . . . . . 10.3 Dimension and Intersections . . . . . 10.4 Complete Intersections . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . .

206 208 209 212 215 217

Complete Algebraic Varieties . . . . . . . The Main Theorem of Elimination Theory 9.2.1 Consequences of the main theorem elimination theory . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . .

11.

. . . . . 199 . . . . . 205

Completeness of Projective Varieties 9.1 9.2

10.

. . . . . 195

225 . . . . . . . . . . . 225 . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

Fiber-Dimension: Semicontinuity

. . . .

. . . .

230 232 235 239 241

11.1 Fibers of a Dominant Morphism . . . . . . . . . . . . . . 241 11.2 Semicontinuity . . . . . . . . . . . . . . . . . . . . . . . 244 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 12.

Tangent Spaces: Smoothness of Algebraic Varieties 12.1 Tangent Space Smoothness . 12.2 Tangent Space Smoothness .

249

at a Point of an Affine Variety: . . . . . . . . . . . . . . . . . . . . . . . . 249 at a Point of a Projective Variety: . . . . . . . . . . . . . . . . . . . . . . . . 253

Contents

xix

12.3 Zariski Tangent Space of an Algebraic Variety: Intrinsic Definition of Smoothness . . . . . . . . . . . . . 255 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Solutions to Exercises

265

Bibliography

295

Index

297

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Chapter 1

Basics on Commutative Algebra

This chapter contains fundamental definitions and results that will be frequently used in the next chapters. This choice allows the reader to have a self-contained book, avoiding to retrieve basic results in Commutative Algebra throughout the vast literature. For further information and details, the reader is referred to, e.g. Atiyah and McDonald (1969), Eisenbud (1995), Lang (2002) and Matsumura (1980). All rings considered in this book will be commutative and with a multiplicative identity and it will be taken from granted once and for all. If R is a ring, 0R and 1R (or simply 0 and 1, if no confusion arises) will respectively denote its additive and multiplicative identities. Given R and S two rings, any non-zero ring homomorphism ϕ : R → S will be such that ϕ(1R ) = 1S . An integral domain (or simply, domain) is a ring with no zero-divisors, i.e. where cancellation law holds. A field K is a domain in which every non-zero element is invertible, i.e. a unit. In symbols, U (K) = K \ {0}, where U (K) denotes the (multiplicative) group of units. For any field K the symbol K will denote its algebraic closure (K always exists and it is uniquely determined up to field isomorphism, cf. M. Artin’s proof in, e.g. Lang (2002, cf. Chapter V, § 2. Theorem 2.5 and Corollaries 2.6 and 2.9)). Z will denote the domain of integers, whereas Q, R and C will denote the fields of rational, real and complex numbers, respectively. Zp will denote the finite field with p elements, p ∈ Z a prime. For the sake of completeness of this preliminary overview, we have to recall the classical result (first proved by Gauss in 1799), whose proof can be found in, e.g. Lang (2002), to which the interested reader is referred. 1

2

A First Course in Algebraic Geometry and Algebraic Varieties

Theorem 1.0.1. The field C is algebraically closed. Let R be a ring. An ideal I ⊆ R is a subset of R such that, for any a, b ∈ I and for any r ∈ R one has a + b ∈ I and ra ∈ I. If R is a domain, Q(R) will denote its quotient field, so that R ⊆ Q(R) where equality holds if and only if R is a field. For any ring R, R[x] denotes the ring of polynomials in one indeterminate x and with coefficients from R. More generally, if x := (x1 , . . . , xn ) are n indeterminates over R, R[x1 , . . . , xn ] denotes the ring of polynomials in the n indeterminates and with coefficients from R; for simplicity, sometimes we will use the symbol R[x] to denote this ring, if no confusion arises. 1.1

Ideals and Operations on Ideals

Let R be a ring. A proper ideal p ⊂ R is said to be prime if rs ∈ p implies that either r ∈ p or s ∈ p; in particular, p is prime if and only if R/p is an integral domain. Analogously, a proper ideal m ⊂ R is said to be maximal if no proper ideal I in R exists such that m ⊂ I ⊂ R, where all the inclusion are strict; in particular, m is maximal if and only if R/m is simple (i.e. with no proper ideals) which, by the assumptions on R, is equivalent to R/m being a field. It is therefore clear that a maximal ideal is also prime, but the converse is not true: e.g. consider the ideal (x) ⊂ Z[x], which is prime but not maximal. We recall some standard operations on ideals, which will be frequently used in the next chapters. Let I be an ideal of R; a set S of elements of R is called a set of generators for I if    ri si | si ∈ S, ri ∈ R , I= i

where in Σi ri si only finitely many ri ’s are non-zero. The ideal I is said to be a finitely generated ideal if I admits a finite set of generators S := {s1 , . . . , sn }; in such a case, we will write I := (s1 , . . . , sn ). In particular, when n = 1, I is called a principal ideal. If I1 , I2 are ideals, their sum is defined as I1 + I2 := {x1 + x2 | x1 ∈ I1 , x2 ∈ I2 }; it is the smallest ideal of R containing both I1 and I2 . More generally, if we have a (possibly infinite) family {Iα }α∈A of ideals of R, their sum is

Basics on Commutative Algebra

defined as  α∈A

 Iα :=



3

 xα | xα ∈ Iα ,

α∈A

where in the sums almost all (i.e. up to a finite number) of the xα ’s equal 0. As above, Σα∈A Iα is the smallest ideal of R containing all the Iα ’s. For a (possibly infinite) family {Iα }α∈A of ideals, their intersection ∩α∈A Iα is an ideal. The product of two ideals I1 and I2 is denoted by I1 ·I2 and defined to be the ideal generated by all the products x1 x2 , for any x1 ∈ I1 and x2 ∈ I2 . The elements of this ideal are finite sums of the forms Σi xi yi , where xi ∈ I1 and yi ∈ I2 , for any i. Similarly, one can define the product of a (possibly infinite) family {Iα }α∈A of ideals in R. As a particular case of products of ideals, for any positive integer n, one has the notion of nth -power of an ideal I, which is denoted by I n (by convention, one poses I 0 := (1) = R). Thus, I n is the ideal generated by all the products of the form x1 x2 · · · xn , where xi ∈ I, for any 1  i  n. For any ideal I of R, one defines the radical of I to be the ideal √ I := {x ∈ R | xn ∈ I, for some positive integer n} (1.1) (some texts denote the radical with the symbol rad(I); see, e.g. Atiyah √ and McDonald (1969)). Note that, for any ideal I, one clearly has I ⊆ I. Definition 1.1.1. I is said to be a radical ideal (or simply radical), if √ I = I, equivalently if R/I is a reduced ring, namely R/I has no nilpotent elements, i.e. there exists no x ∈ R \ {0} such that xn = 0 for some integer n  2). Lemma 1.1.2.

√ (i) For any ideal I ⊆ R, I is radical. (ii) Any prime (respectively, maximal) ideal is radical.

Proof. √ √ √ I. Now, f ∈ I implies there (i) As for any ideal, one has I ⊆ √ positive exists a positive integer n such√that f n ∈ I. Thus, for √some√ I ⊆ I. integer m, (f n )m ∈ I, i.e. f ∈ I, which means that (ii) It directly follows from the definition of prime (respectively, maximal)  ideal.

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A First Course in Algebraic Geometry and Algebraic Varieties

Let ϕ : R → S be any ring homomorphism. Definition 1.1.3. If J ⊆ S is any ideal, ϕ−1 (J) is an ideal in R, called the contracted ideal of J w.r.t. ϕ and denoted by J c (sometimes even by J ∩ R, even if ϕ is not necessarily injective, cf. Atiyah and McDonald (1969)). Conversely, if I ⊆ R is an ideal, the ideal of S generated by the subset ϕ(I) ⊆ S is called the extended ideal of I w.r.t. ϕ and denoted by I e . Remark 1.1.4. (i) Note that ϕ(I) ⊆ S in general is not an ideal: consider, e.g. ι : Z → Q to be the natural inclusion and I any proper ideal of Z. (ii) For any ideal J ⊆ S one has J ⊇ (J c )e and in general the inclusion is ϕ strict; consider, e.g. Z2 → Z2 [x], where ϕ the natural inclusion, and J = (x2 + x + 1), so (J c )e = (0). Vice versa, for any ideal I ⊆ R one has I ⊆ (I e )c and in general the inclusion is strict; consider as in (i) ι Z → Q and I = (p), for some prime p ∈ Z, so (I e )c = (1). (iii) If p ⊂ S is a prime ideal, then pc ⊂ R is also prime; conversely, being p ⊂ R a prime ideal does not imply that pe ⊆ S is necessarily a prime ideal, cf., e.g. (ii) above. (iv) Note that if m ⊂ S is a maximal ideal then mc ⊆ R in general is only ϕ

prime but not maximal. Indeed, consider, e.g. Z[x]  Z defined by ϕ(q(x)) = q(0), for any q(x) ∈ Z[x], and J = (p), for any prime p ∈ Z; in such a case (p)c is prime but not maximal since Z[x]/(p)c ∼ = Zp [x] is an integral domain but not a field. 1.2

UFDs and PIDs

Recall that a ring R is said to be a principal ring if every ideal I of R is principal. If R is an integral domain and a principal ring, then R is said to be a principal ideal domain (PID). Any field K and Z are easy examples of PIDs. Recall that, when K is a field, the ring K[x] is a Euclidean domain, roughly speaking it is an integral domain where Euclidean division algorithm holds. This implies that K[x] is a PID (cf., e.g. Lang, 2002, IV § 1, Theorem 1.2). From now on, let D denote an integral domain. Given a, b ∈ D we say that a divides b, or that b is divisible by a (in symbols a|b) if there exists c ∈ D s.t. b = ca; in particular, a is invertible if and only if a|1D . Elements a, b ∈ D are called associate elements (simply associates) if a|b and b|a, i.e. b = ea, for some e ∈ U (D). An element a ∈ D \ {0} is said to be irreducible

Basics on Commutative Algebra

5

if it is not a unit in D and, whenever one can write a = bc, with b, c ∈ D, then either b or c is a unit in D; in other words a is irreducible if and only if it is divisible only by its associate elements and by the units in D. An element a ∈ D \ {0} is said to have a unique factorization into irreducible elements if there exist u ∈ U (D) and pi ∈ D irreducible elements, 1  i  r (not necessarily all distincts), such that a = up1 · · · pr and, if moreover we have two factorizations a = up1 · · · pr = u q1 · · · qs , we have r = s and, after a permutation of the indexes, we have pi = ui qi for any 1  i  r, for some ui ∈ U (D). D is said to be a unique factorization domain (UFD), if every nonzero element of D has a unique factorization into irreducible elements. In particular, any prime element (i.e. generating a prime ideal) of a UFD is an irreducible element (the converse being always true). At last, in a UFD one can define a greatest common divisor for finite set of elements in D (this is defined up to units of D). Recall that any PID is a UFD (cf., e.g. Lang, 2002, II § 5, Theorem 5.2); in particular, Z, any field K and K[x] are UFD’s. 1.3

Polynomial Rings

Here we collect some useful terminology and results on polynomials in one or more indeterminates, with coefficients from an integral domain, in particular from a UFD. 1.3.1

Polynomials in D[x], where D a UFD

If D0 is an integral domain, definitions in Section 1.2 apply to the integral domain D := D0 [x1 , . . . , xn ] (cf. Theorem 1.3.1 and Remark 1.3.2) giving rise to divisibility among polynomials, associate polynomials, irreducible polynomials, etc. Any polynomial of degree 1 with invertible coefficients is always irreducible. In general, the irreducibility of a polynomial depends on the domain D0 ; e.g. the polynomial x2 + 1 is irreducible as an element of R[x] but it factorizes into irreducible elements (x − i)(x + i) in C[x], where i2 = −1. Recall the following fundamental results, whose proofs can be found in, e.g. Lang (2002, cf. Chapter IV § 2), to which we refer the interested reader.

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A First Course in Algebraic Geometry and Algebraic Varieties

Theorem 1.3.1 (Unique factorization theorem). Let D be a UFD and x be an indeterminate over D. Then D[x] is a UFD. Remark 1.3.2. When K is a field, K[x1 , . . . , xn ] is a UFD for any n  1, as one obtains by recursively applying Theorem 1.3.1: pass from D := K[x1 , . . . , xn−1 ] a UFD to D := K[x1 , . . . , xn ] = D[xn ]. Theorem 1.3.3 (Gauss’ theorem). Let D be a UFD, K := Q(D) be its quotient field and x be an indeterminate over D. Let f := f (x) = Σni=0 di xi ∈ D[x] be a non-constant polynomial. Assume that f is a primitive polynomial, namely the content c(f ) := g.c.d.(d0 , d1 , . . . , dn ) ∈ D of f is such that c(f ) = 1 (where the equality is meant up to a unit in D). Then f is irreducible in D[x] if and only if it is irreducible in K[x]. Suppose now to have f ∈ D[x] and consider its factorization into irreducible elements in D[x]; merging the repeated irreducible factors of f , one gets f = g1e1 · · · gses ,

(1.2)

where the gi ’s are all the irreducible, distinct factors of f ∈ D[x] and ei ’s are non-negative integers, 1  i  s. The exponent ei , 1  i  s, is called the multiplicity of the factor gi in f . If ei > 1 then gi is said to be a multiple factor of f . One obviously has deg(f ) = Σsi=1 ei deg(gi ). Definition 1.3.4. If f ∈ D[x] is a non-constant polynomial then f = a0 + a1 x + a2 x2 + · · · + an xn ∈ D[x], for some aj ∈ D, 0  j  n with an = 0. The integer n  1 is the degree of f and an ∈ D is called the leading coefficient of f , which is also denoted by lc(f ). A non-constant polynomial is said to be monic if lc(f ) = 1. A non-constant polynomial f ∈ D[x], for which lc(f ) ∈ U (D), is always associated to a monic polynomial f  ∈ D[x]. In particular, for D = K a field, any non-constant polynomial is always associated to a monic one. Definition 1.3.5. Let f ∈ D[x] be a non-constant polynomial. An element α ∈ D is said to be a root of f , if f (α) = 0. The following is a straightforward result. Lemma 1.3.6. α ∈ D is a root of f ∈ D[x] if and only if (x − α) divides f in D[x].

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7

Definition 1.3.7. α ∈ D is said to be a multiple root of f ∈ D[x], if (x−α) is a multiple factor of f , i.e. if one has f = (x − α)e g, for some integer e  2 (cf. decomposition (1.2)). The greatest integer e for which a factorization as above exists is called the multiplicity of the root α for f . If e = 1, then α is said to be a simple root of f . If α is not a root of f , then one has e = 0. Note that e is the multiplicity of the root α if and only if one has g(α) = 0 for a decomposition as in Definition 1.3.7; moreover, since deg(g) = deg(f ) − e, one has e  deg(f ), for any root α of f . Corollary 1.3.8. A non-constant polynomial f ∈ D[x] of degree d has at most d roots in D (when the roots are counted with multiplicity). Proof. It directly follows by induction on d.



The notion of derivative of a polynomial can be given formally, with no use of infinitesimal calculus. This allows one to consider derivatives of polynomials in D[x], for D any domain. Precisely, if f := ad xd + ad−1 xd−1 + · · · + a1 x + a0 ∈ D[x], one defines the (first) derivative of f to be df = f  := dad xd−1 + (d − 1)ad−1 xd−2 + · · · + a1 dx

(1.3)

and recursively, the kth derivatives, for any k  2, to be the polynomial k d(f [k−1] ) given by ddxfk = f [k] := , where f [1] := f  . dx Similarly, if f ∈ D[x1 , . . . , xn ], one can define the partial derivative with respect to the indeterminate xj , 1  j  n, denoted by ∂f ∂xj , to be the (first) derivative of f considered as an element of (D[x1 , . . . , xj−1 , xj+1 , . . . , xn ])[xj ]. It is therefore clear that one can also define higher order partial derivatives and that the Schwarz rule holds, as it can be easily verified on monomials. We will not dwell on this. Lemma 1.3.9. α ∈ D is a multiple root of f ∈ D[x] if and only if it is a common root of f and f  . Proof. Since α ∈ D is a root of f , by Lemma 1.3.6, f = (x − α)e g, for some g ∈ D[x] s.t. g(α) = 0 and for some positive integer e. One has f  = e(x − α)e−1 g + (x − α)e g  . If f  = 0 (e.g. if D is of characteristic 0), then e = 1 if and only if f  (α) = g(α) = 0, proving the statement in this

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A First Course in Algebraic Geometry and Algebraic Varieties

case. If otherwise f  = 0 (which can only occur in positive characteristic since f is non-constant by assumption), then f (x) = (h(x))p , for some nonconstant polynomial h ∈ D[x] (cf. Lang, 2002, Proposition 1.12, p. 179), which implies that α is a root of multiplicity at least p.  1.3.2

The case D = K a field

Here we focus on the case D = K a field. Proposition 1.3.10. Let K be any field. Then K[x] contains infinite irreducible monic polynomials. Proof. K[x] contains irreducible polynomials as, e.g. x − α ∈ K[x], for any α ∈ K. If K is an infinite field, we are done. If otherwise K is a finite field, assume by contradiction that f1 , . . . , fn ∈ K[x] are all the monic, irreducible polynomials, for some positive integer n. The polynomial f := 1 + (f1 · · · fn ) ∈ K[x] is monic and it must irreducible, indeed none of the  fi ’s divides f , 1  i  n. This contradicts our assumption. As a direct consequence of Lemma 1.3.6, by induction on the degree of the polynomial one obtains the following. Theorem 1.3.11. Let K be an algebraically closed field. Any non-constant polynomial f ∈ K[x] of degree d contains exactly d roots in K (when roots are counted with multiplicity). In particular, f factorizes in K[x] as f = a(x − α1 ) · · · (x − αd ), where a ∈ K \ {0} and α1 , . . . , αd ∈ K are all the (not necessarily distinct) roots of f . Corollary 1.3.12. If K is an algebraically closed field, then K contains infinitely many elements. Proof. It directly follows from Proposition 1.3.10 and Theorem 1.3.11.  For what concerns polynomial rings K[x1 , . . . , xn ], a fundamental result is the following. Theorem 1.3.13. Let K be any infinite field and let f ∈ K[x1 , . . . , xn ], n  1, be a non-constant polynomial. If there exists a subset J ⊆ K with infinitely many elements s.t. f (a1 , . . . , an ) = 0 for any (a1 , . . . , an ) ∈ J n , where J n := J ×· · ·×J the n-tuple product of J, then f = 0 in K[x1 , . . . , xn ].

Basics on Commutative Algebra

9

Proof. The proof is by induction on n. If n = 1, the statement directly follows from Corollary 1.3.8. Assume therefore n  2 and that the statement holds true for all polynomials in n − 1 indeterminates with coefficients from K. Take f ∈ K[x1 , . . . , xn ] which, by the inductive hypothesis, can be considered to be non-constant with respect to all the indeterminates (in particular with respect to xn ). Let d be the degree of f with respect to xn , i.e. when f is considered as an element of (K[x1 , . . . , xn−1 ])[xn ] ∼ = K[x1 , . . . , xn ]. Thus, one can write f = f0 + xn f1 + · · · + xdn fd ,

(1.4)

with fj ∈ K[x1 , . . . , xn−1 ], 0  j  d, and fd = 0 by the assumption on the degree. With this set-up, assume by contradiction there exists an infinite subset J ⊆ K for which f (a1 , . . . , an ) = 0, ∀ (a1 , . . . , an ) ∈ J n . Since fd = 0, by induction there exists (a1 , . . . , an−1 ) ∈ J n−1 s.t. fd (a1 , . . . , an−1 ) = 0. Thus, (1.4) gives φf (xn ) := f (a1 , . . . , an−1 , xn ) ∈ K[xn ], where φf (xn ) = f0 (a1 , . . . , an−1 ) + xn f1 (a1 , . . . , an−1 ) + · · · + xdn fd (a1 , . . . , an−1 ) is a non-constant polynomial in K[xn ] since fd (a1 , . . . , an−1 ) = 0. This gives a contradiction since φf (xn ) ∈ K[xn ] and φf (a) = f (a1 , . . . , an−1 , a) = 0 for any a ∈ J which is infinite.  Remark 1.3.14. Theorem 1.3.13 is false if K is a finite field; indeed, consider f = xp − x ∈ Zp [x], where p ∈ Z a prime; one has f (α) = 0, for any α ∈ Zp (cf., e.g. Example 2.1.10-(iii) for details), even if f is not the zero-polynomial. 1.3.3

Resultant of two polynomials in D[x]

For the whole section, D will denote a UFD. Here we shall briefly recall some basic results which allow one to establish if two polynomials f, g ∈ D[x] have a non-constant, common factor in D[x] (recall Theorem 1.3.1). Let f (x) = an xn + · · · + a0 ,

g(x) = bm xm + · · · + b0 ,

with an bm = 0 (1.5)

be non-constant polynomials in D[x]. Lemma 1.3.15 (Euler’s lemma). In the above assumptions, f and g have a non-constant common factor in D[x] if and only if there exist non-zero polynomials p(x), q(x) ∈ D[x], with deg(p) < m, deg(q) < n, such that pf = qg.

(1.6)

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A First Course in Algebraic Geometry and Algebraic Varieties

Proof. If φ ∈ D[x] is a non-constant common factor of f and g, then f = φq and g = φp, for some p, q ∈ D[x], which verify the statement. Conversely, suppose (1.6) holds, with deg(p) < m and deg(q) < n. In particular, any irreducible factor of g in D[x] divides pf ; since deg(p) < m = deg(g), there exists at least a non-constant, irreducible factor φ of g which divides f , proving the statement.  With notation and assumptions as in (1.5), one defines the Sylvester matrix of the two polynomials f, g ∈ D[x] to be the (m + n) × (m + n) matrix ⎞ ⎛ a0 a1 . . . an 0 0 . . . 0 ⎜ 0 a0 a1 . . . an 0 . . . 0 ⎟ ⎟ ⎜ ⎟ ⎜ ... ⎟ ⎜ ⎟ ⎜ ⎜ 0 . . . 0 0 a0 a1 . . . an ⎟ (1.7) S(f, g) := ⎜ ⎟, ⎜ b0 b1 . . . bm 0 0 . . . 0 ⎟ ⎟ ⎜ ⎜ 0 b0 b1 . . . bm 0 . . . 0 ⎟ ⎟ ⎜ ⎠ ⎝ ... 0 . . . 0 0 b0 b1 . . . bm where the block containing a0 , . . . , an consists of m = deg(g) rows whereas the block containing b0 , . . . , bm consists of n = deg(f ) rows. One denotes by R(f, g) := det(S(f, g)) ∈ D,

(1.8)

which is called the resultant of the two polynomials. One has the following theorem. Theorem 1.3.16. Let f and g be polynomials as in (1.5). Then f and g have a non-constant common factor in D[x] if and only if R(f, g) = 0. Proof. By Euler’s lemma, the existence of polynomials p, q ∈ D[x] satisfying (1.6), with deg(p) < m and deg(q) < n, is equivalent to the existence of n + m elements in D, say ci , 0  i  m − 1, not all zero, and dj , 0  j  n − 1, not all zero, such that (cm−1 xm−1 + · · · + c0 ) f (x) = (−dn−1 xn−1 − · · · − d0 ) g(x),

(1.9)

holds true. This is equivalent to the fact that the ci ’s and the dj ’s give rise to a non-zero solution of the homogeneous linear system of

Basics on Commutative Algebra

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n + m equations ⎧ = −b0 D0 ⎪ ⎪a0 C0 ⎪ ⎪ ⎨a1 C0 + a0 C1 = −b1 D0 − b0 D1 ⎪ ... ⎪ ⎪ ⎪ ⎩a C n

m−1

...

(1.10)

= −bm Dn−1

in the indeterminates Ci ’s and Dj ’s, respectively, 0  i  m − 1, 0  j  n − 1, whose coefficient matrix is exactly the transpose of the Sylvester matrix S(f, g). One finally concludes by the well-known fact that the existence of a non-zero solution of (1.10) is equivalent to R(f, g) = 0 (by elementary transformations, this condition is equivalent to the existence of a non-zero solution of (1.10) in the field K := Q(D); on the other hand, since (1.10) is homogeneous one has a non-zero solution in D).  Corollary 1.3.17. If D = K is an algebraically closed field and f, g are polynomials as in (1.5), then R(f, g) = 0 if and only if f and g have a common root in K. Proof. It is a direct consequence of Theorem 1.3.16 and of the fact that, if K is algebraically closed, then non-constant irreducible elements in K[x] are all of the form (x − α), α ∈ K.  When f ∈ D[x] is such that n = deg(f )  2, one would like to find a condition similar to that given in Theorem 1.3.16 in order to establish if f has a multiple factor in D[x], i.e. f = g e h, for some integer e  2 and some non-constant polynomial g ∈ D[x]. This question is obviously related to the existence of multiple roots of f , possibly in some field K containing D (cf. Definition 1.3.7 and Lemma 1.3.9). On the other hand, derivatives in positive characteristic not always well-behave (recall, e.g. the proof of Lemma 1.3.9). Indeed when D is a UFD of positive characteristic, there actually exist non-constant polynomials f ∈ D[x] for which, either f  = 0 or f  constant even if deg(f ) > 1 (in both cases one cannot define R(f, f  )). For the scope of this book, we can limit ourselves to the characteristic-zero case and refer the interested reader to, e.g. Lang (2002). Thus, when D is a UFD of characteristic 0 and f ∈ D[x] is a polynomial of degree n  2, then f  ∈ D[x] is a non-constant polynomial of degree n−1. As above, one can define the (2n − 1) × (2n − 1)-Sylvester matrix S(f, f  )

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A First Course in Algebraic Geometry and Algebraic Varieties

and consequently R(f, f  ) ∈ D, which is simply denoted by Δ(f ) and called the discriminant of f . Proposition 1.3.18. Let D be a UFD of characteristic 0 and f ∈ D[x] a polynomial of degree n  2. Then f has a multiple factor in D[x] if and only if Δ(f ) = 0. Proof. (⇒) By the assumptions, there exists a non-constant g ∈ D[x] such that f = g 2 k, for some k ∈ D[x]. Then f  = 2gg  k + g 2 k  , which shows that g|f and g|f  . Since deg(f )  2 and since the characteristic is zero, f  ∈ D[x] is a non-constant polynomial, so that Δ(f ) makes sense and one concludes by Theorem 1.3.16. (⇐) Assume Δ(f ) = 0. From Theorem 1.3.16, it follows that f and f  have a non-constant common factor g ∈ D[x] which we can assume to be irreducible. From f = gh, for some h ∈ D[x], we deduce f  = g  h + gh , where g  = 0. Since g|f  by assumption, then g|g  h. Since deg(g) > deg(g  )  and g is irreducible, then g|h so g 2 |f . Corollary 1.3.19. If D = K is an algebraically closed field of characteristic 0 and f ∈ K[x] is a polynomial of degree n  2, then Δ(f ) = 0 if and only if f has a multiple root. 1.3.4

Resultant in D[x1 , . . . , xn ] and elimination

In the previous section, we introduced the notion of resultant for two polynomials in D0 [x], where D0 any UFD. In particular, if D is a UFD then D[x1 , . . . , xn ] = (D[x1 , . . . , xn−1 ])[xn ], with D0 := D[x1 , . . . , xn−1 ] a UFD (cf. Theorem 1.3.1 and Remark 1.3.2). Thus, we can apply the machinery developed in Section 1.3.3 to any pair of non-constant polynomials f, g ∈ D0 [xn ] to get Rxn (f, g) := R(f, g) ∈ D0 = D[x1 , . . . , xn−1 ], which is called the resultant polynomial of f and g with respect to the indeterminate xn . Note that Rxn (f, g) is a polynomial where the indeterminate xn does not appear; this gives rise to the terminology elimination theory. Proposition 1.3.20. Let D be a UFD and let f, g ∈ D[x1 , . . . , xn ] be non-constant polynomials of the form f = ad xdn + · · · + a0 ,

g(x) = bm xm n + · · · + b0

(1.11)

with ad , bm ∈ D[x1 , . . . , xn−1 ] \ {0}. Then the resultant polynomial R := Rxn (f, g) ∈ D[x1 , . . . , xn−1 ] belongs to the ideal generated by f and g in

Basics on Commutative Algebra

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D[x1 , . . . , xn ]. More precisely there exist polynomials A, B ∈ D[x1 , . . . , xn ], with degrees w.r.t. the indeterminate xn at most m − 1 and d − 1, respectively, such that Af + Bg = R holds. Proof. One constructs the Sylvester matrix S(f, g) as in (1.7). Then one finds the following relations: ⎧ m−1 x f (x) = ad xd+m−1 + · · · + a0 xm−1 n n ⎪ ⎪ ⎪ ⎪ m−2 d+m−2 m−2 ⎪ x f (x) = ad xn + · · · + a0 xn ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ d ⎨ f (x) = ad x + · · · + a0 n

⎪ xd−1 g(x) = bm xd+m−1 + · · · + b0 xd−1 ⎪ n n ⎪ ⎪ ⎪ d−2 d+m−2 ⎪ x g(x) = bm xn + · · · + b0 xd−2 ⎪ n ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎩ m g(x) = bm xn + · · · + b0 . If we first multiply each relation by the co-factor of the corresponding element in the last column of S(f, g) and then add-up all the relations obtained above, we get that a relation of the form Af + Bg = R holds true.  For more details, see, e.g. Hassett (2007, § 5.4, pp. 84–86). 1.4

Noetherian Rings and the Hilbert Basis Theorem

A ring R is said to be a Noetherian ring if any ideal I of R is finitely generated. Z and any field K are easy examples of Noetherian domains. Since K[x] is a Euclidean domain, it is a PID and so in particular it is a Noetherian ring. There actually exist rings which are not Noetherian: e.g. the ring K[x1 , x2 , x3 , . . .] of polynomials with coefficients from a field K and with infinite indeterminates is an integral domain which cannot be Noetherian as the maximal ideal (x1 , x2 , x3 , . . .) cannot be finitely generated. Note that K[x1 , . . . , xn ], with n  2, is not a PID: e.g. the maximal ideal (x1 , . . . , xn ) cannot be principal. On the other hand, when n  2, K[x1 , . . . , xn ] is actually Noetherian. This is a consequence of the following more general result. Theorem 1.4.1 (Hilbert’s basis theorem). Let R be a Noetherian ring and let x be an indeterminate. Then the ring R[x] is Noetherian.

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A First Course in Algebraic Geometry and Algebraic Varieties

Remark 1.4.2. Let R be any ring andlet x1 , . . . , xn be  indeterminates over R. One clearly has R[x1 , . . . , xn ] = R[x1 , . . . , xn−1 ] [xn ]. Thus, as a consequence of Theorem 1.4.1, if R is Noetherian then also R[x1 , . . . , xn ] is, for any positive integer n. In particular, this holds for R = K any field. Proof of Theorem 1.4.1. Let f (x) ∈ R[x] be any polynomial. If f (x) is not the zero-polynomial, then f (x) := a0 + a1 x + a2 x2 + · · · + ai xi ∈ R[x], where aj ∈ R, 0  j  i, with ai = 0, i.e. the integer i  0 is the degree of f (x) and ai := lc(f (x)) ∈ R (recall Definition 1.3.4). If otherwise f (x) is the zero-polynomial, its leading coefficient is 0 ∈ R by definition. Let I ⊂ R[x] be any non-trivial proper ideal (if I is either (0) or (1) there is nothing to prove). For any integer i  0, denote by Ji the subset of R consisting of all leading coefficients of polynomials in I having degrees at most i. Similarly, denote by J the subset of R consisting of all leading coefficients of polynomials in I. It is clear that J0 ⊆ J1 ⊆ J2 ⊆ · · · ⊆ J and that 0 ∈ J0 . By straightforward computations, one verifies that J and Ji , for any i  0, are ideals of R. Being R Noetherian, all these ideals are finitely generated. Thus, for any i  0, there exists a finite set {rih }h∈Hi of elements of R, with Hi a finite set of indexes, such that Ji = (rih )h∈Hi . By the very definition of Ji , one can determine as many polynomials {fih (x)}h∈Hi s.t. fih (x) ∈ I, deg(fih (x))  i and lc(fih (x)) = rih ,

∀ h ∈ Hi .

Similarly, there exists a finite set {rm }m∈M of elements of R, with M a finite set of indexes, such that J = (rm )m∈M , and as many polynomials {fm (x)}m∈M s.t. fm (x) ∈ I and lc(fm (x)) = rm ,

∀ m ∈ M.

Let N := Maxm∈M {deg(fm (x))} and consider the ideal I  generated by the polynomials {fm (x)}m∈M and {fih (x)}h∈Hi for i  N , i.e. I  := ({fm (x)}m∈M , {fih (x)}h∈Hi ,iN ). One has I  ⊆ I and I  is finitely generated. To conclude the proof, it suffices to show that I  = I. Assume by contradiction that the inclusion I  ⊂ I is strict. Thus, there exists an element g(x) ∈ I \ I  of minimal degree with respect to this property.

Basics on Commutative Algebra

15

If deg(g(x)) := d > N , then g(x) = b0 + b1 x + · · · + bd xd , with 0 = bd ∈ J, since g(x) ∈ I. By the assumption on J, we have bd = Σm∈M qm rm , for some qm ∈ R. Consider the polynomial  q(x) := qm fm (x) xd−deg(fm (x)) ∈ I  . m∈M

One has deg(q(x)) = deg(g(x)) and lc(q(x)) = lc(g(x)). Since g(x) − q(x) ∈ I and deg(g(x) − q(x)) < d = deg(g(x)), by the minimality condition on d = deg(g(x)), one must have g(x) − q(x) ∈ I  . This would imply g(x) = (g(x) − q(x)) + q(x) ∈ I  , a contradiction. If otherwise deg(g(x)) := i  N , g(x) = b0 + b1 x + · · · + bi xi , with 0 = bi ∈ Ji . As above, by the assumption on Ji , bi = Σh∈Hi qih rih , for some qih ∈ R. Consider the polynomial  qih fih (x) xi−deg(fih (x)) ∈ I  , qi (x) := h∈Hi

so that deg(qi (x)) = deg(g(x)) and lc(qi (x)) = lc(g(x)). Since g(x) − qi (x) ∈ I and deg(g(x) − qi (x)) < i = deg(g(x)), one concludes as in the previous case.  Remark 1.4.3. (1) If S is a subring of a Noetherian ring R, in general one cannot conclude that S is Noetherian too: consider, e.g. the ring S := K[x1 , x2 , x3 , . . .] of polynomials in infinite indeterminates with coefficients from a field K. From Remark 1.4.2 S is not Noetherian on the other hand, since it is an integral domain, it admits a quotient field Q(S) := R. Thus, S is a subring of a Noetherian ring R. (2) On the contrary, any quotient R/I of a Noetherian ring R is Noetherian too, as it easily follows from the bijective correspondence between ideals of R/I and ideals of R containing I (cf. also Atiyah and McDonald, 1969, Proposition 7.1). Proposition 1.4.4. A ring R is Noetherian if and only if every ascending chain I1 ⊆ I2 ⊆ I3 ⊆ · · · ⊆ In ⊆ In+1 ⊆ · · · ⊂ R

(1.12)

of proper ideals is stationary, i.e. there exists an integer n0 among the indexes s.t. In0 = In0 +h , for any integer h  1.

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A First Course in Algebraic Geometry and Algebraic Varieties

Proof. (⇒) Suppose that R is Noetherian and that (1.12) is an ascending chain of proper ideals. Recalling what discussed in Section 1.1, I := Σn In is a proper ideal of R; indeed if 1 ∈ I, there should exist an index n1 such that 1 ∈ In1 , which contradicts the assumptions on the chain (1.12). Since R is Noetherian, there exist r1 , . . . , rm ∈ I s.t. I = (r1 , . . . , rm ). Let ri ∈ In(i) and let n0 := Max1im {n(i)}. Then, by the ascending condition, In(i) ⊆ In0 so, in particular, ri ∈ In0 , for any 1  i  m. Thus, I ⊆ In0 , i.e. In0 = In0 +h , for any integer h  1. (⇐) Suppose that the second part of the statement holds true and let I be any proper ideal of R. If I were not finitely generated, one could find an infinite sequence a1 , a2 , a3 , . . . of distinct generators of I giving rise to an ascending sequence of proper ideals (a1 ) ⊂ (a1 , a2 ) ⊂ (a1 , a2 , a3 ) ⊂ · · · which could not be stationary, a contradiction. 1.5

R-Modules, R-Algebras and Finiteness Conditions

Let R be a ring. An abelian group (M, +) is called a R-module if there exists a R-multiplication map R × M → M, (a, m) → am, such that for any a, b ∈ R and any m, n ∈ M the following hold: (a + b)m = am + bm, a(m + n) = am + an, (ab)m = a(bm), 1m = m. Easy examples of R-modules are, e.g. M = R, M = I as well as M = R/I for any ideal I ⊂ R, M = Rn := R ⊕ · · · ⊕ R where the R-multiplication is considered componentwise. If R = K is a field, a K-module is nothing but a K-vector space. If N ⊂ M is a subgroup s.t. for any n ∈ N and any r ∈ R one has rn ∈ N , then N is said to be a R-submodule of M . If T ⊂ M is a subset of a R-module M , we denote by T  := {Σi ai ti | ti ∈ T, ai ∈ R} the subset of elements in M of the form Σai ti , where in the sums only finitely many ai ’s are non-zero. The set T  is a submodule of M which is called the R-module generated by T and T is said to be a set of generators of the R-module T . A R-module M is said to be finitely generated, if M = T  for some finite subset T ⊂ M . Let M and N be two R-modules. A map ϕ : M → N is a R-module homomorphism if it is a group homomorphism which is also R-linear, namely ϕ(x + y) = ϕ(x) + ϕ(y), ϕ(r x) = r ϕ(x),

∀ x, y ∈ M, ∀ r ∈ R.

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17

A R-algebra is a ring S together with a ring-homomorphism ϕ : R → S (recall that by assumption ϕ(1R ) = 1S , so Ker(ϕ)  R is always a proper ideal). S  ⊂ S is a R-subalgebra of S, if S  is a subring of S which is a R-algebra. Remark 1.5.1. (i) Note that a R-algebra is also a R-module by r · s := ϕ(r) s, for any r ∈ R and any s ∈ S, where the two structure are compatible. (ii) Easy examples of R-algebras are, e.g. the ring of polynomials R[x1 , . . . , xn ] for any n  1, the quotient ring R/I for any ideal I  R with ϕ = πI the canonical projection. (iii) If R = K is a field and S = 0, then ϕ necessarily is injective, i.e. K can be identified with its image in S. Thus, a K-algebra is nothing but a ring containing the field K as a subring. If in particular S = F is also a field, then K → F is said to be a field extension. The degree of the field extension, denoted by [F : K], is dimK (F) as a K-vector space. (iv) Since R is commutative and with identity, there exits a unique ring homomorphism Z → R, n → n · 1R . In other words, any (commutative and with identity) ring R is automatically a Z-algebra. (v) Let ϕ : R → S and ψ : R → T be ring homomorphisms. A R-algebra homomorphism, say η : S → T , is a ring homomorphism which is also a homomorphism of R-modules. This occurs if and only if η ◦ ϕ = ψ. Definition 1.5.2. In the above notation, S is said to be a finite R-algebra if S is finitely generated as a R-module. When R = K is a field, easy examples of finite K-algebras are, e.g. K[x]/(xn ) as well as any (algebraic) field extension of finite degree. ϕ Let S be a R-algebra, with R −→ S as above. Consider s1 , . . . , sn elements in S and x1 , . . . , xn indeterminates over R. One can define a ringhomomorphism Φ := Φs1 ,...,sn : R[x1 , . . . , xn ] → S

(1.13)

by the rules Φ(xi ) = si , 1  i  n, and Φ(a) = ϕ(a), ∀ a ∈ R. Then Im(Φ) ⊆ S is the smallest subalgebra of S containing all polynomial expressions in the elements s1 , . . . , sn with coefficient from the subring ϕ(R). The subalgebra Im(Φ) is simply denoted by R[s1 , . . . , sn ]. Definition 1.5.3. A R-algebra S is said to be a R-algebra of finite type if S = R[s1 , . . . sn ], for some finitely many elements s1 , . . . , sn ∈ S.

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A First Course in Algebraic Geometry and Algebraic Varieties

In particular, a ring R is said to be finitely generated if it is a Z-algebra of finite type. Example 1.5.4. If S is a finite R-algebra, then it is also a R-algebra of finite type. The converse does not hold in general; indeed, R[x1 , . . . , xn ], where n  1 an integer and xi indeterminates over R, 1  i  n, is a R-algebra of finite type which is not a finite R-algebra. Let K and F be fields such that F is a K-algebra. Let s1 , . . . , sn ∈ F \ K. One denotes by K(s1 , . . . , sn ) the quotient field Q (K[s1 , . . . , sn ]), which is the smallest subfield of F containing K and s1 , . . . , sn ∈ F. Definition 1.5.5. A field extension K ⊂ F is called a finitely generated field extension if F = K(s1 , . . . , sn ), for some finitely many elements s1 , . . . , sn in F. When n = 1, the field extension is called simple. Remark 1.5.6. Note that if K and F are fields such that F is a K-algebra of finite type, then K ⊆ F is a finitely generated field extension. In particular, for a field F, we have (∗) F is a finite K − algebra ⇒ (∗∗) F is a K − algebra of finite type ⇒ (∗ ∗ ∗) K ⊆ F is a finitely generated field extension. The opposite implications do not hold in general: to show that in general “(∗∗) does not imply (∗)”, recall, e.g. Example 1.5.4 whereas, to note that in general “(∗ ∗ ∗) does not imply (∗∗)”, we can deduce it from the next result. Proposition 1.5.7. Let K be any field. Let x1 , . . . , xn be indeterminates over K, for any integer n  1, and let K(x1 , . . . , xn ) := Q (K[x1 , . . . , xn ]) be the field of rational functions with coefficients from K, which is a finitely generated field extension of K. Then K(x1 , . . . , xn ) is not a K-algebra of finite type. Proof. For simplicity, we consider the case n = 1, the general case being similar. Therefore, in what follows we will simply put x1 = x. Assume by contradiction there exist finitely many s1 , . . . , st ∈ K(x) such that K(x) = K[s1 , . . . , st ], and let s ∈ K[x] be the product of all denominators of s1 , . . . , st ∈ K(x). In such a case, for any z ∈ K(x) there would exist a positive integer m (depending on z) such that sm z ∈ K[x]. This is a contradiction; indeed, it suffices to take any irreducible non-constant polynomial c ∈ K[x] which does not divide s in K[x], and consider  z := 1c ∈ K(x).

Basics on Commutative Algebra

1.6

19

Integrality

Let R and S be rings such that R ⊆ S. Definition 1.6.1. An element v ∈ S is said to be integral over the subring R if there exists a monic polynomial f (t) ∈ R[t], where t is an indeterminate over R, s.t. f (v) = 0. Remark 1.6.2. (i) If v ∈ R, then v is integral over R. (ii) If R ⊆ S is a field extension, v ∈ S is integral over R if and only if it is algebraic over R. (iii) v ∈ Q is integral over Z if and only if v ∈ Z. Proof. Indeed, if v = rs ∈ Q, with r and s coprime, then an expression of the form v n + an−1 v n−1 + · · · + a0 = 0, ai ∈ Z, 0  i  n − 1, gives rn +an−1 rn−1 s+· · ·+a0 sn = 0, i.e. s divides rn . This implies that s divides r and so s = ±1.  (iv) Let R be any UFD and let Q(R) be its quotient field. Then v ∈ Q(R) is integral over R if and only if v ∈ R. The proof is similar to that of (iii) above. Note that the previous statement, in particular, applies to the polynomial ring R = K[t], where K any field and t an indeterminate over K. Proposition 1.6.3. Let R and S be rings s.t. R ⊆ S. The following conditions are equivalent: (i) v ∈ S is integral over R; (ii) R[v] is a finite R-algebra, i.e. it is a finitely generated R-module; (iii) there exists a subring B of S, containing R[v], which is a finitely generated R-module. Proof. (i) ⇒ (ii): from (i), there exist a positive integer n and elements ai ∈ R, 0  i  n − 1, such that for any non-negative integer r one has v n+r = −(an−1 v n+r−1 + · · · + a0 v r ). Thus, one gets that, for any integer m  0, v m belongs to the R-submodule of S generated by the elements 1, v, v 2 , . . . , v n−1 , which implies that R[v] is a finitely generated R-module.

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A First Course in Algebraic Geometry and Algebraic Varieties

(ii) ⇒ (iii): it suffices to consider B = R[v]. (iii) ⇒ (i): let R[v] ⊆ B ⊆ S be ring inclusions, such that B is a finitely generated R-module. Let v1 , . . . , vn ∈ B be generators of B as a R-module. Since v, vi ∈ B then vvi ∈ B, for any 1  i  n, as B is a ring. By the assumptions on B, there exist elements aij ∈ R, 1  i, j  n, such that vvi =

n 

aij vi ,

1  i  n.

j=1

Denoting by δij the Kronecker symbol, the previous equality gives det(δij v − aij ) vi = 0,

1  i  n.

Since v1 , . . . , vn generates B, this implies det(δij v − aij ) = 0, i.e. v is a root of the characteristic polynomial of the matrix (aij ), 1  i, j  n, which is a monic polynomial with coefficients from R.  Corollary 1.6.4. Let R and S be rings s.t. R ⊆ S. The set C := {v ∈ S | v integral over R} is a subring of S such that R ⊆ C ⊆ S. Proof. It is clear that as sets R ⊆ C. Take now v, w ∈ C any two elements. Since R ⊆ R[v] and since w is integral over R, then w is also integral over R[v]. From Proposition 1.6.3, R[v, w] is a finitely generated R[v]-module. By transitivity, R[v, w] is a finitely generated R-module. Therefore, from Proposition 1.6.3, R[v, w] ⊆ C; in particular, vw, v ± w ∈ C, which proves  that C is a subring of S containing R as a subring. C as in Corollary 1.6.4 is called the integral closure of R in S. Definition 1.6.5. An integral domain R is said to be integrally closed if it coincides with its integral closure in Q(R), i.e. v ∈ Q(R) and v integral over R implies v ∈ R. From Remark 1.6.2-(iii) and (iv), Z and any UFD are integrally closed. Proposition 1.6.6 (cf. Milne (2017, Proposition 1.18, p. 19)). Let R be any integral domain and F := Q(R) be its quotient field. Let K be any field such that K ⊇ F. If α ∈ K is algebraic over F, then there exists d ∈ R such that dα ∈ K is integral over R.

Basics on Commutative Algebra

21

Proof. By assumptions, there exists a relation of the form αn + an−1 αn−1 + · · · + a0 = 0, ai ∈ F, 0  i  n − 1. Let d ∈ R be the product of all denominators of the ai ’s. Then dai ∈ R, for any 0  i  n − 1. If we multiply the previous relation by dn , we get (dα)n + dan−1 (dα)n−1 + · · · + dn a0 = 0, i.e. dα is a root of a monic polynomial in R[x].



Corollary 1.6.7 (cf. Milne (2017, Corollary 1.19, p. 20)). Let R be an integral domain and F := Q(R) be its quotient field. For any algebraic field extension F ⊆ K, consider C the integral closure of R in K. Then K = Q(C). Proof. The proof of Proposition 1.6.6 shows that any element α ∈ K can  be written as α = βd , where β integral over R and d ∈ R. 1.7

Zariski’s Lemma

In Section 1.5, we introduced finiteness conditions for K-algebras, where K is a field. Remark 1.7.1. Recall that for a field F which is a simple extension of K, i.e. F = K(v) for some v ∈ F \ K, two cases occur: (i) if v ∈ F is transcendental over K, then F ∼ = K(x), with x an indeterminate, and so Proposition 1.5.7 implies that K ⊂ F is a finitely generated field extension which is neither a K-algebra of finite type nor a finite K-algebra; (ii) if v ∈ F is algebraic over K, then F = K[v] is a finite K-algebra (so in particular also a K-algebra of finite type) and the field extension K ⊂ F is finite (a fortiori finitely generated), i.e. [F : K] = deg(fv (x)) < +∞ where fv (x) ∈ K[x] is the minimal polynomial of v ∈ F. In other words, when v is algebraic the three conditions (∗), (∗∗) and (∗ ∗ ∗) in Remark 1.5.6 coincide. The next fundamental result shows that the same occurs for any finitely generated field extension.

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A First Course in Algebraic Geometry and Algebraic Varieties

Lemma 1.7.2 (Zariski’s lemma). Let K ⊂ F be a field extension. Then F is a K-algebra of finite type if and only if [F : K] < +∞. Remark 1.7.3. (i) Note that, in the assumptions of Zariski’s Lemma, K ⊂ F is in any case a finitely generated field extension (recall Remark 1.5.6). (ii) There are field extensions K ⊂ F which do not satisfy any of the finiteness conditions introduced in Section 1.5. Take, e.g. the field extension Q ⊂ Q (respectively, Zp ⊂ Zp , where p ∈ Z a prime) given by algebraic closure. This cannot be a finitely generated field extension otherwise, being an algebraic extension, it would be of finite degree over Q (cf. Lang, 2002, Proposition 1.6, for details), which is a contradiction since Q has infinite dimension as a Q-vector space. In particular, it is neither a finite Q-algebra nor a Q-algebra of finite type, as it easily follows from Zariski’s Lemma. The same discussion holds verbatim for the field extension Zp ⊂ Zp . Proof of Zariski’s Lemma. First of all, we can assume F with infinite elements, otherwise there is nothing else to prove. (⇐) This implication is obvious (cf. Example 1.5.4). (⇒) By assumption we have F = K[v1 , . . . , vn ], for some integer n. The proof proceeds by induction on n. The case n = 1 has already been discussed in Remark 1.7.1; therefore assume n  2 and use inductive hypothesis. We have K ⊂ K1 := K(v1 ) ⊂ F. Since F = K[v1 , . . . , vn ], one also has F = (K(v1 )) [v2 , . . . , vn ] = K1 [v2 , . . . , vn ], i.e. F is a K1 -algebra of finite type. By induction, the extension K1 ⊂ F is algebraic of finite degree. By transitivity of degree extension, one has [F : K] = [F : K1 ] · [K1 : K] so, to deduce the finiteness of [F : K] it suffices to show that v1 is algebraic over K. Assume by contradiction that v1 is transcendental over K. Since F = K1 [v2 , . . . , vn ], with v2 , . . . , vn algebraic over K1 , from Proposition 1.6.6 there exists d ∈ K[v1 ] s.t. dvi is integral over K[v1 ], for any 2  i  n. Let f ∈ K1 be any element. Claim 1.7.4. For any integer N >> 0, dN f ∈ K[v1 , dv2 , . . . , dvn ]. Proof. Since f ∈ K1 ⊂ F, then  αj1 ...jn v1j1 v2j2 . . . vnjn . f= j1 ...jn

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Basics on Commutative Algebra

Thus, for any N sufficiently large, any monomial of dN f is of the form dN αj1 ...jn v1j1 v2j2 · · · vnjn = dN −(j2 +···+jn ) αj1 ...jn v1j1 (dv2 )j2 · · · (dvn )jn , where dN −(j2 +···+jn ) αj1 ...jn v1j1 ∈ K[v1 ].



From Corollary 1.6.4, dN f is therefore integral over K[v1 ]. Since by assumption v1 is transcendental over K, then K[v1 ] is a PID (in particular, a UFD). From Remark 1.6.2-(iv), K[v1 ] is therefore integrally closed, i.e. dN f ∈ K[v1 ]. If we take c ∈ K[v1 ] irreducible, not dividing d, and if we consider f := c−1 , from the previous discussion, for any N >> 0 we would get dN f ∈ K[v1 ], a contradiction. Thus, v1 is algebraic and the theorem is proved.  1.8

Transcendence Degree

Let K ⊂ F be a field extension and let s1 , . . . , sn ∈ F. Similarly to (1.13), consider the K-algebra homomorphism Φ : K[x1 , . . . , xn ] → F, Φ(xj ) = sj ,

1  j  n,

(1.14)

where x1 , . . . , xn are indeterminates over K. Definition 1.8.1. The elements s1 , . . . , sn ∈ F are said to be algebraically independent over K if Ker(Φ) = (0). Otherwise, they are said to be algebraically dependent over K and any non-zero f ∈ Ker(Φ) is said to be a relation of algebraic dependence over K among the si ’s. If s1 , . . . , sn are algebraically independent over K, the homomorphism Φ above extends to an isomorphism Q(n) := K(x1 , . . . xn ) ∼ = K(s1 , . . . , sn ). Definition 1.8.2. Let s1 , . . . , sn ∈ F be elements which are algebraically independent over K. The set {s1 , . . . , sn } is said to be a (finite) transcendence basis of F over K if F is an algebraic extension of K(s1 , . . . , sn ). Lemma 1.8.3. Let {s1 , . . . , sn } be a transcendence basis of F over K and let t1 , . . . , th ∈ F be algebraically independent over K. Then h  n. Proof. Since F is an algebraic extension of K(s1 , . . . , sn ), there exists a non-zero polynomial f ∈ K[y1 , x1 , . . . , xn ] = A(n+1) s.t. f (t1 , s1 , . . . , sn ) = 0. Since s1 , . . . , sn are algebraically independent, the polynomial f is nonconstant with respect to the indeterminate y1 . Since moreover t1 ∈ F is

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transcendent over K, the polynomial f has to be non-constant with respect to at least one of the indeterminates xi , 1  i  n; up to a re-labeling of the indeterminates, we can assume this occurs for x1 . In such a case s1 is algebraic over K(t1 , s2 , . . . , sn ). By transitivity of algebraic extensions, F is an algebraic extension of K(t1 , s2 , . . . , sn ). Moreover t1 , s2 , . . . , sn are algebraically independent over K (otherwise s1 would be algebraically dependent to s2 , . . . , sn ). Thus, {t1 , s2 , . . . , sn } is a transcendence basis of F over K. Consider the following statement, for some 1  i  h − 1: (Si ) Up to a possible permutation of s2 , . . . , sn , the set {t1 , t2 , . . . , ti , si+1 , . . . sn } is a transcendence basis of F over K. (S1 ) has already been proved above. We proceed by induction on i; assume therefore we have proved (Si ), for some 1  i  h − 1, we want to prove (Si+1 ); by the inductive hypothesis, there exists a non-zero polynomial g ∈ K[yi+1 , y1 , . . . , yi , xi+1 , . . . , xn ] such that g(ti+1 , t1 , . . . , ti , si+1 , . . . , sn ) = 0. The polynomial g is non-constant with respect to the indeterminate yi+1 (since t1 , . . . , ti , si+1 , . . . , sn are algebraically independent) as well as with respect to at least one indeterminates xj , e.g. xi+1 (since t1 , . . . , ti , ti+1 are algebraically independent). Thus, si+1 is algebraic over K(t1 , . . . , ti+1 , si+2 , . . . , sn ). Moreover t1 , . . . , ti+1 , si+2 , . . . , sn are algebraically independent: indeed if one had h(t1 , . . . , ti+1 , si+2 , . . . , sn ) = 0 for some non-zero polynomial h ∈ K[y1 , . . . , yi+1 , xi+2 , . . . , xn ], by the inductive hypothesis the polynomial h could not be constant with respect to the indeterminate yi+1 , so ti+1 would be algebraic over K(t1 , . . . , ti , si+2 , . . . , sn ); in such a case si+1 would be algebraic over K(t1 , . . . , ti , si+2 , . . . , sn ), contradicting the inductive hypothesis. It follows that {t1 , . . . , ti+1 , si+2 , . . . , sn } is a transcendence basis of F over K, i.e. (Si+1 ) has been proved. By induction, (Sh ) holds true; in particular h  n.  If F admits a finite transcendence basis over K, we will say that F has finite transcendence degree over K. In such a case, from Lemma 1.8.3, any two transcendence bases have the same cardinality. This non-negative integer is called transcendence degree of F over K and it is denoted by trdegK (F).

(1.15)

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25

Remark 1.8.4. (i) If K ⊆ F is an algebraic extension, trdegK (F) = 0. (ii) If F does not admit a finite transcendence basis over K (e.g. the quotient field of the polynomial ring K[x1 , x2 , . . .] with infinite indeterminates), one poses trdegK (F) = +∞. (iii) If x1 , . . . , xn are indeterminates over K, then for Q(n) := K(x1 , . . . , xn ) one has trdegK (Q(n) ) = n and {x1 , . . . , xn } is a transcendence basis of Q(n) over K. (iv) If F = K(s1 , . . . , sn ) and s1 , . . . , sn ∈ F are algebraically independent, then trdegK (F) = n and F ∼ = Q(n) . In such a case, F is said to be a purely transcendental extension of K. (v) For a finitely generated field extension K ⊂ F = K(s1 , . . . , sn ) with trdegK (F) = n, the field F is called a field of algebraic functions in the si ’s. Proposition 1.8.5. Let K ⊂ L ⊂ F be field extensions and assume trdegK (L) < +∞ and trdegL (F) < +∞. Then also trdegK (F) < +∞ and trdegK (F) = trdegK (L) + trdegL (F). More precisely, if s1 , . . . , sn ∈ L is a transcendence basis of L over K and t1 , . . . , tm ∈ F is a transcendence basis of F over L, then s1 , . . . , sn , t1 , . . . , tm is a transcendence basis of F over K. Proof. Assume by contradiction there exists a non-zero polynomial g ∈ K[x1 , . . . , xn , y1 , . . . , ym ], where xi ’s and yj ’s are indeterminates over K. Consider the polynomial hs (y1 , . . . , yn ) := g(s1 , . . . , sn , y1 , . . . , ym ) ∈ L[y1 , . . . , ym ]. From the fact that s1 , . . . , sn are algebraically independent over K, this polynomial is non-zero. In this case, hs (y1 , . . . , yn ) would give a non-zero algebraic relation over L among the elements t1 , . . . , tm , contradicting the assumption. Thus, s1 , . . . , sn , t1 , . . . , tm are algebraically independent over K. We are left to showing that K(s1 , . . . , sn , t1 , . . . , tm ) ⊆ F is an algebraic extension. Note that L(t1 , . . . , tm ) ⊆ F is an algebraic extension, since {t1 , . . . , tm } is a transcendence basis of F over L. The same occurs to the extension K(s1 , . . . , sn , t1 , . . . , tm ) ⊆ L(t1 , . . . , tm ), since {s1 , . . . , sn } is a transcendence basis of L over K. Therefore, the composition K(s1 , . . . , sn , t1 , . . . , tm ) ⊆ L(t1 , . . . , tm ) ⊆ F is an algebraic extension and we are done. Proposition 1.8.6. Let K be a field and let R be any integral K-algebra, with quotient field Q(R). If trdegK (Q(R)) := n < +∞, then the non– negative integer n equals the maximum number of elements in R which are algebraically independent over K.

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Proof. If A ⊂ R is any set of elements of R which are algebraically independent over K then, from the fact that A ⊂ R ⊆ Q(R) and from Lemma 1.8.3, one has that |A|  n. We want to show that there actually exist subsets A ⊂ R for which their cardinality reaches the maximum n. Take s1 , . . . , sn ∈ Q(R) any transcendence basis of Q(R) over K. Let z ∈ R \ {0} s.t. zs1 , . . . , zsn ∈ R. We want to show that zs1 , . . . , zsn are algebraically independent over K. Assume by contradiction there exists a non-zero polynomial  ai1 ,...,in xi11 · · · xinn ∈ K[x1 , . . . , xn ], f := i1 ,...,in

where ai1 ,...,in ∈ K and x1 , . . . , xn indeterminates over K, such that 0 = f (zs1 , . . . , zsn ) =



ai1 ,...,in z i1 +···+in si11 · · · sinn .

i1 ,...,in

If we expand the previous expression in powers w.r.t. z, this reads (∗) 0 = cr (s1 , . . . , sn )z r + cr−1 (s1 , . . . , sn )z r−1 + · · · + c0 (s1 , . . . , sn ), where cj (x1 , . . . , xn ) ∈ K[x1 , . . . , xn ] for any 0  j  r. Assume first c0 (x1 , . . . , xn ) ∈ K[x1 , . . . , xn ]\ {0}. In such a case equality (∗) reads c0 (s1 , . . . , sn ) = −Σrj=1 cj (s1 , . . . , sn )z j . Since z ∈ R divides the right-side-member of the previous equality whereas the left-side-member is a polynomial expression in s1 , . . . , sn ∈ Q(R) with coefficients from K, it follows that c0 (s1 , . . . , sn ) = 0. This contradicts the fact that s1 , . . . , sn in Q(R) are algebraically independent over K. If otherwise cj (x1 , . . . , xn ) = 0, for any 0  j  − 1 < r, where  r is the minimum integer for which c (x1 , . . . , xn ) ∈ K[x1 , . . . , xn ] \ {0}, equality (∗) reads in this case 0 = −z  · (Σrj= cj (y1 , . . . , yn ) z j− ). Since Q(R) is a field and since z = 0, the previous equality implies therefore 0 = Σrj= cj (y1 , . . . , yn )z j− , where c (x1 , . . . , xn ) = 0, and one can conclude as in the previous case.  Definition 1.8.7. Let K be a field and let R be an integral K-algebra with quotient field Q(R). By small abuse of terminology, one defines the transcendence degree of R over K, denoted by trdegK (R), to be the transcendence degree of Q(R) over K, namely trdegK (R) := trdegK (Q(R)).

(1.16)

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Remark 1.8.8. (i) If x1 , . . . , xn are indeterminates over K, then trdegK (K[x1 , . . . , xn ]) = n in the sense of Definition 1.8.7. (ii) If Φ : R  R is a surjective homomorphism of integral K-algebras, then trdegK (R)  trdegK (R ),

(1.17)

in the sense of Definition 1.8.7. Indeed, set n := trdegK (R). In view of Proposition 1.8.6, for any choice of s1 , . . . , sn+1 ∈ R, these elements are algebraically dependent over K, i.e. there exists g ∈ K[x1 , . . . , xn+1 ] \ {0}, where x1 , . . . , xn+1 indeterminates over K, such that g(s1 , . . . , sn+1 ) = 0. Set fi := Φ(si ) ∈ R , for any 1  i  n + 1; one has therefore g(f1 , . . . , fn+1 ) = Φ(g(s1 , . . . , sn+1 )) = Φ(0) = 0, the first equality following from the fact that Φ is a K-algebra homomorphism. Since Φ is surjective, any element of R is the image via Φ of (at least) an element of R, so this implies that the maximal number of algebraically independent elements of R over K is at most n. From Proposition 1.8.6, it follows that trdegK (R )  n = trdegK (R), in the sense of Definition 1.8.7. (iii) If F = K(s1 , . . . , sn ) is a finitely generated field extension of K, then trdegK (F)  n as it follows from (i), (ii) and the fact that one has a surjective K-algebra homomorphism Φ : K[x1 , . . . , xn ]  K[s1 , . . . , sn ]. 1.9

Tensor Products of R-Modules and of R-Algebras

Let M , N and P be R-modules. A map f : M × N → P is said to be R-bilinear if, for any x ∈ M and any y ∈ N , the maps f (x, −) : N → P, y → f (x, y) and f (−, y) : M → P, x → f (x, y) are R-linear. In what follows we shall construct a R-module T , called the tensor product of the modules M and N , in such a way that, for any R-module P , R-bilinear maps M × N → P turn out to be in bijective correspondence with R-linear maps T → P . Indeed, one has Proposition 1.9.1. Let M and N be R-modules. There exists a pair (T, g), where T a R-module and g : M × N → T a bilinear map, satisfying the following property:

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A First Course in Algebraic Geometry and Algebraic Varieties

(∗) given any R-module P and any R-bilinear map f : M × N → P, there exists a unique linear map f  : T → P such that f = f  ◦ g. Moreover, if (T  , g  ) is another pair satisfying property (∗), then there exists a unique isomorphism of R-modules j : T → T  such that j ◦ g = g  . Proof. Let C be the free R-module whose elements are given by (finite) linear combinations of elements of M × N with coefficients from R, i.e. of the form Σni=1 ri (xi , yi ), where n ∈ N, xi ∈ M , yi ∈ N . Let D be the R-submodule of C generated by elements of the form (x + x , y) − (x, y) − (x , y), (x, y + y  ) − (x, y) − (x, y  ), (r x, y) − r (x, y), (x, r y) − r (x, y) and put T := C/D. For any element (x, y) of the basis of C, we denote with the symbol x ⊗ y its image in T . Thus, T is generated by elements of the form x ⊗ y and, from above, one has (x + x ) ⊗ y = x ⊗ y + x ⊗ y, x ⊗ (y + y  ) = x ⊗ y + x ⊗ y  , (r x) ⊗ y = x ⊗ r y = r x ⊗ y.

Similarly, the quotient map g : M × N −→ T , defined as g(x, y) := x ⊗ y, is R-bilinear. Let P be any R-module; any map f : M × N → P extends by linearity to a R-module homomorphism f : C → P . If in particular f is R-bilinear, from the above relations, f vanishes on all generators of D and so on D. Thus, it determines a well-defined homomorphism f  : T → P of R-modules such that f  (x⊗y) = f (x, y). The map f  is uniquely determined by this condition, so the pair (T, g) as above satisfies condition (*). Let (T  , g  ) be another pair satisfying (*); replacing (P, f ) with (T  , g  ) in the statement of (*), one obtains a unique R-linear map j : T → T  such that g  = j ◦ g. Changing role between T and T  , one gets a R-linear map j  : T  → T such that g = j  ◦ g  . Since j ◦ j  and j  ◦ j are both the identities, it follows that j is an isomorphism.  Definition 1.9.2. The R-module T in Proposition 1.9.1 is called the tensor product of M and N and it is denoted by M ⊗R N (or simply by M ⊗ N , if no confusion arises). By its definition, if (xi )i∈I and (yj )j∈J are families of generators of M and N , respectively, then M ⊗ N is generated by the elements xi ⊗ yj , i ∈ I, j ∈ J; in particular, if M and N are finitely generated R-modules, the same occurs for M ⊗ N . Example 1.9.3. One word of warning; the element x ⊗ y strictly depends on the modules M and N . Take, e.g. R = Z and let M = Z, N = Z2 and

Basics on Commutative Algebra

29

M  = 2Z be the submodule of M generated by 2. Let 1 ∈ Z2 and consider z := 2 ⊗ 1. Viewed as an element of M ⊗ N = Z ⊗ Z2 , z is the zero element indeed 2 ⊗ 1 = 2 · 1 ⊗ 1 = 1 ⊗ 2 · 1 = 1 ⊗ 0 = 0M⊗N . On the other hand, as an element of M  ⊗ N = 2Z ⊗ Z2 , z is non-zero. The proof of Proposition 1.9.1 can be extended to multilinear maps f : M1 × · · · × Mr → P of R-modules, giving rise to multi-tensor product M1 ⊗ · · · ⊗ Mr

(1.18)

of modules. We left to the reader to check straightforward computations. Proposition 1.9.4. Let M, N and P be R-modules. The following are uniquely determined isomorphisms: (i) M ⊗ N ∼ = N ⊗ M, defined by x ⊗ y → y ⊗ x, (ii) (M ⊗ N ) ⊗ P ∼ = M ⊗ (N ⊗ P ) ∼ = M ⊗ N ⊗ P, defined by (x ⊗ y) ⊗ z → x ⊗ (y ⊗ z) → x ⊗ y ⊗ z, (iii) (M ⊕N )⊗P ∼ = (M ⊗P )⊕(N ⊗P ), defined by (x, y)⊗z → (x⊗z, y ⊗z), (iv) R ⊗ M ∼ = M, defined by r ⊗ x → r x. Proof. See Atiyah and McDonald (1969, Proposition 2.14). 1.9.1



Restriction and extension of scalars f

Let R −→ S be a ring homomorphism and let N be a S-module. Then N has a natural structure of R-module defined by: r · x := f (r) x,

for any r ∈ R and any x ∈ N.

This structure of R-module on N is said to be obtained by restriction of scalars. In particular, the homomorphism f naturally defines a structure of R-module on the ring S (cf. Section 1.5). Proposition 1.9.5. With notation as above, assume that N is finitely generated as a S-module and that S is finitely generated as a R-module. Then N is finitely generated as a R-module. Proof. Let y1 , . . . , yn be a system of generators of N as a S-module and x1 , . . . , xm a system of generators of S as a R-module. The set {xi yj }, 1  i  n, 1  j  m, is a system of mn generators of N as a R-module. 

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A First Course in Algebraic Geometry and Algebraic Varieties

Let M be a R-module; from above one can consider the R-module S ⊗R M . This R-module has also a structure of S-module given by: s · (s ⊗ x) = (ss ) ⊗ x,

for any s, s ∈ S and any x ∈ M.

The S-module S ⊗R M is said to be obtained from M by extension of scalars. Proposition 1.9.6. Assume that M is a finitely generated R-module. Then S ⊗R M is a finitely generated S-module. Proof. If x1 , . . . , xm is a system of generators of M as a R-module, the  elements 1 ⊗ xi , 1  i  m, generate S ⊗R M as a S-module. Let now M , N and P be R-modules and f : M × N → P be a R-bilinear map. Since, for any x ∈ M , the induced map f (x, −) : N → P , y → f (x, y) is R-linear, then f induces a map M −→ Hom(N, P ) which is R-linear, since f is R-linear with respect to the variable x ∈ M . Conversely, φ any R-homomorphism M −→ HomR (N, P ) defines a R-bilinear map fφ : M × N → P, (x, y) −→ φ(x)(y). This shows there exists a bijective correspondence between the set of R-bilinear maps M × N → P and Hom(M, Hom(N, P )). At the same time, by definition of tensor product, the set of R-bilinear maps M × N → P bijectively corresponds to Hom(M ⊗R N, P ). Thus, one has a canonical isomorphism of R-modules Hom(M ⊗R N, P ) ∼ = Hom(M, Hom(N, P )). 1.9.2

Tensor product of algebras ϕ

ψ

Let S and T be two R-algebras, with R −→ S and R −→ T the corresponding structural morphisms. From Remark 1.5.1-(i), we can consider the R-module U := S ⊗R T . We will endow U with a R-algebra structure. Consider the map S × T × S × T −→ U, (s, t, s , t ) → ss ⊗ tt ; it is R-linear in each factor. By definition of multi-tensor product (1.18), the previous map induces a homomorphism of R-modules S ⊗T ⊗S ⊗T → U . From Proposition 1.9.4 this homomorphism corresponds to a homomorphism of R-modules U ⊗ U → U which, by Proposition 1.9.1, corresponds to a R-bilinear map: μ : U × U −→ U, μ(s ⊗ t, s ⊗ t ) := s s ⊗ t t .

(1.19)

By linearity one has μ(Σi (si ⊗ ti ), Σj (sj ⊗ tj )) = Σi,j (si sj ⊗ ti tj ). In other words, the map μ defines a product on U .

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Proposition 1.9.7. The product μ endows the R-module U with a structure of commutative ring, with identity 1⊗1. Furthermore, U is a R-algebra. Proof. The first part of the statement is straightforward. For the last part, note that the map R −→ U , r → ϕ(r) ⊗ 1 = 1 ⊗ ψ(r) is a ring homomorphism. Indeed one has a commutative diagram of ring homomorphisms S f

ϕ

U,

R g



ψ

T where f and g are, respectively, defined by f (s) = s ⊗ 1 and g(t) = 1 ⊗ t.  1.10

Graded Rings and Modules, Homogeneous Ideals

Let S be a ring (commutative and with identity, as always) and let moreover G(+) be an abelian group. Definition 1.10.1. S is said to be a G-graded ring (equivalently, endowed with a G-graduation), if S has a decomposition  S= Sg , (1.20) g∈G

where each Sg is a sub-group of the abelian group S(+) such that: (i) 1 ∈ S0 , and (ii) for any (g, h) ∈ G × G one has Sg · Sh ⊆ Sg+h , where for any subsets A, B ⊆ S one denotes A · B := {ab | a ∈ A, b ∈ B} and A + B := {a + b| a ∈ A, b ∈ B}. The group Sg is called the degree-g graded part of S whereas its elements are called homogeneous elements of degree g of S. When in particular G = Z, then S is simply called a graded ring. For any non-empty subset F ⊆ S of a G-graded ring, one poses Fg := F ∩ Sg ,

∀g∈G

(1.21)

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A First Course in Algebraic Geometry and Algebraic Varieties

and H(F ) :=



Fg ,

(1.22)

g∈G

where the latter is called the set of homogeneous elements of F . Notation 1.10.2. From now on in this book, to distinguish homogeneous elements from non-homogeneous ones, capital letters A, B, etcetera will be used to denote homogeneous elements of a graded ring S. When, moreover, we want more precisely specify the degree of a given homogeneous element, we will sometimes use the symbol Ag to stress that it is an element in Sg . On the contrary, elements in S which are not homogeneous will be simply denoted by small letters like a, b, etcetera. Remark 1.10.3. From the previous definitions and remarks, one has that: (a) any f ∈ S can be uniquely written as a finite sum f = Fg1 + · · · + Fgn ,

(1.23)

where Fgi ∈ Sgi , for distinct integers 1  i  n and distinct elements g1 , . . . , gn ∈ G. The elements Fg1 , . . . , Fgn are called the homogeneous components of f and (1.23) is called the decomposition of f into its homogeneous components; (b) S0 is a subring of S, whereas Sg is a S0 -submodule of S, for any g ∈ G. In particular, if S0 ∼ = K is a field, then Sg is a K-vector space; (c) if G = Z, for any integer n we set  Sd . S>n := d>n

If Sd = {0} for any d < 0, then S is more precisely a non-negatively graded ring and S>n is an ideal of S, for any integer n. In particular, one has S = S>−1 . Definition 1.10.4. The ideal S>0 is simply denoted by S+ and is called the irrelevant ideal of S. (cf. Chapter 3 for geometric motivations of this terminology). Definition 1.10.5. Let I be an ideal of a graded ring S. Then I is said  to be a homogeneous ideal if I = g∈G Ig , i.e. f ∈ I if and only if all the homogeneous components of f belong to I.

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Example 1.10.6. It is easy to see that, e.g. the polynomial ring S := K[x1 , x2 ] is a (non-negatively) graded ring, where the graduation is induced by the degree (cf. Section 1.10.1 below for a more general treatment). Then one easily verifies that S+ = (x1 , x2 ) is a homogeneous ideal. On the other hand, I := (x1 + x22 , x1 x2 ) is not homogeneous. To see this, consider, e.g. I  f = (x1 + x22 )(1 + 2x2 ) + (x1 x2 )x21 = (x1 ) + (2x1 x2 + x22 ) + (2x32 ) + (x31 x2 ), where the right-hand side of the previous equality is the decomposition of f into its homogeneous parts; then F1 := x1 ∈ S1 and F2 := (2x1 x2 +x22 ) ∈ S2 do not belong to I, whereas F3 = 2x32 = 2x2 (x1 + x22 ) − 2(x1 x2 ) ∈ I and F4 = x31 x2 ∈ I. Proposition 1.10.7. Let S be a G-graded ring and let I be an ideal. Then I is a homogeneous ideal if and only if I can be generated by a family of homogeneous elements. Proof. (⇒) Let I := (f )∈L . By Remark 1.10.3-(a), any generator f of I can be uniquely decomposed by means of its homogeneous components: f = F,gt + F,gt +1 + · · · + F,gt +k , for suitable integers t , k depending on the choice of ∈ L. Since I is for any ∈ L these components belong to I so that I := homogeneous,  F,gj ∈L,t jt +k , as desired. 





(⇐) Let I := (F )∈L , where {F }∈L is a family of homogeneous generators of I. Then, any a ∈ I is of the form a = Σ∈L a F , for some a ∈ S. Decomposing any a as in (1.23), one easily sees that any homogeneous component of a is in I, i.e. I is homogeneous.  Proposition 1.10.8. (a) Let S be a G-graded ring and let I1 , I2 be homogeneous ideals. Then I1 · I2 , I1 ∩ I2 , I1 + I2 are homogeneous ideals. (b) If S is a non-negatively graded ring and I is a homogeneous ideal, then: √ I is homogeneous; (i) (ii) I is prime if and only if, for any pair (F, G) ∈ H(S) × H(S) s.t. F G ∈ I, then either F ∈ I or G ∈ I.

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Proof. (a) is a straightforward consequence of the definitions. √ To prove (b) − (i), for any f ∈ I consider its decomposition into its homogeneous components f = F + F+1 + · · · + F+m , (1.24) √ for some integers m,  0. By definition of I, there exists a positive integer r  1 s.t. f r ∈ I. From (1.24), the decomposition of f r into homogeneous components is of the form f r = Fr + o(r ),√where o(r ) ∈ r (S+ )r . Since √ I is homogeneous then F ∈ I, i.e. F ∈ I. Thus, also f − F ∈ I. Recursively applying the same reasoning, one can conclude. For (b) − (ii), the implication (⇒) directly follows from the definition of prime and homogeneous ideal. Let us prove the other implication. Take elements a, b ∈ S such that ab ∈ I. Consider the decompositions of a and b in S into their homogeneous components a := As + As+1 + · · · + As+t and b := Bn + Bn+1 + · · · + Bn+m  for some integers t, s  0, m, n  0. Then ab = i Hi ∈ I, where for any index i s.t. n + s  i  n + m + s + t, Hi is the degree-i homogeneous component of ab, with Hi := ΣAs+j Bn+k , where summation is on indices 0  j  t and 0  k  m such that s + j + m + k = i. Since I is homogeneous, then Hi ∈ I for any n+s  i  n+m+s+t. In particular, this holds for Hs+t+n+m = As+t Bn+m ∈ I. By the assumption on I, either As+t ∈ I or Bn+m ∈ I. We can reduce to the case that e.g. / I and Bn+m ∈ I (if indeed both of them are in I, one replaces a As+t ∈ and b with a − As+t and b − Bn+m respectively and proceeds). By induction assume one has proved Bn+m , Bn+m−1 , . . . , Bn+m−r ∈ I, for some r  0. Consider Hm+n+s+t−r−1 ∈ I, where Hm+n+s+t−r−1 = As+t Bn+m−r−1 + As+t−1 Bn+m−r + · · · + As+t−r−1 Bn+m . By inductive hypothesis and As+t ∈ / I, one has Bn+m−r−1 ∈ I. By recursive application of the same strategy, one proves that all homogeneous components of b are in I, i.e. I is prime.  By using decomposition in homogeneous components, one can easily prove the following easy result. Proposition 1.10.9. Let S be a G-graded ring and let I ⊆ S be an ideal. Consider the canonical projection π : S  S/I.

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(i) For any g ∈ G one has π(Sg ) ∼ = Sg /Ig ; in particular  S/I ∼ Sg /Ig . =

(1.25)

g∈G

(ii) Moreover, (1.25) is a direct sum if and only if I is homogeneous. From the previous result, when I is a homogeneous ideal, if one poses (S/I)g := Sg /Ig , for any g ∈ G, then the ring S/I is a endowed with a G-graduation induced by the G-graduation of S. Definition 1.10.10. If S is a contains as a sub-field the set G1 , G2 ∈ H(S) are of the same be denoted by Q0 (S) and called fractions of S. 1.10.1

graded, integral domain, the field Q(S) 1 consisting of all fractions G G2 such that degree, with G2 = 0. This sub-field will the sub-field of degree-zero, homogeneous

Homogeneous polynomials

We now discuss in more details a concrete example of non-negatively graded ring. Let K be any field and let X0 , . . . , Xn be indeterminates over K. A non-zero polynomial F ∈ K[X0 , . . . , Xn ] is said to be homogeneous if all its monomials have the same degree. Definition 1.10.11. For any integer d  0, we will denote by K[X0 , . . . , Xn ]d the set consisting of the zero-polynomial together with all degree-d, homogeneous polynomials in K[X0 , . . . , Xn ]. This is a K-vector space, whose canonical basis is given by monic monomials of degree d in the indeterminates X0 , . . . , Xn . Lemma 1.10.12. For any integer d  0, one has b(n, d) := dimK (K[X0 , . . . , Xn ]d ) =

  n+d , d

(1.26)

where b(n, d) stands for the   binomial coefficient of indexes n and d, with the convention b(n, 0) := n0 = 1. Proof. If d = 0, for any n one has K[X0 , . . . , Xn ]0 = K and we are done in this case. Therefore, we may assume d  1.

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A First Course in Algebraic Geometry and Algebraic Varieties

If n = 1, for any integer d, (1.26) holds as the canonical basis of d−1 d−2 2 d d K[X0 , X1 ]d is  given by X0 , X0 X1 , X0 X1 , . . . , X1 , consisting of d+1 d + 1 = d monomials. We can therefore assume n  2 and proceed by double induction on n and d. Let Mdn denote the canonical basis of K[X0 , . . . , Xn ]d ; its cardinality |Mdn | equals dimK (K[X0 , . . . , Xn ]d ). The set Mdn is a disjoint union of the two non-empty subsets Mdn = M ∪ M , where M is the set of monomials in Mdn containing Xn whereas M = Mdn \ M . On the one hand, M is the canonical  basis  of K[X0 , . . . , Xn−1 ]d ; by the inductive hypothesis on n, |M | = n−1+d . On the other, if we divide each monomial appearing in M by d Xn , we get the canonical basis of K[X0 , . . . , Xn ]d−1 which by construction bijectively corresponds to M and which, by the inductive hypothesis on d, n+d−1 consists of d−1 elements. One concludes by observing that |Mdn |

      n+d−1 n−1+d n+d = |M | + |M | = + = d−1 d d 





and the proof is complete. Remark 1.10.13. (i) Previous results show that the polynomial ring (n)

S(n) := K[X0 , . . . , Xn ] = ⊕d0 Sd

(1.27)

(n)

is a non-negatively graded ring. In particular, S0 = K and each (n) (n) graded piece Sd , for any d  1, is a (free) S0 -module, i.e. K-vector space of finite dimension given by (1.26). Graduation of S(n) is given by the degree of the homogeneous components of a polynomial. (ii) One comment for the reader; we use here the symbol S(n) to denote K[X0 , . . . , Xn ] (and not other notation like A(n+1) which instead will be used later on, cf., e.g. Chapter 2) since we want to shed light onto the graded structure of this ring. Moreover, the index (n) is used instead of (n + 1), the correct number of indeterminates, since as customary S(n) will be identified with the ring of homogeneous coordinates of the projective space Pn in Chapter 3. Notation 1.10.14. Following Notation 1.10.2, homogeneous polynomials in S(n) will be denoted by capital letters like F , G, etcetera. When, in (n) particular, we consider a polynomial in Sd , we will sometimes more precisely denote it by Fd , to explicitly remind its degree. On the contrary,

Basics on Commutative Algebra

37

polynomials which are not homogeneous will be denoted by small letters f , g, etc., as done in the previous sections. If f ∈ S(n) is of degree m, then f can be uniquely decomposed as a linear combination of its degree-d homogeneous parts, for 0  d  m, which we write f=

m 

Fd ,

(1.28)

d=0

where not necessarily all Fd = 0. Proposition 1.10.15. (i) F is homogeneous of degree d if and only if the following identity between polynomials in K[X0 , . . . , Xn , t] holds: F (tX0 , . . . , tXn ) = td F (X0 , . . . , Xn ).

(1.29)

(ii) If F is homogeneous of degree d, then the Euler’s identity n  i=0

Xi

∂F =dF ∂Xi

(1.30)

holds true. (iii) Let F, g ∈ S(n) be non-constant polynomials, where F is homogeneous and g divides F . Then g = G is also homogeneous. In particular, irreducible factors of a homogeneous polynomial are all homogeneous. Proof. (i) Condition (1.29) is obviously necessary. The fact that it is also sufficient directly follows from the decomposition as in (1.28). Indeed, for f ∈ S(n) , condition f (tX0 , . . . , tXn ) = td f (X0 , . . . , Xn ) forces f to (n) be an element of Sd . (ii) Taking the partial derivative with respect to the indeterminate t of (1.29) and then evaluating for t = 1 gives Euler’s identity. (iii) One has F = gh, for some h ∈ S(n) . Assume by contradiction that g is not homogeneous. From (1.28), there exist integers k > 0, j  0 such that g = Gj + Gj+1 + · · · + Gj+k ,

with Gj , Gj+k = 0.

Since F is homogeneous, then h has to be non-homogeneous, i.e. there exist integers r > 0, i  0 such that h = Hi + Hi+1 + · · · + Hi+r ,

with Hi , Hi+r = 0.

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A First Course in Algebraic Geometry and Algebraic Varieties

Now, F = gh would give F = Gj Hi + (Gj Hi+1 + Gj+1 Hi ) + · · · + Gj+r Hi+k , where Gj Hi = 0, Gj+r Hi+k = 0, since S(n) is an integral domain. Thus, i + j = deg(Gj Hi ) < deg(Gj+r Hi+k ) = i + j + r + k would contradict the hypothesis on F .  We now define important operators on homogeneous and nonhomogeneous polynomials, which will be frequently used in the next chapters (cf., e.g. Chapter 3). In what follows, x1 , . . . , xn and X0 , . . . , Xn will denote indeterminates over K. Definition 1.10.16. For any integer i ∈ {0, . . . , n}, let δi : K[X0 , . . . , Xn ] → K[X0 , . . . , Xi−1 , Xi+1 , . . . , Xn ]

(1.31)

be the map defined by δi (f (X0 , . . . , Xi−1 , Xi , Xi+1 , . . . , Xn )) := f (X0 , . . . , Xi−1 , 1, Xi+1 , . . . , Xn ). Viceversa, let hi : K[x1 , . . . , xn ] → K[X0 , . . . , Xn ]

(1.32)

be the map defined in such a way that, if f ∈ K[x1 , . . . , xn ] is of degree d, then   X0 X1 Xn , ,..., hi (f (x1 , . . . , xn )) := Xid f , Xi Xi Xi where in the above definition one has replaced xh , with h and with X Xi for i + 1  h  n.

Xh−1 Xi ,

for 1  h  i,

Lemma 1.10.17. For any integer i ∈ {0, . . . , n}: (i) the map δi is a K-algebra homomorphism; (ii) the map hi is not a K-algebra homomorphism; on the other hand it is multiplicative, i.e. hi (f g) = hi (f )hi (g) and, when deg(f ) = deg(g) = deg(f + g), it is also additive, i.e. hi (f + g) = hi (f ) + hi (g);

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Basics on Commutative Algebra

(iii) (iv) (v) (vi)

for any f ∈ K[x1 , . . . , xn ] of degree d, hi (f ) ∈ K[X0 , . . . , Xn ]d ; δi ◦ hi = idK[x1 ,...,xn ] ; Xi does not divide f if and only if δi (f ) has the same degree of f ; let F ∈ S(n) be any homogeneous polynomial and let m  0 be the multiplicity of Xi as a factor of F (recall (1.2)), then hi (δi (F )) = XFm . i

Proof. Properties (i), (ii), (iv) and (v) are straightforward, whereas (iii) follows by applying Proposition 1.10.15-(i). To prove (vi), it suffices to consider the case where Xi does not divide F and verifying in this case that hi (δi (F )) = F . To prove this, taking into account (v), it suffices to prove it on monomials of degree d and then use linearity on each monomial appearing in F .  Note that, if F ∈ S(n) is homogeneous, in general δi (F ) does not remain homogeneous (it remains homogeneous only if F is constant w.r.t. Xi ). For this reason δi is called the dehomogenizing operator w.r.t. Xi and, correspondingly, δi (F ) is called the dehomogenized polynomial of F w.r.t. Xi . On the contrary, by Lemma 1.10.17-(ii), hi (f ) is called the homogenized polynomial of f w.r.t. Xi and, consequently, hi is called the homogenizing operator w.r.t. Xi . As in Section 1.3.4, one can consider resultant of two homogeneous polynomials with respect to a given indeterminate. One has: Theorem 1.10.18. Let F, G ∈ S(n) be non-constant, homogeneous polynomials, with deg(F ) = d and deg(G) = m. Write F = Ad + Ad−1 Xn + · · · + A0 Xnd and G = Bm + Am−1 Xn + · · · + B0 Xnm , where Aj ∈ K[x0 , . . . , Xn−1 ]j , Bk ∈ K[x0 , . . . , Xn−1 ]k , with 0  j  d, 0  k  m and A0 B0 = 0. Then the resultant polynomial RXn (F, G) is either 0 or it is homogeneous of degree dm. Proof. If F, G ∈ (K[X0 , . . . , Xn−1 ])[Xn ] =: D[Xn ] have a non-constant common factor then, by Theorem 1.3.16, one concludes RXn (F, G) = 0. Assume therefore that F and G have no non-constant common factor, i.e. RXn (F, G) ∈ K[X0 , . . . , Xn−1 ] \ {0}. For simplicity, let R(X0 , . . . , Xn−1 ) := RXn (F, G). By (1.7) and the fact that the polynomials Aj and Bi are homogeneous, for any t ∈ K one has that R(tX0 , . . . , tXn−1 )

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A First Course in Algebraic Geometry and Algebraic Varieties

is given by the determinant of the (m + d) × (m + d) matrix: ⎛ d ... ... A0 0 0 t Ad td−1 Ad−1 d d−1 ⎜ 0 t Ad−1 . . . ... A0 0 t Ad ⎜ ⎜ ... ... ... ... ... ... ... ⎜ ⎜ ... ... . . . td Ad td−1 Ad−1 . . . ⎜ ... C := ⎜ m m−1 ⎜ t Bm t Bm−1 ... .. B0 0 ... ⎜ m m−1 ⎜ 0 Bm−1 . . . ... .. B0 t Bm t ⎜ ⎝ ... ... ... ... ... ... ... ... ... ... ... ... tm B m . . .

... ... ... ... 0 ... ... ...

⎞ 0 0⎟ ⎟ . . .⎟ ⎟ ⎟ A0 ⎟ ⎟. ⎟ ⎟ 0⎟ ⎟ ⎠ B0

If we multiply the ith-row of C by tm−i+1 , 1  i  m, and the (m+j)th-row of C by td−j+1 , 1  j  d, we get the following matrix ⎛

tm+d Ad tm+d−1 Ad−1 . . .

...

...

...

tm A0

...

...

...

...

tm A1

...

...

⎜ 0 tm+d−1 Ad ⎜ ⎜ ⎜ ... ... ⎜ ⎜ ⎜ 0 0  := ⎜ C ⎜tm+d B tm+d−1 B m ⎜ m−1 ⎜ m+d−1 B ⎜ 0 t m ⎜ ⎜ ⎝ ... ... 0

0

...

...

...

. . . td+1 Ad . . .

...

...

. . . td+1 B1 td B0

0

0

. . . td+1 B2 td B1

..

0

...

...

...

...

...

...

...

0

tm+1 Bm tm Bm−1

0

... 0

⎞

tm−1 A0 . . . 0 ⎟ ⎟ ⎟ ... .. .. ⎟ ⎟ ⎟ ... .. tA0 ⎟ ⎟. ... .. 0 ⎟ ⎟ ⎟ ... .. 0 ⎟ ⎟ ⎟ ... .. .. ⎠ ...

.. tB0

 once with respect to rows and then with respect Using linearity for det(C), to columns, we get the following relation:  = tq R(X0 , . . . , Xn ), tp R(tX0 , . . . , tXn ) = det(C)  d+1  + 2 whereas where p := m+ (m− 1)+ · · ·+ 1 + d+ (d− 1)+ · ·· 1 = m+1 2 m+d+1 . Thus, R(tX q := (m + d) + (m + d − 1) + · · · + 1 = 0 , . . . , tXn ) = 2 tq−p R(X0 , . . . , Xn ) and one then concludes by Proposition 1.10.15–(i) and by the fact that q − p = md.  To conclude this section, we note that for homogeneous polynomials in two variables, we have the following useful result. Proposition 1.10.19. Let K be algebraically closed, d a positive integer and F ∈ K[X0 , X1 ]d . Then there exist λ ∈ K \ {0} and d pairs (ai , bi ) ∈ K2 , 1  i  d (not necessarily all distinct pairs but each of them different from (0, 0)) such that F (X0 , X1 ) = λ(a1 X1 − b1 X0 ) · · · (ad X1 − bd X0 ).

Basics on Commutative Algebra

41

The pairs (ai , bi ) are uniquely determined up to order whereas each pair is determined up to proportionality. These pairs are called roots of the homogeneous polynomial F . Proof. Assume that r is the multiplicity of X0 as a factor of F , with 0  r  d. Thus, F = X0r G, where G ∈ K[X0 , X1 ]d−r is not divisible by X0 . If r = d, then G = λ is a constant and the pairs in the statement all coincide with (0, 1), which is therefore counted with multiplicity d. Let us assume therefore that r < d. Then G ∈ K[X0 , X1 ]d−r is a nonconstant, homogeneous polynomial (cf. Proposition 1.10.15-(iii)), which is not divisible by X0 ; thus δ0 (G) ∈ K[X1 ]. One concludes by applying Theorem 1.3.11 to the polynomial δ0 (G) and then by homogenizing via h0 each linear factor.  1.10.2

Graded modules and graded morphisms

If S is a G-graded ring and S is a H-graded ring, for G(+) and H(+) abelian groups, a ring homomorphism f : S → S  is said to be homogeneous if there exists a group homomorphism φ : G → H s.t., for any g ∈ G one has f (Sg ) ⊆ Sφ(g) . If f and φ are both isomorphisms, then f is an isomorphism of φ-graded rings. If moreover G = H, a homomorphism f : S → S is said to be homogeneous of degree 0 if φ = idG . An isomorphism of degree 0 is simply called an isomorphism of graded rings. If G = H = Z and if f : S → S is homogeneous, then φ : Z → Z is given by the multiplication with an integer d, so f (Sg ) ⊆ Sdg , for any g ∈ Z. In this case, f is said to be homogeneous of degree d. If S is a G-graded ring, a S-modulo M is called a G-graded module if  M admits a decomposition M = g∈G Mg , where each Mg is an abelian sub-group of the group M(+), s.t., for any (g, h) ∈ G×G, Sg ·Mh ⊆ Mg+h , where we used notation as above. If G = Z, M is simply said to be a graded module. As an easy example, any homogeneous ideal of a G-graded ring S is a Ggraded S-module. Same terminology and definitions introduced for graded rings can be extended to graded modules, in particular one has the notion of homogeneous homomorphisms of graded modules. Given a G-graded S-module M, one can change its graduation by using the following procedure: fix h ∈ G and define  M(h)g where M(h)g := Mh+g . (1.33) M(h) := g∈G

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A First Course in Algebraic Geometry and Algebraic Varieties

Then, M (h) is a G-graded S-module which is isomorphic to M as an S-module but in general not as a G-graded S-module (cf. Exercise 1.5). Example 1.10.20. Let K be a field and V be a K-vector space of dimension n + 1  1. The symmetric algebra over V is Sym(V ) :=  d d∈N Sym (V ), which is a non-negatively graded ring, simply denoted by S(V ). Its degree d homogeneous part will be therefore denoted by S(V )d . A  similar proof as that for (1.26) shows that dim(S(V )d ) = b(n, d) = n+d d . Moreover, S(V )0 ∼ = K and S(V ) is generated as a K-algebra by S(V )1 . As usual, for F ∈ S(V )d , we write d = deg(F ). The choice of an (ordered) basis (e0 , . . . , en ) for V induces the dual basis (e0 , . . . , en ) on V ∗ ∼ = Hom(V, K), which is defined by ei (ej ) = ej (ei ) = δij , where δij the Kronecker’s symbol. One usually poses ei := Xi and ei := ∂i ,

0  i  n.

In this way, S(V ∗ ) is naturally identified with S(n) = K[X0 , . . . , Xn ] as in Section 1.10.1. Dually, the ring S(V ) is identified with the ring of differential operators, which is denoted by Dn := K[∂0 , . . . , ∂n ] and which is (abstractly) isomorphic to S(n) as a graded ring. The degree-d homogeneous part Dn,d of the ring Dn is called the K-vector space of degreed homogeneous differential operators, i.e. differential operators where only degree-d monomials in ∂0 , . . . , ∂n appear (the zero-differential operator is considered to be homogeneous of any degree). Let X := (X0 , . . . , Xn ) be indeterminates and, for any multi-index I := (i0 , . . . , in ) ∈ Nn+1 , let |I| := i0 + · · · + in be the length of the multi-index. We will pose X I := X0i0 . . . Xnin .

(1.34) (n)

Thus, any degree-d homogeneous polynomial in Sd can be written also as F = Σ|I|=dFI X I , (similar notation can be used for differential operators). In the above notation, from (1.29) F is homogeneous of degree d if and only if F (tX) = td F (X), ∀t ∈ K and Euler’s identity (cf. (1.30)) reads d F (X) = Σni=0 ∂i F (X). 1.11

Localization

Let R be a ring and let S ⊂ R be a subset. S is said to be a multiplicative system if

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Basics on Commutative Algebra

(i) 0 ∈ / S, (ii) 1 ∈ S, (iii) for any s, t ∈ S, one has st ∈ S. In the cartesian product R × S one poses the following relation: (a, s) ≡ (b, t) ⇔ ∃ u ∈ S s.t. (at − bs)u = 0.

(1.35)

It is easy to see that ≡ is an equivalence relation. Remark 1.11.1. When in particular R is an integral domain and S = R \ {0}, then ≡ coincides with the equivalence relation used to construct the field of fractions Q(R), i.e. (a, s) ≡ (b, t) if and only if (at − bs) = 0. will be denoted by RS and Definition 1.11.2. The quotient set R×S ≡ called localization of R with respect to S. The ≡–equivalence class of a pair (a, s) ∈ R × S will be denoted by as . Operations on R endow RS with a structure of a (commutative and with identity) ring by the following rules: at + bs a b + := s t st

and

a b ab := . s t st

One has a natural ring homomorphism j : R → RS ,

j

defined by a −→

a , 1

which is called the localization homomorphism. Proposition 1.11.3. (i) With notation as above, one has that Ker(j) = {a ∈ R | sa = 0 for some s ∈ S}. (ii) For any s ∈ S, j(s) = 1s ∈ RS is invertible. Proof. (i) Take a ∈ R and assume that as = 0 in R, for some s ∈ S. Then j(a) = a1 is such that (a, 1) ≡ (as, s) = (0, s), i.e. j(a) = 0 ∈ RS . Conversely, for any a ∈ Ker(j), one has (a, 1) ≡ (0, s), for some s ∈ S. This means there exists u ∈ S s.t. 0 = u(as − 0) = a(us) in R, i.e. a is a zero-divisor in R. (ii) j(s)−1 := 1s . 

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A First Course in Algebraic Geometry and Algebraic Varieties

Remark 1.11.4. (i) From Proposition 1.11.3, if R is an integral domain then j is injective, in which case R can be identified with its image j(R) and so considered as a subring of RS . Moreover, for any choice of multiplicative system S ⊂ R, RS is identified with a subring of Q(R), the field of fractions of R. (ii) When otherwise R is an arbitrary ring and S coincides with the set of all non-zero divisors in R, RS is called the ring of total fractions of R. Definition 1.11.5. If R is a graded ring and S is a multiplicative system of R, the elements of RS are endowed with a graduation induced by that  of R: deg as := deg(a) − deg(s). The subset a  R(S) := ∈ RS | a, s ∈ H(R), s ∈ S, deg(a) = deg(s) s is a subring of RS which is called the homogeneous localization of R with respect to S. R(S) is therefore the subring of homogeneous, degree-zero elements in RS . Recalling Definition 1.1.3, one has: Proposition 1.11.6. Let R be a ring, S ⊂ R a multiplicative system and j the associated localization homomorphism. (i) Any ideal of RS is an extended ideal. (ii) Let J ⊂ R be a proper ideal. J e ⊆ RS is a proper ideal of RS if and only if J ∩ S = ∅. (iii) Let I1 and I2 be ideals in RS . Then I1 = I2 if and only if I1c = I2c . (iv) If R is Noetherian, then RS is Noetherian. Proof. (i) Let I ⊆ RS be any ideal. For any as ∈ I, one has j(s) as = 1s as = a ∈ I and a ∈ I c . Thus, j(a) 1s = as ∈ (I c )e , i.e. I ⊆ (I c )e . By Remark 1.1.4-(ii) one concludes that I = (I c )e . (ii) 1 ∈ J e if and only if there exist a ∈ R and s ∈ S s.t. 1 = as ∈ J e , i.e. if and only if (a, s) ≡ (d, d), for some d ∈ S. By (1.35), this is equivalent to the existence of u ∈ S s.t. uad = uds ∈ S, since S is a multiplicative system, i.e. if and only if a(ud) ∈ J ∩ S. (iii) The statement follows from (i) as, for any ideal I ⊆ RS , one has I = (I c )e .

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Basics on Commutative Algebra

(iv) Let I1 ⊆ I2 ⊆ · · · ⊆ RS be any ascending chain of ideals of RS . Since R is noetherian, the ascending chain I1c ⊆ I2c ⊆ · · · ⊆ R is stationary (cf. Proposition 1.4.4). This means there exists a positive integer n0 s.t. Inc 0 = Inc 0 +k for any integer k  1. From (i) and (iii) we get In0 = (Inc 0 )e = (Inc 0 +k )e = In0 +k , for any k  1, i.e. the original ascending chain of ideals of RS is stationary too.  Remark 1.11.7. From correspondence

Proposition

1.11.6-(ii)

and

(iii),

the

{Ideals of RS } → {Ideals of R}, defined by I → I c , is injective and it restricts to a bijective correspondence {Proper ideals of RS } → {Ideals of R not intersecting S}. As a matter of notation, if J ⊂ R is any ideal such that J ∩ S = ∅, then the ideal J e of RS is usually denoted by JRS

(1.36)

(cf., e.g. Atiyah and McDonald, 1969). Corollary 1.11.8. (i) If J ⊂ S is an ideal such that J ∩ S = ∅, then one has (RS /JRS ) ∼ = (R/J)S . (ii) If p ⊂ R is a prime ideal s.t. p ∩ S = ∅, then pRS is prime, such that c (pRS ) = p. (iii) Any prime ideal of RS is of the form pRS , for some prime ideal p of R s.t. p ∩ S = ∅. Proof. (i) By assumption, the sequence ι

π

0 → J → R → R/J → 0 is exact, i.e. ι is injective, Im(ι) = Ker(π) and π is surjective. From the fact that localization is an exact operation (cf. for details Atiyah and McDonald, 1969, Proposition 3.3), the previous exact sequence gives rise to the exact sequence 0 → JRS → RS → (R/J)S → 0, which proves the statement.

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A First Course in Algebraic Geometry and Algebraic Varieties

(ii) Assume first that p is a prime ideal in R. If as bt ∈ pRS , there exists u ∈ S s.t. uab = stu ∈ p. Since u ∈ S and S ∩ p = ∅, then ab ∈ p, which implies either a ∈ p or b ∈ p. Thus, either as ∈ pRS or bt ∈ pRS . c Let now b ∈ (pRS ) be any element. By definition of pRS , this means there exist a ∈ p and s, t ∈ S s.t. bt = as ∈ pRS . By (1.35), there exists u ∈ S s.t. u(at − bs) = 0, i.e. ubs = uat ∈ p. On the other hand, since us ∈ S and p ∩ S = ∅, then b ∈ p because p is prime. Thus (pRS )c ⊆ p. c By Remark 1.1.4-(ii), one deduces (pRS ) = p. (iii) Let q ⊂ RS be any prime ideal. From Remark 1.1.4-(iii) and Proposition 1.11.6-(ii), p := qc is a prime ideal of R s.t. p ∩ S = ∅. From Proposition 1.11.6-(i) and (iii), q = p RS .  Remark 1.11.9. Previous results show, in particular, that there exists a bijective correspondence between prime ideals of RS and prime ideals of R not intersecting S. 1.11.1

Local rings and localization

A ring R is said to be a local ring if it contains a unique maximal ideal m. The field R/m is called the residue field of the local ring (R, m). Any field F is a local ring, of maximal ideal (0) and residue field F; for any prime p ∈ Z and any positive integer n, Zpn is a local ring, of maximal ideal (p) and residue field Zp . Proposition 1.11.10. Let R be a ring and m be a maximal ideal of R. Then (R, m) is local if and only if U (R) = R \ m. Proof. If (R, m) is local, then every non-invertible element is contained in m. Conversely, assume that U (R) = R \ m; for any proper ideal I ⊂ R one necessarily has I ⊆ m, i.e. all proper ideals of R are contained in m which is therefore the only maximal ideal of R.  We will now briefly discuss how localization in some cases can produce local rings from a given ring R. Remark 1.11.11. (i) If p is any prime ideal of a ring R, the set S := R \ p is a multiplicative system. In this case, one usually denotes the localization RS by Rp

(1.37)

Basics on Commutative Algebra

47

which, by abuse of terminology, is called the localization of R w.r.t. p (cf., e.g. Atiyah and McDonald, 1969). From Remark 1.11.9, there is therefore a bijective correspondence between prime ideals of Rp and prime ideals q of R contained in p. If moreover R is a graded ring, and p is a homogeneous prime ideal, the homogeneous localization R(S) (cf. Definition 1.11.5) with S := R \ p is denoted by R(p) .

(1.38)

(ii) If f ∈ R is any non-nilpotent element, the set S := {1, f, f 2 , f 3 , . . .} is a multiplicative system and the localization RS will be denoted by Rf .

(1.39)

When moreover R is a graded ring and f is a homogeneous element in R, the homogeneous localization R(S) will be denoted by R([f ]) .

(1.40)

(iii) If R is an integral domain, the ideal (0) is prime and one has R(0) = Q(R). If moreover R is a graded ring, then R((0)) = Q0 (R), where Q0 (R) as in Definition 1.10.10. In the above notation, one has the following. Proposition 1.11.12. (i) For any prime ideal p in R, (Rp , pRp ) is a local ring with residue field isomorphic to Q(R/p). (ii) For any graded ring R and any homogeneous, prime ideal p, (R(p) , pR(p) ) is a local ring with residue field isomorphic to Q0 (R/p). Proof. (i) From Corollary 1.11.8–(ii), pRp is a prime ideal of Rp ; in particular all its elements are not invertible in Rp . Consider any as ∈ Rp \ pRp ; then a∈ / p. Thus, j(a) = a1 is invertible in Rp . One concludes that (Rp , pRp ) is local by Proposition 1.11.10. At last, by definition of localization, it is clear that Rp /pRp ∼ = Q(R/p). (ii) It follows from (i) and the fact that R(p) ⊂ Rp as a subring. 

48

1.12

A First Course in Algebraic Geometry and Algebraic Varieties

Krull-Dimension of a Ring

Let R be a ring. A chain of prime ideals of R is a finite sequence of inclusions p0 ⊂ p1 ⊂ p2 ⊂ · · · pm ,

(1.41)

where pi is a prime ideal of R, for any 0  i  m, and where the inclusions are strict. The length of the chain (1.41) is set to be m by definition. The chain (1.41) is said to be maximal if it cannot be part of a chain of prime ideals of length higher than m. Given a prime ideal p in R, one defines the height of p, denoted by ht(p), as follows: ht(p) := Supm∈Z {m | p0 ⊂ p1 ⊂ · · · ⊂ pm = p} .

(1.42)

Spec(R) := {p | p prime ideal of R}.

(1.43)

Set

Definition 1.12.1. Given R a ring we define the Krull-dimension of R, denoted by K dim(R), as K dim(R) := Supp∈Spec(R) {ht(p)}. Example 1.12.2. (i) If R = K is a field, then K dim(K) = 0. (ii) If R is any PID, e.g. R = Z or R = K[x], where K is a field and x an indeterminate over K, then K dim(R) = 1. Indeed, being a PID, any ideal in R is principal; thus any prime ideal is generated by a prime element of R which is also irreducible. Therefore any (maximal) chain of prime ideals in R is of the form (0) ⊂ (p), where p ∈ R irreducible. (ii) Let K be a field, n  2 be a positive integer and R := K[x1 , . . . , xn ], where xi indeterminates over K, for 1  i  n. The ideal (x1 ) ⊂ R is a prime ideal, since (xR1 ) ∼ = K[x2 , . . . , xn ] is an integral domain. Moreover, (0) ⊂ (x1 ) is a chain of prime ideals in R, thus ht((x1 ))  1. Note moreover that the previous chain of prime ideals is also maximal. Indeed, if by contradiction there were a prime ideal p = (0), (x1 ) such that 0 ⊂ p ⊂ (x1 ), then x1 ∈ / p but any element p ∈ p should be divisible by x1 in R, namely of the form p = xk1 f , with k a positive integer and f ∈ R, where f not divisible by x1 in R. This is a contradiction; indeed, since

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p is a prime ideal and since by assumption x1 ∈ / p, then one must have f ∈ p which would imply that f is divisible by x1 . Therefore one more precisely has ht((x1 )) = 1. Using similar arguments as above, one shows that the chain of prime ideals (0) ⊂ (x1 ) ⊂ (x1 , x2 ) ⊂ (x1 , x2 , x3 ) ⊂ · · · ⊂ (x1 , x2 , . . . , xn ) is maximal, which implies K dim(R)  n. From Corollary 1.12.4-(i) below, we will see that this is actually an equality. (iii) If otherwise K is a field and x1 , x2 , x3 , . . . is a sequence of infinite indeterminates over K, the polynomial ring R := K[x1 , x2 , x3 , . . .] of polynomials in infinite indeterminates and with coefficients from K is such that K dim(R) = +∞. Indeed, for any positive integer n, there exists in R the chain of prime ideals (0) ⊂ (x1 ) ⊂ (x1 , x2 ) ⊂ (x1 , x2 , x3 ) ⊂ · · · ⊂ (x1 , x2 , . . . , xn ), which is not maximal; in other words R contains chains of prime ideals of arbitrarily large length. The next result shows that Krull-dimension extends the definition of transcendence degree for integral K-algebras of finite type given in Definition 1.8.7 to more general rings which are not necessarily integral K-algebras of finite type. Precisely, recalling Definition 1.8.7, one has the following. Theorem 1.12.3. Let K be a field and let R be an integral K-algebra of finite type. Then K dim(R) = trdegK (R). In particular, any integral Kalgebra of finite type has finite Krull-dimension. Corollary 1.12.4. (i) If K is a field and if x1 , . . . , xn are indeterminates over K, then K dim(K[x1 , . . . , xn ]) = n. (ii) If K is a field and if A ⊂ B is an integral extension of integral Kalgebras, then K dim(A) = K dim(B). Proof. (i) follows from the fact that trdegK (K[x1 , . . . , xn ]) = n as in Remark 1.8.8, whereas (ii) follows from the fact that trdegK (A) =  trdegK (B).

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A First Course in Algebraic Geometry and Algebraic Varieties

Remark 1.12.5. One warning for the reader. Theorem 1.12.3 does not hold for more general integral K-algebras. Indeed, let K be a field, n  1 be an integer and x1 , . . . , xn be indeterminates over K. The field of rational functions K(x1 , . . . , xn ) is a K-algebra which is not of finite type over K (cf. Proposition 1.5.7) and, being a field, it is such that K dim(K(x1 , . . . , xn )) = 0; on the other hand, being a purely transcendental extension of K, one has trdegK (K(x1 , . . . , xn )) = n (cf. Remark 1.8.4-(iii) and (iv)). Thus, R := K(x1 , . . . , xn ) is an integral K-algebra for which K dim(R) < trdegK (R). Proof of Theorem 1.12.3. Set r := trdegK (R). Since R is an integral K-algebra of finite type, then r < +∞. We first show that r  K dim(R). By Definition 1.8.7, r = trdegK (Q(R)) and this integer equals the maximum number of elements in R which are algebraically independent over K (cf. Proposition 1.8.6). First consider the case r = 0, i.e. K ⊆ Q(R) is an algebraic field extension. Since K ⊆ R ⊆ Q(R), all elements of R are algebraic over K. Since R is an integral domain, we claim that R is a field; if the claim is true, then equality K dim(R) = trdegK (R) = r = 0 holds. We therefore need to showing that R is a field, namely that U (R) = R \ {0}; it suffices to showing that, for any a ∈ R \ {0}, K[a] is a field. We may therefore assume that R = K[a]. Since R is an integral K-algebra of finite type and a ∈ R is algebraic over K, then K ⊆ R is an algebraic simple field extension (cf. Remark 1.7.1) and we are done. Assume now r  1. For any p ∈ Spec(R), R p is an integral K-algebra of finite type, as it follows from the fact that R is and from the existence of the canonical epimorphism π : R  R p . For any length-m prime ideal chain p0 = (0) ⊂ p1 ⊂ · · · ⊂ pm in R, for some m  0, factoring by p1 yields a prime ideal chain in (0) ⊂

R p1

p2 pm ⊂ ··· ⊂ , p1 p1

  which is of length m − 1. If we can show that trdegK pR1 < r then, by   induction on r, we can deduce that K dim pR1 < r so that K dim(R)  r. Since we have the canonical surjection π : R  pR1 , by Remark 1.8.8-(ii), we   have that trdegK pR1  trdegK (R) = r. We need to showing that equality does not hold. Assume by contradiction that equality holds. Therefore,

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there exist y1 , . . . , yr ∈ R such that their images π(y1 ), . . . , π(yr ) ∈ pR1 are algebraically independent over K. Take any element z ∈ p1 \ {0}; thus, the elements y1 , . . . , yr , z ∈ R are algebraically dependent over K (cf. Proposition 1.8.6) so there exists a polynomial f (x1 , x2 , . . . , xr+1 ) in K[x1 , x2 , . . . , xr+1 ], where x1 , x2 , . . . , xr+1 indeterminates over K, such that f (y1 , . . . , yr , z) = 0. Since R is an integral domain, we may therefore assume that f := f (x1 , x2 , . . . , xr+1 ) is irreducible and that f is not a polynomial in the only indeterminate xr+1 , otherwise z ∈ p1 ⊂ R would be algebraic over K so one would have K[z] = K(z) (cf. Remark 1.7.1) and so z ∈ R would be invertible, which contradicts z ∈ p1 . Thus f (x1 , . . . , xr ) := f (x1 , x2 , . . . , xr , 0) ∈ K[x1 , . . . , xr ] is a non-constant polynomial for which f (π(y1 ), π(y2 ), . . . , π(yr )) = 0 in pR1 , contradicting the algebraic independence of π(y1 ), π(y2 ), . . . , π(yr ) over K. This completes the proof of r  K dim(R). Now, we show that r  K dim(R); we will use induction on r. If r = 0, by the previous step, we already know that in this case K dim(R) = trdegK (R) = r = 0 holds true. Let us assume r  1 and let R = K[s1 , . . . , sn ], for some n > 0 and some si ∈ R, 1  i  n. Up to a possible permutation, we may assume that s1 is transcendental over K. Let us consider the multiplicative system S := K[s1 ] \ {0} ⊂ R. Localizing R by S gives RS ∼ = K(s1 )[s2 , . . . , sn ], i.e. RS is an integral K(s1 )-algebra of finite type. Since we have the field extensions K ⊂ K(s1 ) ⊂ Q(R) ∼ = K(s1 , . . . , sn ) with r = trdegK (R), by Proposition 1.8.5 we have r = trdegK (R) = trdegK (K(s1 )) + trdegK(s1 ) (Q(R)). Therefore, as in Definition 1.8.7, one gets trdegK(s1 ) (Q(R)) = trdegK(s1 ) (RS ) = r − 1. By induction, it follows that there exists a maximal chain of prime ideals in RS of length r − 1, namely (∗) q0 ⊂ q1 ⊂ · · · ⊂ qr−1 . For any 0  j  r − 1, consider the contracted ideal (cf. Definition 1.1.3) pj := qcj = qj ∩R. Any pj is a prime ideal in R (cf. Corollary 1.11.8-(iii))such that pj ∩ S = ∅. The induced chain of prime ideals in R (∗∗) p0 ⊂ p1 ⊂ · · · ⊂ pr−1 is proper, i.e. the inclusions are strict, otherwise pi = pj , for some 0  i = j  r − 1, would give pei = pi RS = qi = pej = pj RS = qj ,

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A First Course in Algebraic Geometry and Algebraic Varieties

contradicting (∗). Since pr−1 ∩ S = ∅, then pr−1 ∩ K[s1 ] = {0}. Thus R injectively maps K[s1 ] into the the canonical epimorphism π : R  pr−1

R quotient ring pr−1 , since Ker(π) = pr−1 . Note that pr−1 ∈ Spec(R) cannot R would be a field; being moreover be a maximal ideal of R; otherwise pr−1 a K-algebra of finite type, since R is, then by Zariski’s lemma (cf. Lemma 

1.7.2) we would have

R pr−1

: K < +∞, so any element of

R pr−1

would be

R algebraic over K, which is forbidden by the facts that K[s1 ] ⊂ pr−1 and that s1 is transcendental over K. Therefore, there exists a maximal ideal pr ⊂ R such that pr−1 ⊂ pr . Using (∗∗) we get therefore a chain

p0 ⊂ p1 ⊂ · · · ⊂ pr−1 ⊂ pr of prime ideals of R which shows that K dim(R)  r, as wanted.



Exercises Exercise 1.1. Let R be a ring (commutative and with identity, as always in this book). (i) Prove that if I is a prime ideal in R then, for any expression I = I  ∩ I  , with I  , I  ideals in R, then either I = I  or I = I  . (ii) Let (0) := {a ∈ R | ∃ n  1 s.t. an = 0} be the set of nilpotents  elements of R, which is  called the nilradical of R. Prove that (0) is an ideal in R such that (0) = ∩p∈Spec(R) p. (iii) Let a ∈ R be a nilpotent element and let u be invertible in R. Show that a + u is invertible in R. Exercise 1.2. Let R be a ring and let I, J ⊂ R be ideals. Show that the set (I : J) := {r ∈ R | r J ⊆ I} is an ideal, which is called the quotient ideal of I by J. Show moreover that, for any ideals I and J in R, one has (cf. Atiyah and McDonald, 1969, p. 26): √ √ √ I ⊆ (I : J), (I : J) J ⊆ I, I ∩ J = I ∩ J. Exercise 1.3. Let R be ring and let x be an indeterminate over R. Let f (x) := a0 + a1 x + a2 x2 + · · · + an xn ∈ R[x].

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Show that f (x) ∈ R[x] is: (i) invertible if and only if a0 is invertible in R and a1 , . . . , an are nilpotent elements in R; (ii) nilpotent if and only if a0 , a1 , . . . , an are nilpotent elements; (iii) a zero-divisor if and only if there exists r ∈ R∗ = R \ {0} such that r · f (x) = 0. Exercise 1.4. Let K be a field and let R ⊂ K be a subring such that K is a finite R-algebra. Show that R is also a field (cf. Hulek, 2003, Lemma 1.22). Exercise 1.5. Let S be a non-negatively graded ring and let M be a non-negatively graded S-module. Let h ∈ N and let M(h) be the S-module defined as in (1.33). Show that M(h) is a graded S-module which is isomorphic to M as a S-module. Give an example where M(h) is not isomorphic to M as a graded S-module.

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Chapter 2

Algebraic Affine Sets

From now on K will always denote a field. To start with, we consider basic facts with no further assumptions on K. We will then discuss some examples showing where difficulties come out in this general setting. Thus, we will focus once and for all on the case of K algebraically closed (unless otherwise explicitly stated), in particular containing infinitely many elements (cf. Corollary 1.3.12). 2.1

Algebraic Affine Sets and Ideals

For any field K, let AnK (or simply An , when the field K is understood), the n–dimensional numerical (standard) affine space over K. This is the set Kn of ordered n-tuples (p1 , . . . , pn ), where pi ∈ K, 1  i  n. The use of the symbol An (instead of Kn ) resides on the fact that we want to take into account the geometric nature of Kn , i.e. considering its elements as points instead as vectors. We will indeed endowed An K with a structure of topological space (cf. Definition 2.1.13), and this structure we want to be kept in mind to be different from that of Kn as a numerical n-dimensional vector space over K. An element P = (p1 , . . . , pn ) in An will be called a point of the affine space and p1 , . . . , pn will be its coordinates in An . Correspondingly, the (numerical) vector (p1 , . . . , pn ) ∈ Kn , i.e. the vector having those components with respect to the canonical basis of Kn , will be simply denoted by P and called coordinate vector of the point P . The point O ∈ An , whose coordinate vector is O = (0, 0, . . . , 0) ∈ Kn , is the origin of An . 55

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A First Course in Algebraic Geometry and Algebraic Varieties

Let x := (x1 , . . . , xn ) be a n–tuple of indeterminates over K. In this (n) chapter, we will denote by AK , or simply by A(n) , the polynomial ring K[x] = K[x1 , . . . , xn ]. An element f (x) = f (x1 , . . . , xn ) ∈ A(n) will be simply denoted by f , if no confusion arises. It is then clear that any f ∈ A(n) can be regarded as a map f : An → K, P → f (P ) = evP (f ),

(2.1)

where evP (f ) denotes the evaluation of f ∈ A(n) at the point P ∈ An . Definition 2.1.1. The subset of An Za (f ) := f −1 (0) = {P ∈ An | f (P ) = 0}, will be called the set of zeroes (or simply the zero-set) of f . The subscript a in Za (−) stands for the word affine, to distinguish from the projective case which will be considered in Chapter 3. Remark 2.1.2. From Definition 2.1.1 it is already clear the reason why, for geometric objects, one is mostly concerned with algebraically closed fields. Indeed, consider the polynomials 1, x21 + 1 ∈ A(1) . Whatever K is, Za (1) = ∅. On the other hand, if, e.g. K = R then Za (1) = ∅ = Za (x21 + 1), since x21 + 1 is irreducible in A(1) . If otherwise K = C, in this case Za (1) = ∅ ⊂ Za (x21 + 1) = {i, −i} and the cardinality of the zero-set of the polynomial x21 + 1 ∈ A(1) equals its degree. More generally, one poses Definition 2.1.3. For any F ⊂ A(n) finite subset of polynomials,  Za (f ) Za (F ) := f ∈F

is called the zero-set of F . More precisely, if F := {f1 , . . . , ft }, then Za (F ) = Za (f1 , . . . , ft ) =

t 

Za (fi ) = {P ∈ An | fi (P ) = 0, ∀ i = 1, . . . , n} .

i=1

Definition 2.1.4. A subset Y ⊆ An is called an Algebraic Affine Set (AAS, for short), if Y = Za (F ), for some finite subset F ⊂ A(n) . When F is explicitly given as F = {f1 , . . . , ft }, the polynomials fi ∈ A(n) , 1  i  t, are said to determine a system of equations defining Y .

Algebraic Affine Sets

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Given F = {f1 , . . . , ft } as above, consider the ideal these polynomials generate in A(n) . Thus, I := (f1 , . . . , ft ) is a finitely generated ideal. Proposition 2.1.5. For any field K, one has Za (F ) = Za (I) := {P ∈ An | f (P ) = 0, ∀ f ∈ I}. Proof. It is clear that Za (I) ⊆ Za (f1 , . . . , ft ). On the other hand, since any f ∈ I is f = Σti=1 gi fi , for suitable gi ∈ A(n) , 1  i  t, it is clear that  P ∈ Za (f1 , . . . , ft ) implies P ∈ Za (I). Remark 2.1.6. From Proposition 2.1.5, an AAS Y does not depend on the system of equations defining it. In particular any finite set of generators of the same ideal I defines the same algebraic affine set Y in An . Example 2.1.7. As above, let K be any field. (i) In A1 , one has {0} = Za (x1 ) = Za ((x1 )) = Za ({xn1 }n∈F ), for F ⊂ N any (possibly infinite) subset. (ii) In A(2) consider the three ideals I = (x1 , x2 ), J = (x1 , x2 − x21 ) and K = (x2 , x2 − x21 ). One has I = J; indeed J is a proper ideal in A(1) , I ⊆ J (since x1 , x2 = 1(x2 − x21 ) + x1 (x1 ) ∈ J) and I is maximal (since A(2) /I ∼ = K is a field). On the other hand, K  I, as it easily follows from the fact that K = (x2 , x21 ). However, Za (I) = Za (J) = Za (K) = O = (0, 0) ∈ A2 . To understand the geometric counterpart of the three ideals, take for simplicity K = R. The origin O is the AAS cut out by the generators of the three ideals I, J and K above. But one has main differences among the three ideals as it follows. The generators of the ideals I give rise to affine lines, which are the two coordinate axes of the Cartesian plane, which transversally intersect along the origin O (cf. Figure 2.1). The ideal J gives rise to the intersection between the vertical axis Za (x1 ) and the parabola Za (x2 − x21 ) with vertex at O (cf. Figure 2.2); once again this intersection is transversal. For what concerns the ideal K, its two generators give rise respectively to the parabola Za (x2 − x21 ) and to its tangent line Za (x2 ) at the origin O;

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A First Course in Algebraic Geometry and Algebraic Varieties

O

Fig. 2.1

Origin as Za (I).

Fig. 2.2

Origin as Za (J).

thus the line Za (x2 ) has intersection multiplicity 2 with the parabola at the origin O (cf. Figure 2.3). In all the three cases, passing to Za (−) discards multiplicities from the picture. We will give more precise algebraic motivations for this phenomenon in Remark 2.1.14-(iv).

Algebraic Affine Sets

Fig. 2.3

59

Origin as Za (K).

A direct consequence of Theorem 1.4.1 is the following. Corollary 2.1.8. Let K be any field. Any AAS in An is of the form Za (I), for some ideal I ⊆ A(n) . Remark 2.1.9. In particular, Definitions 2.1.3 and 2.1.4 make sense for any subset F ⊆ A(n) ; no matter F is finite or not, the ideal I := (F ) is always finitely generated since A(n) is Noetherian (cf. Remark 1.4.2). We discuss easy examples, some of which show basic differences between the cases with either K algebraically closed or not, and either infinite or not. Example 2.1.10. (i) In A(1) , consider the subset F := {xn1 }n∈N . The ideal I generated by F is I = (F ) = (x1 ), since K[x1 ] is a PID. Moreover Za (F ) = Za ((F )) = Za ((x1 )) = {0}. (ii) Consider f ∈ A(1) a non-constant polynomial. If K is not algebraically closed, there exist polynomials f with no root in K (e.g. when f is irreducible of deg(f )  2). In such a case, Za ((f )) = ∅ = Za ((1)) even if the inclusion (f ) ⊂ (1) = A(1) is strict (being (f ) a proper maximal ideal in A(1) ). When otherwise K is algebraically closed, for any f ∈ A(1) , one has ∅ = Za ((f )); more precisely, Za ((f )) = {α1 , . . . , αk }, where k  deg(f ) and αi are all the distinct roots of f , the case k = n occurring if and only if f has simple roots.

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A First Course in Algebraic Geometry and Algebraic Varieties

(iii) Conversely to (ii), taking α1 , . . . , αk ∈ K distinct elements and posing f := Πki=1 (x1 − αi ) ∈ K[x1 ], one has {α1 , . . . , αk } = Za ((f )) ⊂ A1K . (iv) If K is a finite field, there do exist non-zero polynomials f ∈ A(1) for which Za (f ) = A1 . Take, e.g. K = Zp , p ∈ Z a prime, and consider f := xp1 − x1 ∈ Zp [x]. Then Za (f ) = A1 = Za (0); indeed, as a consequence of Lagrange’s theorem and the fact that U (Zp ) is cyclic of order p − 1, f totally splits in Zp [x1 ] as f = Πj∈Zp (x1 − j). Recalling operations on ideals as in Section 1.1, one can discuss first properties of AAS’s in An . Proposition 2.1.11. Let K be any field. Then: (i) for any subsets F and G of A(n) one has F ⊆ G =⇒ Za (F ) ⊇ Za (G).

(2.2)

In particular, for any ideals I1 ⊆ I2 in A(n) , one has Za (I1 ) ⊇ Za (I2 ); (ii) for any I1 , I2 ⊆ A(n) ideals, Za (I1 ∩ I2 ) = Za (I1 ) ∪ Za (I2 ); (iii) for any family {Iα }α∈A of ideals in A(n) , Za (Σα∈A Iα ) = ∩α∈A Za (Iα ); (iv) for any P ∈ An , whose coordinates are (p1 , . . . , pn ), the ideal mP := (x1 − p1 , . . . , xn − pn )

(2.3)

is maximal in A(n) and Za (mP ) = {P √ }. (v) for any ideal I ⊆ A(n) , Za (I) = Za ( I). Proof. (i) For any P ∈ Za (G) and for any f ∈ G, one has f (P ) = 0. Since F ⊆ G, then P ∈ Za (F ) also. The rest of the statement follows from Proposition 2.1.5, and from Remark 2.1.9. (ii) Since I1 ∩ I2 ⊆ I1 , I2 , by (i) we have Za (I1 ) ∪ Za (I2 ) ⊆ Za (I1 ∩ I2 ). On the other hand, if P ∈ / Za (I1 ) ∪ Za (I2 ), there exist f1 ∈ I1 and f2 ∈ I2 s.t. f1 (P ) = 0 and f2 (P ) = 0. Thus, f1 f2 ∈ I1 ∩ I2 is such that (f1 f2 )(P ) := f1 (P )f2 (P ) = 0, since K is a field, which shows that P ∈ / Za (I1 ∩ I2 ).

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(iii) One has Iα ⊆ Σα∈A Iα , for any α ∈ A. Therefore, by (i), Za (Σα∈A Iα ) ⊆ Za (Iα ), for any α ∈ A, which implies that Za (Σα∈A Iα ) ⊆ ∩α∈A Za (Iα ). On the other hand, for any P ∈ ∩α∈A Za (Iα ), one has that fα (P ) = 0, for any fα ∈ Iα and for any α ∈ A. By definition of Σα∈A Iα , this implies that P ∈ Za (Σα∈A Iα ). (iv) For any P ∈ An , we have the K-algebra homomorphism ΦP : A(n) → K defined by the rules ΦP (k) = k, ∀ k ∈ K, and ΦP (xi ) = pi , 1  i  n. In particular, for any f ∈ A(n) , ΦP (f ) = evP (f ), where evP (−) is the evaluation at the point P as above. Clearly, the morphism ΦP is surjective. Thus, Ker(ΦP ) is a maximal ideal of A(n) and, by the definition of mP as in (2.3), one has mP ⊆ Ker(ΦP ). To prove the maximality of mP it therefore suffices to show that equality holds. To do this, denote by y := (y1 , . . . , yn ) a set of new indeterminates in such a way that y := x − P , where P is the coordinate vector associated to the point P ∈ An . For any f = f (x) ∈ Ker(ΦP ), one has f (x) = f (y + P ) =: g(y) ∈ K[y] ∼ = A(n) . By the assumptions on f (x), one has g(y) ∈ (y1 , . . . , yn ) ⊂ K[y], i.e. g(y) = y1 h1 (y) + · · · + yn hn (y), for some hi (y) ∈ K[y], 1  i  n. This implies that f (x) = g(x − P ) = (x1 − p1 )h1 (x − P ) + · · ·+ (xn − pn )h1 (x − P ) ∈ mP , which shows that Ker(ΦP ) ⊆ mP . The last part of the assertion is trivial. √ follows from (i) and the fact (v) The inclusion Za ( I) ⊆ Za (I) directly √ that, for any ideal√I one has I ⊆ I (recall Section 1.1). On the other hand, for any g ∈ I, there exists a positive integer n such that g n ∈ I (g n )(P ) = so that, for any P ∈ Za (I), one has (g n )(P ) = 0. Since √ n (g(P )) , as a power in K, this forces g(P ) = 0, i.e. P ∈ Za ( I), which  proves the other inclusion.

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Note that, for any field K, one has ∅ = Za ((1)) and An = Za ((0)), i.e. they are AAS’s. Therefore, using Proposition 2.1.11-(ii) and (iii), one easily observes: Proposition 2.1.12. The set Can := {AAS’s in An }

(2.4)

is the set of closed subsets of a topology on An . Definition 2.1.13. The topology on An having Can as the set of closed subsets is called the Zariski topology of An and will be denoted by ZarAn , or simply by Zarna . Remark 2.1.14. (i) From now on, unless otherwise specified, An will be always endowed with the topology Zarna . The AAS’s will be called also (Zariski) closed subsets of An (for brevity, the term Zariski will be omitted in the sequel). An open subset in Zarna will be of the form Za (I)c , for some ideal I in A(n) , i.e. the complement of a closed subset of An . Any subset Y ⊆ An will be endowed from now on with the topology induced by Zarna on Y . This topology will be called the Zariski topology of Y and denoted by Zarna,Y (or simply ZarY , when the inclusion Y ⊆ An is clearly understood). (ii) From Proposition 2.1.11-(iv), it follows that Zarna is T1 and that Can contains all finite subsets of An . It is clear that for n = 1 these, together with the empty-set, are the only proper closed subsets of A1 ; if moreover K is finite, these are the only closed subsets at all. If otherwise K is infinite (e.g. when K is algebraically closed, cf. Corollary 1.3.12), any non-empty open subset of A1 contains infinite elements and any two non-empty open subsets intersect, i.e. Zara1 is not T2 . We will show that this more generally holds for any n  1 (cf. Remark 4.1.1). This is false when K is finite: take, e.g. K = Z2 , U1 = Za (x)c and U2 = Za (x − 1)c . (iii) It is then clear that, when K is either R or C, the euclidean topology of AnK is finer than Zarna . (iv) Proposition 2.1.11-(v) shows that we do not have a bijective correspondence between ideals in A(n) and elements in Can , since any ideal I which is not radical, defines the same closed subset of its radical ideal √ I. On the other hand, passing to radical ideals allows one to discard

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non-reduced pieces from the scene. To understand why, consider the following examples. Example 2.1.15. Let K be any field. Consider, e.g. x2 , x22 ∈ A(2) ; the associated closed sets in A2 is the same Za (x2 ) = Za (x22 ),given by the x1 -axis. Algebraically speaking, this is due to the fact that (x22 ) = (x2 ). The point set given by Za (x2 ) = Za (x22 ) does not take into account that the x1 -axis in the second case should have to be considered endowed with multiplicity 2 at any of its points. In terms of ideals, the situation is completely different: (x2 ) is prime, being A(2) /(x2 ) an integral domain, whereas (x22 ) is not prime, since A(2) /(x22 ) is not a reduced ring (recall Definition 1.1.1). This phenomenon occurred also in Example 2.1.7-(ii): the ideal I = J is radical, since maximal (cf. Lemma 1.1.2-(ii)) whereas √ K is / K. On the other hand, K = I, not radical since K = (x2 , y) but x ∈ which motivates O = Za (I) = Za (K). When K is not algebraically closed, the failure of the bijective correspondence between ideals of A(n) and elements in Can is even worse: indeed it fails even if we restrict to radical ideals of A(n) , as some of the examples discussed in the following remark show. Remark 2.1.16. (i) Let K be a non-algebraically closed field. Let f ∈ A(1) be a nonconstant polynomial with no roots in K (e.g. when f is irreducible). Then Za ((f )) = ∅ = Za ((1)), even if (f ) is a proper ideal. If we moreover assume that f ∈ A(1) is irreducible, then (f ) is a maximal ideal (since A(1) is a PID), and so radical (as it has  been proved in Lemma 1.1.2 (ii)). Thus, Za ((f )) = ∅, even if (f ) = (f ) ⊂ (1) = A(1) . (ii) Consider once again Example 2.1.10-(iv), where we found Za (xp −x) = Za (0) = Zp , even if (0) ⊂ (xp − x). Note moreover that the ideal (xp − x) is radical: indeed (xp − x) = ∩a∈Zp (x − a), where each (x − a) is a maximal (so prime ideal) in Zp [x] (cf. the primary decomposition of an ideal Atiyah and McDonald, 1969, Chapter 4). (iii) Let f := x21 + x22 + 1 ∈ K[x1 , x2 ], where we consider K to be either R or C. In both cases, f is irreducible. Since K[x1 , x2 ] is a UFD (but not a PID), the ideal (f ) is prime (so radical) but not maximal (since K[x1 , x2 ]/(f ) is not a field). Thus, from the algebraic point of view, properties of the ideal (f ) remain unchanged, no matter if K is either R or C.

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On the contrary the geometric situation is completely different. When  K = R, one has Za (f ) = ∅ = Za ((1)) even if (f ) = (f ). In the affine classification of real conics, this is a non-degenerate empty affine conic. When otherwise K = C, Za (f ) = ∅ and its support is a non-degenerate conic with a center in the complex affine plane (or a non-degenerate ellipse when we consider the complexification of the affine plane, namely classification of conics in the complex affine plane up to affine transformations with coefficients from the subfield R). Identifying A2C with A4R , one can easily show that the locus Za ((f )) (endowed with the topology induced by the euclidean topology of A4R ) is homeomorphic to a 2-sphere in A3R minus 2 points. This is called the open Riemann surface associated to Za (f ) (cf. Exercise 2.1 for full details). Other examples of AAS’s are the following. Example 2.1.17 (Coordinate subspaces). Let K be any field, which we will assume infinite for simplicity, and let 0 < m  n be integers. For any sequence of integers 0 < i1 < i2 < i3 < · · · < im  n, one has an injective map ϕ := ϕi1 ,i2 ,i3 ,··· ,im : Am → An defined as follows: ϕ

(b1 , b2 , . . . , bm ) −→ (0, . . . , 0, b1 , 0, . . . , 0, b2 , 0, . . . , 0, bm , 0, . . . , 0). i2 im i1 ϕ is a homeomorphism between (Am , Zarm a ) and Im(ϕ) endowed with the topology induced by Zarna . This image is the (Zariski) closed subset of An Za (xj | ∀j = i1 , . . . , im ), which is called a m-dimensional coordinate affine subspace of An . The case m = 1 gives rise to the ith-coordinate axis of An , for 1  i  n. Example 2.1.18 (Affine subspaces). Let K be any field, which we will assume infinite for simplicity, and let 1 , . . . , k ∈ A(n) be linearly independent linear forms, i.e. i = ai1 x1 + · · · + ain xn , where aij ∈ K, 1  i  k, 1  j |, n, and b1 , . . . , bk ∈ K. Consider Y := Za (1 − b1 , . . . , k − bk ), i.e. the set of solutions of the linear system Ax = b, where A is the k × n matrix whose ith-row is filled-up by the coefficients of the linear form i , 1  i  k, and where x and b are the n × 1 and k × 1 column vector of indeterminates and constant terms, respectively. By assumptions on the linear forms, rank(A) = k, so Ax = b is compatible. Compatibility condition is independent from the field K and Rouch´e–Capelli’s theorem ensures that the set of solutions Y is a

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linear variety in An of dimension n − k, i.e. Y has a linear parametric representation of the form x1 = c1 + λ1 (t1 , . . . , tn−k ), . . . , xn = cn + λn (t1 , . . . , tn−k ), where ci ∈ K are constant, λi are linear homogeneous polynomials in the parameters tj and with coefficients from K, 1  i  n, 1  j  n − k. Y is called a (n − k)-dimensional affine subspace of AnK . If I = (1 − b1 , . . . , k − bk ) ⊂ A(n) is the ideal generated by the linear equations defining Y , then I is a radical ideal: consider indeed the K-algebra homomorphism π : K[x1 , . . . , xn ] −→ K[t1 , . . . , tn−k ], π

xi −→ ci + λi (t1 , . . . , tn−k ),

1  i  n.

The map π is surjective, as it follows from Gauss–Jordan elimination theory, and its kernel is given by I, as its generators are representatives of the equivalence class of linear systems whose zero-set is exactly Y . Thus, I is prime, since K[t1 , . . . , tn−k ] is an integral domain, so it is radical (cf. Lemma 1.1.2). Example 2.1.19 (Affine hypersurfaces). Let K be any field and let f ∈ A(n) be a non-constant polynomial. The AAS Y := Za (f ) = Za ((f )) is called the affine hypersurface of equation f = 0 in An . Since A(n) is a UFD then, up to units, f factorizes as f = f1r1 f2r2 · · · fr ,

(2.5)

where f1 , . . . , f ∈ A(n) are non-proportional, irreducible polynomials and r1 , . . . , r are positive integers. From Proposition 2.1.11, it is clear that Za (f ) = Za (f1 · · · f ) =

 

Za (fi );

(2.6)

i=1

 indeed, by the primary decomposition of ideals in a UFD, one has (f ) = (f1 · · · f ) (cf. Atiyah and McDonald, 1969, Chapter 4, for more details). Y is called the affine hypersurface determined by f ∈ A(n) and the polynomial fred = f1 f2 · · · f

(2.7)

is called the reduced equation of Y , which is indeed another equation for Y . The integer deg(fred ) is called the degree of Y . When n = 1, d = 1 gives a

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point, d = 2 gives either the empty-set or two distinct points (it depends on the field K), etcetera. If n = 2, d = 1 gives a line, d = 2 gives a conic, d = 3 gives a plane cubic curve, etcetera. For n = 3, d = 1 is a plane, d = 2 is a quadric, d = 3 is a cubic surface, and so on. Definition 2.1.20. Let K be an infinite field and let f ∈ A(n) be a non– constant polynomial. The open set Ua (f ) := Za (f )c = An \ Za (f ) is called a principal open (affine) set of Zarna . Lemma 2.1.21. Principal open (affine) sets form a basis for the topology Zarna . Proof. Any open set of Zarna is of the form U = Za (I)c , for some ideal I ⊂ A(n) . Since A(n) is Noetherian, I = (f1 , . . . , fm ), for some fi ∈ A(n) , m 1  i  m, and Za (I) = ∩m i=1 Za (fi ). This shows that U = ∪i=1 Ua (fi ).  Example 2.1.22 (Products of AAS’s. Cylinders). (i) Let Z1 ⊆ Ar and Z2 ⊆ As be (Zariski) closed sets. Denote by (r) (s) Ax := K[x1 , . . . , xr ] and Ay := K[y1 , . . . , ys ] the ring of polynomials which are (evaluating) functions operating on the affine spaces Ar and As , respectively. As a set, it is clear that Ar × As = Ar+s , and the ring of functions operating on this affine space will be denoted (r+s) by Ax,y := K[x1 , . . . , xr , y1 , . . . , ys ]. Now, Z1 = Za (I1 ), for some (r)

ideal I1 = (f1,1 , . . . , f1,n ) ⊆ Ax , and Z2 = Za (I2 ), for some ideal (s) I2 = (f2,1 , . . . , f2,m ) ⊆ Ay , as it follows from Noetherianity. Since we have natural ring inclusions (r+s) ← A(s) A(r) x → Ax,y y ,

the polynomials generating the ideals I1 and I2 can be regarded as (r+s) elements in Ax,y . One poses Z1 × Z2 := Za ((f1,1 , . . . , f1,n , f2,1 , . . . , f2,m )) ⊂ Ar+s ,

(2.8)

where the ideal (f1,1 , . . . , f1,n , f2,1 , . . . , f2,m ) is intended as an ideal in (r+s) Ax,y . Z1 × Z2 is called the product of the two AAS’s; it is a closed subset of Ar+s . For example, if p1 ∈ A1x1 and p2 ∈ A1y1 , their product is given by the point P = Za ((x1 − p1 , y1 − p2 )) = (p1 , p2 ) ∈ A2 .

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(ii) Particular cases of products are given by cylinders. If Z1 ⊂ Ar is a (Zariski) closed proper subset as above, then Z1 × A1 ⊂ Ar+1 is called the cylinder over Z1 in Ar+1 whereas Z1 is called a directrix of the cylinder. Since A1 in itself is determined by the ideal (0) (moreover this is the only possibility, when K is algebraically closed), the cylinder is simply given by Z × A1 := Za ((f1,1 , . . . , f1,n , 0)) = Za ((f1,1 , . . . , f1,n )) ⊂ Ar+1 ; (2.9) in other words, equations defining Z1 in Ar coincide with equations defining the cylinder over Z1 in Ar+1 . For example, if p1 ∈ A1x , the cylinder in A2 over p1 is {p1 } × A1y , whose defining equation is Za (x1 − p1 ); this is nothing but the line x1 = p1 in the plane. The closed subsets Za (x21 + x22 − 1), Za (x21 − x22 − 1) and Za (x21 − x2 ) regarded in A3 are the elliptic, hyperbolic and parabolic quadric cylinders in the real affine space A3 (see Figure 2.4). is finer than the (iii) When K is infinite, note that the topology Zarr+s a topology Zra × Zsa . Consider as a set A2 = A1 × A1 ; from Remark 2.1.14-(ii), closed subsets in Zar1a are only ∅, finite number of points and A1 . Therefore, if we endow A2 with the topology Zar1a × Zar1a , the only closed subsets in this topology are ∅, A2 , finite unions of points, finite unions of parallel lines to the x1 -axis and finite unions of parallel lines to the x2 -axis. On the contrary, since K is infinite, the set Za (x1 − x2 ) ⊂ A2 is a closed subset for Zara2 , which is not closed in Zar1a × Zar1a . However, given Z1 × Z2 ⊂ Ar+s , endowed with the (natural) Zariski topology Zarr+s a,Z1 ×Z2 (recall notation as in

Fig. 2.4

Elliptic, hyperbolic and parabolic cylinders in the real affine 3-space.

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Remark 2.1.14-(i)), the topology induced by Zarr+s a,Z1 ×Z2 on {p1 } × Z2 (Z1 × {p2 }, respectively), with p1 ∈ Z1 (p2 ∈ Z2 , respectively), coincides with Zarsa,Z2 (Zarra,Z1 , respectively). Up to now, the “operation” Za (−) allows one to bridge from ideals in A(n) to (Zariski closed) subsets of An . We now want to “drive” from subsets of An to ideals in A(n) . To do this, we introduce the following. Definition 2.1.23. Let K be any field. For any subset Y ⊆ An , one denotes by Ia (Y ) := f ∈ A(n) | f (P ) = 0, ∀ P ∈ Y . It is an ideal, which is called the ideal of Y in A(n) . Remark 2.1.24. Note that for any f ∈ Ia (Y ), one has Y ⊆ Za ((f )); geometrically speaking, if Y = ∅, An , elements in Ia (Y ) give rise to hypersurfaces in An containing Y . From Definition 2.1.23 it follows that Y1 ⊆ Y2 ⇒ Ia (Y1 ) ⊇ Ia (Y2 ),

(2.10)

for any subsets Y1 and Y2 of An . Proposition 2.1.25. Let K be any field and let Y be any subset of An . Then Za (Ia (Y )) = Y , where Y is the closure (in the topology Zarna ) of Y in An . In particular, for any subset Y of An one has Y ⊆ Za (Ia (Y )), where equality holds if and only if Y is (Zariski) closed in An . Proof. By Definition 2.1.23, Za (Ia (Y )) is a closed subset in An such that Y ⊆ Za (Ia (Y )); then Y ⊆ Za (Ia (Y )). Conversely, let W := Za (J) be any closed subset containing Y , J some ideal in A(n) . Since Y ⊆ W , from (2.10) one has J ⊆ Ia (W ) ⊆ Ia (Y ). By Proposition 2.1.11 (i), one has therefore W = Za (J) = Za (Ia (W )) ⊇ Za (Ia (Y )), i.e. any closed subset containing Y  contains also Za (Ia (Y )). In particular, Y ⊇ Za (Ia (Y )). We consider some examples where the inclusion Y ⊂ Y is strict. In all the examples below, we will consider K an infinite field. Example 2.1.26. (i) Let Y := A1 \ {0}. Then Y = Za (x1 )c is an open set of A1 strictly contained in it. On the other hand Ia (Y ) = Ia (A1 ) = (0), since nonzero, non-constant polynomials f ∈ A(1) have at most finitely many roots in K. In particular Y = A1 . The same conclusion holds for e.g. any principal open set Ua (f ) ⊂ A1 , for any f ∈ A(1) \ K.

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(ii) Let T ⊂ K be an infinite subset of K and, for any integer n  2, let T n denote the n-tuple product T × · · · × T ⊂ An . If U  An is a subset containing T n , one has Ia (U ) = (0), as it follows from Theorem 1.3.13, and so U = An as in (i). The same conclusion holds when, e.g. K is either R or C, and U contains an open polydisk of the euclidean topology on An . By Proposition 2.1.11-(i) and (2.10), one is therefore left with reversing– inclusion maps: Ia (−)

{Subsets of An } −→ Y



Ideals of A(n)

 (2.11)

→ Ia (Y )

and Za (−)

{Subsets of An } ←−



Ideals of A(n)

Za (J) ← J.

 (2.12)

Remark 2.1.27. (i) By definition of Za (−), the map (2.12) is neither surjective nor injective. The non-surjectivity follows from the fact that Im(Za (−)) = Can  {subsets of An }. Similarly, for any field K, the non-injectivity of the map (2.12) is a direct consequence of Proposition 2.1.11-(v), as it follows for any choice of ideal I which is not radical. (ii) Note that, when K is not algebraically closed, injectivity of (2.12) fails also on radical ideals. Consider, e.g. K = R and m = (x2 + 1)  A(1) ; m is maximal so radical (cf. Lemma 1.1.2), on the other hand ∅ = Za (m) = Za ((1)). Similarly, for any prime p ∈ Z, the ideal I = (xp − x) ⊂ Zp [x] is a proper, radical ideal (cf. Remark 2.1.16 (ii)) on the other hand Za (I) = Za ((0)) = A1Zp . (iii) Examples 2.1.26 show that (2.11)  is not injective; on the other hand, Can surjects onto Ideals of A(n) . An important task is to understand how “domain” and “target” of the maps Ia (−) and Za (−) have to be modified in order to get bijections. A second task is to understand whether Ia (−) is the inverse map of Za (−). Results of the next sections will give answers to these questions, when K is algebraically closed (cf. Corollary 2.2.5).

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Hilbert “Nullstellensatz”

In Proposition 2.1.11-(iv), we proved that, for any field K and for any P ∈ An , the ideal mP = (x1 − p1 , . . . , xn − pn ) ⊂ A(n) is maximal. The next fundamental result will show that, when K is algebraically closed, all maximal ideals of A(n) are of this form. Theorem 2.2.1 (Hilbert “Nullstellensatz”-weak form). Let K be an algebraically closed field. Then m ⊂ A(n) is a maximal ideal ⇔ m = mP , for some P = (p1 , . . . , pn ) ∈ An . Remark 2.2.2. (i) The assumption K algebraically closed is essential for the (⇒)–part. (1) (1) Indeed, taking, e.g. m1 = (x2 + 1) ⊂ AR and m2 = (x2 + x+ 1) ⊂ AZ2 , one gets maximal ideals in both cases but none of them is of the form prescribed by Theorem 2.2.1. (ii) Theorem 2.2.1 can be viewed as the analogue in A(n) , for any integer n  1 and for any algebraically closed field K, of the Fundamental (1) theorem of Algebra for AC (recall Theorem 1.0.1). Indeed, if I ⊂ A(n) is a proper ideal then I ⊆ m, for some maximal ideal m ⊂ A(n) . From Theorem 2.2.1, m = mP for some P ∈ An , therefore ∅ = Za (I) ⊇ Za (mP ) = {P }, as it follows from Proposition 2.1.11-(i) and (iv). In other words, if K is algebraically closed, Za (I) = ∅ ⇔ I = (1). Both examples in (i) are such that ∅ = Za (m1 ) ⊂ A1R and ∅ = Za (m2 ) ⊂ A1Z2 even if the ideals are maximal. Proof of Hilbert “Nullstellensatz”-weak form. (⇐) This is Proposition 2.1.11-(iv), which holds for any field K. (⇒) Let m ⊂ A(n) be any maximal ideal; then F := A(n) /m is a field. The π ι composition of K-algebra homomorphisms K → A(n)  F, where ι the natural inclusion and π the canonical projection, gives rise to a K-algebra homomorphism ϕ : K → F which is necessarily injective, since K is a field. ϕ Then K → F is a field extension. Since F is, by construction, also a Kalgebra of finite type then, by Zariski’s Lemma (cf. Lemma 1.7.2), the field extension K ⊂ F is algebraic of finite degree. Thus, F = K, since K is algebraically closed. Posing pi := π(xi ) ∈ F = K, for any 1  i  n, then mP := (x1 − p1 , . . . , xn − pn ) ⊆ Ker(π) = m. On the other hand, mP is maximal in A(n) from Proposition 2.1.11-(iv), therefore equality must hold. 

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Theorem 2.2.3 (Hilbert “Nullstellensatz”-strong form). Let K be an algebraically closed field and √ let n  1 be any integer. For any ideal I ⊆ A(n) one has Ia (Za (I)) = I. Remark 2.2.4. The assumption √ that K is algebraically closed is essential for the equality Ia (Za (I)) = I. Indeed, taking, e.g. A1R one has ∅ = (1) Za ((1)) = Za ((x2 + 1)) even if (x2 + 1) is radical since maximal in AR . Similarly, in A1Zp , p ∈ Z a prime, Za ((0)) = Za ((xp − x)) even if (xp − x) is (1)

radical in AZp (recall Remark 2.1.16-(ii)). Proof of Hilbert “Nullstellensatz”-strong form. For simplicity denote √ Y := Za (I). We can assume that I is a proper ideal. Note that I ⊆ Ia (Y ). √ Indeed, for any g ∈ I there exists a positive integer s = s(g) such that g s ∈ I so, for any P ∈ Y , 0 = g s (P ) = (g(P ))s which implies g ∈ Ia (Y ), since K is a field. √ We now prove the other inclusion Ia (Y ) ⊆ I. Let g ∈ Ia (Y ) be any element; by Noetherianity of A(n) , let I = (f1 , . . . , fm ), where fi ∈ A(n) suitable non-constant polynomials. Consider y another indeterminate over (n+1) K. In the polynomial ring Ax,y := K[x1 , . . . , xn , y] ∼ = A(n+1) consider the ideal J := (f1 , . . . , fm , yg − 1). Since g ∈ Ia (Y ), then Za (J) = ∅ ⊂ An+1 . The fact that K is algebraically closed implies that J = (1), as it follows from the Hilbert “Nullstellensatz”-weak form. Thus, there exist polynomials (n+1) q1 , . . . , qm , p ∈ Ax,y such that 1 = q1 f1 + · · · + qm fm + p(yg − 1),

(2.13)

(n+1)

as a polynomial identity in Ax,y . If one poses y = g −1 in (2.13) (this is the so called Rabinowithch’s trick), one gets 1 = q˜1 f1 + · · · + q˜m fm ,

(2.14)

where q˜1 , . . . , q˜m ∈ K[x1 , . . . , xn , g −1 ]. Taking any sufficiently large positive integer N and multiplying both members of (2.14) by g N yields ∗ fm , g N = q1∗ f1 + · · · + qm

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with qi∗ √ ∈ A(n) , for 1  i  m. This implies g N ∈ I so g ∈ Ia (Y ) ⊆ I.

√ I, i.e. 

Important consequences of the previous results are the following. Corollary 2.2.5. Let K be any algebraically closed field. Then: (i) for any subset Y ⊆ An , one has Ia (Y ) = Ia (Y ); (ii) the maps (2.11) and (2.12) induce bijections:   Radical ideals of A(n)  (2.15)  1−1  (n) . ←→ Maximal ideals of A 1−1

{(Zariski) closed subsets of An } ←→  {Points of An } Proof.

(i) By Proposition 2.1.25, for any subset Y ⊆ An , Y = Za (Ia (Y )). On the other hand, one has also Y = Za (Ia (Y )), i.e. Za (Ia (Y )) = Za (Ia (Y )). Since K is algebraically closed,  by the Hilbert “Nullstellensatz”-strong  form we have Ia (Y ) = Ia (Y ) which implies Ia (Y ) = Ia (Y ) since, by definitions, they are both radical ideals. (ii) The correspondence between points in An and maximal ideals is the content of Hilbert “Nullstellensatz”-weak form. The bijective correspondence between radical ideals of A(n) and (Zariski) closed subsets of An follows from the fact that, if I is a radical ideal, by the Hilbert “Nullstellensatz”-strong form one has Ia (Za (I)) = I whereas, if Y is (Zariski) closed then Za (Ia (Y )) = Y , as it follows from Proposition 2.1.25.  Remark 2.2.6. (i) In Chapter 4, we will see that Corollary 2.2.5-(i) more generally holds for any field K. This will be a direct consequence of Theorem 1.3.13 and the fact that Y is dense in its closure Y . (ii) The algebraic counter-part of the Hilbert “Nullstellensatz”-strong form is the following. (n) is any Claim 2.2.7. √ If K is any algebraically closed field and I ⊆ A ideal, then I = ∩m maximal, m = ∩P ∈Za (I) mP . m⊇I

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Proof. The second equality follows from the Hilbert “Nullstellensatz”weak form. ideal |m ⊇ I}. For the first equality, let MI := {m ⊂ A(n) maximal √ √ √ If m ∈ MI then I ⊆ m = m (cf. Lemma 1.1.2), so I ⊆ ∩MI m. To prove the other inclusion, observe that since K is algebraically closed any maximal ideal of A(n) is of the form mP , for P ∈ An , as the Hilbert “Nullstellensatz”-weak form states. Thus, f ∈ ∩MI m if√and only if f (P ) = 0 for any P ∈ Za (I), i.e. if and only if f ∈ Ia (Za (I)) = I, the last equality following from Hilbert “Nullstellensatz”-strong form.  Proposition 2.2.8. Let K be any algebraically closed field. Then (2.15) is a reversing-inclusion bijective correspondence for which the following further properties hold: (i) for any closed subsets Y1 , Y2 ∈ Can , one has Ia (Y1 ∪ Y2 ) = Ia (Y1 ) ∩ Ia (Y2 ). The same holds for any finite union of closed subsets of An ; (ii) for  any collection {Yα }α∈A of closed subsets, one has Ia (∩α∈A Yα ) = Σα∈A Ia (Yα ). Proof. Statement about reversing-inclusion directly follows from Proposition 2.1.11-(i) and formula (2.10).   (i) By definition Ia (Y1 ) ∩ Ia (Y2 ) := f ∈ A(n) | Y1 , Y2 ⊆ Za (f ) , which coincides with Ia (Y1 ∪ Y2 ). (ii) Let Yα := Za (Jα ), for Jα ⊆ A(n) radical ideal for any α ∈ A. By Proposition 2.1.11, Ia (∩α∈A Yα ) = Ia (Za (Σα∈A Jα )). Since K is algebraically closed, by the Hilbert “Nullstellensatz”-strong form, the √ latter equals Σα∈A Jα , as decided.  Remark 2.2.9. (a) Reversing-inclusion in Proposition 2.2.8 holds true for any field K, for any pair of ideals I ⊆ J in A(n) and for any pair of subsets Y1 ⊆ Y2 in An (cf. Proposition 2.1.11-(i) and formula (2.10)). (b) Proposition 2.2.8-(i) more generally holds for any field K and for any pairs Y1 , Y2of subsets of An , as the proof of Proposition 2.2.8 shows. (c) The use of Σα∈A Ia (Yα ) in part (ii) is unavoidable, even if Ia (Yα ) is a radical ideal, for any α ∈ A. For example, for any field K, I1 = (x21 −x2 ) and I2 = (x21 + x2 ) are both radical ideals in A(2) , since they are prime ideals; on the other hand I1 + I2 is not radical as x21 = 12 (x21 − x2 ) + 1 2 / I1 + I2 . 2 (x1 + x2 ) ∈ I1 + I2 but x1 ∈

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(d) From Example 2.1.19, when K is algebraically closed (2.15) implies that hypersurfaces in An are in 1 − 1 correspondence with principle radical ideals in A(n) (which are not maximal when n  2). Recalling the decomposition (2.7), the closed subsets Za (f1 ), . . . , Za (f ) are called the irreducible components of the hypersurface Za (f ) = Za (fred ). These are the hypersurfaces corresponding to the irreducible factors of fred in A(n) . Note that, since A(n) is a UFD, each such irreducible factor is also a prime element in A(n) therefore it generates a prime ideal in A(n) . The closed subset Za (fi ) ⊂ An is also called an irreducible hypersurface in An , for any 1  i  . Thus, (2.15) implies that irreducible hypersurfaces in An are in 1 − 1 correspondence with principal prime ideals in A(n) . 2.3

Some Consequences of Hilbert “Nullstellensatz” and of Elimination Theory

We will discuss some nice consequences of the “machinery” developed up to this point. 2.3.1

Study’s principle

In this section, we will focus on the case of K an algebraically closed field. Theorem 2.3.1 (Study’s principle). Let K be an algebraically closed field and let n  1 be an integer. Let f, g ∈ A(n) be non-constant polynomials, with f irreducible. If g(p1 , . . . , pn ) = 0, for any P = (p1 , . . . , pn ) ∈ Za (f ), then f divides g in A(n) . Proof. By the assumptions on f and g, one has g ∈ Ia (Za (f )). Since K is algebraically closed, from Theorem 2.2.3, Ia (Za ((f ))) = (f ). On the other hand, since f ∈ A(n) is irreducible and A(n) is a UFD, then f is a prime element. This implies that (f ) is a prime ideal and so a radical ideal  (cf. Lemma 1.1.2) therefore g ∈ (f ), i.e. f |g in A(n) . Remark 2.3.2. If K is not algebraically closed, Study’s principle does not hold. Take, e.g. f, g ∈ R[x] any two distinct, non-constant, irreducible polynomials; then Za (f ) = Za (g) = ∅ even if neither f |g or g|f . Similar considerations can be done, e.g. in the ring Z2 [x].

Algebraic Affine Sets

2.3.2

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Intersections of affine plane curves

Interesting application of Elimination Theory, is related to intersection of affine plane curves (i.e. hypersurfaces in A2 ) and classification of proper (Zarisky) closed subsets of A2 . In what follows, the term curve is always meant to be an affine (possibly reducible) plane curve. Theorem 2.3.3. Let K be any infinite field. The intersection of two curves, if not empty, consists of either finitely many points, or of a curve, or of a union of a curve and of finitely many points. Corollary 2.3.4. For any infinite field K, non-empty, proper closed subsets in Zar2a consist only of finitely many points, curves and unions of a curve and finitely many points. Proof of Theorem 2.3.3. Let Za (f ), Za (g) be two curves, with f, g ∈ A(2) non-constant polynomials. If f and g have a common, non-constant factor h ∈ A(2) , the curve Za (h) is contained in the intersection Za (f, g) of the two originary curves. Therefore, from now on we will assume that f and g have no common, non-constant factor. If both f and g are constant with respect to the same indeterminate, e.g. x1 , then Za (f, g) = ∅: if f and g have some powers of linear factors in their own factorizations, these factors represent lines which are parallel to the x1 axis and, by the assumptions on f and g, one concludes. If otherwise f is constant with respect to x1 whereas g is constant with respect to x2 , then Za (f, g) consists of at most finitely many points (Za (f, g) = ∅ if no powers of linear factors appear in the factorization of either f or g), and we are done also in this case. We can therefore assume that f and g are both non-constant with respect to the same indeterminate, e.g. x2 . By the assumption on f and g, the resultant R := Rx2 (f, g) ∈ K[x1 ] is non-zero (cf. Theorem 1.3.16 and Section 1.3.4). Thus, if (p1 , p2 ) ∈ Za (f, g), one must have R(p1 ) = 0 as it follows from Theorem 1.3.16 applied to f  (x2 ) := f (p1 , x2 ), g  (x2 ) = g(p1 , x2 ) ∈ K[x2 ], which must have (x2 − p2 ) ∈ K[x2 ] as a common, nonconstant factor. Since R ∈ K[x1 ] has at most finitely many roots in K, one can have at most finitely many choices for p1 ∈ K. If we make same considerations with respect to the other indeterminate, we arrive at the same conclusion for p2 ∈ K. Thus, Za (f, g) consists of at most finitely many points. 

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A direct consequence of the previous results is the following. Theorem 2.3.5 (Bezout’s theorem (weak form)). Let Z1 = Za (f ) and Z2 = Za (g) be curves in A2 , with f, g ∈ A(2) \ K. If φ := g.c.d.(f, g) in A(2) then Z1 ∩ Z2 = Za (φ) ∪ Z3 , where Z3 is either empty or a finite set of points.

Exercises Exercise 2.1. Let f = x21 + x22 + 1 ∈ C[x1 , x2 ]. Consider Za (f ) ⊂ A2C and identify A2C with the Euclidean topological space R4 , via the identifications x1 = a + ib, x2 = c + id,

a, b, c, d ∈ R, i2 = −1.

Endowing A2C with the Euclidean topology of R4 , show that Za (f ) is homeomorphic to a two-sphere in R3 minus two points. Exercise 2.2. Let K be any infinite field. Let Y := {(t, t2 ) ∈ A2 | t ∈ K}. Show that Y is an AAS. Find explicit generators of Ia (Y ) and show that Ia (Y ) is a prime ideal. Exercise 2.3. In A(3) := R[x1 , x2 , x3 ] consider the ideal I := (x2 −x21 , x3 − x1 x2 ). (3)

(i) Prove that the quotient ring AI is an integral domain. (ii) Show that x1 x3 − x22 ∈ I. (iii) Let J := (x2 − x21 , x1 x3 − x22 ). Show that Za (I)  Za (J) and that is not an integral domain.

A(3) J

Exercise 2.4. Let K be an algebraically closed field. Let Yk ⊂ A3 be the union of the two lines Y  := Za (x2 , x3 ) and Yk = Za (x3 − k, x1 ), where k ∈ K \ {0} a non-zero element. (i) Determine the ideal Ik := Ia (Yk ) ⊂ A(4) , for any k ∈ K \ {0}. (ii) Consider k ∈ K as a movable parameter, so that one has a family of AAS’s Yk , consisting of “pairs” of affine lines, and a family of ideals Ik = Ia (Yk ). Show that the family of AAS’s Yk , k ∈ K, describes pairs of skew lines in A3 approaching each other when k approaches to 0. Show that, for any k = 0, the ideal Ik is radical but not prime.

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(iii) Set I0 the ideal where one poses k = 0 among the generators of the ideal Ik as in (i). Show that I0 is neither prime nor radical. Find an √ element in I0 \ I0 . Exercise 2.5. In A(2) := R[x1 , x2 ] consider the two polynomials f = (x1 + 1)2 + x22 − 1 and g = (x1 − 1)2 + x22 − 1 and let J := (f, g) be the ideal they generate. (i) Determine Za (J) ⊆ A2R . (2) (ii) Prove that the quotient ring AJ is not a reduced ring. √ (iii) Deduce that J cannot be a radical ideal and determine J .

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Chapter 3

Algebraic Projective Sets

From now on, unless otherwise stated, K will be always considered once and for all to be an algebraically closed field. Let V be a (n + 1)-dimensional K-vector space. One can define the following relation on V \ {0}: v ∼ w ⇐⇒ ∃ t ∈ K∗ := K \ {0} s.t. w = tv. It is easy to see that this is an equivalence relation, which is called proportionality. We will denote by [v] the equivalence class of v ∈ V \ {0}. The quotient set P(V ) :=

(V \ {0}) ∼

is called the projective space associated to V . We will denote by πV (or simply π if no confusion arises) the canonical projection π : V \ {0}  P(V ).

(3.1)

The integer n will be called the dimension of the projective space, which is denoted by dim(P(V )) (the empty-set is considered as the projective space of dimension −1, which is associated to the vector space V = {0}). Elements of P(V ) will be called points. When in particular V = Kn+1 is the (n + 1)-dimensional numerical (standard) K-vector space, the associated projective space will be simply denoted by Pn and called the n-dimensional numerical projective space over K. If v = (v0 , . . . , vn ) ∈ Kn+1 is a non-zero numerical vector, its proportionality equivalence class [v] will be also denoted by [v0 , . . . , vn ] 79

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A First Course in Algebraic Geometry and Algebraic Varieties

whereas the (n + 1)-tuple (v0 , . . . , vn ) is called a vector of homogeneous coordinates for [v]. Thus, homogeneous coordinates of a point P ∈ Pn are such that: (a) not all of them equal zero, and (b) they are defined up to proportionality by a factor t ∈ K∗ . For any 0  i  n, we will denote by Pi ∈ Pn the point whose homogeneous coordinates are proportional to the components of the vector ei of the canonical basis of Kn+1 . The points Pi are called the vertices of the fundamental (n+1)-hedron (or pyramid) of Pn . The point Pn+1 = [1, . . . , 1], is called the unit point of Pn whereas the points Pi , with 0  i  n, are called the fundamental points of Pn . 3.1

Algebraic Projective Sets

Consider the projective space P(V ) of dimension n and fix a homogeneous element F ∈ S(V ∗ )d (recall Example 1.10.20 and Section 1.10.1). Remark 3.1.1. Once we fix a (ordered) basis on V , we have a natural identification of Kn+1 with V . This induces a projectivity ϕ : Pn → P(V ), which introduces homogeneous coordinates on P(V ). Recalling notation (1.27), for any d  0, in this case S(V ∗ ) naturally identifies with S(n) by means of the isomorphism ϕ  : S(n) −→ S(V ∗ ) induced by the choice of −1 (F ) is a homogeneous the (ordered) basis in V . If F ∈ S(V ∗ )d , then ϕ polynomial of degree d. Therefore, F vanishes at P ∈ P(V ) (where P the point corresponding to the equivalence class [P ] = [p0 , . . . , pn ] of the vector P ∈ V and the homogeneous coordinates [p0 , . . . , pn ] are w.r.t. the given basis of V ) means that ϕ −1 (F )(p0 , . . . , pn ) = 0 which holds if and only −1 if ϕ  (F )(tp0 , . . . , tpn ) = 0, for any t ∈ K∗ (recall (1.29)). One says that ϕ −1 (F )(X) = 0 is a homogeneous equation of Z(F ) in Pn , where X := (X0 , . . . , Xn ) indeterminates (cf. also Section 3.3.8). If one introduces two different sets of homogeneous coordinates on P(V ), ϕ : Pn → P(V ),

ψ : Pn → P(V ),

S(V ∗ ) is identified to S(n) in two different ways. In other words, there are two isomorphisms ϕ  : S(n) → S(V ∗ ),

ψ : S(n) → S(V ∗ )

determined by the two distinct bases of V . If X = (X0 , . . . , Xn ) and Y := (Y0 , . . . , Yn ) denote the row-vectors of induced indeterminates, respectively,

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and if A denotes the matrix associated to the base change in V , the map ωA := ψ−1 ◦ ϕ  : S(n) → S(n) is an isomorphism of graded rings, which sends ϕ −1 (F )(X) to the polynomial  −1     (F )(X) := ψ−1 (F ) Y At , ωA ϕ where X t = A Y t . This isomorphism depends on the matrix A, so it is determined up to a non-zero proportionality factor. Any such isomorphism of S(n) is said to be a homogeneous linear substitution of the indeterminates. In particular the homogeneous equation ψ−1 (F )(Y ) = 0 can be deduced from ϕ −1 (F )(X) = 0 by means of a homogeneous linear substitution in the indeterminates, which preserves the degrees. From (1.29) and what discussed above, differently from the affine case, an element F ∈ S(V ∗ )d does not determine a function on P(V ). On the other hand, it makes sense to ask for F ∈ S(V ∗ )d to vanish at P : if P = [p] ∈ P(V ), one has F (p) = 0 if and only if F (t p) = 0, for any t ∈ K \ {0}. In such a case, the point P is called a zero of the polynomial F . Given F ∈ S(V ∗ )d , therefore it makes sense to consider Zp (F ) ⊆ P(V ), which is called the zero-set of F in P(V ). Remark 3.1.2. (i) Given any identification of P(V ) with Pn as above, any polynomial f ∈ S(n) of degree d can be uniquely decomposed into its homogeneous (n) components f = F0 + · · · + Fd , where Fi ∈ Si , 0  i  d. Since K has infinitely many elements (Corollary 1.3.12), by Theorem 1.3.13 the polynomial f vanishes at P ∈ Pn (in the above sense) if and only if all of its homogeneous components Fi do, 0  i  d. (ii) When K is finite, what stated in (i) does not hold in general. Take, e.g. K = Zp , x := x0 and f = xp − x ∈ Zp [x]: f vanishes at all a ∈ Zp (cf. Example 2.1.10) but its homogeneous components vanish only at 0 ∈ Zp . Recalling notation (1.22), from Remark 3.1.2-(i) one poses: Definition 3.1.3. For any subset T ⊆ S(n) the subset of Pn , Zp (T ) := ∩F ∈H(T ) Zp (F ), is called the zero-set of T in Pn .

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The subscript p in the above definition stands for the term projective, to make distinction with the affine case in Definition 2.1.1. Note that Zp (T ) = Zp ((H(T ))),

(3.2)

where (H(T )) ⊆ S(n) is a homogeneous ideal (cf. Proposition 1.10.7). Since S(n) is Noetherian, there exist F1 , . . . , Fm ∈ H(T ) s.t. (H(T )) = (F1 , . . . , Fm ). Thus, as in Definition 2.1.4, the set {F1 , . . . , Fm } is called a system of homogeneous equations defining Zp (T ). In particular, one has Definition 3.1.4. A subset Z ⊆ Pn is called an Algebraic Projective Set (APS), if there exist polynomials F1 , . . . , Fm ∈ H(S(n) ) s.t. Z = Zp (F1 , . . . , Fm ); equivalently, Z = Zp (I) for a homogeneous ideal I ⊆ S(n) . More generally, Z ⊆ P(V ) is a APS if there exists a subset T of S(V ∗ ) such that Z = Zp (T ). Recalling Propositions 1.10.8 and 2.1.11, one has Proposition 3.1.5. (i) Let T1 ⊆ T2 be any subsets of S(n) ; then one has Zp (T2 ) ⊇ Zp (T1 ). In particular for any pair of homogeneous ideals I1 ⊆ I2 in S(n) , then Zp (I1 ) ⊇ Zp (I2 ). (ii) For any homogeneous ideals I1 , I2 ⊆ S(n) , one has Zp (I1 ∩ I2 ) = Zp (I1 ) ∪ Zp (I2 ). (iii) For any family {Iα }α∈A of homogeneous ideals in S(n) , one has Zp (Σα∈A Iα ) = ∩α∈A Zp (Iα ). Proof. One applies verbatim strategies as in the proofs of Proposition 2.1.11-(i), (ii) and (iii).  From the previous proposition, Cnp := {APS’s in Pn }

(3.3)

is the set of closed subsets of a topology on Pn . This topology will be called the Zariski topology on Pn , which will be denoted by ZarPn (or simply Zarnp ). Proposition 3.1.5 can be stated in more generality for APS’s in P(V ), using S(V ∗ ) instead of S(n) . Therefore, CpV will denote the set of APS’s of P(V ), which is the set of closed subsets of the Zariski topology on P(V ).

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This topology will be denoted by ZarP(V ) . Open sets in the Zariski topology will be complementary sets of APS’s. In particular, for any F ∈ S(V ∗ )d , Up (F ) = Zp (F )c

(3.4)

is called a principal open set of P(V ). As in Lemma 2.1.21, principal open sets form a basis for ZarP(V ) . Any non-empty subset Y ⊆ P(V ) will be endowed, from now on, with the topology induced on Y by ZarP(V ) . This will be called the Zariski topology of Y and will be denoted by ZarP(V ),Y (respectively, Zarnp,Y or simply ZarY , when the inclusion Y ⊆ Pn is clearly understood). From Remark 3.1.1, in what follows we will focus for simplicity on the case of Pn ; statements can be easily adapted to P(V ), replacing S(n) with S(V ∗ ). From Remark 3.1.2-(i) and the Noetherianity of S(n) , one has: Definition 3.1.6. For any subset Y ⊆ Pn , Ip (Y ) := {f ∈ S(n) | f (P ) = 0, ∀ P ∈ Y } is a homogeneous, finitely generated ideal, which is called the homogeneous ideal of Y in S(n) . Same strategies as in the affine case give the following. Proposition 3.1.7. (i) For any subsets Y1 ⊆ Y2 ⊆ Pn , Ip (Y1 ) ⊇ Ip (Y2 ); (ii) For any subsets Y1 , Y2 ⊆ Pn , Ip (Y1 ∪ Y2 ) = Ip (Y1 ) ∩ Ip (Y2 ). 3.2

Homogeneous “Hilbert Nullstellensatz”

From the previous section, as in the affine case one has natural reversing– inclusion maps  Ip (−)  {Subsets of Pn } −→ Homogeneous ideals of S(n) Y → Ip (Y )

(3.5)

 Zp (−)  {Subsets of Pn } ←− Homogeneous ideals of S(n) Zp (I) ← I.

(3.6)

and

To better understand their behaviour on (Zariski) closed subsets and to prove the homogeneous version of Hilbert “Nullstellensatz”, we need to first introduce some useful definitions. Use notation as in (3.1).

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A First Course in Algebraic Geometry and Algebraic Varieties

Definition 3.2.1. For any subset Y ⊆ Pn , let Ca (Y ) := π −1 (Y ) ∪ {0}   = (p0 , . . . , pn ) ∈ Kn+1 \ {0} | [p0 , . . . , pn ] ∈ Y ∪ {0}. This subset of An+1 is called the affine cone over Y . Remark 3.2.2. (i) It is clear from the definition that Ca (Y ) = Ca (Y  ) ⇔ Y = Y  .

(3.7)

Moreover, for any subset Y ⊆ Pn , by definition of affine cone one has Ia (Ca (Y )) = Ip (Y ).

(3.8) a

(ii) By Proposition 2.1.25 and by (3.8), one has Ca (Y ) = Za (Ia (Ca (Y ))) = a Za (Ip (Y )), where Ca (Y ) denotes the (affine) closure of Ca (Y ) in An+1 . Since Ip (Y ) is a homogeneous ideal, then p

Za (Ip (Y )) = Ca (Zp (Ip (Y )) = Ca (Y ),

(3.9)

p

where Y denotes the (projective) closure of Y ⊆ Pn . In particular one a p has Ca (Y ) = Ca (Y ). From this equality and from (3.7), it follows that Ca (Y ) ∈ Can+1 if and only if Y ∈ Cpn and that the map π in (3.1) is continuous. √ Corollary 3.2.3. For any homogeneous ideal I ⊆ S(n) , Zp (I) = Zp ( I). Proof. It follows from Remark 3.2.2 and from Proposition 2.1.11-(v).  Differently from the affine case (cf. (2.15)), even if K is algebraically closed and even if one restricts the maps (3.5) and (3.6) to the sets of homogeneous radical ideals and of (Zariski) closed subsets of Pn , one still does not have bijective correspondences. Indeed, one always has Zp ((1)) = ∅; on the other hand the following result, which can be considered as the homogeneous analogue of the Hilbert “Nullstellensatz”-weak form Theorem 2.2.1, shows that this is not the only case, even on radical ideals. Theorem 3.2.4 (Homogeneous Hilbert “Nullstellensatz” weak form). Let I ⊂ S(n) be a proper, homogeneous ideal. Then √ Zp (I) = ∅ ⇔ I = S+ , (3.10) where S+ is the irrelevant (cf. Definition 1.10.4).

maximal (so radical)

ideal

of S(n)

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Proof. First √ note that, by the maximality of S+ and by Lemma 1.1.2, condition I = S+ is equivalent to the fact that there exists an integer t  1 s.t. St+ ⊆ I. Thus, (3.10) reduces to showing: Zp (I) = ∅ ⇔ ∃ t > 0 s.t. St + ⊆ I. (⇐) Suppose St+ ⊆ t t t (X0 , X1 , . . . , Xn ) ⊆ St+ ∅ ⊇ Zp (St+ ) ⊇ Zp (I).

(3.11)

I, for some positive integer t. Since ⊆ I, from Proposition 3.1.5-(i), we have

(⇒) Assume Zp (I) = ∅, with I = (F1 , . . . , Fm ), any Fi a homogeneous polynomial. One can assume that not all polynomials are linear forms, otherwise the assumption Zp (I) = ∅ would give I = S+ and we are done. Using notation as in Definition 1.10.16, consider fi := δ0 (Fi ), 1  i  m. Thus Zp (F1 , . . . , Fm ) = ∅ implies Za (f1 , . . . , fm ) = ∅. Since K is algebraically closed, by Theorem 2.2.1, one has therefore (f1 , . . . , fm ) = (1) ⊂ A(n) so there exists polynomials g1 , . . . , gm ∈ A(n) such that 1=

m 

fi gi .

(3.12)

i=1

Recall that h0 (fi )|Fi , for any 1  i  m (more precisely the two polynomials coincide if and only if X0 does not divide Fi ). Hence, there exists a nonnegative integer si s.t. X0si h0 (fi ) = Fi , for any 1  i  m. This implies there exists a positive integer N0 such that (3.12) becomes X0N0 =

m 

Fi Gi ∈ I,

(3.13)

i=1

where Gi = h0 (gi ). Reasoning in the same way for all the other indexes j = 1, . . . , n, one deduces that there also exist positive integers N1 , . . . , Nn such that also X1N1 , X2N2 , . . . , XnNn ∈ I. Let N := Max{N0 , N1 , . . . , Nn }. By the assumptions on the degrees of the Fi ’s, one has N  2. Take t := (n + 1)(N −1)+1 > n+1. Thus, any monomial of the form X0a0 X1a1 · · · Xnan , with Σni=0 ai  t, is such that there exists at least one aj for which aj  N  Nj , for some j ∈ {0, . . . , n}, i.e. X0a0 X1a1 · · · Xnan ∈ I. Since St+ is generated by  monomials X0a0 X1a1 · · · Xnan , where Σni=0 ai = t, one can conclude. To sum-up the ideals (1) and S+ both map to the empty-set via Zp (−)

{Homogeneous ideals of S(n) } −→ Cpn ,

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whereas S+ is not in the image of the map Ip (−)

Cpn −→ {Homogeneous ideals of S(n) }, since Ip (∅) = (1). On the other hand, one has Theorem 3.2.5 (Homogeneous Hilbert “Nullstellensatz”-strong form). Let √ I ⊂ S(n) be a homogeneous ideal such that Zp (I) = ∅. Then Ip (Zp (I)) = I. Proof. From Remark 3.2.2, we have Ip (Zp (I)) = Ia (Ca (Zp (I))) = Ia (Za (I)). √ Since K is algebraically closed, from Theorem 2.2.3, the latter  equals I, as desired. As for Corollary 2.2.5-(ii), one has therefore Corollary 3.2.6. The maps (3.5) and (3.6) induce bijections: 1−1

Cpn ←→ {Radical ideals of S(n) } \ {S+ }. 3.3 3.3.1

Fundamental Examples and Remarks Points

When K is algebraically closed, (2.15) gave bijective correspondence between points in An and maximal ideals mP ⊂ A(n) ; the situation is different for points in Pn . As in the affine case, points are still closed sets in Zarnp ; indeed, if P = [p0 . . . . , pn ] ∈ Pn with e.g. pi = 0, then {P } = Zp (pi X0 − p0 Xi , . . . , pi Xn − pn Xi ). On the other hand, the homogeneous ideal I := (pi X0 − p0 Xi , . . . , pi Xn − pn Xi ) ⊂ S(n) is prime (n) but not maximal, as S I ∼ = K[Xi ] is an integral domain but not a field. Using Theorem 3.2.5 and Example 2.1.18, one can easily show the following more precise result. Proposition 3.3.1. Let I ⊂ S(n) be a radical ideal. Then Zp (I) = {P }, for some point P ∈ Pn , if and only if I is generated by linearly independent (n) linear forms L0 , . . . , Ln−1 ∈ S1 . In particular, Zarnp is T1 and Cnp contains all finite subsets of Pn . Moreover, from Proposition 1.10.19, when n = 1 these (together with the empty-set) are the only proper closed subsets of P1 . As in the affine case, Zar1p is not T2 . This will more generally hold for any Zarnp , n  1 (cf. Remark 4.1.1).

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3.3.2

Coordinate linear subspaces

Let 0  m  n be integers. For any sequence of integers 0 < i0 < i1 < · · · < im  n, one has an injective map ϕ := ϕi1 ,i2 ,i3 ,··· ,im : Pm −→ Pn defined as follows: ϕ

[p0 , p1 , . . . , pm ] −→ [0, . . . , 0, p0 , 0, . . . , 0, p2 , 0, . . . , 0, pm , 0, . . . , 0). i0 i1 im As in Example 2.1.17, ϕ is a homeomorphism between (Pm , Zarm p ) and Im(ϕ) ⊂ Pn endowed with the induced topology by Zarnp . This image is the closed subset Zp (Xj | ∀j = i0 , . . . , im ), which is called a m-dimensional coordinate linear subspace of Pn . For m = 0, we get the points Pi which are the vertices of the fundamental pyramid of Pn ; for m = 1 we get coordinate axes of Pn . For m = n − 1, we get fundamental hyperplanes of Pn . The hyperplane given by Zp (Xj ) will be also denoted by Hj , 0  j  n. 3.3.3

Hyperplanes and the dual projective space

Let V be a (n + 1)-dimensional K-vector space. If we consider V ∗ := Hom(V, K), then P(V ∗ ) is called the dual projective space of P(V ). Recalling Example 1.10.20 and Remark 3.1.1, the choice of an (ordered) basis (e0 , . . . , en ) for V allows one to identify P(V ) with Pn and V ∗ with (n) S1 = K[X0 , . . . , Xn ]1 , where (X0 , . . . , Xn ) the dual basis of (e0 , . . . , en ). (n) For simplicity, P(S1 ) is denoted by (Pn )∗ . Since any hyperplane of Pn is of the form Ha := Zp (a0 X0 + . . . + an Xn ), for some a ∈ Kn+1 \ {0} and since Ha = Hta , for any t ∈ K∗ , it is clear that (Pn )∗ is identified with the set of hyperplanes in Pn , the correspondence given by [a] ↔ Ha . In this correspondence, the fundamental hyperplane Hi ⊂ Pn corresponds to the vertex Pi∗ of the fundamental pyramid of (Pn )∗ , 0  i  n. 3.3.4

Fundamental affine open sets (or affine charts) of Pn

Consider the principal open set Up (Xi ) = Hic = Pn \ Hi , 0  i  n. We denote it by Ui := {[p0 , . . . , pn ] ∈ Pn | pi = 0}.

(3.14)

Since Pn = ∪ni=0 Ui , then {U0 , . . . , Un } is a (finite) open covering of Pn , where each open set is principal (cf. (3.4)). Similarly, Pn can be also written

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as a disjoint union Pn = Ui ∪Hi , 0  i  n, where each Hi is homeomorphic to Pn−1 . For any i ∈ {0, . . . , n}, consider now the map  p0 pi−1 pi+1 pn φi P = [p0 , . . . , pn ] ∈ Ui −→ ,..., , ,..., ∈ An , (3.15) pi pi pi pi which is well-defined. Proposition 3.3.2. For homeomorphism.

any

i ∈ {0, . . . , n},

the

map

φi

is

a

Proof. We refer to the case i = 0, since the case i > 0 can be proved similarly. To ease notation, set φ := φ0 and U := U0 . It is clear that φ is bijective, whose inverse is given by φ−1 (c1 , . . . , cn ) = [1, c1 , . . . , cn ]. It suffices to prove that φ and φ−1 are closed maps. For the map φ, note that for any closed subset Y ⊆ Pn , Y ∩U is closed in U . Therefore, to show that φ is closed, it suffices to prove that φ(Zp (F )∩U ) is closed in An , for any F ∈ H(S(n) ); this is obvious since, by definition of φ, φ(Zp (F ) ∩ U ) = Za (δ0 (F )). Similarly, to show that φ−1 is closed it suffices to showing that, for any non-constant polynomial g ∈ A(n) , the set φ−1 (Za (g)) is closed in U . By the definition of φ one observes that φ−1 (Za (g)) = Zp (ho (g)) ∩ U which is  closed in U , so φ−1 is closed. Remark 3.3.3. (i) Since any Ui is homeomorphic to An , the principal open sets Ui are also called principal open affine sets, or even affine charts, of Pn . Thus, Pn has a finite open covering by affine charts. (ii) The projective space Pn can be therefore viewed as an extension of An , by identifying An with any of the affine charts Ui , i.e. An → Pn (c1 , c2 , . . . , cn ) −→ [c1 , c2 , . . . ci , 1, ci+1 , . . . cn ] (in the sequel, for simplicity, we will usually identify An with the affine chart U0 ). In this identification, the fundamental hyperplane Hi ⊂ Pn will be called the hyperplane at infinity of Ui and each point P ∈ Hi is called a point at infinity (or improper point) of Ui , 0  i  n. (iii) More generally, for any subset Y ⊂ Pn , Yi := Y ∩ Ui is an open subset of Y , 0  i  n, and {Y0 , . . . , Yn } is an open covering of Y . If in particular Y is moreover closed in Pn , say Y = Zp (I) for some

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homogeneous ideal I, then Yi is closed in Ui , 0  i  n. Indeed, by Noetherianity I = (F1 , . . . , Fm ), for some Fj ∈ H(S(n) ). From the proof of Proposition 3.3.2, one has Yi = Za (δi (F1 ), . . . , δi (Fm )).

(3.16)

In particular any Yi is a fundamental affine open set of Y (since it is open in Y and homeomorphic to an affine closed set) and contemporarly it is locally closed in Pn , i.e. it is the intersection of a closed set and of an open set of Pn . 3.3.5

Projective closure of affine sets

As already mentioned above, we will usually identify An with U0 via the map φ0 . In this case, for any subset Y ⊂ An , its closure in Pn , denoted by p Y will be called the projective closure of Y . Moreover, one poses p

Y∞ := Y ∩ H0

(3.17)

and call it the set of points at infinity (or improper points) of Y ⊂ An . Suppose that Y ∈ Can ; let therefore I = (f1 , . . . , fm ) ⊂ A(n) be such that Y = Za (I). Differently from (3.16), in general the m polynomials obtained p by homogenizing the generators of I are not sufficient to determine Y , as the following easy example shows. Example 3.3.4. Identify A2 with the affine chart U0 ⊂ P2 and consider Y = Za (I) ⊂ A2 , where I = (x1 , x2 + x21 ). One has h0 (x1 ) = X1 , h0 (x2 + x21 ) = X0 X2 + X12 and let I = (h0 (x1 ), h0 (x2 + x21 )) = (X1 , X0 X2 + X12 ), which is a homogeneous ideal. Note that Y is the origin O ∈ A2 , indeed I = mO , as x2 = (x2 + x21 ) − x1 (x1 ) ∈ I. The point O is also closed in P2 (corresponding to the fundamental vertex P0 = [1, 0, 0] of P2 ). p Thus, O = O = P0 whereas Zp (I) = {[0, 0, 1], [1, 0, 0]}. In particular, p Za (I)  Zp (I). Proposition 3.3.5. Let Y = Za (J) ⊆ An be a closed subset, for some ideal J ⊂ A(n) . Identify An with the affine chart U0 ⊂ Pn and define the homogeneous ideal   J∗ := h0 (f ), ∀ f ∈ J ⊆ S(n) . p

Then Y = Zp (J∗ ).

(3.18)

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Proof. By definition of J∗ , Zp (J∗ ) is a closed set of Pn containing Y . Then p Y ⊆ Zp (J∗ ). To prove the opposite inclusion, let W ⊆ Pn be any closed subset containing Y . For any F ∈ H(Ip (W )), the polynomial δ0 (F ) vanishes along Y , i.e. Y ⊆ Za (δ0 (F √ )). By Theorem 2.2.3, since K is algebraically closed, one has δ0 (F ) ∈ J, i.e. there exists an integer r  1 such that δ0 (F )r ∈ J. Assume deg(F ) = d and let deg(δ0 (F )) := d − s, for some integer 0  s < d (recall that s = 0 if and only if X0 does not divide F ). Then F = X0s h0 (δ0 (F )) and so F r = X0sr h0 (δ0 (F ))r . From Lemma 1.10.17-(ii), h0 is multiplicative so h0 (δ0 (F ))r = h0 (δ0 (F )r ). By definition of √ J∗ , this implies that F r ∈ J∗ . Thus, Ip (W ) ⊆ J∗ ; from Proposition 3.1.5√ (i) and Corollary 3.2.3, it follows that W = Zp (Ip (W )) ⊇ Zp ( J∗ ) = Zp (J∗ ). In other words, any projective closed subset containing Y contains p also Zp (J∗ ). Since this must hold also for Y , one has proved the other inclusion.  Note the following nice consequences of the previous result. Remark 3.3.6. (i) With same notation and assumptions as in Proposition 3.3.5, take J := (f1 , . . . , fm ) ⊆ A(n) and let Fi := h0 (fi ) ∈ H(S(n) ), 1  i  m. Then, p even if in general Y ⊆ Zp (F1 , . . . , Fm ), one has p

Y = Y ∩ U0 = Zp (F1 , . . . , Fm ) ∩ U0 ,

(3.19)

as it follows from the proof of Proposition 3.3.2 (therein we showed that φ−1 0 (Y ) = Zp (F1 , . . . , Fm ) ∩ U0 ). (ii) If Y ⊆ An is closed, then p

Y = Y ∪ Y∞ .

(3.20)

Proof. By the induced topology Zarnp,An , the closure of Y in U0 = An is p given by Y ∩ U0 . On the other hand, since Y is already closed in U0 = An , p  one has Y = Y ∩ U0 . Note that the previous proof is another way to see that any closed subset of An is locally closed in Pn (cf. Remark 3.3.3-(iii)) (iii) For any subset Y ⊆ Pn , one has p

Ip (Y ) = Ip (Y ).

(3.21)

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Proof. Let indeed F ∈ H(S(n) ) such that Y ⊆ Zp (F ); then Y ⊆ Zp (F ) p p p one has also Y = Zp (Ip (Y )). By so Y = Zp (Ip (Y )). On the other hand, 

p p Theorem 3.2.5, one has Ip (Y ) = Ip (Y ) and so Ip (Y ) = Ip (Y ), since by definition both ideals are radical ideals.  A different proof follows from Remark 3.2.2 and Corollary 2.2.5-(i); a p p indeed one has Ip (Y ) = Ia (Ca (Y )) = Ia (Ca (Y ) ) = Ia (Ca (Y )) = Ip (Y ). (iv) If Y ⊂ U0 is a finite set of points, then clearly Y∞ = ∅. When K is not algebraically closed, it may happen that Y∞ = ∅ even if Y is not reduced to a finite sets of points. Consider, e.g. the real ellipse C = Za (x21 +x22 −1) ⊂ A2R . Its projective closure is the non-degenerate conic Zp (X12 +X22 −X02 ) ⊂ P2R so that C∞ = Zp (X0 , X12 +X22 ) = ∅. The same conic, considered instead in A2C , is such that C∞ = {[0, 1, i], [0, 1, −i]}, with i2 = −1 (cf. Corollary 6.2.6 for more general results). p (v) Identifying An with U0 , let An denote the projective closure of An in p Pn . Then An = Pn , i.e. An is dense in Pn . (n)

Proof. For any integer d  1, let F ∈ Sd be any homogeneous polynomial such that An ⊆ Zp (F ). Up to a permutation of the indeterminates, we can assume that F contains a monomial proportional to X0d . Therefore, we (n−1) = can write F as F = X0d F0 + X0d−1 F1 + . . . + Fd , where Fi ∈ Si K[X1 , . . . , Xn−1 ]i , 0  i  d. We get δ0 (F ) = F0 + F1 + · · · Fd ∈ A(n) ,

(3.22)

which is the decomposition of δ0 (F ) in its homogeneous components. By the assumptions on F , one has An = U0 ⊆ Za (δ0 (F )). Since K is infinite (because algebraically closed), by Theorem 1.3.13, we get that δ0 (F ) = 0. Thus, Fi = 0, for any 0  i  d, and so also F = 0. Therefore Zp (F ) =  Zp ((0)) = Pn , as desired. 3.3.6

Projective subspaces and their ideals

Let V be a (n + 1)-dimensional K-vector space and let W ⊂ V be a subvector space of dimension m + 1 > 0, with m < n. Then P(W ) is called (projective) subspace of P(V ) of dimension m and codimension c := n − m. The empty-set is the only subspace of dimension −1; points in P(V ) are subspaces of dimension 0; codimension-1 subspaces are hyperplanes (cf. Section 3.3.3).

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The following statements are easy consequences of standard Linear Algebra: (i) Z ⊆ P(V ) is a subspace if and only if Z = Zp (L0 , . . . , Lh ), where L0 , . . . , Lh ∈ V ∗ = S(V ∗ )1 ; (ii) If L, L0 , . . . , Lh ∈ V ∗ , then Zp (L0 , . . . , Lh ) ⊆ Zp (L) if and only if L is linearly dependent from L0 , . . . , Lh in V ∗ ; (iii) Zp (L0 , . . . , Lh ) = Zp (G0 , . . . , Gk ) if and only if L0 , . . . , Lh and G0 , . . . , Gk span the same sub-vector space of V ∗ ; (iv) Zp (L0 , . . . , Lh ) = Zp (Fi1 , . . . , Fic ), where Fi1 , . . . , Fic is a basis of W ∗ := Span{L0 , . . . , Lh } ⊆ V ∗ and where c is the codimension of W ∗ in V ∗ ; (v) if Λ1 , Λ2 ⊆ P(V ) are subspaces s.t. Λi := P(Wi ) for i = 1, 2, then Λ1 ∩ Λ2 ⊂ P(V ) is a subspace, which is called the intersection subspace of the two subspaces; more precisely Λ1 ∩ Λ2 = P(W1 ∩ W2 ). (vi) Let S ⊂ P(V ) be any subset. Since the family LS of projective subspaces containing S is not empty, one can consider S := ∩Λ∈LS Λ. By (v), this is a subspace of P(V ); more precisely, it is the smallest subspace of P(V ) containing S. This is called the subspace generated by S (or even the linear envelope of S) in P(V ). S is said to be non-degenerate in P(V ) if S = P(V ), i.e. if S is not contained in any proper projective subspace. If Λ1 , . . . , Λh are subspaces, one uses the symbol Λ1 ∨· · ·∨Λh instead of Λ1 ∪· · ·∪Λh . (vii) (projective) Grassmann formula: for any subspaces Λ1 , Λ2 in P(V ) one has dim(Λ1 ) + dim(Λ2 ) = dim(Λ1 ∨ Λ2 ) + dim(Λ1 ∩ Λ2 ).

(3.23)

The proof is a straightforward application of linear Grassmann formula for subspaces in V . (viii) Let dim(V ) = n + 1. If Λ1 , Λ2 are projective subspaces in P(V ) such that dim(Λ1 ) + dim(Λ2 )  n then Λ1 ∩ Λ2 = ∅. The previous result is a direct consequence of (vii) and it is a natural generalization of the well-known fact that two lines in the projective plane always intersect. It is moreover clear that, by the existence of parallelism, no Grassmann formula can exist for affine subspaces in An defined by non-homogeneous linear systems (cf. Example 2.1.18). (ix) Given (m + 1)-points P0 , P1 , . . . , Pm ∈ P(V ), by Grassmann formula, one has dim(P0 ∨ P1 ∨ · · · Pm )  m. The points are said to be linearly independent (or even in general linear position) if equality holds.

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It is then clear that, for any m-dimensional subspace Λ in P(V ), with m  n, there exist (m + 1) points P0 , P1 , . . . , Pm ∈ Λ which are linearly independent, i.e. such that Λ = P0 ∨ P1 ∨ · · · Pm . Fixing an (ordered) basis of V induces a choice of homogeneous coordinates on P(V ), which identifies P(V ) with Pn and Λ = P0 ∨P1 ∨· · · Pm with an APS of the form Zp (L0 , . . . Lc−1 ) ⊂ Pn , where c is the codimension (n) of Λ in Pn and where L0 , . . . , Lc−1 ∈ S1 . It is easy to see that the natural map φ

Pm −→ Λ ⊂ Pn ,

φ

defined by [λ0 , . . . , λm ] −→

m 

λi Pi , (3.24)

i=0 n is a homeomorphism between (Pm , Zarm p ) and (Λ, Zarp,Λ ) (this extends what discuss in Section 3.3.2). The map φ is called a parametric representation of Λ. (x) If Λ ⊆ P(V ) is a subspace, one defines

Λ⊥ := {[L] ∈ P(V )∗ | Λ ⊆ Zp (L)}. This is a subspace of P(V )∗ , which is called the orthogonal to Λ. One has that dim(Λ⊥ ) = codim(Λ) = c. (xi) One has the following properties: (Λ⊥ )⊥ = Λ,

⊥ (Λ1 ∨ Λ2 )⊥ = Λ⊥ 1 ∩ Λ2 ,

⊥ (Λ1 ∩ Λ2 )⊥ = Λ⊥ 1 ∨ Λ2 .

As for homogeneous ideals defining subspaces, let Λ be a subspace of codimension c in the projective space Pn . By (i) and (iv) above, (n) Λ = Zp (L0 , . . . , Lc−1 ), where L0 , . . . , Lc−1 linearly independent in S1 . Claim 3.3.7. The ideal I := (L0 , . . . , Lc−1 ) is prime. Assuming for a moment the content of the claim, one gets that I is radical (cf. Lemma 1.1.2) thus it coincides with Ip (Λ). In other words, subspaces Λ ⊂ Pn , of codimension c, are APS’s of Pn whose homogeneous ideal Ip (Λ) is generated by c linearly independent linear forms (cf. Example 2.1.18 for the affine case). Proof. Since L0 , . . . , Lc−1 are linearly independent linear forms, there (n) exist Lc , . . . , Ln ∈ S1 such that {L0 , . . . , Lc−1 , Lc , . . . , Ln } is a basis for ϕ (n) (n) (n) S1 (cf. (1.26)). The map ϕ : S1 → S1 , defined by Xi −→ Li , is an (n) automorphism of the vector space S1 which extends to an automorphism

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of the K-algebra S(n) , always denoted by ϕ, by the rule: ϕ(f (X0 , . . . , Xn )) = f (L0 (X), . . . , Ln (X)), where we posed for brevity X = (X0 , . . . , Xn ) (note that ϕ preserves the graduation of S(n) ). In particular, ϕ bijectively maps the maximal ideal S+ to the ideal (L0 (X), . . . , Ln (X)) and, consequently, S(n) ∼ the ideal (X0 , . . . , Xc−1 ) to I. Since (X0 ,...,X = S(n−c) , it follows that I c−1 ) is prime.  By using Example 1.10.20 and Remark 3.1.1, one can more generally state previous results for linear subspaces in P(V ), replacing S(n) with S(V ∗ ). 3.3.7

Projective and affine subspaces

As usual, we will identify An with the affine chart U0 of Pn and we will denote by H0 the hyperplane at infinity for U0 . Let Λ ⊂ Pn be any non-empty subspace of codimension c > 0 and denote by Λ0 := Λ∩U0 . If Λ ⊆ H0 , it is clear that Λ0 = ∅; therefore, we may assume that Λ is not contained in H0 . In such a case, from Section 3.3.6, Λ is defined by c linearly independent, homogeneous, linear equations of the form: ⎧ ⎪ ⎨ a10 X0 + a11 X1 + · · · + a1n Xn = 0 (3.25) ...... ...... ...... ⎪ ⎩ ac0 X0 + ac1 X1 + · · · + acn Xn = 0. The previous linear system can be written as A∗ X t = 0, where A∗ is a c × (n + 1) matrix with entries in K, of maximal rank c, whereas X := (X0 , . . . , Xn ) and 0 the (c × 1)-matrix with zero entries. From (3.16), Λ0 is defined by ⎧ ⎪ ⎨ a11 x1 + · · · + a1n xn + a10 = 0 (3.26) ...... ...... ...... ⎪ ⎩ ac1 x1 + · · · + acn xn + ac0 = 0, Xi , 1  i  n. The non-homogeneous linear where as customary xi := X 0 system (3.26) can be written as

A · xt + a = 0, where A is the (c × n)-matrix obtained by selecting the last n columns and all the rows of A∗ , whereas a is the (c × 1)-matrix defined by the first column of A∗ , whereas x := (x1 , . . . , xn ).

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From the assumptions on A∗ , the rank of A is c; in particular, (3.26) is compatible and Λ0 = ∅ is an affine subspace in the sense of Example 2.1.18. The homogeneous linear system obtained by adding the equation X0 = 0 to (3.25) defines the projective subspace Λ ∩ H0 of dimension m − 1, where m = n − c. This subspace is called the direction of Λ0 . Let ξ 0 be a solution of (3.26), i.e. a point P0 ∈ Λ0 . Let moreover ξ 1 , . . . , ξ m be independent solutions of the homogeneous linear system associated to (3.26). The bijective map φ0 : (λ1 , . . . , λm ) ∈ Am → ξ 0 + λ1 ξ 1 + · · · + λm ξ m ∈ Λ0 is the restriction to Am = U0 ⊂ Pm of the map φ : Pm → Λ as in (3.24), when in Λ we choose the m + 1 points P0 , P1 , . . . , Pm with relative homogeneous coordinates [1, ξ 0 ], [0, ξ i ],

1 ≤ i ≤ m,

respectively. Thus, φ0 is a homeomorphism between (Am , Zarm a ) and (Λ0 , Zarnp,Λ0 ) and it is a parametric representation of Λ0 (cf. Example 2.1.18). The map φ0 can be interpreted as a way to introduce a system of coordinates in Λ0 , in such a way that the point P0 coincides with its origin. Conversely, from Section 3.3.5, for any affine subspace Λ0 as in (3.26) it follows that Λ as in (3.25) is its projective closure. As for Section 3.3.6 (vi), one has therefore a natural notion of non-degenerate subset S of an affine space An . 3.3.8

Homographies, projectivities and affinities

Let V and W be two K-vector spaces. A map ϕ : P(V ) → P(W ) is called a homography if there exists an injective linear map f : V → W s.t. πW ◦ f = ϕ ◦ πV , where πV and πW as in (3.1); i.e. ϕ([v]) = [f (v)] for any v ∈ V \ {0}. Note that the existence of a homography ϕ : P(V ) → P(W ) implies dim(P(V ))  dim(P(W )); moreover, it is straightforward to verify that the composition of homographies is still a homography. As a matter of notation, we will put ϕ = ϕf when we want to stress that the homography ϕ depends on the linear map f . Let HomK (V, W ) be

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the vector space of K-linear maps from V to W . It is easy to see that ϕf = ϕg ⇐⇒ [f ] = [g] in P(HomK (V, W )).

(3.27)

Thus, the set of homographies from P(V ) to P(W ), denoted by O(P(V ), P(W )), identifies with the subset of P(HomK (V, W )) whose elements represent equivalence classes of injective linear maps from V to W . A homography ϕ = ϕf is called projectivity if f : V → W is an isomorphism; in such a case, P(V ) and P(W ) are said to be projectively isomorphic. Note that homographies in O(P(V ), P(V )) are all projectivities of P(V ) onto itself. In particular, O(P(V ), P(V )) forms a group with respect to the composition, which is denoted by PGL(V ). This group is the image of the linear group GL(V ) under the canonical quotient map πEnd(V ) as in (3.1). If in particular P(V ) and P(W ) are projectively isomorphic, then GL(V ) ∼ = GL(W ) and PGL(V ) ∼ = PGL(W ). n+1 , then GL(V ) and PGL(V ) will be denoted by GL(n + 1, K) If V = K and PGL(n + 1, K), respectively; the first group identifies with the group of non-degenerate, (n + 1) × (n + 1) matrices with entries in K whereas the second one with the quotient of GL(n + 1, K) modulo its center, i.e. modulo the group of scalar matrices of the form tIn+1 , where t ∈ K∗ and In+1 the identity matrix of order n + 1. If V has dimension n + 1, a projectivity ϕ : Pn → P(V ) assigns to any point P ∈ P(V ) a proportionality class [p0 , . . . , pn ] of a numerical vector in Kn+1 ; in other words, ϕ : Pn → P(V ) can be viewed as introducing a system of homogeneous coordinates on P(V ). In this correspondence, fundamental points in P(V ) are the images via ϕ of the fundamental points in Pn . In the system of homogeneous coordinates on P(V ) induced by ϕ, the fact that P has homogeneous coordinates [p0 , . . . , pn ] will be denoted by P =ϕ [p0 , . . . , pn ], or simply by P = [p0 , . . . , pn ], if no confusion arises. With the choice of two different systems of homogeneous coordinates ϕ : Pn → P(V ) and ψ : Pn → P(V ) on P(V ), there exists a non-degenerate, (n + 1) × (n + 1) matrix A with entries in K such that, for any P ∈ P(V ) with P =ϕ [x] = [x0 , . . . , xn ] and P =ψ [y] = [y0 , . . . , yn ], then y = A · xt , where A denotes any representative of the proportionality class [A] of matrices determined by A. Note that [A] determines the projectivity ψ −1 ◦ ϕ ∈ PGL(n + 1, k). t

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Theorem 3.3.8 (Fundamental theorem of projectivities). Let P1 and P2 be two projective spaces, both of dimension n. Consider (n+2)-tuples of points (P0 , . . . , Pn+1 ) and (Q0 , . . . , Qn+1 ) in P1 and P2 , respectively, which are in general position (i.e. any (n + 1)-tuple of such points are linearly independent points). Then there exists a unique projectivity ϕ : P1 → P2 such that ϕ(Pi ) = Qi , 0  i  n + 2. Proof. Left to the reader as Exercise 3.1.



Corollary 3.3.9 (Fundamental theorem on systems of homogeneous coordinates). Let P(V ) be a projective space of dimension n and let P0 , . . . , Pn+1 ∈ P(V ) be points in general position. Then, there exists a unique system of homogeneous coordinates on P(V ) in such a way that the ordered (n + 2)-tuple (P0 , . . . , Pn+1 ) corresponds to the ordered (n + 2)-tuple formed by fundamental points and the unit point of Pn . Let P(V ) be a projective space of dimension n. If Z := P(W ) is a mdimensional linear subspace of P(V ), to explicitly construct a homography ϕ : Pm → Z one can proceed as follows. Consider any basis w 0 , . . . , wm of W and then the linearly independent points Pi = [wi ] ∈ P(V ), 0 ≤ i ≤ m. One has Z = P0 ∨ . . . ∨ Pm and one can construct the map ϕ : [λ0 , . . . , λm ] ∈ Pm → [λ0 w 0 + · · · + λm wm ] ∈ Z which is a projectivity sending the (natural) fundamental points of Pm to P0 , . . . , Pm , respectively, and the unit point to [w 0 + · · · + w m ]; the homography ϕ ∈ O(Pm , P(V )) as above is said to be a homogeneous parametric representation of Z. In particular, if Z1 , Z2 are two m-dimensional linear subspaces of P(V ), the previous discussion shows there exists a projectivity of P(V ) into itself which sends Z1 onto Z2 . This also implies that, given any m-dimensional linear subspace Z of P(V ), there always exists a system of homogeneous coordinates on P(V ) in such a way that Z = Zp (Xm+1 , . . . , Xn ). Proposition 3.3.10. Any homography ϕ : P(V ) → P(W ) is a continuous map in the Zariski topologies of the projective spaces P(V ) and P(W ), respectively. If in particular ϕ is a projectivity, then it is a homeomorphism. Proof. Let f : V → W be the injective linear map inducing ϕ; its transpose linear map f t : W ∗ → V ∗ extends to a degree-zero, surjective,

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graded algebra homomorphism f t : S(W ∗ ) → S(V ∗ ). For any G ∈ H(S(W ∗ )), one has ϕ−1 (Zp (G)) = Zp (f t (G)), which implies that ϕ is continuous.  Proposition 3.3.11. A homography ϕ : P(V ) → P(W ) is a homeomorphism of P(V ) onto its image, which is a linear subspace of P(W ) of the same dimension of P(V ). Proof. The image of ϕ = ϕf is the linear subspace Z := P(f (V )) ⊆ P(W ). From Proposition 3.3.10, it therefore suffices to show that ϕ is closed. Consider the surjective algebra homomorphism f t : S(W ∗ ) → S(V ∗ ) introduced in the proof of Proposition 3.3.10. For any F ∈ H(S(V ∗ )), there exists G ∈ H(S(W ∗ )) such that f t (G) = F . Since ϕ−1 (Zp (G)) = Zp (f t (G)) = Zp (F ) (cf. the proof of Proposition 3.3.10), one has  ϕ(Zp (F )) = Zp (G) ∩ Z, which implies that ϕ is closed. Definition 3.3.12. Given APS’s X, Y ⊆ Pn , they are said to be projectively equivalent if X and Y are transformed into each other by a linear change of coordinates in the projective space Pn . Namely, there exists a projectivity ϕ ∈ PGL(n + 1, K), associated to a class [A] of an invertible square-matrix of order n + 1, which establishes an isomorphism between X and Y. Remark 3.3.13. Projective equivalence obviously induces an equivalence relation among APS’s. We will see in Chapter 6 that projective equivalence is a stronger relation than that induced by isomorphism of APS’s. Indeed, in Section 4.1.1 we will introduce the notion of homogeneous coordinate ring for an APS and we will see that homogeneous coordinate rings depend on the embedding into the ambient projective space; therefore isomorphic APS’s might have non-isomorphic homogeneous coordinate rings (cf. Example 6.5.11). This cannot occur for projectively equivalent APS’s, i.e. projectively equivalent APS’s have isomorphic homogeneous coordinate rings (cf. Remark 6.5.10). A map ψ : An → Am , with n  m, is called an affinity if there exists a homography Ψ : Pn → Pm such that, identifying An with the affine chart U0 , the restriction of Ψ to U0 coincides with ψ. In particular, any affinity is a homeomorphism onto its image and it sends affine subspaces of An to affine subspaces of Am of the same dimension. From Section 3.3.7 and what discussed above, ψ : An → Am is an affinity if and only if there exist

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a (n × m)-matrix A of maximal rank n and a (m × 1)-matrix a such that ψ(xt ) = A · xt + a,

(3.28)

where x = (x1 , . . . , xn ). In particular, ψ is a homomorphism of K-vector space if and only if ψ(0t ) = 0t . Affinities of An onto itself form a group under composition, which is called the affine group of An and which is denoted by Aff(An ). It is then clear that elements of Aff(An ) transform affine subspaces to affine subspaces of the same dimension. In particular, if Z is any affine subspace of dimension m of An , there always exists an affinity in Aff(An ) which maps Z to the affine subspace Za (xm+1 , . . . , xn ) ⊂ An . 3.3.9

Projective cones

Let Z ⊂ Pn be a (Zariski) closed subset and let Ca (Z) ⊂ An+1 be its affine cone. We can always identify An+1 as an affine open subset of Pn+1 in such a way that the given Pn coincides with the hyperplane at infinity of An+1 . Thus, we can consider the projective closure of Ca (Z) in Pn+1 , which we will denote by Cp (Z) and call the projective cone over Z with vertex O, where O = [1, 0, . . . , 0] ∈ U0 = An+1 ⊂ Pn+1 . From (3.17), we have that Cp (Z) ∩ Pn = Ca (Z)∞ = Z ⊂ Pn . Therefore, Cp (Z) = Ca (Z) ∪ Ca (Z)∞ = Ca (Z) ∪ Z, so Cp (Z) is the union of (projective) lines passing through O and through a point of Z. Finally one has Ip (Cp (Z)) = Ip (Z),

(3.29)

where the latter ideal is considered as a homogeneous ideal in S(n+1) . 3.3.10

Projective hypersurfaces and projective closure of affine hypersurfaces

Similarly to Example 2.1.19, for any non-constant polynomial F ∈ H(S(n) ) we can consider its decomposition F = F1r1 F2r2 · · · Fr as in (2.5), where F1 , . . . , F ∈ S(n) are all its non-proportional, irreducible factors (all homogeneous, from Proposition 1.10.15-(iii)) and where r1 , . . . , r are positive integers. Then Y := Zp (F ) is called the projective hypersurface determined by F and the polynomial Fred = F1 F2 · · · F

(3.30)

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is the reduced equation of Y . As in the affine case, deg(Fred ) is the degree of Y and Ip (Y ) = (Fred ) is its radical ideal. One can consider principal open (projective) sets Up (F ) and show that they form a basis for Zarnp (the proof is identical to that of Lemma 2.1.21). Moreover, using the Homogeneous Hilbert “Nullstellensatz”-strong form (cf. Theorem 3.2.5), one can easily adapt the proof of Theorem 2.3.1 to prove: Theorem 3.3.14 (Homogeneous Study’s principle). Let F, G ∈ H(S(n) ) be non-constant polynomials and let Y = Zp (F ) be the projective hypersurface defined by F . If Zp (G) ⊇ Y, then G is divided by Fred in S(n) . The previous result allows us to consider projective closures of affine hypersurfaces. Indeed one has: Proposition 3.3.15. Let Y = Za (f ) ⊂ An be an affine hypersurface, An with the affine for some non-constant polynomial f ∈ A(n)  . Identifying p p n chart U0 of P , one has Ip (Y ) = h0 (fred ) , where Y the projective closure of Y and fred the reduced equation of Y . Proof. Since Ia (Y ) = (fred ) (cf. Remark 2.2.9-(d)), we can directly p assume that f = fred. From Proposition (3.3.5), Y = Zp (J∗ ), where J∗ := h0 (g)| g ∈ (f ) . Since any g ∈ (f ) is of the form g = f h, for (cf. Lemma 1.10.17-(ii)), then some h ∈ A(n)  , and sinceh0 is multiplicative p J∗ = h0 (f ) , i.e. h0 (f ) ⊆ Ip (Y ). Note that if f factors as in (2.7), the fact that h0 is multiplicative implies that h0 (f ) = h0 (f1 )h0 (f2 ) · · · h0 (f ) where all the factors are square-free homogeneous polynomials, i.e. h0 (f ) is a reduced homogeneous polynomial. On the other hand, for any homogeneous polynomial G ∈ Ip (Y p ), one p has Z p (G) ⊇ Y . Thus, from  the homogeneous Study’s principle, we have p  G ∈ h0 (f ) , i.e. Ip (Y ) ⊆ h0 (f ) and we are done. 3.3.11

Proper closed subsets of P2

As in Corollary 2.3.4, we can classify all proper closed subsets of the projective plane. In what follows, the term curve is always meant to be any projective hypersurface in P2 . Proposition 3.3.16. Proper closed subsets in Zar2p consist only of finitely many points, curves and unions of a curve and finitely many points.

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Proof. If Z = ∅, there is nothing to prove. Therefore, let Z ⊂ P2 be any proper, non-empty closed subset. We can proceed as follows. One has Z = Z1 ∪ Z2 , where Z1 := Z ∩ U0 is a AAS in U0 ∼ = A2 whereas Z2 := Z ∩ H0 = (Z1 )∞ , with H0 = Zp (X0 ) the line at infinity of U0 . Since Z2 ⊆ H0 and since H0 is homeomorphic to P1 , Z2 is either empty, or a finite number of points or it is the whole line H0 . As for Z1 , since it is closed in U0 ∼ = A2 , we can apply Corollary 2.3.4. Then Z1 = Z1 ∪ Z2 , where Z1 either consists of finitely many points or it is empty and Z2 either consists of finitely many affine curves or it is empty. Then one concludes by using Sections 3.3.1, 3.3.7 and Proposition 3.3.15.  3.3.12

Affine and projective twisted cubics

Let n  2 be an integer, t an indeterminate over K and h2 (t), . . . , hn (t) ∈ K[t], not all of them constant polynomials. Consider C := {(t, h2 (t), . . . , hn (t)) ∈ An | t ∈ K} ⊂ An . It is clear that C is the AAS defined by C = Za (x2 − h2 (x1 ), . . . , xn − hn (x1 )).

(3.31)

  Lemma 3.3.17. One has Ia (C) = x2 − h2 (x1 ), . . . , xn − hn (x1 ) , which is a prime ideal in A(n) . Proof. Consider the K-algebra homomorphism ρn : A(n) → K[t] defined by ρn (x1 ) = t, ρn (xi ) = hi (t), 2  i  n, which is obviously surjective, so Ker(ρn ) is a prime ideal. Now, g ∈ Ker(ρn ) if and only if 0 = ρn (g(x1 , . . . , xn )) = g(t, h2 (t), . . . , hn (t)),

∀t ∈ K,

i.e. if and only if g ∈ Ia (C), which means Ia (C) = Ker(ρn ). The ideal J := x2 − h2 (x1 ), . . . , xn − hn (x1 ) is contained in Ker(ρn ) and is such that A(n) /J ∼ = K[x1 ] ∼ = K[t]. This implies that J = Ker(ρn ), as desired.  Any curve C as above is called an affine rational curve with polynomial parametrization. In what follows, we will study a particular case of the previous set-up.

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Consider the injective map φ : t ∈ A1 → (t, t2 , t3 ) ∈ A3 .

(3.32)

From (3.31), C := Im(φ) is a rational curve with polynomial parametrization, which is called (standard) affine twisted cubic. The image of C under any affinity of A3 will be called affine twisted cubic. From Lemma 3.3.17, the ideal Ia (C) = (f1 , g1 ), where f1 := x21 − x2 , g1 := x31 − x3 , is prime so radical (cf. Lemma 1.1.2). By straightforward computations, if one takes f2 := g1 − x1 f1 , one has Ia (C) = (f1 , f2 ) := (x2 − x21 , x3 − x1 x2 ).

(3.33)

Consider now the map ψ : [λ, μ] ∈ P1 → [λ3 , λ2 μ, λμ2 , μ3 ] ∈ P3

(3.34)

which is well-defined, injective and whose image Z is called the (standard) projective twisted cubic. Note that φ is the restriction of ψ to A1 ∼ = U 0 ⊂ P1 , thus C ⊂ Z. More precisely Z = C ∪ {P } where P = [0, 0, 0, 1] = ψ([0, 1]). The set Z is an APS in P3 ; indeed, Z = Zp (F1 , F2 , F3 ), where F1 := X0 X2 − X12 , F2 := X0 X3 − X1 X2 , F3 := X1 X3 − X22 ,

(3.35)

are three irreducible, homogeneous polynomials which are linearly indepen(3) dent in S2 . These polynomials are determined by the maximal minors of the matrix of linear forms  X0 X1 X2 A := . (3.36) X1 X2 X3 For this reason Z is said to be determinantal and the condition rank(A) = 1 is said to be a matrix equation of Z (cf. also Harris, 1995, Example 1.10, p. 9). Claim 3.3.18. One has C in P3 .

p

= Z, where C

p

the projective closure of C

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Even if this is a consequence of a more general fact (cf. Remark 9.2.2(iii)), we give here a constructive proof. p

Proof. The inclusion C ⊆ Z is trivial. On the other hand, if we consider p any homogeneous G ∈ Ip (C ) and if we identify A3 with U0 ⊂ P3 , one has δ0 (G) ∈ Ia (C), i.e. G(1, t, t2 , t3 ) = 0 for any t ∈ K. Thus, G(λ3 , λ2 μ, λμ2 , μ3 ) = 0, for any μ ∈ K and for any λ ∈ K∗ , which implies  therefore that G(λ3 , λ2 μ, λμ2 , μ3 ) is identically zero, i.e. G ∈ Ip (Z). From (3.33) and (3.35), we note that δ0 (F1 ) = f1 , δ0 (F2 ) = f2 , δ0 (F3 ) = f3 := x1 f2 − x2 f1 ∈ Ia (C). and viceversa h0 (f1 ) = F1 , h0 (f2 ) = F2 , h0 (f3 ) = F3 . Even if Ia (C) is generated by f1 and f2 , the polynomials F1 and F2 do not generate Ip (Z). More precisely, they do not even define Z as an APS, since Zp (F1 , F2 ) = Zp (h0 (f1 ), h0 (f2 )) = Z ∪ Zp (X0 , X1 ),

(3.37)

(cf. also Eisenbud, 1995, § 21.10, pp. 539–540).  p Claim 3.3.19. Ip C = (F1 , F2 , F3 ). Proof. To prove the claim, we use the following notation: B1 := F1 , B2 := −F2 , B3 := F3 , i.e. we consider the maximal minors of the matrix A in (3.36) with a suitable sign. Let us consider the free, graded S(3) -module  ⊕3 M := S(3) (−2) , whose homogeneous degree-d part Md is the K-vector space  (3) ⊕3  (3) = (Sd−2 )⊕3 S (−2) d (cf. Section 1.10.2). One is therefore reduced to proving that the homogeneous, degree-0 homomorphism  p φ (G1 , G2 , G3 ) ∈ M −→ G1 B1 + G2 B2 + G3 B3 ∈ Ip C is surjective. This is equivalent to showing that the induced K-vector space homomorphism  p φd (G1 , G2 , G3 ) ∈ Md −→ G1 B1 + G2 B2 + G3 B3 ∈ Ip C d

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is surjective for any integer d. Let K := Ker(φ), which is still a graded S(3) -module. Its homogeneous, degree-d piece Kd consists of triples (3) (G1 , G2 , G3 ) ∈ (Sd−2 )⊕3 such that Σ3i=1 Gi Bi = 0, i.e. Kd = Ker(φd ). These triples are called syzygies of (F1 , F2 , F3 ) and K is said to be the syzygy module. One trivially has Kd = {0} if d < 2; moreover K2 = {0} since, as observed above, the polynomials F1 , F2 and F3 are linearly independent in (3) S2 . On the other hand, a1 := (X0 , X1 , X2 ),

a2 := (X1 , X2 , X3 )

are two linearly independent elements in K3 , as it clearly follows from the matrix A in (3.36). Consider the free, graded S(3) -module N :=  (3) ⊕2 S (−3) and the degree-0, homogeneous homomorphism ψ

(D1 , D2 ) ∈ N −→ D1 a1 + D2 a2 ∈ K. We first show that ψ is an isomorphism of S(3) - graded modules. To prove this it suffices to show that, for any positive integer d, the induced K-vector space homomorphism φd

(D1 , D2 ) ∈ (Sd−3 )⊕2 −→ D1 a1 + D2 a2 ∈ Kd ⊆ (Sd−2 )⊕3 (3)

(3)

is an isomorphism. To prove this, consider any (G1 , G2 , G3 ) ∈ Kd . By Laplace’s rule and the definition of Bi , 1  i  3, this is equivalent to ⎛ ⎞ G3 G2 G1 det ⎝X0 X1 X2 ⎠ = 0, (3.38) X1 X2 X3 which implies that the three rows in (3.38) are linearly dependent over the field K(X0 , X1 , X2 , X3 ), whereas the rows of A are linearly independent, since the polynomials F1 = B1 , F2 = −B2 and F3 = B3 are non-zero. There exist therefore rational functions b1 a1 , ∈ K(X0 , X1 , X2 , X3 ), a0 b0 with a0 , a1 (respectively, b0 , b1 ) relatively prime elements, such that a1 X2 + a0 a1 G2 = X1 + a0 a1 G3 = X0 + a0 G1 =

b1 a1 b 0 X 2 + a0 b 1 X 3 X3 = , b0 a0 b 0 b1 a1 b 0 X 1 + a0 b 1 X 2 X2 = , b0 a0 b 0 b1 a1 b 0 X 0 + a0 b 1 X 1 X1 = . b0 a0 b 0

(3.39)

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Since the Gi ’s are polynomials in K[X0 , X1 , X2 , X3 ], if p is any prime factor of a0 , then it must divide b0 X0 , b0 X1 , b0 X2 , i.e. p must divide b0 . Recursively applying the same argument and changing the roles between a0 and b0 , one can assume a0 = b0 , so that (3.39) become G1 =

a1 X 2 + b 1 X 3 , b0

G2 =

a1 X 1 + b 1 X 2 , b0

G3 =

a1 X 0 + b 1 X 1 . b0

Thus, b0 divides the polynomials α1 := a1 X2 + b1 X3 ,

α2 := a1 X1 + b1 X2 ,

α3 := a1 X0 + b1 X1 ,

which implies that b0 divides therefore the polynomials X1 α3 − X0 α2 = −b1 B1 ,

X2 α3 − X0 α1 = b1 B2 ,

X2 α2 − X1 α1 = −b1 B3 .

Since B1 , B2 , B3 are all irreducible and distinct, this implies that b0 divides b1 . Since, by assumption, b0 and b1 are relatively prime elements, one concludes that b0 ∈ K∗ . Therefore, we can assume a0 = b0 = 1. For simplicity of notation, let a := a1 , b := b1 . We want to show that (3) (a, b) ∈ (Sd−3 )⊕2 . Indeed, if j = d − 3 and if Aj , Bj are the homogeneous, degree-j components of a, b, from (3.39) one gets Aj X2 + Bj X3 = 0, Aj X1 + Bj X2 = 0, Aj X0 + Bj X1 = 0 so, reasoning as above, one deduces Aj = Bj = 0, for any integer j = d − 3. This proves that ψ is bijective. In particular, if I := (F1 , F2 , F3 ), we proved the existence of an exact sequence of S(3) -graded modules ψ

φ

0 → N −→ M −→ I → 0, where M and N are free S(3) -graded modules. Namely the maps ψ and φ are degree-0, homogeneous homomorphisms such that ψ is injective, φ is surjective and Im(ψ) = Ker(φ). Such an exact sequence is called a homogeneous free resolution of the homogeneous ideal I. From Lemma 1.10.12 and from the exactness of the sequence, for any integer d one has    d+1 d d+3 −2 = − (3d + 1). dim(Id ) = 3 3 2 3 p

We now compute the dimension of Ip (C )d , for any integer d  0 (for d < 0 this dimension is obviously 0). Consider the homogeneous substitution of

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indeterminates of degree 3 ν : f (X0 , X1 , X2 , X3 ) ∈ S(3) → f (λ3 , λ2 μ, λμ2 , μ3 ) ∈ S(1) which induces surjective K-vector space homomorphisms (3)

(1)

νd : Sd → S3d , p

for any non-negative integer d. One has Ker(νd ) = Ip (C )d , for any d  0, so   d+3 p  − (3d + 1) = dim(Id ). dim Ip (C )d = 3 p

This proves that I = Ip (C ) as desired.



Finally, one more generally sets: a projective twisted cubic is any closed subset Z ⊂ P3 for which there exists a projectivity Φ of P3 such that Φ(Z) coincides with the image of (3.34). Thus, Z is a projective twisted cubic if (1) and only if there exists a basis Fi (λ, μ) of S3 , 0  i  3, such that Z is the image of the following map: ψ : [λ, μ] ∈ P1 → [F0 (λ, μ), F1 (λ, μ), F2 (λ, μ), F3 (λ, μ)] ∈ P3 which is called a parametric representation of the projective twisted cubic Z. Exercises Exercise 3.1. Prove Theorem 3.3.8. Exercise 3.2. Prove Proposition 3.3.16 using resultants and Elimination Theory of homogeneous polynomials. Exercise 3.3. Let m < n be positive integers. Let V be a K-vector space of dimension n + 1 and let W ⊂ V be a vector subspace of V of dimension m + 1. Let V ∗ = HomK (V, K) be the dual vector space of V . Consider Ann(W ) := {f ∈ V ∗ | f (w) = 0, ∀ w ∈ W } ⊂ V ∗ . (i) Prove that Ann(W ) is a vector subspace of V ∗ and compute its dimension. (ii) Let Λ := P(W ) be the linear subspace of P(V ) associated to W . Write Λ⊥ in terms of Ann(W ).

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Exercise 3.4. (i) Let Z := Zp (X12 + X22 − X0 X1 ) ⊂ P2R be a projective conic. Give affine classification of the affine conics which are the traces cut-out by Z in the three different affine charts U0 , U1 and U2 of P2R . (ii) Determine the affine (respectively, projective) cone Ca (Z) ⊂ A3R (respectively, Cp (Z) ⊂ P3R ) and Ca (Z)∞ when A3R is identified with the affine chart U3 of P3R . (n)

Exercise 3.5. Let X = Zp (F1 , . . . , Fm ) ⊂ Pn an APS, where Fj ∈ Sdj , dj positive integers for any 1  j  m. Prove that Z can be written as a zero-locus of a set of homogeneous polynomials of the same degree d > 0.

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Chapter 4

Topological Properties and Algebraic Varieties

4.1

Irreducible Topological Spaces

Let (Y, TY ) be any topological space, where TY denotes a given topology on Y . Then Y is said to be irreducible if Y cannot be expressed as Y = Y1 ∪Y2 , where Y1 , Y2 are proper closed subsets of Y . Equivalently, Y is irreducible if and only if any two non-empty, open subsets U1 , U2 ∈ TY are such that U1 ∩ U2 = ∅. When Y is not irreducible, it is called reducible. Remark 4.1.1. (i) If Y contains more than one point and it is irreducible, then it cannot be Hausdorff (i.e. TY is not T2 ). (ii) If K = R or C, AnK endowed with the euclidean topology is reducible for any n  1. (iii) Irreducibility is a topological property, i.e. it is invariant under homeomorphisms of topological spaces. (iv) If Y is irreducible, then it is connected. The converse is not true in general (cf. Example 4.1.5-(iii)). In what follows, any subset W ⊆ Y will be considered as endowed with the induced topology by that of Y . Proposition 4.1.2. Let Y be a topological space and let W ⊆ Y be any subset.

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(i) W is irreducible if and only if for any pair of distinct points P1 , P2 ∈ W there exists an irreducible subset Z ⊆ W such that P1 , P2 ∈ Z; (ii) W is irreducible if and only if any non-empty, open subset U ⊆ W is dense, i.e. U = W where U denotes the closure of U in W ; (iii) W is irreducible if and only if W is irreducible, where W denotes the closure of W in Y ; (iv) W is irreducible if and only if any non-empty, open subset U ⊆ W is irreducible. Proof. (i) The implication (⇒) is trivial. For (⇐), suppose by contradiction that W is reducible and let W = W1 ∪ W2 , where Wi proper closed subsets of W , 1  i  2. Let Pi ∈ Wi \ W3−i , 1  i  2, and let Z be any irreducible subset Z ⊆ W s.t. P1 , P2 ∈ Z. Then Z = Z1 ∪ Z2 , where Zi = Wi ∩ Z, 1  i  2, and Z1 , Z2 irreducible, proper closed subsets of Z, since Pi ∈ Z3−i . This contradicts the irreducibility of Z. (ii) Let us prove the non-trivial implication of (ii). Let W be irreducible and let U be any non-empty open subset of W . If U were not dense in W , then one would have W = (U ∩ W ) ∪ (W \ U ), where U ∩ W and W \ U are both proper closed subsets of W ; this is against the irreducibility assumption on W . (iii) Let Z be any closed subset of Y and denote by Z  := Z ∩ W and Z  := Z ∩ W , where W denotes the closure of W in Y . One has Z  = W ⇔ W ⊆ Z ⇔ W ⊆ Z ⇔ Z  = W . In other words, Z  is a proper closed subset of W if and only if Z  is a proper closed subset of W . From (ii) we know that W is irreducible if and only if any of its proper closed subsets have empty interior set. Thus, if the interior of Z  is empty, the same occurs for the interior of Z  , so that if W is irreducible then W is. Viceversa, if W is irreducible, let W = W1 ∪ W2 , where Wi closed subsets of W , 1  i  2. The irreducibility of W implies that W ⊆ Wi , for either i = 1 or i = 2. Consequently, W ⊆ Wi , i.e. W = Wi . Thus, W is irreducible. (iv) The implication (⇒) is a direct consequence of (ii) and (iii), whereas (⇐) is trivial.  Corollary 4.1.3. For any integer n  1: (i) (Pn , Zarnp ), (ii) any non-empty, open subset U ⊆ Pn ,

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(iii) (An , Zarna ), (iv) any non-empty, open subset U ⊆ An , are irreducible. Proof. From Proposition 4.1.2-(iv), it suffices to prove (i), which is a direct consequence of Proposition 4.1.2-(i). Indeed, for any pair of distinct points P1 , P2 ∈ Pn , the line P1 ∨P2 is irreducible, since homeomorphic to P1 (recall (3.24) and Remark 4.1.1-(iii)).  Remark 4.1.4. From Remark 4.1.1-(i) and Corollary 4.1.3, it follows that for any integer n  1 topologies Zarna and Zarnp are not T2 , i.e. Hausdorff (cf. Remark 2.1.14-(i) and Section 3.3.1 for n = 1). Note further that irreducibility in Corollary 4.1.3 holds true since K is an infinite field; if otherwise K = Zp , for p ∈ Z a prime, then all the topological spaces listed in Corollary 4.1.3 are reducible and both Zarna , Zarnp are the discrete topology. Example 4.1.5. (i) Any affine subspace of An is irreducible, since homeomorphic to Am for some m  n. Same conclusion holds for any projective subspace of Pn . (ii) Any affine twisted cubic as in Section 3.3.12 is irreducible, since it is homeomorphic to A1 . Therefore, any projective twisted cubic is irreducible too. (iii) The affine hypersurface Y = Za (x1 x2 ) ⊂ A2 is reducible. Indeed, Ia (Y ) = (x1 x2 ) = (x1 ) ∩ (x2 ) so Y = Za (x1 ) ∪ Za (x2 ) = Y1 ∪ Y2 , where Y1 , Y2 ⊂ Y are proper closed subsets of Y . In particular, Y is connected but not irreducible. Recalling Proposition 4.1.2, note moreover that: (a) there exist pairs of distinct points in Y not contained in any irreducible subset of Y ; (b) U := Za (x1 )c ∩ Y is an open subset of Y whose closure in Y is U = Za (x2 ) = Y2 , so not dense in Y ; (c) U1 := Za (x1 , x2 )c ∩ Y is a non-connected open subset of Y which is reducible. Corollary 4.1.6. Let Y, W be topological spaces such that Y is irreducible. Assume there exists a continuous map f : Y → W . Then Im(f ) ⊆ W is irreducible. Proof. Since Im(f ) ⊆ W is endowed with the induced topology of W , it suffices to showing that, when f is assumed to be also surjective, then W is irreducible. To prove this, if W = W1 ∪ W2 , where W1 , W2 closed

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subsets of W , then Y = f −1 (W1 ) ∪ f −1 (W2 ). Since Y is irreducible and f is continuous, one must have Y = f −1 (Wi ), for some i ∈ {1, 2}. Consequently, by the surjectivity of f , W = f (Y ) = f (f −1 (Wi )) = Wi .  4.1.1

Coordinate rings, ideals and irreducibility

Here, we focus on the case of Zariski topology. Definition 4.1.7. For any subset Y ⊆ An , consider the Ia (Y ) ⊆ A(n) . The K-algebra of finite type (cf. Definition 1.5.3) A(Y ) :=

A(n) Ia (Y )

ideal

(4.1)

is called the (affine) coordinate ring of Y . Similarly, for any subset Y ⊆ Pn , with homogeneous ideal Ip (Y ) ⊆ S (n) , the graded K-algebra of finite type (recall Proposition 1.10.9-(ii)) S(Y ) :=

S (n) Ip (Y )

(4.2)

is called the homogeneous coordinate ring of Y . Proposition 4.1.8. Let Y ⊆ Pn be any irreducible subset and let U ⊆ Y be any non-empty open subset of Y , then Ip (Y ) = Ip (U ). The same holds replacing above Pn with An and Ip (−) with Ia (−). Proof. We prove the first part, the proof in the affine case being identical. p Let Y be the projective closure of Y in Pn ; then (3.21) gives Ip (Y ) = p  = Y, Ip (Y ). Since Y is irreducible, from Proposition 4.1.2-(ii), we have U p p p  = Y and, from  denotes the closure of U in Y . Thus U = U where U 

(3.21), we get Ip (U ) = Ip (Y p ) = Ip (Y ). a

Corollary 4.1.9. For any subset Y ⊆ An , let Y denote its closure in An . a p Then A(Y ) = A(Y ). Similarly, for any subset Y ⊆ Pn , let Y denote its p projective closure. Then S(Y ) = S(Y ). When it is clear from the context and no confusion arises, for brevity we will sometimes use the symbol I(Y ) instead of Ia (Y ) or Ip (Y ). The rings A(Y ) and S(Y ) are always reduced rings (cf. Definition 1.1.1) since I(Y ) is always radical. On the other hand, in some cases these rings are not integral domains. For example, if Ip (Y ) = (X0 X1 ) ⊂ S (2) , then S(Y ) contains X0 and X1 as zero-divisors. Indeed, Zp (Ip (Y )) = Y = Y0 ∪ Y1 ,

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where Yi = Zp (Xi ), 0  i  1; namely Y is a reducible conic which is the union of two fundamental lines of P2 . The following result gives an algebraic criterion for irreducibility. Proposition 4.1.10. A subset Y ⊆ An (respectively, Y ⊆ Pn ) is irreducible if and only if I(Y ) is a prime ideal, i.e. if and only if A(Y ) (respectively, S(Y )) is an integral domain. Proof. The second part of the statement is obvious by definition of A(Y ) (respectively, S(Y )); therefore we need to prove the first equivalence. From Corollary 4.1.9, it suffices to consider the case with Y closed in the Zariski topology. Assume therefore Y to be irreducible. If f g ∈ I(Y ) then (f g) ⊆ I(Y ) so Y = Z(I(Y )) ⊆ Z(f g) = Z(f ) ∪ Z(g) (cf. Propositions 2.1.11 and 3.1.5). Thus, either Y ⊆ Z(f ) or Y ⊆ Z(g). This means that either f ∈ I(Y ) or g ∈ I(Y ), i.e. I(Y ) is prime. Conversely let I(Y ) be a prime ideal and let Y = Y1 ∪ Y2 , where Y1 , Y2 closed subsets of Y . Then I(Y ) = I(Y1 ) ∩ I(Y2 ) (cf. Propositions 2.2.8 and 3.1.7).  Claim 4.1.11. In the above assumptions, I(Y ) is either I(Y1 ) or I(Y2 ). Proof. If it were I(Y )  I(Yi ) for both i = 1, 2, there would exist fi ∈ I(Yi ) \ I(Y ), 1  i  2, such that f1 f2 ∈ I(Y1 ) · I(Y2 ) ⊆ I(Y1 ) ∩ I(Y2 ) = I(Y ), contradicting that I(Y ) is a prime ideal (cf. Exercise 1.1). From Claim 4.1.11, we have either Y = Y1 or Y = Y2 so Y is irreducible.  Recalling bijective correspondences in Corollaries 2.2.5 and 3.2.6, we have: Corollary 4.1.12. The maps (2.11) and (2.12) induce bijections:  1−1  {irreducible closed subsets of An } ←→ prime ideals of A(n) . Similarly, the maps (3.5) and (3.6) induce bijections: {irreducible closed subsets of Pn }   1−1 ←→ homogeneous prime ideals of S (n) \ {S+ }. Example 4.1.13. (i) We find in this way an alternative proof of Corollary 4.1.3, via purely algebraic approach; indeed one has A(An ) = A(n) and S(Pn ) = S (n)

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which are integral domains, so one concludes by Proposition 4.1.10. For the open subsets, one uses Corollary 4.1.9. (ii) For any subset Z ⊂ Pn we have that Z irreducible ⇔ Ca (Z) irreducible ⇔ Cp (Z) irreducible.

(4.3)

Proof. This follows from (3.8), (3.29) and what discussed above.  (iii) If Z ⊂ Pn is any hypersurface, then Z is irreducible if and only if its reduced equation F ∈ S (n) is an irreducible polynomial, as it follows from (3.30) and Claim 4.1.11. Same occurs in the affine case (cf. Example 2.1.19). (iv) Any affine rational curve with polynomial representation as in (3.31) is irreducible, as it follows from Lemma 3.3.17. (v) Recalling (3.37), the closed subset K := Zp (F1 , F2 ) is reducible consisting of the (standard) projective twisted cubic together with the line at infinity of the affine plane Za (x1 ) ⊂ A3 . From Proposition 4.1.10, I = (F1 , F2 ) cannot be a prime ideal so S(K) is not an integral domain; indeed, if we denote by F3 , X0 ∈ S(K) the images of F3 , X0 ∈ S (3) in S(K), respectively, since Ip (K) = (F1 , F2 ), then (3) F3 , X0 ∈ S(K) \ {0} (F3 is linearly independent from F1 , F2 in S2 ) on the other hand X0 · F3 = 0 in S(K), since X0 · F3 = X1 F2 − X2 F1 ∈ Ip (K). (vi) Corollary 2.3.4 and Proposition 3.3.16 imply that any non-empty, irreducible, proper closed subset of A2 and of P2 is either one point or an irreducible curve. 4.1.2

Algebraic varieties

In the sequel, we will always endow topological spaces with Zariski topology and use the following terminology. Definition 4.1.14. Y is said to be a projective variety (respectively, affine variety) if Y is irreducible and closed in a projective (respectively, affine) space. Y is said to be a quasi-projective variety (respectively, quasi-affine variety) if Y is irreducible and locally closed (i.e. it is the intersection of a closed set and of an open set) in a projective (respectively, affine) space. Remark 4.1.15. From the previous definition, we note that Y is a quasiprojective (respectively, quasi-affine) variety if and only if it is an open subset of a projective (respectively, affine) variety, namely its closure in Pn (respectively, in An ).

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Proposition 4.1.16. Let Z ∈ Cpn . Then Z is a projective variety if and only if for any affine chart Ui ⊂ Pn , 0  i  n, Zi := Z ∩ Ui ⊂ Ui ∼ = An is an affine variety. Proof. (⇒) This is obvious. (⇐) In this case one is reduced to showing that Ip (Z) is a prime ideal. Since Ip (Z) is a homogeneous ideal, the primality of Ip (Z) can be proved by using Proposition 1.10.8-(ii) and the fact that, by assumptions, Ia (Zi ) is prime, for any 0  i  n.  From Definition 4.1.14, any projective (respectively, affine, quasi-affine) variety is also a quasi-projective variety, in other words the notion of quasiprojective variety is the most general one. Definition 4.1.17. The term algebraic variety will be used from now on to indicate any quasi-projective variety. Similarly, with the term closed algebraic set we will indicate any AAS or any APS, whereas with locallyclosed algebraic set any locally-closed subset of either an affine or a projective space.

4.2

Noetherian Spaces: Irreducible Components

Definition 4.2.1. A non-empty topological space (Y, TY ) is said to be Noetherian if it satisfies the following condition: (*) for any sequence {Yn }n∈N of closed subsets of Y such that, for any n ∈ N one has Yn ⊇ Yn+1 , there exists m ∈ N such that, for any n  m, one has Yn = Ym . Condition (*) is called the descending chain conditions (denoted by d.c.c. for short) on closed subsets of Y . Remark 4.2.2. By passing to complementary sets, one clearly has that Y is Noetherian if and only if it satisfies the ascending chain conditions (a.c.c.) on its open sets, that is: (**) for any sequence {Un }n∈N of open subsets of Y such that, for any n ∈ N one has Un ⊆ Un+1 , there exists m ∈ N s.t., for any n  m, one has Un = Um . In what follows, any subset W ⊆ Y of a topological space Y will be endowed with the induced topology, i.e. with TY,W .

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Proposition 4.2.3. Let Y be a Noetherian topological space. (i) Any non-empty subset W ⊆ Y is Noetherian; (ii) Y , as well as any non-empty subset W ⊆ Y , is compact. Proof. (i) is an easy consequence of the following observation: if W ⊆ Y and if W1 ⊇ W2 are closed subsets of W , there exist Yi closed subsets of Y such that Wi = Yi ∩ W , 1  i  2, and Y1 ⊇ Y2 ; indeed it suffices to replace Y2 with Y1 ∩ Y2 if necessary. As for (ii), by contradiction let {Ui }i∈I be an open covering of Y from which one cannot extract a finite sub-covering. Thus, there exist sequences {in }n∈N of indexes in I and of distinct points {Pn }n∈N of Y such that, for Un := ∪nh=1 Uih , one has Pn−1 ∈ Un but Pn ∈ Un for any n ∈ N. This would contradict the a.c.c. on open subsets of Y .  Example 4.2.4. An easy example of Noetherian topological space is the following: consider Y an infinite set, P(Y ) the power-set of Y and let CY ⊂ P(Y ) be the subset consisting of ∅, Y and all finite subsets of Y . Then CY can be taken as a family of closed subsets for a topology on Y , which turns out to be Noetherian. The next result shows that also Zar is Noetherian. Corollary 4.2.5. (i) For any integer n  1, (An , Zarna ) and (Pn , Zarnp ) are Noetherian. (ii) Any locally-closed algebraic set is Noetherian. Proof. (i) directly follows from the Noetherianity of the rings A(n) and S (n) (cf. Proposition 1.4.4) and from reversing inclusions (cf. (2.10) and Proposition 3.1.7-(i)). To prove (ii) observe that, from Proposition 4.2.3-(i), statement (i) implies (ii).  The following result bridges Noetherianity with irreducibility. Theorem 4.2.6 (Irredundant decomposition). Let Y be a Noetherian topological space and let W ⊆ Y be a non-empty, closed subset. Then: (i) W can be expressed as a finite union W := W1 ∪ · · · ∪ Wn , where each Wi is a closed, irreducible subset of W ;

(4.4)

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(ii) decomposition (4.4) is uniquely determined (up to reordering the Wi ’s) if it is irredundant, i.e. if Wi ⊆ Wj for any i = j ∈ {1, . . . , n}. Proof. (i) If W is irreducible, we are done. Otherwise let W = W1 ∪ W  , where W1 , W  proper closed subsets of W . If both of them are irreducible, we are done (observe moreover that neither W1 ⊆ W  nor W  ⊆ W1 , otherwise W would be irreducible against the assumption). Otherwise, at least one of W1 , W  has to be reducible. Assume that W  is. Repeating the same argument, we can construct a sequence {Wn }n∈N of closed subsets of W s.t., for any n ∈ N, Wn+1  Wn . By Noetherianity of Y and by Proposition 4.2.3, we conclude.  be another decomposition, satisfying the (ii) Let W = W1 ∪ · · · ∪ Wm same assumptions of that in (4.4); assume moreover that both of them are irredundant. One has  ), Wi = (Wi ∩ W1 ) ∪ · · · ∪ (Wi ∩ Wm

for any

i ∈ {1, . . . , n}.

Since Wi is irreducible, there exists j ∈ {1, . . . , m} s.t. Wi = Wi ∩ Wj , i.e. Wi ⊆ Wj . Similarly, there exists h ∈ {1, . . . , n} s.t. Wj ⊆ Wh , i.e. Wi ⊆ Wh . Since (4.4) is irredundant, then i = h and Wi = Wj . Repeating the same arguments, one can conclude.  Definition 4.2.7. When (4.4) is an irredundant decomposition, W1 , . . . , Wn are called the irreducible components of W and (4.4) is said to be the decomposition of W into its irreducible components. Corollary 4.2.8. If (Y, TY ) is a Noetherian and Hausdorff topological space, then Y is finite and TY is the discrete topology. Proof. By Theorem 4.2.6, Y can be decomposed as a finite union of its irreducible components. Therefore, to prove the statement, we can assume Y to be irreducible and conclude by using Remark 4.1.1-(i).  In the case of Zariski topology we have: Corollary 4.2.9. (i) Any algebraic locally-closed subset can be decomposed into finitely many algebraic varieties.

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(ii) Any radical ideal I ⊂ A(n) (respectively, homogeneous radical ideal I ⊂ S (n) ) can be uniquely written as I = ∩ni=1 Ij , where Ij prime ideals in A(n) (respectively, homogeneous prime ideals in S (n) ). Proof. (i) is a consequence of Theorem 4.2.6, whereas (ii) follows from the bijective correspondences proved in Corollary 2.2.5, in the nonhomogeneous case, and in Corollary 3.2.6, in the homogeneous one. In this latter situation, the only left case deals with the ideal S+ , which obviously  is S+ = (X0 ) ∩ (X1 ) ∩ · · · ∩ (Xn ). Remark 4.2.10. In general, the intersection of algebraic varieties is not necessarily an algebraic variety. Take, e.g. F1 , F2 ∈ S (3) homogeneous polynomials as in (3.35). These polynomials are irreducible in S (3) so the quadric (hyper)surfaces Zp (F1 ), Zp (F2 ) ⊂ P3 are projective varieties (cf. Example 4.1.13-(iii)). On the other hand, Zp (F1 ) ∩ Zp (F2 ) = Zp (F1 , F2 ) is reducible, consisting of the union of a projective twisted cubic and a line (cf. Example 4.1.13-(v)). 4.3

Combinatorial Dimension

Let (Y, TY ) be a topological space. Definition 4.3.1. The combinatorial (or topological) dimension of the topological space (Y, TY ), denoted by dimc (Y ) (or even dimt (Y )) is defined to be: dimc (Y ) := Supn∈Z0 {Z0 ⊂ Z1 ⊂ · · · ⊂ Zn } , where each Zi is a non-empty, irreducible, closed subset of Y , 0  i  n, and where all the inclusions are strict. Example 4.3.2. (i) One obviously has dimc (A0 ) = dimc (P0 ) = 0. (ii) Let K be an infinite field. Considering as customary A1 := A1K endowed with the Zariski topology, one has dimc (A1 ) = 1, since irreducible, nonempty closed subsets are simply points and A1 , cf. Remark 2.1.14-(ii). Therefore, for any P ∈ A1 , one has Z0 = {P } ⊂ Z1 = A1 . The same holds for P1 (cf. Section 3.3.1). (iii) Similarly as in (ii), dimc (A2 ) = 2, respectively, dimc (P2 ) = 2 (cf. Corollary 2.3.4, respectively, Proposition 3.3.16), since in this case irreducible curves enter into the game. In Corollary 4.3.8, we will

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show that more generally dimc (An ) = dimc (Pn ) = n holds, for any non-negative integer n. (iv) If (Y, TY ) is an irreducible, Noetherian and T1 (i.e. points in Y are closed) topological space, then dimc (Y ) = 0 if and only if Y is a singleton, whereas dimc (Y ) = 1 if and only if proper closed subsets of Y are finite subsets of Y . (v) The following is an example of a Noetherian topological space having infinite combinatorial dimension. Let Y = [0, 1] be the real interval on Y whose and let TY be the topology   set of closed subsets is given by CY := {Y } ∪ {∅} ∪ [ n1 , 1], n ∈ N . Thus, Y is Noetherian and all closed subsets of Y are irreducible. For any n ∈ N, the following:       1 1 1 ,1 ⊂ ,1 ⊂ ··· ⊂ ,1 {1} = [1, 1] ⊂ 2 3 n is a chain of non-empty, irreducible closed subsets, where each inclusion is strict. The supremum on n is therefore infinite. From the very definition of combinatorial dimension, if Y is Noetherian one has that dimc (Y ) := maxi {dimc (Yi ) | Yi irreducible component of Y } .

(4.5)

Definition 4.3.3. Let (Y, TY ) be a topological space. Y is said to have pure dimension (or even that Y is pure) if each of the irreducible components of Y have the same combinatorial dimension equal to dimc (Y ). Example 4.3.4. (i) Any irreducible topological space is obviously pure. There are also reducible topological spaces which are pure, as, e.g. the affine reducible conic Y = Za (x21 − x22 ) ⊂ A2 . Nonetheless there are also reducible topological spaces which are not pure. Consider, e.g. Y = Za ((x1 x2 − x1 , x22 − x2 )) ⊂ A2 . Since Ia (Y ) = (x1 , x2 ) ∩ (x2 − 1), from Proposition 2.1.11-(ii), Y turns out the be the union of the origin {O} = Za ((x1 , x2 )) := Y1 and of the line Y2 = Za ((x2 − 1)) not passing through O. Thus, Y1 and Y2 are the irreducible components of Y , dimc (Y ) = dimc (Y2 ) = 1 > 0 = dimc (Y0 ) and Y is not pure. (ii) If Y is pure and dimc (Y ) = 1, then Y is called curve; affine (or projective) plane curves are closed subsets in the Zariski topology which are pure of combinatorial dimension 1. If Y is pure and dimc (Y ) = 2, then Y is called surface.

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Proposition 4.3.5. Let (Y, TY ) be a topological space. Then: (i) if Z ⊆ Y is a subset, then dimc (Z)  dimc (Y ). If dimc (Y ) < +∞ then all subsets of Y have finite combinatorial dimension and the quantity codimc,Y (Z) := dimc (Y ) − dimc (Z)  0

(4.6)

is called the combinatorial (or even topological) codimension of Z in Y ; (ii) if {Uj }j∈J is an open covering of Y , then dimc (Y ) = Supj∈J {dimc (Uj )}; (iii) if Y is irreducible, with dimc (Y ) < +∞, and if Z ⊆ Y is closed with dimc (Z) = dimc (Y ), then Y = Z (and so Z is irreducible). Proof. (i) If dimc (Y ) = +∞, there is nothing else to prove. Assume therefore dimc (Y ) < +∞; in this case Y is Noetherian and so Z ⊆ Y is also Noetherian. With no loss of generality, we may assume Z to be irreducible (otherwise Z has finitely many irreducible components). If Z2 ⊂ Z1 ⊂ Z is any collection of proper closed subsets of Z, there exist closed subsets Y1 , Y2 ⊂ Y such that Zi = Yi ∩ Z, 1  i  2, so in particular Y2 ⊂ Y1 . With no loss of generality, we may assume Z1 and Z2 to be irreducible. In such a case, also Y1 and Y2 can be assumed to be irreducible; indeed, if by contradiction, e.g. Y1 were reducible, say Y1 = Y1 ∪ Y1 with Z1 ∩ Y1 = ∅ = Z1 ∩ Y1 , then Z1 = Z1 ∩ Y1 = (Z1 ∩ Y1 ) ∪ (Z1 ∩ Y1 ), where Z1 ∩ Y1 and Z1 ∩ Y1 are closed subsets of Z1 for the induced topology, then Z1 would be reducible. To complete the proof of (i), one simply applies the previous reasoning to any chain of closed, non-empty, proper subsets of Z and Y . (ii) Assume to have a chain of proper inclusions of non-empty, closed irreducible subsets of Y (∗) Z0 ⊂ Z1 ⊂ · · · ⊂ Zn and let P0 ∈ Z0 be a point. There exists j0 ∈ J such that P0 ∈ Uj0, since {Uj }j∈J is an open covering of Y . Thus, Uj0 ∩ Zi = ∅, for any 0  i  n. Since Uj0 ∩ Zi is an open set of Zi which is irreducible, then Uj0 ∩ Zi is dense in Zi , for any 0  i  n (cf. Proposition 4.1.2-(ii)). Therefore, Uj0 ∩ Zi ⊂ Uj0 ∩ Zi+1 where each inclusion is strict; indeed, if it were Uj0 ∩ Zi = Uj0 ∩ Zi+1 for some index i, then by density and

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irreducibility we would have Zi = Zi+1 , contradicting (∗). Thus, from (∗) we have Uj0 ∩ Z0 ⊂ Uj0 ∩ Z1 ⊂ · · · ⊂ Uj0 ∩ Zn , which is a chain of proper inclusions of non-empty, closed subsets of Uj0 , which can be assumed to be irreducible with no loss of generality. Considering the supremum, one finds dimc (Uj0 ). Considering the supremum on j ∈ J, one concludes. (iii) Let Z ⊆ Y be closed and set dimc (Y ) := n < +∞. Since by assumption we have dimc (Z) = n, there exists a maximal chain of irreducible closed subsets of Z, say (∗) Z0 ⊂ Z1 ⊂ · · · ⊂ Zn , where all inclusions are strict. Since Z is closed in Y and each Zj is closed in Z, then each Zj is also closed and irreducible in Y , 0  j  n. Then (∗) is also a maximal chain of irreducible closed subsets of Y , therefore Zn = Y . Thus, Y ⊆ Z, since Zn is closed in Z, which implies Z = Y and that Z is irreducible.  Remark 4.3.6. (i) Note that the combinatorial codimension of affine (respectively, projective) linear subspaces of the affine (respectively, projective) space coincides with the usual notion of codimension. (ii) Since An ∼ = U0 ⊂ Pn (cf. Remark 3.3.3-(ii)), from Proposition 4.3.5-(ii) we get dimc (An ) = dimc (Pn ). Recalling the definition of Krull-dimension and of affine coordinate ring of an affine variety (cf. Definitions 1.12.1 and 4.1.7), one has the following result. Proposition 4.3.7. Let K be an algebraically closed field and n  1 be an (n) integer. Let Y ⊆ AnK be any affine variety and let A(Y ) = IAa (Y ) be its affine coordinate ring. Then dimc (Y ) = K dim(A(Y )). Proof. Since Y ⊆ AnK is an affine variety, then Y is irreducible, Ia (Y ) ⊂ A(n) is a prime ideal and A(Y ) is an integral K-algebra of

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finite type (cf. Proposition 4.1.10). Moreover, from Corollary 4.2.5, Y is Noetherian, therefore dimc (Y ) < +∞. Let (∗)

Y0 ⊂ Y1 ⊂ · · · ⊂ Ym = Y

be any chain of non-empty, irreducible, closed subsets of Y . Since Y is closed in An and each Yj is closed in Yj+1 , for any 0  j  m − 1, each Yj is closed and irreducible in An . Therefore, any Yj is an affine variety of An so each ideal Ia (Yj ) is a prime ideal in A(n) . From the bijective correspondence between irreducible, closed subsets of An and prime ideals in A(n) , together with the reversing inclusion among ideals, respectively closed subsets (cf. Corollary 4.1.12 and Proposition 2.2.8), (∗) gives rise to the chain of prime ideals in A(n) (∗∗)

Ia (Y0 ) ⊃ Ia (Y1 ) ⊃ · · · ⊃ Ia (Ym ) = Ia (Y ), (n)

each of which contains the prime ideal Ia (Y ). Let π : A(n)  A(Y ) = IAa (Y ) be the canonical epimorphism, which maps Ia (Y ) to the prime ideal (0) in I (Y ) A(Y ) whereas, for any 0  j  m − 1, Ia (Yj )e = π (Ia (Yj )) = Iaa (Yj) := pj . Any pj is a prime ideal in A(Y ), as it directly follows from the fact that Ia (Yj ) is a prime ideal in A(n) and from the epimorphism π. Taking (∗) and  (∗∗) as maximal chains, one deduces that dimc (Y ) = K dim(A(Y )). Corollary 4.3.8. For any integer n  0, dimc (An ) = dimc (Pn ) = n. Proof. The fact that dimc (An ) = n follows from Proposition 4.3.7 and from Corollary 1.12.4-(i), whereas dimc (Pn ) = n follows from Remark 4.3.6-(ii).  More generally, for projective varieties one has the following. Proposition 4.3.9. Let K be an algebraically closed field and n  1 be an (n) integer. Let Y ⊆ PnK be any projective variety and let S(Y ) = ISp (Y ) be its homogeneous coordinate ring. Let Ca (Y ) ⊂ An+1 the affine cone over Y . Then K dim(S(Y )) = dimc (Ca (Y )). Proof. Since Ip (Y ) = Ia (Ca (Y )), we have an isomorphism of integral K– (n) (n+1) algebras of finite type S(Y ) = ISp (Y ) ∼ = IaA(Ca (Y )) = A(Ca (Y )) so Ca (Y ) is irreducible since Y is. Thus, Ca (Y ) ⊂ An+1 is an affine variety so, from  Proposition 4.3.7, dimc (Ca (Y )) = Kdim(A(Ca (Y ))) = Kdim(S(Y )).

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As a consequence of some other definitions of dimension (cf. Corollary 10.2.2), in the above set-up we will more precisely show that K dim(S(Y )) = dimc (Ca (Y )) = dimc (Y ) + 1.

(4.7)

Exercises (2)

Exercise 4.1. Find an example of an irreducible polynomial f ∈ AR whose zero set Za (f ) is not irreducible in A2R (cf. Hartshorne, 1977, Exercise 1.12, p. 18). (3)

Exercise 4.2. Let K be an algebraically closed field. Consider in AK = K[x1 , x2 , x3 ] the ideal J := (x1 x3 − x22 , x31 − x2 x3 ). Show that J is not a prime ideal and find irreducible components of Za (J) in A3K (cf. Reid, 1988, p. 57). Exercise 4.3. Let K be an algebraically closed field. Show that if X ⊂ PnK is a quasi-projective variety and if Y ⊂ X is irreducible and locally closed in X, then Y ⊂ PnK is a quasi-projective variety. Exercise 4.4. Let K be an algebraically closed field and let I = (x21 − (3) x32 , x22 − x33 ) ⊂ AK be an ideal. Consider the K-algebra homomorphism (3) α : AK → K[t], defined by α(x1 ) = t9 , α(x2 ) = t6 , α(x3 ) = t4 . Show that Ker(α) = I and deduce that Za (I) is irreducible, with Ia (Za (I)) = I (cf. Fulton, 2008, Exercise 1.40, p. 12). Exercise 4.5. Let K be an algebraically closed field and let C be the set parametrizing conics in the projective plane P2K . (i) Show that C can be identified with the projective space P5K . (ii) Let D ⊂ C be the subset parametrizing degenerate conics in P2K . Show that D can be identified with an cubic hypersurface Σ3 in P5K . (iii) Show that Σ3 is an irreducible hypersurface.

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Chapter 5

Regular and Rational Functions on Algebraic Varieties

Here we study functions which are defined on algebraic varieties. To do this, we need first to introduce some basic facts concerning pre-sheaves and sheaves on a topological space. 5.1

Basics on Sheaves

Let (Y, TY ) be a topological space. Definition 5.1.1. A pre-sheaf F (of sets) on Y is the datum of: (i) a non-empty set F(U ), for any open set U ⊆ Y , and (ii) a map ρU V : F(U ) → F(V ), for any V ⊆ U ⊆ Y open sets, such that the following hold: (F1) ρU U = IdF(U) , for any open set U ⊆ Y ; V V (F2) for any open sets W ⊆ U ⊆ V , one has ρU W ◦ ρU = ρW . The maps ρU V are called restrictions and, for any open set U ⊆ Y , the elements of F(U ) are called sections of F over U . In particular, elements of F(Y ) are called global sections of F. In the above definition, to any open set U of a topological space it is associated a non-empty set F(U ); in particular, for any pre-sheaf F, we have

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F(∅) = ∅. In the sequel, we shall always consider pre-sheaves F such that F(∅) = {·} is a fixed singleton. This convention will be taken for granted and not mentioned anymore. Definition 5.1.2. A sheaf (of sets) on Y is the datum of a pre-sheaf F on Y as above, satisfying moreover the following condition: (F3) for any family {Ui }i∈I of open sets of Y , where I a set of indexes, and for any collection of sections (fi ) ∈ Πi∈I F(Ui ) s.t. U

j i ρU Ui ∩Uj (fi ) = ρUi ∩Uj (fj ), ∀ i, j ∈ I,

there exists a unique f ∈ F(U ), where U := ∪i∈I Ui , such that ρU Ui (f ) = fi , ∀ i ∈ I. Remark 5.1.3. Not all pre-sheaves are sheaves. Consider, e.g. (Y, TY ) a topological space and let S be any non-empty set. For any non-empty open subset U ⊆ Y set CS (U ) := {S} whereas CS (∅) = {·}, where {·} = S. For any ∅ = V ⊆ U open set inclusion, set moreover ρU V = IdS , whereas if U V = ∅ set ρV to be the constant map to {·}. Then CS defined in this way is called constant pre-sheaf associated to S on Y . It is easy to observe that if Y contains two open sets U and U  such that U ∩ U  = ∅ (e.g. when Y is Hausdorff), then CS is a pre–sheaf but not a sheaf. If otherwise Y is an irreducible topological space, any constant pre-sheaf is a sheaf. When the F(U )’s are more precisely, e.g. abelian groups, rings, K-algebras, R-modules for a given ring R, etc., one requires restrictions ρU V to be homeomorphims of abelian groups, rings, K-algebras, R-modules, etcetera, respectively. In this case, F is called a pre-sheaf (respectively a sheaf) of abelian groups, rings, K-algebras, R-modules, etcetera, on Y . Remark 5.1.4. Let F be a sheaf on the topological space Y and let U be any non-empty, open subset of Y ; one can define the pre-sheaf F|U by F|U (V ) := F(V ), for any non-empty, open set V of U . Since F is a sheaf, the F|U is a sheaf too. This is called the sheaf obtained by restriction of F to U . For further details on sheaves, see, e.g. Hartshorne (1977, § II.1, pp. 60–69; Mumford, 1988, § I.4, pp. 24–30).

Regular and Rational Functions on Algebraic Varieties

5.2

127

Regular Functions

If we take a polynomial f ∈ A(n) , we can consider the associated function f : An → K (recall (2.1)). Accordingly, if g1 , g2 ∈ A(n) are polynomials, with g2 = 0, the rational function gg12 can be considered as a function defined on the principal open subset Ua (g2 ) := An \ Za (g2 ) and with values in K. By contrast, given a projective space P(V ), an element g ∈ S(V ∗ ) cannot be considered as function on P(V ) with values in K, even if g is homogeneous. On the other hand, if we take homogeneous elements G1 , G2 ∈ H(S(V ∗ )) of the same degree and P = [p] ∈ Up (G2 ) := P(V ) \ Zp (G2 ), where p ∈ V , it makes sense to consider G1 (tp) G1 (p) G1 = , (P ) := G2 G2 (p) G2 (tp)

∀t ∈ K \ {0}.

1 Thus, as in the affine case, G G2 can be considered as a function defined on the principal open subset Up (G2 ) and with values in K. We sometimes will drop the indices a and p in Ua (g2 ), Up (G2 ), respectively, when this does not create ambiguity.

Remark 5.2.1. In what follows, the field of fractions Q(A(n) ) will be simply denoted by Q(n) and its elements will be simply called rational functions in the indeterminates x1 , . . . , xn (or even rational functions on An ). Similarly, the field Q(S(V ∗ )) contains the subfield Q(V ∗ ) which ∗ 1 consists of all fractions G G2 such that G1 , G2 ∈ H(S(V )) are of the same degree, where G2 = 0. These will be called degree-zero rational functions. If we introduce in P(V ) homogeneous coordinates [X0 , . . . , Xn ], then Q(V ∗ ) identifies with the subfield Q0 (S(n) ) of Q(n+1) consisting of all fractions G1 G2 where G1 , G2 are homogeneous polynomials of the same degree in the indeterminates X0 , . . . , Xn , with G2 non-zero (recall Definition 1.10.10). (n) This sub-field of Q(n+1) will be simply denoted by Q0 . Let now Y ⊆ P(V ) be a locally closed subset. Let f : Y → K be a function and let P ∈ Y be a point. Then f is said to be regular at the point P if there exist an open neighborhood U ⊆ Y of P and G1 , G2 ∈ H(S(V ∗ )) of the same degree, such that Zp (G2 ) ∩ U = ∅, i.e. U ⊆ Up (G2 ), and such 1 that the restriction of f to U coincides with the restriction of G G2 to U . A function f : Y → K is said to be regular on Y , if it is regular at any point of Y (in the above sense).

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Remark 5.2.2. For f : Y → K being regular at a point P is both a local and open property, i.e. it only depends on a (Zariski) open neighborhood of P and it holds for all points in the (Zariski) open neighborhood of P . If, as customary, we identify An with the affine open set U0 of Pn , a straightforward application of maps in Definition 1.10.16 gives the following. Lemma 5.2.3. Let Y ⊆ An be any locally closed subset and let f : Y → K be a function. Then f is regular at P ∈ Y (in the sense of the above definition) if and only if there exist a (Zariski) open neighborhood U ⊆ Y of P and polynomials g1 , g2 ∈ A(n) , such that Za (g2 ) ∩ U = ∅ and the restriction of f to U coincides with the restriction of gg12 to U . The same strategy also shows that     (n) Q0 = Q0 S(n) ∼ = Q(n) = Q A(n) .

(5.1)

Let therefore Y be any algebraic variety. For any non-empty, open subset U ⊆ Y we denote by OY (U ) the set of regular functions on U . Constant functions on Y are obviously regular functions on any open set U ⊆ Y . If moreover f, g ∈ OY (U ), then the functions f + g : P ∈ U → f (P ) + g(P ),

f · g : P ∈ U → f (P ) · g(P )

(5.2)

are also regular on U . Thus, for any algebraic variety Y and for any nonempty open subset U ⊆ Y , OY (U ) is a K-algebra with respect to the operations (+, ·) as in (5.2), which is called the algebra of regular functions on U . If moreover U, U  ⊆ Y are non-empty open subsets, with U  ⊆ U , one has an obvious restriction map  ρU U  : f ∈ OY (U ) → f |U  ∈ OY (U ),

(5.3)

which is well-defined and which clearly is a K-algebra homomorphism with respect to the operations (5.2). We will sometimes abuse notation and denote ρU U  (f ) simply by f |U  . Since K-algebras OY (U ) and homomorphisms (5.3) satisfy conditions (F1) and (F2) in Definition 5.1.1, we will denote by OY the pre-sheaf of regular functions on Y as the datum of these K-algebras and morphisms (in Remark 5.2.5 below we will see that it is a pre-sheaf of integral Kalgebras and in Proposition 5.3.4 we will more precisely prove that OY is actually a sheaf).

Regular and Rational Functions on Algebraic Varieties

129

For any non-empty open subset U ⊆ Y , if f ∈ OY (U ) is non-zero we will denote by ZU (f ) the zero locus of f in U , i.e. the set ZU (f ) := f −1 (0).

(5.4)

Proposition 5.2.4. Let Y be any algebraic variety. (i) Any f ∈ OY (Y ) is continuous, if f is considered as a function with values in (A1K , Zar1a ). (ii) If f, g ∈ OY (Y ) are such that there exists a non-empty open subset U of Y such that f |U = g|U , then f = g on Y . Proof. (i) One has to show that ZY (f − a) is closed for all a ∈ K. By replacing f with f − a, it suffices to verify that ZY (f ) is closed; this can be done locally. If ZY (f ) = ∅, let P ∈ ZY (f ) be any point. Since f is regular at P , there exists an open neighborhood UP ⊆ Y of P such (n) 1 ) of the same degree, that f |UP = G G2 |UP , for suitable G1 , G2 ∈ H(S c G2 = 0, such that UP ⊆ Up (G2 ) = Zp (G2 ) . Thus, ZY (f ) ∩ UP = Zp (G1 ) ∩ UP , where Zp (G1 ) ⊂ Pn a projective hypersurface. In other words we have shown that, for any P ∈ ZY (f ), there exists an open neighborhood UP of P in Y s.t. ZY (f ) ∩ UP is closed in UP . This implies that ZY (f ) is closed in Y . Indeed, denote by ZY (f ) the closure in Y of ZY (f ); if we had ZY (f )  ZY (f ), any point Q ∈ ZY (f ) \ ZY (f ) would give that, for any open neighborhood V of Q in ZY (f ), V ∩ ZY (f ) = ∅ but Q ∈ / V ∩ ZY (f ). On the other hand, all the open sets of ZY (f ) are induced by opens sets of ZY (f ) thus we would contradict that ZY (f ) ∩ UP is closed in UP , for some P ∈ V ∩ ZY (f ). (ii) Note that ZY (f − g) is closed in Y and contains U , which is dense (see Proposition 4.1.2).  Remark 5.2.5. (a) Proposition 5.2.4-(i) more generally holds for any locally closed subset in P(V ), on the other hand we will not use it. (b) If f ∈ OY (Y ) is non-zero, then ZY (f ) is a proper closed subset of Y . Therefore UY (f ) := Y \ ZY (f ), also denoted by U (f ) for brevity, is a non-empty open subset of Y which is called the principal open subset of

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Y associated to f . Note that, by construction, in U (f ) one can consider the function f1 which is regular therein. (c) A proof similar to that of Proposition 5.2.4-(ii) shows that, for any nonempty open subset U ⊆ Y , the K-algebra OY (U ) is integral. Indeed if 0 = f · g ∈ OY (U ) then ZU (f · g) = U , i.e. U = ZU (f ) ∪ ZU (g). Since U is irreducible, then either U = ZU (f ) or U = ZU (g), i.e. either f = 0 on U or g = 0 on U . (d) If f : Y → K is a function which is regular at a point P ∈ Y , then the value f (P ) depends only on f and P , i.e. it does not depend on the choice of the open neighborhood U = UP of P and on the choice of representative homogeneous polynomials G1 , G2 of the same degree 1 such that f |U = G G 2 |U . (n)

1 (e) For any point P ∈ Y , let ΦP := G G2 ∈ Q0 be a rational function and let UP ⊆ Y be an open neighborhood of P s.t. UP ⊆ Up (G2 ) = Zp (G2 )c . Then ΦP |UP ∈ OY (UP ). For any pair of points P = Q ∈ Y , and for any choice of open neighborhoods UP and UQ , respectively, one has UP ∩ UQ = ∅ since Y is irreducible. Thus, if one has ΦP |UP ∩UQ = ΦQ |UP ∩UQ , the datum   (UP , ΦP |UP ), (UQ , ΦQ |UQ )

defines an element in OY (U ), where U := UP ∪ UQ open set in Y .

5.3

Rational Functions

For Y any algebraic variety, we set H(Y ) := {(U, f ) | U ⊆ Y non-empty, open subset, f ∈ OY (U )} and we define a relation R on H(Y ) in this way: (U, f ) R (U  , f  ) ⇔ f |U∩U  = f  |U∩U  . Since Y is irreducible, R is a well-defined equivalence relation. ) will be denoted by K(Y ) and Definition 5.3.1. The quotient set H(Y R the equivalence class of (U, f ) will be denoted by [U, f ] (sometimes also by Φf ) and called a rational function on Y .

Lemma 5.3.2. K(Y ) is endowed with a structure of a field, which gives a field extension of the ground field K.

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131

Proof. One poses (i) [U, f ] + [V, g] := [U ∩ V, f + g], (ii) [U, f ] · [V, g] := [U ∩ V, f g], (iii) one has an injective map K → K(Y ) defined by λ → [Y, λ], for any λ ∈ K, (iv) if [V, g] is such that g = 0, then [V, g]−1 := [V \ ZV (g), g1 ] = [UV (g), 1g ], as in Remark 5.2.5. It is straightforward to check that the above conditions are well-defined, i.e. they do not depend on the representatives, and moreover that in (ii) one has [U ∩ V, f g] = 0 if and only if [U, f ] and [V, g] are both non-zero (cf. Remark 5.2.5-(c)).  For this reason, K(Y ) is called the field of rational functions on Y . Lemma 5.3.3. For any non-empty, open subset U ⊆ Y, one has an injective homomorphism of integral K-algebras jU

jU : OY (U ) → K(Y ), defined by f −→ [U, f ].

(5.5) 

Proof. Obvious. In particular, one has OY (Y ) ⊆ K(Y ).

(5.6)

Note that the previous inclusion in general is strict: e.g. x1 ∈ K(A1 ) is not / OA1 (A1 ). On the other hand, if we replace A1 with regular on A1 so x1 ∈ the open subset W := A1 \ {0} ⊂ A1 , which is also an algebraic variety, then x1 ∈ OW (W ). Proposition 5.3.4. For any algebraic variety Y, OY is a sheaf (of integral K-algebras) which is called the structural sheaf of Y . Proof. We already know that OY is a pre-sheaf of integral K-algebras on Y . We need to show that condition (F3) in Definition 5.1.2 holds. Let {Ui }i∈I be any family of non-empty, open subsets of Y and correspondingly, consider any collection of regular functions (fi ) ∈ Πi∈I OY (Ui ) s.t. U

j i ρU Ui ∩Uj (fi ) = ρUi ∩Uj (fj ),

∀ i, j ∈ I.

This condition implies jUi (fi ) = jUj (fj ), i.e. there exists a rational function Φf ∈ K(Y ) such that Φf = [Ui , fi ] = [Uj , fj ], On the other hand, since its

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restriction to any Ui is fi ∈ OY (Ui ), for any i ∈ I, then Φf |U = f ∈ OY (U ), where U = ∪i∈I Ui . The fact that f ∈ OY (U ) is uniquely determined follows from the fact that restriction maps ρU Ui are injective, for any i ∈ I; indeed, assume there exist f, g ∈ OY (U ) s.t., for any i ∈ I, one has U ρU Ui (f ) = f |Ui = fi = g|Ui = ρUi (g).

Since f and g are both regular on U and since Ui ⊆ U is open in U , then one concludes by Proposition 5.2.4-(ii).  Definition 5.3.5. For any rational function Φ ∈ K(Y ), we consider the open set UΦ := ∪i Ui , where the union is over all the representatives [Ui , fi ] of Φ in K(Y ). This is called the domain of Φ (or even the open-set of definition of Φ), denoted also by Dom(Φ), and it is the biggest open subset of Y over which Φ is regular. With this set-up, for any open set U ⊆ Y , one simply has OY (U ) = {Φ ∈ K(Y ) | U ⊆ UΦ } . Definition 5.3.6. Let Y be any algebraic variety. Any locally-closed and irreducible subset W ⊆ Y is said to be an (algebraic) subvariety of Y . In other words, W is an algebraic variety on its own which is a locallyclosed subset of Y . Remark 5.3.7. (i) Take any algebraic variety Y ⊆ An and consider its (affine) coordinate ring A(Y ), as in Definition 4.1.7. For any subvariety W ⊆ Y , the ideal Ia (W ) , (5.7) Ia,Y (W ) := Ia (Y ) defined as the image of the ideal Ia (W ) ⊆ A(n) via the canonical quotient morphism A(n)  A(Y ), is a prime ideal of A(Y ) (when otherwise W = Y , one obviously has Ia,Y (W ) = (0)). (ii) Similarly, let Y ⊆ Pn be any algebraic variety and let S(Y ) be its (homogeneous) coordinate ring. For any subvariety W ⊆ Y , the homogeneous ideal Ip (W ) , (5.8) Ip,Y (W ) := Ip (Y ) defined as the image of the homogeneous ideal Ip (W ) ⊆ S(n) via the surjective homomorphism S(n)  S(Y ), is a homogeneous prime ideal of S(Y ).

Regular and Rational Functions on Algebraic Varieties

133

Definition 5.3.8. Let W be a subvariety of an algebraic variety Y . A rational function Φ ∈ K(Y ) is said to be defined in W if there exists a representative (U, f ) of Φ = [U, f ] s.t. U ∩ W = ∅, i.e. Φ is regular in an open, dense subset of W . The set of rational functions defined in W will be denoted by OY,W ; in symbols OY,W := {Φ ∈ K(Y ) | UΦ ∩ W = ∅}. Operations (i)–(iii) as in the proof of Lemma 5.3.2 endow OY,W with an integral K-algebra structure, for any subvariety W ⊆ Y . The following properties are easy consequences of the previous definitions. Proposition 5.3.9. Let Y be an algebraic variety and let W ⊆ Y be a subvariety. Then (i) (ii) (iii) (iv)

OY,Y = K(Y ); OY,W is a subring of the field K(Y ); OY,W contains OY (Y ) as a subring; the set mY,W := {Φ ∈ OY,W | Φ(P ) = 0, ∀ P ∈ UΦ ∩ W } is an ideal of OY,W .

When in particular W = {P } is a point, OY,P is called the ring of germs of regular functions at P and mY,P is the ideal of germs of regular functions vanishing at P . For any point P ∈ Y , one has OY (Y ) ⊆ OY,P ⊆ K(Y ).

(5.9)

Thus, for any non-empty, open set U ⊆ Y , OY (U ) = {Φ ∈ K(Y ) | Φ regular at P , ∀ P ∈ U } =



OY,P ,

(5.10)

P ∈U

whereas K(Y ) = {Φ ∈ K(Y ) | Φ regular at some point P ∈ Y } =



OY,P .

(5.11)

P ∈Y

Recalling definitions as in Section 1.11, we can prove the following result. Theorem 5.3.10. Let Y be an algebraic variety and let W be any subvariety of Y . Then (OY,W , mY,W ) is a local ring of residue field K(W ). Proof. Take any Φ := [U, f ] ∈ OY,W \ mY,W where, up to a change of representative (U, f ) for Φ, we can assume U ∩ W = ∅. Since f ∈ OY (U ), ZU (f ) is a proper closed subset of U (cf. the proof of Proposition 5.2.4). Moreover, since f ∈ / mY,W , then also ZU (f ) ∩ W is a proper, closed subset of U ∩ W . Let U  := UU (f ) = U \ ZU (f ) be the principal open set in

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U associated to f . From Remark 5.2.5-(b), one has f  := moreover

1 f

∈ OY (U  );

U  ∩ W = (U ∩ W ) \ (ZU (f ) ∩ W ) = ∅. Thus, Φ is invertible with Φ−1 := [U  , f  ] ∈ OY,W . From Proposition 1.11.10 we conclude that (OY,W , mY,W ) is a local ring. O be the residue field of (OY,W , mY,W ). Consider Let now KY,W := mY,W Y,W the map ϕY,W :

KY,W −→ K(W ) [U, f ] + mY,W −→ [U ∩ W, f |U∩W ].

(5.12)

This is well-defined and it is easy to check that it is a (non-zero) homomorphism of K-algebras; since KY,W and K(W ) are both fields, ϕY,W is injective. We now prove that ϕY,W is also surjective. Let [U  , f  ] ∈ K(W ) and let P ∈ U  be any point. The fact that f  is regular at P implies there exists an open neighborhood U  of P in U  and two homogeneous polynomials G1 , G2 ∈ H(S(V ∗ )) of the same degree, with G2 = 0, such that f  |U  =

G1 |U  G2

(5.13)

and U  ⊆ Up (G2 ) (recall definitions as in Section 5.2). One can always find  ∩ Zp (H) = ∅ (otherwise  ⊂ P(V ) s.t. U  ∩ W = U  and U an open subset U  ∩ Y . By the choice   one can always replace U with U ∩ Up (H)). Let U := U  of U , one has U ⊆ Up (H). Define f :=

G1 |U ∈ OY (U ) ⊆ K(Y ). G2

(5.14)

 ∩ W = U  ⊂ W , then [U, f ] ∈ OY,W . From (5.13) and Since U ∩ W = U (5.14) it follows that [U  , f |U  ] R [U  , f  ], i.e. ϕY,W ([U, f ] + mY,W ) = [U  , f  ], proving the surjectivity of ϕY,W .



Definition 5.3.11. For any algebraic variety Y and any subvariety W ⊆ Y , the ring (OY,W , mY,W ) is called the local ring of W in Y .

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For any open subset U ⊆ Y s.t. U ∩ W = ∅ the map jU as in (5.5) induces an injective K-algebra homomorphism γU

γU : OY (U ) → OY,W , defined by f −→ [U, f ].

(5.15)

Definition 5.3.12. The ideal IU (W ) := γU−1 (mY,W ) = {f ∈ OY (U ) | ZU (f ) ⊇ U ∩ W } will be called the ideal of W in U . Lemma 5.3.13. Let Y be any algebraic variety and let U ⊆ Y be any non-empty, open subset. (i) For any subvariety W ⊆ Y s.t. W  := U ∩ W = ∅, one has OU,W  ∼ = OY,W . (ii) In particular, K(U ) ∼ = K(Y ). Proof. It is clear that (ii) is a particular case of (i), taking W = Y . Therefore, we only need to prove (i). Let [U  , f  ] ∈ OU,W  , where U  ⊆ U an open set s.t. U  ∩ W  = U  ∩ U ∩ W = ∅. Since U  is open in Y , f  ∈ OY (U  ) and U  ∩ W = ∅, then [U  , f  ] ∈ OY,W . Thus, one has an injective K-algebra homomorphism ψ : OU,W  → OY,W , defined by [U  , f  ] → [U  , f  ]. Note that ψ is also surjective; indeed for any [U  , f  ] ∈ OY,W , where the representative (U  , f  ) is s.t. U  ∩ W = ∅, one has [U  , f  ] = ψ ([U  ∩ U, f  |U  ∩U ]) since U  ∩ U ∩ W = ∅.



Recalling notation as in Remarks 1.11.11, 5.3.7, we can prove the following fundamental result. Theorem 5.3.14 (Fundamental theorem on regular and rational functions). Let Y ⊆ An be any affine variety. Then: (a) OY (Y ) = A(Y ). (b) For any subvariety W ⊆ Y , the ideal IY (W ) (cf. Definition 5.3.12) is a prime ideal in OY (Y ) which is isomorphic to the prime ideal Ia,Y (W ) of A(Y ) as in (5.7). Conversely any prime ideal of OY (Y ) is of this type; furthermore, IY (W ) is a maximal ideal if and only if W is a point P ∈Y.

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(c) For any subvariety W ⊆ Y one has OY,W ∼ = OY (Y )IY (W ) ∼ = A(Y )Ia,Y (W ) . (d) K(Y ) = Q(A(Y )) whereas, for any subvariety W ⊆ Y one has K(W ) ∼ =

OY (W )IY (W ) IY (W )OY (W )IY (W )

∼ =

A(Y )Ia,Y (W ) . Ia,Y (W )A(Y )Ia,Y (W )

Let Y ⊆ Pn be any projective variety. Then (e) OY (Y ) = K. (f) For any subvariety W ⊆ Y, the ideal Ip,Y (W ) as in (5.8), is a homogeneous, prime ideal of S(Y ). Conversely any homogeneous, nonirrelevant, prime ideal of S(Y ) is of this type. (g) For any subvariety W ⊆ Y, one has OY,W ∼ = S(Y )(Ip,Y (W )) . (h) K(Y ) = S(Y )((0)) whereas, for any subvariety W ⊆ Y, K(W ) ∼ =

S(Y )(Ip,Y (W )) . Ip,Y (W )S(Y )(Ip,Y (W ))

Proof. Let us first focus on the affine case. (a) By definition of A(Y ), there is a natural injective K-algebra homomorphism α : A(Y ) → OY (Y ). From the Hilbert “Nullstellensatz”-weak form in An (cf. Theorem 2.2.1) and (5.7), there is a bijective correspondence between points P ∈ Y and maximal ideals mY (P ) := Ia,Y (P ) of A(Y ). Identifying A(Y ) with its image α(A(Y )), we can therefore interpret mY (P ) = {f ∈ A(Y ) ⊆ OY (Y ) | f (P ) = 0} ⊂ OY (Y ), i.e. mY (P ) ⊆ IY (P ), where the latter ideal is as in Definition 5.3.12. Claim 5.3.15. For any P ∈ Y , one has A(Y )mY (P ) ∼ = OY,P . Proof. For any fg ∈ A(Y )mY (P ) , the set UY (g) := Y \ZY (g) is a non-empty, open subset of Y containing P , indeed g ∈ A(Y ) \ mY (P ) by definition of A(Y )mY (P ) so fg is regular over UY (g) (cf. the proof of Proposition 5.2.4-(i) and Remark 5.2.5-(b)). We can therefore consider the following map:  f αP f UY (g), αP : A(Y )mY (P ) −→ OY,P , defined by −→ . (5.16) g g This is a K-algebra homomorphism. Note moreover that αP is an isomor



 f  f phism: first UY (g), g = UY (g ), g if and only if on UY (g) ∩ UY (g  ) = ∅

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one has f g  −f  g = 0. Since UY (g)∩UY (g  ) is an open, dense subset of Y , by Proposition 5.2.4-(ii), one has f g  −f  g = 0 on Y , i.e. f g  −f  g = 0 ∈ A(Y ).  Since g, g  ∈ A(Y ) \ mY (P ), from (1.35) we get that fg = fg ∈ A(Y )mY (P ) , so αP is injective. At last, the surjectivity of αP directly follows from Lemma 5.2.3.  By construction, αP is a local isomorphism of local rings, i.e. the maximal ideals mY (P )A(Y )mY (P ) and mY,P = IY (P ) bijectively correspond under αP . In particular, this proves points (b), (c) and (d) for the case W = {P }. From (5.10) and Claim 5.3.15 we get   OY,P ∼ A(Y )mY (P ) , A(Y ) ∼ = = α(A(Y )) ⊆ OY (Y ) = P ∈Y

mY (P )∈Specm(A(Y ))

where Specm(A(Y )) := {maximal ideals of A(Y )}.

(5.17)

Since A(Y ) is integral, any A(Y )mY (P ) is a subring of Q(A(Y )) and, from the definition of localization (cf. also Matsumura, 1980, Lemma 2, p. 8) one has ∩mY (P )∈Specm(A(Y )) A(Y )mY (P ) = A(Y ), proving (a). (b) Since Ia (W ) = Ia (W ), we can consider W a closed subvariety. By definition of A(Y ), we have a bijective correspondence between prime ideals p ⊆ A(Y ) and prime ideals p ⊂ A(n) containing Ia (Y ). From Proposition 2.1.11-(i) and (2.15), any such prime ideal p corresponds to a closed subvariety W := Za (p) ⊆ Za (Ia (Y )) = Y ⊆ An . Since p is prime, it is radical so p = Ia (W ) and p = Ia,Y (W ). Using (a), the prime ideal Ia,Y (W ) by definition coincides with the ideal IY (W ) ⊂ OY (Y ). At last, from the proof of part (a), Ia,Y (W ) ∼ = IY (W ) is maximal if and only if W is a point. (c) On the one hand, for any subvariety W ⊆ Y , the ring OY,W is local, with maximal ideal mY,W (cf. Theorem 5.3.10). On the other hand, from part (a) above, we have A(Y ) ∼ = IY (W ). Thus, = OY (Y ) and Ia,Y (W ) ∼ one has that A(Y )Ia,Y (W ) ∼ = OY (Y )IY (W ) is a local ring of maximal ideal Ia,Y (W )A(Y )Ia,Y (W ) ∼ = IY (W )OY (Y )IY (W ) (cf. Proposition 1.11.12). Since A(Y )Ia,Y (W ) ⊆ K(Y ), as in the proof of Claim 5.3.15, one constructs a map αW : A(Y )Ia,Y (W ) −→ OY,W which is a local isomorphism of local K-algebras. (d) From Proposition 5.3.9-(i) and from (c), we get K(Y ) = OY,Y ∼ = A(Y )(0) = Q(A(Y )), the last equality following from Remark 1.11.11.

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Similarly, from Theorem 5.3.10, Proposition 1.11.12-(i) and from (c) above, one gets the assertions on K(W ). We now consider the case of Y ⊆ Pn a projective variety. Recalling notation as in (1.40) and (3.14), a preliminary step is given by the following result. Lemma 5.3.16. Let i ∈ {0, . . . , n} be any integer for which Yi := Y ∩ Ui = ∅. Then A(Yi ) ∼ = S(Y )([Xi ]) , where Xi ∈ S(Y ) denotes the image via the quotient morphism S(n)  S(Y ) of the indeterminate Xi ∈ S(n) . Proof. Assuming that the affine variety Y0 = ∅, for simplicity we consider only the case i = 0, the other cases being similar. One has a natural K-algebra isomorphism

X1 Xn ϕ0 ϕ0 (n) ,..., A(n) −→ S([X0 ]) , defined by f (x1 , . . . , xn ) −→ f . X0 X0 If f ∈ Ia (Y0 ), by (1.32) and the definition of Ip (Y ), we get that h0 (f ) = deg(f ) (n) (n) ϕ0 (f ) ∈ Ip (Y ), i.e. ϕ0 (f ) ∈ Ip (Y )S([X0 ]) . Note that Ip (Y )S([X0 ]) is a X0 (n)

proper ideal of S([X0 ]) : indeed if we denote by S the multiplicative system given by 1 and by all the powers of the indeterminate X0 , by the fact that Y0 = ∅ it follows that Ip (Y ) ∩ S = ∅, so one concludes by Proposition 1.11.6-(ii). In particular, ϕ0 induces a homomorphism of K-algebras (n)

ϕ0 : A(Y ) −→

S([X0 ]) (n)

Ip (Y )S([X0 ])

∼ = S(Y )([X0 ]) ,

where the isomorphism on the right directly follows from the fact that (n) S(Y ) = ISp (Y ) and from the definition of localization with respect to the multiplicative system S = {1, X0 , X02 , X03 , . . .}. Note that ϕ0 is an isomorphism: it is injective since ϕ0 (f ) = 0 if and only if h0 (f ) = 0, i.e. if and only if f = 0; the surjectivity follows from the fact that, for any positive integer n and for any F ∈ S(Y )n , there always exists a polynomial  f ∈ A(Y ) such that XFn = ϕ0 (f ). 0

We can proceed with the proof of the “projective” part of the statement. (g) Let W ⊆ Y be any (closed) subvariety be any integer for which Yi = ∅ and Wi simplicity, assume that it occurs for, e.g. i = (i), we have OY,W ∼ = OY0 ,W0 . Since Y0 and W0

and let i ∈ {0, . . . , n} := W ∩ Ui = ∅. For 0. From Lemma 5.3.13are affine varieties, from

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(c) above we have OY0 ,W0 ∼ = OY0 (Y0 )IY0 (W0 ) . Moreover, from (a), one has also OY0 (Y0 ) ∼ = A(Y0 ) and IY0 (W0 ) ∼ = Ia,Y0 (W0 ). From the proof of Lemma 5.3.16, the isomorphism ϕ0 bijectively maps the ideal Ia,Y0 (W0 ) ∼ ∼ to  Ip,Y (W )S(Y )([X0 ]) so we get OY,W = A(Y0 )Ia,Y0 (W0 ) =  the ideal . Since X0 ∈ / Ip,Y (W ) (because W0 = ∅ S(Y )([X0 ]) I (W )S(Y ) p,Y

([X0 ])

by reasoning as in Lemma 5.2.3, one easily notices that   assumption), ∼ S(Y )([X0 ]) Ip,Y (W )S(Y ) = S(Y )(Ip,Y (W )) , proving (g). ([X ]) 0

(h) The statement about K(Y ) follows from Proposition 5.3.9-(i) and from (g) above, whereas statement about K(W ) follows from Theorem 5.3.10, Proposition 1.11.12-(ii) and from (g) above. (f) As in the affine case, it directly follows from bijective correspondence as in Corollary 4.1.12 for Pn . (e) Let f ∈ OY (Y ) be any non-zero element. Since for any algebraic variety we have OY (Y ) ⊆ K(Y ), from (h) above any such f can be considered as an element of K(Y ) ∼ = S(Y )((0)) = Q0 (S(Y )) ⊂ Q(S(Y )), where Q0 (S(Y )) as in Definition 1.10.10. Notice first that an index j ∈ {0, . . . , n} is such that Yj = ∅ if and only if Y ⊆ Zp (Xj ) = Hj , where Hj ∼ = Pn−1 the hyperplane at infinity of the affine open set Uj ; this occurs if and only if (n) S(n−1) (n−1) ∼ S(n) Xj ∈ Ip (Y ), in which case S(Y ) = ISp (Y ) ∼ = Ip,H = (Xj ) (Y ) , where S j

and Ip,Hj (Y ) =

Ip (Y ) (Xj ) .

In other words, for any such j, we can replace S(n)

with S(n−1) , i.e. Y ⊂ Pn is degenerate as it is contained into the hyperplane Hj ∼ = Pn−1 and one can work directly therein. From now on we will therefore assume Yi = ∅, for any 0  i  n. Since f ∈ OY (Y ) then f |Yi ∈ OYi (Yi ), for any i. Moreover, as Yi is affine, from (a) and Lemma 5.3.16, f |Yi ∈ OYi (Yi ) ∼ = A(Yi ) ∼ = S(Y )([Xi ]) . Thus, for any i ∈ {0, . . . , n}, there exist a positive integer Ni and a polynomial Gi ∈ S(Y )Ni such that f |Yi := GNii , i.e. XiNi f |Yi ∈ S(Y )Ni , Xi for any 0  i  n. Since f ∈ OY (Y ) ⊂ K(Y ) ∼ = S(Y )((0)) =

Q0 (S(Y )) ⊂ Q(S(Y )) and f |Yi ∈ S(Y )([Xi ]) ⊂ Q0 (S(Y )) ⊂ Q(S(Y )), for any 0  i  n, considering f as an element of Q(S(Y )), we have XiNi f ∈ S(Y )Ni , ∀ i ∈ {0, . . . , n}.

(5.18)

Let N be any positive integer such that N  Σni=1 Ni . Since S(Y )N is generated as a K-vector space by monomials X0α0 X1α1 · · · Xnαn such that

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Σni=0 αi = N , from (5.18), we get S(Y )N f ⊆ S(Y )N .

(5.19)

Recursively applying (5.19), for any positive integer q one has S(Y )N f q ⊆ S(Y )N . In particular, X0N f q ∈ S(Y )N , for any integer q  1, i.e. S(Y )[f ] ⊆ S(Y ) ·

1 . X0N

(5.20)

Since S(Y ) is a noetherian ring and since S(Y ) · X1N is a finitely generated 0 S(Y )-module, then S(Y )[f ] is a finitely generated S(Y )-module (cf. Atiyah and McDonald, 1969, Propositions 6.2, 6.5). From Proposition 1.6.3, it follows that f ∈ Q0 (S(Y )) is therefore integral over S(Y ), i.e. there exist a1 , . . . , am ∈ S(Y ) s.t. f m + a1 f m−1 + · · · + am = 0.

(5.21)

Since f ∈ Q0 (S(Y )) there exist homogeneous polynomials 1 G1 , G2 ∈ H(S(Y )) of the same degree such that f = G G2 , so (5.21) becomes m−1 m m G1 + a1 G1 G2 + · · · + am G2 = 0. Let aj = Aj0 + Aj1 + · · · + Ajj be the decomposition of the coefficients aj ∈ S(Y ) in their homogeneous components, 1  j  n. Decomposing the previous equality into homogeneous pieces we get in particular Gm 1 + m G + · · · + A G = 0, i.e. A10 Gm−1 2 m0 2 1 f m + A10 f m−1 + · · · + Am0 = 0,

(5.22)

where Aj0 ∈ S(Y )0 = K, for any 1  j  m. In particular, f ∈ K(Y ) is algebraic over K. Since K is algebraically closed, f ∈ K proving (e).  From Theorem 5.3.14-(a) and (e) it follows that for Y either an affine or a projective variety, OY (Y ) is an integral K-algebra of finite type. This has already been observed to more generally hold for any algebraic variety Y (cf. Remark 5.2.5-(c)). Corollary 5.3.17. Let Y ⊂ Pn be any projective variety and let S(Y ) be its homogeneous coordinate ring. Let Q(S(Y )) be the quotient field of the integral K-algebra S(Y ). Let x0 , . . . , xn be the images via the canonical epimorphism π : S(n)  S(Y ) of the homogeneous indeterminates X0 , X1 , . . . , Xn ∈ S(n) . Then, for any 0  j  n, one has: (i) Q(S(Y )) ∼ = K(Y )(xj ); (ii) xj is transcendental over K(Y ).

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Proof. It suffices to prove the statements for j = 0. Let us consider any non-zero element fg ∈ Q(S(Y )); thus one can write f = F0 + F1 + · · · + Fh and g = G0 + G1 + · · · + Gk , where each Fi , respectively, Gj , is either zero or homogeneous of degree i, respectively, j, for 1  i  h, 0  j  k. Therefore,   h x1 xn i i=0 Fi 1, x0 , . . . , x0 x0 F0 + F1 + · · · + Fh f   . = =  k j x1 xn g G0 + G1 + · · · + Gk j=0 Gj 1, x0 , . . . , x0 x0 Thus, fg ∈ S(Y )((0)) (x0 ). Since Y is projective, from Theorem 5.3.14-(h), we know that K(Y ) ∼ ) so (i) is proved. = S(Y  ((0))   x1 xn s Assume now = 0, where fs ∈ K(Y ) s fs x0 , . . . , x0 x0 = deg(f ). If we set d := max with d s (ds ) and Fs (x0 , . . . , xn ) :=  s  s fs

x1 xn x0 , . . . , x0

0=

 s

=

x−d 0

xd0s , then we get

s Fs (x0 , . . . , xn )xs−d = x−d 0 0





s Fs (x0 , . . . , xn )xd+s−d 0

s

Gs (x0 , . . . , xn )xs0 ,

s s with Gs = xd−d Fs ∈ S(Y )d , i.e homogeneous of degree d, for any s. 0 Since S(Y ) is an integral domain, the previous equality imposes Σs Gs (x0 , . . . , xn )xs0 = 0. Moreover, since S(Y ) is a graded ring, it then follows that Gs (x0 , . . . , xn ) = 0 for any s. This implies that fs = 0 for any  s, i.e. that x0 is transcendental over K(Y ).

5.3.1

Consequences of the fundamental theorem on regular and rational functions

We want to discuss some fundamental consequences of Theorem 5.3.14. Corollary 5.3.18. For any integer n  0, one has OAn (An ) = A(n) , OPn (Pn ) = K, and K(An ) ∼ = K(Pn ) ∼ = Q(n) . Proof. The first two equality are given by (a) and (e) of Theorem 5.3.14. The fact that K(An ) ∼ = Q(n) is point (d) of the same result. The fact that n ∼ (n) follows either from Lemma 5.3.13-(ii) or from point (h) also K(P ) = Q of Theorem 5.3.14, Proposition 1.11.12-(ii) and isomorphism (5.1), namely (n)  one has K(Pn ) ∼ = S((0)) ∼ = Q(A(n) ) = Q(n) . = Q0 (S(n) ) ∼

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Corollary 5.3.19 (cf. also Mumford, 1988, Proposition 2, pp. 20–21). If Y is any affine variety, the ring A(Y ) completely determines the ringed space (Y, OY ). In other words, A(Y ) determines the topological space Y since: • as a set of points one has Y = Specm(A(Y )), where Specm(A(Y )) as in (5.17), whereas, • all irreducible closed subsets of Y are in bijective correspondence with elements in Spec(A(Y )), where Spec(A(Y )) defined in (1.43), moreover, • all closed subsets of Y are unions of finitely many irreducible closed subsets, • finally, for any open set U ⊆ Y and for any point P ∈ Y, all integral K-algebras OY (U ) and OY,P giving rise to the structural sheaf of Y are suitable localizations of the affine coordinate ring A(Y ). Proof. From the bijective correspondence of Theorem 5.3.14-(b) between points P ∈ Y and maximal ideals of A(Y ), as a set of points Y can be identified with Specm(A(Y )). In this identification, the correspondence between prime ideals p of A(Y ) and irreducible closed subvarieties ZY (p) ⊆ Y in Theorem 5.3.14-(b), reads as ZY (p) = {m ∈ Specm(A(Y )) | p ⊆ m}.

(5.23)

Since Y is a noetherian topological space, any arbitrary closed subset Z is a finite union of its irreducible components, say Z1 , . . . , Zk . From i) Proposition 2.2.8 applied to ideals Ia,Y (Zi ) = IIaa(Z (Y ) of A(Y ), we get k Ia,Y (Z) = ∩i=1 Ia,Y (Zi ), so identification (5.23) can be extended to any radical ideal of A(Y ). This implies that A(Y ) recover the whole topology of Y , since it determines the family of all its closed subsets. Finally, from OY,P ∼ = A(Y )mY,P and from (5.10), where any OY (U ) and any OY,P is viewed as a subring of K(Y ) as in (5.11), one realizes that A(Y ) determines  also the structural sheaf OY . Corollary 5.3.20. Let Y be any algebraic variety, and let U ⊂ Y be any non-empty, open subset. Then OU ∼ = OY |U (cf. Remark 5.1.4). In particular, if Y is any affine variety and if Y denotes its projective closure, then for any P ∈ Y one has OY,P ∼ = OY ,P and mY,P ∼ = mY ,P . Proof. From Definition 5.3.5 and (5.10) one has that, for any non-empty, open set V ⊆ U ⊂ Y , OU (V ) = {Φ ∈ K(U ) | V ⊂ UΦ } and OY (V ) = {Φ ∈  K(Y ) | V ⊂ UΦ }. Then, one concludes by Lemma 5.3.13-(ii).

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Recalling Definition 1.5.5, one has also the following fundamental result. Corollary 5.3.21. Let Y be any algebraic variety. Then K(Y ) is a finitely generated field extension of the base field K. Proof. From Lemma 5.3.13-(ii), we can assume Y to be projective. Thus, from Theorem 5.3.14-(h), K(Y ) ∼ = S(Y )((0)) = Q0 (S(Y )). Since S(Y ) is a quotient of S(n) , it is a K-algebra of finite type generated by the images of the indeterminates X0 , . . . , Xn via the quotient morphism. Thus, Q(S(Y )) is a finitely generated field extension of K. Since Q0 (Y ) is the subfield of degree-zero fractions in Q(S(Y )), one can conclude.  5.3.2

Examples

We discuss here some applications of the previous results. Example 5.3.22 (Points). (i) If Y = {P } is a point, from the first part of Theorem 5.3.14 we get that OP (P ) ∼ = K(P ) ∼ = K. More generally, if Y is an algebraic variety and P ∈ Y is a point then K(P ) ∼ = K, i.e. the field of rational functions of a point is of intrinsic nature for P and it is independent from the inclusion of P as a subvariety of Y . To prove this, with no loss of generality we may assume Y to be affine. Indeed, if Y is any quasi-projective variety, consider Y ⊆ Pn its projective closure and, for any index i ∈ {0, . . . , n} s.t. Y ∩ Ui = ∅, take the affine variety Yi := Y ∩ Ui ; from Lemma 5.3.13-(ii), K(Y ) = K(Y ) = K(Yi ) so we can replace Y with Yi . Thus, since Y can be considered affine, from Theorem 5.3.14-(c), OY,P ∼ = A(Y )mY,P where mY,P denotes both the ideal of germs of regular P functions vanishing at P and the ideal Iam(Y ) , via identification in Theorem 5.3.14-(a). From Theorem 5.3.14-(d) and Proposition 1.11.12  A(Y )mY,P A(Y ) ∼ ∼ ∼ (i), we therefore get K(P ) = = K since = Q mY,P A(Y )mY,P

mY,P

∼ = K. (ii) If Y ⊂ Pn is any projective variety and if P ∈ Y is any point, from Theorem 5.3.14-(f), one has that Ip,Y (P ) is a homogeneous prime ideal of S(Y ) which does not coincide with the irrelevant ideal of S(Y ); in particular it is not maximal (as it occurs also for Ip (P ) ⊂ S(n) , cf. Section 3.3.1). The ideal Ip,Y (P ) is the prime ideal generated by all homogeneous elements F ∈ H(S(Y )) vanishing at P . A(Y ) mY,P

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Example 5.3.23 (Irreducible conics). (i) Let Y := Za (x1 x2 − 1) ⊂ A2 , which we call a hyperbola (as in the real case). Since x1 x2 − 1 ∈ A(2) is irreducible, Ia (Y ) ⊂ A(2) is a prime ideal. (1) The (affine) coordinate ring A(Y ) is s.t. A(Y ) ∼ = K[x1 ][x−1 1 ] = Ax1 (1) (cf. notation as in (1.39)). From Theorem 5.3.14, OY (Y ) ∼ = Ax1 . 1 (1) is identified with a (proper) subring In particular OA1 (A ) ∼ = A of A(Y ). On the other hand Theorem 5.3.14 ensures also K(A1 ) ∼ = K(Y ) ∼ = Q(1) . Geometric motivations about these function rings are suggested in Exercise 5.1 and discussed in full details in its solution. (ii) Let V := Za (x2 − x21 ) ⊂ A2 be a parabola. This conic has “more similarity” with A1 than what the hyperbola above does. Indeed A(V ) ∼ = = K[x1 ] = A(1) = A(A1 ) and, from Theorem 5.3.14, K(V ) ∼ K(x1 ) ∼ = Q(1) ∼ = K(A1 ), the latter as occurred for the hyperbola. For geometric motivations, see Exercise 5.2 and its solution. (iii) At last, consider Z := Za (x21 + x22 − 1) ⊂ A2 , with e.g. K = C, and A2 considered as the complexification of a real affine plane. Thus, classification of conics is up to affine transformations with coefficients from the subfield R; then Z is a non-degenerate ellipse, since its points at infinity are the cyclic points [0, 1, i] and [0, 1 − i]. Similar computations as in Remark 2.1.16-(iii) show that one has a rational parametrization A1 ⊃

φ3

W −→ Z ⊂ A2 2 2 +1 t −1 t → ( t 2it , 2t ) (x1 , x2 ) , x2 − ix1 ←

where i2 = −1 and W := A1 \ {0}. The map φ−1 3 is given by the pencil of (parallel) lines in A2 , t : x2 − ix1 − t = 0, t ∈ W . This is actually given by the pencil of (projective) lines through one of the cyclic points, (1) namely from [0, i, 1]. One then computes that A(Z) ∼ = A(W ) = Ax1 ∼ 1 ∼ (1) and K(Z) = K(W ) = K(A ) = Q , as it occurs for the hyperbola above. Example 5.3.24 (Semi-cubic parabola). Let Y := Za (x31 − x22 ) ⊂ A2 , which is called a semi-cubic parabola, or even a cuspidal plane cubic (cf. Figure 5.1). In this case, Y has a polynomial parametrization as in (3.31) given by: φ

⊂ A2 A1 −→ Y 2 3 t → (t , t ) , and A(Y ) ∼ = K[t2 , t2 ] ⊂ K[t].

Regular and Rational Functions on Algebraic Varieties

Fig. 5.1

145

Semi-cubic parabola or cuspidal plane cubic (real part).

In particular, A(Y ) is not isomorphic to A(A1 ); on the other hand, from Theorem 5.3.14 one has K(Y ) ∼ = K(t) ∼ = K(A1 ). The pencil of lines 1 ∼ x2 = t x1 , t ∈ A = K, through the origin gives a bijective correspondence between Y and A1 by the following rule: ψ

Y −→ A1 (0, 0) = (x1 , x2 ) →

x2 x1

(0, 0) = (x1 , x2 ) → 0. The map ψ is bijective so it is a homeomorphism between (Y, Zar2a,Y ) and (A1 , Zar1a ); moreover ψ −1 = φ, i.e. φ is a homeomorphism. In Example 6.2.5-(v), we will show that φ, on the other hand, is not an isomorphism of algebraic varieties. Corollary 5.3.21 ensures that, for any algebraic variety Y , the field extension K ⊆ K(Y ) is finitely generated. Note that all fields of rational functions appearing in Examples 5.3.23 and 5.3.24 are more precisely isomorphic to K(t), where t an indeterminate, so they all are purely transcendental extensions of K with transcendence degree 1 (cf. (1.15)). The following example shows that this is not always the case, i.e. not all algebraic varieties have fields of rational functions which are purely transcendental extension of K.

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Example 5.3.25 (Some elliptic affine plane cubics). Let K = C and Ya := Za (x22 − x1 (x1 − 1)(x1 − a)) ⊂ A2 ,

a ∈ C.

When a is either 0 or 1, Ya has a node (i.e. an ordinary double point) at the origin and for this reason it has a rational parametrization. To see this consider for simplicity the case a = 0, the other case being similar. In such a case, Y0 is an affine rational cubic with polynomial parametrization φ

φ

A1 −→ Y0 , defined by t −→ (t2 + 1, t3 + t). Note that the map φ is injective only on W := A1 \ {i, −i}, i2 = −1, whereas φ(i) = φ(−i) = (0, 0). The non-injectivity of φ produces the nodal singularity of Y0 at the origin. Conversely, the pencil of lines x2 = tx1 , t ∈ C, through the origin gives bijective correspondence between Y0 \ {(0, 0)} and W . Using Theorem 5.3.14 one can compute that K(Y0 ) ∼ = K(t), as we expected from the existence of the polynomial parametrization φ and the fact that φ|W will be an isomorphism of algebraic varieties (cf. Chapter 6). To sum up, in terms of fields of rational functions, Y0 and Y1 behave exactly as the previous examples of conics and cuspidal plane cubics. Let us consider therefore the case a = 0, 1. Since Ya is affine, from Theorem 5.3.14-(a) we have that OYa (Ya ) ∼ = A(Ya ), where   A(Ya ) ∼ = f + x2 g | f, g ∈ A(1) and x22 = x1 (x1 − 1)(x1 − a) , where A(1) = C[x1 ]. From Theorem 5.3.14-(d), we get   K(Ya ) ∼ = φ + x2 ψ | φ, ψ ∈ Q(1) and x22 = x1 (x1 − 1)(x1 − a) , 2 a1 where Q(1) = C(x1 ); indeed, any element of Q(A(Ya )) is of the form ab00 +x +x2 b1 , with a0 , a1 , b0 , b1 ∈ A(1) and x22 = x1 (x1 −1)(x1 −a), which can be therefore b0 −x2 b1 (1) 2 a1 , as written as ab00 +x +x2 b1 b0 −x2 b1 = φ(x1 ) + x2 ψ(x2 ), for some φ, ψ ∈ Q claimed. Thus, x2 ∈ K(Ya ) is a root of the polynomial Px2 (t) := t2 − x1 (x1 − 1)(x1 − a) ∈ Q(1) [t], where t is an indeterminate over Q(1) . A key remark is now the following.

Lemma 5.3.26 (cf. also Reid, 1988, § 2.2). Let K be a field of characteristic different from 2 and let λ ∈ K, with λ = 0, 1. Let t be an indeterminate and Φ, Ψ ∈ K(t) rational functions s.t. Φ2 = Ψ(Ψ−1)(Ψ−λ). Then Φ, Ψ ∈ K.

Regular and Rational Functions on Algebraic Varieties

Fig. 5.2

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Complex Torus or elliptic plane cubic.

The proof of this result is of pure algebraic nature and uses the fact that K[t] is a UFD and Fermat’s method of infinite descent. The interested reader is referred to Reid (1988, Chapter 2, § 2.2). In our set-up, Lemma 5.3.26 ensures it cannot exist a non-constant map from a non-empty open set U ⊆ A1 to Ya given by rational functions in the coordinate t ∈ U (i.e. for a = 0, 1, Ya admits no rational parametrization). Note moreover that it also implies that the polynomial Px2 (t) is irreducible over Q(1) (otherwise, it would admit a root in Q(1) = C(x1 ) contradicting Lemma 5.3.26). Thus, for a = 0, 1, the rational function x2 ∈ K(Ya ) is algebraic over Q(1) (but (1) ∼ Q [t] is an algebraic extension of degree 2 of not in Q(1) ) and K(Ya ) = (Px2 (t))

Q(1) , i.e. K(Ya ) is a finitely generated field extension of C of transcendental degree 1 over C, but not purely transcendental over C. Identifying the complex affine plane with the real affine four-dimensional space, in the Euclidean topology any Ya with a = 0, 1 has a form as in Figure 5.2. Exercises Exercise 5.1. Let K be an algebraically closed field. Let Y := Za (x1 x2 − 1) ⊂ A2 be the hyperbola as in Example 5.3.23-(i). Consider the quasi-affine variety W := A1 \ {0}. Show that the map φ1

W −→ Y, defined as φ1

t −→



1 t, t

is a homeomorphism of algebraic varieties. Compare the ring OW (W ) and the field K(W ) with OY (Y ) and K(Y ). Show that the map φ1 is the

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restriction of a homeomorphism Φ1 : P 1 → Y , where Y denotes the projective closure in P2 of Y , when A2 is identified with the affine chart U0 of P2 . Exercise 5.2. Let K be an algebraically closed field. Let V := Za (x2 − x21 ) ⊂ A2 be the parabola as in Example 5.3.23-(ii). Taking the map φ2

A1 −→ V, defined as  φ2  t −→ t, t2 , answer to the same questions as those in Exercise 5.1. Exercise 5.3. Let K be an algebraically closed field. Let d  2 be any integer and let V := {(t, t2 , . . . , td ) | t ∈ A1 } ⊂ Ad . Show that V is an affine variety in Ad . Determine OV (V ), K(V ) and trdegK (K(V )). Exercise 5.4. Let K be an algebraically closed field. Consider the affine variety V := Za (x1 x4 − x2 x3 ) ⊂ A4 and let f := xx12 ∈ K(V ). Describe Dom(f ) as defined in Definition 5.3.5 (cf. Reid, 1988, Ex. 4.9, p. 83). Exercise 5.5. Let K be an algebraically closed field. Consider Z := Zp (X12 − X0 X2 ) ⊂ P3 . Prove that Z is a projective variety and describe K(Z).

Chapter 6

Morphisms of Algebraic Varieties

In this chapter, we will study suitable maps which preserve the structure of algebraic varieties. 6.1

Morphisms

Let V and W be algebraic varieties. Definition 6.1.1. A map ϕ : V → W is said to be a morphism of algebraic varieties, or simply a morphism, if: (i) ϕ is a continuous map between the topological spaces V and W , and (ii) for any non-empty open set U ⊆ W and for any f ∈ OW (U ), one has fϕ := f ◦ ϕ ∈ OV (ϕ−1 (U )). The set of all morphisms from V to W will be denoted by Morph(V, W ). For any algebraic variety V , one has IdV ∈ Morph(V, V ). Moreover, if V, W Z are algebraic varieties and if ϕ ∈ Morph(V, W ) and ψ ∈ Morph(W, Z), then ψ ◦ ϕ ∈ Morph(V, Z). Definition 6.1.2. ϕ ∈ Morph(V, W ) is said to be an isomorphism if there exists ψ ∈ Morph(W, V ) such that ϕ ◦ ψ = IdW and ψ ◦ ϕ = IdV . If such a ψ exists, it is uniquely determined and will be denoted by ϕ−1 . The set of all isomorphisms from V to W will be denoted by Isom(V, W ). If Isom(V, W ) = ∅, then V and W are said to be isomorphic algebraic varieties, which will be denoted by V ∼ = W . The set Isom(V, V ) will be denoted by Aut(V ) and called the set of automorphisms of V .

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Remark 6.1.3. (i) For any ϕ ∈ Morph(V, W ) and for any non-empty open subset U ⊆ W , from the previous definitions one has a natural map ϕU : OW (U ) −→ OV (ϕ−1 (U )) f

−→ fϕ := f ◦ ϕ,

(6.1)

which is a homomorphism of integral K-algebras. (ii) When U = W , the morphism ϕW will be simply denoted by ϕ# : OW (W ) −→ OV (V ).

(6.2)

(iii) If ϕ ∈ Isom(V, W ) then, for any non-empty open subset U ⊆ W , ϕU −1 is an isomorphism of K-algebras; indeed (ϕU )−1 := (ϕ−1 )ϕ (U) . In particular V ∼ = OW (W ). = W ⇒ OV (V ) ∼

(6.3)

Remark 6.1.4. Recalling Example 5.3.23 and Exercises 5.1, 5.2, isomorphisms in (6.3) imply that the hyperbola Y = Za (x1 x2 − 1) and the ellipse Z = Za (x21 + x22 − 1) cannot be isomorphic to A1 . Below (cf. Example 6.2.5-(iv)), we will show that they both are isomorphic to the quasi-affine variety W := A1 \ {0} whereas the parabola V = Za (x2 − x21 ) is instead isomorphic to the whole A1 . Definition 6.1.5. If ϕ ∈ Morph(V, W ) is such that Im(ϕ) = W , then ϕ is said to be a dominant morphism, or simply dominant. Parametrizations φ1 , in Exercise 5.1, and φ3 , in Example 5.3.23-(iii), are dominant (but not surjective) morphisms with target A1 . Remark 6.1.6. If ϕ ∈ Morph(V, W ) is dominant then, for any non-empty open subset U ⊆ W , one has U ∩ Im(ϕ) = ∅. Indeed, if otherwise there existed a non-empty open set U1 ⊂ W such that U1 ∩ Im(ϕ) = ∅ one would have Im(ϕ) ⊆ U1c ⊂ W , with U1c a proper, non-empty, closed subset of W , contradicting the dominance of ϕ. Later on (cf. Corollary 11.1.2), we will more precisely show that any dominant morphism ϕ ∈ Morph(V, W ) is such that Im(ϕ) always contains a non-empty open subset of the target variety W . Proposition 6.1.7. If ϕ ∈ Morph(V, W ) is dominant, then K(W ) ⊆ K(V ) is a field extension. In particular V ∼ = W ⇒ K(V ) ∼ = K(W ).

(6.4)

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151

Proof. Recalling notation as in Definition 5.3.1 and taking into account Remark 6.1.6, the morphism ϕ induces a map H(W ) −→ H(V ) (U, f ) −→ (ϕ−1 (U ), fϕ ), which is compatible with the equivalence relation R. Thus, it defines a map K(V ) ϕ∗ : K(W ) −→ −1 [U, f ] −→ [ϕ (U ), fϕ ],

(6.5)

which is a K-algebra homomorphism. Since ϕ∗ is a non-zero field homomorphism it is therefore injective, proving the first part of the statement. When in particular ϕ ∈ Isom(V, W ), ϕ∗ is a field isomorphism between K(W ) and K(V ).  Remark 6.1.8. The converse of (6.4) does not hold: take, e.g. the hyperbola Y = Za (x1 x2 − 1), for which we found K(Y ) ∼ = K(A1 ) 1 (cf. Example 5.3.23-(i)), even if Y cannot be isomorphic to A as observed in Remark 6.1.4. Definition 6.1.9. Let V be an algebraic variety and let W ⊆ V be any subvariety. The natural inclusion map ιW : W → V is a morphism which is called an open (closed, locally-closed, respectively) immersion if W is an open (closed, locally closed, respectively) subvariety of V . Remark 6.1.10. If V is an algebraic variety and φ ∈ Morph(V, W ) then, in general, φ(V ) is neither open nor closed in W . In general φ(V ) is a constructible set, i.e. a union of finitely many locally closed subsets of W (cf. Corollary 11.1.3). To have an example, take, e.g. the hyperbolic ιV paraboloid V := Za (x1 x3 − x2 ) → A3 . Let π : A3 → A2 be the projection defined by π(x1 , x2 , x3 ) = (x1 , x2 ), which is easily seen to be a morphism (cf. also Example 6.2.5-(i)). Then φ := π ◦ ιV ∈ Morph(V, A2 ) and φ(V ) = (A2 \ Za (x1 )) ∪ {(0, 0)}, which is neither closed nor open in A2 but it is a constructible set in A2 . In the next sections, we shall give some useful criteria ensuring that a given map between algebraic varieties is actually a morphism. 6.2

Morphisms with (Quasi) Affine Target

In this section, we focus on cases where the target of ϕ : V → W is a quasi-affine (or affine) variety. We start with the following basic result.

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Proposition 6.2.1. Let V be an algebraic variety. There is a bijective correspondence between Morph(V, A1 ) and OV (V ). Proof. Any f ∈ OV (V ) is a continuous map from V to A1 , as it follows from Proposition 5.2.4-(i). Let now U ⊆ A1 be any non-empty open set. From Lemma 5.3.13-(ii) and Corollary 5.3.18, one has K(U ) ∼ = K(A1 ) = c(t) (1) ∼ Q = K(t), where t an indeterminate over K. Let b(t) ∈ OA1 (U ) be any regular function on U , i.e. c(t), b(t) ∈ A(1) = K[t] and U ⊆ Ua (b) = Za (b)c . Using notation as in (6.1), one has   c(t) c(f ) c(t) U ◦f = ∈ OV (f −1 (U )), f = b(t) b(t) b(f ) i.e. f defines a morphism from V to A1 . Conversely, any ϕ ∈ Morph(V, A1 ) gives rise to ϕ# (IdA1 ) = (IdA1 )ϕ ∈ OV (V ) and, from the previous step, it turns out that ϕ is the morphism defined by this regular function.  Since An is affine, from Theorem 5.3.14-(a), any indeterminate xi ∈ A(n) = A(An ) = OAn (An ), 1  i  n, can be identified with the regular function x

i An −→ A1 , (a1 , a2 , . . . , an ) → ai

(6.6)

which, in this interpretation, is called the projection πi of An onto its ithaxis. If W ⊆ An is any quasi-affine variety, by abuse of notation, we will always denote by xi ∈ A(W ) the restriction of the regular function (6.6) to the subvariety W . Proposition 6.2.2. Let V be an algebraic variety and let W ⊆ An be a quasi-affine variety. Then ϕ ∈ Morph(V, W ) ⇔ ϕi := xi ◦ ϕ ∈ OV (V ),

∀ 1  i  n.

In particular, for any n  1, there is a bijective correspondence between Morph(V, An ) and OV (V )⊕n . Proof. (⇒) Since xi ∈ A(W ) ⊆ OW (W ), by composition of morphisms we have ϕi = xi ◦ ϕ ∈ Morph(V, A1 ) = OV (V ), as it follows from Lemma 6.2.1. (⇐) Let ϕ : V → W be any map s.t. ϕi = xi ◦ ϕ ∈ OV (V ), for any 1  i  n. Then, for any f ∈ A(An ) = A(n) , one has f  := f (ϕ1 , . . . , ϕn ) ∈ OV (V ), since f  is a polynomial expression in the

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ϕi ’s. From the proof of Proposition 5.2.4-(i), ZV (f  ) ⊆ V is a closed subset of V and moreover ϕ−1 (Za (f ) ∩ W ) = ZV (f  ), i.e. ϕ is continuous. Let us show that ϕ is also a morphism. Take any non-empty open subset U ⊆ W and any f ∈ OW (U ); by definition of regularity and by Lemma 5.2.3, for any P  ∈ U there exist an open neighborhood UP  ⊆ U and two polynomials gP  , hP  ∈ A(n) such that hP  = 0, UP  ⊆ Ua (hP  ) = Za (hP  )c , such that f |UP  = hgP  |UP  and moreover, for any Q ∈ U different from P  one has P

(∗)

gP  gQ |UP  ∩UQ = f |UP  ∩UQ = |U ∩U . hP  hQ P  Q

Consider ϕU (f ) = f ◦ ϕ as in (6.1); we need to show that ϕU (f ) is regular is a local on the open set ϕ−1 (U ) ⊆ V . From the facts that: regularity  −1 condition, {UP  }P  ∈U is an open covering of U , ϕ (UP  ) P  ∈U is an open covering of ϕ−1 (U ) and from (∗), it suffices to check that for any P  ∈ U one has ϕUP  (f ) ∈ OV (ϕ−1 (UP  )). Note that ϕUP  (f ) = f ◦ ϕ|ϕ−1 (UP  ) =

gP  (ϕ1 , . . . , ϕn ) | −1 . hP  (ϕ1 , . . . , ϕn ) ϕ (UP  )

Since ϕi ∈ OV (V ) for any 1  i  n and since UP  ⊆ Ua (hP  ), one has gP  (ϕ1 ,...,ϕn ) −1 (UP  )), as desired. h  (ϕ1 ,...,ϕn ) ∈ OV (ϕ P

For what concerns Morph(V, An ), the case n = 1 is Lemma 6.2.1 whereas  the case n  2 follows from the first part applied to W = An . Similarly to (5.4), for any f1 , . . . , fn ∈ OV (V ) one can define ZV (f1 , . . . , fn ) := {P ∈ V | f1 (P ) = f2 (P ) = · · · = fn (P ) = 0} ⊆ V. (6.7) Thus, from Proposition 6.2.2, for any ϕ ∈ Morph(V, An ) one can define ZV (ϕ) := ZV (ϕ1 , . . . , ϕn ) = ϕ−1 ((0, . . . , 0)). In the sequel, for any K-algebras R and S, we will denote by HomK (R, S)

(6.8)

the set of all K-algebra homomorphisms from R to S. Recalling (6.2), one has: Proposition 6.2.3. Let V be an algebraic variety and let W ⊆ An be an affine variety. The map α

Morph(V, W ) −→ HomK (A(W ), OV (V )) ϕ −→ ϕ# is bijective.

(6.9)

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Proof. By Theorem 5.3.14-(i), one has OW (W ) = A(W ) = now any h ∈ HomK (A(W ), OV (V )) and let

A(n) Ia (W ) .

ξi := h(xi ) ∈ OV (V ),

Take

(6.10) ϕh

for any xi ∈ A(W ), 1  i  n. Consider the map V −→ An , defined by P −→ (ξ1 (P ), . . . , ξn (P )). From Proposition 6.2.2, ϕh ∈ Morph(V, An ). If we show that ϕh (V ) ⊆ W , we will get that ϕh ∈ Morph(V, W ). To prove this, consider any polynomial f ∈ Ia (W ). For any P ∈ V we have ϕh (f )(P ) = (f ◦ ϕh )(P ) = f (ξ1 (P ), . . . , ξn (P )), which means ϕh (f ) = f ◦ ϕh = f (h(x1 ), . . . , h(xn )). Since h ∈ HomK (A(W ), OV (V )), the latter equals h(f (x1 , . . . , xn )) = h(0) = 0, as xi ∈ A(W ) whereas f ∈ Ia (W ). Thus ϕh (V ) ⊆ W , as desired. At last, the map HomK (A(W ), OV (V )) −→ Morph(V, W ) h −→ ϕh is the inverse of α, which is therefore a bijection as desired.



Corollary 6.2.4. Let V and W be affine varieties. Then V ∼ = W as affine varieties if and only if A(V ) ∼ = A(W ) as integral K-algebras of finite type. Proof. One implication is (6.3), the other follows from Proposition 6.2.3.  Example 6.2.5. (i) Let 1  m < n be integers and consider the set inclusion I := {i1 , i2 , . . . , im } ⊂ {1, 2, . . . , n}. The injective, K-algebra homomorπI#

phism K[xi1 , . . . , xim ] → K[x1 , . . . , xn ] corresponds to the surjective morphism π

I πI : An  Am , (a1 , . . . , an ) −→ (ai1 , . . . , aim ),

(6.11)

which is called the projection of A onto the coordinates I = {i1 , i2 , . . . , im }. (ii) Let (b1 , . . . , bn ) = (0, . . . , 0) be non-negative integers. The morphism ϕ ϕ : A1 → An , defined by t −→ (tb1 , . . . , tbn ), corresponds to the Kalgebra homomorphism n

ϕ#

ϕ# : K[x1 , . . . , xn ] → K[t], xi −→ tbi ,

1  i  n.

The image of ϕ is an affine rational curve with polynomial parametrization (cf. (3.31)).

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(iii) In particular, for any non-negative integer b, ϕ : A1 → A2 , t −→ (t, tb ) defines an isomorphism between A1 and the affine plane curve V = Za (x2 − xb1 ) as ϕ corresponds to the K-algebra isomorphism A(V ) = K[x1 ,x2 ] (x2 −xb1 )

ϕ#

−→ K[t] = A(A1 ), defined by ϕ# (x1 ) = t and ϕ# (x2 ) = tb

(cf. Corollary 6.2.4). Specifically, the inverse morphism ϕ−1 is given by π1 |V , where π1 the projection of the affine plane A2 onto the x1 -axis, which is a morphism since it is a restriction to V of a morphism from A2 to A1 . (iv) The map φ2 in Example 5.3.23-(ii) and Exercise 5.2 is an isomorphism. Indeed the parametrization φ2

φ2 : A1 → V, t −→ (t, t2 ) of the parabola V = Za (x2 − x21 ) ⊂ A2 is given by the regular functions t, t2 ∈ OA1 (A1 ) so, from Proposition 6.2.2, φ2 is a morphism is given by the restriction to V of the (cf. Figure 6.1). The map φ−1 2 projection π1 : A2 → A1 , (x1 , x2 ) → x1 onto the first coordinate as in (6.6), which is therefore a morphism. Since V and A1 are both affine varieties, Corollary 6.2.4 gives intrinsic motivation of A(V ) ∼ = A(A1 ) as observed in Example 5.3.23-(ii) and in Exercise 5.2.

Fig. 6.1

Projection of the parabola Za (x2 − x21 ) onto the x1 -axis.

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Fig. 6.2

Projection of the hyperbola Za (x1 x2 − 1) to the x1 -axis.

Different situation occurs for, e.g. the hyperbola Y = Za (x1 x2 − 1) as in Example 5.3.23-(i) and in Exercise 5.1 (cf. Figure 6.2). From Proposition 6.2.2 the parametrization φ1 : A1 \ {0} := W → Y is an isomorphism, i.e. Y is isomorphic to the (quasi-affine) variety W ⊂ A1 . From (6.3) it follows that OW (W ) ∼ = OY (Y ) ∼ = A(Y ), the latter isomorphism coming from the fact that Y is affine (cf. Theorem (1) 5.3.14). Recall that in Example 5.3.23-(i) we showed that A(Y ) ∼ = Ax1 ; the fact that φ1 is an isomorphism in particular proves that OW (W ) ∼ = (1) Ax1 (as stated without proof in Example 5.3.23-(i) and in Exercise 5.1). In particular, this is another way to prove that K(W ) ∼ = Q(1) . Considerations about the hyperbola hold verbatim for the ellipse Z := Za (x21 + x22 − 1) with the map φ3 as in Example 5.3.23-(iii), namely Z turns out to be isomorphic to W = A1 \ {0}. (v) By definition of isomorphism, any φ ∈ Isom(V, W ) is also a homeomorphism between V and W as topological spaces; on the other hand, the converse does not hold. Consider, e.g. the semi-cubic parabola Y = Za (x31 − x22 ) in Example 5.3.24. The map φ : A1 → Y therein is a morphism, as it follows from Proposition 6.2.2. In Example 5.3.24, we proved that φ is also a homeomorphism. On the other hand,

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157

we also showed that A(Y ) ∼ = K[t, t2 ], which is a (proper) subring of 1 ∼ A(A ) = K[t]. Since Y and A1 are both affine varieties, from Corollary 6.2.4, φ cannot be an isomorphism. In fact that the map ψ constructed in Example 5.3.24 is the inverse of φ as a homeomorphism, but it is not a morphism, as ψ is not a polynomial map, i.e. a map given by a collection of polynomials, as prescribed by Proposition 6.2.2. Indeed, for any open set U ⊆ A1 containing 0, one has t ∈ OA1 (U ) but / OY (ψ −1 (U )) as (0, 0) ∈ ψ −1 (U ). ψ −1 (t) = xx21 ∈ Interesting consequences of previous results are the following. Corollary 6.2.6. If V is an affine variety which is isomorphic to a projective variety, then V is a point. ∼ Proof. From Theorem 5.3.14-(a), if V is an affine variety, then OV (V ) = A(V ). On the other hand, since V is isomorphic to a projective variety, by (6.3) and Theorem 5.3.14-(e), we have also OV (V ) ∼ = K. Thus, for some non-negative integer n, one has that Ia (V ) ⊂ A(n) must be a maximal ideal. By Theorem 2.2.1, V is a point.  Corollary 6.2.7. Any morphism ϕ : V → W, where V a projective variety and W an affine variety, is constant. Proof. Since V is projective, OV (V ) = K (cf. Theorem 5.3.14-(e)). At the same time, since W is affine then it is a closed, irreducible subset of some n affine space An , for some non-negative integer n, so A(W ) = IaA(W ) . For any ϕ ∈ Morph(V, W ), consider ϕ# ∈ HomK (A(W ), OV (V )). As in (6.10), for any generator xi ∈ A(W ) one has ξi = ϕ# (xi ) := ai ∈ K, for any 1  i  n. This implies that for any point P ∈ V one has ξi (P ) := ai so  ϕ(P ) = (a1 , . . . , an ), i.e. ϕ is constant. Corollary 6.2.8. Let V and W be affine varieties and ϕ ∈ Morph(V, W ). Then, ϕ is dominant if and only if the map ϕ# ∈ HomK (A(W ), A(V )) is injective. Proof. (⇒) If by contradiction Ker(ϕ# ) = 0, for any non-zero polynomial f ∈ Ker(ϕ# ) one has ϕ# (f ) = 0 ∈ A(V ), i.e. ϕ# (f ) ∈ Ia (V ). Since f ∈ A(W ) = OW (W ), by the proof of Proposition 5.2.4-(i), ZW (f ) is a proper closed subset of W (as 0 = f ∈ A(W )). Now, ϕ# (f ) ∈ Ia (V ) is equivalent to f ◦ ϕ identically zero on W , i.e. ϕ(V ) ⊆ ZW (f )  W , contradicting the dominance of ϕ.

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(⇐) Assume by contradiction that ϕ is not dominant, i.e. ϕ(V ) = K  W , where K is an irreducible, proper closed subset of W (recall Corollary 4.1.6 and Proposition 4.1.2-(iii)). Then, with notation as in (5.7), for any nonzero f ∈ Ia,W (K) we would have ϕ(V ) ⊆ ZW (f ). Any such f determines a non-zero regular function f ∈ A(W ) = OW (W ) s.t. f ◦ ϕ(P ) = 0 for any P ∈ V , i.e. a non-zero element of Ker(ϕ# ), contradicting the injectivity assumption.  The previous results are also consequences of more general facts which will be discussed later on (cf. Remark 9.2.2). −1 Example 6.2.9. Morphisms φ−1 1 and φ3 in Examples 5.3.23-(i), (iii) and 6.2.5-(iv) and in Exercise 5.1 are dominant but not surjective morphisms with target A1 . Indeed, in both cases we have the injection A(A1 ) = (1) A(1) → Ax1 ∼ = A(Y ) ∼ = A(Z), where Y the hyperbola and Z the ellipse, respectively.

The use of isomorphisms allows to extend some previous definitions. Definition 6.2.10. Let V be any algebraic variety. V is said to be an affine variety if V is isomorphic to an irreducible closed subset in some affine space. If U ⊆ V is an open subset which is an affine variety, then U will be called an affine open set of V . Lemma 6.2.11. For any 0  i  n, the principal open set Ui ⊂ Pn is an affine open set of Pn isomorphic to An ; in other words the homeomorphism φi as in Proposition 3.3.2 is actually an isomorphism, for any 0  i  n. More generally, any projective variety V ⊆ Pn has a finite open covering {Vi }0in , where Vi := V ∩ Ui is an affine open set of V . Proof. One is reduced to showing that the map φi as in (3.15) is an isomorphism, for any i ∈ {0, . . . , n}. From Proposition 3.3.2, any such φi is a homeomorphism. The fact that φi is a morphism follows from the expression of φi in (3.15) and from Proposition 6.2.2. To prove that is a morphism too, one uses the definition of regular function and φ−1 i Lemma 5.2.3.  Another example of affine open set of an algebraic variety is given by W := A1 \ {0} ⊂ A1 : W is isomorphic to, e.g. the hyperbola Y = Za (x1 x2 − 1) ⊂ A2 as in Example 6.2.5-(iv). This is a particular case of the following more situation.

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Lemma 6.2.12. For any positive integer n and for any non-constant polynomial f ∈ A(n) , let Z := Za (f ) ⊂ An be the affine hypersurface defined by f . Then U := An \ Z ⊂ An is an affine open set, being isomorphic to the irreducible hypersurface Z  := Za (xn+1 f − 1) ⊂ An+1 . Moreover, one has (n) OU (U ) ∼ = Af . Proof. First note that (xn+1 f − 1) ∈ A(n+1) is irreducible so Z  is an (n+1) (n) ∼ affine variety: this follows from A(Z  ) = (xA = Af ⊂ Q(A(n) ), n+1 f −1) which therefore has to be an integral domain, and from Proposition 4.1.10. Consider now the map ψ

ψ : Z  → An \ Z, (a1 , a2 , . . . , an+1 ) −→ (a1 , a2 , . . . , an ). Since ψ = πI |Z  , for the multi-index I := (1, 2, . . . , n), then ψ is a morphism. Conversely, the map   1 ϕ n  ϕ : A \ Z → Z , (a1 , a2 , . . . , an ) −→ a1 , a2 , . . . , an , f (a1 , . . . , an ) is a morphism by Proposition 6.2.2 which is the inverse of the morphism ψ, proving that Z  and U = An \ Z are isomorphic. At last, since Z  is an affine hypersurface, by Theorem 5.3.14-(a) and by (6.3) we have (n) OU (U ) ∼ = A(Z  ) ∼ = Af , where the last isomorphism has been proved above.  Corollary 6.2.13. An admits a basis for the topology Zarna consisting of affine open sets. Proof. It directly follows from Lemma 2.1.21 and the previous result.  The next example shows that there actually exist quasi-affine varieties which cannot be affine varieties. Example 6.2.14. From Lemma 6.2.12, if Z ⊂ A2 is any curve then A2 \ Z is an affine open set of A2 . On the contrary, for any point P ∈ A2 , we now show that A2 \ {P } is a quasi-affine variety which cannot be affine. With the use of affine transformations, it is not a restriction to consider P to be the origin O = (0, 0). Then W := A2 \ {O} = Za ((x1 , x2 ))c is quasiaffine. Let ιW : W → A2 be the open immersion of W as a subvariety, so ιW ∈ Morph(W, A2 ). Since A2 is affine, from Proposition 6.2.3, ιW 2 corresponds to the K-algebra homomorphism ι# W ∈ HomK (A(A ), OW (W )). 2 (2) # (2) Since A(A ) = A , ιW simply sends any polynomial f ∈ A (viewed as a regular function on A2 ) to its restriction to W . Since W is a dense open set in A2 , it is clear that ι# W is an injective homomorphism.

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Claim 6.2.15. ι# W is an isomorphism. Proof. From above, we need to show that ι# W is surjective. From Lemma 5.3.13 and Theorem 5.3.14-(d), we have OW (W ) ⊂ K(W ) ∼ = K(A2 ) = (2) (2) Q(A ) = Q , i.e. any regular function on W can be identified with a rational function φ = hg ∈ Q(2) such that g, h ∈ A(2) , h = 0 and W ⊆ Ua (h) = Za (h)c . Thus, to prove the surjectivity of ι# W is equivalent to showing that any φ ∈ OW (W ) is actually a polynomial, i.e. that h ∈ K∗ . Take therefore any φ = hg ∈ OW (W ) and assume by contradiction that h ∈ A(2) \ K. Since W ⊆ Ua (h), then one has either Za (h) = ∅ the first case we would have or Za (h) = {O}. From Theorem 2.2.3, in   (h) = (1) whereas in the second case (h) = mO = (x1 , x2 ), a contradiction in both cases. This concludes the proof of Claim 6.2.15.  Thus, if W were affine, from Theorem 5.3.14-(a) we would have OW (W ) ∼ = A(W ) so, from Claim 6.2.15 and Corollary 6.2.4, ιW would be an isomorphism which is a contradiction, since ιW is injective but not surjective onto A2 . 6.3

Morphisms with (Quasi) Projective Target

We start with the following general result. Proposition 6.3.1. Let n, m be non-negative integers and let V ⊆ Pn and W ⊆ Pm be quasi-projective varieties. A non-constant map ψ : V → W is a morphism if and only if, for any P ∈ V, there exist an open neighborhood UP of P in V, a non-negative integer k and m+ 1 homogeneous polynomials (n) F0 , . . . , Fm ∈ Sk such that: (a) there exists i ∈ {0, . . . , m} for which Fi (P  ) = 0, for any P  ∈ UP , and (b) ψ(P  ) = [F0 (P  ), . . . , Fm (P  )], for any P  ∈ UP . Proof. (⇒) For P ∈ V , let Q := ψ(P ) ∈ W ⊆ Pm . Since W = ∅, there exists an index i ∈ {0, . . . , m} for which Wi := W ∩ Ui = ∅, Ui the fundamental affine open set of Pm . For simplicity, assume this occurs for i = 0. Thus, W0 is an open set of W , so UP := ψ −1 (W0 ) ⊆ V is an open neighborhood of P in V , being ψ continuous. The map ψ  := ψ|UP : UP → W0 ιW0

is a morphism; up to composing with the inclusion W0 → U0 and φ0 with the isomorphism U0 −→ Am , ψ  has quasi-affine target W0 . From Proposition 6.2.2, ψ  is therefore given by a m-tuple of regular functions over UP ⊆ V ⊆ Pn . Up to suitably restricting the open neighborhood UP ,

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this means there exist m non-negative integers di , 1  i  m, and m pairs of homogeneous polynomials (F1 , F1,0 ), (F2 , F2,0 ), . . . , (Fm , Fm,0 ), (n)

where Fi , Fi,0 ∈ Sdi , 1  i  m, such that UP ⊆ ∩m i=1 Up (Fi,0 ) = c   Z (F ) (i.e. F (P ) =  0 for any P ∈ U and for any i ∈ {1, . . . , m}) ∩m p i,0 i,0 P i=1 and   F1 (P  ) F2 (P  ) Fm (P  )   , ,..., ψ (P ) = , ∀ P  ∈ UP . F1,0 (P  ) F2,0 (P  ) Fm,0 (P  ) If we put F := F1,0 F2,0 · · · Fm,0 and Fi := F1,0 F2,0 · · · Fi−1,0 Fi Fi+1,0 · · · Fm,0 , (n) for any i ∈ {1, . . . m} we have F, Fi ∈ Sk , where k := Σm i=1 di . Moreover, for any P  ∈ UP , we have F(P  ) = 0 and  m (P  ) 1 (P  ) F2 (P  ) F F ψ  (P  ) = . , ,..., F (P  ) F (P  ) F (P  ) (n)

Thus, we can assume that F0 := F1,0 = F2,0 = · · · = Fm,0 ∈ Sk , so (n) F1 , F2 , . . . , Fm ∈ Sk and that for any P  ∈ UP one has F0 (P  ) = 0 and   F1 (P  ) F2 (P  ) Fm (P  )   , ,..., ψ (P ) = . F0 (P  ) F0 (P  ) F0 (P  ) m Using the isomorphism φ−1 0 : A → U0 , the previous equality reads as

 Fm (P  ) F1 (P  ) F2 (P  ) , , . . . , ψ  (P  ) = 1, = [F0 (P  ), F1 (P  ), . . . , Fm (P  )], F0 (P  ) F0 (P  ) F0 (P  )

for any P  ∈ UP , as desired. (⇐) Take P ∈ V and assumptions as in the statement. Composing with the isomorphism φi : Ui → Am , from Proposition 6.2.2 the map ψ|UP is a morphism with target Wi := W ∩ U0 a quasi-affine variety. Since {UP }P ∈V is an open covering of V , it follows that ψ is a morphism.  Example 6.3.2. For any integer n  1, consider the projective space Pn . (n)

(i) Let d  1 be an integer and let F0 , . . . , Fr ∈ Sd polynomials. Put B := Zp (F0 , . . . , Fr ) ⊂ Pn

be homogeneous

(6.12)

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and U := Pn \ B.

(6.13)

B is a proper (possibly empty) closed subset of Pn , whereas U is an open (so dense and irreducible) subset of Pn . Define the map ν

ν : U −→ Pr , [P ] = [p0 , . . . , pn ] −→ [F0 (P ), . . . , Fr (P )].

(6.14)

From Proposition 6.3.1, ν is a morphism. The open set U is called the domain of ν, i.e. the set where ν is defined, whereas B is called the indeterminacy locus of ν. Take homogeneous coordinates [Y0 , . . . , Yr ] in the target and let Hi := Zp (Yi ) ⊂ Pr be the fundamental hyperplane. Note that Im(ν) is not contained in any of the hyperplanes Hi ’s, 0  i  r: indeed ν(U ) ⊆ Hi , for some i, means that U ⊆ Zp (Fi ); since U ⊂ Pn is open and dense, the latter condition would imply Fi = 0 as a polynomial, contradicting that any Fi is homogeneous of some degree d > 0. Note moreover that, for any 0  i  r, one has ν −1 (Hi ) = U ∩ Zp (Fi ) ⊂ Pn ,

(6.15)

i.e. the hypersurface U ∩ Zp (Fi ) = ZU (Fi ) is the pre-image via the map ν of Hi ∩ Im(ν), which is called a hyperplane section of Im(ν). (n) The polynomials F0 , . . . , Fr are linearly independent in Sd if and r only if Im(ν) is non-degenerate in P : any possible hyperplane  r ai Yi ⊂ Pr H := Zp i=0

for which Im(ν) ⊆ H would give U ⊆ Zp (Σri=0 ai Fi ) ⊂ Pn . In such a case Σri=0 ai Fi is identically zero which, by the linear independence of the Fi ’s, implies (a0 , . . . , ar ) = (0, . . . , 0). If otherwise F0 , . . . , Fr are linearly dependent, any non-trivial linear combination Σri=0 ai Fi = 0 corresponds, via ν, to a hyperplane of Pr containing Im(ν). (n) Thus, if F0 , . . . , Fr are linearly independent in Sd , we pose L := Span{F0 , . . . , Fr } and denote by νL : U −→ Pr

(6.16)

the associated morphism as in (6.14). The target Pr of νL can be identified with the linear subspace P(L) of P(Snd ); this linear subspace

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will be called linear system of hypersurfaces of degree d in Pn , of (projective) dimension r = dim(L) − 1. (n) When L = Sd the associated map will be simply denoted by νn,d (n) (cf. also Section 6.5) and P(Sd ) will be called the complete linear system of hypersurfaces of degree d in Pn ; it has projective dimension N (n, d) := b(n, d) − 1, where b(n, d) the binomial coefficient as in (1.26). (ii) Consider P1 and take d = 2. If we take the complete linear system (1) P(S2 ), with F0 = X02 , F1 = X0 X1 , F2 = X12 the canonical basis of (1) S2 , then B = ∅, U = P1 and ν1,2 : P1 → P2 is such that Im(ν1,2 ) = Zp (Y0 Y2 −Y12 ) ⊂ P2 , where [Y0 , Y1 , Y2 ] are homogeneous coordinates for P2 . Any linear section of the conic Zp (Y0 Y2 −Y12 ) is given by Zp (Y0 Y2 − Y12 , a0 Y0 + a1 Y1 + a2 Y2 ), where (a0 , a1 , a2 ) = (0, 0, 0), and it bijectively corresponds to the two roots (counted with multiplicity) in P1 of the (1) homogeneous polynomial a0 X02 +a1 X0 X1 +a2 X22 ∈ S2 in the sense of Proposition 1.10.19, i.e. to a pair (counted with multiplicity) of points in P1 . Note that ν1,2 coincides with the map Φ2 considered in Example 5.3.23-(ii) and Exercise 5.2, which is an isomorphism from P1 onto the conic Zp (Y0 Y2 − Y12 ), i.e. the isomorphism φ2 between affine varieties in Example 6.2.5-(iv) naturally extends to an isomorphism between their projective closures. (iii) With notation as in (ii), if otherwise we take L = Span{F0 = X02 , F1 = (1) X0 X1 } a proper subspace of S2 , then B = {[0, 1]} is a base point of the linear system P(L), U = P1 \{[0, 1]} = U0 ⊂ P1 and the morphism νL is νL P1 , [X0 , X1 ] → [X02 , X0 X1 ]. Since X0 = 0, on points given by U0 −→ of U0 one has [X02 , X0 X1 ] = [X0 , X1 ] so νL extends to the identity of P1 (this follows from more general facts which will be discussed later on; cf. Example 8.1.12). (1) (iv) If we consider instead d = 3, L = S3 , F0 , F1 , F2 , F3 the canonical basis (1) of S3 , one easily deduces that Im(ν1,3 ) is the (standard) projective twisted cubic as in Section 3.3.12 and that ν1,3 is an isomorphism onto its image.

Example 6.3.3. With notation as above, consider the case of F0 , . . . , Fr (n) linearly independent linear forms in S1 . (n)

(i) If r = n, then F0 , . . . , Fn is a basis of S1 simply a projectivity of Pn .

and the morphism νL is

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(ii) If otherwise r < n then B = Zp (F0 , . . . , Fr ) ⊂ Pn is a linear subspace of dimension n − r − 1, the open set of definition is U := Pn \ B and the morphism νL : U → Pr is called projection of Pn to Pr with center the linear subspace B. A geometric interpretation of the morphism νL is given by the following construction: the target Pr is identified with any r-dimensional linear subspace Λ ⊂ Pn which is skew with respect to the center of projection B and, for any point P ∈ U , νL (P ) ∈ Λ is given by the intersection of Λ with the linear subspace P ∨ B ⊂ Pn (recall Section 3.3.6, in particular (3.23)). For more details on projections of a projective space from a non-empty linear subspace, see Example 8.1.14. 6.4

Local Properties of Morphisms: Affine Open Coverings of an Algebraic Variety

Before treating some particularly interesting examples (from the geometric point of view) of morphisms with projective target (cf., e.g. Section 6.5), we give here two fundamental results which will be frequently used later on (cf., e.g. Theorem 6.5.1 as well as Chapters 7 and 8). The first result gives a general criterion to verify when a continuous map between general algebraic varieties is actually a morphism. Proposition 6.4.1. Let V and W be any algebraic varieties and let ϕ : V → W be any continuous map. Then ϕ is a morphism if and only if there exists an open covering {Wj }j∈J of W such that, for any j ∈ J, there exists an open covering {Vij }i∈I(j) of ϕ−1 (Wj ) such that the continuous map ϕij : Vij → Vj induced by ϕ is a morphism, for any j ∈ J and any i ∈ I(j). The previous proposition states that for a continuous map ϕ between algebraic varieties to be a morphism is a local property, i.e. ϕ globally is a morphism if and only if it locally is a morphism on open sets of suitable open coverings of the algebraic varieties which are the domain and the target of ϕ, respectively. Proof. It is clear that if ϕ is a morphism, then each ϕij = ι−1 Wi ◦ ϕ ◦ ιVij is a morphism, being a composition of morphisms. Conversely, assume that any ϕij is a morphism and let U ⊆ W be any non-empty open set and let f ∈ OW (U ) be any regular function on it. Let fij be the restriction of the continuous map f ◦ ϕ to the open set ϕ−1 (U ) ∩ Vij of V . Note that fij coincides with the restriction to ϕ−1 (U ) ∩ Vij of f |U∩Wj ◦ ϕij , i.e. fij ∈ OV (ϕ−1 (U ) ∩ Vij ). Since {ϕ−1 (U ) ∩ Vij }i∈I(j) is an

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open covering of ϕ−1 (U ), it follows that ϕU (f ) is regular of ϕ−1 (U ), i.e. ϕ is a morphism.  The second result extends Corollary 6.2.13 to any algebraic variety. Proposition 6.4.2 (Affine open coverings of an algebraic variety). On any algebraic variety V there exists a basis for the topology ZarV consisting of open affine subsets. Proof. We must show that, for any point P ∈ V and for any open set U ⊂ V containing P , there exists an open affine subset UP with P ∈ UP ⊆ U . Since U is also a variety, we may assume V = U . Moreover, since any variety can be covered by quasi-affine varieties, we can assume V to be quasi-affine in An , for some non-negative integer n. a a Let Z := V \ V = ∅, where V the closure of V which is quasi-affine a in An . Then Z is a proper closed subset of V and let Ia (Z) ⊂ A(n) be its ideal. Then, since Z is closed and P ∈ / Z, there exists f ∈ Ia (Z) such that f (P ) = 0. Now Z ⊆ Za (f ) and P ∈ / Za (f ), so P ∈ UP := V \ (V ∩ Za (f )) which is an open subset of V , since V ∩ Za (f ) is a closed subset of V by induced topology; in particular UP is irreducible. On the other hand a

UP = V ∩ (An \ Za (f )) = (V ∪ Z) ∩ (An \ Za (f )) = V ∩ (An \ Za (f )), the latter equality following from Z ⊆ Za (f ); thus UP is also a closed subset of An \ Za (f ) which, by Lemma 6.2.12, is an affine open set of An . Hence,  UP is affine, as desired. Corollary 6.4.3. Let V be any algebraic variety and let W ⊆ Y be any subvariety. Then: (i) there is a bijective correspondence between prime ideals in OY,W and closed subvarieties of Y containing W . (ii) Q(OY,W ) = K(Y ). Proof. Recall that, by its very definition, OV,W ⊆ K(V ). Moreover, from Lemma 5.3.13-(ii), for any non-empty open subset U ⊆ V one has K(V ) ∼ = K(U ) and W  := W ∩ U is a subvariety of U . Therefore, by Proposition 6.4.2, up to replacing V with any of its affine open set, we may assume V to be affine. Since V is affine, statement (i) directly follows Theorem 5.3.14-(b) and (c) and from Remark 1.11.9. As for (ii), by Theorem 5.3.14-(a) and (d) one has OY (Y ) = A(Y ) ⊆ OY,W ⊆ K(Y ) and Q(A(Y )) = K(Y ), which concludes the proof. 

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Veronese Morphism: Divisors and Linear Systems

Keep notation as in Example 6.3.2-(i) and, for simplicity, set N := N (n, d). (n) If we let F0 , . . . , FN be any basis of the K-vector space Sd , we can consider the induced complete linear system of hypersurfaces of degree d in Pn . By Theorem 3.2.4, in such a case the base locus is (n)

(n)

B = Zp (F0 , . . . , FN ) = Zp ((S+ )d ) = Zp (S+ ) = ∅. Therefore, from Proposition 6.3.1 and from (6.14), the map νn,d

νn,d : Pn → PN , [X0 , . . . , Xn ] −→ [F0 (X0 , . . . , Xn ), . . . , FN (X0 , . . . , Xn )] (6.17) is a morphism, which will be called Veronese morphism of indexes n and (n) d induced by the chosen basis F0 , . . . , FN of Sd . We also pose Vn,d := Im(νn,d ), which we call Veronese variety of indexes n and d. In Theorem 6.5.1 we will justify the use of the term “variety” for such an image. In view of the proof of Lemma 1.10.12, one can restrict the analysis by focusing on a more specific case which, in a certain sense, “generates” all Veronese varieties of given indexes n and d. Precisely, let X denote the row– vector of the homogeneous coordinates of Pn , i.e. X = (X0 , . . . , Xn ), and, n for any multi-index I := (i0 , . . . , in ) ∈ Zn+1 0 such that |I| := Σj=0 ij = d, set (n)

X I := X0i0 X1i1 · · · Xnin , which is a homogeneous, degree-d monomial in Sd in the given homogeneous indeterminates. Letting I ∈ Zn+1 0 vary among all the multi-indexes such that |I| = d, with lexicographic ordering among the multi-indexes, the collection  I (6.18) X I∈Zn+1 , |I|=d 0

(n)

is the canonical (or standard) basis of the K-vector space Sd , which is formed by all homogeneous, degree-d monomials in the (n+1)-homogeneous indeterminates X0 , . . . , Xn . The Veronese morphism induced by the standard basis (6.18) will be denoted by st νn,d

st νn,d : Pn → PN , X −→ [· · · , X I , · · · ]

and called the standard Veronese morphism. Similarly its image st st ) ⊂ PN will be denoted by Vn,d and called the standard Veronese Im(νn,d variety of indexes n and d.

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(n)

Any basis F0 , . . . , FN of Sd arises from the standard basis (6.18) (n) via a linear transformation in GL(Sd ) which in turn induces, up to scalar multiplication, an automorphism of PN , i.e. a projectivity in PGL(N + 1, K) giving rise to a change of homogeneous coordinates of PN . st , where Thus, any Veronese morphism νn,d is such that νn,d = Φ ◦ νn,d Φ a suitable projective transformation of the ambient space PN and, consequently, any Veronese variety Vn,d is projectively equivalent to the st , in the sense of Definition 3.3.12. Thus, to standard Veronese variety Vn,d study Veronese varieties Vn,d ⊂ PN one can reduce to standard models st ⊂ PN ; moreover we can obviously avoid the study of the case d = 1 Vn,d since, from Example 6.3.3, any map νn,1 is simply a projectivity of Pn . With this set-up, we are now in position to prove the following. Theorem 6.5.1. For any integers n  1 and d  2, any Vn,d is a non-degenerate, projective variety in PN and any Veronese morphism νn,d is an isomorphism; in particular Vn,d is isomorphic to Pn . Furthermore, the homogeneous ideal Ip (Vn,d ) is generated by homogeneous quadratic polynomials, i.e. any Vn,d is cut-out by finitely many quadric hypersurfaces of PN . Proof. From the previous observation, we can reduce to prove the st st st and Vn,d . Since νn,d is a morstatement for the standard cases νn,d st phism, irreducibility of Vn,d directly follows from Corollary 4.1.6. Moreover (n)

st st Vn,d is non-degenerate in PN , since νn,d is defined by a basis of Sd (cf. Example 6.3.2-(i)). Observe now that the homogeneous coordinates (n) of PN are indexed by all homogeneous, degree-d monomials X I in Sd , taken with the lexicographic order; correspondingly denote by ZI the homogeneous coordinates of PN . If I = (i0 , . . . , in ) and J = (j0 , . . . , jn ) are two multi-indexes such that |I| = |J| = d, then set I + J := (i0 + j0 , . . . , in + jn ). If we moreover take multi-indexes I, J, K, L such that |I| = |J| = |K| = |L| = d and that I + J = K + L, the quadric hypersurface defined by the homogeneous equation

ZI ZJ − ZK ZL = 0 st contains Vn,d , as X I X J − X K X L = X I+J − X K+L = 0. Thus,

st Vn,d ⊆ W := Zp ZI ZJ − ZK ZL | I, J, K, L ∈ Zn+1 0 ,

 |I| = |J| = |K| = |L| = d and I + J = K + L .

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We need to show that W is irreducible and that the other inclusion holds, st surjects onto W . namely that νn,d To prove the irreducibility of W , consider the K-algebra homomorphism θn,d

θn,d : K[· · · , ZI , · · · ] → K[X], f (· · · , ZI , · · · ) −→ f (· · · , X I , · · · ), where K[· · · , ZI , · · · ] = S(N ) is interpreted as the homogeneous coordinate ring of PN whereas K[X] = S(n) that of Pn . The image of θn,d is the subring of S(n) defined as  S(n) (d) := S(n) (d)k , (6.19) k0 (n)

where, as in (1.33), one sets S(n) (d)k := Sd+k for any k  0. Note that S(n) (d) is a graded integral domain, θn,d is a homogeneous homomorphism of degree d and ker(θn,d ) = Ip (W ). Since Ip (W ) is a homogeneous prime ideal, this shows that W ⊂ PN is irreducible. st , we claim that for any point of W at least To prove that W ⊆ Vn,d one of the coordinates Z(d,0,...,0) , Z(0,d,...,0) , . . . , Z(0,0,...,d) is non-zero: indeed assume by contradiction they are all zero, on the other hand since W ⊂ PN there must be a coordinate Z(i0 ,...,in ) = 0 which, with no loss of generality, we may assume such that i0 > 0 with the index i0 maximal w.r.t. this property, namely Zj0 ,...,jn = 0 for all j0 > i0 . Note that, by assumption, one has i0 < d so there is another index, say i1 , such that 0 < i1 < d. From the equations ZI ZJ = ZK ZL defining W , one gets 2 0 = Z(i = Z(i0 +1,i1 −1,...,in ) Z(i0 −1,i1 +1,...,in ) , 0 ,i1 ,...,in )

i.e. Z(i0 +1,i1 −1,...,in ) = 0 contradicting the maximality of i0 . With this for granted, let therefore H0 := Zp (Z(d,0,...,0) ), H1 := Zp (Z(0,d,...,0) ), . . . , Hn := Zp (Z(0,0,...,d) ) be fundamental hyperplanes in PN and let Ui := PN \ Hi , 0  i  n, be the corresponding affine open sets of PN . Then, the open sets Wi := W ∩ Ui of W are not empty, dense (since W is irreducible) and cover W . For any φi

0  i  n, we can consider the map Wi −→ Pn defined as φi

[. . . , ZI , . . .] −→ [Z(1,0,...,d−1i ,...,0) , Z(0,1,...,d−1i ,...,0) , . . . , Z(0,0,...,d−1i ,...,1) ], where the symbol d − 1i stands for “the integer d − 1 is at the ith position of the corresponding multi-index” and where the right-side-member equals [X0 Xid−1 , X1 Xid−1 , . . . , Xn Xid−1 ] = [X0 , X1 , . . . , Xn ],

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the second equality following from the fact that on Wi one has Xid = 0 which therefore implies Xi = 0. From Proposition 6.3.1, it follows that φi is a morphism, for any 0  i  n. Note also that, for 0  i = j  n, one has Wi ∩Wj = ∅ by the irreducibility of W and, from the equations defining W , Z(0,0,...,1t ,...,d−1i ,...,0) =

Z(0,0,...,di ,...,0) Z t j Z(0,0,...,1i ,...,d−1j ,...,0) (0,0,...,1 ,...,d−1 ,...,0)

holds true, for any 0  i = j = t  n. Therefore, on Wi ∩ Wj one has [Z(1,0,...,d−1i ,...,0) , Z(0,1,...,d−1i ,...,0) , . . . , Z(0,0,...,d−1i ,...,1) ] = [Z(1,0,...,d−1j ,...,0) , Z(0,1,...,d−1j ,...,0) , . . . , Z(0,0,...,d−1j ,...,1) ], namely the morphisms φi and φj agree on the overlaps Wi ∩ Wj , for any 0  i = j  n. From Proposition 6.4.1, these maps patch together to define a global morphism φ : W → Pn which clearly is the inverse of the st st . This last assertion not only shows that νn,d surjects onto W , morphism νn,d st N st is actually an i.e. Vn,d = W is a projective variety in P , but also that νn,d st n st ) = Ip (W ) isomorphism, thus Vn,d is isomorphic to P , and also that Ip (Vn,d is generated by homogeneous quadratic polynomials.  Remark 6.5.2. The fact that Vn,d is Zariski closed in PN will more easily follow from some general properties related to completeness of projective varieties (cf. Theorem 9.2.1 and Remark 9.2.2-(iii)). From the fact that νn,d is an isomorphism onto its image, one has the following. Corollary 6.5.3. Let n  1 and d  2 be integers. If Y ⊂ Pn is a projective variety, then νn,d (Y ) is a subvariety of Vn,d and so of PN . The importance and the significance of the Veronese morphisms νn,d (n) allows us to specify in more details Corollary 6.5.3. If F := ΣI aI X I ∈ Sd , the hypersurface Y := Zp (F ) ⊂ Pn is mapped by νn,d to the intersection of Vn,d = νn,d (Pn ) with the hyperplane HF := Zp (ΣI aI ZI ) ⊂ PN , namely the Veronese morphism allows to reduce the study of some problems concerning hypersurfaces of given degree d  2 in Pn to hyperplane sections of the (n) projective variety Vn,d ⊂ PN . If more generally F ∈ Skd , for some integer n k  2, then Y = Zp (F ) ⊂ P is mapped by νn,d to the intersection of Vn,d with the degree-k hypersurface νn,d (Y ) ⊂ PN obtained by the equation of (n) F where one replaces the monomials X I with ZI . Similarly, for F ∈ Sm ,

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where m not a multiple of d, note that one can always find a positive (n) integer a such that Xia−m F ∈ Skd , for some k and for any 0  i  n. Since Y = Zp (F ) = Zp (X0a−m F, X1a−m F, . . . , Xna−m F ) ⊂ Pn (cf. Exercise 3.5) then νn,d (Y ) is cut-out on Vn,d by the corresponding n + 1 hypersurfaces of PN . This extends also to the most general case of Y = Zp (F1 , . . . , Fr ) ⊂ Pn , (n) for some Fi ∈ Smi , 1  i  r (cf. also Harris, 1995, Example 2.7, p. 24). Below we discuss in more details some particular cases. Example 6.5.4 (Rational normal curves of degree d). This is the case V1,d , for any d  2. From Theorem 6.5.1, any V1,d is a non-degenerate curve in Pd isomorphic to P1 . Such a curve is called a rational normal curve of degree d in Pd . Up to a projective transformation of Pd , any such curve st , so we can focus on standard cases. is projectively equivalent to V1,d We have already met some standard rational normal curves; the conic st ⊂ P2 , Zp (X0 X2 − X12 ) ⊂ P2 in Example 6.3.2-(ii) is nothing but V1,2 st 3 whereas the projective twisted cubic in Section 3.3.12 is V1,3 ⊂ P . Any st V1,d is irreducible and non-degenerate in Pd ; by the correspondence among st st and hypersurfaces of degree d in P1 , V1,d turns hyperplane sections of V1,d d out to be a curve of degree d in P . If we restrict the Veronese morphism st to the affine chart U0 ⊂ P1 , we get the morphism ν1,d 0 0 st ν1,d : A1 → V1,d := V1,d ∩ U0 ⊂ U0 ∼ = Ad , t → (t, t2 , t3 , . . . , td ). 0 Therefore V1,d is an affine rational curve with polynomial parametrization, 0 is as in (3.31). With no use of Theorem 6.5.1, one directly gets that V1,d 0 irreducible and isomorphic to A1 , since ν1,d is given by a d-tuple of regular functions on A1 (cf. Proposition 6.2.2) and its inverse morphism simply is 0 of the first projection π1 : Ad → A1 . As for the the restriction to V1,d st is the projective closure in Pd of projective twisted cubic in (3.34), V1,d 0 0 st . Moreover, the isomorphism ν1,d in the affine chart extends to ν1,d as V1,d 1 an isomorphism between P and V1,d . As done in Section 3.3.12 for the st is given by projective twisted cubic, by the proof of Theorem 6.5.1, V1,d the zero-locus of all the quadratic polynomials coming from the maximal minors of the matrix of linear forms   X0 X1 X2 . . . Xd−1 , (6.20) A := X1 X2 X3 . . . Xd st ). which also generate Ip (V1,d

For further readings, see, e.g. Harris (1995, Examples 1.14 and 1.15, pp. 10–11).

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Example 6.5.5 (Veronese surface). Another well-known classical example of Veronese variety is given by V2,2 , which is called Veronese surface. This is a non-degenerate, projective surface in P5 , which is isomorphic to P2 . Furthermore one deduces that its degree is 4, indeed by using the bijective correspondence among conics in P2 and hyperplane sections of V2,2 , two general such hyperplane sections intersect along four distinct points as two general plane conics do. The two-dimensional complete linear system of lines in P2 gives rise to a two-dimensional family of conics contained in V2,2 , whereas the five-dimensional complete linear system of conics in P2 (cf. Exercise 4.5) gives rise to the five-dimensional linear system of hyperplane section curves in V2,2 whose general element (i.e. the image of any conic in C \ D ∼ = P5 \ Σ3 in notation of Exercise 4.5) is a rational normal quartic curve in P4 and so on (see also, e.g. Harris, 1995, Exercise 2.5, p. 23). 6.5.1

Veronese morphism and consequences

One can use the Veronese isomorphism νn,d to prove some interesting general properties of arbitrary algebraic varieties. For example, the next result is the extension to the projective case of what proved in Lemma 6.2.12 for An . Proposition 6.5.6. For any non-negative integers n, d, let W ⊂ Pn be any hypersurface of degree d. Then Pn \ W is an affine variety. Proof. Using the Veronese isomorphism νn,d , Pn \W ∼ = Vn,d \(HW ∩ Vn,d ), where HW ⊂ PN denotes the hyperplane which cuts out the hyperplane section HW ∩ Vn,d isomorphic to the hypersurface W ⊂ Pn of degree d. One concludes by observing that Vn,d \ (HW ∩ Vn,d ) ∼ = Vn,d ∩ (Pn \ HW ) ∼ = N  Vn,d ∩ A . Corollary 6.5.7. If V ⊂ Pn is a projective variety, which is not a point, and W ⊂ Pn is any hypersurface then V ∩ W = ∅. In particular, any two projective plane curves intersect. Proof. If by contradiction one had V ∩W = ∅, choosing d := deg(W ), from the proof of Proposition 6.5.6, we would get V ∼ = νn,d (V ) ⊂ Zp (HW )c ∼ = N A . Then V would be isomorphic to an irreducible, closed subset in AN i.e. to an affine variety. From Corollary 6.2.6, V would be a point against the assumptions. 

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From Corollary 6.2.7, we already know that P2 and A2 cannot be isomorphic; the second part of Corollary 6.5.7 more generally shows that they are not even homeomorphic, as in A2 there actually exist curves with empty intersections. Similarly to Example 6.2.14, with the use of the Veronese isomorphism νn,d we can show that there exist quasi-projective varieties which cannot be either projective, or affine or quasi-affine. To do this, consider first the following preliminary result. Lemma 6.5.8. For any integer n  2, let Λ ⊂ Pn be a linear subspace s.t. dim(Λ)  n − 2. Let W := Pn \ Λ. Then OW (W ) ∼ = K. Proof. Since W ⊂ Pn is an open dense subset of Pn , from Lemma (n) 5.3.13 and Theorem 5.3.14 (h) we have K(W ) ∼ = K(Pn ) = S((0)) . Since OW (W ) ⊂ K(W ), then any regular function φ ∈ OW (W ) is of the form (n) 1 φ = G G2 , for some G1 , G2 ∈ Sd , with d  0 an integer, G2 = 0 and c W ⊂ Up (G2 ) = Zp (G2 ) . Since dim(Λ)  n − 2, reasoning as in the proof  of Claim 6.2.15, we get that G2 ∈ K∗ and so also G1 ∈ K. Example 6.5.9. We can now show that, for any point P ∈ P2 , the algebraic variety W := P2 \{P } is a quasi-projective variety which is neither projective, nor affine nor quasi-affine. By using projective transformations, with no loss of generality we may assume P to be the fundamental point [1, 0, 0]. We first note that W cannot be a projective variety: otherwise, for any line not passing through P , W \ would be affine from Proposition 6.5.6, on the other hand W ∩ (P2 \ ) ∼ = A2 \ {(0, 0)}, contradicting Example 6.2.14. By Lemma 6.5.8, W can be neither an affine variety: otherwise A(W ) = OW (W ) ∼ = K, i.e. Ia (W ) would be maximal i.e. W would be a point, a contradiction. Similarly, W cannot be quasi-affine: otherwise, if W denotes its affine closure, always by Lemma 6.5.8, we would have A(W ) ⊆ OW (W ) ∼ = K and we can argue as before. We conclude this section by observing that, from the proof of Theorem 6.5.1, another fundamental difference among affine and projective varieties enter explicitly into the game. Remark 6.5.10. Corollary 6.2.4 establishes that the coordinate ring A(V ) of any affine variety V is invariant for the isomorphism class represented by V ; in other words, if V ⊂ An and W ⊂ Am are isomorphic affine varieties, then A(V ) ∼ = A(W ), no matter the embedding in different affine spaces. Instead, projective varieties behave differently from this point

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of view; namely for V ⊂ Pn any projective variety, the homogeneous coordinate ring S(V ) in general is not invariant under isomorphism. To be more precise, if V, W ⊆ Pn are projective varieties which are isomorphic via a projectivity Φ ∈ PGL(n + 1, K) sending V to W (as, e.g. st ) it occurs between any Veronese variety Vn,d and its standard model Vn,d then, by Definition 3.2.1, Φ gives rise to an affinity Φ ∈ GL(n + 1, K) of An+1 inducing an isomorphism Ca (V ) ∼ = Ca (W ). Since these are both affine varieties, by Corollary 6.2.4 we have A(Ca (V )) ∼ = A(Ca (W )). Thus, since (n) S as integral K-algebras and since, from (3.8), one has that for A(n+1) ∼ = n any projective variety Z ⊆ P S(Z) =

S(n) ∼ A(n+1) = A(Ca (Z)) = Ip (Z) Ia (Ca (Z))

holds, one deduces that S(V ) ∼ = S(W ), as it occurs for coordinate rings of affine varieties. On the contrary, if V ⊆ Pn and W ⊆ Pm are projective varieties, embedded in different ambient projective spaces, but which are isomorphic as algebraic varieties (as it occurs for Pn and Vn,d ⊂ PN , with d  2) S(V ) is in general not isomorphic to S(W ). This has been implicitly encountered in the proof of Theorem 6.5.1. To let things be easily understood, we discuss in full details the following example. Example 6.5.11. For any d  2, we know that any rational normal curve V1,d ⊂ Pd is isomorphic to P1 . On the one hand, we have S(P1 ) = S(1) := S; on the other, since any V1,d is projectively equivalent inside Pd to the (d) st standard rational normal curve V1,d , we have that S(V1,d ) = IpS(V1,d ) is st isomorphic to the homogeneous coordinate ring S(V1,d ). Now, by definition st st ) := R is of the standard Veronese morphism ν1,d , the graded ring S(V1,d the image of the homogeneous homomorphism θ1,d

θ1,d : S(d) −→ S(1) = S, Z(d−i,i) −→ X0d−i X1i ,

0  i  d,

where Zd,0 , . . . , Z0,d the indeterminates in S(d) whereas X0 , X1 those in S(1) , and the kernel of the previous homomorphism is exactly the ideal generated by the maximal minors of the matrix of linear forms (6.20). Note that θ1,d is not surjective: focusing on the graded summands we get that S0 = R0 but R1 ∼ = Sd and more generally, for any k  1, Rk ∼ = Skd . In other words,

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R is isomorphic to the graded subring of S = S(1) given by   S(1) (d) k , S(1) (d) := k0

  (1) where, as in (1.33), each graded summand is defined as S(1) (d) k := Skd . 6.5.2

Divisors and linear systems

The study of Veronese morphisms suggests a more general approach. Let W be either an affine or a projective space. We will denote by Div(W ) the free abelian group generated by the set Ω of all irreducible hypersurfaces in W . Any element of Div(W ) is of the form D := ΣY ∈Ω rY Y , where rY ∈ Z are integers which are almost all zero except for finitely many of them. Such a D is called a divisor of W , the integer rY is called the multiplicity of Y in D and the hypersurfaces for which rY = 0 are called the irreducible components of D. The hypersurface supp(D) := ∪rY =0 Y is called the support of D. If rY ∈ {−1, 0, 1} for any Y , then D is said to be a reduced divisor ; if rY  0 for any Y in D, then D is said to be an effective divisor ; if D = Y , where Y an irreducible hypersurface, then D is said to be an irreducible divisor. For any D ∈ Div(W ) one defines the degree of D as deg(D) := ΣY ∈Ω rY deg(Y ). If fY = 0 is an equation of Y in Ω (in a chosen system of coordinates of W and moreover homogeneous if W a projective space), given D = ΣY ∈Ω rY Y , one sets fD := ΠY ∈Ω fYrY so that fD = 0 is an equation of D (it is homogeneous in the projective case). By the Hilbert “Nullstellensatz”, two equations of a divisor D differ by a non-zero constant multiplicative factor. Set W = P(V ) of dimension n, endowed with a system of homogeneous coordinates. Let D be a divisor and let Λ ⊂ P(V ) be a linear subspace of dimension m < n. Let fD = 0 be a homogeneous equation of D and assume that Λ has a parametric representation given by X = Aλ, where A a (m+1)×(n+1)-matrix of rank m+1, X the column-vector of homogeneous coordinates of W and λt = [λ0 , . . . , λm ] ∈ Pm homogeneous parameters. The polynomial fD (Aλ) is identically zero if and only if Λ ⊆ supp(D); in such a case, one states that D contains Λ and set Λ ⊆ D. Otherwise, the equation fD (Aλ) = 0 defines a divisor DΛ in Λ, w.r.t. the homogeneous coordinates [λ0 , . . . , λm ] given by the parametrization of Λ. The divisor DΛ is called the intersection of Λ with D. The previous definition is well-posed and deg(DΛ ) = deg(D).

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When instead W = An one has similar definitions and considerations; the main difference is that in general, for an affine subspace Λ, one has deg(DΛ )  deg(D). One can define in an obvious way the projective closure of a divisor D in An . Therefore, one has: Proposition 6.5.12 (Bezout’s theorem for linear sections). If D is a divisor of either an affine or a projective space and if Z is a subspace, then either Z is contained in D or D intersects Z along a divisor DZ of degree at most deg(D); moreover deg(DZ ) = deg(D) in the projective case. Example 6.5.13 (Intersection multiplicity of a line at a point). If W = Pn and if deg(D) = d. Take Z a line not contained in D, then DZ = m1 P1 + · · · + mh Ph , where Pi are distinct points of Z and mi are positive integers, with 1  i  h, such that d = m1 + · · · + mh . The positive integer mi is called the intersection multiplicity between Z and D at the point Pi and it is denoted by μ(D, Z; Pi ). On the other hand, one poses μ(D, Z; P ) = 0 if P ∈ / supp(DZ ) and μ(D, Z; P ) = ∞ for any P ∈ Z, when Z ⊆ D. For simplicity, from now on let us refer to the case W = P(V ) of dimension n. Fix a positive integer d and consider the set LP(V ),d of effective divisors of degree d in W . When W = Pn , we will simply write Ln,d . One has LP(V ),d = P(Symd (V ∗ )). A r-dimensional subspace of LP(V ),d is called a linear system of divisors of degree d in P(V ). Fixing a homogeneous system of coordinates in P(V ) by means of a projectivity φ : Pn → P(V ), one has a homogeneous isomorphism (n) S(n) ∼ = Sym(V ∗ ) which determines isomorphisms Sd ∼ = Symd (V ∗ ), for any d  0. In this set-up one determines bijective projectivities φd : Ln,d → LP(V ),d inducing homogeneous coordinates on LP(V ),d , namely the coordinates [· · · , fI , · · · ]|I|=d (where the multi-indexes I are taken, e.g. with the lexicographic order) of a divisor D are simply the coefficients, up to a nonzero constant multiplicative factor, of the polynomial fD (X) = Σ|I|=d fI X I giving an equation of D in the given system of homogeneous coordinates for P(V ). Let L ⊆ LP(V ),d be a linear system of dimension r. If D0 , . . . , Dr ∈ L are linearly independent divisors, whose homogeneous equations in the given system of homogeneous coordinates are fDi = 0, 0  i  r, then a divisor

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D sits in L if and only if fD = λ0 fD0 + · · · + λr fDr = 0 (up to a non-zero constant multiplicative factor), where [λ0 , . . . , λr ] ∈ Pr . A linear system of dimension 0 consists of only one effective divisor, if dim(L) = 1, respectively, 2, then L is called a pencil, respectively, a web. If L = ∅, one sets dim(L) = −1. In any case, in this set-up,  D ⊂ P(V ) Zp (fD0 , . . . , fDr ) = D∈L

is called the base locus of the linear system L and it is denoted by Bs(L). If P ∈ P(V ), one denotes by L(−P ) the set of all divisors D ∈ L such that P ∈ supp(D) (i.e. effective divisors in L containing the point P ). One has L(−P ) = L if and only if P ∈ Bs(L), namely P is a base point of L. If P ∈ / Bs(L) then L(−P ) is a sub-linear system of L of dimension r − 1. Example 6.5.14 (Linear series on P1 ). A linear system L of dimension r in LP(V ),d , where P(V ) a projective line, is called linear series of degree d and dimension r and will be denoted by grd . If P ∈ P(V ) is not a base point for grd , we know that dim(grd (−P )) = r − 1; therefore one has r  d. Note that LP(V ),d is nothing but the linear series associated to a Veronese morphism ν1,d . This is called complete linear series and it is a gdd which is st . uniquely determined up to projectivities, i.e. gdd corresponds to ν1,d 2 Consider the complete linear series g2 on P(V ) and take a pencil in it, i.e. a g12 . Choosing a homogeneous system of coordinates on P(V ), there exist two linearly independent homogeneous polynomials f0 , f1 , in the homogeneous indeterminates [X0 , X1 ] ∈ P1 ∼ = P(V ) such that any element of the g12 is of the form λ0 f0 +λ1 f1 = 0, where [λ0 , λ1 ] ∈ P1 . The divisors D0 and D1 , associated to f0 and f1 , cannot have the same support otherwise f0 and f1 would be proportional, a contradiction. Therefore, we have two possibilities: if P is a common point of D0 and D1 then P is a base point of the g12 so g12 (−P ) = g12 , i.e. any divisor in the g12 is of the form P + Q, where Q varies in the g11 parametrizing points of P(V ) whereas P is fixed. If otherwise supp(D0 ) ∩ supp(D1 ) = ∅, then the g12 is base point free and any divisor of the g12 is of the form P + QP , where P ∈ g11 varying and, for any such P there exists one and only one QP ∈ P(V ) such that P + QP ∈ g12 . Any base point free g12 induces a natural map σ

σ : P(V ) → P(V ), P −→ QP , which is a bijection and moreover it is an involution, namely σ 2 = IdP(V ) .

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Exercises Exercise 6.1. Let ϕ : A2 → A2 be the map defined by ϕ (x1 , x2 ) −→ (x1 , x1 x2 ). Prove that ϕ is a morphism, find its image and conclude that Im(ϕ) is neither open nor closed in A2 . Exercise 6.2. Let C := Za (x31 + x21 − x22 ) ⊂ A2 . Prove that C is an affine variety and find a parametrization ϕ : A1 → C which is a morphism of affine varieties. Establish if ϕ can be an isomorphism. st ⊂ P5 . Find Exercise 6.3. Consider the standard Veronese surface V2,2 st 2 5 explicit parametric equations of the image via ν2,2 : P → P of the conic st (C) is a rational normal quartic C := Zp (X0 X2 −X12 ) ⊂ P2 . Deduce that ν2,2 4 in P . st : P2 → P9 be the standard Veronese morphism Exercise 6.4. Let ν2,3 of indexes 2, 3. Determine parametric equations of the image of the line := Zp (X1 − X2 ) ⊂ P2 and of the conic C = Zp (X0 X2 − X12 ) ⊂ P2 . st st ( ) and of ν2,3 (C). Determine degrees of ν2,3

Exercise 6.5. Consider any complete linear series gdd on P1 . Take any point P ∈ P1 and consider the sublinear series gdd (−P ) having P as base point. Describe the image of P1 via gdd (−P ) using suitable projections of Im(gdd ).

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Chapter 7

Products of Algebraic Varieties

In this chapter, we will study products of algebraic varieties. Namely, we want to describe how to give an algebraic variety structure to the settheoretical cartesian product V × W , when both V and W are algebraic varieties. 7.1

Products of Affine Varieties

Note that Example 2.1.22-(i) already describes the product of two affine spaces Ar × As , where r and s are non-negative integers. As a set of points, Ar × As has been identified with Ar+s ; in particular, the cartesian product of the two affine spaces has a natural structure of affine variety, the affine space Ar+s , where the Zariski topology on Ar × As = Ar+s is Zarr+s a , which is finer than the product Zarra × Zarsa of the two Zariski topology (recall Example 2.1.22-(iii)). From Corollary 5.3.19, we know that any affine variety is completely determined, together with its structural sheaf, by its ring of coordinates. (r) The coordinate ring of Ar , respectively, of As , is the polynomial ring Ax := (s) K[x1 , . . . , xr ], respectively, Ay := K[y1 , . . . , ys ]. These rings of polynomials are those formed by (evaluating) regular functions which operates on the affine spaces Ar and As , respectively. Similarly, for Ar × As = Ar+s the coordinate ring of regular functions operating on this affine space simply (r+s) is the polynomial ring Ax,y := K[x1 , . . . , xr , y1 , . . . , ys ]. Note further that (r+s) (r) (s) Ax,y ∼ = Ax ⊗K Ay , where the isomorphism is as integral K-algebras of finite type (cf. Section 1.9.2). From Proposition 6.2.3, the two natural K-algebra monomorphisms (r) (r+s) (s) Ax → Ax,y ← Ay correspond to the natural two projections onto 179

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the first and the second factor, respectively, π

1 Ar × As −→ Ar

π

2 and Ar × As −→ As .

(7.1)

These projections are morphisms since they respectively coincide with the projections πI and πJ as in (6.11), with multi-indexes I = (1, 2, . . . , r) and J = (r + 1, . . . , r + s), respectively (cf. Example 6.2.5-(i)). Once the product of two affine spaces is clearly described, one would like to extend previous considerations to the more general case of a product of two affine varieties. Concerning the topological structure, we have the following. Proposition 7.1.1. Let V and W be affine varieties. Then V × W is an affine variety. Proof. Up to isomorphism, we can assume there exist non-negative integers r and s such that V ⊆ Ar and W ⊆ As , where the inclusions are closed immersions as affine varieties (cf. Definition 6.1.9). We will show that the cartesian product V × W is endowed with a structure of Zariski closed, irreducible subset of the affine space Ar+s . The fact that V × W is Zariski closed in Ar+s has been already met in Example 2.1.22-(i). Recall indeed that, if V := Za (f1 , . . . , fn ) = Za (Ia (V )) and W := Za (g1 , . . . , gm ) = Za (Ia (W )), then (2.8) establish that V × W = Za (f1 , . . . , fn ) ∩ Za (g1 , . . . , gm ) = Za (Ia (V )) ∩ Za (Ia (W )), where Ia (V ), Ia (W ) are considered as ideals in A(r+s) , via the natural K-algebra inclusions A(r) → A(r+s) and A(s) → A(r+s) reminded above and where Za (Ia (V )) = V × As is the closed s-dimensional cylinder over V in Ar+s whereas Za (Ia (W )) = Ar × W is the closed r-dimensional cylinder over W in Ar+s . Moreover one has Ia (V × W ) = (f1 , . . . , fn , g1 , . . . , gm ) = Ia (V ) + Ia (W ),

(7.2)

where the ideals Ia (V ) and Ia (W ) are meant as ideals in Ar+s via the previous inclusions. To complete the proof, we are left to showing that V × W is irreducible as topological space. Assume that V × W = Z1 ∪ Z2 where Zi closed subsets of V × W , for 1  i  2. For any point P ∈ V , WP := {P } × W is a subset of V × W which is homeomorphic to W and so irreducible. Then WP ⊆ Zi , for i ∈ {1, 2}. Consider Vi := {P ∈ V : WP ⊆ Zi },

for 1  i  2.

For any point Q ∈ W , pose Vi (Q) := {P ∈ V | (P, Q) ∈ Zi }, for 1  i  2. Then (V × {Q}) ∩ Zi = Vi (Q) × {Q}, so Vi (Q) is closed, for any Q ∈ W

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and for any 1  i  2. Since one has Vi = ∩Q∈W Vi (Q), 1  i  2, then V1 , V2 are closed in V . From V = V1 ∪ V2 and from the irreducibility of V , therefore one has either V = V1 (in which case V × W = Z1 ) or V = V2 (consequently, V × W = Z2 ). In all cases V × W is irreducible.  Since V × W is an affine variety, from Proposition 6.2.3, we are left with the description of its coordinate ring. Proposition 7.1.2. Let V and W A(V × W ) ∼ = A(V ) ⊗K A(W ).

be two affine varieties. Then

Proof. As in the proof of Proposition 7.1.1, up to isomorphism we can assume there exist non-negative integers r and s such that V ⊆ Ar and W ⊆ As , where the inclusion are closed immersions as affine varieties. From Example 2.1.22-(iii) and the fact that V × W = (V × As ) ∩ (Ar × W ) in Ar+s , the natural K-bilinear map α

α : A(r) × A(s) → A(r+s) , (f, g) −→ f g is such that α(Ia (V ) × A(s) ) ⊆ Ia (V × W ) ⊇ α(A(r) × Ia (W )). Therefore, α induces a K-bilinear map β

β : A(V ) × A(W ) → A(V × W ), (f, g) −→ f g. Since the generators of A(V ) and A(W ) as K-algebras of finite type are contained in the image of β and these generate A(V × W ), then β is surjective and any element of A(V × W ) is of the form Σi,j ci,j fi gj , where fi ∈ A(V ), gj ∈ A(W ), ci,j ∈ K. From (7.2), Ia (V × W ) ⊂ A(r+s) is generated by the elements in α(Ia (V ) × A(s) ) and in α(A(r) × Ia (W )), in other words A(V × W ) ∼ =

A(r+s) , Ia (V ) + Ia (W )

where Ia (V ) and Ia (W ) considered as ideals in A(r+s) . We are left to showing that A(V × W ) is isomorphic to A(V ) ⊗K A(W ); to do so, let U be any K-vector space and let γU : A(V )×A(W ) → U be any K-bilinear map with target U . Consider the map δU : A(V × W ) → U defined as follows: δU (Σi,j ci,j fi gj ) := Σi,j ci,j γU (fi , gj ). Note that γU = δU ◦ β, where β : A(V ) × A(W ) → A(V × W ) the K-bilinear map defined above.

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Moreover, δU is well-posed, K-linear and uniquely determined by γU and β. From Proposition 1.9.1, it follows that A(V × W ) ∼ = A(V ) ⊗K A(W ) as K-vector spaces. Since A(V ) and A(W ) are integral K-algebras of finite type, from Proposition 1.9.7, A(V × W ) is also a K-algebra of finite type. Finally, the integrality of A(V × W ) directly follows from the fact that V × W is irreducible.  Remark 7.1.3. As for the product of two affine spaces, the two natural projections π

V V V × W −→

π

W and V × W −→ W

(7.3)

are morphisms of affine varieties since πV = π1 ◦ιV ×W and πW = π2 ◦ιV ×W where π1 and π2 as in (7.1) whereas ιV ×W the closed immersion of V × W in Ar+s (cf. Definition 6.1.9). 7.2

Products of Projective Varieties

Let us consider more generally V and W algebraic varieties. Thus, for some integers n and m, we have V ⊆ Pn , W ⊆ Pm , where the inclusions are locally closed morphisms. To endow the (set theoretical) cartesian product V × W with a structure of algebraic variety, it will suffice to define a set-theoretical injective map Ψ

V × W → PN ,

(7.4)

for some positive integer N , such that: (i) ψ(V × W ) ⊂ PN is a quasi-projective variety, (ii) for any affine open subsets UV ⊆ V and UW ⊆ W (recall Proposition 6.4.2), Ψ(UV × UW ) is an affine open subset of Ψ(V × W ); moreover such open sets determine an open covering of Ψ(V × W ), (iii) for any choice of affine open subsets UV ⊆ V and UW ⊆ W , the map Ψ|UV ×UW : UV × UW −→ Ψ(UV × UW ) is an isomorphism of affine varieties. Note that (iii) is nothing but a compatibility condition between the structure of algebraic variety on Ψ(V × W ) as in (i) and the one given in Section 7.1 on its affine open subsets as in (ii). More precisely, the

Products of Algebraic Varieties

183

subset Ψ(UV × UW ), which is an affine open subset w.r.t. the structure of algebraic variety of ψ(V × W ) ⊂ PN as in (i), has to be isomorphic (as affine variety) to the affine variety UV × UW , whose structure has been defined in Section 7.1. Lemma 7.2.1. Given a set-theoretical injective map Ψ as in (7.4), with Ψ(V ×W ) ⊂ PN a quasi-projective variety, then Ψ satisfies the compatibility condition (iii) above if and only if, for any points P ∈ V and Q ∈ W, there exist UP ⊆ V and UQ ⊆ W affine open neighborhoods of P ∈ V and Q ∈ W, respectively, s.t. Ψ|UP ×UQ defines an isomorphism of the affine variety UP × UQ onto its image Ψ(UP × UQ ) ⊆ PN and Ψ(UP × UQ ) is an affine open subset of ψ(V × W ). Proof. The implication (⇒) is trivial. For the converse, let UV ⊆ V and UW ⊆ W be any affine open subsets. Since UV = ∪P ∈UV UP and UW = ∪Q∈UW UQ , from Section 7.1 we have that {UP × UQ }(P,Q)∈UV ×UW is an open covering of UV × UW , formed by affine open sets by assumptions. The map Ψ|UV ×UW is therefore continuous and it is an isomorphism as it follows Proposition 6.4.1.  A map Ψ for which conditions in Lemma 7.2.1 hold true is said to satisfy the property of being local. Lemma 7.2.2. If a map Ψ as above exists, it is uniquely determined (up to isomorphism) by the property of being local. Proof. Assume there exist a map Ψ as above and another injective map Φ : V ×W → PM , for some integer M , both of them satisfying the property of being local. Thus Φ◦Ψ−1 : Ψ(V ×W ) → Φ(V ×W ) is bijective. It suffices to show that Φ ◦ Ψ−1 is a morphism. The proof of this is identical to that of Lemma 7.2.1, using the fact that both Ψ and Φ satisfy the property of being local.  The previous result states that, to give a structure of algebraic variety on V × W , one is reduced to showing the existence of a map Ψ satisfying conditions (i), (ii) above and the property of being local and that, once such a map has been constructed, the structure on V × W is uniquely determined up to isomorphisms. The aim of the next section is to construct such a map Ψ.

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Segre morphism and the product of projective spaces

7.2.1

Let us consider first V = Pn and W = Pm , for some non-negative integers n and m. Definition 7.2.3. We define the Segre map of indexes n and m by σn,m :

Pn × Pm

−→

PN

([x0 , . . . xn ], [y0 , . . . , ym ]) −→ [x0 y0 , x0 y1 , . . . , xi yj , . . . , xn ym ], where N := (n + 1)(m + 1) − 1, with 0  i  n and 0  j  m. This map is well-defined and the image Σn,m := Im(σn,m ) is called the Segre variety of indexes n and m. The use of the term “variety” is justified by the following result. Lemma 7.2.4. The Segre map σn,m is bijective and its image Σn,m is a projective variety in PN . Proof. Take homogeneous coordinates [Wij ] in PN , with 0  i  n and 0  j  m. These coordinates are lexicographically ordered so that they are compatible with σn,m . Thus, points in Σn,m satisfy homogeneous quadratic equations Wij Wkr − Wir Wkj = 0,

0  i, k  n, 0  j, r  m.

(7.5)

Let Z be the closed subset of PN cut out by quadric hypersurfaces as in (7.5); it is clear that Σn,m ⊆ Z. We claim that, for any point R ∈ Z, there is a unique pair (P, Q) ∈ Pn × Pm such that R = σn,m (P, Q); this will imply that Σn,m = Z and that σn,m bijectively maps Pn × Pm onto 0 ] ∈ Z be any point. Without loss of Σn,m . To prove the claim, let R := [wij 0 generality, we can assume w00 = 0 and so, by rescaling all the coordinates 0 = 1 (the other cases can up to this multiplicative non-zero factor, that w00 be handled analogously); since R ∈ Z then 0 0 0 0 0 w0j = wij w00 = wij . wi0

Setting 0 0 , . . . , wn0 ] ∈ Pn P := [1, w10

0 0 and Q := [1, w01 , . . . , w0m ] ∈ Pm ,

this means that R = σn,m ((P, Q)) and that P and Q are uniquely determined.

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We are left to showing that Σn,m is irreducible. For any 0  i  n and 0  j  m, denote by UijN := PN \ Zp (Wij ) the principal affine open set in PN isomorphic to AN and, similarly, An ∼ = Zp (X0 )c := U0n ⊂ Pn and m∼ c m m N = A = Zp (Y0 ) := U0 ⊂ P . Previous computations show that Σn,m ∩ U00 n m n m σn,m (U0 × U0 ) and that σn,m |U0 ×U0 is an isomorphism from the affine N , variety U0n × U0m ∼ = An × Am ∼ = An+m onto the affine closed set Σn,m ∩ U00 which is therefore an affine variety. Similar conclusion holds for Σn,m ∩ UijN , for any pair of indexes (i, j) with 0  i  n and 0  j  m.   One has an open covering of Σn,m , Σn,m = ∪i,j Σn,m ∩ UijN , where each Σn,m ∩ UijN is an affine open subset of Σn,m . Moreover, W := ∩i,j Σn,m ∩ UijN = ∅ is an open subset of Σn,m , which is also open in any affine open subset Σn,m ∩ UijN , for any i, j as above; in particular, W is irreducible. Since W is contained in any open set of an affine open covering of Σn,m as a dense open subset, this forces Σn,m to be irreducible.  As for the Veronese varieties, the fact that Σn,m is closed in PN is also a consequence of the completeness of projective varieties, which will be discussed later on (cf. Theorem 9.2.1 and Remark 9.2.2-(iii)). At last, we have: Lemma 7.2.5. The bijection σn,m : Pn × Pm → Σn,m satisfies condition (ii) as in (7.4) and the property of being local. Proof. The proof of Lemma 7.2.4 shows that {σn,m (Uin × Ujm )}i,j is an open covering of the projective variety Σn,m , where each σn,m (Uin × Ujm ) is isomorphic to the affine variety Uin × Ujm , i.e. condition (ii) of (7.4) for the map σn,m holds. Moreover, for any pair of indexes (i, j), one has that Σn,m ∩ UijN is an affine open neighborhood of any of its points over which the Segre map is an isomorphism with Uin × Ujm . From Lemma 7.2.1, σn,m satisfy the property of being local.  From Lemma 7.2.2, previous results determine (up to isomorphism) a structure of projective variety on Pn × Pm , which from now on, will be always identified with Σn,m by means of the Segre morphism σn,m . Remark 7.2.6. (i) The two natural projections π

π

1 2 Pn and Pn × Pm −→ Pm Pn × Pm −→

(7.6)

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are morphisms of projective varieties. This follows from Proposition 6.4.1 and the fact that their restrictions to the affine open subsets U0n × U0m are morphisms (cf. Remark 7.1.3). −1 N ∪j Σn,m ∩ U0j , therefore it is an open (ii) Note that U0n × Pm = σn,m n m subset of P × P . Namely, identifying An with the affine chart U0n ⊂ Pn , the Segre map σn,m defines a structure of quasi-projective variety on An × Pm . (iii) Identify An with the affine open set U0n ⊂ Pn and similarly Am with the affine open set U0m ⊂ Pm . Therefore, U0n × U0m is isomorphic to An × Am = An+m . With explicit computations, if we consider Pn × Pm as the Segre variety Σm,n ⊂ PN , where N = (n + 1)(m + 1) − 1, then U0n × U0m coincides with Σm,n \ (Σm,n ∩ Zp (W00 )), therefore, for any [w] ∈ U0n × U0m ⊂ PN , the following map   w10 wn0 w01 w0m ,..., , ,..., U0n × U0m → An+m , [w] → w00 w00 w00 w00 is the required isomorphism. 7.2.2

Products of projective varieties

Consider the following preliminary result. Proposition 7.2.7. (i) A subset T ⊆ Pn ×Pm is closed if and only if it is defined by polynomials Gk (X0 , . . . , Xn , Y0 , . . . , Ym ), 1  k  s, where each polynomial Gk is bi-homogeneous, i.e. it is separately homogeneous with respect to the two set of indeterminates (X0 , . . . , Xn ) and (Y0 , . . . , Ym ). (ii) A subset T ⊆ An ×Pm is closed if and only if it is defined by polynomials Fk (x1 , . . . , xn , Y0 , . . . , Ym ), 1  k  s, where each polynomial Fk is homogeneous with respect to the set of indeterminates (Y0 , . . . , Ym ). Proof. (i) Take any closed subset Z ⊂ PN , where N = (n + 1)(m + 1)− 1. Then Z := Zp (A1 (W00 , . . . , Wnm ), . . . , As (W00 , . . . , Wnm )), for some homogeneous polynomials Ak ∈ H(S (N ) ), 1  k  s. The closed subset −1 (Z) ⊂ Pn × Pm is defined by the equations σnm Gk (X0 , . . . , Xn , Y0 , . . . , Ym ),

1  k  s,

obtained by applying to the polynomials Ak ’s the indeterminate substitutions Wij = Xi Yj , 0  i  n, 0  j  m. Each of the polynomials

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G1 , . . . , Gs are separately homogeneous with respect to the two set of indeterminates (X0 , . . . , Xn ) and (Y0 , . . . , Ym ). Conversely, any polynomial G(X0 , . . . , Xn , Y0 , . . . , Ym ) which is homogeneous of degree d in the set of indeterminates (X0 , . . . , Xn ) and of degree r in the set of indeterminates (Y0 , . . . , Ym ) determines the same closed subset T ⊂ Pn × Pm defined by the vanishing locus of the set of polynomials Yjd−r G(X0 , . . . , Xn , Y0 , . . . , Ym ), 0  j  m,

(7.7)

where we assumed d  r. All polynomials in (7.7) are now homogeneous of the same degree d with respect to both set of indeterminates (X0 , . . . , Xn ) −1 (Z), where Z is the closed subset and (Y0 , . . . , Ym ). Therefore T = σn,m in PN defined by homogeneous polynomials in the indeterminates Wij obtained by polynomials in (7.7) via the indeterminate substitutions Wij = Xi Yj , 0  i  n, 0  j  m. (Note that the indeterminate substitutions are not uniquely determined, so Z is not uniquely determined; on the other hand, two different choices differ by elements of the ideal generated by polynomials as in (7.5), so Z ∩ Σn,m is independent from the choices considered.) (ii) An × Pm is open in Pn × Pm since it is Zp (X0 )c ; therefore, the assertion follows by considering suitable dehomogenization of a system of  equations for T ⊂ Pn × Pm , which is a closed subset as proved in (i). Let now V ⊂ Pn and W ⊂ Pm be projective varieties. We will show that V × W is a closed subvariety of Pn × Pm ; from Section 7.2.1, this will endow V × W with a structure of projective variety. Suppose that Ip (V ) is generated by homogeneous polynomials Ph (X0 , . . . , Xn ) = 0,

h = 1, . . . , s

and that Ip (W ) is generated by homogeneous polynomials Qk (Y0 , . . . , Ym ) = 0,

k = 1, . . . , t.

Then V × W is defined by the system of homogeneous polynomials Ph (X0 , . . . , Xn ) = 0 = Qk (Y0 , . . . , Ym ),

h = 1, . . . , s, k = 1, . . . , t.

From Proposition 7.2.7 (i), V × W is closed in Pn × Pm . We are therefore left to showing that V × W is irreducible. If either V or W is a point, then V × W is irreducible since homeomorphic to an irreducible topological space. Assume therefore that neither V nor W are points; without loss

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of generality, we can assume that V0 := V ∩ U0n = ∅ and W0 := V ∩ U0m = ∅. Now V0 × W0 is an affine variety so it is irreducible. Any polynomial G(X0 , . . . , Xn , Y0 , . . . , Ym ) which is separately homogeneous with respect to the two sets of indeterminates (X0 , . . . , Xn ) and (Y0 , . . . , Ym ) and which vanishes along V0 × W0 is such that, for any [q0 , . . . , qm ] ∈ W0 , G(X0 , . . . , Xn , q0 , . . . , qm ) ∈ Ip (V0 ) = Ip (V ) (recall (3.21)), i.e. G(X0 , . . . , Xn , Y0 , . . . , Ym ) vanishes also along V × W0 . Similarly, it vanishes along V0 × W , so it also vanishes along V × W . In p other words, V0 × W0 = V × W , which implies that V × W is irreducible. Remark 7.2.8. The two natural projections πV : V × W −→ V and πW : V × W −→ W

(7.8)

are morphisms of projective varieties since they are compositions of morphisms, i.e. πV = π1 ◦ ιV ×W and πW = π2 ◦ ιV ×W where π1 and π2 projections as in (7.6) whereas ιV ×W the closed immersion of V × W into Pn × Pm ∼ = Σn,m ⊂ PN , N = (n + 1)(m + 1) − 1 (cf. Definition 6.1.9). 7.3

Products of Algebraic Varieties

Previous considerations allow to easily extend similar reasoning to the more general case of algebraic varieties, i.e. quasi-projective varieties. Let V ⊂ Pn and W ⊂ Pm be any quasi-projective varieties. Denote by V and W their projective closures, respectively. Then C := V \V is a closed subset of V as well as D := W \ W is a closed subset of W . Therefore, one has   V × W = (V \ C) × (W \ D) = (V × W ) \ (V × D) ∪ (C × W ) . The previous equality shows that V × W is an open set of the projective variety V × W ; thus V × W has a natural structure of a quasi-projective variety. The fact that the natural projections πV : V × W → V

and πW : V × W → W

are morphisms of quasi-projective varieties follows from the same reasoning as in Remark 7.2.8, where ιV ×W is a locally closed immersion of V × W into Pn × Pm ∼ = Σn,m ⊂ PN , N = (n + 1)(m + 1) − 1 (cf. Definition 6.1.9).

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7.4

Products of Morphisms

Let V, W, Z be algebraic varieties and ϕ ∈ Morph(Z, V ) and ψ ∈ Morph(Z, W ). Define set-theoretically the map ϕ × ψ : Z → V × W,

ϕ×ψ

P −→ (ϕ(P ), ψ(P )),

∀ P ∈ Z.

(7.9)

This is the unique map which let the diagrams ϕ×ψ

Z −→ V × W πV ϕ V

ϕ×ψ

and

Z −→ V × W πW ψ W

to be commutative, where πV , respectively, πW , the projection onto the first, respectively, the second, factor of the product V × W . In this set-up, one has the following. Proposition 7.4.1. Let V, W, Z be algebraic varieties and ϕ ∈ Morph(Z, V ) and ψ ∈ Morph(Z, W ). Then ϕ× ψ ∈ Morph(Z, V × W ). Proof. Up to isomorphisms, we may assume that Z ⊆ Pr , V ⊆ Pn and W ⊆ Pm are quasi-projective varieties, for some non-negative integers r, n, m. By Proposition 6.3.1, for any point P ∈ Z there exist an open neighborhood UP of P , positive integers h and k and homogeneous (r) (r) polynomials F0 , . . . , Fn ∈ Sk , G0 , . . . , Gm ∈ Sh such that, for any P  ∈ UP there exist i ∈ {0, . . . , n} and j ∈ {0, . . . , m}, for which Fi (P  ) = 0 and Gj (P  ) = 0, such that ϕ(P  ) = [F0 (P  ), . . . , Fn (P  )] ∈ V ⊆ Pn ψ(P  ) = [G0 (P  ), . . . , Gm (P  )] ∈ W ⊆ Pm , ∀ P  ∈ UP . Therefore, for any P  ∈ UP , one has (ϕ × ψ)(P  ) = [. . . , Fi (P  ) · Gj (P  ), . . .] ∈ V × W ⊆ Σm,n ⊂ PN , (r)

where N = (n + 1)(m + 1) − 1 and where Fi · Gj ∈ Sk+h , 0  i  n, 0  j  m. Once again from Proposition 6.3.1, it follows that ϕ × ψ is a morphism.  As a consequence we have the following useful. Theorem 7.4.2. Let V, W be algebraic varieties. If X is an algebraic variety for which there exist ρ1 ∈ Morph(X, V ), ρ2 ∈ Morph(X, W ) such

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A First Course in Algebraic Geometry and Algebraic Varieties

that, for any algebraic variety Z and for any pair (ϕ, ψ) ∈ Morph(Z, V ) × Morph(Z, W ), there exists a unique η ∈ Morph(Z, X) which let the diagrams η

Z −→ X ρ1 ϕ V

η

and

Z −→ X ρ2 ψ W

to be commutative, then there exists a unique isomorphism α : X → V × W for which the diagrams α

X −→ V × W πV ρ1 V

α

and

X −→ V × W πW ρ2 W

(7.10)

are commutative, where πV and πW the projections onto the factors of V × W. Proof. From Proposition 7.4.1, α := ρ1 × ρ2 ∈ Morph(X, V × W ) is uniquely defined by ρ1 and ρ2 and it is a morphism which let the diagrams (7.10) to be commutative. We need to show that α is an isomorphism. From the assumptions, there exists a unique β ∈ Morph(V × W, X) for which the diagrams β

V × W −→ X ρ1 πV V

β

and

V × W −→ X ρ2 πW W

are commutative. Then β ◦ α ∈ Morph(X, X) let the diagrams β◦α

X −→ X ρ1 ρ1 V

β◦α

and

X −→ X ρ2 ρ2 V

to be commutative. By uniqueness, β ◦ α = IdX ; similarly one shows that  α ◦ β = IdV ×W , so α is an isomorphism whose inverse is β. Corollary 7.4.3. (i) If V ∼ = V  and W ∼ = W  are algebraic varieties, then V × W ∼ = V  ×W  as algebraic varieties. (ii) If V and W are algebraic varieties, then V × W ∼ = W × V as algebraic varieties.

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(iii) If V, W and Z are algebraic varieties, then V ×(W ×Z) ∼ = (V ×W )×Z; in particular one can simply writes V × W × Z. More generally, if V1 , V2 , . . . , Vn are algebraic varieties, then V1 × V2 × · · · Vn , defined as ((((V1 × V2 ) × V3 ) × · · · ) × Vn ), is isomorphic to Vi1 × · · · × Vin , where (i1 , . . . , in ) is any permutation of {1, . . . , n}. (iv) For any point P ∈ V, the subset of V × W given by πV−1 (P ) := {(P, Q) ∈ V × W | ∀ Q ∈ W } is a closed subvariety of V × W which is isomorphic to W . Similarly, for any point Q ∈ W, −1 (Q) := {(P, Q) ∈ V × W | ∀ P ∈ V } πW

is a closed subvariety of V × W which is isomorphic to V . (v) If V, W and Z are algebraic varieties, a map ϕ : Z → V × W is a morphism if and only if πV ◦ ϕ and πW ◦ ϕ are morphisms, where πV and πW the projections onto the factors of V × W . Proof. Left as an exercise to the reader (cf. Exercise 7.2). 7.5



Diagonals, Graph of a Morphism and Fiber-Products

From the previous analysis, one can deduce further interesting properties of algebraic varieties. Let V be an algebraic variety; the subset Δ(V ) := {(P, P ) | P ∈ V } ⊂ V × V

(7.11)

is called the diagonal of V × V . Proposition 7.5.1. For any algebraic variety V, Δ(V ) is a closed subvariety of V × V which is isomorphic to V . Proof. Up to isomorphism, we may assume that V ⊆ Pn , for some integer n, as a quasi-projective variety. Thus Δ(V ) = Δ(Pn ) ∩ (V × V ). To show that Δ(V ) is closed in V × V it is therefore enough to prove that Δ(Pn ) is closed in Pn × Pn . To show this, it suffices to noticing that Δ(Pn ) = Zp (X1 Y0 − X0 Y1 , . . . , Xn Y0 − X0 Yn ) and then one concludes by using Proposition 7.2.7-(i). To show that Δ(V ) is irreducible and isomorphic to V observe that we have a morphism of algebraic varieties IdV × IdV : V −→ V × V, P

IdV ×IdV

−→

(P, P ),

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whose image is ΔV . Therefore, Δ(V ) is irreducible by Corollary 4.1.6 and so it is a closed subvariety of V × V . Let now ιΔ(V ) be the closed immersion of Δ(V ) into V × V and let π1 : V × V → V be the projection onto, e.g. the first factor. Then α := π1 ◦ ιΔ(V ) ∈ Morph(Δ(V ), V ) which is the inverse  of IdV × IdV , showing that Δ(V ) ∼ =V. Recalling Proposition 6.4.2 and using a similar strategy as in Proposition 7.5.1, one can also prove the following. Lemma 7.5.2. Let V be an algebraic variety and U1 and U2 be non-empty affine open subsets of V . Then U1 ∩ U2 is an affine open subset of V . Proof. One needs only to show that U1 ∩ U2 is isomorphic to a closed subset in some affine space. By assumption Ui is isomorphic to an affine variety Vi ⊆ Ani , for some non-negative integer ni , 1  i  2. Thus, U1 × U2 ⊂ An1 +n2 is an affine variety. One concludes by observing that U1 ∩ U2 = (U1 × U2 ) ∩ Δ(An1 +n2 ) and that Δ(An1 +n2 ) is closed in An1 +n2 .



Let now ϕ : V → W be a morphism of algebraic varieties; the set Γϕ := {(P, ϕ(P )) | P ∈ V } ⊆ V × W,

(7.12)

is called the graph of the morphism ϕ. Proposition 7.5.3. For any algebraic varieties V and W and for any ϕ ∈ Morph(V, W ), Γϕ is a closed subvariety of V × W which is isomorphic to V . Proof. From Proposition 7.5.1, Δ(W ) is closed in W × W . Now ϕ × IdW : V × W → W × W is a morphism since it is the product of the two morphisms ϕ and IdW (cf. Proposition 7.4.1). Since Γϕ = (ϕ × IdW )−1 (Δ(W )), the continuity of (ϕ × IdW ) ensures that Γϕ is closed in V × W . Let ιΓϕ be the closed immersion of Γϕ into V × W , then the morphism πV ◦ ιΓϕ : Γϕ → V is actually an isomorphism, whose inverse is given by the morphism IdV × ϕ : V → V × W . This at once shows that Γϕ is irreducible, so it is a subvariety of V × W , which is isomorphic to V . 

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193

Let now V, W and Z be algebraic varieties and let ϕ ∈ Morph(V, Z) and ψ ∈ Morph(W, Z). Consider the subset of V × W defined as follows: V ×Z W := {(P, Q) ∈ V × W | ϕ(P ) = ψ(Q)} ⊆ V × W ;

(7.13)

this is called the fiber-product of V and W over Z (w.r.t. the morphisms ϕ and ψ). Proposition 7.5.4. Let V, W and Z be algebraic varieties let ϕ ∈ Morph(V, Z) and ψ ∈ Morph(W, Z). Then the fiber-product V ×Z W is a closed subset of V × W . Proof. Since ϕ and ψ are morphisms, from Proposition 7.4.1 it follows that ϕ × ψ is a morphism. Moreover, from Proposition 7.5.1, Δ(Z) is closed in Z × Z. Thus, by the continuity of (ϕ × ψ), one concludes since V ×Z W =  (ϕ × ψ)−1 (Δ(Z)). Remark 7.5.5. In general V ×Z W is not irreducible; for example if V and W are subvarieties of Z and if ϕ = ιV : V → Z and ψ = ιW : W → Z are respective immersions as subvarieties, then V ×Z W = V ∩ W and we know that in general intersections of subvarieties may be reducible (cf. Remark 4.2.10). On the other hand, in some cases, it can happen that V ×Z W is irreducible and so a closed subvariety of V × W ; e.g. it occurs when W = Z and ψ = IdW indeed in such a case V ×Z W = Γϕ the graph of the morphism ϕ : V → W , which is irreducible by Proposition 7.5.3. Exercises Exercise 7.1. Describe the Segre variety Σ1,1 and show that it is doubly ruled, namely it contains two distinct one-dimensional families of lines where lines belonging to the same family are skew whereas any line of the first family intersects any line of the second family. Exercise 7.2. Prove Corollary 7.4.3. Exercise 7.3. Let K be an algebraically closed field and let A and B be integral K-algebras of finite type. Show that A⊗K B is an integral K-algebra of finite type. Deduce in an algebraic way that if V and W are affine varieties defined over an algebraically closed field, then V × W is an affine variety.

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Exercise 7.4. Find a counterexample to Exercise 7.3 when K is not algebraically closed. Exercise 7.5. Find examples, different from those in Remark 7.5.5, of algebraic varieties V , W and Z and of morphisms ϕ ∈ Morph(V, Z) and ψ ∈ Morph(W, Z) such that V ×Z W turns out to be irreducible (respectively, reducible).

Chapter 8

Rational Maps of Algebraic Varieties

In this chapter, we introduce the notion of rational maps, birational maps and birational equivalence of algebraic varieties, which are mile-stone concepts for the classification of algebraic varieties. As we will see, a rational map is a morphism which is defined only on some non-empty, open subset of an algebraic variety. On the other hand, since any non-empty, open set in an algebraic variety is dense, this already carries a lot of information about the map. 8.1

Rational and Birational Maps

We start with the following preliminary result. Lemma 8.1.1. Let V and W be algebraic varieties, let ϕ, ψ ∈ Morph(V, W ) and assume there exists a non-empty, open subset U ⊆ V s.t. ϕ|U = ψ|U . Then ϕ = ψ on V . Proof. ϕ and ψ determine the morphism ϕ × ψ : V → W × W (cf. Proposition 7.4.1). From the assumption, one has (ϕ × ψ)(U ) ⊆ Δ(W ), where Δ(W ) the diagonal in W × W (cf. (7.11)). Since U is dense in V and ϕ × ψ is continuous, then (ϕ × ψ)(U ) = (ϕ × ψ)(V ) ⊆ (ϕ × ψ)(U ) ⊆ Δ(W ), the latter inclusion following from the fact that Δ(W ) is closed in W × W (cf. Proposition 7.5.1). This implies ϕ = ψ. 

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Definition 8.1.2. Let V and W be algebraic varieties. A rational map Φ between V and W , denoted by Φ : V  W , is an equivalence class of pairs (U, ϕU ), where U ⊆ V is a non-empty, open subset of V , ϕU ∈ Morph(U, W ) and where (U1 , ϕU1 ) and (U2 , ϕU2 ) are said to be equivalent if ϕU1 and ϕU2 agree on U1 ∩ U2 . Thus, Φ = [U, ϕU ], where [U, ϕU ] the equivalence class of the pair (U, ϕU ). In such a case, ϕU is said to be a representative morphism of Φ over the open set U . If Φ is a rational map and if (U1 , ϕU1 ), (U2 , ϕU2 ) are representative morphism of Φ, then (U1 ∪U2 , ϕU1 ∪U2 ) is another representative of Φ, where ϕU1 ∪U2 |U := ϕUi for i = 1, 2. The previous observation shows that there i exists a non-empty, open subset of V , say UΦ , which is the biggest open subset of V where Φ|UΦ is a morphism. This open set is called the open set of definition (or even the domain) of the rational map Φ; sometimes UΦ is also denoted by Dom(Φ). From Examples 5.3.23-(i), (iii), 6.2.5-(iv) and Exercise 5.1, the maps φ1 and φ3 are rational maps s.t. Uφ1 = Uφ3 = W = A1 \ {0}. Similarly, from Example 6.3.2-(i), when B = ∅, the map νL as in (6.16) defines a rational map νL : Pn  Pr such that UνL = Pn \ B. In general the composition of rational maps is not well-defined. For example, consider the map ϕ : A1 → A2 , x1 → (x1 , 0), which is a morphism, and Φ : A2  A1 , (x1 , x2 ) 

x1 , x2

which is a rational map whose domain is UΦ = A2 \ Za (x2 ). From the fact that ϕ(A1 ) ∩ UΦ = ∅, the composition Φ ◦ ϕ is not defined. To avoid this kind of phenomena, one gives the following definition. Definition 8.1.3. A rational map Φ : V  W is said to be dominant if for some (and so every) representative (U, ϕU ) is a dominant morphism in the sense of Definition 6.1.5. To have some examples, note that on any algebraic variety V , dominant rational maps Φ : V  A1 are nothing but rational functions Φ ∈ K(V ) \ K. Furthermore, if V is a (n + 1)-dimensional K-vector space and if we identify it with the affine space An+1 , the map π in (3.1) is a morphism (as it follows from Proposition 6.3.1) so it defines a rational surjective (so dominant) map π : An+1  Pn such that Dom(π) = An+1 \ {0}.

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Lemma 8.1.4. Let V, W, Z be algebraic varieties and let Φ : V  W and Ψ : W  Z be dominant rational maps. Then Ψ ◦ Φ : V  Z is a dominant rational map. Proof. By assumptions on Φ and Ψ, the representatives (UΦ , ϕUΦ ) and (UΨ , ψUΨ ) are dominant morphisms (which for simplicity will be simply denoted in the sequel by ϕ and ψ, respectively). Then Im(ϕ) ∩ UΨ = ∅ and it contains a dense open subset U0 of UΨ , which is therefore also an open subset of W . Then ϕ−1 (U0 ) ⊆ V is an open subset where the composition ψ ◦ ϕ is defined. On this open set ψ ◦ ϕ is a dominant morphism. Indeed, by the dominance of Ψ, for any proper closed subset K  Z one has that Im(ψ) is not contained in K and ψ −1 (K) is a proper closed subset of W . For the same reason, Im(ϕ) is not contained in ψ −1 (K), so Im(ψ ◦ ϕ) is not contained in K  Z; since this holds for any proper closed subset K  Z, this implies that Ψ ◦ Φ is a dominant rational map.  Definition 8.1.5. A birational map Φ : V  W is a dominant rational map which admits an inverse rational map, namely a rational map Ψ : W  V s.t. Ψ ◦ Φ = IdV and Φ ◦ Ψ = IdW , where the previous equalities are intended as rational maps. If there exists a birational map from V to W , then V and W are said to be birationally equivalent (or simply birational). By the very definition of birational maps, V and W are birational varieties if and only if there exist non-empty, open subsets UV ⊆ V and UW ⊆ W which are isomorphic. For this reason, birational maps are sometimes called also birational isomorphisms. Theorem 8.1.6. Let V and W be algebraic varieties. There is a bijective correspondence between: (i) the set of dominant rational maps V  W, and (ii) the set of K-algebra monomorphisms K(W ) → K(V ). In this bijective correspondence, birational maps correspond to field isomorphisms. Proof. Let Φ : V  W be any dominant rational map; since (UΦ , ϕ) is a dominant morphism, from (6.5) we get a K-algebra monomorphism ϕ∗ : K(W ) → K(UΦ ). We then conclude by K(UΦ ) ∼ = K(V ). Conversely, let θ

K(W ) → K(V ) be any K-algebra monomorphism. From Proposition 6.4.2,

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W is covered by affine open subsets. Thus, by Lemma 5.3.13-(ii) we can assume W to be affine. Let therefore y1 , . . . , yn be generators of the affine coordinate ring A(W ) as a K-algebra of finite type. Since A(W ) ⊆ K(W ), then θ(y1 ), . . . , θ(yn ) ∈ K(V ). There exists a non-empty, open subset U ⊆ V s.t. θ(y1 ), . . . , θ(yn ) ∈ OV (U ) ⊂ K(V ) (it suffices to take the intersection of all the domains of definition of the rational functions θ(yi ), 1  i  n). θ|

This defines a K-algebra homomorphism A(W ) −→ OV (U ), yi → θ(yi ) which is injective, since θ is. Since W is affine, from Proposition 6.2.3, the homomoprhism θ| corresponds to ϕθ ∈ Morph(U, W ). By the injectivity of θ| and by Corollary 6.2.8, we get that ϕθ is dominant. This correspondence defines therefore a dominant rational map Φ : V  W , with representative ϕθ , which is such that (ϕθ )∗ = θ, i.e. this correspondence is the inverse of that reminded at the beginning of the proof.  Corollary 8.1.7. (i) V and W are birationally equivalent if and only if K(V ) ∼ = K(W ). (ii) Any algebraic variety V is birationally equivalent to any of its nonempty, open subset. (iii) Any algebraic variety V is birationally equivalent to an affine variety and to a projective variety. Proof. (i) This directly follows from Theorem 8.1.6. (ii) Since for any non-empty, open subset U ⊆ V one has K(U ) ∼ = K(V ), the statement follows from (i). (iii) From Proposition 6.4.2, any algebraic variety has an affine open set U . Moreover, since U is affine, its projective closure U is a projective variety for which U is an open dense subset. Then one concludes  by (ii). Remark 8.1.8. (i) Birationality is an equivalence relation among algebraic varieties. For any algebraic variety V , the symbol [V ]bir will denote the equivalence class consisting of all algebraic varieties birationally equivalent to V . Any representative of [V ]bir will be called a model of the birational class; it is clear that [V ]bir contains all algebraic varieties which are isomorphic to V on the other hand it contains also several other models as the following examples show.

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(ii) The rational map φ1 : A1  Y := Za (x1 x2 − 1) in Examples 5.3.23(i), 6.2.5-(iv) and in Exercise 5.1 is birational, since Y is isomorphic to A1 \ {0} via φ1 . On the other hand, we already observed that Y is not isomorphic to A1 . Same occurs for, e.g. the ellipse in Examples 5.3.23(iii), 6.2.5-(iv) and for the semi-cubic parabola in Examples 5.3.24, 6.2.5-(v), since it has been showed therein that both Za (x21 + x22 − 1) and Za (x31 − x22 ) \ {(0, 0)} are isomorphic to A1 \ {0}. (iii) Recalling examples discussed in the previous chapters, we also get that A1 , P1 , A1 \{0}, the ellipse, the hyperbola, the parabola, the semi-cubic parabola, the cubic Za (x22 − x21 (x1 − 1)) (recall Example 5.3.25), any affine and projective rational normal curve of degree d, etc. these are all models of [P1 ]bir : indeed, in all these cases, we proved that the field of rational functions is isomorphic to K(x1 ) (these are all examples of rational curves, cf. Section 8.2). (iv) On the contrary the affine curve Ya : Za (x22 − x1 (x1 − 1)(x1 − a)), with 1, 0 = a ∈ K, cannot be a model of [P1 ]bir : in example Example 5.3.25 we showed that K(Ya ) is an algebraic extension of degree 2 of the field K(x1 ) (Ya is a smooth plane cubic, as such it is called elliptic curve). For V any algebraic variety, Bir(V ) will denote the group of birational transformations of V , which contains Aut(V ) as a subgroup. When V = Pn , for some n  1, Bir(Pn ) is also called Cremona group in honor of Luigi Cremona, who was a pioneer in the study of birational maps of projective spaces. As a consequence of Example 8.1.12, we will deduce that Bir(P1 ) = Aut(P1 ) = PGL(2, K). On the other hand, for n  2, Aut(Pn ) = PGL(n + 1, K) is a proper subgroup of the Cremona group of Pn , whose structure turns out to be in general quite intricate (cf., e.g. Verra, 2005). 8.1.1

Some properties and some examples of (bi)rational maps

In this section, we shall discuss interesting examples of rational and birational maps. Let V be any algebraic variety; if one has a rational map Ψ : V  An , for some positive integer n, composition with the isomorphism φ0 : An → U0 and with the open inclusion ιU0 : U0 → Pn  := ιU0 ◦ φ0 ◦ Ψ : V  Pn . determines a rational map Ψ  : V  Pn . Let U  be its domain and Conversely, take a rational map Ψ Ψ   ) ∩ H0 = let ψ be its representative morphism over the open set UΨ  . If ψ(UΨ n  ∅, then Ψ corresponds to a rational map Ψ : V  A .

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Corollary 8.1.9. Let V be any algebraic variety. Any non-constant morphism φ : V → P1 determines a rational map Φ : V  A1 , i.e. an element Φ ∈ K(V ). If moreover φ is not surjective, the map Φ is a morphism, i.e. Φ ∈ OV (V ). Proof. Only the last assertion needs some comment: if φ(V )  P1 , for any choice of P ∈ P1 \ φ(V ) one has that P1 \ {P } ∼ = A1 , from which one concludes.  One basic question is: what about the converse of Corollary 8.1.9? In other words, since any Φ ∈ K(V ) induces a morphism φ : UΦ → A1 and so a rational map Φ : V  A1 , is it true that Φ always extends to a morphism to P1 ? The answer in general is negative, as the following example shows. 2 1 Example 8.1.10. Let Φ := X X0 ∈ K(P ). This rational function is regular 2 on the principal affine open set U0 ⊂ P2 , so it defines a rational map Φ : P2  A1 with a representative morphism φ : A2 ∼ = U02 → A1 , which is the projection π

1 x1 = [1, x1 ] π1 : A2 → A1 , [1, x1 , x2 ] = (x1 , x2 ) −→

onto the first coordinate (when one identifies A1 with U01 ⊂ P1 ). Note that π1 is the restriction to the open set U02 of the projection πΛ

πΛ : P2  P1 , [X0 , X1 , X2 ]  [X0 , X1 ] from the linear subspace Λ := {[0, 0, 1]} ⊂ P2 . Thus, πΛ is a rational map whose domain is P2 \ Λ. In other words, Φ extends to a rational map from P2 to P1 but not to a morphism defined on the whole P2 with target P1 . On the other hand, in some cases we have affirmative answer to the previous question, as the following examples show. Example 8.1.11. Any rational map Ψ : A1  An extends to a unique morphism ψ : P1 → Pn . Proof. It is enough to show it for n = 1, the general case being similar (using Proposition 6.2.2). The rational map Ψ : A1  A1 corresponds to a rational function Ψ ∈ K(A1 ) ∼ = Q(1) ∼ = K(t), where t is an indeterminate f over K. Thus, there exist f, g ∈ A(1) ∼ = K[t] s.t. Ψ = g , where UΨ = Ua (g) = Za (g)c , i.e Ψ ∈ OA1 (Ua (g)). We can assume that g.c.d.(f, g) = 1. The map Ψ defines a morphism ψ : Ua (g) → A1 . We first show that the morphism ψ uniquely extends

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to a morphism A1 → P1 . To do this, identify the target A1 of ψ with U0 ⊂ P1 and consider the map ψ  : A1 → P 1 ,

P → [g(P ), f (P )].

This map is well-defined as, for points P ∈ A1 \ Ua (g), one has [g(P ), f (P )] = [0, f (P )] = [0, 1] since f (P ) = 0 (otherwise (t − P ) would be a common divisor of f and g against the assumption on g.c.d.(f, g)). Observe moreover that ψ  is a morphism; indeed let f (t) = f0 + f1 t + . . . fn tn

and g(t) = g0 + g1 t + . . . gm tm ,

with fi , gj ∈ K, fn , gm = 0, 0  i  n, 0  j  m, i.e. deg(f ) = n and deg(g) = m. Assume, e.g. n  m (the other case being similar). Identify 1 once again A1 (the domain of ψ  ) with U0 ⊂ P1 ; thus t = X X0 and the homogeneous polynomials   X1 n F (X0 , X1 ) = X0 f = h0 (f ) and X0   X1 G(X0 , X1 ) := X0n g = X0n−m h0 (g) X0 are such that [g(P ), f (P )] = [G(1, P ), F (1, P )] for any P ∈ A1 . This implies that ψ  is a morphism (cf. Proposition 6.3.1); moreover its uniqueness follows from Lemma 8.1.1. The next step is to show that ψ  uniquely extends to a morphism P1 → P1 . Since P1 = U0 ∪ {[0, 1]}, from the expression of F and G above, one has [G(0, 1), F (0, 1)] = [0, fn ] = [0, 1], since fn = 0. In particular, ψ  is the restriction to U0 of the map ψ : P1 → P1 , [X0 , X1 ] → [G(X0 , X1 ), F (X0 , X1 )], which is a morphism from Proposition 6.3.1 (its uniqueness once again follows from Lemma 8.1.1). Similarly, as in the previous example, one has the following. Example 8.1.12. Any rational map Φ : P1  Pn is always a morphism. Proof. We can assume Φ to be non-constant, otherwise there is nothing to prove. Let UΦ be the domain of Φ and let φ : UΦ → Pn be the

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representative morphism. Since being a morphism is a local property (cf. Proposition 6.4.1), from Proposition 6.3.1 we can directly assume that φ is globally defined on UΦ by a collection of homogeneous polynomials (1) F0 , . . . , Fn ∈ Sk , for some integer k  1, with Zp (F0 , . . . , Fn ) = ∅. Then φ(P ) = [F0 (P ), . . . , Fn (P )],

∀ P ∈ UΦ ,

where, with no loss of generality, we can assume g.c.d(F0 , . . . , Fn ) = 1. Thus, for any point P = [p0 , p1 ] ∈ P1 \ UΦ , (F0 (P ), . . . , Fn (P )) = (0, . . . 0), otherwise the polynomials F0 , . . . , Fn would have as common divisor the homogeneous polynomial p1 X0 −p0 X1 , against the assumption. This means that Φ is wherever defined, so it is a morphism. To have concrete applications of previous statements, consider: Example 8.1.13. Let K = C and let Ψ : A1  A2 be the rational map defined by  2  t − 1 −2t , 2 t  2 . t +1 t +1 Its domain is UΨ = A1 \ {i, −i} and its representative morphism ψ over UΨ is such that ψ(UΨ ) ⊂ C = Za (x21 + x22 − 1) ⊂ A2 , the first inclusion being strict since (1, 0) ∈ C is not contained in Im(ψ) as one can easily check. The map Ψ is more precisely birational; indeed, consider the pencil of lines through (1, 0) ∈ C, i.e. x2 = t(x1 − 1), where t ∈ K. This pencil defines a morphism C \ {(1, 0)} → A1 ,

(p1 , p2 ) →

p2 =t p1 − 1

which is the inverse of the morphism ψ over the open set C \ {(1, 0)}, as for any t ∈ UΨ the intersection Za (x2 − t(x1 − 1), x21 + x22 − 1) (off the point of coordinates (1, 0), which is the fixed point of the pencil of lines) 2 −2t determines the point of coordinates tt2 −1 +1 , t2 +1 ∈ C\{(1, 0)}. In particular Ψ is another rational parametrization of the ellipse, which differs from the parametrization considered in Examples 5.3.23-(iii) and 6.2.5-(iv). As in Example 8.1.11, ψ uniquely extends to a morphism ψ  : A1 → P1 , t → [t2 + 1, t2 − 1, −2t]. Note indeed that ψ  is well-defined since ψ  (±i) = [0, 1, ±i] ∈ C ∩H0 , where C denotes the projective closure of C in P2 and where we identified A2 with

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the affine chart U02 ⊂ P2 . Moreover, identifying A1 with U01 ⊂ P1 , one has 1 t= X X0 so ψ

[X0 , X1 ] −→ [X02 + X12 , X12 − X02 , −2X0 X1 ], which shows that the previous map is a morphism since (1)

Zp (X02 + X12 , X12 − X02 , −2X0 X1 ) = Zp (X02 , X12 , X0 X1 ) = Zp ((S+ )2 ) = ∅. Note that Im(ψ  ) = C \ {[1, 1, 0]} nonetheless, as in Example 8.1.11, ψ  extends to a morphism ψ : P1 → C which maps [0, 1] to [1, 1, 0] ∈ C (this is the point where originally the inverse of ψ was not defined). Example 8.1.10 shows that the projection of P2 from one of its point gives rise to a rational map which cannot be extended to a morphism. Below, we consider in more generalities these examples of rational maps, namely the projections of a projective space from a linear subspace. Example 8.1.14 (Projections). Consider the projective space Pn with homogeneous coordinates [X0 , . . . , Xn ]. Let us denote with X the columnvector whose entries are these homogeneous coordinates. Let A be a (n+1)×(m+1)-matrix with entries in K and such that rank(A) := r  n+1. Then P(Ker(A)) is either empty, when r = n + 1, or it is a linear subspace Λ := ΛA ⊂ Pn of dimension n − r. In the empty case, i.e. r = n + 1, the πΛ [AX] is a homography as in map Pn → Pm , given by [X0 , . . . , Xn ] −→ Section 3.3.8, in particular n  m, the map is injective and everywhere defined. From now on we therefore shall focus on the case when Λ = ∅; in such a case the map πΛ : Pn \ Λ → Pm ,

π

Λ [X0 , . . . , Xn ] −→ [AX]

(8.1)

is well-defined and it is a morphism on the open set U := Pn \ Λ of Pn . So, πΛ defines a rational map πΛ : Pn  Pm which is also called degenerate homography or even projection of Pn from the linear subspace Λ to the projective space Pm . The subspace Λ is also called the center of the projection πΛ . The domain of πΛ is the open set U , where it is defined as a morphism. If, e.g. r = 1, then Λ is a hyperplane in Pn and πΛ is a constant map. A typical example of degenerate homography is the following: let Λ ⊂ Pn be a linear subspace of dimension n − r, with r  1 and let Σ ⊂ Pn be

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another linear subspace of dimension r−1 which is skew to Λ, i.e. Λ∩Σ = ∅. From the projective Grassmann formula (3.23) one has that Λ ∨ Σ = Pn . Therefore, for any point P ∈ Pn \ Λ := U , the linear space P ∨ Σ is of dimension n−r+1 which, once again by the projective Grassmann formula, intersects Σ at a unique point P  . The map τ : Pn \ Λ → Σ, P −→ P  := (P ∨ Λ) ∩ Σ τ

is a degenerate homography; indeed up to projectivities, we may fix homogeneous coordinates on Pn in such a way that Λ = Zp (X0 , . . . , Xr−1 ) and Σ = Zp (Xr , . . . , Xn ). In this set-up the map τ is simply given by τ

[X0 , . . . , Xn ] −→ [X0 , . . . , Xr−1 ],

(8.2)

i.e. τ = πΛ . From the previous expression for τ we note that the map is surjective and, moreover, that τ (P ) = τ (Q) if and only if P ∨ Λ = Q ∨ Λ as linear subspaces of Pn . Any projection of Pn from a linear subspace Λ of dimension n − r onto a skew subspace Σ of dimension r − 1 can be obtained as the map τ above after a projective transformation, i.e. any such projection is the composition of τ and of a projectivity of Pn . Identify An with the affine chart U0 ⊂ Pn and consider the degenerate homography in (8.2). Since Σ is not contained in the hyperplane H0 = Zp (X0 ), then Σ \ Σ∞ ∼ = Ar−1 is an affine subspace of An = U0 . The restriction of τ to U0 \ (U0 ∩ Λ) ∼ = An \ Λ0 , where Λ0 the affine subspace cut-out by Λ in U0 , is a morphism. If Λ0 = ∅, then τ0 := τ|An \Λ0 is called projection of An with center Λ0 onto Ar−1 . If otherwise Λ0 = ∅, then τ0 is called projection of An onto Ar−1 with direction parallel to Λ. In the case of parallel projections, τ0 is a morphism. When otherwise the center is not empty, τ0 is a rational map from An to Ar−1 but not a morphism. Consider now Λ ⊂ Pn a linear subspace, with Λ ∼ = Pn−k−1 , and let n k πΛ : P  P be the projection with center Λ. If V is any algebraic variety not contained in Λ, the restriction of πΛ to V defines a rational map πV : V  Pk ,

(8.3)

which is called projection of V on the subspace Pk ; πV is a morphism on V \ (V ∩ Λ); in particular, in V ∩ Λ = ∅, then πV ∈ Morph(V, Pk ). On the

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other hand, in some cases the set V \ (V ∩ Λ) is strictly contained in the whole domain of the map πV , as the following example shows. Example 8.1.15. Let C ⊂ P2 be the conic as in Example 8.1.13. The pencil of lines therein defines a projection of A2 onto the x2 -axis and so a linear projection πP : P2  P1 , where P1 identified with the line H1 = Zp (X1 ) ⊂ P2 and where the center of πP is the base point of the pencil of lines P = [1, 1, 0]. Since P is on C, πP defines a rational map πC which is a projection of C to the linear subspace P1 . This is a morphism certainly over C \ {P } and it is an example of stereographic projection of an irreducible conic onto P1 . For any point Q = (0, y0 ) on the x2 -axis in A2 = U0 , with y0 = ±i, the unique line of the pencil P ∨ Q intersects  P at  2 C outside y −1 0 a unique further point, say PQ ∈ C, which is PQ = y02 +1 , y2y 2 +1 . From 0

0

Example 8.1.13, this is a morphism. Identifying the line H1 with P1 , πC is therefore a birational map C  P1 whose domain certainly contains C \ {P }. On the other hand, C ∼ = P1 via a Veronese morphism ν1,2 . Up to the composition with ν1,2 , from Example 8.1.12 πC extends to a unique morphism defined on the whole C. The geometric interpretation of such an extension is given by considering the tangent line X0 − X1 = 0 to C at P and then intersecting it with the line H1 , which gives the point [0, 0, 1]. 8.2

Unirational and Rational Varieties

We start with some important definitions. Definition 8.2.1. An algebraic variety V is said to be unirational if there exist a positive integer r, a dominant rational map Φ : Pn  V and a dense open subset U ⊆ Im(Φ) s.t., for any P ∈ U , Φ−1 (P ) is a finite set. Such a map Φ is said to be a generically finite dominant rational map. From Theorem 8.1.6, the map Φ is associated with a field extension K(V ) ⊆ K(x1 , . . . , xr ) = Q(n) . Definition 8.2.2. An algebraic variety V is said to be rational if it is birational to Pr (or Ar ), for some non-negative integer r. In particular, if V is rational then it is unirational. Moreover, from Corollary 8.1.7-(i), we get: Corollary 8.2.3. V is a rational algebraic variety if and only if K(V ) ∼ = Q(r) , i.e. if and only if the field K(V ) is obtained as a purely transcendental

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extension of the ground field K of some transcendental degree r, for some non-negative integer r. All the curves listed in Remark 8.1.8-(ii) and (iii) are rational curves, whereas the curve Ya in (iv) therein is not rational (recall indeed it has been called elliptic curve). The so-called L¨ uroth problem is a fundamental problem in Algebraic Geometry: L¨ uroth problem Does unirationality imply rationality? In 1876, L¨ uroth proved that in the curve case the answer to the previous question is yes, i.e. any unirational curve is also rational (cf., e.g. Cohn, 1991, cf. p. 148). In 1894, Castelnuovo showed that also unirational surfaces are rational (cf., e.g. Barth et al., 2004; Beauville, 1996). Only after almost one century, e.g. Clemens and Griffiths (1972) showed that a (smooth) cubic threefold (i.e. an hypersurface of degree 3 in P4 with no singular points) is unirational but in general not a rational variety, providing an example for three dimensions that unirationality does not imply rationality. In what follows we shall give several examples of rational varieties which are birational but not isomorphic to Pn , for n  2. 8.2.1

Stereographic projection of a rank-four quadric surface

A nice example of a surface which is birational (but not isomorphic) to P2 is given by an irreducible, doubly ruled quadric Q ⊂ P3 . This is a rankfour quadric in P3 , i.e. the homogeneous quadratic polynomial defining Q is a quadratic form in the indeterminates X0 , . . . , X3 whose associated symmetric matrix has non-zero determinant. Since K is algebraically closed, all these quadrics are projectively equivalent. Therefore, with no loss of generality, we may assume that Q = Zp (X0 X3 − X1 X2 ), which is the isomorphic image of P1 × P1 embedded, via the Segre morphism σ2,2 , as an irreducible doubly ruled quadric in P3 (cf. Exercise 7.1). Take P0 = [1, 0, 0, 0] ∈ Q. From (8.3), the rational map πP0 : P3  P2 , [X0 , X1 , X2 , X3 ]  [X1 , X2 , X3 ] induces the projection πQ : Q  P2 which is called stereographic projection of the rank-four quadric surface to a plane. This rational map is a morphism over the open subset Q \ {P0 }. Differently from the stereographic projection in Example 8.1.15, the expression of πQ shows that the map cannot be extended (as a morphism) at

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the point P0 , moreover that its target space is the plane H0 = Zp (X0 ) ⊂ P3 . Our aim is to show that πQ is a birational map to such a plane. To do this consider H3 = Zp (X3 ) ⊂ P3 and the open set U := Q \ (Q ∩ H3 ) ⊂ Q. Note that Q ∩ H3 = Zp (X1 , X3 ) ∪ Zp (X2 , X3 ) = 13 ∪ 23 , where 13 and 23 are the two lines passing through P0 of the two different rulings of Q. Thus, ϕ := πQ |U : U → U  ⊂ H0 ∼ = P2 is a morphism (since P0 ∈ 13 ∪ 23 ) whose image is the open set U  := H0 \ (H0 ∩ H3 ), which is isomorphic to P2 minus a line (such a line is given by 03 = Zp (X0 , X3 ) ⊂ H0 ). Claim 8.2.4. ϕ : U → U  is an isomorphism. Proof. For any point Q = [0, q1 , q2 , q3 ] ∈ U  , consider the line P0 ∨ Q, which has parametric equations X0 = s, X1 = q1 t, X2 = q2 t, X3 = q3 t, [s, t] ∈ P1 . Such a line intersects U at the point [ q1q3q2 t, q1 t, q2 t, q3 t], / U . In other words, which is [ q1q3q2 , q1 , q2 , q3 ], as t = 0 since P0 ∈ ϕ−1 (Q) = [q1 q2 , q1 q3 , q2 q3 , q32 ] which is a morphism, as it follows from its expression and the fact that q3 = 0 since Q ∈ U  . In particular, the map πQ is birational (but not a morphism) so Q is a rational surface. Identifying the plane H0 with P2 , with homogeneous coordinates [X1 , X2 , X3 ], from Claim 8.2.4, the inverse of ϕ is induced by the rational map ν : P2  P3 , [X1 , X2 , X3 ]  [X1 X2 , X1 X3 , X2 X3 , X32 ], (2)

where ν = νL as in (6.16) with L = Span{X1 X2 , X1 X3 , X2 X3 , X32 } ⊂ S2 . In particular νL is not defined at B := {P1 = [1, 0, 0], P2 = [0, 1, 0]} ⊂ P2 , which is the base locus of the linear system of plane conics P(L), and Im(νL ) ⊆ Q. To sum up, πQ and νL are two birational maps, one is the inverse of the other (as rational maps), none of them is a morphism. Note, moreover, that 03 ⊂ H0 is the line joining P1 , P2 , the two points of indeterminacy of νL , that νL contracts 03 \ {P1 , P2 } to P0 ∈ Q and, conversely, that πQ contracts 13 \ P0 := {[s, 0, t, 0], s, t ∈ K, t = 0} to P1 ∈ H0 and 23 \ P0 := {[s, t, 0, 0], s, t ∈ K, t = 0} to P2 ∈ H0 . This also explains the choice of the open set U and U  to construct the isomorphism ϕ.  The previous construction shows that Q is birational to P2 ; to conclude that they cannot be isomorphic it suffices to observing that Q contains pairs of skew lines, which cannot occur in P2 .

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8.2.2

Monoids

Other examples of rational varieties in any dimension n  1 can be easily constructed as follows. Let Fd (x1 , . . . , xn ), Fd−1 (x1 , . . . , xn ) ∈ K[x1 , . . . , xn ] be reduced homogeneous polynomials of degree d and d − 1, respectively. Consider the hypersurface Z := Za (Fd (x1 , . . . , xn ) + Fd−1 (x1 , . . . , xn )) ⊂ An ,

(8.4)

which is called (affine) monoid of degree d with vertex the origin O ∈ An . The projective closure of Z is the hypersurface Z ⊂ Pn given by Z = Zp (Fd (X1 , . . . , Xn ) + X0 Fd−1 (X1 , . . . , Xn )),

(8.5)

which is called (projective) monoid of degree d with vertex P0 = [1, 0, 0, . . . , 0] ∈ Pn . More generally, any hypersurface V ⊂ An which is the transform of Z via a linear transformation of An as well as its projective closure V will be called monoid of degree d. Example 8.2.5. (i) The parabola Z = Za (x2 − x21 ) ⊂ A2 is a (affine) monoid of degree two with vertex O = (0, 0). (ii) The semi-cubic parabola Z = Za (x22 − x31 ) ⊂ A2 is a (affine) monoid of degree three with vertex O. Identifying A2 with the affine chart U0 of P2 , then Z is given by Zp (X0 X22 − X13 ) which is a (projective) monoid with vertex the fundamental point P0 = [1, 0, 0] ∈ P2 . (iii) Similarly, the plane nodal cubic Z := Za (x31 +x21 −x22 ) ⊂ A2 is a (affine) monoidal curve, whose vertex is O = (0, 0) and whose projective closure is the (projective) monoid Zp (X13 + X12 X0 − X22 X0 ) ⊂ P2 with vertex the fundamental point P0 = [1, 0, 0]. (iv) The rank-four quadric surface Q = Zp (X0 X3 − X1 X2 ) ⊂ P3 is a (projective monoid) of degree two with vertex P0 = [1, 0, 0, 0], being Q0 = Q ∩ U0 = Za (x3 − x1 x2 ) an affine monoid of vertex O. Similarly, the quadric cone Zp (X0 X3 − X12 ) ⊂ P3 (i.e. a rank-three quadric) is a monoidal surface too, always with vertex the point P0 . As in Section 8.2.1, we can prove the following result. Proposition 8.2.6. For any n  2, any monoid of degree d  2 is rational.

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Proof. With no loss of generality, we can focus on the case of Z ⊂ Pn a monoid with vertex P0 = [1, 0, . . . , 0], i.e. whose reduced equation is given by (8.5). In such a case, we will more precisely show that the projection πZ : Z  H0 = Zp (X0 ) ⊂ Pn from the point P0 ∈ Z is birational onto H0 ∼ = Pn−1 . The restriction of πZ at Z \ {P0 } is a surjective morphism ϕ : Z \ {P0 } → H0 ∼ = Pn−1 , [X0 , . . . , Xn ] → [X1 , . . . , Xn ]. Similarly to the quadric case, the dominant rational map πZ is called stereographic projection of the monoid from its vertex P0 . It suffices to find a dominant rational map inverting πZ . To do this observe that for any [a1 , . . . , an ] ∈ H0 \ (Zp (Fd−1 ) ∩ H0 ), i.e. s.t. Fd−1 (a1 , . . . , an ) = 0, one has −1

ϕ



 Fd (a1 , . . . , an ) , a1 , . . . , an , ([a1 , . . . , an ]) = Fd−1 (a1 , . . . , an )

i.e. the rational map Ψ : Pn−1 ∼ = H0  Pn defined by Ψ

[X1 , . . . , Xn ]  [Fd (X1 , . . . , Xn ), X1 Fd−1 (X1 , . . . , Xn ), . . . , Xn Fd−1 (X1 , . . . , Xn )]



is the desired rational inverse. 8.2.3

Blow-up of Pn at a point

Here, we use products of projective varieties to introduce an important birational transformation. Consider Pn with homogeneous coordinates [X] := [X0 , . . . , Xn ] and Pn−1 with homogeneous coordinates [Y ] := [Y1 , . . . , Yn ]. In Pn × Pn−1 consider the closed subset (cf. Proposition 7.2.7-(i)): n := Zp (Xi Yj − Xj Yi ), P

for 1  i, j  n.

(8.6)

n → Pn denote the morphism defined by the restriction to P n Let σ : P n n−1 n of the first projection π1 : P × P → P . The map σ is called the blow-up of Pn at the point P0 , where P0 = [1, 0, . . . , 0] the fundamental point of Pn . Similar approach holds for an arbitrary point P ∈ Pn (cf., e.g. Hartshorne, 1977); for simplicity in what follows we will focus only on the case of P0 . n \ σ −1 (P0 ) is a quasi-projective variety isomorphic Proposition 8.2.7. P n n is rational, being birational to Pn . (via σ) to P \ {P0 }. In particular, P Proof. For any Q = [q0 , . . . , qn ] ∈ Pn \ {P0 } there exists i ∈ {1, . . . , n} for which qi = 0. Thus, ([q0 , . . . , qn ], [y1 , . . . , yn ]) ∈ σ −1 (Q) if and only if

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yj = yi qji , for any 1  j  n; in other words n ⊂ Pn × Pn−1 σ −1 (Q) = ([q0 , . . . , qn ], [q1 , . . . , qn ]) ∈ P is a unique point. Consider the map Pn \ {P0 }

τ

−→

Pn × Pn−1

Q = [q0 , . . . , qn ] −→ σ −1 (Q) = ([q0 , . . . , qn ], [q1 , . . . , qn ]).

(8.7)

Composing τ with the Segre isomorphism σn,n−1 gives a map γ = σn,n−1 ◦τ defined by γ([q0 , . . . , qn ]) = [q0 q1 , q0 q2 , . . . , qn2 ], which is therefore a morphism by Proposition 6.3.1. Since σn,n−1 is an isomorphism, one deduces that τ is a morphism. In particular, one gets n \ σ −1 (P0 ) is a quasi-projective variety. Indeed τ (Pn \ {P0 }) = that P n \ σ −1 (P0 ). The latter is locally closed in Pn × Pn−1 : indeed P n is closed P −1 n \ σ −1 (P0 ) and also σ (P0 ) is closed, being σ a morphism. Moreover, P is also irreducible, since Pn \ {P0 } is irreducible and τ is a morphism (cf. Corollary 4.1.6). At last, by its expression, τ = (σ|Pn \{P0 } )−1 , i.e. τ is an isomorphism.  From (8.6), one gets σ −1 (P0 ) = π1−1 (P0 ) ∼ = Pn−1 ,

(8.8)

n . which is called the exceptional divisor of σ in P Remark 8.2.8. A geometric interpretation of the isomorphism in (8.8) can be given as follows. Let H be any hyperplane in Pn not passing through P0 ; up to a projectivity of Pn fixing P0 , we can assume with no loss of generality that this hyperplane coincides with H0 = Zp (X0 ). Any point Q ∈ H uniquely determines the line rQ := P0 ∨ Q in Pn . Conversely, any line r ⊂ Pn passing through P0 intersects H at a unique point Qr = r ∩ H such that P0 ∨ Qr = r. In other words, the set LP0 := {lines in Pn through P0 } bijectively corresponds to the hyperplane H ∼ = Pn−1 . For any point Q = [0, q1 , . . . , qn ] ∈ H, the line rQ = P0 ∨ Q has parametric equations X0 = λ, Xi = μ qi ,

1  i  n,

with [λ, μ] ∈ P1 .

(8.9)

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By (8.7), at the points of rQ \ {P0 } the map τ restricts to the morphism: τ |rQ \{P0 } :

rQ \ {P0 }

−→

n \ σ −1 (P0 ) ⊂ Pn × Pn−1 P

[λ, μq1 , . . . , μqn ] −→ ([λ, μq1 , . . . , μqn ], [q1 , . . . , qn ]),

(8.10)

where the previous expression follows from the fact that ([λ, μq1 , . . . , μqn ], [μq1 , . . . , μqn ]) = ([λ, μq1 , . . . , μqn ], [q1 , . . . , qn ]), as μ = 0 on rQ \ {P0 }. Composing τ |rQ \{P0 } with the Segre morphism σn,n−1 , one gets a rational map τ |rQ : rQ  Pn(n+1)−1 . Since rQ ∼ = P1 , from Example 8.1.12, τ |rQ is a morphism extending τ |rQ \{P0 } at the point P0 ; by (8.9), this extension is given by τ |rQ (P0 ) := ([1, 0, . . . , 0], [q1 , . . . , qn ]).

(8.11)

n Consider r Q := τ (rQ ) ⊂ P . From Proposition 8.2.7 and (8.11), r Q is isomorphic to P1 in particular it is an irreducible rational curve. Moreover, by its definition −1 r (P0 ) = τ |rQ (P0 ) ∩ σ −1 (P0 ) = [q1 , . . . , qn ] ∈ σ −1 (P0 ). Q∩σ

In particular, the map H Q [0, q1 , . . . , qn ]

∼ σ−1(P0 ) = LP0 −→ −1 ∼ (P0 ) = rQ −→ r Q ∩σ ←→

[q1 , . . . , qn ]

can be interpreted as the isomorphism (8.8). n is irreducible. Proposition 8.2.9. P n = (P n \ σ −1 (P0 )) ∪ σ −1 (P0 ). From Proposition Proof. Note that P n \ σ −1 (P0 )) is irreducible and σ −1 (P0 ) is closed and irreducible 8.2.7, (P n n \ σ −1 (P0 ) is dense in P n . in P ; it therefore suffices to show that P

To prove this, it is enough to show that any point of σ −1 (P0 ) is in the n \ σ −1 (P0 ). In n of some algebraic subset contained in P closure in P −1 Remark 8.2.8, it has been proved that any point of σ (P0 ) is given by −1 τ |rQ (P0 ) = r (P0 ), for some point Q ∈ H, and moreover for any Q ∩ σ ∼ 1 n Q ∈ H the map τ |rQ is an isomorphism. Thus, r Q = P is the closure in P −1 1 n ∼ of τ (rQ \ {P0 }) ⊂ P \ σ (P0 ), where τ (rQ \ {P0 }) = A . 

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Example 8.2.10. 1 ∼ (i) By definition of blow-up, it is clear that P = P1 . 2 is the rational surface in P2 × P1 given by the equation (ii) For n = 2, P Zp (X1 Y2 − X2 Y1 ). Repeating verbatim the proof of Proposition 8.2.7, 2 ∩ (U 2 × U 1 ) is isomorphic to the one easily realizes that, e.g. P 0 0 hyperbolic paraboloid Σ := Za (x2 − yx1 ) ⊂ A3 , where A3 is U02 × U01 ∼ = A2 × A1 and where affine coordinates (x1 , x2 , y) are given by X1 Y2 −1 2 (P0 )∩(U02 ×U01 ) x1 = X0 , x2 = X X0 , y = Y1 . The exceptional divisor σ is the line Za (x1 , x2 ) contained in Σ. Its points (0, 0, y), with y varying in A1 , bijectively correspond to direction coefficients y of lines through the origin of A2 (i.e. P0 in the chart U02 ) in the pencil {x2 − yx1 = 0} ⊂ A2 . 8.2.4

Blow-ups and resolution of singularities

Here, we discuss some examples which show how blow-ups can be used to resolve singularities of algebraic varieties (see also Mumford, 1995, Example 2.21, p. 32). As in the previous section, things more generally hold for a blow-up at any arbitrary point P of Pn . On the other hand, up to projectivities, one can always reduce to the basic case of the blow-up at the fundamental point P0 = [1, 0, . . . , 0]. Let V ⊆ Pn be any algebraic variety passing through P0 . Consider the n → Pn as above and let blow-up σ : P WP0 := σ −1 (V \ {P0 }),

(8.12)

n . where the closure is taken in P Claim 8.2.11. WP0 is a projective variety which is birational to V . Proof. V \ {P0 } is an algebraic variety, being an open dense subset of V ; furthermore, since σ|V \{P0 } is an isomorphism between V \ {P0 } and its n , i.e. it is a image, then σ(V \ {P0 }) is irreducible and locally closed in P quasi-projective variety. This implies that WP0 is irreducible and birational n is closed in Pn × Pn−1 , WP0 is also projective.  to V . Moreover, since P σ −1 (V ) is called the total transform of V via σ . We let −1

V (V ); P0 := WP0 ∩ σ

(8.13)

since V is quasi-projective and σ is a morphism, V P0 is locally closed in

WP0 (so irreducible) and birational to V . VP0 is called the proper transform

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of V via σ. The restriction of σ at V P0 is a birational morphism which will

be denoted by σV : VP0 → V and called blow-up of V at P0 . Example 8.2.12. (i) Let V := Zp (X0 X12 − X23 ) ⊂ P2 ; note that V is the projective closure of the semi-cubic parabola V0 = V ∩ U0 = Za (x21 −x32 ) ⊂ A2 ∼ = U0 , so V is a monoid in P2 with vertex the fundamental point P0 = [1, 0, 0] (cf. Example 8.2.5-(ii)). Since V is projective, then T := V × P1 is a projective subvariety of P2 × P1 , whose defining equation 2 → P2 be the blow-up of P2 at is simply Zp (X0 X12 − X23 ). Let now σ : P 2 is defined by Zp (X1 Y2 − X2 Y1 ). The total P0 . As in Example 8.2.12-(ii), P transform of V is therefore: 2 = Zp (X0 X 2 − X 3 , X1 Y2 − X2 Y1 ). σ −1 (V ) = T ∩ P 1 2

(8.14)

X2 1 Take affine charts U02 ⊂ P2 , with affine coordinates x1 = X X0 , x2 = X0 , and U21 ⊂ P1 , with affine coordinate t = YY12 . From (8.14), the corresponding open subset of the total transform is:

σ −1 (V ) ∩ (U02 × U21 ) = Za (x21 − x32 , x1 − x2 t) = Za (x22 (x2 − t2 ), x1 − x2 t), which reads also Za (x1 − t3 , x2 − t2 ) ∪ Za (x2 , x1 ). In the affine chart U02 × U21 ∼ = A2 × A1 ∼ = A3 , with affine coordinates (x1 , x2 , t), the set Za (x1 −t3 , x2 −t2 ) is an affine twisted cubic in A3 , precisely the image of the morphism A1 −→ A3 given by t −→ (t3 , t2 , t), whereas Za (x2 , x1 ) is simply the t-axis in A3 , which coincides with the affine part of the exceptional divisor σ −1 (P0 ) ∩ (U02 × U21 ) (cf. Example 8.2.10-(ii)). 2 ∩ (U 2 × U 1 ), the proper transform To sum up, in the affine chart P 0 2

(V0 )P0 of the semi-cubic parabola V0 ⊂ A2 is isomorphic to an affine twisted 2 of (V

is the proper transform cubic C ⊂ A3 . The projective closure in P 0) P0

V P0 (which in this case coincides with WP0 , since V is already projective). From Claim 3.3.18, VP0 is therefore isomorphic to the standard projective twisted ν1,2 : P1 −→ P3 , [Y1 , Y2 ] −→ [Y13 , Y12 Y2 , Y1 Y22 , Y23 ]. −1

(P0 ), where The total transform σ −1 (V ) coincides with V P0 ∪ σ −1 1 ∼

σ (P0 ) = P . As for the blow-up morphism σV : VP0 → V , previous computations show that σV |VP ∩(U 2 ×U 1 ) is nothing but the restriction to 0 2 0 C of the projection πI : A3 → A2 onto the first two coordinates, i.e. with multi-index I = (1, 2).

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With a more geometrical perspective, πI is the projection onto the plane Za (t) ⊂ A3 from the point at infinity of the t-axis; under this projection one therefore has πI |C : C −→ V0 , (t3 , t2 , t) −→ (t3 , t2 ), i.e. C ⊂ A3 maps onto the semi-cubic parabola V0 ⊂ A2 . We will see that V0 is singular at the origin (cf. Example 12.1.3-(ii)), whereas the affine twisted cubic C ⊂ A3 is a smooth curve, i.e. it is non-singular at any of its points (cf. Example 12.1.3). In other words, the semi-cubic parabola acquires its cuspidal singularity at O because of the projection πI |C ; indeed, the intersection between C ⊂ A3 (i.e. the proper transform of V0 ) and the t-axis (i.e. the affine part of the exceptional divisor) is given by Za (x1 − t3 , x2 − t2 , x1 , x2 ). This intersection is set-theoretically supported at the unique point O = {(0, 0, 0)} ∈ C, which corresponds to the coefficient direction m = 0 of the line Za (x2 ) ⊂ A2 . On the other hand, the intersection multiplicity at O of the system of equations Za (x1 − t3 , x2 − t2 , x1 , x2 ) is 2 (cf. Definition 12.1.4) as the t-axis is the tangent line to C ⊂ A3 at O. Thus, the cuspidal singularity of V0 at O deals with the fact that πI |C is a tangential projection and the singular point is where the parametrization is not regular (i.e. where the Jacobian matrix vanishes). Conversely, the affine twisted cubic C ⊂ A3 can be viewed as a non-singular birational model of V0 , since C is the proper transform of V0 under the blow-up πI |C and one says that the morphism πI |C resolves the cuspidal singularity of V0 at O (cf. also Harris, 1995, Example 1.26, pp. 14–15). (ii) Let V := Zp (X13 + X0 (X12 − X22 )) ⊂ P2 . This is a (projective) monoid of vertex P0 = [1, 0, 0] being the projective closure of the (affine) monoid V0 = V ∩ U02 = Za (x31 + x21 − x22 ) ⊂ A2 (cf. Example 8.2.5-(iii)). Computations as in (i) show that the total transform of V is given by σ −1 (V ) = Zp (X13 + X0 (X12 − X22 ), X1 Y2 − X2 Y1 ).

(8.15)

X2 1 Take affine charts U02 ⊂ P2 , with affine coordinates x1 = X X0 , x2 = X0 , and U11 ⊂ P1 , with affine coordinate s = YY21 . From (8.15), we get that σ −1 (V ) ∩ (U02 × U11 ) is given by

Za (x31 + x21 − x22 , x2 − x1 s) = Za (x21 (x1 + 1 − s2 ), x2 − x1 s), which is Za (x1 − s2 + 1, x2 − s3 + s) ∪ Za (x1 , x2 ). The first algebraic subset Za (x1 − s2 + 1, x2 − s3 + s) is the image of the morphism A1 −→ A3 , defined by s → (s2 − 1, s3 − s, s), i.e. it is an affine twisted cubic in A3 so the proper transform V

0 P0 is irreducible. The second algebraic subset Za (x1 , x2 )

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is the s-axis, i.e. the affine part of the exceptional divisor. The proper

transform V P0 of the monoid V0 coincides with WP0 and it is isomorphic to the projective closure of C.

The blow-up morphism σV : V P0 → V in the above affine charts reads as the restriction to C of the projection onto the plane Za (s) ⊂ A3 from the point at infinity of the s-axis. In this case, the intersection between C and the s-axis is given by Za (x1 − s2 + 1, x2 − s3 + s, x1 , x2 ); this system of equations gives {Q1 = (0, 0, 1), Q−1 = (0, 0, −1)} ∈ C, where the points of intersection correspond to the coefficient directions m = ±1 of the two lines Za (x2 − x1 ) and Za (x2 + x1 ) in the plane Za (s). The nodal singularity of the curve V0 at O appears since σV is a secant projection of C to the s-plane. This projection C −→ V0 is given by (s2 − 1, s3 − s, s) → (s2 − 1, s3 − s); the s-axis is a secant line to C at the points Q1 and Q2 which are identified under the projection, i.e. they both map to O ∈ V0 . This creates the nodal singularity at O (i.e. a point where the parametrization is not injective). To sum-up, the affine twisted cubic C ⊂ A3 is a non-singular birational model of the plane nodal cubic V0 and σV resolves its (nodal) singularity at O (see also Harris, 1995, Example 1.26, pp. 14–15).

Exercises Exercise 8.1. Let X := Za (x1 x2 · · · xn − 1) ⊂ An , for any integer n  2. Show that X is an affine variety which is birational to An−1 . Exercise 8.2. For any integer n  1, give an example of an affine variety which is birational but not isomorphic to An . Exercise 8.3. Consider the affine irreducible curve C := Za (x41 + x31 − x32 ) ⊂ A2 . Show that C is a rational curve determining an explicit birational Φ

parametrization A1  C. Exercise 8.4. With the use of blow-ups, determine a birational model of the curve C ⊂ A2 as in Exercise 8.3 which is isomorphic to A1 . Exercise 8.5. Consider the map q : P2  P2 defined by q [X0 , X1 , X2 ]  [ X10 , X11 , X12 ] which is called elementary quadratic (or Cremona) transformation of P2 . Show that q is a birational automorphism of P2 , explicitly finding the open set U ⊂ P2 where q|U is an isomorphism. Describe the indeterminacy locus of q and of its rational inverse. Finally, describe the image via q of any line of P2 (cf. Mumford, 1995, Example 2.20, p. 31; Ciliberto, 2021, § 18.2.2, pp. 270–271).

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Chapter 9

Completeness of Projective Varieties

Here, we discuss an important topological property of projective varieties. 9.1

Complete Algebraic Varieties

Complete varieties are, in the category of algebraic varieties, the analogues of compact topological spaces in the category of Hausdorff topological spaces. Recall that the image of a compact topological space under a continuous map is compact and, hence, it is closed if the target space is Hausdorff. Moreover, a Hausdorff topological space X is compact if and only if, for all topological spaces Y , the projection πY : X × Y → Y (with X × Y endowed with the product topology) is a closed map, i.e. it maps closed subsets of X × Y to closed subsets of Y (cf. Bourbaki, 1966, 10.2). For algebraic varieties, some of the previous requests are empty some others make no sense. Indeed: • compactness (in the Zariski topology) is a property which is satisfied by all algebraic varieties (cf. Section 4.2); • any algebraic variety V , which is not reduced to a single point, is never a Hausdorff space; • the Zariski topology ZV ×W of the product V ×W of two algebraic varieties V and W never coincides with the product topology ZarV × ZarW , unless at least one of them is reduced to a single point.

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Thus, in the category of algebraic varieties we will consider the following. Definition 9.1.1. An algebraic variety V is said to be complete if, for all algebraic varieties W , the projection morphism πW : V × W → W onto the second factor is a closed map. Note that, in the previous definition, V × W has a structure of algebraic variety as discussed in Chapter 7, i.e. it is endowed with the Zariski topology ZarV ×W . Corollary 9.1.2. Let V be a complete algebraic variety. Let W be an algebraic variety and let ϕ : V → W be a morphism. Then ϕ(V ) ⊆ W is a closed subvariety. In particular, if ϕ is dominant then it must be surjective, i.e. ϕ(V ) = W . Proof. For the first part of the statement, note first that ϕ(V ) is irreducible since V is irreducible and ϕ is a morphism (cf. Corollary 4.1.6). Thus, we are left to prove that ϕ(V ) is closed in W . From Proposition 7.5.3, the graph Γϕ is closed in V × W . Then one concludes by using that V is complete and that πW (Γϕ ) = ϕ(V ). The second assertion follows from the fact that ϕ(V ) is a closed subset of W containing an open (so dense) subset of W .  The previous corollary gives motivation for the terminology complete: if V and W are algebraic varieties, with V a subvariety of W , and V is complete then, if one has dim(V ) = dim(W ), one necessarily must have V = W . This behavior is in contrast with, e.g. the open immersion ιU0 An ∼ = U0 → Pn , for n  1. In other words, for any n  1, An cannot be a complete variety. The next example more generally show other affine varieties which are not complete. Example 9.1.3. Let V := Za (x1 x2 − x3 ) be the hyperbolic paraboloid in A3 . From Exercise 7.1, V is the image of A1 × A1 ∼ = U01 × U01 in A3 via the 1 1 restriction to U0 × U0 of the Segre morphism σ1,1 : P1 × P1 → P3 . Take the multi-index I = (1, 2) and consider the projection morphism πI : A3 → A2 . The map φ := (πI )|V : V → A2 is the morphism φ in Remark 6.1.10, where we showed that φ(V ) is neither closed nor open in A2 (it is a constructible set). In particular, Corollary 9.1.2 does not hold so V cannot be complete.

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9.2

The Main Theorem of Elimination Theory

The core of this section is to show that, among algebraic varieties, projective varieties play a special role namely they are complete varieties. Theorem 9.2.1. If V is a projective variety, then V is complete. Proof. We have to show that, for any algebraic variety W , the projection πW : V × W → W is a closed morphism. Since V is projective, up to an isomorphism, we can assume V to be a closed, irreducible subset of Pr , for some integer r. From the fact that V × W is therefore closed in Pr × W , it suffices to consider V = Pr . From Proposition 6.4.2, ZarW has a basis consisting of affine open sets. Since the property of being closed is local (recall, e.g. the proof of Proposition 5.2.4-(i)), we can verify the property of being closed in any open set of an affine open covering of W ; with no loss of generality, we can therefore reduce to the case W to be affine. In such a case, W can be considered as an irreducible, closed subset of An , for some integer n. By the induced topology ZarW , we can reduce to the case W = An . To sum up, we need to show that the second projection π2 : Pr × An → An is closed. To prove this, let Z ⊆ Pr × An be any closed subset. From Proposition 7.2.7, Z is defined by polynomial equations gj (X, y) = 0,

1  j  s,

for some integer s, where X = (X0 , . . . , Xr ), y = (y1 , . . . , yn ) and where the polynomials gj are homogeneous of degree dj with respect to the set of indeterminates X, 1  j  s. A point P = (p1 , . . . pn ) ∈ An is such that P ∈ π2 (Z) if and only if −1 π2 (P ) ∩ Z = ∅, i.e. if and only if Zp (g1 (X, P ), . . . , gs (X, P )) = ∅,

(9.1)

(r)

where gj (X, P ) := gj (X0 , . . . , Xr , p1 , . . . , pn ) ∈ Sdj , 1  j  s. Since K is algebraically closed, by the Homogeneous Hilbert “Nullstellensatz”-weak form (cf. Theorem 3.2.4), (9.1) is equivalent to (r)

(S+ )d  (g1 (X, P ), . . . , gs (X, P )),

∀ d  1. (r)

Let Zd := {P ∈ An | (g1 (X, P ), . . . , gs (X, P ))  (S+ )d }. Since π2 (Z) = ∩d1 Zd , it suffices to show that Zd is closed for any d  1.

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For any point P ∈ An and for any d  1 consider the vector space homomorphism (r)

(r)

(r)

ρd (P ) : Sd−d1 ⊕ · · · Sd−ds −→ Sd s defined by ρd (P ) (F1 (X), . . . , Fs (X)) := j=1 Fj (X) gj (X, P ), where (r)

Sd−dt = (0) for those dt > d. With this set-up, P ∈ Zd if and only if   r+d (r) = dim(Sd ). rk(ρd (P )) < d

(9.2)

Condition (9.2) is equivalent to the vanishing of all the minors of order r+d of the matrix associated to the linear map ρd (P ) in the canonical d bases of these vector spaces. These minors are polynomial expressions in the pi ’s, 1  i  n. Replacing the coordinate pi with the indeterminate yi , 1  i  n, one gets that Zd is the closed subset of An defined as the vanishing locus of such polynomial minors, as desired.  The previous result is called the Main Theorem of Elimination Theory. Motivation for the terminology is clearly described by the strategy of the proof: we started with some equations gj (X, y) = 0, 1  j  s, and we asked can be written as for the image of the projection map (X, y) → y, which   y ∈ An | ∃ X ∈ Pr s.t. gj (X, y) = 0, ∀ 1  j  s . In other words, the indeterminates X have been “eliminated” from the problem. Theorem 9.2.1 states that the set of all such y ∈ An can itself be written as the zero-set of suitable polynomial equations. Elimination Theory is concerned with providing algorithms for passing from the equations defining Z ⊆ Pr × An to equations defining π2 (Z) ⊂ An . For example, let Z ⊆ P1 × An be the closed subset defined by two polynomials g1 (X0 , X1 , y) := s0 (y)X0m + s1 (y)X0m−1 X1 + · · · + sm X1m and g2 (X0 , X1 , y) := t0 (y)X0n + t1 (y)X0n−1 X1 + · · · + tn X1n , with si (y), tk (y) ∈ K[y] = A(n) , 0  i  m, 0  k  n. For any P ∈ An , (1)

the two polynomials g1P (X0 , X1 ) := g1 (X0 , X1 , P ) ∈ Sm , g2P (X0 , X1 ) := (1) g1 (X0 , X1 , P ) ∈ Sn have a common zero in P1 if and only if the resultant is such that R(δ0 (g1P ), δ0 (g2P )) = 0. Indeed, if this common zero is [0, 1] ∈ P1 then g1P (0, 1) = g2P (0, 1) = 0 gives sm (P ) = tn (P ) = 0, i.e. the last column of the Sylvester matrix (1.7) of δ0 (g1P ), δ0 (g2P ) is zero. If otherwise the common zero is of the form [1, a] ∈ P1 , for some a ∈ K, the assertion follows from Theorem 1.3.16 (cf. also the proof of Proposition 1.10.19)).   From this observation, since δ0 (g1 (X0 , X1 , y)), δ0 (g2 (X0 , X1 , y)) ∈ K[y] [X1 ], then

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π2 (Z) ⊂ An is given by Za (RX1 (δ0 (g2 (X0 , X1 , y)), δ0 (g2 (X0 , X1 , y))) ⊆ An , as RX1 (δ0 (g2 (X0 , X1 , y)), δ0 (g2 (X0 , X1 , y))) ∈ K[y]. Elimination Theory runs this procedure in general, with the use of elimination ideals (for details, see, e.g. Cohn et al., 2015, Chapter 8, § 5). 9.2.1

Consequences of the main theorem of elimination theory

In the previous section, we have just seen that every projective variety is complete; the converse is not true. On the other hand, it is quite hard to exhibit explicit examples of complete varieties which are not projective; there are examples in dimension 2 due to Nagata and in dimension 3 to Hironaka (cf., e.g. Hartshorne, 1977, Exercise 7.13, p. 171, and Example 3.4.1, p. 443). We will certainly do not treat such examples, so for practical purposes the terms “projective variety” and “complete variety” can be considered almost synonymous. Remark 9.2.2. Theorem 9.2.1 gives different (and shorter) proofs of some results already encountered in the previous chapters, as well as some new consequences. For example: (i) If V is a projective variety and W is an affine variety, any morphism ϕ : V → W is constant; in particular OV (V ) ∼ = K. Proof. This has been already proved in Theorem 5.3.14-(e) and in Corollary 6.2.7. We can give an alternative proof, using the completeness of V . Indeed, since W is affine, we can assume that W is a closed subvariety of An , for some n  0. If we identify An with the affine chart U0 ⊂ Pn and if we consider the morphism ϕ := ιU0 ◦ ιW ◦ ϕ : V → Pn , by the completeness of V we get that ϕ (V ) is closed in Pn . On the other hand, ϕ (V ) = ϕ(V ) ⊆ W ⊆ An is closed in W which is closed in An ; then one concludes by Corollary 6.2.6.  (ii) If V is a projective variety and ϕ : V → Pn is a morphism, then ϕ(V ) ⊂ Pn is closed; in particular it is a projective variety. Recall that this does not occur for arbitrary algebraic varieties (cf., Example 9.1.3). (iii) From (ii) one easily deduces that any Veronese variety Vn,d , any Segre variety Σn,m , etc., are always projective varieties. (iv) Let Λ ⊂ Pn be a linear space inducing a projection πΛ : Pn  Pk , for suitable k < n (cf. Example 8.1.14). We know that if V ⊂ Pn is a

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projective variety s.t. V ∩ Λ = ∅, then the projection πV : V → Pk is a morphism. Thus, πV (V ) ⊂ Pk is again a projective variety which is called the projection of V on the given Pk . (v) Let V ⊂ Pn be a projective variety which contains more than one point and let F ∈ H(S (n) ) be a non-constant homogeneous polynomial. Then V ∩ Zp (F ) = ∅. Proof. This has already been proved in Corollary 6.5.7. An alternative proof which uses completeness of V runs as follows. Assume by contradic(n) tion there exists a homogeneous polynomial F ∈ Sd , for some integer d  1, such that F (P ) = 0 for any P ∈ V . Let P, Q ∈ V be two distinct points and let G ∈ Ip (P ) be homogeneous, of degree d, such that G(Q) = 0. Consider the morphism ϕ : V → P1 , R → [F (R), G(R)],

∀ R∈V

(this is well-defined by the assumption on F ). Then, ϕ(V ) is closed and irreducible in P1 ; therefore ϕ is either constant or it is surjective. Since [0, 1] ∈ / ϕ(V ), this would imply ϕ is constant. This is a contradiction as ϕ(P ) = [F (P ), G(P )] = [1, 0] and ϕ(Q) = [F (Q), G(Q)] = [1, 0] by the choice of G.  Exercises Exercise 9.1. Let V = W = A1 with A(V ) = K[t] and A(W ) = K[s]. Let Z := Za (ts − 1) ⊂ A1 × A1 . (i) Show that Z is a closed variety in A1 × A1 . (ii) Let π2 : A1 × A1 → A1 be the projection onto the second factor. Using (i), deduce in an alternative way that A1 cannot be a complete variety. to be the closure of Z in (iii) If otherwise V = P1 and W = A1 , set Z 1 1 1 P × A . Show that πA1 (Z) is closed in A . Exercise 9.2. Consider in P3R the line := Zp (X1 + X2 − X0 , X3 − X0 ). Find F ∈ S (3) such that Zp (F ) ∩ = ∅ deducing that, in Remark 9.2.2-(v), the assumption for the field K to be algebraically closed is essential.

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Exercise 9.3. For any integers n  1 and d  2, consider the projective (n) space P(Sd ). Let 

(n) (n) (n) R := [F ] ∈ P(Sd ) | F = F  · F  , F  ∈ Sh , F  ∈ Sd−h , 1  h  d − 1 . (n)

Show that R is a proper closed subset of P(Sd ) (cf. Shafarevich, 1994, Proposition p. 60). Exercise 9.4. (i) Let V and W be algebraic varieties and ϕ ∈ Morph(V, W ). The morphism ϕ is said to be a finite morphism if ϕ is dominant and, for any point P ∈ W , there exists an affine open neighborhood UP of P s.t. ϕ−1 (UP ) is an affine open set of V and moreover ϕUp : OW (UP ) → OV (ϕ−1 (UP )) is an integral extension of K–algebras. Show that if ϕ is a finite morphism then, for any point P ∈ Im(ϕ), one has ϕ−1 (P ) is a finite set of V . (ii) Consider the hypersurface Z = Za (x23 + x3 (x1 − x2 ) + x21 − 1) ⊂ A3 . Show that Z is irreducible and that the restriction to Z of the projection πI : A3 → A2 , where I = (1, 2), defines a morphism ϕ ∈ Morph(Z, A2 ) which is finite and such that ϕ(Z) is closed. Exercise 9.5. Let P ∈ Pn be a point and let V ⊂ Pn be an irreducible hypersurface not containing P . Let ϕ : V → Pn−1 be the restriction to V of the projection from P onto Pn−1 . Show that ϕ is a finite morphism.

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Chapter 10

Dimension of Algebraic Varieties

Here, we first define the dimension of an algebraic variety V with the use of its field of rational functions K(V ). Then, we will compare the given general definition with, e.g. the combinatorial dimension introduced in Section 4.3 for topological spaces and with some other notions of “dimension” (cf. Section 10.2 and, e.g. Theorem 12.3.9). 10.1

Dimension of an Algebraic Variety

Let V be any algebraic variety. In Corollary 5.3.21, we proved that K(V ) is a finitely generated field extension of the base field K. In particular, trdegK (K(V )) < +∞, as this number equals the maximal number of algebraically independent elements among the generators of K(V ) over K (recall the proof of Corollary 5.3.21). In this set-up, one sets: Definition 10.1.1. Let V dim(V ) := trdegK (K(V )).

be

any

algebraic

variety.

One

poses

Immediate consequences of the previous definition are the following. Proposition 10.1.2. (i) dim(An ) = dim(Pn ) = n. (ii) If V is any algebraic variety and if U ⊆ V is any non-empty open subset, then dim(U ) = dim(V ). (iii) Let n  1 be an integer and let Λ be a projective (respectively, affine) subspace of Pn (respectively, of An ) s.t. Λ ∼ = Ph (respectively, Λ ∼ = Ah ), with h  0. Then dim(Λ) = h. 225

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(iv) If V is any algebraic variety, then dim(V ) = 0 if and only if V is a point. (v) If V and W are algebraic varieties and if Φ : V  W is a dominant rational map, then dim(W )  dim(V ). (vi) If W is a closed, proper subvariety of an algebraic variety V, then dim(W ) < dim(V ). (vii) If V ⊂ Pn is a projective variety and if Ca (V ) ⊂ An+1 is the affine cone over V, then dim(Ca (V )) = dim(V ) + 1. Proof. (i) It follows from Corollary 5.3.18 and from Remark 1.8.4-(iii). (ii) It follows from Lemma 5.3.13-(ii). (iii) If Λ ⊂ An is an affine subspace s.t. Λ ∼ = Ah , identifying An with the affine chart U0 , the projective closure Λ of Λ in Pn is a linear subspace isomorphic to Ph and Λ is a dense open subset of Λ (cf. Section 3.3.5). From (ii) above, we can therefore focus on the case Λ ⊂ An an affine subspace. Since Λ ∼ = A(h) = K[x1 , . . . , xh ], where = Ah , then A(Λ) ∼ the xi ’s are indeterminates over K. Since Λ is affine, from Theorem 5.3.14-(d), K(Λ) = Q(A(Λ)) ∼ = Q(A(h) ) = Q(h) and we are done. (iv) If V is a point, then V ∼ = A0 and we conclude using (iii). Conversely, assume dim(V ) = 0. From Proposition 6.4.2 and (ii) above, with no loss of generality, we can assume V to be affine so V can be considered as an irreducible, closed subset of An , for some positive integer n. From Theorem 5.3.14, we have K ⊆ A(V ) ⊆ K(V ) = Q(A(V )). Since dim(V ) = 0, it follows that K ⊆ K(V ) is an algebraic extension. On the other hand, since K is algebraically closed, we must have K = K(V ). This forces A(V ) = K and so Ia (V ) ⊂ A(n) to be a maximal ideal. From the Hilbert “Nullstellensatz”-weak form (cf. Theorem 2.2.1), V is a point. (v) If Φ : V  W is dominant, from Theorem 8.1.6, we have a field extension K(W ) ⊆ K(V ). Since trdegK (K(V )) = dim(V ) < +∞, from the field extensions K ⊆ K(W ) ⊆ K(V ) we get also trdegK(W ) (K(V )) < +∞. We conclude by Proposition 1.8.5. (vi) Using (ii) and Proposition 6.4.2, up to replace V with an affine open subset U intersecting W , with no loss of generality we may assume V to be affine, with coordinate ring A(V ), and W to be an affine subvariety of V , with coordinate ring A(W ). Since W ⊂ V is a proper subvariety, from the bijective correspondence between closed

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subvarieties of V and prime ideals in A(V ), one deduces that there A(V ) exists a prime ideal (0) = p ⊂ A(V ) such that A(W ) ∼ = p . Since both A(V ) and A(W ) are integral K-algebras of finite type, from the proof of Theorem 1.12.3, we get that trdegK (A(W )) < trdegK (A(V )), which therefore concludes the proof of (vi). (vii) As in the proof of Proposition 4.3.9, one has a K-algebra isomorphism S(V ) ∼ = A(Ca (V )). Moreover, since V is a projective variety, then Ca (V ) is an affine variety in An+1 . From Theorem 5.3.14-(d) one has K(Ca (V )) ∼ = Q(A(Ca (V ))) ∼ = Q(S(V )). Since A(Ca (V )) is an integral K-algebra of finite type, from Definition 1.8.7, one has also dim(Ca (V )) = trdegK (A(Ca (V )) = trdegK (S(V )). On the other hand, from Corollary 5.3.17 one has that Q(S(V )) ∼ = K(V )(t) where t is an indeterminate over K(V ), i.e. K(V ) ⊂ K(V )(t) is a simple, purely transcendental field extension. Thus, from the definition of dim(V ) := trdegK (K(V )) and from the transitivity of transcendence degree as in Proposition 1.8.5, the proof is complete.  Similarly as for the combinatorial dimension (cf. Section 4.3), when dim(V ) = 1 then V is called an irreducible curve, if otherwise dim(V ) = 2 then V is called an irreducible surface, if dim(V ) = 3 then V is an irreducible threefold and so on. More generally, any algebraic set Y is said to be of pure dimension (or simply, pure, cf. also Definition 4.3.3) if all its irreducible components have the same dimension; in particular, a curve is an algebraic set of pure dimension 1, a surface is an algebraic set of pure dimension 2 and so on. When otherwise Y is not pure, since Y has finitely many irreducible components then dim(Y ) is defined to be the maximum among all the dimensions of its irreducible components (cf. (4.5)). Important consequences are the following. Remark 10.1.3. (i) From Proposition 10.1.2-(v), the dimension is a birational invariant of algebraic varieties. (ii) In particular if V is a rational variety of dim(V ) = n, then it is birational to Pn (and even to An ). Moreover, K(V ) is a purely transcendental extension of K. Indeed, in such a case, K(V ) is isomorphic to Qn so trdegK (K(V )) = n. (iii) Revisiting Examples 5.3.23, 5.3.24, 6.2.5 (ii)–(v), 8.1.13 and 8.2.12 (i)– (ii) and taking into account, e.g. any affine (respectively, projective)

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rational normal curve of degree d  1, these are all irreducible rational curves since they are all birational, sometimes isomorphic, to P1 . (iv) The elliptic curve Ya in Example 5.3.25, with a = 0, 1, is an example of an irreducible curve which is not rational. Indeed, therein we proved that K(Ya ) is an algebraic extension of degree 2 of K(x1 ) ∼ = Q(1) , as 2 x2 ∈ K(Ya ) has minimal polynomial Px1 (t) = t − x1 (x1 − 1)(x1 − a) in Q(1) [t], where t an indeterminate over K. Thus, the field extensions K ⊂ K(x1 ) ⊂ K(Ya ) are such that the first extension is purely transcendental (of transcendence degree 1) whereas the second one is algebraic of degree two, so trdegK (K(Ya )) = trdegK (Q(1) ) = 1 = dim(Ya ). Instead, for a = 0, 1 we obtain plane (nodal) irreducible cubic curves which are monoids, so they are irreducible rational curves. (v) For any subvariety W of an algebraic variety V , one can define its codimension in V , which is denoted by codimV (W ) := dim(V ) − dim(W ).

(10.1)

From Proposition 10.1.2-(ii) and (vi), this integer is always nonnegative; it is zero when W is a non-empty, open subset of V whereas it is strictly positive when W is a closed and proper subvariety of V (or more generally a locally closed subset of a proper, closed subvariety of V ). Proposition 10.1.4. For any algebraic varieties V and W, one has dim(V × W ) = dim(V ) + dim(W ). Proof. From Propositions 6.4.2 and 10.1.2-(ii), we can reduce to the case that V and W are both affine varieties. In particular, we can consider them as irreducible, closed subsets V ⊂ An and W ⊂ Am , for some integers n and m. By Proposition 7.1.1, V × W ⊂ An × Am ∼ = An+m is an affine variety. Let v := dim(V ) and w := dim(W ). By assumptions on V and n] m] and A(W ) = K[yI1a,...,y W , we have A(V ) = K[xI1a,...,x (V ) (W ) . From (1.16), we have trdegK (A(V )) = v and trdegK (A(W )) = w. Since A(V ) is generated by (the images of) x1 , . . . xn as an integral K-algebra of finite type, the previous equality implies that {x1 , . . . , xn } contains a transcendence basis over K determined by v  n elements. For simplicity, assume this basis to be {x1 , . . . , xv }, i.e. Q(v) ∼ = K(x1 , . . . , xv ) ⊂ Q(A(V )), where the field extension is algebraic. As for A(W ), assume that {y1 , . . . , yw } is a transcendence basis over K.

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From Corollary 6.2.8, the two projection morphisms πV πW V  V ×W  W give rise to injective K-algebra homomorphisms # πV

# πW

A(V ) → A(V × W ) ← A(W ) (cf. (6.2) and Theorem 5.3.14-(a)). Since {x1 , . . . , xv } is a set of elements of A(V ) which are algebraically independent over K, these elements remain algebraically independent over K when viewed as elements of A(V × W ). Same occurs for the set {y1 , . . . , yw }. We want to show that {x1 , . . . , xv , y1 , . . . , yw } is a set of elements in A(V × W ) which are algebraically independent over K. Reasoning recursively, it suffices to show that {x1 , . . . , xv , y1 } is formed by algebraically independent elements over K. Assume there exists a polynomial f (t1 , . . . , tv+1 ) ∈ K[t1 , . . . , tv+1 ] such that f (x1 , . . . , xv , y1 ) = 0. Since we have K[t1 , . . . , tv+1 ] = (K[t1 , . . . , tv ]) [tv+1 ], the polynomial f (t1 , . . . , tv+1 ) can be viewed as a polynomial in the indeterminate tv+1 and with coefficients from K[t1 , . . . , tv ], namely f (t1 , . . . , tv+1 ) = c0 (t1 , . . . , tv ) + c1 (t1 , . . . , tv )tv+1 + · · · + cd (t1 , . . . , tv )tdv+1 , where d the degree w.r.t. the indeterminates tv+1 of the polynomial f (t1 , . . . , tv+1 ) and where cj (t1 , . . . , tv ) ∈ K[t1 , . . . , tv ], 0  j  d. Since V is an affine variety, then A(V ) ∼ = OV (V ) (cf. Theorem 5.3.14-(a)). Thus, since x1 , . . . , xv ∈ A(V ), in particular any polynomial expression evaluated at x1 , . . . xv gives an element in the ground field K. Thus, cj (x1 , . . . , xv ) ∈ K, for any 0  j  d, so g(tv+1 ) := f (x1 , . . . , xv , tv+1 ) ∈ K[tv+1 ]. From the hypothesis one has g(y1 ) = f (x1 , . . . , xv , y1 ) = 0. Thus, since y1 ∈ A(W ) is by assumption algebraically independent over K, the polynomial g(tv+1 ) ∈ K[tv+1 ] has to be identically zero. This means that any coefficient cj (x1 , . . . , xv ) has to be zero, 0  j  d. On the other hand, the elements x1 , . . . , xv ∈ A(V ) are by assumption algebraically independent over K, therefore all the polynomials cj (t1 , . . . , tv ) ∈ K[t1 , . . . , tv ], 0  j  d, have to be identically zero (otherwise any non-zero polynomial among them would give a non-trivial algebraic relation among x1 , . . . , xv , which cannot occur from their algebraic independence over K). This shows that f (t1 , . . . , tv+1 ) is identically zero, i.e. it is the zero-polynomial and so that x1 , . . . , xv , y1 are algebraically independent elements over K. K[x1 ,...,xn ,y1 ,...,ym ] so Finally, from (2.8) one has A(V × W ) ∼ = Ia (V )+Ia (W ) A(V ×W ) is generated as an integral K-algebra of finite type by (the images of) x1 , . . . , xn , y1 , . . . , ym . Thus, K(x1 , . . . , xv , y1 , . . . , yw ) ⊂ Q(A(V × W )) is an algebraic extension and so one concludes. 

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Proposition 10.1.5. Let V be any closed subvariety of An (equivalently, Pn ). Then dim(V ) = n − 1 if and only if V is an irreducible hypersurface. In particular any hypersurface of An (equivalently, Pn ) is of pure dimension n − 1. Proof. As usual, by Propositions 6.4.2 and 10.1.2-(ii), it suffices to consider V to be affine. (⇒) If dim(V ) = n − 1 < n = dim(An ), V is a proper, closed subvariety of An so Ia (V ) = (0). Take any non-zero f ∈ Ia (V ); since Ia (V ) is prime, we can moreover assume f to be irreducible. Then V ⊆ Za (f ) ⊂ An , where the second inclusion is strict. Therefore, dim(Za (f )) < n so necessarily n − 1 = dim(V ) = dim(Za (f )). Since V is closed in An , then V is also closed in Za (f ). By Proposition 10.1.2-(vi), we must have V = Za (f ). (⇐) Let V = Za (f ), for some non-zero irreducible polynomial f ∈ K[x1 , . . . , xn ]. Up to re-labeling the indeterminates, we can assume f to be non-constant with respect to the indeterminate x1 ; then any g ∈ (f ) is non-constant with respect to x1 , i.e. (f ) does not contain any non-zero ,...,xn ] is generated by (the polynomial h(x2 , . . . , xn ). Since A(V ) = K[x1(f ) images of) x1 , . . . , xn , in particular (the images of) x2 , . . . , xn ∈ A(V ) are algebraically independent over K; thus dim(V )  n − 1. By Proposition 10.1.2-(vi) equality must hold. For the last part of the statement, any (possibly reducible) affine hypersurface is given by Za (f ) = ∪i=1 Za (fi ), where f = f1 · · · f is its reduced equation (recall (2.6) and (2.7)). Any Za (fi ) is an irreducible  component of Za (f ) to which one applies the first part of the proof. 10.2

Comparison on Various Definitions of “Dimension”

Up to now, we have introduced several concepts related to the word “dimension”, e.g. the Krull-dimension for (commutative and with identity) rings, the combinatorial dimension for topological spaces and the dimension of algebraic varieties. We want to compare all these notions. We start with the following easy result. Proposition 10.2.1. Let V be any affine variety and let A(V ) denote its coordinate ring. Then dim(V ) = dimc (V ) = K dim(A(V )). Proof. The second equality has been proved in Proposition 4.3.7. Using Definition 1.8.7, we have that dim(V ) = trdegK (A(V )) so, since A(V )

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is an integral K-algebra of finite type, by Theorem 1.12.3, we have that  trdegK (A(V )) = K dim(A(V )), and the proof is complete. Corollary 10.2.2. (i) Let V be any algebraic variety. Then dim(V ) = dimc (V ). In particular, if dim(V ) = d, then: (a) all maximal chains of closed subvarieties of V have length d; (b) for any closed subvariety ∅ = W ⊂ V, there exists a maximal chain of closed subvarieties of V containing W ; (c) for any integer 0  m  d, there exists a non-empty, closed subvariety Wm ⊂ V such that dim(Wm ) = m. (ii) If V ⊂ Pn is in particular a projective variety and if S(V ) denotes its homogeneous coordinate ring, then (4.7) holds true. Proof. (i) From Proposition 10.1.2-(ii), for any non-empty, open subset U ⊆ V one has dim(U ) = dim(V ). Since, by Proposition 6.4.2, any algebraic variety V has an affine open covering, with no loss of generality, one can reduce to the case of V an affine variety and conclude that dim(V ) = dimc (V ) by using Proposition 10.2.1. Since V can be assumed to be affine, by Theorem 1.12.3 applied to the coordinate ring A(V ), (a) directly follows from Proposition 4.3.7. To prove (b), it suffices to observe that W ⊂ V is a chain of length 1 and that any chain of closed subvarieties of V can be always extended to a maximal chain of length d, by definition of dimc (V ) = d. As for (c), if ∅ = W0 ⊂ W1 ⊂ · · · Wd = V is a maximal chain of closed subvarieties then, by Proposition 10.1.2-(vi), for any 0  m  d − 1 one has dim(Wm ) < dim(Wm+1 ), therefore one more precisely has dim(Wm ) = m. Since, by (a), all maximal chains of closed subvarieties of V have precisely length d, one concludes. (ii) Since V ⊂ Pn is a projective variety, then Ca (V ) ⊂ An+1 is an affine variety. From Proposition 10.2.1 we have that dim(Ca (V )) = dimc (Ca (V )). Proposition 4.3.9 shows that K dim(S(V )) = dimc (Ca (V )). Moreover, by Proposition 10.1.2-(vi), we also have dim(Ca (V )) = dim(V ) + 1. Since, by (i), we have dim(V ) = dimc (V ), in particular (4.7) holds  true.

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Another important identification, which also provides a local interpretation of dimension, is given by the following. Proposition 10.2.3. Let V be any algebraic variety and let P ∈ V be any point. Then dim(V ) = K dim(OV,p ), where OV,P the local ring of V at P . Proof. From Corollary 6.4.3-(i), there is a bijective correspondence between prime ideals of OV,p and closed subvarieties of V containing the point P . Therefore, prime ideal chains in OV,p bijectively correspond to chains of closed subvarieties of V containing the point P . From Corollary 10.2.2-(i), (a), any such maximal chain has length dim(V ).  Remark 10.2.4. Recall that, for any point P ∈ V , OV,p is an integral K-algebra such that OV,p ⊆ K(V ). From Corollary 6.4.3-(ii) one has Q(OV,p ) ∼ = K(V ) whereas, from Definition 1.8.7, trdegK (OV,p ) = trdegK (K(V )), the latter being equal to dim(V ). On the other hand, Proposition 10.2.3 is not a consequence of Theorem 1.12.3 since, in general, OV,p is not a K-algebra of finite type (recall Remark 1.12.5). 10.3

Dimension and Intersections

Here, we discuss some results related to the dimension of the locus obtained by intersecting an algebraic variety V with a certain finite number of irreducible projective hypersuperfaces. These results will have significant consequences on the structure of algebraic varieties. Theorem 10.3.1. Let V ⊂ Pn be an algebraic variety and let W1 , . . . , Ws , s  1, be irreducible hypersurfaces of Pn . Then: (i) if V ∩ W1 ∩ · · · ∩ Ws = ∅, any irreducible component X of V ∩ W1 ∩ · · · ∩ Ws is such that dim(X)  dim(V ) − s; (ii) if V is projective with dim(V )  s, then V ∩ W1 ∩ · · · ∩ Ws = ∅. Proof. (i) Using induction on s, we can reduce to the case s = 1. Since X is closed in V , up to replacing V with an affine open set intersecting X, we may assume that V is an affine variety and that X is an irreducible component of ZV (f ), for some f ∈ A(V ). If f = 0 in A(V ) this means that W1 is defined by a polynomial lying in Ia (V ), i.e. V ⊆ W1 so that V ∩ W1 = V and X = V since V is irreducible; in this case

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dim(X) = dim(V ) > dim(V ) − 1. If otherwise f ∈ A(V ) \ {0}, then f is not invertible in A(V ) since ∅ = X ⊆ V ∩ W1 . Let p ⊂ A(V ) be the prime ideal defining X. Since X ⊆ ZV (f ), by reversing inclusion we have p ⊃ (f ). Moreover, since X is an irreducible component of V ∩W1 , then p is a minimal prime ideal containing (f ) (in general, the minimal prime ideals of an ideal I are defined to be the minimal elements of the set of all prime ideals containing I). Since f = 0 and not invertible in A(V ), from Krull’s principal ideal theorem (also Hauptidealsatz, cf., e.g. Milne, 2017, Theorem 15.3), it follows that ht(p) = 1. From the bijective correspondence prime ideals-closed subvarieties, this implies that it cannot exist any closed subvariety Z ⊂ V satisfying X ⊂ Z ⊂ V . Thus X ⊂ V is a maximal chain of closed subvarieties of V containing Z so dim(X) = dim(V ) − 1. (n) (ii) We use induction on s. Consider first s = 1 and let F1 ∈ Sd be the homogeneous polynomial such that W1 = Zp (F1 ), where d = deg(W1 )  1. Set S(V ) the homogeneous coordinate ring of V and let F 1 ∈ S(V ) be the image of F1 under the canonical epimorphism π : S (n)  S(V ); thus W1 ∩ V = ZV (F 1 ). If F 1 = 0 in S(V ), as in (i), this implies that V ⊆ W1 so V ∩ W1 = V = ∅ since dim(V )  1. If otherwise F 1 = 0 in S(V ), the F 1 is a homogeneous element of positive degree in S(V ). Let Ca (V ) ⊂ An+1 be the affine cone over V ; since A(Ca (V )) ∼ = S(V ) as integral K-algebras, then F 1 ∈ A(Ca (V )) defines a closed subset Y ⊂ Ca (V ) which certainly is not empty since, being F 1 homogeneous of positive degree, its vanishing locus Y contains the vertex O of the affine cone Ca (V ). Applying (i), we get that any irreducible component of Y has dimension at least dim(Ca (V ))− 1  1, the latter inequality following from Proposition 10.1.2-(vii) and from the assumption dim(V )  1. Thus, V ∩ W1 = ∅. Assume now s  2. By inductive hypothesis, V ∩W1 ∩· · ·∩Ws−1 = ∅; thus, from (i), dim(X)  dim(V ) − (s − 1)  1 for any irreducible component X of V ∩ W1 ∩ · · · ∩ Ws−1 , the second inequality following from dim(V )  s, as in the assumptions. Applying reasoning for the case s = 1 as above to X and Ws−1 , one concludes.  As a consequence of the previous result, we also find a different proof of Corollary 6.2.6. Corollary 10.3.2. Let V ⊂ An be an affine variety which coincides with its projective closure in Pn . Then V is a point.

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Proof. Identifying An with the affine chart U0 of Pn , we have inclusions p V ⊂ U0 ⊂ Pn . From the assumption, V ∩ H0 = ∅, where H0 = Zp (X0 ) the fundamental hyperplane of Pn . From Theorem 10.3.1-(ii), it follows that p  dim(V ) < 1 which implies dim(V ) = 0. Another fundamental property, which will also be used later on in the proof of Theorem 11.1.1, is the following. Proposition 10.3.3. Let V ⊆ Pr be an algebraic variety and let Z ⊂ V be a closed subvariety such that codimV (Z) = s  1. Then, there exist hypersurfaces W1 , . . . , Ws ⊂ Pr such that any irreducible component of V ∩ W1 ∩ · · · ∩ Ws has codimension s in V and moreover Z is one of such irreducible components. Proof. We argue by induction on the integer s. Let s = 1; since Z is a proper, closed subvariety of V , there exists F1 ∈ Ip (Z) \ Ip (V ) and, by Theorem 10.3.1-(i), any irreducible component Y of Zp (F1 ) ∩ V satisfies codimV (Y ) = 1. Since Z ⊆ Zp (F1 ) ∩ V and Z is irreducible, then Z must be one of such irreducible components. Assume now s  2. From Corollary 10.2.2-(i), (c), there exists a closed subvariety Z  ⊂ V such that Z ⊂ Z  ⊂ V and that dim(Z  ) = dim(Z) + 1. From the inductive hypothesis, there exist W1 , . . . , Ws−1 hypersurfaces of Pr such that V ∩ W1 ∩ · · · ∩ Ws−1 = Z1 ∪ · · · ∪ Zk , where Zj is irreducible with dim(Zj ) = dim(V ) − (s − 1), for any 1  j  k, and moreover there exists j0 ∈ {1, . . . , k} such that Z  = Zj0 . Note that since Z is a proper, closed irreducible subset of Z  , in particular Ip (Z) is not contained in Ip (Zj ), for any 1  j  k. From the fact that any Ip (Zj ) is a prime ideal, it follows that Ip (Z) is not contained there exists a homogeneous polyin the union ∪kj=1 Ip(Zj ). Therefore,  nomial Fs ∈ Ip (Z) \ ∪kj=1 Ip (Zj ) which gives rise to the hypersurface Ws := Zp (Fs ) ⊂ Pr such that Ws contains Z but it does not contain any of the Zj , 1  j  k. Therefore, V ∩ W1 ∩ · · · ∩ Ws = (Z1 ∪ · · · ∪ Zk ) ∩ Ws , so any irreducible component of this intersection has exactly codimension s in V and Z, being irreducible and closed, is one of such irreducible components. 

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10.4

235

Complete Intersections

We conclude this chapter by discussing special cases of intersections of an algebraic variety with finitely many hypersurfaces of an affine, respectively, projective space. We start with the following. Definition 10.4.1. Let R be an integral K-algebra of finite type and let f1 , . . . , fs ∈ R. The elements f1 , . . . , fs form a regular sequence in R if, for any 1  i  s, the multiplication map R R · fi −→ (f1 , . . . , fi−1 ) (f1 , . . . , fi−1 ) is injective. In other words, f1 , . . . , fs form a regular sequence if and only if, for any 1  i  s, the image of fi via the quotient epimorphism R R is not a zero-divisor in (f1 ,...,f (equivalently, πi−1 : R  (f1 ,...,f i−1 ) i−1 ) fi ∈ R is not a zero-divisor modulo the ideal (f1 , . . . , fi−1 ), for any 1  i  s). We have the following. Proposition 10.4.2. (i) Let V be an affine variety and let A(V ) be its coordinate ring. If f1 , . . . , fs ∈ A(V ) form a regular sequence then, for any 1  i  s, any irreducible component of ZV (f1 , . . . , fi ) has dimension exactly dim(V ) − i. (ii) Let V ⊂ Pn be a projective variety and let S(V ) be its homogeneous coordinate ring. If F1 , . . . , Fs ∈ S(V ) are homogeneous elements forming a regular sequence then, for any 1  i  s, any irreducible component of ZV (F1 , . . . , Fi ) has dimension exactly dim(V ) − i. Proof. We focus on (i), since the proof of (ii) is similar. Note that the statement is equivalent to saying that, for any 1  i  n, ZV (fi ) does not contain any irreducible component of ZV (f1 , . . . , fi−1 ). By assumptions (f1 , . . . , fi−1 ) ⊂ A(V ) moreover, since fi is not a is not a zerozero-divisor modulo  (f1 , . . . , fi−1 ), it follows that fi  (f1 , . . . , fi−1 ). In particular, fi ∈ / (f1 , . . . , fi−1 ). divisor modulo If, by contradiction, ZV (fi ) contained an irreducible component of  fi−1 ) such that ZV (f1 , . . . , fi−1 ), there would exist gi ∈ A(V ) \ (f1 , . . . ,  ZV (f1 , . . . , fi−1 ) ⊂ ZV (fi ) ∪ ZV (gi ) = ZV (fi gi ), i.e. fi gi ∈ (f1 , . . . , fi−1 ) which is a contradiction. 

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Definition 10.4.3. (i) Let V ⊂ An be an affine variety. Assume that Ia (V ) = (f1 , . . . , fs ), where f1 , . . . , fs ∈ A(n) is a regular sequence. Then V is said to be (affine) complete intersection of the s hypersurfaces Za (f1 ), . . . , Za (fs ) of An . If V = Za (f1 , . . . , fs ),where f1 , . . . , fs ∈ A(n) is a regular (f1 , . . . , fs )) then V is said to be sequence (namely Ia (V ) = (affine) set-theoretically complete intersection of the s hypersurfaces Za (f1 ), . . . , Za (fs ) of An . (ii) Let V ⊂ Pn be a projective variety such that Ip (V ) = (F1 , . . . , Fs ), where F1 , . . . , Fs ∈ S (n) is a collection of homogeneous elements forming a regular sequence. Then V is said to be (projective) complete intersection of the s hypersurfaces Zp (F1 ), . . . , Zp (Fs ) of Pn . If V = Zp (F1 , . . . , Fs ), where F1 , . . . , Fs ∈ S (n) is a collection ofhomogeneous elements forming a regular sequence (namely Ip (V ) = (F1 , . . . , Fs )) then V is said to be (projective) set-theoretically complete intersection of the s hypersurfaces Zp (F1 ), . . . , Za (Fs ) of Pn . Note that, if V is either set-theoretically complete intersection or complete intersection as above, in both cases dim(V ) = n − s, i.e. V has codimension s in the ambient space. Moreover, it is clear from the definition that if V is a complete intersection then it is also set-theoretically complete intersection. The converse does not hold in general, as we shall discuss in the following clarifying examples. Example 10.4.4. (i) Consider C ⊂ A3 the affine twisted cubic as in Section 3.3.12. From (3.33) we know that Ia (C) = (x2 − x21 , x3 − x1 x2 ). Set f1 := x2 − x21 , f2 := x3 − x1 x2 ∈ A(3) and note that f1 , f2 is (3) ∼ a regular sequence: indeed, R := A (f1 ) = K[x1 , x3 ] and the image (

π1 (f2 ) under the quotient epimorphism π1 : A(3)  A(f13)) = R is ∼ π1 (f2 ) = x3 − x31 which is a non-zero divisor in R since (π1R (f2 )) = K[x1 ] is an integral domain. Therefore, the affine twisted cubic is (affine) complete intersection of the two quadrics Za (f1 ) and Za (f2 ) in A3 ; it is therefore also set-theoretically complete intersection of the given two quadrics and codimA3 (C) = 2 which equals the number of generators of its radical ideal and the number of hypersurfaces which cut-out C in A3 .

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(ii) Let us consider Γ := C ⊂ P3 the projective twisted cubic, defined as the projective closure in P3 of the affine twisted cubic C as in (i), when A3 is identified with the affine chart U0 ⊂ P3 (cf. Claim 3.3.18). We proved therein that Ip (Γ) = (F1 , F2 , F3 ), where F1 := X0 X2 − X12 ,

F2 := X0 X3 − X1 X2 ,

F3 := X1 X3 − X22 ,

are three irreducible, homogeneous polynomials of degree 2 which are (3) linearly independent in S2 and which as the maximal minors  0 Xarise 1 X2 (cf. (3.35), (3.36) and of the matrix of linear forms A := X X1 X2 X3 Claim 3.3.19). Note that, dehomogenizing w.r.t. the indeterminates X0 , one has δ0 (F1 ) = f1 , δ0 (F2 ) = f2 , δ0 (F3 ) = f3 := x1 f2 − x2 f1 ∈ Ia (C). From Proposition 10.4.2-(ii), F1 , F2 , F3 ∈ S (3) cannot be a regular sequence since dim(Γ) = 1 > dim(P3 ) − 3, so Γ is not complete intersection in P3 . On the other hand, consider the homogeneous cubic (3) polynomial G := X2 F3 − X3 F2 = 2X1 X2 X3 − X23 − X0 X32 ∈ S3 . Note that, F1 , G ∈ S (3) is a regular sequence: indeed Zp (F1 ) ⊂ P3 S (3) is a quadric cone which is irreducible (cf. Section 8.2.2) thus (F is 1) an integral domain and therefore G is not a zero-divisor modulo (F1 ). From the definition of G, it is clear that Γ ⊆ Zp (F1 , G); we claim that one actually has Γ = Zp (F1 , G). This is a consequence of the following Linear Algebra result.  a11 a12 a13  Lemma 10.4.5. Let K be a field and let A := aa21 aa22 aa23 be a 31 32 33 square matrix of order 3, where aij ∈ K. Set   a11 a12 A1 := a21 a22

  a11 a12 a13 A2 := a21 a22 a23

⎞ ⎛ a11 a12 A3 := ⎝a21 a22 ⎠ a31 a32

as minors of A. If det(A) = det(A1 ) = 0, then either rank(A2 ) < 2 or rank(A3 ) < 2. Proof. From the assumptions rank(A)  2. If rank(A2 ) < 2, there is nothing else to prove. Assume therefore rank(A2 ) = 2; thus, since det(A) = 0, the homogeneous linear system having A as coefficient matrix is equivalent to that having A2 as coefficient matrix. Solutions of A2 x = 0 depend on one parameter in K and are proportional to

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the determinants, up to suitable signs ±, of the 2 × 2 minors of A2 . is by On the other hand, one of this minors is A1 whosedeterminant  assumption equal to 0. Therefore, one has x = t

b1 b2 0

where t ∈ K

a parameter and where b1 and b2 the determinants of the other two minors (with sign) of A2 and at least one of the bi ’s is non-zero. One has therefore that ⎧ ⎪ ⎨ a11 b1 + a12 b2 = 0 a21 b1 + a22 b2 = 0 ⎪ ⎩ a31 b1 + a32 b2 = 0 holds, which implies that the homogeneous linear system of three equations in two indeterminates having A3 as coefficient matrix has  non-zero solutions, therefore rank(A3 ) < 2. With above set-up, consider the following matrices: ⎞ ⎛   X0 X1 X2 X0 X1 ⎠ ⎝ A := X1 X2 X3 , A1 := , X1 X2 X2 X3 0 ⎛ ⎞   X0 X1 X0 X1 X2 A2 := , A3 := ⎝X1 X2 ⎠ X1 X2 X3 X2 X3 (3)

containing linear forms in S1 . Note that det(A) = G, det(A1 ) = F1 and that A2 = At3 . For any point P ∈ P3 , denote by A(P ) (equivalently, Ai (P )) the matrix obtained by evaluating the linear forms at the homogeneous coordinates of P . Therefore, P ∈ Zp (F1 , G) if and only if det(A(P )) = 0 = det(A1 (P )) which, from Lemma 10.4.5, implies rank(A2 (P )) = rank(A3 (P )) < 2; since the vanishing of the maximal minors of A2 (equivalently of A3 ) are exactly the generators of Ip (Γ) (cf. (3.36)), this means that P ∈ Γ, i.e. Γ = Zp (F1 , G) which shows that Γ is set-theoretically complete intersection of a quadric and a cubic in P3 . (iii) Any irreducible hypersurface Zp (f ) ⊂ Pn obviously is a complete intersection. In particular any line in P2 is. The same holds for any line  ⊂ Pn , for any n  2, since its homogenous ideal is generated by n − 1 linearly independent linear forms. Thus, take the line  in P3 such that Ip () = (X0 , X1 ). Let Q = Zp (X0 X3 − X1 X2 ) the rank-four quadric in P3 which is the isomorphic image under the Segre morphism

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σ2,2 of P1 × P1 (cf. Section 8.2.1). From the equation of Q one note that  ⊂ Q and  is a hypersurface in Q. Then  is neither complete intersection nor set-theoretically complete intersection in Q even if  is complete intersection in P3 (cf. Exercise 10.4). Exercises Exercise 10.1. Let V and W be algebraic varieties and let ϕ ∈ Morph(V, W ) be a finite morphism (cf. Exercise 9.4). Show that dim(V ) = dim(W ). Exercise 10.2. Let V ⊂ Pn be a projective variety. Let s be the maximum among the dimension of linear subspaces Λ ⊂ Pn such that dim(Λ) = s and Λ ∩ V = ∅. Deduce that dim(V )  n − s − 1. Exercise 10.3. Let V := Za (x1 x3 − x2 ) ⊂ A3 . Show that V is an irreducible affine hypersurface. Then, let ϕ ∈ Morph(V, A2 ) be the restriction of the projection πI : A3 → A2 , with I = (1, 2). Prove that the dimension of the fibers of ϕ is not constant. Find the maximal open set where the fiber dimension is constant. Deduce that ϕ cannot be a finite morphism. Exercise 10.4. Show that the line  = Zp (X0 , X1 ) ⊂ P3 is contained in the rank-four quadric surface Q = Zp (X0 X3 − X1 X2 ) ⊂ P3 and it is neither complete intersection nor set-theoretically complete intersection in Q. Exercise 10.5. Show that the line  = Zp (X0 , X1 ) ⊂ P3 is contained in the rank-three quadric surface (i.e. quadric cone) Q = Zp (X0 X2 − X12 ) ⊂ P3 and it is not complete intersection but set-theoretically complete intersection in Q .

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Chapter 11

Fiber-Dimension: Semicontinuity

In this chapter, we study dimensional behavior of the fibers of a given dominant morphism ϕ ∈ Morph(V, W ) of algebraic varieties, namely the dimension of the subsets ϕ−1 (P ) ⊆ V , for P ∈ Im(ϕ) varying. For further reading, we refer to, e.g. Shafarevich (1994, § I.6.3, pp. 76–78). 11.1

Fibers of a Dominant Morphism

If V and W are algebraic varieties and if ϕ ∈ Morph(V, W ) is a dominant morphism, recall that from Proposition 10.1.2-(v) the integer r := dim(V )− dim(W ) is non-negative. Theorem 11.1.1. Let V and W be algebraic varieties and ϕ ∈ Morph(V, W ) be a dominant morphism. Set r := dim(V ) − dim(W ). For any point P ∈ Im(ϕ) ⊆ W consider the set ϕ−1 (P ) := {Q ∈ V | ϕ(Q) = P } ⊆ V, which is called the fiber of the morphism ϕ over the point P . Then ϕ−1 (P ) is a closed algebraic subset of V . Moreover: (i) for any irreducible component FP of ϕ−1 (P ), one has dim (FP )  r; (ii) there exists a non-empty, open subset U of W such that U ⊆ Im(ϕ) and, for any point P ∈ U and for any irreducible component FP ⊆ ϕ−1 (P ), one more precisely has dim (FP ) = r. Proof. ϕ−1 (P ) is closed in V since P ∈ Im(ϕ) ⊆ W is closed and ϕ is a continuous map. 241

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To prove (i), since V is irreducible then Im(ϕ) is an algebraic subvariety of W . From Proposition 6.4.2, let U ⊆ Im(ϕ) be any affine open set of Im(ϕ). Similarly, let U  ⊆ ϕ−1 (U ) be any affine open set of ϕ−1 (U ). Note that, by the dominance of ϕ and the fact that U is an open set of Im(ϕ), one has U = Im(ϕ) = W so the composition ϕ := ϕ ◦ ιU  ∈ Morph(U  , U ) is dominant too. Therefore, with no loss of generality, we may assume V and W to be affine varieties. In this set-up, since ϕ ∈ Morh(V, W ) is dominant and V and W are affine varieties, by Corollary 6.2.8 the map ϕ# ∈ HomK (A(W ), A(V )) is an injective K-algebra homomorphism. Set d := dim(W ); from Proposition 10.3.3 it follows that there exist f1 , . . . , fd ∈ A(W ) such that {P } is an irreducible component of ZW (f1 , . . . , fd ). Then ϕ−1 (P ) ⊆ ZV (ϕ# (f1 ), . . . , ϕ# (fd )), where 0 = ϕ# (fi ) ∈ A(V ) by injectivity of ϕ# and moreover, from the fact that {P } is an irreducible component of ZW (f1 , . . . , fd ), it follows that ϕ−1 (P ) is a connected component of ZV (ϕ# (f1 ), . . . , ϕ# (fd )). Thus any irreducible component Fp of ϕ−1 (P ) is also an irreducible component of ZV (ϕ# (f1 ), . . . , ϕ# (fd )). From Theorem 10.3.1-(i) and from the injectivity of ϕ# we have that dim(Fp )  dim(V ) − d = dim(V ) − dim(W ) = r. To prove (ii), as above we may assume that V and W are affine varieties. By the dominance of ϕ, A(W ) is a subalgebra of A(V ) via the monomorphism ϕ# . We may assume dim(V ) > 0 otherwise dim(W ) = dim(V ) = 0 and ϕ constant and there is nothing else to prove. Let v1 , . . . , vn ∈ A(V ) such that A(V ) = A(W )[v1 , . . . , vn ] namely, taking x1 , . . . , xn indeterminates over A(W ) and setting x := (x1 , . . . , xn ), there exist polynomials f1 (x), . . . , fk (x) ∈ A(W )[x] such that A(W )[x] A(V ) ∼ = (f1 (x),...,fk (x)) . For any point P ∈ Im(ϕ) ⊆ W and for any irreducible component FP ⊆ ϕ−1 (P ), the coordinate ring A(FP ) is a quotient ring of the K-algebra K[x] := K[ v1 , . . . , vn ]. (f1 (P )(x), . . . , fk (P )(x)) ϕ∗

Since ϕ# extends to a field homomorphism K(W ) → K(V ), by the transitivity of transcendence degree (cf. Proposition 1.8.5), we have trdegK(W ) (K(V )) = trdegK (K(V )) − trdegK (K(W )) = dim(V ) − dim(W ) = r  0.

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Thus, from the fact that K(V ) ∼ = K(W )(v1 , . . . , vn ), up to reordering generators, we may assume that v1 , . . . , vr ∈ A(V ) are a transcendence basis of K(V ) over K(W ) so they are algebraically independent over A(W ). This implies that any element of A(V ) is algebraic over A(W )[v1 , . . . , vr ]. Therefore, for any r + 1  j  n, there exist polynomials gj (x1 , . . . , xr , xj ) ∈ A(W )[x1 , . . . , xr , xj ] which are non-constant w.r.t. the indeterminate xj and such that gj (v1 , . . . , vr , vj ) = 0, i.e. gj (x1 , . . . , xr , xj ) ∈ (f1 (x), . . . , fk (x)) for any r + 1  j  n. Let l.c.xj (gj ) ∈ A(W )[x1 , . . . , xr ] be the leading-coefficient polynomial of gj (x1 , . . . , xr , xj ) w.r.t. the indeterminate xj , r + 1  j  n. By the assumptions Uj := ZW (l.c.xj (gj ))c is a non-empty, principal open subset of W , for any r + 1  j  n. Set U :=

n 

Uj ,

(11.1)

j=r+1

which is a non-empty open subset of W . For any point P ∈ U , the polynomials gj (P )(x1 , . . . , xr , xj ) ∈ K[x1 , . . . , xr , xj ] are not identically zero (because of the leading coefficients w.r.t. the indeterminate xj and the definition of v1 , . . . ,  vr ,  vj ) = 0, Uj , for any r + 1  j  n) and are such that gj (P )( i.e. gj (P )(x1 , . . . , xr , xj ) ∈ (f1 (P )(x), . . . , fk (P )(x)) ⊂ K[x], for any r + 1  j  n. This implies that, for any irreducible component FP ⊆ ϕ−1 (P ) the elements vr+1 , . . . , vn ∈ A(FP ) are algebraically dependent over K[ v1 , . . . , vr ]. Thus dim(FP )  r. From Theorem 11.1.1-(i)  one concludes that dim(FP ) = r. Another fundamental property, which directly follows from the previous results, deals with what mentioned in Remark 6.1.6. Indeed, we are now in position to deduce the following. Corollary 11.1.2. Let V and W be algebraic varieties and let ϕ ∈ Morph(V, W ) be a dominant morphism. Then Im(ϕ) always contains a non-empty open subset U of W . Moreover for all the points P in such an open set U, setting r = dim(V ) − dim(W ), the fiber ϕ−1 (P ) is a non-empty, pure, r-dimensional algebraic subset of V . Proof. The open set U is that in (11.1).



Similarly, recalling Remark 6.1.10, we have the following. Corollary 11.1.3 (Chevalley’s theorem). Let V and W be algebraic varieties and let ϕ ∈ Morph(V, W ) be any morphism. The image of ϕ is a

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constructible set, namely a union of finitely many locally-closed subsets of W . More generally, any ϕ ∈ Morph(V, W ) maps constructible sets in V to constructible sets in W . Proof. The second part of the statement follows directly from the first. To prove the first part of the statement, we use induction on dim(W ). If dim(W ) = 0, the morphism ϕ is constant and surjective. Therefore, we may assume dim(W ) := m > 0 and that the statement holds true in dimension less than m. To proceed with the proof, there are two cases to be considered. If ϕ is not dominant, since Im(ϕ) is irreducible in W then Z := Im(ϕ) is a closed subvariety properly contained in W , ϕ ∈ Morph(V, Z) is dominant and dim(Z) < dim(W ) from Proposition 10.1.2-(vi). By induction, we are done in this case. Assume therefore ϕ ∈ Morph(V, W ) to be dominant. Let U be a nonempty subset of W such that U ⊆ Im(ϕ), as in (11.1). If W = U , i.e. ϕ is surjective, there is nothing else to prove. If otherwise U ⊂ W , then U c is a non-empty, proper closed subset of W . Denote by Z1 , . . . , Zk the irreducible components of U c and let Yi,1 , . . . , Yi,ti be irreducible components of ϕ−1 (Zi ), 1  i  k. For any (i, ji ), 1  i  k, 1  ji  ti , let ιYi,ji be the closed immersion of Yi,ji into V and let ϕi,ji := ϕ ◦ ιYi,ji ∈ Morph(Yi,ji , Zi ). Since dim(Zi ) < dim(W ) by Proposition 10.1.2-(vi), for any 1  i  k, by inductive assumptions ϕi,ji (Yi,ji ) is constructible in Zi , which is closed in  W . Thus, Im(ϕ) = U ∪ (∪i,ji ϕi,ji (Yi,ji )) is constructible in W . 11.2

Semicontinuity

Focusing on projective varieties, another interesting and very useful consequence of Theorem 11.1.1 is given by the following result. Proposition 11.2.1. Let V, W be projective varieties, ϕ ∈ Morph(V, W ) be a surjective morphism and r = dim(V ) − dim(W ). For any integer j such that r  j  dim(V ), the sets   Wj := P ∈ W | ϕ−1 (P ) has an irreducible component FP s.t. dim(FP )  j

are closed subsets of W . Proof. From Theorem 11.1.1 and from the surjectivity of ϕ, it follows that Wr = W and moreover there exists a closed subset X ⊂ W for which Wj ⊆ X, for any j > r. Assume therefore, j > r and Wj = ∅. To prove that Wj is closed in W it suffices to proving that the intersection of Wj

Fiber-Dimension: Semicontinuity

245

with any of the irreducible component of X, which is closed in W , is closed in X. We therefore may assume X to be irreducible and let Y1 , . . . , Yh be the irreducible components of ϕ−1 (X) for which ϕi := ϕ ◦ ιYi ∈ Morph(Yi , X) is dominant, for any 1  i  h. Since V is projective and Yi ⊂ V is closed in V , then Yi is a projective variety, for any 1  i  h. From Theorem 9.2.1 and Corollary 9.1.2, it follows that any ϕi is a surjective morphism. If j  dim(Yi ) − dim(X), for some i, then from Theorem 11.1.1-(i) we get that Wj = X which is closed in W . If otherwise j > dim(Yi ) − dim(X), for any 1  i  h, then Wj is contained in a proper, closed subset of X. Applying once again the same arguments, one concludes.  To sum-up, for projective varieties the content of the previous results is the following: for any surjective morphism between projective varieties V and W , there exists a non-empty open set U of W over which all the fibers of the morphism are pure algebraic subsets of V of minimal possible dimension, which is dim(V )−dim(W ) (cf. Theorem 11.1.1-(ii)). There could also exist possible subsets of W , which are in the complementary of the open set U , where the dimension of the fibers may jump and these subsets are closed in W (cf. Proposition 11.2.1). This behavior is known as (upper)semicontinuity of fiber dimension according to the following. Theorem 11.2.2 (Upper-semicontinuity of the fiber dimension). Let V and W be projective varieties, ϕ ∈ Morph(V, W ) be a surjective morphism. For all Q ∈ V define   e(Q) := max(dim(Z)) | Z is an irreducible component of ϕ−1 (ϕ(Q)) containing Q .

The function e : V → Z0 is upper-semicontinuous, i.e. for all j ∈ Z0 , the set Σj (ϕ) := {Q ∈ V | e(Q)  j} is closed in W . Proof. The proof is based on Proposition 11.2.1. Indeed, set r := dim(V )− dim(W ) which is non-negative. If j ∈ Z0 is such that j  r then, from Theorem 11.1.1-(i), one has that Σj (ϕ) = ϕ−1 (W ) = V and we are done. For those j ∈ Z0 for which j > r one has either Σj (ϕ) = ∅ (e.g. for those j > dim(V )) or Σj (ϕ) = ϕ−1 (Wj ), for those ∅ = Wj ⊂ W defined as in Proposition 11.2.1. Since Wj is closed in W and ϕ is continuous, we deduce  that Σj (ϕ) is a proper, closed subset of V . A nice consequence of the previous result is the following. Corollary 11.2.3 (Irreducibility of a fibration with irreducible fibers of constant dimension and with irreducible base variety). Let

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V be a projective algebraic set, W be a projective variety with dim(W ) = m and let ϕ ∈ Morph(V, W ) be a surjective morphism. Assume that, for any P ∈ W, the fiber ϕ−1 (P ) is irreducible and dim(ϕ−1 (P )) = d. Then V is irreducible, i.e. V is a projective variety, of dimension dim(V ) = m + d. Proof. Write V = ∪ni=1 Vi as an irredundant union of irreducible components of V . To prove irreducibility of V , we want to show that i = 1 and that V = V1 . Since V is projective, so any Vi is, and ϕ is a surjective morphism there is at least one i ∈ {1, . . . , n} for which ϕ(Vi ) = W otherwise, from Corollary 9.1.2, W would be reducible against assumption. Up to reordering the irreducible components of V , assume that this occurs for, e.g. i = 1. Let ιVi be the closed immersion of Vi in V and let ϕi := ϕ◦ιVi ∈ Morph(Vi , W ), 1  i  n. For any P ∈ W set λi (P ) := dim(ϕ−1 i (P )). Thus, for any P ∈ W , one has d = dim(ϕ−1 (P )) = max1in {λi (P )}. Since any λi is uppersemicontinuous, from Theorem 11.2.2 there exists an i0 ∈ {1, . . . , n} such that λi0 (P ) = d, for any P ∈ W ; in particular ϕi0 : Vi0 → W is −1 (P ) ∩ Vi surjective. For such an index i0 one has that ϕ−1 i0 (P ) = ϕ −1 is a closed subset of ϕ (P ) of the same dimension. Therefore, one has −1 (P ), for any P ∈ W , as it follows from Proposition 10.1.2-(vi) ϕ−1 i0 (P ) = ϕ and from the irreducibility of the fiber ϕ−1 (P ) for any P ∈ W . Therefore, Vi0 = V1 = V and the irredundant decomposition consists of only V , namely V is irreducible. Finally since the open set U , as in Theorem 11.1.1-(ii), where the uppersemicontinuous function λ := λ1 reaches its minimum d coincides with the whole variety W , by Theorem 11.1.1-(ii) one concludes that dim(V ) =  m + d = dim(W ) + dim(ϕ−1 (P )), for any P ∈ W . Exercises Exercise 11.1. Let V and W be algebraic varieties. Give an example of a morphism ϕ ∈ Morph(V, W ) which is dominant, where all the ϕ-fibers have the same dimension and moreover there exists a suitable non-empty open set U ⊆ W such that, for each point P ∈ U the fiber ϕ−1 (P ) is reducible nonetheless with V irreducible. Exercise 11.2. Give an example of an algebraic variety V ⊂ P2 × A1 such that, if ϕ denotes the restriction to V of the second projection onto A1 , then ϕ is surjective, not all the ϕ-fibers are irreducible nonetheless V is irreducible and all the ϕ-fibers have the same dimension.

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Exercise 11.3. Let ϕ : V → W be a morphism between projective varieties of positive dimensions for which there exists a point P ∈ Im(ϕ) such that ϕ−1 (P ) consists of finitely many points. Show that dim(Im(ϕ)) = dim(V ). Exercise 11.4. Let V be an algebraic variety and let W ⊂ V be a proper subvariety. Prove that K dim(OV,W ) = codimV (W ). Exercise 11.5. Let V be an algebraic variety and let P = Q ∈ V be any pair of distinct points. Show that there always exists an irreducible hypersurface W ⊂ V such that P ∈ W but Q ∈ / W.

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Chapter 12

Tangent Spaces: Smoothness of Algebraic Varieties

In this chapter, we consider the local study of an algebraic variety V , namely we examine the structure of V locally at a given point P ∈ V . Throughout this chapter the field K is taken to be not only algebraically closed but also of characteristic 0. 12.1

Tangent Space at a Point of an Affine Variety: Smoothness

We start as usual with the affine case. Up to isomorphism, we may assume V ⊆ Ar is an affine variety, for some non-negative integer r. Let Ia (V ) := (f1 , . . . , fs ) ⊆ A(r) be its radical ideal and let P = (p1 , . . . , pr ) ∈ V be a point. Definition 12.1.1. The affine tangent space at the point P of V is the affine subspace of Ar defined by the system of linear equations ∂f Σri=1 ∂xji (P )(xi − pi ) = 0, 1  j  s. It will be denoted by TV /Ar ,P . By Leibniz’s rule, it is easy to observe that the previous definition depends only on the point P ∈ V and on the ideal Ia (V ), i.e. it does not depend on the choice of generators of Ia (V ). When in particular V is an affine subspace of Ar , then at any P ∈ V , we have TV /Ar ,P = V .

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A First Course in Algebraic Geometry and Algebraic Varieties

For V ⊂ Ar an affine variety with Ia (V ) := (f1 , . . . , fs ) one considers   ∂fj J := J(f1 , . . . , fs ) = (12.1) ∂xi 1ir, 1js the Jacobian matrix of the equations defining V , which is a s × r matrix with polynomial entries. For any integer h ∈ {0, . . . , r}, let Jh ⊂ A(r) be the ideal generated by the minors of order r − h + 1 of J. One can consider the set V ∩ Za (Jh ) := {P ∈ V | rk(J(P ))  r − h}, i.e.   V ∩ Za (Jh ) = P ∈ V | dim(TV /Ar ,P )  h , (12.2) where dim(TV /Ar ,P ) denotes the dimension as a (classical) affine subspace, i.e. the dimension of the K-vector space determined by the direction of TV /Ar ,P . From above, the right-hand side member of (12.2) is in particular a closed subset of V . Thus, dim(TV /Ar ,P ) reaches its minimum over a nonempty, open subset U of V (possibly U = V , as we will see below). The open set U consists of points P ∈ V where the Jacobian matrix evaluated at P , namely J(P ), has maximal rank. From now on we will denote such an open set U as Sm(V ) := U according to the following. Definition 12.1.2. A point P ∈ Sm(V ) is said to be a smooth point (also simple or non-singular point) for V . Otherwise, P ∈ V \ Sm(V ) is said to be a singular point for V . The non-empty open set Sm(V ) is called the smooth locus of V whereas the (possibly empty) proper, closed subset V \ Sm(V ) will be denoted by Sing(V ) and called the singular locus of V . If Sing(V ) = ∅, then V is said to be a smooth affine variety (or also a non-singular affine variety), otherwise V is said to be singular. Example 12.1.3. (i) Let V = Za (x2 − x21 ) ⊂ A2 be the parabola. Setting f := x2 − x21 , its Jacobian matrix is J = (−2x1 1). Note that Za (J0 ) = Za (J1 ) = A2 . This implies that dim(TV /A2 ,P )  1 at any point P ∈ V . Since one has J2 = (2x1 , 1), then Za (J2 ) = ∅ so the previous inequality is an equality for any P ∈ V . It follows that V is a smooth irreducible curve and that at any point P ∈ V one has dim(TV /A2 ,P ) = 1 = dim(V ). Similarly, one can easily check that the “standard” affine twisted cubic V = {(t, t2 , t3 ) | t ∈ A1 } ⊂ A3 (as well as more generally the “standard” affine rational normal curve of degree d in

Tangent Spaces: Smoothness of Algebraic Varieties

251

Ad , V = {(t, t2 , t3 , . . . , td ) | t ∈ A1 } ⊂ Ad ) is smooth at any of its points. (ii) Consider the semi-cubic parabola, i.e. the cuspidal plane cubic V = Za (x22 −x31 ) ⊂ A2 (cf., Example 5.3.24). We have J = (−3x21 2x2 ) and, as above, Za (J0 ) = Za (J1 ) = A2 so any P ∈ V is s.t. dim(TV /A2 ,P )  1. For h = 2, J2 = (−3x21 , 2x2 ) = (x21 , x2 ) so V ∩ Za (J2 ) = {O = (0, 0)}. Thus, for any P ∈ V \ {O} one has dim(TV /A2 ,P ) = 1 = dim(V ). On the contrary, at the origin one has dim(TV /A2 ,O )  2. Since TV /A2 ,O has to be an affine subspace of A2 , we must have TV /A2 ,O = A2 and the previous inequality is an equality. Thus, V is singular; more precisely Sing(V ) = {O = (0, 0)} whereas its smooth locus is Sm(V ) = V \ {O}. (iii) Identical conclusions as in (ii) hold for the plane nodal cubic V = Za (x31 + x21 − x22 ) ⊂ A2 which is monoid of degree 3 in A2 (cf. Example 8.2.12-(ii)). In this case J = (3x21 + 2x1 − 2x2 ) and once again V ∩ Za (J2 ) = {O = (0, 0)} which is the singular locus of V . (iv) More generally, if V is an (irreducible) hypersurface V = Za (f ) ⊂ Ar , ∂f ∂f ∂f the Jacobian matrix is simply J = ( ∂x ∂x2 . . . ∂xr ). For any 1 r 0  h  r − 1, one has Za (Jh ) = A therefore any P ∈ V is such that dim(TV /Ar ,P )  r − 1 = dim(V ). The only non-trivial information is contained in V ∩ Za (Jr ) = Sing(V ). In particular, V is smooth if and only if J(P ) = (0 0 . . . 0) for any P ∈ V . Otherwise any P ∈ V ∩ Za (Jr ) is a singular point. Note that Sing(V ) = V ∩ Za (Jr ) ∂f ∂f , ∂f , . . . , ∂x ); this confirms that Sing(V ) is defined by the ideal (f, ∂x 1 ∂x2 r is a proper, closed subset in V , since Ia (V ) = (f ) is a principal ideal. To sum-up, at any smooth point Q ∈ Sm(V ) one has dim(TV /Ar ,Q ) = r − 1 = dim(V ).

(12.3)

If otherwise P ∈ Sing(V ), then dim(TV /Ar ,P ) = r = dim(Ar ). (v) One word of warning concerning the singular locus of an affine hypersurface. Recall that Sing(V ) = V ∩Za (Jr ). Indeed, it can happen that Za (Jr ) = ∅ even if V ∩ Za (Jr ) = ∅ and in such a case V is a smooth hypersurface. Consider, e.g. the quadric hypersurface V = Za (x21 +x22 +· · ·+x2r −1) ⊂ Ar . In this case, Za (Jr ) = {O = (0, 0, . . . , 0)} but Sing(V ) = ∅ as O ∈ / V. Recalling Example 6.5.13, we can give a geometric interpretation of the affine tangent space at a point of an affine variety. Let P ∈ V and let b := (b1 , . . . , br ) ∈ Kr be a non-zero vector. Consider parametric equations

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of the line P,b passing through P and with direction b, i.e. P,b : xi = bi t + pi ,

1  i  r, t ∈ K.

Points of V ∩ P,b corresponds to the solutions of the system of equations φ1 (t) = · · · = φs (t) = 0, where φi (t) := fi (b1 t+p1 , . . . , br t+pr ), 1  i  s, and where Ia (V ) = (f1 , . . . fs ) as above. If one takes φ(t) = g.c.d.(φ1 (t), . . . , φs (t)),

(12.4)

the points of P,b ∩ V bijectively correspond to the roots of the polynomial φ(t) ∈ K[t]. The root t = 0 of φ(t), corresponding to P ∈ P,b ∩ V , will occur with a certain multiplicity μ  1. Definition 12.1.4. The intersection multiplicity between  := P,b and V at the point P is the multiplicity μ of the root t = 0 of the polynomial φ(t). It is denoted by μ(V, ; P ). It is easy to check that the previous definition is independent from the chosen parametric equations of P,b and from the chosen generators of Ia (V ). Remark 12.1.5. (i) φ(t) is identically zero if and only if P,b is entirely contained in V ; in such a case we will set μ(V, ; P ) = +∞. (ii) If a line  does not pass through P , we will put μ(V, ; P ) = 0. (iii) It t = 0 is a simple root of φ(t), i.e. it has algebraic multiplicity 1, then μ(V, ; P ) = 1 and we will say that the intersection at P is simple or even transverse. When otherwise μ(V, ; P )  2, we will say that the line has contact of order greater than 1 at P with V . Proposition 12.1.6. Let V ⊂ Ar be an affine variety and let P ∈ Sm(V ) be a smooth point. The affine tangent space TV /Ar ,P is the union of all the lines in Ar which have contact of order greater that 1 at P with V . Proof. As above, let Ia (V ) = (f1 , . . . , fs ). To simplify notation, let x := (x1 , . . . xr ) be the vector of indeterminates and let p = (p1 , . . . , pr ) be the coordinate vector of the point P ∈ V . For any j ∈ {1, . . . , s}, by Taylor

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253

expansion, we set fj (x) = fj ((x − p) + p) = hj (x − p) + gj (x − p),

(12.5)

where hj is a linear form and where gj is a polynomial whose homogeneous factors have at least degree two. For any 1  j  s, one has fj (b t + p) = hj (b t) + gj (b t) = t hj (b) + t2 qj (t), for suitable qj ∈ K[t]. The line P,b has contact of order greater than 1 at the point P with V if and only if h1 (b) = h2 (b) = · · · = hs (b) = 0.

(12.6)

On the other hand, from Taylor’s formula applied to (12.5), one has hj (x − p) =

r  ∂fj i=1

∂xi

(p)(xi − pi ),

so (12.6) establishes that b belongs to the direction of TV /Ar ,P , which is the sub-vector space of Kr defined by a homogeneous linear system whose coefficient matrix is precisely J(P ).  12.2

Tangent Space at a Point of a Projective Variety: Smoothness

The case of a projective variety is similar to the affine case; we shall briefly mention to it. Up to isomorphism, let V ⊂ Pr be a projective variety and let P = [p0 , p1 , . . . , pr ] ∈ V be a point. Let Ip (V ) = (F1 , . . . , Fs ) be the homogeneous radical ideal of V . Definition 12.2.1. The projective tangent space to V at the point P is the linear subspace of Pr defined by the homogeneous linear system ∂F Σri=0 ∂Xji (P )Xi = 0, 1  j  s. It will be denoted by TV /Pr ,P . As in the affine case, the definition is independent from the choice of the generators of Ip (V ) and one can similarly define the set Sm(V ), Sing(V ) as well as the concepts of smooth projective variety (or also non-singular projective variety) and of singular projective variety. If p0 = 0, i.e. if P ∈ V ∩ U0 := V0 , the projective hyperplanes defining TV /Pr ,P are the projective closures of the affine hyperplanes in Definition 12.1.1 defining TV0 /Ar ,P . Thus, from Section 3.3.5, the linear projective

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subspace TV /Pr ,P turns out to be the projective closure of the affine space TV0 /Ar ,P . Same conclusion holds for any other non-empty affine chart Vi = V ∩ Ui = ∅, 1  i  r. Let LP ⊂ Pr be any line passing through the point P ∈ V ; let moreover Q = [q0 , q1 , . . . , qr ] = P be any other point of LP . To simplify notation, we pose p := (p0 , p1 , . . . , pr ), q = (q0 , q1 , . . . , qr ) and X = (X0 , . . . , Xr ) the row-vector of homogeneous coordinates in Pr . Thus, parametric equations of LP are given by X = λp + μq, [λ, μ] ∈ P1 . Points of V ∩ LP corresponds to the solutions [λ, μ] ∈ P1 of the system of equations F1 (λ p + μ q) = F2 (λ p + μ q) = · · · = Fs (λ p + μ q) = 0. The previous system of equations is equivalent to a unique homogeneous equation F (λ, μ) = 0, where F (λ, μ) := g.c.d.(F1 (λ p + μ q), . . . , Fs (λ p + μ q)). The multiplicity of the root [1, 0] (corresponding to the point P ) will be called intersection multiplicity between LP and V at the point P and will be denoted by μ(V, LP ; P ). Similarly to the affine case, μ(V, L; P ) = 0 if the line L does not pass through P , whereas μ(V, L; P ) = +∞ when the line L passes through P and it is entirely contained in V . Moreover, the definition of μ(V, L; P ) is compatible with that given in Definition 12.1.4, i.e. if P ∈ V0 = V ∩ U0 = ∅ and if  := L ∩ U0 , then μ(V, L; P ) = μ(V0 , ; P ) and TV /Pr ,P is the union of all projective lines L ⊂ Pr having contact of order greater than 1 at the point P with V , i.e. such that μ(V, L; P )  2. Example 12.2.2. (i) When V = Zp (F ) is an (irreducible) hypersurface in Pr , with (r) F ∈ Sd for some positive integer d, its Jacobian matrix is ∂F ∂F ∂F . . . ∂X ). As in the affine case, for any 0  h  r − 1, J = ( ∂X 0 ∂X1 r r one has Zp (Jh ) = P therefore any P ∈ V is such that dim(TV /Pr ,P )  r − 1 = dim(V ). When h = r, Jr is a homogeneous ideal generated by homogeneous polynomials of degree (d − 1) and Sing(V ) is simply given by Zp (Jh ), as it follows from Euler’s identity (cf. Proposition 1.10.15-(i)). For any smooth point P ∈ Sm(V ) one has therefore dim(TV /Pr ,P ) = r − 1 = dim(V ). On the contrary, any P ∈ Za (Jr ) belongs to Sing(V ) and dim(TV /Pr ,P ) = r = dim(Pr ). In particular, V √ (r) is smooth if and only if Zp (Jh ) = ∅, i.e. Jh = S+ .

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(ii) For V = Zp (F ) an irreducible hypersurface in Pr as above, if L ⊂ Pr is a line not contained in V one has 

μ(V, L; P ) = d,

(12.7)

P ∈L

since the left-hand side member above equals the sum of the (algebraic) multiplicities of the roots of the homogeneous polynomial F (λ p + μ q) ∈ K[λ, μ]d , for any pair of points P, Q ∈ L. 12.3

Zariski Tangent Space of an Algebraic Variety: Intrinsic Definition of Smoothness

The previous definitions of tangent spaces take into account the embedding of the variety V either in an affine or in a projective space; therefore these definitions are not intrinsic. It is possible to intrinsically associate a K-vector space to any point P of any algebraic variety V in such a way that, under this association, one recovers the affine tangent space in Definition 12.1.1 (viewed as a vector space whose origin is P ) when V is affine (cf. also Mumford, 1995, Chapter 1, pp. 3–5). Take V any algebraic variety and let P ∈ V be any point. From Theorem 5.3.10 and Example 5.3.22, we know that (OV,P , mV,P ) is a local ring, with residue field K. To simplify notation, in what follows we will simply denote by m the maximal ideal mV,P . The OV,P -module mm2 is annihilated by m, OV,P -module, i.e. it is a K-vector space. therefore it is a m Lemma 12.3.1. For any P dimensional.

∈ V, the K-vector space

m m2

is finite

Proof. From Corollary 5.3.20, for any open set U ⊆ V containing P one has an isomorphism of local rings OU,P ∼ = OV,P . Thus, by Proposition 6.4.2, with no loss of generality we may assume that V is an affine variety. In particular we can consider V ⊆ Ar , for some integer r, as an irreducible, closed subset. In such a case, from Claim 5.3.15, we have OV,P ∼ = A(V )mV (P ) mV (P )A(V )mV (P ) m ∼ ∼ and m = mV (P )A(V )mV (P ) . Thus, m2 = mV (P )2 A(V )m (P ) . On the V other hand, from Theorem 5.3.14-(b), mV (P ) ∼ = Ia,V (P ) = mP , where Ia (V )

mP = (x1 − p1 , . . . , xr − pr ) is the maximal ideal in A(r) corresponding to P ∈ Ar and Ia (V ) ⊆ A(r) the radical ideal of V ⊆ Ar . Therefore,   P = r, as desired. dim mm2  dim m  m2 P

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12.3.2. Definition   m For any point P ∈ V , the K-vector space m ∗ := Hom 2 K m m2 , K is called the Zariski tangent space of V at P and it is denoted by TV,P . Note that the definition of TV,P is of local nature, i.e. it depends only on the local ring OV,P . In particular, for any open neighborhood U ⊆ V containing P one has TV,P ∼ = TU,P (cf. Shafarevich, 1994, § II.1.3, pp. 86–92). Definition 12.3.3. A K-linear map D : OV,P → K is called a K-derivation from OV,P to K if, for any Φ, Ψ ∈ OV,P and for any λ ∈ K, one has D(λ) = 0 and D(ΦΨ) = Φ(P ) D(Ψ) + Ψ(P ) D(Φ). The set of all K-derivations from OV,P to K determines a sub-vector space of HomK (OV,P , K), which will be denoted by DerK (OV,P , K). Proposition 12.3.4. (i) For any algebraic variety V and any point P ∈ V, there exists a canonical isomorphism TV,P ∼ = DerK (OV,P , K) of K-vector spaces. (ii) If V ⊆ Ar is an affine variety and P ∈ V, there exists a canonical isomorphism of K-vector spaces TV,P ∼ = TV /Ar ,P , where TV /Ar ,P is considered as a K-vector space with origin at the point P . Proof. (i) If D ∈ DerK (OV,P , K), for any Φ, Ψ ∈ m we have D(Φ Ψ) = 0, i.e. D|m2  = 0. This implies that  m ∗D induces a K-linear homomorphism m LD : m2 → K, i.e. LD ∈ m2 = TV,P . Conversely, given L ∈ TV,P , define DL : OV,P → K, Φ → L(Φ − Φ(P )), where Φ − Φ(P ) ∈ mm2 . For any λ ∈ K, one has DL (λ) = L(λ − λ) = L(0) = 0; for any Φ, Ψ ∈ OV,P , one has DL (Φ Ψ) = DL ((Φ − Φ(P ) + Φ(P )) ((Ψ − Ψ(P ) + Ψ(P )) = DL ((Φ − Φ(P )) (Ψ − Ψ(P )) + (Φ − Φ(P )) Ψ(P ) + (Ψ − Ψ(P )) Φ(P ) + Φ(P ) Ψ(P )). Since L is K-linear, by its definition DL is also K-linear. Using that Φ(P ) Ψ(P ) ∈ K and (Φ−Φ(P )) (Ψ−Ψ(P )) ∈ m2 , the latter expression

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equals DL ((Φ − Φ(P )) Ψ(P ) + (Ψ − Ψ(P )) Φ(P )) which, by definition of DL , is

L (Φ − Φ(P )) Ψ(P ) + (Ψ − Ψ(P )) Φ(P )



= Ψ(P ) L Φ − Φ(P ) + Φ(P ) L Ψ − Ψ(P ) ; this reads Ψ(P ) DL (Φ) + Φ(P ) DL (Ψ). In other words, DL ∈ DerK (OV,P , K). Since the maps D → LD and L → DL are one the inverse of the other, we conclude. (ii) Let V ⊂ Ar be an affine variety; from Theorem 5.3.14-(c), for any P ∈ V one has OV,P ∼ = A(V )mV (P ) . With small abuse of notation, set x1 , . . . xr ∈ OV,P to be the images of the indeterminates of A(r) . For any D ∈ DerK (OV,P , K), let λi = D(xi ),

1  i  r.

(12.8)

For any f ∈ Ia (V ), by the K-linearity of D one has ∂f 0 = D(f (x1 , . . . , xr )) = Σri=1 ∂xji (P )λi ,, i.e. (λ1 , . . . , λr ) ∈ Kr lies in the direction of TV /Ar ,P , i.e. (p1 , . . . , pr ) + (λ1 , . . . , λr ) ∈ TV /Ar ,P . Conversely, for (p1 , . . . , pr ) + (λ1 , . . . , λr ) ∈ TV /Ar ,P , posing as in (12.8) D(xi ) := λi , we can define a derivation D ∈ DerK (OV,P , K) as follows: D(g) :=

r  ∂fj i=1

∂xi

(P )λi ,

∀ g ∈ A(V )

and D

g

h

=

D(g) h(P ) − D(h) g(P ) , h(P )2

∀

g ∈ OV,P . h

The map TV /Ar ,P → DerK (OV,P , K) just constructed is welldefined, K-linear and it is the inverse of the previous one; so TV /Ar ,P ∼ = DerK (OV,P , K). We then conclude by part (i).  As in the affine case, a point P ∈ V is said to be smooth (or simple, or non-singular) if dim (TV,P ) = minQ∈V dim (TV,Q ). Otherwise P is said to be a singular point. Applying Proposition 12.3.4-(ii) to any affine open set of an affine open covering of V and using (12.2) for any such affine open set, one deduces that the set of non-singular points in V is a nonempty, dense open set in V which is denoted by Sm(V ). Thus, Sing(V ) :=

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V \ Sm(V ), which is the set of singular points of V , is a proper closed subset of V . The algebraic variety V is said to be singular (respectively, smooth) if Sing(V ) = ∅ (respectively, Sing(V ) = ∅). There is another interpretation of the Zariski tangent space at a point of an algebraic variety, which is extremely useful when, e.g. one is concerned with the study of local properties of algebraic varieties, like Hilbert schemes, which are also “parameter spaces” for families of algebraic varieties (cf., e.g. Sernesi, 2006), or like moduli spaces, parametrizing isomorphism classes of projective varieties (cf., e.g. Caporaso, 2004, for moduli spaces of smooth projective curves, whereas Catanese, 1986, § 19, for moduli spaces of some families of surfaces with given invariants). To discuss this further definition, we first need some algebraic preliminaries. Let (A, m) be a local K-algebra with residue field K (cf. Section 1.11.1). Similarly as above, one observes that mm2 is a K-vector space. If furthermore m A is Noetherian,  m ∗then m2 obviously has finite dimension over K. Thus, one to be the Zariski tangent space of the local Noetherian can define m2 K-algebra (A, m). Recall that, for any algebraic variety V and any P ∈ V the local K-algebra OV,P is Noetherian (cf. Proposition 10.2.3); therefore, we can think about (A, m) in the above definitions as it were the local ring at a point of an algebraic variety. Applying verbatim the same strategy as in the proof of Proposition 12.3.4-(i), one shows that m ∗ ∼ = DerK (A, K). m2

(12.9)

Consider now the polynomial K-algebra K[t], where t an indeterminate K[t] over K, and the quotient ring (t 2 ) which is called the ring of dual numbers over K and denoted by K[], where  denotes the class of t modulo the ideal (t2 ); in particular 2 = 0 in K[]. By its definition, K[] is a finite K-algebra of dimension 2, whose elements are formal expressions a + b, a, b ∈ K, with the following operations: (a + b) + (c + d) := (a + c) + (b + d), λ(a + b) := λa + λb (a + b) · (c + d) := ac + (ad + bc), ∀ a, b, c, d, λ ∈ K, where the last equality follows from the fact that 2 = 0. Furthermore, K[] is a Noetherian local ring, of maximal ideal () and residue field K. Let (A, m) be any Noetherian local K-algebra with residue field K and let φ : A → K[] be any local homomorphism of local rings. Then, for any

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a ∈ A, one has φ(a) := a0 + dφ (a),

(12.10)

A is the class of a modulo m and where dφ (a) ∈ K is where a0 ∈ m determined by φ. Conversely given a d ∈ HomK (A, K) then it recovers a local homomorphism φ = φd as in (12.10). Given φ : A → K[] a local homomorphism as above, we want to show that d := dφ is a K-derivation from A to K, i.e. an element in DerK (A, K). Since φ is a local homomorphism of K-algebras, for any a, b ∈ A and any λ ∈ K, one has

a0 + b0 + d(a + b)  = φ(a + b) = φ(a) + φ(b) = a0 + d(a)  + b0 + d(b) , λ a0 + d(λ a) = φ(λ a) = λ φ(a) = λ a0 + λ d(a) , a0 b0 + d(ab) = φ(a b) = φ(a) φ(b) = (a0 b0 ) + (a0 d(b) + b0 d(a)) . Thus, for any a, b ∈ A and any λ ∈ K, one has d(a + b) = d(a) + d(b), d(λ a) = λ d(a), d(ab) = a0 d(b) + b0 d(a), which proves that d = dφ is a K-derivation. With this set-up and recalling the definition of the vector space of K-algebra homomorphisms as in (6.8), one has the following Proposition 12.3.5. Let (A, m) be a Noetherian  ∗ local K-algebra with residue field K. Then the Zariski tangent space mm2 of (A, m) is isomorphic to the vector space of K-algebra homomorphisms HomK (A, K[]). In particular, if V is an algebraic variety and if P ∈ V is any point, then the Zariski tangent space TV,P of V at P is isomorphic to HomK (OV,P , K[]).  ∗ ∼ Proof. From (12.9), we know that mm2 = DerK (A, K). Previous computation shows that DerK (A, K) ∼ = HomK (A, K[]). The last part of the  statement follows by replacing A with OV,P . Due to the local nature of the Zariski tangent space at a point, we describe here how this local definition reflects on morphisms among algebraic varieties. Definition 12.3.6. Let V and W be algebraic varieties, P ∈ V be a point and ϕ : V → W be a morphism. The local K-algebra homomorphism ϕP : OW,ϕ(P ) → OV,P induces a homomorphism of K-vector spaces, denoted by dϕP : TV,P −→ TW,ϕ(P ) , which is called the differential of ϕ at the point P .

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If ϕ : V → W and ψ : W → Z are morphisms, for any point P ∈ V one has d(ψ ◦ ϕ)P = (dψϕ(P ) ) ◦ (dϕP ) and (d IdV )P = IdTV,P .

(12.11)

Corollary 12.3.7. If ϕ : V → W is an isomorphism of algebraic varieties then, for any point P ∈ V, dϕP is an isomorphism of K-vector spaces. 

Proof. It immediately follows from (12.11).

Using differentials, Theorem 12.3.9 gives an extra alternative definition for the “dimension” of an algebraic variety, with no use of either function or Krull-dimension or topological dimension. We first need an auxiliary result. Proposition 12.3.8. Any algebraic variety V is birationally equivalent to a hypersurface in some affine or projective space. Proof. It suffices to prove, e.g. the second part of statement. Let dim(V ) = v and let {s1 , . . . , sv } be a transcendence basis of K(V ) over K. Since char(K) = 0, by the Primitive Element Theorem (cf., e.g. Lang, 2002, Theorem 4.6, p. 243), there exists θ ∈ K(V ) such that K(V ) ∼ = K(s1 , . . . , sv , θ). By the assumption on {s1 , . . . , sv }, the elements s1 , . . . , sv , θ ∈ K(V ) are algebraically dependent over K. There exists therefore a non-zero polynomial f ∈ K[x1 , . . . , xv , xv+1 ] (we may assume to be irreducible) s.t. f (s1 , . . . , sv , θ) = 0. Since {s1 , . . . , sv } is a transcendence basis of K(V ) over K, f is not constant with respect to the indeterminates xv+1 . Let W := Za (f ) ⊂ Av+1 . Then the affine coordinate ring of W is an (v+1) K[x1 ,x2 ,...,xv+1 ] integral K-algebra of the form A(W ) = A , where Ia (W ) = Ia (W ) xi ’s indeterminates. Since Ia (W ) = (f ) is principal, as in the proof of Proposition 10.1.5, the elements x1 , . . . , xv ∈ A(W ) ⊂ Q(A(W )) ∼ = K(W ) are algebraically independent, where xi denotes the image of the indeterminate xi , 1  i  v + 1. Therefore, K(W ) ∼ = K(x1 , . . . , xv , xv+1 ) ∼ = K(x1 , . . . , xv )[xv+1 ], where the second isomorphism follows from the fact that xv+1 is algebraic over K(x1 , . . . , xv ). If we let y to be an indetermiK(x ,...,x )[y] nate, then K(x1 , . . . , xv )[xv+1 ] ∼ = (f (x11 ,...,xvv ,y)) . Since {s1 , . . . , sv } is a ∼ K(s1 ,...,sv )[y] . transcendence basis over K, then one has K(x1 ,...,xv )[y] = (f (x1 ,...,xv ,y))

(f (s1 ,...,sv ,y))

K(s1 ,...,sv )[y] ∼ K(s1 , . . . , sv )[θ] ∼ By the definition of θ, (f = = (s1 ,...,sv ,y)) ∼ ∼ K(s1 , . . . , sv , θ) = K(V ). To sum up, K(V ) = K(W ) and one concludes by using Corollary 8.1.7-(i). 

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We remark that the assumption char(K) = 0 can be replaced by the weaker assumption that K is a perfect field (see, e.g. Dolgachev, 2013, Remark 4.11, p. 27). Theorem 12.3.9. Let V be an algebraic variety and let P ∈ Sm(V ). Then dim(V ) = dim (TV,P ) .

(12.12)

Proof. Note first that if V and W are birational algebraic varieties then equality (12.12) holds true for V if and only if it holds true for W : indeed there exist non-empty, open sets UV ⊆ V and UW ⊆ W which contain smooth points of V and W , respectively, and which are isomorphic so we can we apply Corollary 12.3.7. Moreover, from (12.3), equality (12.12) holds true for V a hypersurface in Ar . Thus, from Proposition 12.3.8, any algebraic variety V is birationally equivalent to an affine hypersurface in some affine space for which the equality (12.12) holds true, therefore the proof is complete.  The above results show that the concept of Zariski tangent space at a point P of an algebraic variety V deals with the local ring OV,P . It is natural to expect that the same occurs for the concept of smoothness; this is what we actually show below. Let (A, m) be any Noetherian local K-algebra with residue field K. It is a standard result in Commutative Algebra (cf., e.g. Matsumura, 1980, p. 78) that m

 K dim(A), (12.13) dim m2 where the dimension on the left is meant as K-vector space whereas that on the right is the Krull-dimension of the ring. The local ring (A, m) is said to be regular local ring if and only if equality in (12.13) holds. Theorem 12.3.10. Let V be an algebraic variety and let P ∈ V be a point. Then P is a smooth point of V if and only if OV,P is a regular local K-algebra. Proof. From Proposition 6.4.2, there always exists an affine open neighborhood U ⊂ V of the point P ; since OV,P ∼ = OU,P then, with no loss of generality, we may assume V to be an affine variety. Moreover, up to an isomorphism, we may assume that V ⊂ Ar , for some nonnegative integer r. Let therefore P = (p1 , p2 , . . . , pr ) ∈ Ar and let mP = (x1 − p1 , x2 − p2 , . . . , xr − pr ) ⊂ A(r) be its corresponding maximal ideal.

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θ

θ

P P Consider the K-linear map A(r) −→ Kr , defined as f −→ J(P )t , where J = J(f ) is the Jacobian row-matrix of f as in (12.1). Note that θP (xi −pi ), 1  i  r, determine the canonical basis of Kr and that θP (m2P ) = 0. P → Kr . Therefore, θP induces a K-linear map θP : m m2 P

Let now Ia (V ) = (f1 , . . . , fs ) ⊂ A(r) and let J = J(f1 , . . . , fs ) be the Jacobian matrix of the equations defining V as in (12.1). From the previous definition, one has dim (θP (Ia (V ))) = dim(θP (f1 , . . . , fs )) = rank (J(P )). Since P ∈ V , then Ia (V ) ⊆ mP and moreover 

dim (θP (Ia (V ))) = dim θP



m2P + Ia (V ) m2P

 ,



2 mP +Ia (V ) = rank (J(P )). Let now m denote the maximal i.e. dim θP 2 mP ideal of the local ring OV,P . We claim that there exists an isomorphism m ∼ mP . = 2 m2 mP + Ia (V )

(12.14)

If (12.14) holds true then, from (12.13), one has   m

mp = dim m2 m2P + Ia (V )    2  mp mP + Ia (V ) = dim − dim = r − rank (J(P )) . m2P m2P

K dim (OV,P )  dim

Therefore, we have rank (J(P ))  r − K dim (OV,P ) = r − dim(V ), where the second equality in the previous formula follows from Proposition 10.2.3. Now, from (12.12), the first inequality is an equality if and only if P ∈ Sm(V ) equivalently if and only if OV,P is a regular local ring. We are therefore left to showing the existence of an isomorphism as in (12.14). Since V is affine, by Theorem 5.3.14-(c), we have P OV,P ∼ = A(V )mV (P ) , where mV (P ) = Iam(V ) the maximal ideal in (r) A ∼ A(V ) = . Thus, m = mV (P )·A(V )m (P ) and m2 = m2 (P )·A(V )m (P ) , so

Ia (V )

V

V

V

m ∼ mV (P ) · A(V )mV (P ) ∼ mV (P ) ∼ mP , = 2 = 2 = 2 m2 mV (P ) · A(V )mV (P ) mV (P ) mP + Ia (V )

which completes the proof.



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263

Exercises Exercise 12.1. Let C ⊂ A3 be the (reducible) affine curve given by C = Za (x1 x2 , x1 x3 , x2 x3 ). Show that C cannot be isomorphic to any affine plane curve. Exercise 12.2. Let C := Zp (X0 X1 − X22 ) ⊂ P2 which is a smooth projective conic. Set F := X0 X1 − X22 ∈ S (2) and consider the morphism

∂F ∂F ∂F ϕ 2 (P ), (P ), (P ) . ϕ : C → P , P −→ ∂X0 ∂X1 ∂X2 Then ϕ(C) ⊂ P2 is said to be the dual curve of C. Show that ϕ(C) is a smooth curve. Compute its degree. Exercise 12.3. Show that any projective monoid Z ⊂ Pn of degree d as in (8.5) has a point of multiplicity d − 1 at P0 = [1, 0, . . . , 0], i.e. for any line L ⊂ Pn passing through P0 one has μ(Z, ; P0 )  d − 1. Exercise 12.4. A hypersurface Z ⊂ Pn of degree d is said to be a cone if it has at least a point of multiplicity d. The set of points in Z of multiplicity d is called the vertex of the cone and denoted by v(Z). Show that v(Z) is a linear subspace of Pn . Exercise 12.5. If Z ⊂ Pn is a cone of degree d, show that Z is the union of lines passing through P ∈ v(Z), for any point P ∈ v(Z).

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Solutions to Exercises

Here, we report solutions of all the exercises proposed at the end of each chapter. Solution 1.1 (i) Assume that I = I  ∩ I  . Note that I ⊆ I  and I ⊆ I  . If by contradiction there existed f  ∈ I  \ I and f  ∈ I  \ I, then f  · f  ∈ I  · I  ⊆ I  ∩ I  = I which would contradict that I is a   prime ideal.  Therefore, eithern I = I or I = I . (ii) Any a ∈ (0) is such that a = 0, for some positive integer n. Thus, for any p ∈ Spec(R), a ∈ / R\p since R\p is a multiplicative system. This  (0) ⊆ ∩p∈Spec(R) p. implies that a ∈ p, for any p ∈ Spec(R), namely  To prove the other inclusion, take any f ∈ R \ (0); we need to show that there exists a prime ideal p such that f ∈ / p. To prove this note that, since f is not nilpotent, then S := {1, f, f 2, f 3 , . . .} is a multiplicative system in R. From Corollary 1.11.8-(iii), any prime ideal q ⊂ RS is of the form q = p · RS , for some p ∈ Spec(R) such that  p ∩ S = ∅ and we are done. The previous arguments show that (0) = ∩p∈Spec(R) p. Since the intersection of a family of ideals is an ideal, in particular one has also proved that (0) is an ideal in R. (iii) For a ∈ (0) let k := ord(a), i.e. the smallest integer for which ak = 0. Then, for any u ∈ U (R) the element b := u−1 − au−2 + a2 u−3 − a3 u−4 + · · · + (−1)k−1 ak−1 uk−1 ∈ R is such that b = (u + a)−1 in R, namely u + a ∈ U (R).

265

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Solution 1.2 The fact that (I : J) is an ideal is straightforward. Now, for any i ∈ I and any j ∈ J, i · j ∈ I since I is an ideal; thus i ∈ (I : J), i.e. I ⊆ (I : J). Furthermore, (I : J) · J  r · j ∈ I, i.e. (I : J) · J ⊆ I. √ Finally, √ for any √ f ∈ I ∩ J, there exists m ∈ N such that f m ∈ I ∩ J ⊆ I ∩ J. Thus, √ mk1 such that f mk2 √ ∈ J. Therefore, f ∈ I there exist √ f √ ∈ I and √ √ k1 , k2 ∈ N √ ∩ J ⊆ I ∩ J. To prove the other and f ∈ J, i.e. f ∈ I ∩ J, so √ I√ inclusion, note that for any f ∈ I ∩ J there exist k√ 1 , k2 ∈ N such that k2 k1 +k2 f ∈ J. Thus f ∈ I ∩ J and so f ∈ I ∩ J. This shows f k1 ∈√I and √ √ that I ∩ J ⊆ I ∩ J. Solution 1.3 (i) Note that f (x) ∈ R[x] is invertible if and only if there exists g(x) = b0 + b1 x + · · · + bm xm ∈ R[x] such that a0 b0 = 1, a0 b1 = −a1 b0 , . . . , a0 bk = −(bk−1 a1 +· · ·+b0 ak ), 2  k  n. Thus, a0 ∈ U (R) and b0 = a−1 0 ∈ U (R); moreover −1 2 (∗) b1 = −a1 a−2 0 , b2 = a1 a0 −2 − a2 a0 , . . . .

Therefore, given f (x) = a0 + a1 x + · · · + an xn ∈ R[X] such that a0 ∈ U (R) and a1 , . . . , an are nilpotent elements in R, then g(x) ∈ R[x] such that f (x) · g(x) = 1 can be explicitly and uniquely determined as a solution of the system of equations (∗) which, by the nilpotency of the element ai , 1  i  n, has finitely many equations. Conversely, let f (x) ∈ U (R[x]). From above, one has a0 ∈ U (R). If deg(f (x)) = n, then for n = 0 we are done. Assume therefore n  1 and use induction on n. If g(x) = b0 +b1 x+· · ·+bm xm ∈ R[x] such that f (x)·g(x) = 1, in particular an bm = 0. Moreover, multiplying f (x) · g(x) = 1 by f (x)r , one gets f (x)r+1 · g(x) = f (x)r so ar+1 n bm−r = 0. As b0 is a unit, then = 0, i.e. a is nilpotent in R. This implies that an xn ∈ R[x] is am+1 n n nilpotent. Let f (x) − an xn = a0 + a1 x + · · · + an−1 xn−1 . Since f (x) is invertible and an xn is nilpotent, from Exercise 1.1-(iii), f (x) − an xn is invertible in R[x] of degree less than n; by induction, a0 ∈ U (R) and a1 , . . . , an−1 are nilpotent elements in R.

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(ii) Since f (x) ∈ R[x] is nilpotent and 1 ∈ U (R[x]), then 1 + f (x) is invertible in R[x] as it follows from Exercise 1.1-(iii). Therefore, by (i), a1 , . . . , an ∈ R are nilpotent elements whereas 1 + a0 ∈ U (R). On the other hand, for t large enough, one has f t = 0 thus in particular at0 = 0, i.e. a0 is nilpotent in R. If conversely, a0 , a1 , . . . , an ∈ R are nilpotent elements, let kj := ord(aj ) for 0  j  n. Set k := Σnj=0 kj , so that one gets f (x)k = 0. Indeed, f (x)k is a linear combination with integral coefficients of products of the form ar00 ar11 · · · arnn xks such that Σnj=0 rj = k, for each ks . Since we cannot simultaneously have rj < kj , each of these products is zero. (iii) f (x) ∈ R[x] is a zero-divisor if there exists g(x) ∈ R[x] \ {0} such that f (x) · g(x) = 0. Take g(x) of smallest degree w.r.t. this property, say g(x) = b0 + b1 x + · · · , bm xm , where we may assume b0 = 0. One has in particular an · bm = 0. Since deg(an g(x)) < deg(g(x)), by minimality of g(x) it follows that an · g(x) = 0 in R[x]. In particular an · b0 = 0, with b0 = 0. Similarly, an−t · g(x) = 0, for 1  t  n, so an−t · b0 = 0, for any 1  t  n, with b0 = 0. Therefore, it suffices to considering r := b0 ∈ R \ {0}. Solution 1.4 For any b ∈ R \ {0}, there exists b−1 ∈ K. We must show that b−1 is in R. Since K is a finite R-algebra and R ⊆ R[b−1 ] ⊆ K, then R[b−1 ] is a finite R-algebra. From Proposition 1.6.3, b−1 ∈ K is integral over R. Thus, there exist r0 , . . . , rn−1 ∈ R such that b−n + rn−1 b1−n + · · · + r0 = 0. Multiplying by bn−1 the previous equality gives b−1 + rn−1 + · · · + r0 bn−1 = 0, n−1−j ∈ R. i.e. b−1 = −Σn−1 j=0 rj b

Solution 1.5 Note that for any a, b ∈ Z, one has Sa · M(h)b ∼ = Sa · Mb+h ⊆ M(a+b)+h ∼ = M(h)a+b . Therefore, M(h) is a graded S-module. Moreover, M and M(h) are obviously isomorphic as S-modules since they have the same elements but the

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graduation is simply shifted by h. On the other hand, M(h)d ∼ = Md+h is not  isomorphic to Md , if, e.g. S = M = K[X0 , . . . , Xn ] = d0 K[X0 , . . . , Xn ]d . Solution 2.1 First, consider the pencil of lines t : x2 − ix1 − t = 0 in A2C , where t = t1 + it2 ∈ C. With the use of this pencil, one defines a bijective map ψ : C \ {0} → Za (f ) ⊂ A2C , where   t2 + 1 t2 − 1 ψ(t) := x1 (t) = − , x2 (t) = , 2it 2t which is a (complex) rational parametrization of the conic Za (f ). Now, let us set-theoretically identify the affine plane A2C with R4 via x1 (t) := a(t1 , t2 ) + ib(t1 , t2 ) and x2 (t) := c(t1 , t2 ) + id(t1 , t2 ),

i2 = −1,

where a(t1 , t2 ), b(t1 , t2 ), c(t1 , t2 ), d(t1 , t2 ) ∈ R. The equation defining Za (f ) reads in the given coordinates of R4 as  2 a + c2 − b2 − d2 + 1 = 0 ab + cd = 0. The map ψ, given by the pencil of complex lines t , can be considered as a map ψ ∗ : R2 \ {(0, 0)} → Za (f ) ⊂ R4 . Considering Euclidean topologies both in the domain and the target of the previous map, one observes that ψ ∗ is a homeomorphism. Consider now the Euclidean topological space R3 , with coordinates (t1 , t2 , t3 ), in such a way that R2 above is identified with the coordinate plane t3 = 0. In R3 consider the two-sphere Σ, centered at the point C = (0, 0, 12 ) with radius r = 12 . Its North-pole is the point N = (0, 0, 1) whereas its South-pole is O = (0, 0, 0). Let π : Σ \ {N } → A2R be the stereographic projection of Σ from its North-pole N , which is well-known to be a homeomorphism (in the Euclidean topologies) between Σ \ {N } and the coordinate plane t3 = 0; its restriction π| to Σ \ {N, O} is therefore a homeomorphism onto R2 \ {(0, 0)}. Thus the map ψ ∗ ◦ π| : Σ \ {N, O} → Za (f ) ⊂ R4 is the desired homeomorphism (in Euclidean topologies).

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It is clear that Σ is homeomorphic, in the Euclidean topology of R3 , to the unit 2-sphere S 2 ⊂ R3 . This, endowed with a suitable two-chart atlas (cf., e.g. Miranda, 1995, Example 1.13, p. 3), has a natural structure of complex compact manifold of (complex) dimension one and it is called the Riemann sphere (cf. Miranda, 1995, Example 1.20, p. 4). The latter is homeomorphic to the projective complex line P1C (see Miranda, 1995, Problem I.8.C, p. 12), thus Za (f ) is homeomorphic, in the Euclidean topology, to P1C minus two points. Solution 2.2 Note that Y ⊆ Za (x2 − x21 ). On the other hand, any P ∈ Za (x2 − x21 ) is of the form P = (p1 , p2 ) where p2 = p21 , i.e. P = (p1 , p21 ), p1 ∈ K varying. Thus, Y = Za (x2 − x21 ) so it is an AAS. One has Ia (Y ) = (x2 − x21 ) is a (2) principal ideal; moreover I is a prime ideal since AI ∼ = K[x1 ] is an integral domain. Solution 2.3 (i) Consider the R-algebra epimorphism ϕ

R[x1 , x2 , x3 ] −→ R[t], ϕ(x1 ) = t, ϕ(x2 ) = t2 , ϕ(x3 ) = t3 . 1 ,x2 ,x3 ] ∼ Since R[xKer(ϕ) = R[t] is an integral domain, then Ker(ϕ) is a prime ideal; moreover I ⊆ Ker(ϕ). On the other hand, g(x1 , x2 , x3 ) ∈ Ker(ϕ) if and only if g(t, t2 , t3 ) = 0, for any t ∈ R, i.e. Ker(ϕ) ⊆ I. Therefore, R[x1 ,x2 ,x3 ] is an integral domain. I (ii) Since x1 (x3 − x1 x2 ) − x2 (x2 − x21 ) = x1 x3 − x22 then x1 x3 − x22 ∈ I. (iii) Note that Za (I) ⊆ Za (J). On the other hand, the line Za (x1 , x2 ) is contained in Za (J) but not in Za (I). Therefore, Za (I)  Za (J). For 2 ,x3 ] one has x2 = x21 and x1 x3 = x22 , the last part, observe that in R[x1 ,x J 2 ,x3 ] in particular x1 x3 = x41 , namely x1 (x3 − x31 ) = 0 in R[x1 ,x even if J R[x ,x ,x ] 1 2 3 3 x1 = 0 and x3 − x1 = 0 in . Therefore, the quotient ring J cannot be integral.

Solution 2.4 (i) Ia (Yk ) = Ia (Y  ) ∩ Ia (Yk ) = (x1 x2 , x1 x3 , x2 x3 − kx2 , x23 − kx3 ). (ii) Note that, for k = 0, Y  ∩ Yk = ∅. On the other hand, Y  ∩ Y0 = {O = (0, 0, 0)}. As for the ideals, Ik is radical by definition, for any

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k = 0. On the other hand, Ik is not a prime ideal since x1 , x2 ∈ / Ik but x1 x1 ∈ Ik , for any k = 0. (iii) By definition of I0 , one has I0 = (x1 x2 , x1 x3 , x2 x3 , x23 ). Note that √ / I0 , since its generators are all x3 ∈ I0 , since x23 ∈ I0 but x3 ∈ quadratic polynomials. Therefore, I0 is not radical, in particular it √ cannot be a prime ideal, and moreover x3 ∈ I0 \ I0 . Solution 2.5 (i) Note that f = x21 + x22 + 2x1 and g := x21 + x22 − 2x1 . One has Za (J) = Za (mO ) = {O = (0, 0)}, where mO is a maximal ideal (cf. Proposition 2.1.11-(iv)). / J. (ii) and (iii) Note that J ⊂ mO , where the inclusion is proper, since x2 ∈ 1 2 = (f − g) ∈ J and x = f − x (x + 2) ∈ J, On the other hand, x 1 1 1 2 4√ √ i.e. mO ⊆ √ J and so mO = J, by the maximality of mO and the fact that 1 ∈ / J. Therefore J is not radical so R[x1J,x2 ] is not a reduced ring, since x2 is nilpotent in R[xJ1 ,x2 ] . The geometric counterpart of the previous ideal analysis is that Za (f ) and Za (g) are two real conics passing through O and sharing the same tangent line at O, which is the x2 -axis. The intersection at O between the two conics is said to be not transverse. The radical ideal replaces the system of equations of O cut-out by the to tangential conics with the two coordinate axes. Solution 3.1 There exist V and W , K-vector spaces both of dimension n + 1, such that P1 = P(V ) and P2 = P(W ). Set Pi := [v i ] and Qi := [w i ]. Since the points are in general position, then v 0 , . . . , v n is a basis for V and w 0 , . . . , wn is a basis for W . For any choice of λ0 , . . . , λn ∈ K \ {0} there always exists an isomorphism ψ : V → W , such that ψ(v i ) = λi w i , 0  i  n. The problem is to determine suitable λ0 , . . . , λn ∈ K \ {0} such that there also exists λn+1 ∈ K \ {0} for which ψ(v i ) = λi w i , 0  i  n + 1 and moreover (λ0 , . . . , λn , λn+1 ) ∈ (K \ {0})n+2 is uniquely determined up to a constant multiplication factor. By the assumptions, v n+1 (respectively, wn+1 ) linearly depends on v 0 , . . . , v n (respectively, on w 0 , . . . , wn ). Therefore, there exist unique a0 , . . . , an ∈ K and b0 , . . . , bn ∈ K, such that n n   v n+1 = ai v i and w n+1 = bi w i . i=0

i=0

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Since the points are in general position, ai = 0 (respectively, bi = 0) for any 0  i  n. On the one hand, we have  n  bi w i , ψ(v n+1 ) = λn+1 wn+1 = λn+1 i=0

on the other

 ψ(v n+1 ) = ψ

n  i=0

ai v i

=

n 

λi ai wi .

i=0

Since w 0 , . . . , wn is a basis for W , the previous equality gives ⎧ a0 λ0 − b0 λn+1 = 0 ⎪ ⎪ ⎪ ⎪ a1 λ1 − b1 λn+1 = 0 ⎨ ··· · · . ⎪ ⎪ ⎪ · · · · · ⎪ ⎩ an λn − bn λn+1 = 0 By standard Linear Algebra, the previous homogeneous linear system in the indeterminates (λ0 , . . . , λn+1 ) is compatible with solutions which depend on a free parameter in K. Solution 3.2 One has Z = Zp (I), for some homogeneous ideal I ⊂ S(2) . By the “Hilbert basis theorem”, one has I = (F1 , . . . , Fm ), where Fi homogeneous polynomials, 1  i  m, since I is a homogeneous ideal (cf. Proposition 1.10.7). If m = 1, Z is a projective curve and we are done in this case. If m = 2 and F1 , F2 have a non-constant greatest common divisor G ∈ H(S(2) , then Fi = GAi , for some Ai ∈ H(S(2) ), 1  i  2. Then Zp (I) = Zp (G) ∪ Zp (A1 , A2 ), where Zp (G) is a projective curve, as above, and where A1 and A2 have no non-constant common factor. Therefore, we can reduce to F1 and F2 with no non-constant common factor. In such a case, a necessary condition for a point P = [p0 , p1 , p2 ] ∈ P2 to belong to Zp (F1 , F2 ) is that [p0 , p1 ] ∈ Zp (RX2 (F1 , F2 )), where RX2 (F1 , F2 ) is the resultant polynomial of F1 and F2 as in Theorem 1.10.18. By the assumption on F1 and F2 , this is a homogeneous polynomial of degree deg(F1 ) deg(F2 ) in the indeterminates X0 and X1 . By Proposition 1.10.19, one has only finitely many choices for [p0 , p1 ]. Applying the same procedure also with respect to the other indeterminates, one deduces that Zp (F1 , F2 ) consists of at most finitely many points. Recursively applying the previous arguments, one proves the general case for any possible m.

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Solution 3.3 (i) For any α, β ∈ K and any f, g ∈ Ann(W ) one has (αf + βg)(w) = αf (w) + βg(w) = 0, so Ann(W ) is a K-subvector space of V ∗ . Let BW = {w1 , . . . , wm+1 } be a basis of W and let BV = {w 1 , . . . , wm+1 , v m+2 , . . . , v n+1 } a basis of V which extends BW . Let B∗V = {w ∗1 , . . . , w ∗m+1 , v ∗m+2 , . . . , v ∗n+1 } be the dual basis of BV . Thus, V ∗ = Span(B∗V ) and any f ∈ V ∗ can be written n+1 ∗ ∗ as f = Σm+1 i=1 ai wi + Σj=m+2 aj v j , ai , aj ∈ K. Note that f ∈ Ann(W ) if and only if f (wi ) = 0, for any 1  i  m + 1, i.e. if and only if a1 = · · · = am+1 = 0. Therefore, Ann(W ) = Span(v ∗m+2 , . . . , v ∗n+1 ) so dim(Ann(W )) = n − m = dim(V ) − dim(W ). (ii) Λ⊥ = {[L] ∈ P(V )∗ | Λ ⊆ Zp (L)}, where Λ = P(W ). Thus, [L] ∈ Λ⊥ if and only if, for any t ∈ K, tL ∈ Ann(W ), namely Λ⊥ = P(Ann(W )). Solution 3.4 X2 1 (i) In the affine chart U0 take affine coordinates x1 = X X0 and x2 = X0 , so 2 2 that the trace Z0 := Z ∩ U0 of Z in U0 is simply Za (x1 + x2 − x1 ) which is a real ellipse since the intersection with the line at infinity of U0 is X2 0 empty. Let us denote by z0 = X X1 and z2 = X1 the affine coordinates in the affine chart U1 ; then the trace Z1 := Z ∩ U1 in this chart is the affine conic given by Za (1 + z22 − z0 ) which is a parabola, since Z is tangent to the line X1 = 0. Finally, in the affine chart U2 , with X1 0 affine coordinates w0 = X X2 and w1 = X2 , Z cuts out the affine conic Z2 = Z ∩ U2 = Za (w1 + 1 − w0 w1 ) which is a hyperbola, being the line X2 = 0 a secant line to the conic Z. (ii) The equation of Ca (Z) in A3 , with affine coordinates X0 , X1 , X2 , is given by Za (X12 + X22 − X0 X1 ) and it is a cone with vertex at the origin O = (0, 0, 0) ∈ A3 which projects, at its plane at infinity, the conic Z ⊂ P2 . In particular Ca (Z)∞ = Z. Similarly the equation of Cp (Z) in P3 , with homogeneous coordinates [X0 , X1 , X2 , X3 ], is once again given by Zp (X12 + X22 − X0 X1 ) ⊂ P3 . This is a cone with vertex at the fundamental point P3 = [0, 0, 0, 1] ∈ P3 which projects at the fundamental plane H3 = Zp (X3 ) ⊂ P3 the conic Z = Zp (X3 , X12 + X22 − X0 X1 ).

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Solution 3.5 Note that for any hypersurface Zp (F ) ⊂ Pn , one has Zp (F ) = Zp (X0 F, X1 F, . . . , Xn F ). If Ip (X) = (F1 , . . . , Fm ), set d := max1jm {dj }, where dj := deg(Fj ), 1  j  m. Thus, X = Zp (Ip (X)) = Zp (X0d−d1 F1 , . . . , Xnd−d1 F1 , . . . , X0d−dm Fm , . . . , Xnd−dm Fm )

and the latter are homogeneous polynomials all of the same degree d. Solution 4.1 Consider f (x1 , x2 ) := x41 − 2x21 + x22 + 1 ∈ R[x1 , x2 ]. Since R[x1 , x2 ] ⊂ C[x1 , x2 ], note that in C[x1 , x2 ] the polynomial f splits as f = (x21 − 1 + ix2 )(x21 − 1 − ix2 ), where each factor is irreducible in C[x1 , x2 ]. Indeed the corresponding AAS’s, say X1 := Za (x21 − 1 + ix2 ) and X2 := Za (x21 − 1 − ix2 ), are both parabolas in A2C , for which A(X1 ) ∼ = C[t] ∼ = A(X2 ), where t is an indeterminate. Since C[x1 , x2 ] is a UFD, factorization is uniquely determined up to invertible elements. Therefore, as an element of R[x1 , x2 ], the polynomial f is irreducible. On the other hand, Za (f ) ⊂ A2R consists of (real) points given by X1 ∩ X2 , namely Za (f ) = {P1 = (1, 0), P2 = (−1, 0)} which is therefore reducible. Solution 4.2 Note that x3 (x1 x3 − x22 ) − x2 (x31 − x2 x3 ) = x1 (x23 − x21 x2 ) which therefore / J as well as x23 − x21 x2 ∈ / J, therefore, J is belongs to J. Nonetheless x1 ∈ not a prime ideal. Let us consider now the ideals J1 := (x1 x3 −x22 , x31 −x2 x3 , x1 ) and J2 := (x1 x3 −x22 , x31 −x2 x3 , x23 −x21 x2 ). Since J ⊆ J1 and J ⊆ J2 , by reversing inclusion one has Za (J1 ) ⊆ Za (J) and Za (J2 ) ⊆ Za (J), so Za (J1 ) ∪ Za (J2 ) ⊆ Za (J). Note that J1 = (x22 , x2 x3 , x1 ) so Za (J1 ) is the line  = Za (x1 , x2 ) ⊂ A3 , which is irreducible being homeomorphic to A1 . As for J2 , consider the map ϕ : A1 → A3 given ϕ by t −→ (t3 , t4 , t5 ) and let C := Im(ϕ) ⊂ A3 . The map ϕ gives rise to the K-algebra homomorphism ϕ∗ : K[x1 , x2 , x3 ] → K[t], ϕ∗ (x1 ) = t3 , ϕ∗ (x2 ) = t4 , ϕ∗ (x3 ) = t5 .

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1 ,x2 ,x3 ] Denote by A(C) := K[xKer(ϕ) ; since A(C) is a subring of K[t] then A(C) is an integral domain, i.e. the ideal Ker(ϕ) is a prime ideal. Moreover, by definition of ϕ∗ , one has that J2 ⊆ Ker(ϕ). To prove the other inclusion, since any f ∈ Ker(ϕ) is such that f (t3 , t4 , t5 ) = 0, for any t ∈ K, one easily observes that necessarily f ∈ J2 . Therefore, J2 = Ker(ϕ) so C = Za (J2 ) is irreducible. We are left to showing that Za (J) =  ∪ C; if this is the case, C and  will be the irreducible components of Za (J). To prove this, from the generators of J note that, when x1 = 0 then one finds the line . When otherwise x1 = 0 then also x2 x3 = 0; therefore on Za (J) \ Za (x1 , J) we can  2 set t := xx21 = xx32 , therefore t2 = xx21 · xx32 = xx31 , thus, t4 = xx31 = x2 since

on Za (J) one has x23 = x21 x2 ; similarly t3 = from

x22

x22

= x1 x3 , we have x3 x1 =

8

t t3

t4 t

=

x2 x2 /x1

= x1 and moreover,

5

= t , which gives C as above.

Solution 4.3 Assume by contradiction that Y = Y1 ∪ Y2 is reducible in PnK , where each Yi is a closed proper subset of Y , 1  i  2. By induced topology, there exist closed subsets Ki ⊆ PnK , such that Yi = Y ∩ Ki , 1  i  2. Since Y ⊂ X, then Xi := X ∩ Ki = ∅ is a closed subset of X such that Yi = Xi ∩ Y , 1  i  2. Thus, Y would be reducible in X, a contradiction. Furthermore, since X is locally closed in PnK , there exist an open set U of PnK and a closed subset K of PnK such that X = U ∩ K. Similarly, Y is locally closed in X so there exist an open set UX of X and a closed subset KX of X such that Y = UX ∩KX . By induced topology, there exist an open set U  and a closed set K  of PnK such that UX = U  ∩ X and KX = K  ∩ X. Therefore, Y = (U  ∩ (U ∩ K)) ∩ (K  ∩ (U ∩ K)) = (U ∩ U  ) ∩ (K ∩ K  ), where U ∩ U  is an open set of PnK whereas K ∩ K  is a closed subset of PnK , so Y is locally closed in PnK . Solution 4.4 (3)

Let A := AI denote the quotient ring. Any element of A is the residue, modulo I, of an element of the form a(x3 ) + b(x3 )x1 + c(x3 )x2 + d(x3 )x1 x2 in A(3) , where a(x3 ), b(x3 ), c(x3 ), d(x3 ) ∈ K[x3 ]. Define the K-algebra

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homomorphism α : A(3) → K[t], α(x1 ) = t9 , α(x2 ) = t6 , α(x3 ) = t4 . Its image gives rise to a subring R := Im(α) ⊂ K[t] and Ker(α) is such that I ⊆ Ker(α). Moreover, the morphism α induces a K-algebra homomorphism ϕ : A → R for which ϕ(f ) = 0 if and only if f = 0 in A. This implies that ϕ is injective; thus ϕ is an isomorphism, so I = Ker(α). Since R ⊂ K[t], then R is an integral domain which implies that I = Ker(ϕ) is a prime ideal in A(3) . Thus, Za (I) is irreducible and I is radical, i.e. Ia (Za (I)) = I.

Solution 4.5 (i) Let Sym(3×3; K) be the K-vector space of symmetric matrices of order 3 with entries from K. If X denotes the column vector of homogeneous coordinates of P2 , then any Γ ∈ C has a cartesian equation of the form X t AX = 0, for some A ∈ Sym(3 × 3; K). Two different cartesian equations of a given conic Γ ∈ C, say X t AX = 0 and X t A X = 0, are such that A = λA, for some λ ∈ K \ {0}. Then C can be identified with P(Sym(3 × 3; K)), where any Γ ∈ C is identified to the class of proportionality of matrices [A] defining equivalent cartesian equations for Γ. Since dimK (Sym(3 × 3; K)) = 6, then C is a projective space of dimension 5. To fix natural homogeneous coordinates for P(Sym(3×3; K)), which therefore give isomorphism with the numerical projective space P5K , take A := (aij ), 0  i  j  2, a symmetric matrix whose entries are indeterminates over K; homogeneous coordinates for P(Sym(3 × 3; K)) are given by the proportionality class [. . . , aij , . . .] of the entries of A, set with lexicographic order on the indexes 0  i  j  2. (ii) Note that Γ ∈ D if and only if its (and so all) cartesian equation is determined by [A] ∈ P(Sym(3 × 3; K)) such that det(A) = 0. Note that det(A) = det(aij ) is a cubic polynomial in the entries aij ’s of the symmetric matrix A; moreover if A = λA then det(A ) = λ3 det(A) = 0. Therefore, D corresponds to the cubic hypersurface Σ3 := Zp (det(A)) ⊂ P5K . (iii) Let S(5) = K[. . . , aij , . . .] be the homogeneous coordinate ring of P5K ; (5) since F := det(A) ∈ S3 then F is certainly reduced. We need to show

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that F is irreducible. Note that F := det(A) = a00 a11 a22 + 2a01 a02 a12 − a202 a11 − a00 a212 − a201 a22 . Take Λ := Zp (a11 , a02 ), which is a linear subspace of P5K of dimension 3, i.e. Λ ∼ = P3K ; note that Σ3 ∩ Λ = Zp (F, a11 , a02 ) = Zp (a00 a212 + 2 a01 a22 , a11 , a02 ). Taking homogeneous coordinates [a00 , a01 , a12 , a22 ] = [Z0 , Z1 , Z2 , Z3 ] on Λ, then ΣΛ := Σ3 ∩ Λ is a cubic hypersurface of Λ given by ΣΛ = Zp (Z0 Z22 + Z3 Z12 ). Considering the affine open set U0 ⊂ P3K ∼ = Λ, which is the complement of the hyperplane H0 := Zp (Z0 ), then ΣΛ ∩ U0 is the image in A3K ∼ = U0 of the continuous map A2K → ΣΛ ∩ U0 ⊂ U0 ∼ = A3K , (u, v) → (u, uv, −v 2 ), which implies that ΣΛ ∩ U0 is irreducible, and so ΣΛ is irreducible too. This implies that Σ3 is irreducible: otherwise if it were Σ3 = H ∪ Q, where H a hyperplane and Q a (possibly reducible) hyperquadric in P5K , any linear section of Σ3 would contain a linear subspace, contradicting the irreducibility of ΣΛ . Solution 5.1 The given map φ1 is a homeomorphism, since one has φ1

A1 ⊃ W −→ Y ⊂ A2 1 t → (t, t ) x1 ← (x1 , x2 ). Moreover, since Y is an affine variety, then OY (Y ) = A(Y ) = (1)

K[x1 , x2 ] ∼ (1) = K[x1 , x−1 1 ] = Ax1 . (x1 x2 − 1)

Note that Ax1 can be identified with a subring of OW (W ), since 1 1 x1 ∈ OW (W ) and moreover K[x1 ] = A(A ) ⊂ OW (W ). Therefore, OY (Y ) ⊆ OW (W ) (in Example 6.2.5-(iv), we will more precisely show that OY (Y ) ∼ = OW (W )). As for function fields, one has K(W ) = K(A1 ), since W a principal open set of A1 (cf. Lemma 5.3.13). Thus, Y and W are not only homeomorphic as topological spaces, but moreover the function rings OY (Y ) = A(Y ) and K(Y ) are isomorphic to OW (W ) and K(W ), respectively (in Remark 6.1.4 we will show that φ1 is indeed much more than

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a homeomorphism between topological spaces; it is actually an isomorphism of algebraic varieties). The map φ1 is a rational parametrization of the is nothing but the restriction to Y of the first hyperbola Y whereas φ−1 1 projection of A2 , i.e. the projection of the affine plane onto its x1 -axis. The map φ1 is the restriction of a natural projective homeomorphism. Indeed, identifying A2 with the affine chart U0 of P2 , the projective closure Y of Y is given by Zp (X1 X2 − X02 ). The x1 -axis becomes the projective line Zp (X2 ), which is homeomorphic to P1 . In this way, one has the homeomorphism Φ

1 Y ⊂ P2 P1 −→ 2 2 [λ, μ] → [λμ, μ , λ ] [a0 , a1 ] ← [a0 , a1 , a2 ],

where [a0 , a1 ] = [λμ, μ2 ] = [λ, μ] as it occurs on points of Y ⊂ U0 (cf. Example 8.1.11 for more general motivations). The map Φ−1 naturally 1 and moreover it sends the two points at infinity of Y , restricts to φ−1 1 i.e. Q1 = [0, 0, 1] and Q2 = [0, 1, 0], respectively to [1, 0] ∈ A1 \ W and to [0, 1], the point at infinity of A1 . In the next chapter, we will show that Φ1 is an isomorphism of projective varieties, which ensures as above that regular and rational functions on Y will be the same as those on P1 . One can easily give a geometric interpretation of the map Φ−1 1 , observing that this is obtained by simply projecting Y to the line Zp (X2 ) ⊂ P2 from the point Q1 = [0, 0, 1] ∈ Y . Consider indeed the pencil of lines in P2 through the point Q1 ; equation for this pencil is given by Zp (λX1 − μX0 ) ⊂ P2 , for [λ, μ] ∈ P1 . For any point P ∈ Y , there is only one line of the pencil passing through P and conversely, since Y is a conic, points P ∈ Y are in bijective correspondence with the subset of lines in the pencil consisting of all lines except for that passing through Q2 and for the tangent line to Y at Q1 . The line of the pencil passing through Q2 is given by Zp (X0 ), whose trace over the X1 -axis in P2 is the point [0, 1, 0], identified with [0, 1] ∈ P1 via Φ−1 1 , i.e. Q2 corresponds via this projection to the point at infinity of the originary A1 . Finally, the tangent line to Y at Q1 is Zp (X1 ), which intersects Y only at the point Q1 with multiplicity 2. The trace of this tangent line over the X1 -axis is the point [0, 0, 1], identified with [1, 0] ∈ P1 1 via Φ−1 1 , i.e. Q1 corresponds via the projection to the origin O ∈ A \ W , where the originary parametrization φ1 was not defined.

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Solution 5.2 The map φ2 is a homeomorphism, since one has φ2

A1 −→ V ⊂ A2 2 t → (t, t ) x1 ← (x1 , x2 ). Thus, V is homeomorphic to A1 (φ2 will be more precisely an isomorphism of algebraic varieties, cf. Remark 6.1.4). The map φ2 is a polynomial is the restriction to V of parametrization of V (recall (3.31)) and φ−1 2 projection of A2 onto the first axis x1 . As for the hyperbola, φ2 is the restriction of the projective homeomorphism Φ

2 V ⊂ P2 P1 −→ 2 2 [λ, μ] → [λ , λμ, μ ] [a0 , a1 ] ← [a0 , a1 , a2 ],

where V ⊂ P2 the projective closure of V and where [a0 , a1 ] = [λ2 , λμ] = −1 [λ, μ], as it occurs for points on V . The map Φ−1 2 naturally restricts to φ2 and, similarly as for the hyperbola, it is nothing but the projection of V to P1 as one can show by using the pencil of lines through [0, 0, 1] which is the point at infinity of V , i.e. where V is tangent to the line Zp (X0 ). As for the hyperbola above, Φ2 will be an isomorphism of projective varieties (cf. Chapter 6). Solution 5.3 Since V is an affine rational curve with polynomial parametrization, then V = Za (x2 − x21 , x3 − x31 , . . . , xd − xd1 ) as in (3.31). From Lemma 3.3.17, one has Ia (V ) = (x2 − x21 , x3 − x31 , . . . , xd − xd1 ), which is also prime, i.e. V is an (d) affine variety in Ad . Since V is affine, then OV (V ) = A(V ) ∼ . From = IA a (V ) (1) ∼ the generators of Ia (V ), one deduces that A(V ) = K[t] = A . Therefore, K(V ) = Q(A(V )) ∼ = K(t) = Q(1) . Thus, K(V ) is a purely transcendental field extension of K, with transcendence degree 1. Solution 5.4 Note that Za (x2 ) ∩ V = Za (x2 , x1 x4 ) = Za (x2 , x1 ) ∪ Za (x2 , x4 ) = π1 ∪ π2 ,

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where π1 and π2 are planes contained in V . Therefore, if we set U1 := V \ (π1 ∪ π2 ), then U1 ⊂ V is a dense open subset of V and certainly f ∈ OV (U1 ). On the other hand, from the equation defining V , one has x1 x3 x2 = x4 . As above, Za (x4 ) ∩ V = Za (x2 x3 , x4 ) = Za (x2 , x4 ) ∪ Za (x3 , x4 ) = π2 ∪ π3 , where π3 is another plane contained in V . The open set U2 := V \ (π2 ∪ π3 ) is dense in V , xx34 ∈ OV (U2 ), moreover U1 ∩ U2 = ∅ and     x1 x3 U1 U2 ρU1 ∩U2 = ρU1 ∩U2 . x2 x4 Therefore, U1 ∪ U2 = V \ π2 , Φ = [U1 , f ] ∈ OV (U1 ∪ U2 ) in particular Dom(f ) contains V \ π2 . We want to show that this is an equality. Assume by contradiction that V \ π2  U  ⊆ Dom(f ) and let g  h ∈ K(V ) which extends on U as a regular function the given function f . With no loss of generality, we may assume that g and h have no common non-constant irreducible factor. Thus, on U1 one must have hg = xx12 , namely x2 g − x1 h = 0 on U1 and so on V , as U1 is dense in V . Since x1 does not divide x2 and x2 does not divide x1 in A(V), then x2 divides h and x1  divides g, therefore hg = xx21hg  where x2 does not divide g  , otherwise g and h would have x2 as a common irreducible factor, a contradiction. Thus, hg cannot be defined on an open set U  such that V \ π2  U  . Therefore, one can conclude that Dom(f ) = V \ π2 . Solution 5.5 Let P0 = [1, 0, 0, 0] ∈ P3 be the fundamental point. Note that P0 ∈ Z and the projection of Z from P0 gives the map πP0 πP0 : Z \ {P0 } → Zp (X0 ) = H0 ∼ = P2 , [X0 , X1 , X2 , X3 ] −→ [X1 , X2 , X3 ].

This is a homeomorphism, whose inverse on H0 \ (H2 ∩ H0 ) is given by  2  a1 [a1 , a2 , a3 ] → , a1 , a2 , a3 = [a21 , a1 a2 , a22 a2 a3 ]. a2 Thus, Z is a projective variety since it is irreducible, being homeomorphic to an open dense subset of a projective plane, and closed in P3 . Since

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Z0 := Z ∩ U0 = Za (x21 − x2 ) ⊂ A3 is an open dense subset of Z, one has K(Z) ∼ = K(Z0 ). From the fact that Z0 is an affine variety, one 1 ,x2 ,x3 ] ∼ has OZ0 (Z0 ) = A(Z0 ) = K[x = K[x1 , x3 ], therefore, K(Z0 ) = (x21 −x2 ) ∼ Q(A(Z0 )) = K(x1 , x3 ), which is a purely transcendental extension of K with transcendence degree two. Solution 6.1 The map ϕ is a morphism since it is a polynomial map with target an affine variety. Any point P = (p1 , p2 ) ∈ A2 such  that p1 · p2 = 0 is such that P ∈ Im(ϕ): namely ϕ−1 (P ) =

p1 , pp21 . Note moreover that the

line Za (x1 ) ⊂ A2 is mapped via ϕ to the origin O = (0, 0) whereas any P = (p1 , 0) ∈ Za (x2 ) is such that ϕ(P ) = P . In other words Im(ϕ) = (A2 \ Za (x1 )) ∪ {O}, where A2 \ Za (x1 ) is a principal open set whereas O is Zariski closed. Thus, Im(ϕ) is neither closed nor open in A2 but it is a constructible set (cf. Remark 6.1.10 and Corollary 11.1.3). Solution 6.2 Note that O = (0, 0) ∈ C. Consider the pencil of lines through O, namely {t := Za (x2 − tx1 )}t∈A1 . The intersection with the curve C of any line of the pencil such that t = ±1 consists of the point O, with multiplicity μ(C, t ; O) = 2, and of an extra point Pt := (t2 − 1, t3 − t). Whereas lines 1 and −1 intersect C only at O with μ(C, ±1 ; O) = 3. This defines a map ϕ

ϕ : A1 → C ⊂ A2 , t −→ (t2 − 1, t3 − t), which is surjective and moreover it is a morphism, since it is a polynomial map with affine target. This proves that C is irreducible and so that C is an affine variety. Moreover, ϕ is the required polynomial parametrization. Observe that A(C) is isomorphic to a proper subring of A(1) , since on A(C) one has t2 = 1 and t3 = t. Therefore, the natural map ϕ# : A(C) → K[t] = A(1) is injective but not an isomorphism so ϕ cannot be an isomorphism (indeed ϕ is surjective onto C but not injective). Solution 6.3 For [s, t] ∈ P1 , one can easily deduce homogeneous parametric equations of st gives the conic C ⊂ P2 as X0 = s2 , X1 = st, X2 = t2 . Composing with ν2,2 st ν2,2 (C) = {[s4 , s3 t, s2 t2 , s2 t2 , st3 , t4 ] | [s, t] ∈ P1 } ⊂ P5 .

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Denoting by [W0 , W1 , . . . , W5 ] homogeneous coordinates for the target st (C) lies in the hyperplane section of the Veronese space P5 , note that ν2,2 st surface V2,2 with the hyperplane Zp (W2 − W3 ) ∼ = P4 and it has therein a homogeneous parametric representation of a standard rational normal quartic. Solution 6.4 Similarly as in Exercise 6.3, the line  has homogeneous parametric equations given by  : X0 = s, X1 = t, X2 = t, [s, t] ∈ P1 so, composing st , gives with ν2,3 st () = [s3 , s2 t, s2 t, st2 , st2 , st2 , t3 , t3 , t3 , t3 ], [s, t] ∈ P1 , ν2,3

Denoting by [W0 , W1 , . . . , W9 ] homogeneous coordinates for the target st () is a standard projective twisted cubic space P9 , one notices that ν2,3 in the linear subspace Zp (W1 − W2 , W3 − W4 , W3 − W5 , W6 − W7 , W6 − W8 , W6 − W9 ) ∼ = P3 in P9 . Similarly, st ν2,3 (C) = [s6 , s5 t, s4 t2 s4 t2 , s3 t3 , s3 t3 s2 t4 , s3 t3 , s2 t4 , st5 , t6 ], [s, t] ∈ P1 , st (C) is a standard projective rational normal sextic in which shows that ν2,3 the linear subspace Zp (W2 − W3 , W4 − W6 , W5 − W7 ) ∼ = P6 in P9 .

Solution 6.5 st Since, up to projectivities, any gdd on P1 is projectively equivalent to ν1,d , with no loss of generality we can reduce to this case with the choice of st st ) = V1,d is the standard P = [1, 0] a fundamental point of P1 . Thus, Im(ν1,d rational normal curve of degree d in Pd and the point P maps to P0 := st st . Divisors in the gdd = ν1,d on P1 bijectively correspond to [1, 0, . . . , 0] ∈ V1,d st d hyperplane sections of V1,d ⊂ P . Thus, the sublinear series gdd (−P ) consists st with hyperplanes of divisors which bijectively correspond to sections of V1,d passing through P0 . Consider the projection πP

0 πP0 : Pd \ {P0 } → Pd−1 , [X0 , X1 , . . . , Xd ] −→ [X1 , . . . , Xd ].

st is Its restriction π to V1,d π

st π : V1,d \ {P0 } → Pd−1 , [sd , sd−1 t, . . . , td ] −→ [sd−1 t, . . . , td ], [s, t] ∈ P1 . st \{P0 } one has t = 0 therefore, dividing by t the expression Note that on V1,d of points on Im(π), one has Im(π) = {[sd−1 , . . . , td−1 ] | [s, t] ∈ P1 } which

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makes sense for any [s, t] ∈ P1 . In other words Im(gdd (−P )) = Im(π) = st ⊂ Pd−1 , i.e. the standard rational normal curve of degree d − 1 is an V1,d−1 st internal projection of the standard rational normal curve V1,d ⊂ Pd from its point P0 = [1, 0, . . . , 0]. Solution 7.1 The Segre variety Σ1,1 is the rank-four quadric surface given by Zp (W00 W11 − W01 W10 ) ⊂ P3 . Any fiber of the projection of P1 × P1 onto its ith factor, 1  i  2, is mapped via σ1,1 to a line in Σ1,1 . Since σ1,1 is an isomorphism onto its image, lines coming from fibers of the same projection do not meet, so they are skew in P3 , whereas two lines coming from fibers of the two different projections are incident at just one point. More precisely, for any P, Q ∈ P1 , Σ1,1 contains lines P := σ1,1 ({P } × P1 )

and rQ := σ1,1 (P1 × {Q});

any two lines {P }P ∈P1 are skew, the same holds for any two lines {rQ }Q∈P1 , whereas for any point R ∈ Σ1,1 , the lines R and rR meet at R. These two families of lines in Σ1,1 are called rulings of the quadric surface, for this 3 = reason Σ1,1 is said to be a doubly ruled surface. Note that Σ1,1 ∩ U00 3 Za (x3 − x1 x2 ) is the hyperbolic paraboloid in A (see figure below).

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Solution 7.2 ∼ =

∼ =

(i) Let γ  : V  −→ V and δ  : W  −→ W be isomorphisms, which exist by the assumptions. With notation as in Theorem 7.4.2, take X := V  × W  , ρ1 := γ  ◦ πV  and ρ2 := δ  ◦ πW  . Since X is a product, it satisfies assumptions as in Theorem 7.4.2 from which one deduces the isomorphism V  × W  ∼ = V × W. (ii) Letting X := W × V , with ρ1 = πV and ρ2 = πW , since X is a product of algebraic varieties, the existence of an isomorphism W ×V ∼ = V ×W is granted from Theorem 7.4.2. (iii) This follows by simply recursively applying the same procedures as in (i) and (ii) above. −1 (Q) is similar. Note (iv) We focus on πV−1 (P ), since the proof for πW −1 that, by definition of πV (P ), it is isomorphic to the algebraic variety {P } × W . Moreover, since πV is a morphism and P ∈ V is closed in V , then πV−1 (P ) is a closed subvariety of V × W . The projection πW : {P } × W → W is an isomorphism which, composed with that between πV−1 (P ) and {P } × W , gives rise to an isomorphism πV−1 (P ) ∼ = W. (v) If ϕ is a morphism, then πV ◦ ϕ and πW ◦ ϕ are obviously morphisms, since they are both compositions of morphisms. Conversely, assume that α := πV ◦ϕ ∈ Morph(Z, V ) and that β := πW ◦ϕ ∈ Morph(Z, W ). Then α×β is a morphism from Proposition 7.4.1. Moreover, since α×β is the unique map which let the diagrams α×β

Z −→ V × W πV α V

α×β

and

Z −→ V × W πW β W

to be commutative, as ϕ does, one concludes that ϕ = α × β and so that ϕ is a morphism. Solution 7.3 The fact that A ⊗K B is a K-algebra of finite type directly follows from the definition of tensor product of K-algebras as in Section 1.9.2. We need to show that it is an integral K-algebra. Let c, d ∈ A⊗K B be such that c·d = 0. We can write c = Σi ai ⊗ bi and d = Σj αj ⊗ βj , where the sets {bi } and {βj } of B can be taken to be linearly independent over K. Let h : A → K

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be any K-algebra homomorphism. Then h ⊗ IdB : A ⊗K B → B = K ⊗K B is a K-algebra homomorphism. Therefore, one has ⎞ ⎛ ⎞ ⎛   0 = (h ⊗ IdB )(c · d) = (h ⊗ IdB ) ⎝ ai αj ⊗ bi βj⎠ = ⎝ h(ai αj ) ⊗ bi βj⎠ i,j

= (h ⊗ IdB )(c) · (h ⊗ IdB )(d) =

  i

i,j

⎞ ⎛  h(ai ) ⊗ bi · ⎝ h(αj ) ⊗ βj⎠ ∈ B. j

Since B is integral, one of the two factors has to be zero e.g. (Σi h(ai ) ⊗ bi ) = 0. Since {bi } is linearly independent over K, this means that h(ai ) = 0, for any i. Therefore, the same holds for h(ai · αj ) = h(ai ) · h(αj ), for any i and j. Since this holds for any K-algebra homofor some ideal I in a suitable morphism h : A → K and since A = K[x] I polynomial ring K[x], this implies that ai · αj = 0 in A: indeed, by the Hilbert “Nullstellensatz”-weak form, to give any K-algebra homomorphism h : A → K is equivalent to giving a maximal ideal mP ∈ Specm(A) corresponding to a point P ∈ Za (I), such that A=

K[x] h A →K∼ , h(a) = evP (a), = I mP

where a a residue class of a polynomial in K[x] modulo I, thus a polynomial a which vanishes at each point of Za (I) is the zero-polynomial of A by the Hilbert “Nullstellensatz”-strong form. Since A is an integral K-algebra, ai · αj = 0, for any i and j, implies that either ai = 0 for any i or αj = 0 for any j. Finally, if V ⊆ An and W ⊆ Am are affine varieties defined over an algebraically closed field, we already know that V × W ⊆ An × Am ∼ = An+m is a closed immersion. Moreover, we proved that A(V × W ) ∼ = A(V ) ⊗K A(W ). Since A(V ) and A(W ) are both integral K-algebras of finite type and since K is algebraically closed, from the previous part A(V ) ⊗K A(W ) is an integral K-algebra of finite type, so V × W is irreducible therefore it is an affine variety such that, as a set, V × W = Specm(A(V ) ⊗K A(W )). Solution 7.4 Let K = R and let C = (xR[x] 2 +1) . Thus, C is an integral R-algebra of finite type. Nonetheless C ⊗R C is a R-algebra of finite type which is not integral, indeed (1 ⊗ i − i ⊗ 1) · (1 ⊗ 1 − i ⊗ i) = 0.

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Solution 7.5 Let V = A2 and Z = A1 , with ϕ = πx the projection onto the first coordinate, whereas let W = {0} and ψ = ιW be the closed immersion of the point {0} in A1 . In this case   A2 ×A1 {0} = (P, {0}) ∈ A2 × {0} | πx (P ) = 0 ∼ = πx−1 (0), which is isomorphic to A1 as πx−1 (0) = {(0, p2 ) | p2 ∈ A1 }. In this case the fiber-product turns out to be the fiber over 0 ∈ A1 of the projection of A2 onto the first coordinate. On the contrary, if we take V = Za (x21 + x22 − 1) ⊂ A2 , W = {0}, Z = A1 , ϕ = πx ◦ ιV whereas ψ as above, then V ×Z W = ϕ−1 (0) = {(0, 1)}∪{(0, −1)} ⊂ V , which is a reducible algebraic subset of V . Solution 8.1 Let U := An−1 \ Za (x1 x2 · · · xn−1 ). On the open set U we can consider the map  φ : U → A , (x1 , x2 , . . . , xn−1 ) −→ x1 , x2 , . . . , xn−1 , n

φ

 1 , x1 x2 · · · xn−1

which is a morphism, by Proposition 6.2.2, whose image is X. Thus, X is irreducible therefore it is an affine variety. Moreover φ defines a rational map Φ : An−1  X. Consider now the multi-index I = (1, . . . , n − 1) and the corresponding projection πI : An → An−1 , then ψ := πI ◦ ιX ∈ Morph(X, U ), where ιX denotes the closed immersion of X in An . The morphism ψ defines the rational inverse of the rational map Φ. Solution 8.2 For n = 1, we already know that, e.g. the hyperbola Y1 = Za (x1 x2 −1) ⊂ A2 is birationally equivalent but not isomorphic to A1 . Therefore, for any n  2, we can consider Yn := (A1 \ {0}) × An−1 . Recall that A1 \ {0} is an affine variety, being isomorphic to the hyperbola Y1 above, therefore, Yn is an affine variety which is birational to A1 × An−1 ∼ = An , for any n  2. On the other hand Yn is not isomorphic to An , since A(An ) = A(n) whereas 1 (n) . x1 ∈ OYn (Yn ) \ A

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Solution 8.3 Note that O ∈ C. The pencil of lines {Za (x2 − mx1 )}m∈A1 through O gives a parametrization A1 → C, m → (m3 − 1, m4 − m) which is injective off the closed subset Za (m3 − 1) ⊂ A1 . Its rational inverse C \ {O} → A1 is given by (x1 , x2 ) → xx21 . On the other hand the previous maps are not isomorphisms, since xx21 is not regular at O. Solution 8.4 From the previous exercise, we already know that C is birational but not isomorphic to A1 . Intersecting C with the lines of the pencil gives x2 − mx1 = 0 = x31 (x1 + 1 − m3 ). In A3 , with coordinates (x1 , x2 , x3 ), the proper transform of C after blowing-up A2 at the origin O is given by the curve C  ⊂ A3 of parametric equations x1 = m3 − 1, x2 = m4 − m, x3 = m. This gives a polynomial parametrization A1 → C  , namely m → (m3 − 1, m4 − m, m), which is therefore a morphism and whose inverse morphism is given by the projection onto the x3 -axis. Thus, C  is isomorphic to A1 and birational (but not isomorphic) to C. Solution 8.5 From the expression of q, one notices that q is an involution on the open set U := U0 ∩U1 ∩U2 ⊂ P2 , namely q|U is an isomorphism such that q|−1 U = q|U , in particular q is a birational transformation of P2 into itself. Note that q can also be written as q([X0 , X1 , X2 ]) = [X1 X2 , X0 X2 , X0 X1 ], which shows that q coincides with the rational map νL as in (6.16), given (2) by the linear system associated to L = Span{X1 X2 , X0 X2 , X0 X1 } ⊂ S2 . Thus, the indeterminacy locus of q coincides with the base locus B := {P0 = [1, 0, 0], P1 = [0, 1, 0], P2 = [0, 0, 1]} of the linear system of conics P(L). The points in B, which are the fundamental points of P2 , are called fundamental points of q. Denoting by Hi := Zp (Xi ), 0  i  2, the fundamental lines of P2 , note that q contracts Hi \ (Hi ∩ B) to the fundamental point Pi ,

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for any i ∈ {0, 1, 2}. For this reason, the lines Hi are called exceptional lines of q, 0  i  2. Since q is an involution as a birational map, one says that q blows-up the fundamental points to the exceptional lines. Any line  = Zp (Σ2i=0 ai Xi ) not intersecting B maps via q to the conic Zp (a0 X1 X2 + a1 X0 X2 +a2 X0 X1 ) ⊂ P2 . Whereas any line through one of the fundamental points is instead mapped to another line; more precisely, take, e.g.  = Zp (λX1 + μX2 ) any line through the point P0 , with [λ, μ] ∈ P1 . Consider \{P0 }, whose points are given by [μ, μ, −λ] with μ = 0; then q([μ, μ, −λ]) = [−μλ, −μλ, μ2 ] = [λ, λ, −μ], i.e. the projective closure of q( \ {P0 }) is the line r = Zp (μX1 + λX2 ), which cuts out on H0 the point [0, λ, μ]. Thus the points on the exceptional line H0 are in one-to-one correspondence with the directions of lines through the fundamental point P0 ; the same occurs for the other fundamental points (exceptional lines, respectively). Solution 9.1 (i) Since A1 × A1 ∼ = A2 , then Z is the hyperbola in the affine plane which is therefore a closed affine subvariety of A2 . (ii) One has π2 (Z) = A1 \ {0} which is not closed in W = A1 . From Definition 9.1.1, it follows that A1 is not a complete variety. (iii) Take V = P1 in such a way that A1 is identified with the affine open 1 chart U0 so that t = X X0 , where [X0 , X1 ] homogeneous coordinate on 1  of Z in P1 ×A1 is given by Z(X1 s−X0 ) and the P . Then the closure Z projection π2 is given by the restriction to Z of πA1 : P1 × A1 → A1 ,  since π −1  which reads ([X0 , X1 ], s) → s. Note π2 extends to Z A1 (0) ∩ Z =  surjectively maps {([X0 , X1 ], 0) | X1 · 0 − X0 = 0} = {([0, 1], 0)} so Z  = A1 is closed. onto A1 , i.e. πA1 (Z)

Solution 9.2 ∼ A3 , the trace of  in U0 is the affine line Considering the affine chart U0 =  passing through P = (1, 0, 1) ∈ A3 with direction v = (−1, 1, 0), namely  : x1 = 1 − t, x2 = t, x3 = 1, t ∈ A1 . Take now f = x1 x2 − x3 ∈ A(3) . Then  ∩ Za (f ) = ∅, since t2 − t + 1 ∈ R[t] has no roots in R. Moreover, p p Za (f ) = Zp (X0 X3 − X1 X2 ) ⊂ P3 is such that  ∩ Za (f ) = Zp (X1 + X2 − X0 , X3 − X0 , X0 X3 − X1 X2 ). By the previous discussion about the affine part of this intersection, we only need to check for possible intersections when X0 = 0; however in this case, we have Zp (X1 +X2 , X3 , X1 X2 , X0 ) = ∅.

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Solution 9.3 (n)

One can P(Sd ) with the numerical projective space PN , where  n+didentify N = d − 1. Let Rk ⊂ PN be the locus of [F ] ∈ PN s.t. there exist (n) an integer k, 1  k  d − 1, and homogeneous polynomials Fk ∈ Sk (n)   and Fd−k ∈ Sd−k s.t. [F ] = [Fk · Fd−k ]. One has R = ∪d−1 k=1 Rk . For any n+j  (n) Nj 1  j  d − 1, set Nj := j − 1 and P := P(Sj ). One has a natural multiplication map k k   PN , ([Fk ], [Fd−k ]) −→ [Fk · Fd−k ], PNk × PNd−k −→

m

m

where Im(mk ) = Rk . Since PNk × PNd−k ∼ = ΣNk ,Nd−k is the Segre variety of indexes Nk and Nd−k , which is a projective variety, and since mk is a morphism, by completeness of projective varieties, Rk is a projective variety in PN . Then R is a closed algebraic set in PN and any Rk is one of its (n) irreducible component, 1  k  d − 1. Since in Sd exist infinitely many irreducible polynomials, then R is a proper closed subset of PN . Solution 9.4 (i) Up to replacing V and W , respectively, with the affine open sets ϕ−1 (UP ) and UP of the definition, with no loss of generality we may assume that V and W are affine varieties and therefore that ϕ is surjective. Up to isomorphism, we may assume that V ⊆ An , for some non-negative integer n so A(V ) is a quotient, modulo some prime ideal, of K[x1 , . . . , xn ]. Let A(V ) = K[x1 , . . . , xn ] where we denote by x1 , . . . , xn the images in A(V ) of the indeterminates in K[x1 , . . . , xn ]. By the assumption on ϕ, any xj ∈ A(V ) satisfies an identity of the form (xj )dj + aj,dj −1 (xj )dj −1 + · · · + aj,0 = 0, aj,i ∈ A(W ). Let Q ∈ W ; points in ϕ−1 (Q) x1 (Q), . . . , xn (Q) ∈ V ⊆ An , such that

have

coordinates

(xj (Q))dj + aj,dj −1 (Q)(xj (Q))dj −1 + · · · + aj,0 (Q) = 0, aj,i (Q) ∈ K. Since any polynomial T dj + aj,dj −1 (Q)(T )dj −1 + · · · + aj,0 (Q) ∈ K[T ] has finitely many roots in K, then ϕ−1 (Q) is a finite set.

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(ii) Consider the plane section of Z with the plane Za (x1 − x2 ); it is the K[x ,x3 ] conic C := Za (x1 − x2 , x23 + x21 − 1) for which A(C) ∼ = (x2 +x1 2 −1) 1 3 is an integral K-algebra thus C is irreducible; since Z is a quadric surface and since C is an irreducible conic plane section of Z, one deduces that Z is irreducible. Set now f := x23 + x3 (x1 − x2 ) + x21 − 1, which is the polynomial defining Z. We observe that ϕ is surjective onto A2 , indeed for any P = (p1 , p2 ) ∈ A2 , ϕ−1 (P ) contains points Q = (p1 , p2 , p3 ) ∈ Z, where p3 ∈ K is determined by the finitely many solutions of the polynomial equation FP (T ) = 0, where FP (T ) := T 2 + (p1 − p2 )T + (p21 − 1) ∈ K[T ]. In particular, ϕ(Z) = A2 which is closed in A2 . The surjectivity of ϕ implies that we have an ϕ#

injective K-algebra homomorphism K[x1 , x2 ] → A(Z). Let x3 be the image of the indeterminate x3 in A(Z). Then for x3 one has that x23 +x3 (x1 −x2 )+x21 −1 = 0, i.e. x3 is integral over K[x1 , x2 ]. Therefore, the ring inclusion is an integral extension of K-algebras so ϕ is a finite morphism. Solution 9.5 Up to a projective transformation, we may assume that P = Pn = [0, 0, . . . , 0, 1], so that the projection from P is simply [X0 , X1 , . . . , Xn−1 , Xn ]  [X0 , X1 , . . . , Xn−1 ]. From the assumption on V , we may write V = Zp (Xnd + F1 Xnd−1 + · · · + Fd ), where Fj ∈ K[X0 , . . . , Xn−1 ]j , 1  j  d. Consider Hi = Zp (Xi ) ⊂ Pn−1 and Ui = Pn−1 \ Hi the affine chart of Pn−1 , 0  i  n − 1. Now, ϕ−1 (Ui ) is a hypersurface in an affine space Ani , with affine coordinates (y0 , y1 , . . . , yi−1 , yi+1 , . . . , yn−1 , yn ), given by ϕ−1 (Ui ) = Za (ynd + δi (F1 )ynd−1 + · · · + δi (Fd )), where δi the dehomogeneization w.r.t. the indeterminate Xi , 0  i  n − 1. The hypersurfaces ϕ−1 (Ui ) determine an affine open covering of V and the restriction of ϕ to ϕ−1 (Ui ) is simply the restriction to ϕ−1 (Ui ) of the projection from Ani to given by An−1 i π

i (y0 , y1 , . . . , yi−1 , yi+1 , . . . , yn−1 ), (y0 , y1 , . . . , yi−1 , yi+1 , . . . , yn−1 , yn ) −→

i.e. the projection onto the first (n − 1) coordinates of Ani . Therefore, we are reduced to the case of an affine hypersurface Za (f ) ⊂ An , where + · · · + fd (x1 , . . . , xn−1 ) ∈ A(n) , f = xdn + f1 (x1 , . . . , xn−1 )xd−1 n and where ϕ : V → An−1 is the restriction to V of the projection πI : An → An−1 , with I = (1, 2, . . . , n − 1). From the expression of f ,

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the morphism ϕ is surjective, therefore it induces an injective K-algebra (n) (n) homomorphism ϕ# : A(n−1) → A(f ) . Note that xn ∈ A(f ) is integral over A(n−1) as it is clear from the expression of f . Therefore, the ring inclusion is an integral K-algebra extension which implies that ϕ is a finite morphism (cf. Ciliberto, 2021, § 10.2, pp. 125–128). Solution 10.1 Up to replacing V and W with the affine open sets ϕ−1 (UP ) and UP , respectively, we may assume V and W to be affine variety. Since A(W ) ⊆ A(V ) is an integral extension, then K(W ) ⊆ K(V ) is an algebraic extension. Therefore, trdegK (K(V )) = trdegK (K(W )) and we conclude. Solution 10.2 If Λ ⊂ Pn is a linear subspace of dimension s, then it is cut-out by n − s independent hyperplanes W1 , . . . , Wn−s in Pn . Since Λ ∩ V = ∅, then dim(V )  n − s − 1; indeed if by contradiction dim(V )  n − s, from Theorem 10.3.1-(ii), we would have V ∩ Λ = ∅. Solution 10.3 The hypersurface V is isomorphic to A2 : indeed the map ψ : A2 → A3 , ψ defined as (u, v) −→ (u, uv, v), maps A2 onto V and it is a morphism, since it is a polynomial map, moreover the restriction to V of the projection πJ : A3 → A2 , where J = (1, 3), defines ψ −1 ; thus V is irreducible. Now ϕ : V → A2 simply reads (u, uv, v) → (u, uv); in particular ϕ is not surjective onto A2 as points (0, k) with k = 0 are not in the image of ϕ. On points (p1 , p2 ) ∈ A2 such that p1 · p2 = 0, one has ϕ−1 (p1 , p2 ) = (p1 , p2 , pp21 ) ∈ V so dim(ϕ−1 (p1 , p2 )) = 0. If P = (h, 0) then ϕ−1 (h, 0) = (h, 0, 0) ∈ V so dim(ϕ−1 (h, 0)) = 0. However, (0, 0) ∈ Im(ϕ) and ϕ−1 (0) = {(0, 0, k) ∈ A3 | ∀ k ∈ K} is a line. Therefore, the open set where the fibers have constant dimension equal to 0 is (A2 \ Za (x1 )). From Exercise 9.4-(i), we deduce that ϕ cannot be a finite morphism. Solution 10.4 From the equation of , it is clear that the line is contained in Q. Let S(Q) be the homogeneous coordinate ring of Q, which is a graded, integral

291

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S(3) -module and a K-algebra of finite type. Note that the homogeneous ideal Ip,Q () ⊂ S(Q) of  in Q is generated by the images of the two linear generators of the homogeneous ideal Ip () = (X0 , X1 ) of the line  in P3 under the quotient epimorphism π : S(3)  S(Q). Therefore,  is not complete intersection in Q. The line  is not even set-theoretically complete intersection in Q; indeed PGL(4, K) contains Aut(Q) which is an algebraic group of dimension 6, thus chosen any two distinct lines  and  is Q there always exists a projectivity in Aut(Q) sending  in  and viceversa. Therefore, if  were set-theoretically complete intersection in Q then also  would be set-theoretically complete intersection in Q. On the other hand, if  and  are lines of the same ruling of Q we know that  ∩  = ∅ (cf. Exercise 7.1), contradicting Proposition 10.4.2-(ii). Solution 10.5 Replacing Q with Q = Zp (X0 X2 − X12 ), the same reasoning as in Exercise 10.4 shows that the line  = Zp (X0 , X1 ) is contained in Q and that it is not complete intersection in the quadric cone. On the other hand, taking the image of X0 ∈ S(3) via the quotient epimorphism π : S(3)  S(Q ), we see that  = ZQ (π(X0 )) so  is set-theoretically complete intersection in Q . Solution 11.1 Let V = Zp (X0 X2 −X12 ) ⊂ P2 which is an irreducible conic and let W = P1 . Let ϕ be the restriction to V of the projection πP2 : P2  P1 , where πP2

P2 = [0, 0, 1] ∈ V , πP2 : [X0 , X1 , X2 ]  [X0 , X1 ]. For any Q ∈ P1 \ {[0, 1]}, the fiber ϕ−1 (Q) consists of two distinct points, one of which is P2 , whereas ϕ−1 ([0, 1]) = {P2 }. In particular this also shows that ϕ is surjective. Solution 11.2 Let V = Z(2X0 X1 + tX22 ) ⊂ P2 × A1 , which is a closed subset (cf. Proposition 7.2.7-(ii)). First we show that V is irreducible: taking U02 ⊂ P2 X2 1 the affine chart with affine coordinates x1 = X X0 and x2 = X0 , one has 2 1 2 2 1 ∼ 3 V0 := V ∩ (U0 × A ) = Za (2x1 + tx2 ) ⊂ U0 × A = A and A(V0 ) ∼ = K[t, x2 ] which implies that V0 and so V = V0 are irreducible. Now the map ϕ

ϕ : V → A1 , ([X0 , X1 , X2 ], t) −→ t

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is surjective and, for any t ∈ A1 , ϕ−1 (t) = Zp (2X0 X1 + tX22 ) =: Ct ⊂ P2 is a conic. Therefore, all the fibers of ϕ are of pure dimension 1. For t = 0, the symmetric matrix of the conic Ct is of maximal rank, therefore, Ct is irreducible for t = 0. On the other hand, C0 consists of two incident lines Zp (X0 ) ∪ Zp (X1 ) ⊂ P2 . Solution 11.3 Note first that neither W can be a point, because dim(W ) > 0 by assumption, nor ϕ can be a constant morphism, since otherwise ϕ−1 (P ) = V contradicting the assumptions on ϕ−1 (P ) and on dim(V ) > 0. Since V is irreducible and ϕ is a morphism, then Im(ϕ) is irreducible and dim(Im(ϕ)) = dim(Im(ϕ)). Therefore, we need to show that dim(Im(ϕ)) = dim(V ). Since W is projective, Im(ϕ) is a projective variety and therefore ϕ ∈ Morph(V, Im(ϕ)) is a surjective morphism between projective varieties (cf. Corollary 9.1.2). Thus, from Theorem 11.2.2-(ii), there exists a nonempty open set U ⊆ Im(ϕ) where all the fibers have constant dimension equal to 0, since ϕ−1 (P ) consists of finitely many points. Thus, one has dim(V ) − dim(U ) = 0. One can conclude from the fact that dim(U ) = dim(Im(ϕ)). Solution 11.4 The arguments are similar as in the proof of Proposition 10.2.3. Indeed, from Corollary 6.4.3, there is a bijective correspondence between closed subvarieties Z ⊂ V containing W and prime ideals of OV,W . The length of maximal chains of such subvarieties is dim(V ) − dim(W ) = codimV (W ) which equals therefore the Krull-dimension of OV,W . Solution 11.5 Since any algebraic variety has an affine open covering, we can reduce to the affine case. Therefore, up to isomorphism, let V ⊆ An be an affine variety and let A(V ) be its affine coordinate ring. Any irreducible element f ∈ A(V ) gives rise to an irreducible hypersurface ZV (f ) ⊂ V . Thus, if V is an affine curve, P is the requested irreducible hypersurface not containing Q. Assume therefore dim(V )  2. Let mV,P = mV,Q be the maximal ideals in A(V ) corresponding to the points P = Q. Since they are distinct maximal ideals, there exists f ∈ mV,P \mV,Q ; for any such a f , by reversing inclusion,

Solutions to Exercises

293

ZV (mV,P ) ⊂ ZV (f ), i.e. P ∈ ZV (f ) whereas Q ∈ / ZV (f ). If f is irreducible, then take W = ZV (f ) and we are done. If otherwise f = f1 · · · fr , where fi ’s the irreducible factors of f , then Q ∈ / ZV (fi ) for any 1  i  r whereas there exists i0 ∈ {1, . . . , r} such that P ∈ ZV (fi0 ) and one sets W = ZV (fi0 ). Solution 12.1 Note that C is the union of the three coordinate axes in A3 ; in particular O = (0, 0, 0) ∈ C. Thus, TC/A3 ,O ∼ = K3 whereas, for any affine plane curve 2 D ⊂ A and for any point Q ∈ D, one has dim(TD/A2 ,Q )  dim(TA2 ,Q ) = 2. Solution 12.2 ∂F ∂F ∂F = X1 , ∂X = X0 and ∂X = −2X2 . Let [U0 , U1 , U2 ] be Note that ∂X 0 1 2 2 homogeneous coordinates in P , the target of ϕ; so we have

U 0 = X1 ,

U 1 = X0 ,

U2 = −2X2 .

Using the equation of C, one finds ϕ(C) = Zp (4U0 U1 − U22 ) ⊂ P2 , i.e. the dual curve of C is once again an irreducible conic which is therefore smooth. Solution 12.3 Z = Zp (X0 Fd−1 + Fd ), where Fj ∈ K[X1 , . . . , Xn ]j , j = d − 1, d. Any line passing through P0 has homogeneous parametric equations L : X0 = λ, X1 = μv1 , . . . , Xn = μvn , [λ, μ] ∈ P1 , (v1 , . . . , vn ) = (0, . . . , 0). Thus, Z ∩ L is determined by the polynomial equation μd−1 (λFd−1 (v1 , . . . , vn ) + μFd (v1 , . . . , vn )) = 0. The solution μ = 0, i.e. [1, 0] ∈ P1 , is of multiplicity d − 1 and corresponds to the point P0 on Z. Solution 12.4 By Euler’s formula, v(Z) is defined by the vanishing of the partial (n) derivatives of order (d − 1) of the homogeneous polynomial F ∈ Sd , such that, Z = Zp (F ). These partial derivatives are linear forms.

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Solution 12.5 For any P ∈ v(Z), let Q ∈ Z s.t. Q = P . Let L = P ∨ Q ⊂ Pn be the line through the two distinct points of Z. Then μ(Z, L; P )  d and μ(Z, L; Q)  1. From (12.7), one must have L ⊂ Z. This holds for any P ∈ v(Z) and any Q ∈ Z \ {P }.

Bibliography

Atiyah M. F. and Macdonald I. G., Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969. Barth W. P., Hulek K., Peters C. A. M. and Van de Ven A., Compact Complex Surfaces, Second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3 Folge. A Series of Modern Surveys in Mathematics, 4. Springer-Verlag, Berlin, 2004. Beauville A., Complex Algebraic Surfaces, Second edition, London Mathematical Society Student Texts 34. Cambridge University Press, Cambridge, 1996. Bourbaki N., Elements of Mathematics. General Topology, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. Caporaso L., Introduction to Moduli of Curves, School Pragmatic 2004 (course notes) Universit` a di Catania, August/September 2004, available at http:// www.mat.uniroma3.it/users/caporaso/modari.pdf. Catanese F., Moduli of Algebraic Surfaces, Pubbl. Dipartimento di Matematica dell’Univ. di Pisa, Pisa, 1986. Ciliberto C., An Undergraduate Primer in Algebraic Geometry, Unitext, 129. La Matematica per il 3 + 2. Springer, Cham, 2021. Clemens C. H. and Griffiths P. A., The intermediate Jacobian of the cubic threefold, Ann. Math., 95(2), 281–356, 1972. Cohn P. M., Algebraic Numbers and Algebraic Functions, Chapman and Hall Mathematics Series. Chapman & Hall, London, 1991. Cohn D., Little J. and O’Shea D., Ideals, Varieties and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra. Fourth edition. Undergraduate Texts in Mathematics. Springer, Cham, 2015. Dolgachev I., Introduction to Algebraic Geometry, Lecture Notes 2013, University of Michigan at Ann Arbor, available at http://www.math.lsa.umich.edu/ ∼idolga/631.pdf.

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Eisenbud D., Commutative Algebra. With a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. Fulton W., Algebraic Curves. An Introduction to Algebraic Geometry, 2008 (slightly modified version of the text Algebraic Curves. An Introduction to Algebraic Geometry, Advanced Book Classics. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989), available at http://www.math.lsa.umich.edu/∼ wfulton/CurveBook.pdf. Harris J., Algebraic Geometry. A First Course, Graduate Texts in Mathematics, 133. Springer–Verlag, New York, 1995. Hartshorne J., Algebraic Geometry, Graduate Texts in Mathematics, 52. Springer–Verlag, New York-Heidelberg, 1977. Hassett B., Introduction to Algebraic Geometry, Cambridge University Press, Cambridge, 2007. Hulek K., Elementary Algebraic Geometry, Student Mathematical Library, 20. American Mathematical Society, Providence, RI, 2003. Lang S., Algebra, Revised third edition. Graduate Texts in Mathematics, 211. Springer-Verlag, New York, 2002. Matsumura H., Commutative Algebra, Second edition. Mathematics Lecture Note Series, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. Milne J. S., (Basic First Course in) Algebraic Geometry, v. 6.02, 2017, available at http://www.jmilne.org/math/CourseNotes/ag.html. Miranda R., Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics, 5. American Mathematical Society, Providence, RI, 1995. Mumford D., The Red Book of Varieties and Schemes, Lecture Notes in Mathematics, 1358. Springer-Verlag, Berlin, 1988. Mumford D., Algebraic Geometry I. Complex Projective Varieties, Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reid M., Undergraduate Algebraic Geometry, London Mathematical Society Student Texts, 12. Cambridge University Press, Cambridge, 1988. Sernesi E., Appunti del corso di Geometria Algebrica, a.a. 1991/92, Dip. di Matematica Guido Castelnuovo, Univ. di Roma La Sapienza, 1992. Sernesi E., Deformations of Algebraic Schemes, A Series of Comprehensive Studies in Mathematics, 334. Springer–Verlag, 2006. Shafarevich I. R., Basic Algebraic Geometry 1. Varieties in Projective Space, Second edition, Springer–Verlag, Berlin, 1994. Ueno K., An Introduction to Algebraic Geometry, Translations of Mathematical Monographs, 166. American Mathematical Society, Providence, RI, 1997. Verra A., Lectures on Cremona Transformations, School in Algebraic Geometry (course notes), Universit` a di Catania, Politecnico di Torino, September 19–26, 2005, available at http://www.mat.uniroma3.it/users/verra/other. html.

Index

A

algebraically independent elements over a field, 23 a.c.c. on open sets, 115 ascending chain conditions (a.c.c.), 115 automorphism of an algebraic variety, 149

Aff(An ), 99 Aut(V ), 149 (affine) monoid of degree d, 208 affine charts of Pn , 88 affine cone over Y ⊂ Pn , 84 affine coordinate ring, 112 affine group of An , 99 affine hypersurface, 65 affine open covering, 165 affine open set of an algebraic variety, 158 affine rational curve with polynomial parametrization, 101, 157 affine space An K , 55 affine subspaces, 64 affine tangent space at a point of an affine variety, 249 affine twisted cubic, 102 affine variety, 114, 158 affinity, 98 algebra of regular functions on an open set, 128 algebraic affine set (AAS), 57 algebraic closure of a field, 1 algebraic projective set (APS), 82 algebraic subvariety, 132 algebraic variety, 115

B b(n, d), 36 Bs(L), 176 base locus of a linear system, 176 base locus of a linear system of hypersurfaces in Pn , 163 base point free linear series, 176 base point of a linear system, 176 Bezout’s theorem in A2 –weak form, 76 birational class of birationally equivalent algebraic varieties, 199 birational isomorphism, 197 birational maps, 197 birational varieties, 197 birationally equivalent algebraic varieties, 197 blow-up of Pn at P0 , 209 blow-up of a subvariety at a point, 213 297

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C codimc,Y (Z), 120 center of a projection, 203 chain of prime ideals, 48 closed algebraic set, 115 closed immersion, 151 closure of a subset in An , 68 codimension of a subvariety, 228 combinatorial codimension, 120 combinatorial dimension, 118 complete algebraic variety, 218 complete intersection, 236 complete linear series, 176 complete linear system of hypersurfaces of degree d is Pn , 163 constant pre-sheaf, 126 constructible set, 151, 244 contact of order greater than 1 at a point, 252 contracted ideal w.r.t. a ring homomorphism, 4 coordinate affine subspace, 64 coordinate affine subspaces, 64 coordinate axes of Pn , 87 coordinate axis of an affine space, 64 coordinate linear subspace of Pn , 87 coordinate vector of a point in An K , 55 coordinates of a point in An K , 55 Cremona group, 199 cuspidal plane cubic, 145 cuspidal singularity, 214 cyclic points, 144 cylinders, 67 D Δ(V ), 191 dimc (Y ), 118 dimK (K[X0 , . . . , Xn ]d ), 36 Div(W ), 174 decomposition of a polynomial in homogenous parts, 37 degenerate homography, 203 degree of a divisor, 174 degree of a field extension, 17

degree of a projective hypersurface, 100 degree of an affine hypersurface, 65 degree–g graded part of a G-graded ring, 31 degree-zero rational functions in X0 , . . . , Xn , 127 dehomogenized polynomial w.r.t. Xi , 39 dehomogenizing operator w.r.t. Xi , 39 dense open subset, 110 derivatives of polynomials in D[x], 7 descending chain conditions (d.c.c.), 115 descending chain condition on closed sets, 115 diagonal of V × V , 191 differential of a morphism at a point of an algebraic variety, 260 dimension of P(V ), 79 dimension of an algebraic variety, 225 direction of an affine subspace, 95 directrix of a cylinder, 67 discriminant in D[x], 12 divisor, 174 Dom(Φ), 196 domain of a linear system of hypersurfaces in Pn , 163 domain of a rational function, 132 domain of a rational map Φ, 196 dominant morphism, 150 dominant rational map, 196 doubly ruled quadric surface, 282 dual projective plane curve, 263 dual projective space, 87 E effective divisor, 174 elementary Cremona transformation of P2 , 215 elementary quadratic transformation of P2 , 215 ellipse, 144 elliptic curve, 199 elliptic plane cubic, 147

299

Index

equation of a divisor, 174 Euler’s identity, 37 Euler’s Lemma, 10 evaluation at P , 56 exact sequence of graded modules, 105 exceptional divisor of a blow-up at a point, 210 exceptional lines of elementary quadratic transformation of P2 , 287 extended ideal in localization, 45 extended ideal w.r.t. a ring homomorphism, 4 extension of scalars, 30 F fiber of a morphism over a point, 241 fiber-product, 193 field, 1 field extension, 17 field of rational functions K(x1 , . . . , xn ), 18 field of rational functions on an algebraic variety, 131 finite R-algebra, 17 finite morphism, 223 finite transcendence degree over a field, 24 finitely generated R-module, 16 finitely generated field extension, 18 finitely generated ideal, 2 fundamental affine open set of Y , 89 fundamental hyperplanes of Pn , 87 fundamental points of Pn , 80 fundamental points of elementary quadratic transformation of P2 , 287 fundamental theorem on regular and rational functions, 135 G grd , 176 Γϕ , 192 G–graded ring, 31 G-graded module, 41 generated R-module, 16

generators of a R-module, 16 generically finite rational map, 205 global sections of a pre-sheaf, 125 graded module, 41 graded ring, 31 graph of a morphism of algebraic varieties, 192 H ht(p), 48 HomK (R, S), 153 height of a prime ideal, 48 Hilbert “Nullstellensatz”-weak form, 70 Hilbert “Nullstellensatz”-strong form, 71 Hilbert’s basis theorem, 14 homogenized polynomial w.r.t. Xi , 39 homogeneous component of an element, 32 homogeneous coordinate ring, 112 homogeneous differential operators, 42 homogeneous elements of degree g, 31 homogeneous free resolution of the ideal of the standard projective twisted cubic, 105 homogeneous Hilbert “Nullstellensatz”-weak form, 85 homogeneous Hilbert “Nullstellensatz”-strong form, 86 homogeneous homomorphism of graded rings, 41 homogeneous homomorphism of rings of degree d, 41 homogeneous ideal of a subset of Pn , 83 homogeneous localization of a graded ring w.r.t. a multiplicative system, 44 homogeneous localization w.r.t. a homogeneous, non-nilpotent element, 47

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A First Course in Algebraic Geometry and Algebraic Varieties

homogeneous localization with respect to a homogeneous, prime ideal, 47 homogeneous morphism of degree 0, 41 homogeneous parametric representation of a projective subspace, 97 homogeneous polynomial, 35 homogeneous Study’s principle, 100 homogenizing operator w.r.t. Xi , 39 homography, 95 hyperbola, 144 hyperbolic paraboloid, 151, 282 hyperplane at infinity of an affine chart, 88 I (I : J), 52 IY (W ), 135 Isom(V, W ), 149 ideal, 2 ideal of a subset of An , 68 ideal of a subvariety of an algebraic variety, 135 ideal of germs of regular functions vanishing at a point, 133 improper points of An , 88 improper points of Y ⊂ An , 89 indeterminacy locus of a morphism, 163 integral closure of a subring in a ring, 20 integral domain, 1 integral element over a subring, 19 internal projection, 282 intersection multiplicity, 175 intersection multiplicity at a point P with a line, 252 intersection multiplicity of a line at a point, 175 intersection of of ideals, 3 intersection of projective subspaces, 92 involution, 176

irreducibility of fibrations with irreducible fibers of constant dimension and with irreducible base, 246 irreducible affine hypersurface, 74 irreducible components, 117 irreducible components of a divisor, 174 irreducible components of an affine hypersurface, 74 irreducible divisor, 174 irreducible topological space, 109 irredundant decomposition, 117 irrelevant ideal of a non-negatively graded ring, 32 isomorphic algebraic varieties, 149 isomorphism of algebraic varieties, 149 J JAS , 45 Jacobian matrix, 250 K K-algebra homomorphisms, 153 K-algebra of rational functions defined in a subvariety, 133 K-derivation from OV,P to K, 256 K[], 258 K dim(R), 48 Krull–dimension of a ring, 48 L Ln,d , 175 L(−P ), 176 LP(V ),d , 175 leading coefficient of f (x), 6 length of a chain of prime ideals, 48 linear envelope of a subset in P(V ), 92 linear series on P1 , 176 linear system of dimension r of hypersurfaces of degree d in Pn , 163 linear system of divisors, 175

301

Index

linearly independent points in P(V ), 92 local definition of dimension, 232 local isomorphism of local rings, 137 local property for morphisms, 164 local ring, 46 local ring of a subvariety W in a variety Y , 135 localization homomorphism, 43 localization of a ring w.r.t. a multiplicative system, 43 localization w.r.t. a non-nilpotent element, 47 localization with respect to a prime ideal, 47 locally closed algebraic set, 115 locally closed subset, 114 locally closed subset of Pn , 89 locally-closed immersion, 151 L¨ uroth problem, 206 M Morph(V, W ), 149 main theorem of elimination theory, 220 map satisfying the property of being local, 183 matrix equation of the standard projective twisted cubic, 102 maximal chain of prime ideals, 48 maximal ideal, 2 minimal prime ideals, 233 model of a birational class of algebraic varieties, 199 monic polynomial, 6 morphism of algebraic varieties, 149 multi-tensor product, 29 multiple factor of a polynomial, 6 multiple factor of a polynomial in D[x], 11 multiplicative system in a ring, 42 multiplicity of a root of a polynomial f ∈ D[x], 7

multiplicity of an irreducible hypersurface in a divisor, 174 N nilpotent element, 3 nilradical of a ring, 52 nodal singularity, 215 Noetherian ring, 13 Noetherian topological space, 115 non-degenerate subset of An , 95 non-degenerate subset of P(V ), 92 non-negatively graded ring, 32 non-singular affine variety, 250 non-singular point, 250, 257 non-singular projective variety, 253 numerical projective space, 79 O O(P(V ), P(W )), 96 OY , 128 OY,W , 133 (OY,W , mY,W ), 135 open immersion, 151 open Riemann surface of a complex conic, 64 open set of definition of a linear system of hypersurfaces in Pn , 163 open set of definition of a rational function, 132 open set of definition of a rational map Φ, 196 open subset in ZarAn , 62 orthogonal subspace to a subspace of P(V ), 93 P PGL(V ), 96 (projective) Grassmann formula, 92 (projective) monoid of degree d, 208 parabola, 144 parallel projections of An , 204 parametric representation of a projective twisted cubic, 106

302

A First Course in Algebraic Geometry and Algebraic Varieties

parametric representation of a subspace in Pn , 93 parametric representation of an affine subspace, 65, 95 partial derivatives of polynomials in D[x1 , . . . , xn ], 7 pencil of effective divisors, 176 plane nodal cubic, 208 point in An K , 55 points at infinity of An , 88 points at infinity of Y ⊂ An , 89 points in general linear position in P(V ), 92 points in general position, 97 polynomial map of affine varieties, 157 polynomial parametrization of the parabola, 278 power of an ideal, 3 pre-sheaf on a topological space, 125 prime ideal, 2 principal ideal, 2 principal ideal domain (PID), 4 principal open affine set, 66 principal open affine sets of Pn , 88 principal open projective sets, 100 principal open set of P(V ), 83 principal open subset associated to a regular function, 130 principal ring, 4 product of ideals, 3 product of morphisms, 189 product of two AAS’s, 66 projection of An onto one of its coordinate axis, 152 projection of An onto the coordinates I = {i1 , i2 , . . . , im }, 154 projection of An with center an affine subspace, 204 projection of Pn to Pr with center a linear subspace, 164 projection of a projective space from a linear subspace, 203 projection of a variety on a given subspace, 222 projection of an algebraic variety, 204

projective closure, 89 projective cone of degree d, 263 projective cone over Z ⊂ Pn , 99 projective hypersurface, 99 projective space P(V ), 79 projective subspace generated by a subset of P(V ), 92 projective subspace of P(V ), 91 projective tangent space, 253 projective twisted cubic, 106 projective variety, 114 projectively equivalent subsets, 98 projectively isomorphic projective spaces, 96 projectivities of Pn , 164 projectivity, 96 proper transform of a subvariety via blow-up at a point, 213 proportionality relation, 79 pure dimension, 119, 227 pure topological space, 119 purely trascendental extension of a field, 25 Q Q0 (S), 35 Q0 (S(Y )), 139 Q(n) , 23, 127 (n) Q0 , 127 quadric cone surface, 239 quadric cylinders in A3 , 67 quasi-affine variety, 114 quasi-projective variety, 114 quotient field, 2 quotient ideal, 52 R R-algebra, 17 R-algebra homomorphism, 17 R-algebra of finite type, 18 R-bilinear map of R-modules, 27 R-module, 16 R-module homomorphism, 16 R-submodule, 16 Rf , 47

303

Index

R([f ]) , 47 R(p) , 47 Rp , 47 Rabinowithch’s trick, 71 radical ideal, 3 radical of an ideal, 3 rank-four quadric surface, 206, 282 rank-three quadric surface, 239 rational functions in x1 , . . . , xn , 127 rational functions on An , 127 rational functions on Y , 130 rational map of algebraic varieties, 196 rational normal curve, 170 rational parametrization of a conic, 76 rational parametrization of the hyperbola, 277 rational variety, 205 reduced divisor, 174 reduced equation of a projective hypersurface, 100 reduced equation of an affine hypersurface, 65 reduced ring, 3 reducible topological space, 109 regular function at a point, 127 regular function of a locally-closed algebraic set, 127 regular local ring, 261 regular sequence, 235 relation of algebraic dependence over a field, 23 representative morphism of a rational map over an open set, 196 restricted sheaves to open subsets, 126 restriction of scalars, 29 resultant of two polynomials in D[x], 10 resultant polynomial w.r.t. an indeterminate, 12 reversing-inclusion maps, 69 Riemann sphere, 269 ring homomorphism, 1 ring of differential operators, 42

ring of dual numbers over a field, 258 ring of germs of regular functions at a point, 133 ring of polynomials R[x], 2 ring of polynomials R[x1 , . . . , xn ], 2 ring of total fractions of a ring, 44 ringed space, 142 root of a polynomial f ∈ D[x], 7 roots of a homogeneous polynomial F (X0 , X1 ), 41 rulings of a rank-four quadric surface, 282 S σn,m , 184 Σ1,1 , 282 Σn,m , 184 Sing(V ), 250, 253, 257 Sm(V ), 250, 253, 257 secant projection, 215 sections of a pre-sheaf on an open set, 125 Segre map of indexes n and m, 184 Segre variety of indexes n and m, 184 semi-cubic parabola, 145 set of APS’s of P(V ), 82 set of generators for an ideal, 2 set of homogeneous elements of a subset, 32 set-theoretically complete intersection, 236 sheaf on a topological space, 126 simple field extension, 18 simple intersection at a point with a line, 252 simple point, 250, 257 simple ring, 2 simple root of a polynomial f ∈ D[x], 7 singular affine variety, 250 singular algebraic variety, 258 singular locus, 250 singular point, 250, 257 singular projective variety, 253 smooth affine variety, 250

304

A First Course in Algebraic Geometry and Algebraic Varieties

smooth algebraic variety, 258 smooth locus, 250 smooth plane cubic, 199 smooth point, 250, 257 smooth projective variety, 253 Specm of a ring, 137 Spec of a ring, 48 standard affine twisted cubic, 102 standard projective twisted cubic, 102 standard projective twisted cubic is determinantal, 102 st standard Veronese morphism νn,d , 166 st , 166 standard Veronese variety Vn,d standard Veronese variety of indexes n and d, 166 stereographic projection of a monoid from its vertex, 209 stereographic projection of a rank-four quadric surface to a plane, 207 stereographic projection of an irreducible conic on P1 , 205 structural sheaf of an algebraic variety, 131 Study’s principle, 74 sub-field of degree-zero, homogeneous fractions of a graded ring, 35 sum of ideals, 2 support of a divisor, 174 Sylvester matrix of two polynomials, 10 symmetric algebra over V , 42 system of equations for an AAS, 57 system of homogeneous equations for an APS, 82 syzygies for the standard projective twisted cubic, 104 syzygy module of the standard projective twisted cubic, 104 T trdegK (F), 24 tangential projection, 214 tensor product of R-modules, 28

topological codimension, 120 topological dimension, 118 total transform of a subvariety via blow-up at a point, 212 transcendence degree of a field extension, 24 transcendence basis of a field extension, 23 transcendence degree of an integral algebra over a field, 26 transverse intersection, 252 U UΦ , 196 unique factorization domain (UFD), 5 unirational variety, 205 unit, 1 upper-semicontinuity of the fiber dimension, 245 V v(Z), 263 st , 166 Vn,d vector of homogeneous coordinates, 80 Veronese morphism νn,d , 166 Veronese surface, 171 Veronese variety Vn,d , 166 Veronese variety of indexes n and d, 166 vertex of a projective cone, 263 vertices of the pyramid of Pn , 80 W web of effective divisors, 176 X x ⊗ y, 28 Z Zarn p , 82 ZarP(V ),Y , 83 ZarP(V ) , 83 (Zariski) closed subsets of An , 62

Index

Zariski tangent space at a point of an algebraic variety, 256 Zariski tangent space of a local Noetherian K-algebra, 258 Zariski topology of An , 62 Zariski topology of a subset of An , 62 Zariski topology of a subset of P(V ), 83

305

Zariski topology on P(V ), 82 Zariski topology on Pn , 82 zero-locus of a regular function, 129 zero-set, 56 zero-set in Pn , 82 zero-set of a homogenous polynomial in P(V ), 81