GRADUATE STUDIES IN MATHEMATICS 188 Introduction to Algebraic Geometry Steven Dale Cutkosky 10.1090/gsm/188 GRADUATE STUDIES IN MATHEMATICS 188 Introduction to Algebraic Geometry Steven Dale Cutkosky EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed (Chair) Gigliola Staffilani Jeff A. Viaclovsky 2010 Mathematics Subject Classification. Primary 14-01. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-188 Library of Congress Cataloging-in-Publication Data Names: Cutkosky, Steven Dale, author. Title: Introduction to algebraic geometry / Steven Dale Cutkosky. Other titles: Algebraic geometry Description: Providence, Rhode Island : American Mathematical Society, [2018] | Series: Gradu- ate studies in mathematics ; volume 188 | Includes bibliographical references and index. Identifiers: LCCN 2017045552 | ISBN 9781470435189 (alk. paper) Subjects: LCSH: Geometry, Algebraic. | AMS: Algebraic geometry – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA564 .C8794 2018 | DDC 516.3/5–dc23 LC record available at https://lccn.loc.gov/2017045552 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2018 by the author. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 232221201918 To Hema, Ashok, and Maya Contents Preface xi Chapter 1. A Crash Course in Commutative Algebra 1 §1.1. Basic algebra 1 §1.2. Field extensions 6 §1.3. Modules 8 §1.4. Localization 9 §1.5. Noetherian rings and factorization 10 §1.6. Primary decomposition 13 §1.7. Integral extensions 16 §1.8. Dimension 19 §1.9. Depth 20 §1.10. Normal rings and regular rings 22 Chapter 2. Affine Varieties 27 §2.1. Affine space and algebraic sets 27 §2.2. Regular functions and regular maps of affine algebraic sets 33 §2.3. Finite maps 40 §2.4. Dimension of algebraic sets 42 §2.5. Regular functions and regular maps of quasi-affine varieties 48 §2.6. Rational maps of affine varieties 58 Chapter 3. Projective Varieties 63 §3.1. Standard graded algebras 63 v vi Contents §3.2. Projective varieties 67 §3.3. Grassmann varieties 73 §3.4. Regular functions and regular maps of quasi-projective varieties 74 Chapter 4. Regular and Rational Maps of Quasi-projective Varieties 87 §4.1. Criteria for regular maps 87 §4.2. Linear isomorphisms of projective space 90 §4.3. The Veronese embedding 91 §4.4. Rational maps of quasi-projective varieties 93 §4.5. Projection from a linear subspace 95 Chapter 5. Products 99 §5.1. Tensor products 99 §5.2. Products of varieties 101 §5.3. The Segre embedding 105 §5.4. Graphs of regular and rational maps 106 Chapter 6. The Blow-up of an Ideal 111 §6.1. The blow-up of an ideal in an affine variety 111 §6.2. The blow-up of an ideal in a projective variety 120 Chapter 7. Finite Maps of Quasi-projective Varieties 127 §7.1. Affine and finite maps 127 §7.2. Finite maps 131 §7.3. Construction of the normalization 135 Chapter 8. Dimension of Quasi-projective Algebraic Sets 139 §8.1. Properties of dimension 139 §8.2. The theorem on dimension of fibers 141 Chapter 9. Zariski’s Main Theorem 147 Chapter 10. Nonsingularity 153 §10.1. Regular parameters 153 §10.2. Local equations 155 §10.3. The tangent space 156 §10.4. Nonsingularity and the singular locus 159 §10.5. Applications to rational maps 165 Contents vii §10.6. Factorization of birational regular maps of nonsingular surfaces 168 §10.7. Projective embedding of nonsingular varieties 170 §10.8. Complex manifolds 175 Chapter 11. Sheaves 181 §11.1. Limits 181 §11.2. Presheaves and sheaves 185 §11.3. Some sheaves associated to modules 196 §11.4. Quasi-coherent and coherent sheaves 200 §11.5. Constructions of sheaves from sheaves of modules 204 §11.6. Some theorems about coherent sheaves 209 Chapter 12. Applications to Regular and Rational Maps 221 §12.1. Blow-ups of ideal sheaves 221 §12.2. Resolution of singularities 225 §12.3. Valuations in algebraic geometry 228 §12.4. Factorization of birational maps 232 §12.5. Monomialization of maps 236 Chapter 13. Divisors 239 §13.1. Divisors and the class group 240 §13.2. The sheaf associated to a divisor 242 §13.3. Divisors associated to forms 249 §13.4. Calculation of some class groups 249 §13.5. The class group of a curve 254 §13.6. Divisors, rational maps, and linear systems 259 §13.7. Criteria for closed embeddings 264 §13.8. Invertible sheaves 269 §13.9. Transition functions 271 Chapter 14. Differential Forms and the Canonical Divisor 279 §14.1. Derivations and K¨ahler differentials 279 §14.2. Differentials on varieties 283 §14.3. n-forms and canonical divisors 286 Chapter 15. Schemes 289 §15.1. Subschemes of varieties, schemes, and Cartier divisors 289 §15.2. Blow-ups of ideals and associated graded rings of ideals 293 viii Contents §15.3. Abstract algebraic varieties 295 §15.4. Varieties over nonclosed fields 296 §15.5. General schemes 296 Chapter 16. The Degree of a Projective Variety 299 Chapter 17. Cohomology 307 §17.1. Complexes 307 §17.2. Sheaf cohomology 308 §17.3. Cechˇ cohomology 310 §17.4. Applications 312 §17.5. Higher direct images of sheaves 320 §17.6. Local cohomology and regularity 325 Chapter 18. Curves 333 §18.1. The Riemann-Roch inequality 334 §18.2. Serre duality 335 §18.3. The Riemann-Roch theorem 340 §18.4. The Riemann-Roch problem on varieties 343 §18.5. The Hurwitz theorem 345 §18.6. Inseparable maps of curves 348 §18.7. Elliptic curves 351 §18.8. Complex curves 358 §18.9. Abelian varieties and Jacobians of curves 360 Chapter 19. An Introduction to Intersection Theory 365 §19.1. Definition, properties, and some examples of intersection numbers 366 §19.2. Applications to degree and multiplicity 375 Chapter 20. Surfaces 379 §20.1. The Riemann-Roch theorem and the Hodge index theorem on a surface 379 §20.2. Contractions and linear systems 383 Chapter 21. Ramification and Etale´ Maps 391 §21.1. Norms and Traces 392 §21.2. Integral extensions 393 §21.3. Discriminants and ramification 398 Contents ix §21.4. Ramification of regular maps of varieties 406 §21.5. Completion 408 §21.6. Zariski’s main theorem and Zariski’s subspace theorem 413 §21.7. Galois theory of varieties 421 §21.8. Derivations and K¨ahler differentials redux 424 §21.9. Etale´ maps and uniformizing parameters 426 §21.10. Purity of the branch locus and the Abhyankar-Jung theorem 433 §21.11. Galois theory of local rings 438 §21.12. A proof of the Abhyankar-Jung theorem 441 Chapter 22. Bertini’s Theorems and General Fibers of Maps 451 §22.1. Geometric integrality 452 §22.2. Nonsingularity of the general fiber 454 §22.3. Bertini’s second theorem 457 §22.4. Bertini’s first theorem 458 Bibliography 469 Index 477 Preface This book is an introductory course in algebraic geometry, proving most of the fundamental classical results of algebraic geometry. Algebraic geometry combines the intuition of geometry with the preci- sion of algebra. Starting with geometric concepts, we introduce machinery as necessary to model important ideas from algebraic geometry and to prove fundamental results. Emphasis is put on developing facility with connect- ing geometric and algebraic concepts. Examples are constructed or cited illustrating the scope of these results. The theory in this book is developed in increasing sophistication, giving (and refining) definitions as required to accommodate new geometric ideas. We work as much as possible with quasi-projective varieties over an algebraically closed field of arbitrary characteristic. This allows us to inter- pret varieties through their function fields. This approach and the use of methods of algebraic number theory in algebraic geometry have been cen- tral to algebraic geometry at least since the time of Dedekind and Weber (Theorie der algebraischen Functionen einer Ver¨anderlichen [46], translated in [47]). By interpreting the geometric concept of varieties through their regular functions, we are able to use the techniques of commutative algebra. Differences between the theory in characteristic 0 and positive charac- teristic are emphasized in this book. We extend our view to schemes, al- lowing rings with nilpotents, to study fibers of regular maps and to develop intersection theory. We discuss the cases of nonclosed ground fields and nonseparated schemes and some of the extra considerations which appear in these situations. A list of exercises is given at the end of many sections and chapters. xi xii Preface The classic textbooks Basic Algebraic Geometry [136] by Shafarevich, Introduction to Algebraic Geometry [116] by Mumford, and Algebraic Geometry [73] by Hartshorne, as well as the works of Zariski, Abhyankar, Serre, and Grothendieck, have been major influences on this book. The necessary commutative algebra is introduced and reviewed, begin- ning with Chapter 1, “A Crash Course in Commutative Algebra”. We state definitions and theorems, explain concepts, and give examples from com- mutative algebra for everything that we will need, proving some results and giving a few examples, but mostly giving references to books on commuta- tive algebra for proofs.