52 Graduate Texts in Mathematics Editorial Board S Axler F.W Gehring K.A Ribet Graduate Texts in Mathematics most recent titles in the GTM series 129 FULTON/HARRIS Representation Theory: A First Course Readings in Mathematics 130 DODSON/PoSTON Tensor Geometry 131 LAM A First Course in Noncommutative Rings 132 BEARDON Iteration of Rational Functions 133 HARRIS Algebraic Geometry: A First Course 134 RoMAN Coding and Information Theory 135 ROMAN Advanced Linear Algebra 136 ADKINS/WEINTRAUB Algebra: An Approach via Module Theory 137 AxLERIBouRDoN/RAMEY Harmonic Function Theory 138 CoHEN A Course in Computational Algebraic Number Theory 139 BREDON Topology and Geometry 140 AUBIN Optima and Equilibria An Introduction to Nonlinear Analysis 141 BECKERIWEISPFENNING/KREDEL Grtibner Ba~es A Computational Approach to Commutative Algebra 142 LANG Real and Functional Analysis 3rd ed 143 DOOB Measure Theory 144 DENNISIFARB Noncommutative Algebra 145 VICK Homology Theory An Introduction to Algebraic Topology 2nd ed 146 BRIDGES Computability: A Mathematical Sketchbook 147 ROSENBERG Algebraic K-Theory and Its Applications 148 ROTMAN An Introduction to the Theory of Groups 4th ed 149 RATCLIFFE Foundations of Hyperbolic Manifolds 150 EISENBUD Commutative Algebra with a View Toward Algebraic Geometry 151 SILVERMAN Advanced Topics in the Arithmetic of Elliptic Curves 152 ZIEGLER Lectures on Polytopes 153 FULTON Algebraic Topology: A First Course 154 BROWN/PEARCY An Introduction to Analysis 155 KASSEL Quantum Groups 156 KECHRIS Cla~sical Descriptive Set Theory 157 MALLIAVIN Integration and Probability 158 ROMAN Field Theory 159 CONWAY Functions of One Complex Variable II 160 LANG Differential and Riemannian Manifolds 161 BORWEINIERDEL Yl Polynomials and Polynomial Inequalities 162 ALPERIN/BELL Groups and Representations 163 DIXON/MORTIMER Permutation Groups 164 NATHANSON Additive Number Theory: The Classical Bases 165 NATHANSON Additive Number Theory: Inverse Problems and the Geometry of Sumsets 166 SHARPE Differential Geometry: Cartan's Generalization of Klein's Erlangen Program 167 MORANDI Field and Galois Theory 168 EWALD Combinatorial Convexity and Algebraic Geometry 169 BHATIA Matrix Analysis 170 BREDON Sheaf Theory 2nd ed 171 PETERSEN Riemannian Geometry 172 REMMERT Classical Topics in Complex Function Theory 173 DIESTEL Graph Theory 174 BRIDGES Foundations of Real and Abstract Analysis 175 LICKORISH An Introduction to Knot Theory 176 LEE Riemannian Manifolds 177 NEWMAN Analytic Number Theory 178 CLARKE/LEDYAEV/STERN/W OLENSKI Nonsmooth Analysis and Control Theory 179 DOUGLAS Banach Algebra Techniques in Operator Theory 2nd ed 180 SRIVASTAVA A Course on Borel Sets 181 KRESS Numerical Analysis 182 WALTER Ordinary Differential Equations 183 MEGGINSON An Introduction to Banach Space Theory 184 BOLLOBAS Modem Graph Theory 185 COXILITILEIO'SHEA Using Algebraic Geometry 186 RAMAKRISHNAN/V ALENZA Fourier Analysis on Number Fields 187 HARRIS/MORRISON Moduli of Curves 188 GOLDBLATT Lectures on the Hyperreals: An Introduction to Nonstandard Analysis 189 LAM Lectures on Modules and Rings 190 ESMONDEIMURTY Problems in Algebraic Number Theory Robin Hartshorne Algebraic Geometry ~Springer Robin Hartshorne Department of Mathematics University of California Berkeley, California 94720 USA Editorial Board S Axler Mathematics Department San Francisco State Universi~y San Francisco, CA 94132 USA K.A Ribet Department of Mathematics University of California at Berkeley Berkeley, CA 94720-3840 USA ribet@ math berkeley edu Mathematics Subject Classification (2000): 13-xx, 14Al0, 14A15, 14Fxx, 14Hxx, 14Jxx Library of Congress Cataloging-in-Publication Data Hartshorne, Robin Algebraic geometry (Graduate texts in mathematics: 52) Bibliography: p Includes index Geometry, Algebraic I Title II Series 77-1177 516'.35 QA564.H25 ISBN 978-1-4757-3849-0 (eBook) ISBN 978-1-4419-2807-8 DOI 10.1007/978-1-4757-3849-0 Printed on acid-free paper © 1977 Springer Science+Business Media, Inc Softcover reprint of the hardcover I st edition 1977 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights (ASC/SBA) 15 14 springeronline.com For Edie, Jonathan, and Berifamin Preface This book provides an introduction to abstract algebraic geometry using the methods of schemes and cohomology The main objects of study are algebraic varieties in an affine or projective space over an algebraically closed field; these are introduced in Chapter I, to establish a number of basic concepts and examples Then the methods of schemes and cohomology are developed in Chapters II and III, with emphasis on applications rather than excessive generality The last two chapters of the book (IV and V) use these methods to study topics in the classical theory of algebraic curves and surfaces The prerequisites for this approach to algebraic geometry are results from commutative algebra, which are stated as needed, and some elementary topology No complex analysis or differential geometry is necessary There are more than four hundred exercises throughout the book, offering specific examples as well as more specialized topics not treated in the main text Three appendices present brief accounts of some areas of current research This book can be used as a textbook for an introductory course in algebraic geometry, following a basic graduate course in algebra I recently taught this material in a five-quarter sequence at Berkeley, with roughly one chapter per quarter Or one can use Chapter I alone for a short course A third possibility worth considering is to study Chapter I, and then proceed directly to Chapter IV, picking up only a few definitions from Chapters II and Ill, and assuming the statement of the RiemannRoch theorem for curves This leads to interesting material quickly, and may provide better motivation for tackling Chapters II and III later The material covered in this book should provide adequate preparation for reading more advanced works such as Grothendieck [EGA], [SGA], Hartshorne [5], Mumford [2], [5], or Shafarevich [1] Vll Preface Acknowledgements In writing this book, I have attempted to present what is essential for a basic course in algebraic geometry I wanted to make accessible to the nonspecialist an area of mathematics whose results up to now have been widely scattered, and linked only by unpublished "folklore." While I have reorganized the material and rewritten proofs, the book is mostly a synthesis of what I have learned from my teachers, my colleagues, and my students They have helped in ways too numerous to recount I owe especial thanks to Oscar Zariski, J.-P Serre, David Mumford, and Arthur Ogus for their support and encouragement Aside from the "classical" material, whose origins need a historian to trace, my greatest intellectual debt is to A Grothendieck, whose treatise [EGA] is the authoritative reference for schemes and cohomology His results appear without specific attribution throughout Chapters II and III Otherwise I have tried to acknowledge sources whenever I was aware of them In the course of writing this book, I have circulated preliminary versions of the manuscript to many people, and have received valuable comments from them To all of these people my thanks, and in particular to J.-P Serre, H Matsumura, and Joe Lipman for their careful reading and detailed suggestions I have taught courses at Harvard and Berkeley based on this material, and I thank my students for their attention and their stimulating questions I thank Richard Bassein, who combined his talents as mathematician and artist to produce the illustrations for this book A few words cannot adequately express the thanks I owe to my wife, Edie Churchill Hartshorne While I was engrossed in writing, she created a warm home for me and our sons Jonathan and Benjamin, and through her constant support and friendship provided an enriched human context for my life For financial support during the preparation of this book, I thank the Research Institute for Mathematical Sciences of Kyoto University, the National Science Foundation, and the University of California at Berkeley August 29, 1977 Berkeley, California viii ROBIN HARTSHORNE Contents Introduction Xlll CHAPTER I Varieties I Affine Varieties Projective Varieties Morphisms Rational Maps Nonsingular Varieties Nonsingular Curves Intersections in Projective Space ·what Is Algebraic Geometry? 14 24 31 39 47 55 CHAPTER II Schemes Sheaves Schemes First Properties of Schemes Separated and Proper Morphisms Sheaves of Modules Divisors Projective Morphisms Differentials Formal Schemes 60 60 69 82 95 108 129 149 172 190 CHAPTER III Cohomology Derived Functors Cohomology of Sheaves Cohomology of a Noetherian Affine Scheme 201 202 206 213 ix Contents 10 11 12 Cech Cohomology The Cohomology of Projective Space Ext Groups and Sheaves The Serre Duality Theorem Higher Direct Images of Sheaves Flat Morphisms Smooth Morphisms The Theorem on Formal Functions The Semicontinuity Theorem CHAPTER IV Curves Riemann-Roch Theorem Hurwitz's Theorem Embeddings in Projective Space Elliptic Curves The Canonical Embedding Classification of Curves in P" 218 225 233 239 250 253 268 276 281 293 294 299 307 316 340 349 CHAPTER V Surfaces Geometry on a Surface Ruled Surfaces Monoidal Transformations The Cubic Surface in P" Birational Transformations Classification of Surfaces APPENDIX A Intersection Theory I Intersection Theory Properties of the Chow Ring Chern Classes The Riemann-Roch Theorem Complements and Generalizations APPENDIX B Transcendental Methods X The Associated Complex Analytic Space Comparison of the Algebraic and Analytic Categories When is a Compact Complex Manifold Algebraic? Kahler Manifolds The Exponential Sequence 356 357 369 386 395 409 421 424 425 428 429 431 434 438 438 440 441 445 446 Index Cubic curve (cont.) through points determines a 9th, 400 Cubic surface in pJ, 356, 395-409 ample divisors on, 405 as P with points blown up, 400 canonical sheaf of, 184, 401 curves on, 401, 406-409 Picard group of, 136, 401 27 lines on, 402-406 Cubic threefold is not rational, 184 Cup-product, 453, 454, 456 Curve, 105, 136 See also Abstract nonsingular curve; Nonsingular curve; Plane curve affine, 4, 7, 8, 47, 297, 298, 385 ample divisor on, 156, 307, 308, 372 any two homeomorphic, 31 behavior under monoidal transformation, 388-394 birational to a plane curve with nodes, 314 can be embedded in p3, 310 classification of, 56, 341, 345-347 complete intersection, 38, 342, 346, 352, 355 complete~ projective, 44, 136, 232, 294 cubic See Cubic curve definition for Ch II, 105 definition for Ch IV, 294 definition for Ch V, 357 divisors on, 129, 136-140, 294 dual, 54, 304 elliptic See Elliptic curve elliptic quartic, 38, 315, 353 equivalence of singularities, 393 exceptional See Exceptional curve existence for all g ;; 0, 294, 385, 394 genus bounded by degree, 315,351,407, 408 genus of See Genus genus of normalization, 393 hyperelliptic See Hyperelliptic curve in P 3, classification of, 349-355, 354 (Fig), 409 invariant ~P of a singularity, 298, 393-395 locally free sheaf on, 369, 370, 372, 376, 378, 379, 384, 385 of degree 2, 315, 353 See also Conic of degree 3, 159, 315, 353 See also Cubic curve of degree 4, 159, 309, 315, 342, 353, 355, 407 See also Quartic curve of degree 5, 348, 353, 355 of degree 6, 342, 350, 353 of degree 7, 353, 406 of degree 8, 346, 409 of degree 9, 354, 355, 409 of genus 0, 297, 345 See also Rational curve of genus I See Elliptic curve 482 of genus 2, 298, 304, 309, 315, 341, 347, 355 See also Hyperelliptic curve of genus 3, 342, 346-349 of genus 4, 342, 346, 348 of genus 5, 346, 348, 353, 406 of genus 6, 348, 409 of genus 10, 354, 409 of genus II, 355, 409 on a cubic surface, 401, 406-409 on a quadric surface See Quadric surface, curves on over a finite field, 339, 368, 458 See also Riemann hypothesis over R, product of, 44, 338, 367, 368 rational See Rational curve ruled surface over See Ruled surface singularities, 35, 36 (Fig), 38, 386, 393 See also Cusp; Node; Tacnode singular, Picard group of, 148 strange, 311, 316 trigonal, 345 twisted cubic See Twisted cubic curve twisted quartic See Rational quartic curve zeta function of, 458 Cusp, 36 (Fig), 37, 39, 298, 305, 392 (Fig), 394 See also Cuspidal cubic curve higher order, 37 Cuspidal cubic curve, 21, 171, 276 as a projection, 22, 266 blown up, 31, 392 (Fig) divisor class group of, 142, 148 is not normal, 23 is rational, 30 Cycle, 425 associated to a closed subscheme, 425 cohomology class of, 435, 454 homological equivalence of, 435, 444 of dimension zero, 437 rational equivalence of, 425, 426, 436, 454 Decomposable locally free sheaf, 376, 378, 383, 384 Dedekind domain, 40, 41, 58, 132 Deformation, 89, 90 (Fig), 188, 267 See also Family Deformation theory, 265 Degree of a coherent sheaf on a curve, 149, 372 of a divisor on a curve, 137, 142, 294 of a divisor on a projective variety, 132, 146 of a finite morphism of curves, 137, 298 of a hypersurface, 52 of a linear system on a curve, 159 of an intersection, 53 of a plane curve, 4, 54 of a projective variety, 47, 52, 57, 250, 309, 366 of a zero-cycle, 426, 428 Index Deligne, Pierre, 217,249, 289, 317, 347, 449, 452 Del Pezzo surface, 401, 408 6-functor, 205, 219, 234, 238, 240, 243, 282 Demazure, M., 452 Density of a set of primes, 334, 339 Depth, 184, 237, 243, 264 cohomological interpretation of, 217 Derivation, 172, 189 Derivative, 31, 300 Derived functor, 201-206 cohomology, 207, 211, 439 See also Cohomology Ext, 233 higher direct images, R'i., 250 See also Higher direct image sheaf local cohomology modules, H~(M), 217 Determinant of a coherent sheaf, 149, 306 Deuring, M., 331, 334 Diagonal, 24, 48 closed ~ scheme separated, 96 homomorphism, 96, 173 morphism, 96, 99, 107, 175, 427 reduction to the, 427, 428 self-intersection of, 367, 368, 437, 450 Diagonalized bilinear form, 364 Diagram-chasing, 203 Difference polynomial, 49 Differentiable structures on a sphere, 421 Differential form See Differentials Differential geometry, 311, 438, 445 Differentials, 57, 172-190 See also Canonical sheaf Kahler, 172-175 module free ~ regular local ring, 174 of a polynomial ring, 173 on An, 176 on a product, 187 on P", 176 residues of, 247 sheaf locally free~ nonsingular variety, 177, 178, 276 sheaf of, 175-177, 219, 247, 268, 295, 300 sheaf of q-forms, 01/k• 190, 247, 249 Dilatation, 386 Dimension equal to transcendence degree, of a linear system, 157, 295, 357, 424 See also Riemann-Roch problem of An is n, of a projective variety, 10, 57 of a ring, 6, 86 of a scheme, 86, 87, 94 of a special linear system, 341 See also Clifford's theorem of a topological space, 5, 8, 208 of fibres of a morphism, 95, 256, 257, 269 of intersections, 48 of P" is n, 12 relative, 95 Diophantine equation, 335, 340 Direct image cycle, 425 divisor, 306, 436 sheaf, f.:F, 65, 109, 115, 123, 124, 250 See also Higher direct image sheaf Direct limit, lim, 66, 72, 109, 208, 209 > Direct product, 82, 109 See also Product Direct sum, $, 66, 109 Dirichlet's minimum principle, 441 Dirichlet's theorem, 335, 339 Discrete valuation ring, 40, 42, 45, 107, 108, 258, 325 See also Valuation; Valuation ring center of, on a curve, 137 of a prime divisor, 130 set of See Abstract nonsingular curve spectrum of, 74, 93, 95, 124 Disjoint union, 80 Divisor, 57, 129-149 See also Cartier divisor; Divisor class group; Invertible sheaf; Wei) divisor algebraic equivalence of, 140, 367, 369 associated to an invertible sheaf, 144, 145, 157, 294, 425 cohomology class of, 367, 418 degree of, 132, 137, 142, 146, 294 effective, 130 group of all, Div(X), 130, 357 inverse image of, j*, 135, 137, 299 linear equivalence of, 57, 131, 141, 294, 367, 425, 426 locally principal, 142 numerical equivalence of, 364, 367, 369 of an elliptic function, 327 of a rational function, 130, 131, 294 on a curve, 129, 136-140, 294 on a surface, 135, 357 prime, 130 principal, 131, 132, 138, 141 special, 296 very ample See Very ample divisor with normal crossings, 391 Divisor class group, CI(X), 131, 145 See also Picard group exact sequence of an open subset, 133 is zero~ UFO, 131 of a cone, 146 of a cubic surface, 136 of a curve, 139, 140, 142, 148 of a Dedekind domain, 132 of a product, 134, 146 of a quadric hypersurface, 147 of a quadric surface, 133, 135 of a variety in P", 146 of P", 132 Dominant morphism, 23, 81, 91, 137 Dominant rational map, 24, 26 Domination (of local rings), 40, 98 Double line, 36 (Fig), 90 (Fig) 483 Index Double point, 36 (Fig), 37, 38, 393 See also Cusp; Tacnode ordinary See Node Doubly periodic function, 327 Dual curve, 54, 304 Duality, 239-249 See also Serre duality for a finite flat morphism, 239, 306 Dualizing sheaf, 239, 241, 242, 246, 249, 298 Dual locally free sheaf, if, 123, 143, 235, 430 Dual numbers, ring of, 80, 265, 267, 324 Dual projective space, (P")•, 54, 55, 130, 304, 316 d-Uple embedding See -Uple embedding Dwork, Bernard M., 451 Dynkin diagram, 420 Effaceable functor, 206 Effective divisor, 130, 145, 157, 294, 363 Elements de Geometrie Algebrique (EGA), 89, 100, 462 Elimination theory, 35, 39 Elliptic curve, 46, 56, 293, 316-340 See also Cubic curve as a plane cubic curve, 309, 319 automorphisms of, 318, 321, 336 canonical divisor, 297 classified by }-invariant, 317, 345 complex multiplication on, 330-332, 334, 337-339 defined over Q, 335 dual of a morphism, 337 group structure, 297, 316, 321-323 See also Group, law on cubic curve group structure over C, 329 Hasse invariant of, 317, 322, 332-335, 339, 340 in characteristic p, 317, 332-335 isogeny of, 338 Jacobian variety of, 316, 323-326, 338 }-invariant, 316-321, 331, 336, 345, 347 locally free sheaves on, 378 over C, 326-332 Picard group of, 297, 323 points of order n, 322, 323, 329, 337, 340 points of order p, 339 points with integer coordinates, 340 quartic, 38, 315, 353 rational points over Fq• 339 rational points over Q, 317,335,336 (Fig) ring of endomorphisms, 323, 329, 330, 338, 340 supersingular, 332 withj=O, 320, 321, 331, 334 with}= 1728, 320, 321, 331, 334 zeta function of, 458 Elliptic function, 316, 326-332, 338 Elliptic ruled surface, 369, 375, 384, 385, 440 classification of, 377 484 Elliptic scroll, 385 Elliptic surface, 422 Embedded point, 85, 259 Embedded resolution of singularities, 390, 391, 392 (Fig), 419 Embedding a curve in projective space, 307-316 a variety in a complete variety, 168 Enough injectives, 204, 217 Enough locally frees, 238, 239 Enough projectives, 235 Enriques, Federigo, 348 Enriques-Severi-Zariski, lemma of, 244 Enriques surface, 422 Equidimensional, 243 J?:space etale of a presheaf, 67 Etale cohomology, 307, 452, 453 covering, 303, 306, 338, 340, 442 equivalence relation, 445 morphism, 268, 275, 276 (Fig), 299 neighborhood, 275 topology, 452, 453 Euler characteristic, 230, 295, 360, 362, 366, 424 Euler-Poincare characteristic, topological, 456 Euler's lemma, 37 Exact in the middle, 204, 282 Exact sequence of sheaves, 64, 66, 68, 109 Exceptional curve, 29, 31, 108, 386, 392, 395, 408, 437 contraction of, 410, 414-416 infinitely many, 409, 418 of the first kind, 410, 414, 418 self-intersection of, 386 Excision, 212 Exotic sphere, 421 Exponential Chern character, 431, 435 Exponential sequence, 446, 447 Extending a function to a normal point, 23, 217 a morphism, 43, 44, 97, 370 a section of a sheaf, 67, 112, 118 a sheaf by zero, 68, Ill, 149 coherent sheaves, 126 Extension of invertible sheaves, 372, 375, 376, 383, 430 of £T!x-modules, 237 of quasi-coherent sheaves, 114 Exterior algebra, 1\M, 127 Exterior power, NM, 127, 149, 181, 430 Ext group, 233-240,375,376 Ext sheaf, 233-239, 241 Faithful functor, 290 Family flat, 253, 256-266, 260 (Fig), 289, 315 of curves of genus g, 347 Index of divisors, 261, 367, 384 of elliptic curves, 340, 347 of hypersurfaces, 291 of invertible sheaves, 323 of locally free sheaves, 379 of plane curves, 39 of plane curves with nodes, 314 of schemes, 89, 90 (Fig), 202, 250, 253 of varieties, 56, 263 Fermat curve, 320, 335, 339 Fermat hypersurface, 451 Fermat's problem, 58, 335 Fermat surface, 409 Fibre cohomology of, 250, 255, 281, 290 dimension of, 95, 256, 257, 269 is connected, 279, 280 of a morphism, 89, 92 with nilpotent elements, 259, 277, 315 Fibred product, 87, 100 See also Product Field algebraically closed, I, 4, 22, 152 of characteristic p See Characteristic p of complex numbers See Complex numbers, C of elliptic functions, 327 of meromorphic functions, 442 of quadratic numbers, 330-332, 334, 340 of rational numbers See Rational numbers, Q of real numbers, R, 4, 8, 80, 106 of representatives, 187, 275 perfect, 27, 93, 187 separable closure of, 93 spectrum of, 74 transcendence degree of, 6, 27 uncountable, 409, 417 Field extension abelian, 332 purely inseparable, 302, 305, 385 pure transcendental, 303 separable, 27, 300, 422 separably generated, 27, 174, 187, 271 Final object, 79 Fine moduli variety, 347 Fine resolution, 201 Finite field, 80, 339 See also Characteristic p number of solutions of polynomial equations over, 449 Finite morphism, 84, 91, 124, 280, 456 a projective, quasi-finite morphism is, 280, 366 is affine, 128 is closed, 91 is proper, 105 is quasi-finite, 91 of curves, 137, 148, 298-307, 299 (Fig) Finite type, morphism locally of, 84, 90 Finite type, morphism of, 84, 91, 93, 94 Fixed point of a morphism, 451,453,454 Flasque resolution, 201, 208, 212, 248 Flasque sheaf, 67, 207 cohomology vanishes, 208, 221, 251 direct limit of, 209 injective sheaf is, 207 Flat base extension, 255, 287 · family, 253,256-266, 260 (Fig), 289, 315 See also Family module, 253 morphism, 239, 253-267, 269, 299, 340, 436 morphism is open, 266 sheaf, 254, 282 Flatness is an open condition, 266 local criterion of, 270 Formal completion See Completion Formal functions, 276-281, 290, 387, 415 Formal neighborhood, 190 Formal power series, 35 Formal-regular functions, 199,279 See also Holomorphic functions Formal scheme, 190-200, 279 Picard group of, 281 Fractional linear transformation, 46, 328 Free module, 174 Freshman calculus, 408 Freyd, Peter, 203 Frobenius morphism, 21,272, 301,332, 340 See also Characteristic p fixed points of, 454 k-linear, 302, 339, 368, 385 1-adic representation of, 451 trace on cohomology, 455 Frohlich, A., 331 Fulton, William, 297, 436 Functional equation, 450, 458 Function field, 16 determines birational equivalence class, 26 of an integral scheme, 91 of a projective variety, 18, 69 of dimension I, 39, 44 transcendence degree of, 17 valuation rings in, 106, 108 Functor additive, 203 adjoint, 68, 110, 124 coeffaceable, 206, 238, 240, 243 derived See Derived functor effaceable, 206 exact in the middle, 204, 282 faithful, 290 left exact, 113, 203, 284, 286 of global sections See Global sections representable, 241, 324 right exact, 204, 286 satellite, 206 485 Index Fundamental group, '17 1, 190, 338, 420, 442 Fundamental point, 410 Funny curve in characteristic p, 305, 385 GAGA, 330, 440 Galois extension, 318 Galois group, 147, 320, 338, 442 Gaussian integers, 331, 335 General position, 409, 418 Generically finite morphism, 91, 436 Generic point, 74, 75 (Fig), 80, 294 in a Zariski space, 93 local ring of, 91, 425 Generic smoothness, 272 Generization, 94 Genus arithmetic See Arithmetic genus bounded by degree, 315, 351,407,408 geometric See Geometric genus of a curve, 54, 56, 140, 183, 188, 294, 345, 421 of a curve on a surface, 361, 362, 393, 401, 407, 408 Geometrically integral scheme, 93 Geometrically irreducible scheme, 93 Geometrically reduced scheme, 93 Geometrically regular, 270 Geometrically ruled surface See Ruled surface Geometric genus,p8 , 181, 190, 246,247, 294, 421 is a birational invariant, 181 of a complete intersection, 188 Geometry on a surface, 357-368 Germ, 62, 438 Global deformation, 265, 267 See also Deformation; Family Global sections See also Section finitely generated, 122, 156, 228 functor of, 66, 69, 113 restricted to open set D(f), 112 sheaf generated by, 121, 150-156, 307, 358, 365 Glueing analytic spaces, 439 morphisms, 88, 150 schemes, 75, 80, 91, 171, 439, 444 sheaves, 69, 175 Godement, Roger, 61, 172, 201 Graded module, 50 associated to a sheaf, r (JF), 118 quasi-finitely generated, 125 sheaf M associated to, 116 Graded ring, 9, 394, 426 associated to an elliptic curve, 336 graded homomorphism of, 80, 92 Proj of, 76 Graph morphism, 106, 107 of a birational transformation, 410 486 of a morphism, 368, 426 Grauert, Hans, 249, 252, 288, 291, 369, 417, 438, 442, 445 Griffiths, Phillip A., 184, 304, 423, 445 Grothendieck, A., 57, 59, 60, 87, 89, 100, 115, 120, 172, 190, 192, 201, 208, 217, 249,252,259,279,281,282,291,324, 338,363,366,368,429,432,435,436, 441, 442, 449, 451-453 Grothendieck group, K(X), 148, 149,230, 238, 385, 435 Group additive, G0 , 23, 142, 148, 171 fundamental, '17 1, 190, 338, 420, 442 Galois, 147, 320, 338, 442 general linear, GL, 151 Grothendieck See Grothendieck group law on cubic curve, 139 (Fig), 142, 147, 148, 297, 299, 407, 417 multiplicative, Gm, 23, 148, 149 Neron-Severi, See Neron-Severi group of automorphisms See Automorphisms of cycles modulo rational equivalence See Chow ring of divisor classes, See Divisor class group of divisors, Div(X), 130, 357 of divisors modulo algebraic equivalence, See Neron-Severi group of divisors modulo numerical equivalence, Num X, 364, 367-369 of invertible sheaves See Picard group of order 6, 318, 321 of order 12, 321 of order 60, 420 of order 168, 349 of order 51840, 405 projective general linear, PGL, 46, 151, 273, 347 scheme, 324 symmetric, 304, 318, 408 variety, 23, 139 (Fig), 142, 147, 148, 272, 321, 323, 324 Weyl, 405, 408 Gunning, Robert C., 69, 201, 249, 438,442 Halphen, G., 349 Harmonic function, 442 Harmonic integrals, 435, 445 Hartshorne, Robin, 14, 105, 140, 144, 190, 193, 195, 199,224, 249, 281, 366, 367, 383, 419, 428, 440 Hasse, H., 339 Hasse invariant, 317, 322, 332-335, 339, 340 Hausdorff toplogy, 2, 8, 95, 439 Height of a prime ideal, Hermitian metric, 445 Higher direct image sheaf, Ri.(JF), 250, 276, 282, 290, 371, 387, 436 locally free, 288, 291 Hilbert, David, 51, 403, 441 Index Hilbert function, 51 Hilbert polynomial, 48, 49, 52, 57, 170, 230, 231, 294, 296, 366 constant in a family, 256, 261, 263 Hilbert-Samuel polynomial, 394 Hilbert scheme, 258, 349 Hilbert's Nullstellensatz, 4, II Hironaka, Heisuke, 105, 168, 195, 264, 391, 413, 417, 442, 443, 445, 447 Hinebruch, Friedrich,57,363,421,431,432 Hodge index theorem, 356, 364, 366, 435, 452 Hodge manifold, 445 Hodge numbers, hP,q, 190, 247 Hodge spectral sequence, 289 Hodge theory, 414, 435, 445, 452 Holomorphic functions, 60, 190, 279, 330, 438, 447 See also Formal-regular functions Homeomorphism, 21, 31 Homogeneous coordinate ring, 10, II, 18, 23, 49, 128, 132 criterion to be UFD, 147 depends on embedding, I Proj of, 81 Homogeneous coordinates, Homogeneous element, Homogeneous ideal, 9, 10, 92, 125 Homogeneous space, 273 Homological dimension, 238 Homological equivalence of cycles, 435, 444 Homotopy of maps of complexes, 203 Homotopy operator, 203, 221 Hopf map, 386 Horrocks, G., 437 Hurwitz, Adolf, 301, 305, 326 Hurwitz's theorem, 293, 299-307, 31 I, 313, 317, 337, 382 Husemoller, Dale, 356, 421 Hyperelliptic curve, 298, 306, 341, 345, 384 See also Nonhyperelliptic curve canonical divisor of, 342, 343 existence of, 298, 394 moduli of, 304, 347 not a complete intersection, 315, 348 Hyperelliptic surface, 422 Hyperosculating hyperplane, 337 Hyperosculation point, 337, 348 Hyperplane, 10, 429 corresponds to sheaf Ill (I), I 45 Hyperplane section, 147, 179 (Fig) Hypersurface, 4, 7, 8, 12 any variety is birational to, 27 arithmetic genus of, 54 canonical sheaf of, I 83, 184 complement of is affine, 21, 25 existence of nonsingular, 183 Icosahedron, 420 Ideal class group, 132 Ideal of a set of points, 3, 10 Ideal of definition, 196 Ideal sheaf, fy, 109 blowing up, 163, 171 of a closed subscheme, I 15, I 16, 120, 145 of a subvariety, 69 of denominators, 167 Idempotent, 82 Igusa, Jun-ichi, 265, 334 Image direct See Direct image inverse See Inverse image of a morphism of sheaves, 63, 64, 66 of a proper scheme is proper, I 06 scheme-theoretic, 92 Immersion, 120 closed, See Closed immersion open, 85 Indecomposable locally free sheaf, 376, 384 Index of a bilinear form, 364 Index of speciality, 296 Induced structure of scheme, 79, 86, 92 of variety, 21 Inequality of Castelnuovo and Severi, 368, 451 Infinitely near point, 392, 395 Infinitesimal deformation See Deformation Infinitesimal extension, 189, 225, 232, 265 Infinitesimal lifting property, 188 Infinitesimal neighborhood, 86, 190, 276, 393 Inflection point, 139, 148, 304, 305, 335, 337 cubic curve has 9, 305, 322 Initial object, 80 Injective module, 206, 207, 213, 214, 217 Injective object (of a category), 204, 217, 233 Injective resolution, 204, 242 Injective sheaf, 207, 213, 217 Inseparable morphism, 276, 311, 312 Integers, Z, 79, 340 Integral closure, 40, 91, 105 finiteness of, 20, 38, 123 Integrally closed domain, 38, 40, 126, 132, 147, See also Normal Integral scheme, 82, 91 Intersection divisor, 135, 146, 261 Intersection multiplicity, 36, 47, 53, 233, 304, 357, 360, 394, 427 Intersection number, 357-362, 366, 394, 445 Intersection of affines is affine, I06 Intersection of varieties, 14, 21, 47-55 Intersection, proper, 427 Intersection, scheme-theoretic, 171, 358 Intersection theory, 47-55, 58, 424-437 axioms, 426 on a cubic surface, 401 on a monoidal transform, 387 on a nonsingular quasi-projective variety, 427 487 Index Intersection theory (cont.) on a quadric surface, 361, 364 on a ruled surface, 370 on a singular variety, 428 on a surface, 356-362, 366, 425 on p2, 361 See also Bezout's theorem Invariant theory, 420 Inverse image ideal sheaf,j- 1·1Vx, 163, 186 of cycles, 426 of divisors,j*, 135, 137, 299 sheaf, f- 1$"", 65 sheaf, j*$"", I 10, 115, 128, 299 Inverse limit, lim in a category':I92 of abelian groups, 190-192, 277 of rings, 33 of sheaves, 67, 109, 192 Inverse system, 190 Invertible sheaf, 109, 117, 118, 124, 143-146, 169 ample See Ample invertible sheaf associated to a divisor, 144, 145, 157, 294, 425 determines a morphism toP", 150-153, 158, 162, 307, 318, 340 extension of, 372, 375, 376, 383, 430 generated by global sections, 150-156, 307, 358, 365 group of See Picard group on a family, 291 very ample See Very ample invertible sheaf Involution, 106, 306 Irreducible closed subset, 78, 80 component, 5, 7, II, 47, 365 scheme, 82 topological space, 3, 4, 8, II Irregularity, 247, 253, 422 Irrelevant ideal, II Iskovskih, V.A., 184, 304 Isogeny of elliptic curves, 338 Isolated singularity, 420 Italians, 391 Jacobian matrix, 32 Jacobian polynomial, 23 Jacobian variety, 105, 140, 316, 323-326, 338, 445, 447 See also Abelian variety j-Invariant of an elliptic curve, 316-321, 331, 345, 347 Jouanolou, J P., 436 Kahler differentials See Differentials Kahler manifold, 445, 446, 452 Kernel, 63, 64, I 09 Kleiman, Steven L., 144, 238, 268, 273, 345, 434, 435, 452, 453 Klein, Felix, 147, 349, 420 488 Knutson, Donald, 445 Kodaira dimension, 421, 422 Kodaira, Kunihiko, 249, 266,409,414,422, 442,443 Kodaira vanishing theorem, 248, 249, 408, 423, 424, 445 Koszul complex, 245, 389 Krull-Akizuki, theorem of, 108 Krull dimension, 6, 86 See also Dimension Krull's Hauptidealsatz, 7, 48 Krull, Wolfgang, 213, 279 K3 surface, 184, 422, 423, 437, 452 Kunz, E., 249 Kuyk, W., 317, 334 1-Adic cohomology, 435, 449, 452-457 1-Adic integers, 338, 453 Laksov, D., 345 Lang, Serge, 140, 334, 364, 367, 452 Lascu, A.T., 431 Lattice, 326 (Fig) Lefschetz fixed-point formula, 453, 454 Lefschetz pencil, 457 Lefschetz, Solomon, 451 Lefschetz theorem, 190, 281 strong, 452 Left derived functor, 205 Left exact functor, 113, 203, 284, 286 Length of a module, 50, 290, 360, 394 Leray spectral sequence, 252 Leray, theorem of, 211 Lichtenbaum, Stephen, 185, 267 Lie group, 328 Line, 22, 28, 129, 183 See also Linear variety; Secant line; Tangent line on a surface, 13, 136, 367, 402-406, 408 Linear equivalence, 57, 131, 141, 294, 425, 426 See also Divisor =>algebraic equivalence, 367 Linear projection See Projection Linear system, 130, 150, 156-160, 274 complete, 157, 159, 170, 294 determines a morphism toP", 158, 307,318 dimension of, 157, 295, 357, 424 not composite with a pencil, 280 of conics, 170, 396-398 of plane cubic curves, 399, 400 on a curve, 307 separates points, 158, 308, 380 separates tangent vectors, 158, 308, 380 very ample, 158, 307, 308, 396 with assigned base points, 395 without base points, 158, 307, 341 Linear variety, 13, 38, 48, 55, 169, 316 See also Line Lipman, Joseph, 249, 391, 420 Local cohomology See Cohomology, with supports Local complete intersection, 8, 184-186, 245, 428 See also Complete intersection Index Local criterion of flatness, 270 Local homomorphism, 72, 74, 153 Locally closed subset, 21, 94 Locally factorial scheme, 141, 145, 148, 238 Locally free resolution, 149, 234, 239 Locally free sheaf, 109, 124, 127, 178 as an extension of invertible sheaves, 372, 375, 376, 383, 430 Chern classes of See Chern class decomposable, 376, 378, 383, 384 dual of, 123, 143, 235, 430 indecomposable, 376, 384 of rank 2, 356, 370, 376, 378, 437 on a curve, 369, 370, 379, 384 on an affine curve, 385 projective space bundle of See Projective space bundle resolution by, 149, 234, 239 stable, 379, 384 trivial subsheaf of, 187 vector bundle V(lf) of, 128, 170 zeros of a section, 157, 431 Locally noetherian scheme, 83 Locally principal closed subscheme, 145 Locally principal Weil divisor, 142 Locally quadratic transformation, 386 Locally ringed space, 72, 73, 169 See also Ringed space Local parameter, 137, 258, 299 Local ring complete, 33-35, 187, 275, 278, 420 local homomorphism of, 72, 74 of a point, 16-18, 22, 31, 41, 62, 71, 80 of a subvariety, 22, 58 regular See Regular local ring Local space, 213 Logarithmic differential, dlog, 250, 367, 457 Lubkin, Saul, 452 Liiroth's theorem, 303, 422 Macbeath, A.M., 306 MacPherson, R.D., 436 Manifold, 31 See also Complex manifold Manin, Ju 1., 149, 184, 304, 401, 405, 408, 436,452 Maruyama, Masaki, 383 Matsumura, Hideyuki, 123, 172, 184, 195, 250, 268, 418 Mattuck, Arthur, 451 Maximal ideal, Mayer, K.H., 421 Mayer-Vietoris sequence, 212 Meromorphic function, 327, 442 Minimal model, 56, 410, 418, 419, 421 Minimal polynomial, 147 Minimal prime ideal, 7, 50, 98 Mittag-Leffler condition, (ML), 191, 192, 200, 278, 290 Module flat, 253 free, 174 graded See Graded module of finite length, 50, 290, 360, 394 Moduli, variety of, 56, 58, 266, 317, 346, 421 Moishezon, Boris, 366, 434, 442, 443, 446 Moishezon manifold, 442-446 nonalgebraic, 444 (Fig) Monodromy, 457 Monoidal transformation, 356, 386-395, 409, 410, 414, 443 See also Blowing-up ample divisor on, 394 behavior of a curve, 388-394 behavior of arithmetic genus, 387-389 behavior of cohomology groups, 387, 419 canonical divisor of, 387 Chow ring of, 437 intersection theory on, 387 local computation, 389 Picard group of, 386 Morden, L J., 335 Morphism affine, 128, 222, 252 closed, 91, 100 determined by an open set, 24, I 05 dominant, 23, 81, 91 etale, 268, 275, 276 (Fig), 299 finite See Finite morphism flat See Flat, morphism Frobenius See Frobenius morphism generically finite, 91, 436 glueing of, 88, 150 injective, of sheaves, 64 inseparable, 276, 311, 312 locally of finite type, 84, 90 of finite type, 84, 91, 93, 94 of Spec K to X, 80 projective, 103, 107, 123, 149-172, 277, 281,290 proper See Proper morphism quasi-compact, 91 quasi-finite, 91, 280, 366 quasi-projective, 103 ramified, 299, 312 separated See Separated morphism smooth, 268-276, 303 smjective, of sheaves, 64, 66 toP", determined by an invertible sheaf, 150-153, 158, 162, 307, 318, 340 universally closed, I 00 unramified, 275, 299 See also Etale morphism Morrow, James, 266, 442 Moving lemma, 425, 427, 434 Moving singularities, 276 Multiple tangent, 305 Multiplicity, 36, 51, 388, 393, 394 See also Intersection multiplicity Multisecant, 310, 355 Mumford, David, 140, 249, 291, 324, 336, 347, 356, 366, 379,417,419-421, 431,437 489 Index Nagata, Masayoshi, 58, 87, 105, 108, 168, 381, 383, 384, 394, 401, 409, 419 Nakai-Moishezon criterion, 356, 365, 382, 405, 434 Nakai, Yoshikazu, 144, 366, 434 Nakano, Shigeo, 249, 445 Nakayama's Lemma, 125, 153, 175, 178, 288 Narasimhan, M.S., 379 Natural isomorphism, 240 Negative definite, 368, 419 Neron, Andre, 367 Neron-Severi group, 140, 367, 418 is finitely generated, 447 Neron-Severi theorem, 364, 367, 368 Nilpotent element See also Dual numbers, ring of in a ring, 79-81 in a scheme, 79, 85, 190, 259, 277, 315 Nilradical of a ring, 82 Nodal cubic curve, 259, 263, 276 blown up, 29 (Fig) divisor class group of, 148 is rational, 30 Node, 36 (Fig), 37, 258, 293, 298, 392, 394 See also Nodal cubic curve analytic isomorphism of, 34, 38 plane curve with, 310-316 Noetherian formal scheme, 194 Noetherian hypotheses, 100, 194, 201, 213-215, 218 Noetherian induction, 93, 94, 214 Noetherian ring, 80 Noetherian scheme, 83 Noetherian topological space, 5, 8, II, 80, 83 See also Zariski space Noether, Max, 349 Nonalgebraic complex manifold, 444 (Fig) Nonhyperelliptic curve, 340 See also H yperelliptic curve existence of, 342, 345, 385 Nonprojective scheme, 232 Nonprojective variety, 171, 443 (Fig) Nonsingular cubic curve ample sheaves on, 156 canonical sheaf of, 183 divisor class group of, 139 group law on, 139 (Fig), 147, 297, 417 has inflection points, 305, 322 is not rational, 46, 139, 183, 230 Nonsingular curve, 39-47, 136 abstract See Abstract nonsingular curve divisors on, 129 existence of, 37, 231, 352, 406 Grothendieck group of, 149 morphism of, 137 projective ~ complete, 136 Nonsingular in codimension one, 130 Nonsingular points, 32, 37 form an open subset, 33, 178, 187 Nonsingular variety, 31-39, 130, 177-180, 490 268, 424 See also Regular scheme; Smooth morphism hyperplane section of See Bertini's theorem infinitesimal lifting property, 188 fl is locally free, 177, 178, 276 Nonspecial divisor, 296, 343, 349 Norm (of a field extension), 46 Normal See also Projectively normal bundle See Normal, sheaf crossings, divisor with, 391 cuspidal cubic curve is not, 23 point, 23, 38 quadric surface is, 23, 147 ring, 185, 264 ~ R +S2, 185 scheme, 91, 126, 130, 244, 280 sheaf, X y /X' 182, 361, 386, 431, 433, 436 See also Conormal sheaf variety, 23, 263, 410 Normalization, 23, 91, 426 of a curve, 148, 232, 258, 298, 343, 365, 382 Number theory, 58, 316, 451, 452 Numerical equivalence, 364, 367, 369, 435 Numerical invariant, 56, 256, 361, 372, 379, 425, 433 Numerical polynomial, 49 Olson, L., 140 Open affine subset, 25, 106 Open immersion, 85 Open set X1, 81, 118, 151 Open subscheme, 79, 85 Ordinary double point See Node Ordinary inflection point, 305 See also Inflection point Ordinary r-fold point, 38, 305 Osculating hyperplane, 337 p-Adic analysis, 451 p-Adic cohomology, 452 Parameter space, 56 See also Variety, of moduli Parametric representation, 7, 8, 22, 260 Paris seminar, 436, 449 Pascal's theorem, 407 (Fig) Pencil, 280, 422 Perfect field, 27, 93, 187 Period mapping, 445 Period parallelogram, 326 (Fig) Petri, K., 348 Picard group, Pic X, 57, 143, 232, 250, 357, 428 See also Divisor class group as H 1(X, £!! 1), 143, 224, 367, 446 Lefschetz theorem, 190 of a blowing-up, 188 of a cubic surface in P 3, 401 of a family, 323 of a formal scheme, 200, 281 Index of a line with point doubled, 169 of a monoidal transformation, 386 of an elliptic curve, 297, 323 of a nonprojective variety, 171 of a P(