Igor R. Shafarevich: Basic Algebraic Geometry 2

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1 Igor R. Shafarevich: Basic Algebraic Geometry 2

2 Igor R. Shafarevich Basic Algebraic Geometry 2 Second, Revised and Expanded Edition Springer-Verlag Berlin Heidelberg GmbH

3 Igor R. Shafarevich Steklov Mathematical Institute VI. Vavilova 42, Moscow, Russia Translator: Miles Reid Mathematics Institute, Vniversity of Warwick Coventry CV4 7AL, England With 12 Figures The title of the original Russian edition: Osnovy algebraicheskoj geometrii, tom 2 Nauka, Moscow 1988 Third Printing 1997 ISBN ISBN (ebook) DOI / Library of Congress Cataloging-in-Publication Data. Shafarevich, I. R. (lgor' Rostislavovich), [Osnovy algebraicheskoi geometrii. English) Basic algebraic geometryligor R. Shafarevich; [translator, Miles Reid). - 2nd. rev. and expanded ed. p. cm. "Springer study edition." Includes bibliographical references and indexes. ISBN Geometry, Aigebraic. 1. Title. QA564.S '5. - dc20 Thi. work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in anyother way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1977, 1994 Originally published by Springer-Verlag Berlin Heidelberg New York in 1994 This book was typeset by the translator using the ~ STEX -macro package and typefaces, together with Springer-Verlag's lex macro package CPMonoOI SPIN / Printed on acid-free paper

4 Preface to Books 2 and 3 Books 2 and 3 correspond to Chap. V-IX of the first edition. They study schemes and complex manifolds, two notions that generalise in different directions the varieties in projective space studied in Book 1. Introducing them leads also to new results in the theory of projective varieties. For example, it is within the framework of the theory of schemes and abstract varieties that we find the natural proof of the adjunction formula for the genus of a curve, which we have already stated and applied in Chap. IV, 2.3. The theory of complex analytic manifolds leads to the study of the topology of projective varieties over the field of complex numbers. For some questions it is only here that the natural and historical logic of the subject can be reasserted; for example, differential forms were constructed in order to be integrated, a process which only makes sense for varieties over the (real or) complex fields. Changes from the First Edition As in the Book 1, there are a number of additions to the text, of which the following two are the most important. The first of these is a discussion of the notion of the algebraic variety classifying algebraic or geometric objects of some type. As an example we work out the theory of the Hilbert polynomial and the Hilbert scheme. I am very grateful to V. I. Danilov for a series of recommendations on this subject. In particular the proof of Chap. VI, 4.3, Theorem 3 is due to him. The second addition is the definition and basic properties of a Kahler metric, and a description (without proof) of Hodge's theorem. Prerequisites Varieties in projective space will provide us with the main supply of examples, and the theoretical apparatus of Book 1 will be used, but by no means all of it. Different sections use different parts, and there is no point in giving exact indications. References to the Appendix are to the Algebraic Appendix at the end of Book 1. Prerequisites for the reader of Books 2 and 3 are as follows: for Book 2, the same as for Book 1; for Book 3, the definition of differentiable manifold,

5 VI Preface to Books 2 and 3 the basic theory of analytic functions of a complex variable, and a knowledge of homology, cohomology and differential forms (knowledge of the proofs is not essential); for Chap. IX, familiarity with the notion of fundamental group and the universal cover. References for these topics are given in the text. Suggestions for Further Reading Some references for further reading are included in the text. The reader who would like to continue the study of algebraic geometry is recommended the following books. For scheme theory, the cohomology of algebraic coherent sheaves and its applications, see: Hartshorne [35], especially Chap. III. For the Riemann-Roch theorem. An elementary proof for curves is given in the book: W. Fulton, Algebraic curves, Springer. For the general case, see any of the following: A. Borel and J.-P. Serre, Le theoreme de Riemann-Roch, Bull. Soc. Math. France, 86 (1958), Or Yu. I. Manin, Lectures on the K-functor in algebraic geometry, Uspekhi Mat. Nauk 24:5 (1969), English translation in: Russian Math. Surveys, 24:5 (1969), Or W. Fulton and S. Lang, Riemann-Roch algebra, Springer, 1985.

6 Table of Contents Volume 2 BOOK 2. Schemes and Varieties Chapter V. Schemes The Spec of a Ring Definition of Spec A Properties of Points of Spec A The Zariski Topology of Spec A Irreducibility, Dimension Exercises to Sheaves Presheaves The Structure Presheaf Sheaves Stalks of a Sheaf Exercises to Schemes Definition of a Scheme Glueing Schemes Closed Subschemes Reduced Schemes and Nilpotents Finiteness Conditions Exercises to Products of Schemes Definition of Product Group Schemes Separatedness Exercises to Chapter VI. Varieties Definitions and Examples Definitions Vector Bundles... 53

7 VIII Table of Contents Volume Vector Bundles and Sheaves f.4. Divisors and Line Bundles.... Exercises to Abstract and Quasiprojective Varieties Chow's Lemma Blowup Along a Subvariety Example of Non-Quasiprojective Variety Criterions for Projectivity.... Exercises to Coherent Sheaves Sheaves of Ox-modules Coherent Sheaves Devissage of Coherent Sheaves The Finiteness Theorem.... Exercises to Classification of Geometric Objects and Universal Schemes Schemes and Functors The Hilbert Polynomial Flat Families The Hilbert Scheme.... Exercises to BOOK 3. Complex Algebraic Varieties and Complex Manifolds Chapter VII. The Topology of Algebraic Varieties The Complex Topology Definitions Algebraic Varieties as Differentiable Manifolds; Orientation Homology of Nonsingular Projective Varieties Exercises to Connectedness Preliminary Lemmas The First Proof of the Main Theorem The Second Proof Analytic Lemmas Connectedness of Fibres Exercises to The Topology of Algebraic Curves Local Structure of Morphisms Triangulation of Curves Topological Classification of Curves

8 Table of Contents Volume 2 IX 3.4. Combinatorial Classification of Surfaces The Topology of Singularities of Plane Curves Exercises to Real Algebraic Curves Complex Conjugation Proof of Harnack's Theorem Ovals of Real Curves Exercises to Chapter VIII. Complex Manifolds Definitions and Examples Definition Quotient Spaces Commutative Algebraic Groups as Quotient Spaces Examples of Compact Complex Manifolds not Isomorphic to Algebraic Varieties Complex Spaces Exercises to Divisors and Meromorphic Functions Divisors Meromorphic Functions The Structure of the Field M(X) Exercises to Algebraic Varieties and Complex Manifolds Comparison Theorems Example of Nonisomorphic Algebraic Varieties that Are Isomorphic as Complex Manifolds... : Example of a Nonalgebraic Compact Complex Manifold with Maximal Number of Independent Meromorphic Functions The Classification of Compact Complex Surfaces Exercises to Kahler Manifolds Kahler Metric Examples Other Characterisations of Kahler Metrics Applications of Kahler Metrics Hodge Theory Exercises to Chapter IX. U niformisation The Universal Cover

9 X Table of Contents Volume The Universal Cover of a Complex Manifold Universal Covers of Algebraic Curves Projective Embedding of Quotient Spaces Exercises to Curves of Parabolic Type Theta functions Projective Embedding Elliptic Functions, Elliptic Curves and Elliptic Integrals Exercises to Curves of Hyperbolic Type Poincare Series Projective Embedding Algebraic Curves and Automorphic Functions Exercises to Uniformising Higher Dimensional Varieties Complete Intersections are Simply Connected Example of Manifold with 11"1 a Given Finite Group Remarks Exercises to Historical Sketch Elliptic Integrals Elliptic Functions Abelian Integrals Riemann Surfaces The Inversion of Abelian Integrals The Geometry of Algebraic Curves Higher Dimensional Geometry The Analytic Theory of Complex Manifolds Algebraic Varieties over Arbitrary Fields and Schemes 249 References References for the Historical Sketch 256 Index

10 Table of Contents Volume 1 BOOK 1. Varieties in Projective Space Chapter I. Basic Notions Algebraic Curves in the Plane Plane Curves Rational Curves Relation with Field Theory Rational Maps Singular and Nonsingular Points The Projective Plane Exercises to Closed Subsets of Affine Space Definition of Closed Subsets Regular FUnctions on a Closed Subset Regular Maps Exercises to Rational FUnctions Irreducible Algebraic Subsets Rational FUnctions Rational Maps Exercises to Quasiprojective Varieties Closed Subsets of Projective Space Regular FUnctions Rational FUnctions Examples of Regular Maps Exercises to Products and Maps of Quasiprojective Varieties Products The Image of a Projective Variety is Closed Finite Maps... 61

11 XII Table of Contents Volume Noether Normalisation Exercises to Dimension Definition of Dimension Dimension of Intersection with a Hypersurface The Theorem on the Dimension of Fibres Lines on Surfaces.... Exercises to Chapter II. Local Properties Singular and Nonsingular Points The Local Ring of a Point The Tangent Space Intrinsic Nature of the Tangent Space Singular Points The Tangent Cone.... Exercises to Power Series Expansions Local Parameters at a Point Power Series Expansions Varieties over the Reals and the Complexes Exercises to Properties of Nonsingular Points Codimension 1 Subvarieties Nonsingular Subvarieties Exercises to The Structure of Birational Maps Blowup in Projective Space Local Blowup Behaviour of a Subvariety under a Blowup Exceptional Subvarieties Isomorphism and Birational Equivalence Exercises to Normal Varieties Normal Varieties Normalisation of an Affine Variety Normalisation of a Curve Projective Embedding of Nonsingular Varieties Exercises to

12 Table of Contents Volume 1 XIII 6. Singularities of a Map Irreducibility Nonsingularity Ramification Examples Exercises to Chapter III. Divisors and Differential Forms Divisors The Divisor of a Function Locally Principal Divisors Moving the Support of a Divisor away from a Point Divisors and Rational Maps The Linear System of a Divisor Pencil of Conics over pi Exercises to Divisors on Curves The Degree of a Divisor on a Curve Bezout's Theorem on a Curve The Dimension of a Divisor Exercises to The Plane Cubic The Class Group The Group Law Maps Applications Algebraically Nonclosed Field Exercises to Algebraic Groups Algebraic Groups Quotient Groups and Chevalley's Theorem Abelian Varieties The Picard Variety Exercises to Differential Forms Regular Differential I-forms Algebraic Definition of the Module of Differentials Differential p-forms Rational Differential Forms Exercises to

13 XIV Table of Contents Volume 1 6. Examples and Applications of Differential Forms Behaviour Under Maps Invariant Differential Forms on a Group The Canonical Class Hypersurfaces Hyperelliptic Curves The Riemann-Roch Theorem for Curves Projective Embedding of a Surface Exercises to Chapter IV. Intersection Numbers Definition and Basic Properties Definition of Intersection Number Additivity Invariance Under Linear Equivalence The General Definition ofintersection Number Exercises to Applications of Intersection Numbers Bezout's Theorem in Projective and Multiprojective Space Varieties over the Reals The Genus of a Nonsingular Curve on a Surface The Riemann-Roch Inequality on a Surface The Nonsingular Cubic Surface The Ring of Cycle Classes Exercises to Birational Maps of Surfaces Blowups of Surfaces Some Intersection Numbers Resolution of Indeterminacy Factorisation as a Chain of Blowups Remarks and Examples Exercises to Singularities Singular Points of a Curve Surface Singularities Du Val Singularities Degeneration of Curves Exercises to Algebraic Appendix Linear and Bilinear Algebra Polynomials

14 Table of Contents Volume 1 XV 3. Quasilinear Maps Invariants Fields Commutative Rings Unique Factorisation Integral Elements Length of a Module References Index

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