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1 Igor R. Shafarevich Basic Algebraic Geometry 1 Second, Revised and Expanded Edition Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest

2 Table of Contents Volume 1 BOOK 1. Varieties in Projective Space Chapter I. Basic Notions 1 1. Algebraic Curves in the Plane Plane Curves Rational Curves Relation with Field Theory Rational Maps Singular and Nonsingular Points The Projective Plane 16 Exercises to Closed Subsets of Affine Space Definition of Closed Subsets Regulär Functions on a Closed Subset Regulär Maps 27 Exercises to Rational Functions Irreducible Algebraic Subsets Rational Functions Rational Maps 37 Exercises to Quasiprojective Varieties Closed Subsets of Projective Space Regulär Functions Rational Functions Examples of Regulär Maps 52 Exercises to Products and Maps of Quasipro jective Varieties Products The Image of a Projective Variety is Closed Finite Maps Noether Normalisation 65 Exercises to Dimension 67

3 XII Table of Contents Volume Definition of Dimension Dimension of Intersection with a Hypersurface The Theorem on the Dimension of Fibres Lines on Surfaces 78 Exercises to 6 81 Chapter II. Local Properties Singular and Nonsingulax Points The Local Ring of a Point The Tangent Space Intrinsic Nature of the Tangent Space Singular Points The Tangent Cone 95 Exercises to Power Series Expansions Local Parameters at a Point Power Series Expansions Varieties over the Reals and the Complexes 104 Exercises to Properties of Nonsingular Points Codimension 1 Subvarieties Nonsingular Subvarieties 111 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 121 Exercises to Normal Varieties Normal Varieties Normalisation of an Affine Variety Normalisation of a Curve Projective Embedding of Nonsingular Varieties 136 Exercises to Singularities of a Map Irreducibility Nonsingularity Ramification Examples 146 Exercises to 6 148

4 Table of Contents Volume 1 XIII 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 P Exercises to Divisors on Curves The Degree of a Divisor on a Curve Bezout's Theorem on a Curve The Dimension of a Divisor 173 Exercises to The Plane Cubic The Class Group The Group Law Maps Applications Algebraically Nonclosed Field 185 Exercises to Algebraic Groups Algebraic Groups Quotient Groups and Chevalley's Theorem Abelian Varieties The Picard Variety 192 Exercises to Differential Forms Regulär Differential 1-forms Algebraic Definition of the Module of Differentials Differential p-forms Rational Differential Forms 202 Exercises to 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 218 Exercises to 6 220

5 XIV Table of Contents Volume 1 Chapter IV. Intersection Numbers Definition and Basic Properties Definition of Intersection Number Additivity Invariance Under Linear Equivalence The General Definition of Intersection Number 232 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 249 Exercises to Birational Maps of Surfaces Blowups of Surfaces Some Intersection Numbers Resolution of Indeterminacy Factorisation as a Chain of Blowups Remarks and Examples 258 Exercises to Singularities Singular Points of a Curve Surface Singularities Du Val Singularities Degeneration of Curves 270 Exercises to Algebraic Appendix Linear and Bilinear Algebra Polynomials Quasilinear Maps Invariants Fields Commutative Rings Unique Factorisation Integral Elements Length of a Module 286 References 289 Index 293

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