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本書作者是當代著名的前蘇聯代數幾何學家，是一位有獨創性，知識極為淵博的數學家。本書問世（俄文版1972年初版，英文版1977年初版）40多年來，一直被視為一部重要的代數幾何經典名著．與同類書相比，本書內容全面，詳盡，注重給出抽象理論的幾何背景和起源，並配有充分反映幾何本質的實例和圖解。本書所需預備知識僅限於代數基礎，是高年級本科生和研究生學習代數幾何的首選教材。
Book 1： Varieties in Projective Space
1 Basic Notions
1 Algebraic Curves in the Plane
1.1 Plane Curves
1.2 Rational Curves
1.3 Relation with Field Theory
1.4 Rational Maps
1.5 Singular and Nonsingular Points
1.6 The Projective Plane
1.7 Exercises to Section 1
2 Closed Subsets of Affine Space
2.1 Definition of Closed Subsets
2.2 Regular Functions on a Closed Subset
2.3 Regular Maps
2.4 Exercises to Section 2
3 Rational Functions
3.1 Irreducible Algebraic Subsets
3.2 Rational Functions
3.3 Rational Maps
3.4 Exercises to Section 3
4 Quasiprojective Varieties
4.1 Closed Subsets of Projective Space
4.2 Regular Functions
4.3 Rational Functions
4.4 Examples of Regular Maps
4.5 Exercises to Section 4
5 Products and Maps of Quasiprojective Varieties
5.1 Products
5.2 The Image of a Projective Variety is Closed
5.3 Finite Maps
5.4 Noether Normalisation
5.5 Exercises to Section 5
6 Dimension
6.1 Definition of Dimension
6.2 Dimension of Intersection with a Hypersurface
6.3 The Theorem on the Dimension of Fibres
6.4 Lines on Surfaces
6.5 Exercises to Section 6
2 Local Properties
1 Singular and Nonsingular Points
1.1 The Local Ring of a Point
1.2 The Tangent Space
1.3 Intrinsic Nature of the Tangent Space
1.4 Singular Points
1.5 The Tangent Cone
1.6 Exercises to Section 1
2 Power Series Expansions
2.1 Local Parameters at a Point
2.2 Power Series Expansions
2.3 Varieties ovef the Reals and the Complexes
2.4 Exercises to Section 2
3 Properties of Nonsingular Points
3.1 Codimension 1 Subvarieties
3.2 Nonsingular Subvarieties
3.3 Exercises to Section 3
4 The Structure of Birational Maps
4.1 Blowup in Projective Space
4.2 Local Blowup
4.3 Behaviour of a Subvariety Under a Blowup
4.4 Exceptional Subvarieties
4.5 Isomorphism and Birational Equivalence
4.6 Exercises to Section 4
5 Normal Varieties
5.1 Normal Varieties
5.2 Normalisation of an Affine Variety
5.3 Normalisation of a Curve
5.4 Projective Embedding of Nonsingular Varieties
5.5 Exercises to Section 5
6 Singularities of a Map
6.1 Irreducibility
6.2 Nonsingularity
6.3 Ramification
6.4 Examples
6.5 Exercises to Section 6
3 Divisors and Differential Forms
1 Divisors
1.1 The Divisor of a Function
1.2 Locally Principal Divisors
1.3 Moving the Support of a Divisor away from a Point
1.4 Divisors and Rational Maps
1.5 The Linear System of a Divisor
1.6 Pencil of Conics over P1
1.7 Exercises to Section 1
2 Divisors on Curves
2.1 The Degree of a Divisor on a Curve
2.2 Bezout’’s Theorem on a Curve
2.3 The Dimension of a Divisor
2.4 Exercises to Section 2
3 The Plane Cubic
3.1 The Class Group
3.2 The Group Law
3.3 Maps
3.4 Applications
3.5 Algebraically Nonclosed Field
3.6 Exercises to Section 3
4 Algebraic Groups
4.1 Algebraic Groups
4.2 Quotient Groups and Chevalley’’s Theorem
4.3 Abelian Varieties
4.4 The Picard Variety
4.5 Exercises to Section 4
5 Differential Forms
5.1 Regular Differential 1—Forms
5.2 Algebraic Definition of the Module of Differentials
5.3 Differential p—Forms
5.4 Rational Differential Forms
5.5 Exercises to Section 5
6 Examples and Applications of Differential Forms
6.1 Behaviour Under Maps
6.2 Invariant Differential Forms on a Group
6.3 The Canonical Class
6.4 Hypersurfaces
6.5 Hyperelliptic Curves
7 The Riemann—Roch Theorem on Curves
7.1 Statement of the Theorem
7.2 Preliminary Form of the Riemann—Roch Theorem
7.3 The Residue of a 1—Form
7.4 Linear Algebra in Infinite Dimensional Vector Spaces
7.5 The Residue Theorem
7.6 The Duality Theorem
7.7 Exercises to Sections 6—7
8 Higher Dimensional Generalisations
4 Intersection Numbers
1 Definition and Basic Properties
1.1 Definition of Intersection Number
1.2 Additivity
1.3 Invariance Under Linear Equivalence
1.4 The General Definition of Intersection Number
1.5 Exercises to Section 1
2 Applications of Intersection Numbers
2.1 Bezout’’s Theorem in Projective and Multiprojective Space
2.2 Varieties over the Reals
2.3 The Genus of’’a Nonsingular Curve on a Surface
2.4 The Riemann—Roch Inequality on a Surface
2.5 The Nonsingular Cubic Surface
2.6 The Ring of Cycle Classes
2.7 Exercises to Section 2
3 Birational Maps of Surfaces
3.1 Blowups of Surfaces
3.2 Some Intersection Numbers
3.3 Resolution of Indeterrninacy
3.4 Factorisation as a Chain of Blowups
3.5 Remarks and Examples
3.6 Exercises to Section 3
4 Singularities
4.1 Singular Points of a Curve
4.2 Surface Singularities
4.3 Du Val Singularities
4.4 Degeneration of Curves
4.5 Exercises to Section 4
Algebraic Appendix
1 Linear and Bilinear Algebra
2 Polynomials
3 Quasilinear Maps
4 Invariants
5 Fields
6 Commutative Rings
7 Unique Factorisation
8 Integral Elements
9 Length of a Module
References
Index
……
Book 2： Schemes and Varieties
Book 3：Complex Algebraic Varieties and Complex Manifolds
Historical Sketch
References
Index
