本书旨在深层次讲述代数几何原理、20世纪的一些重要进展和数学实践中正在探讨的问题。该书的内容对于对代数几何不是很了解或了解甚少，但又想要了解代数几何基础的数学工作者是非常有用的。

本书适用于数学专业高年级本科生、研究生和与该领域有关的工作者。

Notes for the Second Printing

Preface

Acknowledgments

Index of Notation

1 Affine Algebraic Varieties

　1.1 Definition and Examples

　1.2 The Zariski Topology

　1.3 Morphisms of Affine Algebraic Varieties

　1.4 Dimension

2 Algebraic Foundations

　2.1 A Quick Review of Commutative Ring Theory

　2.2 Hilbert's Basis Theorem

　2.3 Hilbert's NuUstellensatz

　2.4 The Coordinate Ring

　2.5 The Equivalence of Algebra and Geometry

　2.6 The Spectrum of a Ring

3 Projective Varieties

　3.1 Projective Space

　3.2 Projective Varieties

　3.3 The Projective Closure of an Affine Variety

　3.4 Morphisms of Projective Varieties

　3.5 Automorphisms of Projective Space

4 Quasi-Projective Varieties

　4.1 Quasi-Projective Varieties

　4.2 A Basis for the Zariski Topology

　4.3 Regular Functions

5 Classical Constructions

　5.1 Veronese Maps

　5.2 Five Points Determine a Conic

　5.3 The Segre Map and Products of Varieties

　5.4 Grassmannians

　5.5 Degree

　5.6 The Hilbert Function

6 Smoothness

　6.1 The Tangent Space at a Point

　6.2 Smooth Points

　6.3 Smoothness in Families

　6.4 Bertini's Theorem

　6.5 The Gauss Mapping

7 Birational Geometry

　7.1 Resolution of Singularities

　7.2 Rational Maps

　7.3 Birational Equivalence

　7.4 Blowing Up Along an Ideal

　7.5 Hypersurfaces

　7.6 The Classification Problems

8 Maps to Projective Space

　8.1 Embedding a Smooth Curve in Three-Space

　8.2 Vector Bundles and Line Bundles

　8.3 The Sections of a Vector Bundle

　8.4 Examples of Vector Bundles

　8.5 Line Bundles and Rational Maps

　8.6 Very Ample Line Bundles

A Sheaves and Abstract Algebraic Varieties

　A.1 Sheaves

　A.2 Abstract Algebraic Varieties

References

Index