代数几何入门

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 Hilberts Basis Theorem

2.3 Hilberts 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 Bertinis 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

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　　读者对象：《代数几何入门(英文版)》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。