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书名 代数几何入门
分类 科学技术-自然科学-数学
作者 (美)史密斯
出版社 世界图书出版公司
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简介
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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。

目录

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

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