代数几何入门

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。目次：仿射代数变量；代数基础；射影变量；Quasi 射影变量；经典结构；光滑；双有理几何学；映射到射影空间。 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 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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