平面代数曲线导引(影印版)

本书是对平面代数曲线 的一个非正式且通俗易懂的 介绍，也是代数几何的一个 自然切入点。这本书有一个 统一的主题：给曲线足够的 生存空间，美丽的定理就会 随之而来。这本书通过具体 的例子和图片介绍抽象的概 念，为读者提供了对主题的 坚实直觉，同时保持了阐述 的简单易懂。它可以作为平 面代数曲线本科课程的教材 ，也可以作为研究生代数几 何的配套教材。数学背景有 限的人可以阅读这本书。这 是因为对于数学之外的人来 说，对代数几何的入门需求 越来越大，代数几何在从生 物学到化学、机器人到密码 学等领域发挥着越来越大的 作用。

Preface 1 A Gallery of Algebraic Curves 1.1 Curves of Degree One and Two 1.2 Curves of Degree Three and Higher 1.3 Six Basic Cubics 1.4 Some Curves in Polar Coordinates 1.5 Parametric Curves 1.6 The Resultant 1.7 Back to an Example 1.8 Lissajous Figures 1.9 Morphing Between Curves 1.10 Designer Curves 2 Points at Infinity 2.1 Adjoining Points at Infinity 2.2 Examples 2.3 A Basic Picture 2.4 Basic Definitions 2.5 Further Examples 3 From Real to Complex 3.1 Definitions 3.2 The Idea of Multiplicity; Examples 3.3 A Reality Check 3.4 A Factorization Theorem for Polynomials in C [x, y] 3.5 Local Parametrizations of a Plane Algebraic Curve 3.6 Definition of Intersection Multiplicity for Two Branches 3.7 An Example 3.8 Multiplicity at an Intersection Point of Two Plane Algebraic Curves 3.9 Intersection Multiplicity Without Parametrizations 3.10 B6zout's theorem 3.11 B6zout's theorem Generalizes the Fundamental Theorem of Algebra 3.12 An Application of B6zout's theorem: Pascal's theorem 4 Topology of Algebraic Curves in ]?2(C) 4.1 Introduction 4.2 Connectedness 4.3 Algebraic Curves are Connected 4.4 Orientable Two-Manifolds 4.5 Nonsingular Curves are Two-Manifolds 4.6 Algebraic Curves are Orientable 4.7 The Genus Formula 5 Singularities 5.1 Introduction 5.2 Definitions and Examples 5.3 Singularities at Infinity 5.4 Nonsingular Projective Curves 5.5 Singularities and Polynomial Degree 5.6 Singularities and Genus 5.7 A More General Genus Formula 5.8 Non-Ordinary Singularities 5.9 Further Examples 5.10 Singularities versus Doing Math on Curves