代数曲线几何初步

List of Illustrations
List of Tables
Preface
1 Real Algebraic Curves
　1.1 Parametrized and Implicit Curves
　1.2 Introductory Examples
　1.3 Curves in Planar Kinematics
2 General Ground Fields
　2.1 Two Motivating Examples
　2.2 Groups, Rings and Fields
　2.3 General Affine Planes and Curves
　2.4 Zero Sets of Algebraic Curves
3 Polynomial Algebra
　3.1 Factorization in Domains
　3.2 Polynomials in One Variable
　3.3 Polynomials in Several Variables
　3.4 Homogeneous Polynomials
　3.5 Formal Differentiation
4 Atfine Equivalence
　4.1 Affine Maps
　4.2 Affline Equivalent Curves
　4.3 Degree as an Affine Invariant
　4.4 Centres as Affine Invariants
5 Affline Conics
　5.1 Affline Classification
　5.2 The Delta Invariants
　5.3 Uniqueness of Equations
6 Singularities of Afline Curves
　6.1 Intersection Numbers
　6.2 Multiplicity of a Point on a Curve
　6.3 Singular Points
7 Tangents to Afline Curves
　7.1 Generalities about Tangents
　7.2 Tangents at Simple Points
　7.3 Tangents at Double Points
　7.4 Tangents at Points of Higher Multiplicity
8 Rational Afline Curves
　8.1 Rational Curves
　8.2 Diophantine Equations
　8.3 Conics and Integrals
9 Projective Algebraic Curves
　9.1 The Projective Plane
　9.2 Projective Lines
　9.3 Atfine Planes in the Projective Plane
　9.4 Projective Curves
　9.5 Affine Views of Projective Curves
10 Singularities of Projective Curves
　10.1 Intersection Numbers
　10.2 Multiplicity of a Point on a Curve
　10.3 Singular Points
　10.4 Delta Invariants viewed Projectively
11 Projective Equivalence
　11.1 Projective Maps
　11.2 Projective Equivalence
　11.3 Projective Conics
　11.4 Afline and Projective Equivalence
12 Projective Tangents
　12.1 Tangents to Projective Curves
　12.2 Tangents at Simple Points
　12.3 Centres viewed Projectively
　12.4 Foci viewed Projectively
　12.5 Tangents at Singular Points
　12.6 Asymptotes
13 Flexes
　13.1 Hessian Curves
　13.2 Configurations of Flexes
14 Intersections of Proiective Curves
　14．1 The Geometric Idea
　14．2 Resultants in One Variable
　14．3 Resultants in Severa!Variables
　14．4 B6zout’S Theorem
　14．5 Thc Multiplicity Inequality
　14．6 Invariance of the Intersection Number
15 Proiective Cubics
　15．1 Geometric Types 0f Cubics
　15．2 Cubics of General Type
　15．3 Singular Irreducible Cubics
　15．4 Reducible Cubics
16 Linear Systems
　16．1 Projective Spaces of Curves
　16．2 Pcncils of CuiNes
　16．3 Solving Quartic Equations
　16．4 Subspaces or Projective Spaces
　16．5 Linear Systems of Culwes
　16．6 Dual CulNes
17 The Group Structure on a Cubic
　17．1 The Nine Associated Points
　17．2 The Star Operation
　17．3 Cubics as Groups
　17．4 Group Computations
　17．5 Determination of the Groups
18 Rational Projective Curves
　18．1 Thc Projective Concept
　18．2 Quartics with Three Double Points
　18．3 Thc Deficiency of a CHIve
　18．4 Some Rational Curves
　18．5 Some Non-Rational Curves
Index