基本介绍

内容简介

《代数曲线几何初步》由世界图书出版公司出版。

作者简介

作者:（英）菌吉布森（Gibson、C.G）

图书目录

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

文摘

插图：

序言

For some time I have felt there is a good case fob raising the profile o! undergraduate geometry The case can be argued on academic grounds alone Geometry represents a way of thinking within mathematics，quite distinct from algebra and analysis，and So offers a fresh perspective on the subject It can also be argued on purely practical grounds My experience is that there is a measure of concern in various practical disciplines where geometry plays a substantial role（engineering science for instance） that their students no longer receive a basic geometric training And thirdly，it can be argued on psychological grounds Few would deny that substantial areas of mathematics fail to excite student interest：yet there are many students attracted to geometry by its sheer visual content The decline in undergraduate geometry is a bit of a mystery It probably has something to do with the fashion for formalism which seemed to permeate mathematics some decades ago But things are changing The enormous progress made in studying non.1inear phenomena by geometrical methods has certainly revived interest in geometry And for material reasons， tertiary institutions are ever more conscious of the need to offer their students more attractive courses. 0.1 General Background I first became involved in the teaching of geometry about twenty years ago，when my department introduced an optional second year course on the geometry of plane curves，partly to redress the imbalance in the teaching of the subject。It Was mildly revolutionary，since it went back to an earlier sct of precepts where the differential and algebraic geometry of cuwes were pursued simultaneously,to their mutua！advantage. In the final year of study students could pursue this kind of geometry