代数曲线几何初步
2009-1
世界图书出版公司
菌吉布森
250
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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
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.
List of IllustrationsList of TablesPreface1 Real Algebraic Curves 1.1 Parametrized and Implicit Curves 1.2 Introductory Examples 1.3 Curves in Planar Kinematics2 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 Curves3 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 Differentiation4 Atfine Equivalence 4.1 Affine Maps 4.2 Affline Equivalent Curves 4.3 Degree as an Affine Invariant 4.4 Centres as Affine Invariants5 Affline Conics 5.1 Affline Classification 5.2 The Delta Invariants 5.3 Uniqueness of Equations6 Singularities of Afline Curves 6.1 Intersection Numbers 6.2 Multiplicity of a Point on a Curve 6.3 Singular Points7 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 Multiplicity8 Rational Afline Curves 8.1 Rational Curves 8.2 Diophantine Equations 8.3 Conics and Integrals9 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 Curves10 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 Projectively11 Projective Equivalence 11.1 Projective Maps 11.2 Projective Equivalence 11.3 Projective Conics 11.4 Afline and Projective Equivalence12 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 Asymptotes13 Flexes 13.1 Hessian Curves 13.2 Configurations of Flexes14 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 Number15 Proiective Cubics 15．1 Geometric Types 0f Cubics 15．2 Cubics of General Type 15．3 Singular Irreducible Cubics 15．4 Reducible Cubics16 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 CulNes17 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 Groups18 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 CurvesIndex
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代数曲线几何初步是一本为本科生编写的教材，但是其中很多内容也适合更高层次的读者。特别是作者对于代数曲线的历史和背景的介绍，以及其中大量的描述性语言以及丰富的图片，都使得本书非常适合自学，对于增进读者对于代数曲线中很多复杂深刻概念的理解，有着很形象和直观的帮助。
经典代数曲线