代数曲线几何初步

出版时间：2009-1  出版社：世界图书出版公司  作者：菌吉布森  页数：250  
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前言

　　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

内容概要

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.

书籍目录

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

用户评论 (总计2条)

•   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