Elementary Geometry of Algebraic Curves: An Undergraduate Introduction

268 Pages·1999·4.96 MB·English

by  Jamal T. Manassah

Elementary Geometry of Algebraic Curves: an Undergraduate
Introduction
This is a genuine introduction to plane algebraic curves from a geometric
viewpoint, designed as a first text for undergraduates in mathematics, or
for postgraduate and research workers in the engineering and physical
sciences. It is well illustrated, and contains several hundred worked
examples and exercises, making it suitable for adoption as a course text.
From the familiar lines and conics of elementary geometry the reader
proceeds to general curves in the real affine plane, with excursions to
more general fields to illustrate applications, such as number theory. By
adding points at infinity the affine plane is extended to the projective
plane, yielding a natural setting for curves and providing a flood of
illumination into the underlying geometry. A minimal amount ofalgebra
leads to the famous theorem ofBezout, whilst the ideas oflinear systems
are used to discuss the classical group structure on the cubic.
c. G. Gibson is Senior Fellow in Mathematical Sciences, University of
Liverpool.
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Elementary Geometry of Algebraic
Curves: an Undergraduate Introduction
c. G. Gibson
!~'N CAMBRIDGE
~~~ UNIVERSITY PRESS
PUBLISHEDBYTHEPRESSSYNDICATEOFTHE UNIVERSITYOFCAMBRIDGE
The Pitt Building, TrumpingtonStreet,Cambridge CB2 1RP,United Kingdom
CAMBRIDGEUNIVERSITYPRESS
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© CambridgeUniversity Press 1998
This book is incopyright. Subject to statutoryexception
and to the provisions ofrelevant collectivelicensingagreements,
no reproduction ofany part may take place without
the written permission ofCambridgeUniversity Press
First published 1998
Printed in the United Kingdom at the UniversityPress, Cambridge
TypesetinTimes 10/13pt,in Jb.TEX
A catalogue recordfor this book is availablefrom the British Library
Library ofCongress Cataloguing in Publication data
Gibson, ChristopherG., 1940- .
Elementary geometry ofalgebraiccurves / C. G. Gibson.
p. cm.
ISBN0 521 64140 3(hardbound)
1. Curves, Algebraic. I. Title.
QA565.G5 1998
516.3'52-dc21 98-36910 CIP
ISBN0 521 64140 3hardback
ISBN0 521 64641 3paperback
To Dorle
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Contents
List ofIllustrations page x
List ofTables xii
Preface xiii
1 Real Algebraic Curves 1
1.1 Parametrized and Implicit Curves 1
1.2 Introductory Examples 3
1.3 Curves in Planar Kinematics 12
2 General Ground Fields 20
2.1 Two Motivating Examples 20
2.2 Groups, Rings and Fields 22
2.3 General Affine Planes and Curves 25
2.4 Zero Sets ofAlgebraic Curves 28
3 Polynomial Algebra 33
3.1 Factorization in Domains 33
3.2 Polynomials in One Variable 35
3.3 Polynomials in Several Variables 38
3.4 Homogeneous Polynomials 42
3.5 Formal Differentiation 44
4 Affine Equivalence 47
4.1 Affine Maps 48
4.2 Affine Equivalent Curves 49
4.3 Degree as an Affine Invariant 51
4.4 Centres as Affine Invariants 53
5 Affine Conics 60
5.1 Affine Classification 61
5.2 The Delta Invariants 64
5.3 Uniqueness ofEquations 68
VB
vIn Contents
6 Singularities of Affine Curves 71
6.1 Intersection Numbers 71
6.2 Multiplicity ofa Point on a Curve 77
6.3 Singular Points 80
7 Tangents to Affine Curves 85
7.1 Generalities about Tangents 85
7.2 Tangents at Simple Points 86
7.3 Tangents at Double Points 88
7.4 Tangents at Points ofHigher Multiplicity 91
8 Rational Affine Curves 95
8.1 Rational Curves 96
8.2 Diophantine Equations 101
8.3 Conics and Integrals 106
9 Projective Algebraic Curves 108
9.1 The Projective Plane 108
9.2 Projective Lines 110
9.3 Affine Planes in the Projective Plane 115
9.4 Projective Curves 117
9.5 Affine Views ofProjective Curves 118
10 Singularities of Projective Curves 125
10.1 Intersection Numbers 125
10.2 Multiplicity ofa Point on a Curve 129
10.3 Singular Points 130
10.4 Delta Invariants viewed Projectively 134
11 Projective Equivalence 137
11.1 Projective Maps 137
11.2 Projective Equivalence 140
11.3 Projective Conics 142
11.4 Affine and Projective Equivalence 144
12 Projective Tangents 148
12.1 Tangents to Projective Curves 148
12.2 Tangents at Simple Points 149
12.3 Centres viewed Projectively 150
12.4 Foci viewed Projectively 152
12.5 Tangents at Singular Points 156
12.6 Asymptotes 159
13 Flexes 162
13.1 Hessian Curves 163
13.2 Configurations ofFlexes 169
Contents IX
14 Intersections of Projective Curves 173
14.1 The Geometric Idea 173
14.2 Resultants in One Variable 175
14.3 Resultants in Several Variables 177
14.4 Bezout's Theorem 179
14.5 The Multiplicity Inequality 184
14.6 Invariance ofthe Intersection Number 187
15 Projective Cubics 190
15.1 Geometric Types ofCubics 190
15.2 Cubics ofGeneral Type 194
15.3 Singular Irreducible Cubics 196
15.4 Reducible Cubics 198
16 Linear Systems 201
16.1 Projective Spaces ofCurves 201
16.2 Pencils ofCurves 203
16.3 Solving Quartic Equations 207
16.4 Subspaces ofProjective Spaces 208
16.5 Linear Systems ofCurves 210
16.6 Dual Curves 213
17 The Group Structure on a Cubic 217
17.1 The Nine Associated Points 217
17.2 The Star Operation 221
17.3 Cubics as Groups 222
17.4 Group Computations 226
17.5 Determination ofthe Groups 229
18 Rational Projective Curves 234
18.1 The Projective Concept 234
18.2 Quartics with Three Double Points 236
18.3 The Deficiency ofa Curve 240
18.4 Some Rational Curves 242
18.5 Some Non-Rational Curves 245
Index 247