Elementary Geometry of Algebraic Curves: An Undergraduate Introduction

by C. G. Gibson

• ISBN13: 9780521641401
• ISBN10: 0521641403
• Format: Hardcover
• Copyright: 1999-01-13
• Publisher: Cambridge University Press

Summary

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. The book contains several hundred worked examples and exercises, making it suitable for adoption as a course text. From the 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 of algebra leads to the famous theorem of Bezout, whilst the ideas of linear systems are used to discuss the classical group structure on the cubic.

Table of Contents

List of Illustrations	x
List of Tables	xii
Preface	xiii
Real Algebraic Curves	1	(19)
Parametrized and Implicit Curves	1	(2)
Introductory Examples	3	(9)
Curves in Planar Kinematics	12	(8)
General Ground Fields	20	(13)
Two Motivating Examples	20	(2)
Groups, Rings and Fields	22	(3)
General Affine Planes and Curves	25	(3)
Zero Sets of Algebraic Curves	28	(5)
Polynomial Algebra	33	(14)
Factorization in Domains	33	(2)
Polynomials in One Variable	35	(3)
Polynomials in Several Variables	38	(4)
Homogeneous Polynomials	42	(2)
Formal Differentiation	44	(3)
Affine Equivalence	47	(13)
Affine Maps	48	(1)
Affine Equivalent Curves	49	(2)
Degree as an Affine Invariant	51	(2)
Centres as Affine Invariants	53	(7)
Affine Conics	60	(11)
Affine Classification	61	(3)
The Delta Invariants	64	(4)
Uniqueness of Equations	68	(3)
Singularities of Affine Curves	71	(14)
Intersection Numbers	71	(6)
Multiplicity of a Point on a Curve	77	(3)
Singular Points	80	(5)
Tangents to Affine Curves	85	(10)
Generalities about Tangents	85	(1)
Tangents at Simple Points	86	(2)
Tangents at Double Points	88	(3)
Tangents at Points of Higher Multiplicity	91	(4)
Rational Affine Curves	95	(13)
Rational Curves	96	(5)
Diophantine Equations	101	(5)
Conics and Integrals	106	(2)
Projective Algebraic Curves	108	(17)
The Projective Plane	108	(2)
Projective Lines	110	(5)
Affine Planes in the Projective Plane	115	(2)
Projective Curves	117	(1)
Affine Views of Projective Curves	118	(7)
Singularities of Projective Curves	125	(12)
Intersection Numbers	125	(4)
Multiplicity of a Point on a Curve	129	(1)
Singular Points	130	(4)
Delta Invariants Viewed Projectively	134	(3)
Projective Equivalence	137	(11)
Projective Maps	137	(3)
Projective Equivalence	140	(2)
Projective Conics	142	(2)
Affine and Projective Equivalence	144	(4)
Projective Tangents	148	(14)
Tangents to Projective Curves	148	(1)
Tangents at Simple Points	149	(1)
Centres viewed Projectively	150	(2)
Foci viewed Projectively	152	(4)
Tangents at Singular Points	156	(3)
Asymptotes	159	(3)
Flexes	162	(11)
Hessian Curves	163	(6)
Configurations of Flexes	169	(4)
Intersections of Projective Curves	173	(17)
The Geometric Idea	173	(2)
Resultants in One Variable	175	(2)
Resultants in Several Variables	177	(2)
Bezout's Theorem	179	(5)
The Multiplicity Inequality	184	(3)
Invariance of the Intersection Number	187	(3)
Projective Cubics	190	(11)
Geometric Types of Cubics	190	(4)
Cubics of General Type	194	(2)
Singular Irreducible Cubics	196	(2)
Reducible Cubics	198	(3)
Linear Systems	201	(16)
Projective Spaces of Curves	201	(2)
Pencils of Curves	203	(4)
Solving Quartic Equations	207	(1)
Subspaces of Projective Spaces	208	(2)
Linear Systems of Curves	210	(3)
Dual Curves	213	(4)
The Group Structure on a Cubic	217	(17)
The Nine Associated Points	217	(4)
The Star Operation	221	(1)
Cubics as Groups	222	(4)
Group Computations	226	(3)
Determination of the Groups	229	(5)
Rational Projective Curves	234	(13)
The Projective Concept	234	(2)
Quartics with Three Double Points	236	(4)
The Deficiency of a Curve	240	(2)
Some Rational Curves	242	(3)
Some Non-Rational Curves	245	(2)
Index	247