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Complex Algebraic Curves : London Mathematical Society Student Texts - Frances Kirwan

Complex Algebraic Curves

London Mathematical Society Student Texts


Published: 20th April 1992
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This development of the theory of complex algebraic curves was one of the peaks of nineteenth century mathematics. They have many fascinating properties and arise in various areas of mathematics, from number theory to theoretical physics, and are the subject of much research. By using only the basic techniques acquired in most undergraduate courses in mathematics, Dr. Kirwan introduces the theory, observes the algebraic and topological properties of complex algebraic curves, and shows how they are related to complex analysis.

"Kirwan should succeed admirably with her goal of showing senior undergraduates who perhaps do not plan to study mathematics beyond the bachelor's degree 'how the basic ideas of pure mathematics they had studied in previous years could be brought together in one of the showpieces of mathematics.'" Choice "...worth serious consideration for those planning to teach such a course, and perhaps even more importantly, as an inspiration for mathematicians to offer such a course. The book offers us plenty to work with, and is an excellent contribution." S. Zdravkovska and Daniel M. Burns, Jr., Mathematical Reviews "At last an introduction to the theory of algebraic curves that is understandable by undergraduates, and that does not shy away from the various aspects of the theory: algebraic, topological, complex analytic, etc. Good bedside reading." Gian-Carlo Rota, The Bulletin of Mathematical Books

Introduction and backgroundp. 1
A brief history of algebraic curvesp. 2
Relationship with other parts of mathematicsp. 9
Number theoryp. 9
Singularities and the theory of knotsp. 10
Complex analysisp. 15
Abelian integralsp. 17
Real Algebraic Curvesp. 20
Hilbert's Nullstellensatzp. 21
Techniques for drawing real algebraic curvesp. 22
Real algebraic curves inside complex algebraic curvesp. 24
Important examples of real algebraic curvesp. 24
Foundationsp. 29
Complex algebraic curves in C[superscript 2]p. 29
Complex projective spacesp. 34
Complex projective curves in P[subscript 2]p. 40
Affine and projective curvesp. 42
Exercisesp. 46
Algebraic propertiesp. 51
Bezout's theoremp. 51
Points of inflection and cubic curvesp. 70
Exercisesp. 78
Topological propertiesp. 85
The degree-genus formulap. 87
The first method of proofp. 87
The second method of proofp. 90
Branched covers of P[subscript 1]p. 94
Proof of the degree-genus formulap. 98
Exercisesp. 110
Riemann surfacesp. 111
The Weierstrass [weierp]-functionp. 111
Riemann surfacesp. 124
Exercisesp. 138
Differentials on Riemann surfacesp. 143
Holomorphic differentialsp. 143
Abel's theoremp. 152
The Riemann-Roch theoremp. 159
Exercisesp. 177
Singular curvesp. 185
Resolution of singularitiesp. 185
Newton polygons and Puiseux expansionsp. 203
The topology of singular curvesp. 213
Exercisesp. 222
Algebrap. 227
Complex analysisp. 229
Topologyp. 235
Covering projectionsp. 235
The genus is a topological invariantp. 240
Spheres with handlesp. 249
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521423533
ISBN-10: 0521423538
Series: London Mathematical Society Student Texts
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 276
Published: 20th April 1992
Country of Publication: GB
Dimensions (cm): 22.8 x 15.2  x 1.8
Weight (kg): 0.44