Complex Algebraic Curves

Front Cover
Cambridge University Press, Feb 20, 1992 - Mathematics - 264 pages
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.
 

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Contents

Introduction and background
1
11 A brief history of algebraic curves
2
12 Relationship with other parts of mathematics
9
122 Singularities and the theory of knots
10
123 Complex analysis
15
124 Abelian integrals
17
13 Real Algebraic Curves
20
131 Hilberts Nullstellensatz
21
42 Branched covers of P₁
94
43 Proof of the degreegenus formula
98
44 Exercises
110
Riemann surfaces
111
52 Riemann surfaces
124
53 Exercises
138
Differentials on Riemann surfaces
143
62 Abels theorem
152

132 Techniques for drawing real algebraic curves
22
133 Real algebraic curves inside complex algebraic curves
24
Foundations
29
22 Complex projective spaces
34
23 Complex projective curves in P₂
40
24 Affine and projective curves
42
25 Exercises
46
Algebraic properties
51
32 Points of inflection and cubic curves
70
33 Exercises
78
Topological properties
85
41 The degreegenus formula
87
412 The second method of proof
90
63 The RiemannRoch theorem
159
64 Exercises
177
Singular curves
185
72 Newton polygons and Puiseux expansions
203
73 The topology of singular curves
213
74 Exercises
222
Algebra
227
Complex analysis
229
Topology
235
C2 The genus is a topological invariant
240
C3 Spheres with handles
249
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