Complex Algebraic Curves
Douban
Frances Kirwan
overview
Complex algebraic curves were developed in the nineteenth century. They have many fascinating properties and crop up 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 by 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. This book grew from a lecture course given by Dr Kirwan at Oxford University and will be an excellent companion for final year undergraduates and graduates who are studying complex algebraic curves.
contents
1 Introduction and background
1.1 A brief history of algebraic curves
1.2 Relationship with other parts of mathematics
1.2.1 Number theory
1.2.2 Singularities and the theory of knots
1.2.3 Complex analysis
1.2.4 Abelian integrals
1.3 Real Algebraic Curves
1.3.1 Hilbert's Nullstellensatz
1.3.2 Techniques for drawing real algebraic curves
1.3.3 Real algebraic curves inside complex algebraic curves
1.3.4 Important examples of real algebraic curves
2 Foundations
2.1 Complex algebraic curves in Cs
2.2 Complex projective spaces
2.3 Complex projective curves in Ps
2.4 Affine and projective curves
2.5 Exercises
3 Algebraic properties
3.1 Bezout's theorem
3.2 Points of inflection and cubic curves
3.3 Exercises
4 Topological properties
4.1 The degree-genus formula
4.1.1 The first method of proof
4.1.2 The second method of proof
4.2 Branched covers of PI
4.3 Proof of the degree-genus formula
4.4 Exercises
5 Riemann surfaces
5.1 The Weierstrass function
5.2 Riemann surfaces
5.3 Exercises
6 Differentials on Riemann surfaces
6.1 Holomorphic differentials
6.2 Abel's theorem
6.3 The Riemann-Roch theorem
6.4 Exercises
7 Singular curves
7.1 Resolution of Singularities
7.2 Newton polygons and Puiseux expansions
7.3 The topology of singular curves
7.4 Exercises
A Algebra
B Complex analysis
C Topology
C.1 Covering projections
C.2 The genus is a topological invariant
C.3 Spheres with handles
