Contents
Preface
1 Introduction and background
1 A brief history of algebraic curves
2 Relationship with other parts of mathematics
1 Number theory
2 Singularities and the theory of knots
3 Complex analysis
4 Abelian integrals
3 Real Algebraic Curves
1 Hilbert's Nullstellensatz
2 Techniques for drawing real algebraic curves
3 Real algebraic curves inside complex algebraic curves
4 Important examples of real algebraic curves
2 Foundations
1 Complex algebraic curves in C^2
2 Complex projective spaces
3 Complex projective curves in P_2
4 Affine and projective curves
5 Exercises
3 Algebraic properties
1 Bรฉzout's theorem
2 Points of inflection and cubic curves
3 Exercises
4 Topological properties
1 The degree-genus formula
1 The first method of proof
2 The second method of proof
2 Branched covers of P_1
3 Proof of the degree-genus formula
4 Exercises
5 Riemann surfaces
1 The Weierstrass p-function
2 Riemann surfaces
3 Exercises
6 Differentials on Riemann surfaces
1 Holomorphic differentials
2 Abel's theorem
3 The Riemann-Roch theorem
4 Exercises
7 Singular curves
1 Resolution of singularities
2 Newton polygons and Puiseux expansions
3 The topology of singular curves
4 Exercises
A Algebra
B Complex analysis
C Topology
1 Covering projections
2 The genus is a topological invariant
3 Spheres with handles