Complex Algebraic Curves

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 Bézout\'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