This book on complex algebraic curves is intended to be accessible to any third year mathematics undergraduate who has attended courses on algebra,topology and complex analysis. It is an expanded version of notes written to accompany a lecture course given to third year undergraduates at Oxford.It has usually been the case that a number of graduate students have also attended the course, and the lecture notes have been extended somewhat for the sake of others in their position. However this new material is not intended to daunt undergraduates, who can safely ignore it. The original lecture course consisted of Chapters 1 to 5 (except for some of §3.1 including the definition of intersection multiplicities) and part of Chapter 6, although some of the contents of these chapters (particularly the introductory material in Chapter 1) was covered rather briefly.

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 C2

 2.2 Complex projective spaces

 2.3 Complex projective curves in P2

 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 degreo-genus formula

4.1.1 The first method of proof

4.1.2 The second method of proof

 4.2 Branched covers of P1

 4.3 Proof of the degree-genus formula

 4.4 Exercises

5 Riemann surfaces

 5.1 The Weierstrass p-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