Complex Algebraic Curves

Description

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