cover -- copyright page -- title page -- Preface -- It's for. . . -- What this book is, and what it isn't. -- What are the prerequisites for this book? -- Why should I be interested,in algebraic curves? -- A Bit of Perspective. -- The Book's Story Line . . . -- Many thanks to . . . -- Contents -- CHAPTER 1 A Gallery of Algebraic Curves -- 1.1 Curves of Degree One and Two -- Degree One -- Degree Two -- 1.2 Curves of Degree Three and Higher -- Degree Three -- Higher Degrees -- 1.3 Six Basic Cubics -- 1.4 Some Curves in Polar Coordinates -- Rectangular versus Polar Coordinates -- Algebraic versus Not Algebraic -- The Oppositeness Idea -- 1.5 Parametric Curves -- 1.6 The Resultant -- 1.7 Back to an Example -- 1.8 Lissajous Figures -- 1.9 Morphing Between Curves -- 1.10 Designer Curves -- Linkages -- CHAPTER 2 Points at Infinity -- 2.1 Adjoining Points at Infinity -- 2.2 Examples -- 2.3 A Basic Picture -- 2.4 Basic Definitions -- 2.5 Further Examples -- CHAPTER 3 From Real to Complex -- 3.1 Definitions -- 3.2 The Idea of Multiplicity -- Examples -- 3.3 A Reality Check -- 3.4 A Factorization Theorem for Polynomials in C[x, y] -- 3.5 Local Parametrizations of a Plane Algebraic Curve -- 3.6 Definition of Intersection Multiplicity for Two Branches -- 3.7 An Example -- 3.8 Multiplicity at an Intersection Point of Two Plane Algebraic Curves -- 3.9 Intersection Multiplicity Without Parametrizations -- 3.10 Bézout's theorem -- 3.11 Bézout's theorem Generalizes the Fundamental Theorem of Algebra -- 3.12 An Application of Bézout's theorem: Pascal's theorem -- CHAPTER 4 Topology of AlgebraicCurves in P^2(C) -- 4.1 Introduction -- 4.2 Connectedness -- 4.3 Algebraic Curves are Connected -- 4.4 Orientable Two-Manifolds -- 4.5 Nonsingular Curves are Two-Manifolds -- 4.6 Algebraic Curves are Orientable -- 4.7 The Genus Formula -- CHAPTER 5 Singularities.

5.1 Introduction -- 5.2 Definitions and Examples -- 5.3 Singularities at Infinity -- 5.4 Nonsingular Projective Curves -- 5.5 Singularities and Polynomial Degree -- 5.6 Singularities and Genus -- 5.7 A More General Genus Formula -- 5.8 Non-Ordinary Singularities -- 5.9 Further Examples -- Curves of the Form y^m = x^ n -- An Example with Repeated Tangent Lines -- 5.10 Singularities versus Doing Math on Curves -- 5.11 The Function Field of an Irreducible Curve -- 5.12 Birational Equivalence -- 5.13 Examples of Birational Equivalence -- 5.14 Space-Curve Models -- 5.15 Resolving a Higher-OrderOrdinary Singularity -- 5.16 Examples of Resolving an Ordinary Singularity -- 5.17 Resolving Several Ordinary Singularities -- 5.18 Quadratic Transformations -- CHAPTER 6 The Big Three: C, K, S -- 6.1 Function Fields -- 6.2 Compact Riemann Surfaces -- 6.3 Projective Plane Curves -- 6.4 f_1, f_2, f : Curves and Function Fields -- 6.5 g_1, g_2, g: Compact Riemann Surfaces and Curves -- 6.6 h_1, h_2, h: Function Fields and Compact Riemann Surfaces -- 6.7 Genus -- 6.8 Genus 0 -- 6.9 Genus One -- 6.10 An Analogy -- 6.11 Equipotentials and Streamlines -- 6.12 Differentials Generate Vector Fields -- 6.13 A Major Difference -- Some Perspective: A Little about Calculus -- 6.14 Divisors -- 6.15 The Riemann-Roch theorem -- Bibliography -- Index -- About the Author.