## Complex Algebraic CurvesThis development of the theory of complex algebraic curves was one of the peaks of nineteenth century mathematics. They have many fascinating properties and arise in various areas of mathematics, from number theory to theoretical physics, and are the subject of much research. By using only the basic techniques acquired in most undergraduate courses in mathematics, Dr. Kirwan introduces the theory, observes the algebraic and topological properties of complex algebraic curves, and shows how they are related to complex analysis. |

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### Contents

Introduction and background | 1 |

11 A brief history of algebraic curves | 2 |

12 Relationship with other parts of mathematics | 9 |

122 Singularities and the theory of knots | 10 |

123 Complex analysis | 15 |

124 Abelian integrals | 17 |

13 Real Algebraic Curves | 20 |

131 Hilberts Nullstellensatz | 21 |

42 Branched covers of P₁ | 94 |

43 Proof of the degreegenus formula | 98 |

44 Exercises | 110 |

Riemann surfaces | 111 |

52 Riemann surfaces | 124 |

53 Exercises | 138 |

Differentials on Riemann surfaces | 143 |

62 Abels theorem | 152 |

132 Techniques for drawing real algebraic curves | 22 |

133 Real algebraic curves inside complex algebraic curves | 24 |

Foundations | 29 |

22 Complex projective spaces | 34 |

23 Complex projective curves in P₂ | 40 |

24 Affine and projective curves | 42 |

25 Exercises | 46 |

Algebraic properties | 51 |

32 Points of inflection and cubic curves | 70 |

33 Exercises | 78 |

Topological properties | 85 |

41 The degreegenus formula | 87 |

412 The second method of proof | 90 |

63 The RiemannRoch theorem | 159 |

64 Exercises | 177 |

Singular curves | 185 |

72 Newton polygons and Puiseux expansions | 203 |

73 The topology of singular curves | 213 |

74 Exercises | 222 |

Algebra | 227 |

Complex analysis | 229 |

Topology | 235 |

C2 The genus is a topological invariant | 240 |

C3 Spheres with handles | 249 |

### Common terms and phrases

applied assume atlas bijection called Chapter chart choose closed coefficients compact complex algebraic curves connected component consider contains continuous map coordinates corollary cubic curve curve defined curve in P2 Deduce defined definition distinct edges equation equivalent exactly example exercise exists face fact factors figure finite follows formula function f given gives hence holomorphic function homeomorphism homogeneous polynomial identically integral inverse irreducible least lemma line in P2 meromorphic function Moreover multiplicity nonsingular projective curve nonzero Note open neighbourhood open subset pair path point of inflection pole polynomial P(x projective curve projective transformation Proof proposition prove rational Remark respect restriction result Riemann surface satisfying sense Sing(C singular point space sphere Suppose takes tangent line theorem topologically triangulation unique vertices zero