## Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and SchemesThis book consists of two parts. The first is devoted to the theory of curves, which are treated from both the analytic and algebraic points of view. Starting with the basic notions of the theory of Riemann surfaces the reader is lead into an exposition covering the Riemann-Roch theorem, Riemann's fundamental existence theorem, uniformization and automorphic functions. |

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### Contents

Riemann Surfaces and Algebraic Curves | 5 |

Introduction by I R Shafarevich | 10 |

Riemann Surfaces | 16 |

Copyright | |

9 other sections not shown

### Common terms and phrases

affine variety algebraic variety analytic assume automorphism base bundle called canonical Chap characteristic closed complex connected consider consists construction contains coordinates Corollary corresponding covering cubic curve cycles defined definition degree denoted differential dimension divisor easy effective elements embedding equal equations equivalent Example exists fact fibre field finite formula function Further genus g geometry given Hence holomorphic hyperplane ideal instance intersection inverse irreducible isomorphic linear system locally manifold mapping means meromorphic function morphism multiplicity natural neighbourhood nonsingular normal notion obtain open subset parameter particular plane polynomial precisely principal projective proof Proposition prove rational function regular relation Remark respectively Riemann surface ring scheme Sect sheaf simple singular smooth space structure subvariety Suppose tangent theorem theory topology unique vector zeros