## Algebraic GeometryRobin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. After receiving his Ph.D. from Princeton in 1963, Hartshorne became a Junior Fellow at Harvard, then taught there for several years. In 1972 he moved to California where he is now Professor at the University of California at Berkeley. He is the author of "Residues and Duality" (1966), "Foundations of Projective Geometry (1968), "Ample Subvarieties of Algebraic Varieties" (1970), and numerous research titles. His current research interest is the geometry of projective varieties and vector bundles. He has been a visiting professor at the College de France and at Kyoto University, where he gave lectures in French and in Japanese, respectively. Professor Hartshorne is married to Edie Churchill, educator and psychotherapist, and has two sons. He has travelled widely, speaks several foreign languages, and is an experienced mountain climber. He is also an accomplished amateur musician: he has played the flute for many years, and during his last visit to Kyoto he began studying the shakuhachi. |

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### Contents

II | 1 |

III | 8 |

IV | 14 |

V | 24 |

VI | 31 |

VII | 39 |

VIII | 47 |

IX | 55 |

XXXIV | 299 |

XXXV | 307 |

XXXVI | 316 |

XXXVII | 340 |

XXXVIII | 349 |

XXXIX | 356 |

XL | 357 |

XLI | 369 |

X | 60 |

XI | 69 |

XII | 82 |

XIII | 95 |

XIV | 108 |

XV | 129 |

XVI | 149 |

XVII | 172 |

XVIII | 190 |

XIX | 201 |

XX | 202 |

XXI | 206 |

XXII | 213 |

XXIII | 218 |

XXIV | 225 |

XXV | 233 |

XXVI | 239 |

XXVII | 250 |

XXVIII | 253 |

XXIX | 268 |

XXX | 276 |

XXXI | 281 |

XXXII | 293 |

XXXIII | 294 |

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1-module abelian groups algebraically closed field automorphism base points birational Cartier divisor closed immersion closed point closed subscheme closed subset coherent sheaf cohomology corresponding curve of degree curve of genus define definition denote dimension elements elliptic curve embedding exact sequence Example fibre finite morphism finite type flasque flat follows function field functor genus g gives global sections Grothendieck Hence homogeneous homomorphism induced injective integral invertible sheaf isomorphism Lemma linear system linearly equivalent locally free sheaf maximal ideal module monoidal transformation morphism f:X multiplicity natural map noetherian ring noetherian scheme nonsingular curve nonsingular projective nonsingular variety open affine subset open set open subset plane polynomial prime ideal Proj projective space projective variety PROOF Proposition quasi-coherent quasi-coherent sheaf quotient rational regular functions Riemann-Roch ringed space ruled surface sheaf of ideals sheaves singular Spec surjective tangent theorem topological space unique valuation ring Zariski

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