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Read e-book online Algebraic geometry I. Algebraic curves, manifolds, and PDF

By I. R. Shafarevich

ISBN-10: 0387519955

ISBN-13: 9780387519951

This quantity of the Encyclopaedia comprises components. the 1st is dedicated to the speculation of curves, that are handled from either the analytic and algebraic issues of view. beginning with the elemental notions of the idea of Riemann surfaces the reader is lead into an exposition protecting the Riemann-Roch theorem, Riemann's primary lifestyles theorem, uniformization and automorphic services. The algebraic fabric additionally treats algebraic curves over an arbitrary box and the relationship among algebraic curves and Abelian forms. the second one half is an advent to higher-dimensional algebraic geometry. the writer offers with algebraic types, the corresponding morphisms, the idea of coherent sheaves and, eventually, the speculation of schemes. This booklet is a really readable advent to algebraic geometry and should be immensely worthwhile to mathematicians operating in algebraic geometry and intricate research and particularly to graduate scholars in those fields.

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Extra resources for Algebraic geometry I. Algebraic curves, manifolds, and schemes

Example text

Soit C une courbe lisse sur F, soit A l’anneau local de C en un point ferm´e de corps r´esiduel E, soit X un A-sch´ema r´egulier, propre et plat sur A, a` fibre g´en´erique X /K lisse, a` fibre sp´eciale un diviseur Y /E a` croisements normaux stricts. Si X est une K-vari´et´e s´eparablement rationnellement connexe, alors il existe une composante r´eduite Yi de Y qui est g´eom´etriquement irr´eductible sur E. 10 vaut plus g´en´eralement lorsque la fibre g´en´erique est une vari´et´e rationnellement connexe.

De Jong et Starr (travaux en cours) ont obtenu une s´erie de r´esultats sur les intersections compl`etes lisses dans l’espace projectif. Pour simplifier, je cite leurs r´esultats pour les hypersurfaces. 6 (de Jong-Starr) [43] Une hypersurface lisse de degr´e d ≥ 2 dans PnC avec n ≥ d2 − 1 est rationnellement simplement connexe, a` l’exception des quadriques dans P3C . 7 (de Jong-Starr) [43] Une hypersurface lisse de degr´e d ≥ 2 dans PnC avec n ≥ 2d 2 − d − 1 est fortement rationnellement simplement connexe.

On est sur un corps de nombres k, X et Y sont deux k-vari´et´es lisses g´eom´etriquement int`egres, la k-vari´et´e Y est projective, on a un k-morphisme projectif f : X → Y de fibre g´en´erique g´eom´etriquement int`egre. On note U ⊂ X l’ouvert de lissit´e du morphisme f . On demande quels sont les liens entre la g´eom´etrie du morphisme f et les propri´et´es de surjectivit´e des applications induites X(kv ) → Y (kv ) pour presque toute place v, ou d´ej`a pour une infinit´e de places v du corps de nombres k.

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Algebraic geometry I. Algebraic curves, manifolds, and schemes by I. R. Shafarevich


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