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Complex algebraic surfaces - Beauville A.

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Название: Complex algebraic surfaces
Автор: Beauville A.
Категория: Математика
Тип: Книга
Дата: 30.12.2008 16:24:30
Скачано: 52
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Описание: This book is a modified version of a course given at Orsay in 1976-77. The !aim of the course was to give a comparatively elementary proof of the Enriques classification of complex algebraic surfaces, accessible to a student familiar with the basic language of algebraic geometry (divisors, differential forms, ...) as well as sheaf cohomology. I have, however, preferred to assume along the way various hard theorems from algebraic jgeometry, rather than resort to complicated and artificial proofs. v Here is an outline of the course. The first two chapters introduce the basic tools for the study of surfaces: in Chapter I we define the intersection form on the Picard group, and establish its properties; assuming the duality theorem we deduce the fundamental results (the Riemann-Roch theorem, the genus formula). Chapter II is devoted to the structure of birational maps; we show in particular that every surface is obtained from a minimal surface by a finite number of blow-ups. The chapter ends with Castelnuovo's contractibility criterion, which characterizes exceptional curves by their numerical properties. The classification begins in Chapter III with ruled surfaces, that is, surfaces birational to P1 x C. We show that (except in the rational case) their minimal models are P1-bundles over a base curve C, and we study their geometry. Chapter IV gives some examples of rational surfaces; we take a stroll through the huge menagerie collected by the geometers of the 19th century (the Veronese surface, del Pezzo surfaces, ...). The next two chapters are perhaps the keystone of the classification; they give the characterization of ruled surfaces by their numerical properties - more precisely, by the vanishing of the 'plurigenera' Pn. Surfaces with q = 0 are treated in Chapter V, where we prove Castelnuovo's theorem: a surface with q = P2 = 0 is rational. We deduce two important consequences: the structure of minimal rational surfaces and the
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