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Über dieses Buch

Lectures: A. Beauville: Surfaces algébriques complexes.- F.A. Bogomolov: The theory of invariants and its applications to some problems in the algebraic geometry.- E. Bombieri: Methods of algebraic geometry in Char. P and their applications.- Seminars: F. Catanese: Pluricanonical mappings of surfaces with K² =1,2, q=pg=0.- F. Catanese: On a class of surfaces of general type.- I. Dolgacev: Algebraic surfaces with p=pg =0.- A. Tognoli: Some remarks about the "Nullstellensatz".



Surfaces Algébriques Complexes

Cet exposé comprend deux parties. La première est un survol assez rapide de la classification d'Enriques des surfaces algébriques. On s'est inspiré, bien entendu, de la littérature classique sur le sujet, et en particulier du séminaire Chafarevitch [Ch.2]. On a essayé d'être aussi élémentaire que possible, en supposant toutefois connue la cohomologie des faisceaux cohérents. On renvoie à [Be] pour une exposition plus détaillée ainsi que pour des exemples.
La seconde partie comprend des indications sur la démonstration par Chafarevitch et Piatechki-Chapiro du théorème de Torelli pour les surfaces K 3 ([Ch.P]).
Arnaud Beauville

Methods of Algebraic Geometry in Char, P and Their Applications

I. The aim of these lectures is to illustrate some of the algebraic techniques needed for algebraic geometry in char, p, with a particular view to the theory of algebraic surfaces and Enriques' classification. We shall study the new char, p features of Kodaira's vanishing theorem, the completeness of the characteristic system and the theory of the Picard variety, together with the study of elliptic and quasi-elliptic fibrations, the study of Enriques' surfaces in char. 2 and the characterization of abelian surfaces by means of their numerical invariants.
Enrico Bombieri

Algebraic Surfaces with q = pg = 0

1. Notations. Let F be a complex algebraic surface. We will use the following standard notations:
  • O F : the structure sheaf of F.
  • O F(D) : the invertible sheaf associated with a divisor D on F.
  • KF, = − c1(F) : minus the first Chern class of F or a canonical divisor on F.
  • ωF = O F(KF) : the canonical sheaf of F.
  • hi (D) ; the dimension of the space Hi(F,O F(D)).
  • pg (F) = h0(KF) = h2(O F) ; the geometric genus of F.
  • q(F) = h1(KF) = h1(0 F) ; the irregularity of F.
  • \({\text{K}}_{\text{F}}^2 \) : the self-intersection index of KF.
  • \({\text{P}}^{(1)} \left( {\text{F}} \right) = {\text{K}}_{{\text{F'}}}^2 + 1\), where F is a minimal model of a non-rational surface F ; the linear genus of F.
  • c2(F) : the topological Euler-Poincare characteristic of F.
  • Pn(F) = h0(nKF : the n-genus of F.
  • NS(F) : the Neron-Severi group of F, the quotient of the Picard group Pic(F) by the subgroup of divisors algebraically equivalent to zero (= Pic(F) if q = 0).
  • Tors(F) = Tors(NS(F)) = Tors(H1(F,Z)).
If not stated otherwise F will be always assumed to be non-singular and projective.
Igor Dolgachev

The Theory of Invariants and Its Applications to Some Problems in the Algebraic Geometry

First I want to recall the classical construction of a locally trivial fibre bundle. In the simplest case, the fibre bundle Fγ with fibre F and base X may be constructed as an associated bundle to any principal fibration Xγ with a group G as a fibre: the grouT) G acts on the fibre F and \({\text{F}}_{{\gamma }} = {\text{X}}_{{\gamma }} \times _{\text{G}} {\text{F}}\) F, where G acts on F on the right, and on Xγ on the left. A simple, but important, remark: this construction is twice functoriai - it is functorial on the base and functoriai on the fibre, in the sense that to any mo r phi sin h : F → Q of G - spaces (such that f o g = g o h for any gϵG) we have the morphism of corresponding fibre bundles hγ : Fγ → Qγ. The parameter γ here is a cocycle γ ϵ H1 (X, ΓG) where ΓG is some subbundle of the sheaf of functions on X with values in G. These facts are proved directly from the diagram:
$$ \begin{array}{*{20}c} {{\text{X}}_{{\gamma }} \times {\text{F}}\xrightarrow{{{\text{id}}\,\,\,\,\,{\text{h}}}}{\text{X}}_{{\gamma }} \times {\text{Q}}} \hfill \\ {{\text{X}}_{{\gamma }} \times _{\text{G}} {\text{G}}\xrightarrow{{\text{h}}}{\text{X}}_{{\gamma }} \times _{\text{G}} {\text{Q}}\,{\text{;}}} \hfill \\ \end{array} $$
h is well defined because (1 × h) ∘ g = g ∘ (1 × h) and hγ is the orbit space.
F. A. Bogomolov

Pluricanonical Mappings of Surfaces with K2= 1,2, q=pg =0

This lecture is a continuation of Dolgacev's ones on surfaces with q=pg =0, and considers those minimal models of such sur faces for which K2=2 (numerical Campedelli surfaces) and those for which K2=1 (numerical Godeaux surfaces): they are of general type by classification of surfaces.
F. Catanese

On a Class of Surfaces of General Type

This lecture contains an exposition, without many details and proofs, (they will appear in a future paper), of a joint research of E. Bombieri-F. Catanese, dealing whith surfaces having the following numerical invariants; K2 =2, pg =q=1 (of course they are of general type).
F. Catanese

Some Remarks about the “Nullstellensatz”

Let K be a field and \( \Gamma _{{\text{K}}^{\text{n}} } \) ring of the regular functions (in the sense of F.A.C.) on Kn.
Two problems are now natural:
to carachterize the ideals of definition of \( \Gamma _{{\text{K}}^{\text{n}} } \)
to carachterize the ideals of definition of K [X1,…,Xn].
A. Tognoli
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