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This is the first of two volumes representing the current state of knowledge about Enriques surfaces which occupy one of the classes in the classification of algebraic surfaces. Recent improvements in our understanding of algebraic surfaces over fields of positive characteristic allowed us to approach the subject from a completely geometric point of view although heavily relying on algebraic methods. Some of the techniques presented in this book can be applied to the study of algebraic surfaces of other types. We hope that it will make this book of particular interest to a wider range of research mathematicians and graduate students. Acknowledgements. The undertaking of this project was made possible by the support of several institutions. Our mutual cooperation began at the University of Warwick and the Max Planck Institute of Mathematics in 1982/83. Most of the work in this volume was done during the visit of the first author at the University of Michigan in 1984-1986. The second author was supported during all these years by grants from the National Science Foundation.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
Enriques surfaces bear the name of one of the founders of the theory of algebraic surfaces, the Italian geometer Federigo Enriques. He constructed some of them to give the first examples of nonrational algebraic surfaces on which there are no regular differential forms. At the same time a different construction of such surfaces was given by another Italian geometer, no less famous, Guido Castelnuovo. The original construction of Enriques gives a birational model of an Enriques surfaces represented by a surface of degree 6 in P3 passing doubly through the edges of the coordinate tetrahedron (see [En 1,En 4,AS,G-H 1]). Castelnuovo’s model is a surface of degree 7 in P3 with a triple line and other singularities (see [Cas,En 4]. Later on Enriques [En 2,En 4] gave a construction of his surfaces as double planes with some branch curve of degree 8. We discuss these models in great detail in Chapter 4. Subsequently G. Fano [Fa] constructed a family of Enriques surfaces represented by congruences of lines in P3 associated to webs of quadrics (see [Cos 2,[G-H 1]). We will discuss this construction in Part II of this book. In [En 3] Enriques observed that every Enriques surface can be obtained as a quotient of a surface of type K3 (in modern terminology) by a fixed-point-free involution. The first modern treatment of Enriques surfaces over fields of characteristic different from 2 was given by M. Artin [Art 1] and B. Averbukh (see [AS] Chapter IX, [Av]).
François R. Cossec, Igor V. Dolgachev

Chapter 0. Preliminaries

Abstract
A morphism f: X → Y of integral schemes over an algebraically closed field K is called a double cover if f is finite and of degree 2. A double cover is said to be separable (resp. inseparable) if the corresponding extension of the fields of rational functions is separable (resp. inseparable).
François R. Cossec, Igor V. Dolgachev

Chapter I. Enriques Surfaces: Generalities

Abstract
Let K be an algebraically closed field of arbitrary characteristic p. In this section we recall the main results of the classification of nonsingular projective surfaces over K. We refer to [Mu 2,B-M 1,B-M 2] for the proofs of all the assertions peculiar to the case of positive characteristic and to general textbooks [B-P-vdV,Bea 2,G-H 1] for the case of characteristic zero.
François R. Cossec, Igor V. Dolgachev

Chapter II. Lattices and Root Bases

Abstract
A lattice is a free abelian group M of finite rank rk(M) equipped with a symmetric bilinear form ϕ:MxM → Z. The value of this form on a pair (x,y)∈MxM will be denoted by x•y. We write x2 to denote x•x.
François R. Cossec, Igor V. Dolgachev

Chapter III. The Geometry of the Enriques Lattice

Abstract
Let
$${\rm D}=\sum_{{\rm i}\in{\rm l}}{\rm m}_{\rm i}{\rm R}_{\rm i}$$
be an effective divisor on a nonsingular projective surface X with irreducible components Ri. We say that D is of canonical type if
$${K_{x}}\bullet {R_{i}} = D\bullet {R_{i}} = 0for{\text{ }}all{\text{ }}i.$$
François R. Cossec, Igor V. Dolgachev

Chapter IV. Projective Models

Abstract
As in the previous chapters F will denote an Enriques surface over an algebraically closed field of characteristic p ≥ 0.
François R. Cossec, Igor V. Dolgachev

Chapter V. Genus One Fibration

Abstract
Let S be a regular integral scheme of dimension 1, η be its generic point and K = K(η) be its residue field. A projective morphism f: X → S is said to be a genus 1 fibration if X is regular and irreducible, and the general fibre Xη is a geometrically integral regular algebraic curve of arithmetic genus 1.
François R. Cossec, Igor V. Dolgachev

Backmatter

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