Enriques Surfaces I
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
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Series Editors
J. Oesterle A. Weinstein
R. Cos sec Igor V. Dolgachev Fran~ois
Enriques Surfaces I
1989
Birkhauser Boston . Basel . Berlin
FranO. where Kx is the
2 canonical divisor class of X. The first class consists of surfaces for which f(n) = 0 for all mO. This includes rational and ruled surfaces. The second class consists of surfaces for which f(n) is not identically zero but is bounded.
This
includes
surfaces
birationally
isomorphic
to
either
Enriques
surfaces. or surfaces of type K3. or abelian surfaces. or hyperelliptic surfaces. The third class consists of surfaces for which f(n) is asymptotically
linear. This class consists of so-called properly elliptic
surfaces. Finally.
the remaining surfaces are of general
type;
the function
f(n) is asymptotically quadratic in this case. Enriques surfaces can be distinguished from other surfaces by the properties that 2Kx is
linearly
equivalent to 0 and the second Betti number b2 (X) is equal to 10. There are other characterizations if the ground field is of characteristic p"2. For example. X is an Enriques surfaces if and only if it belongs to the second class and has no regUlar differential forms. Or X is an Enriques surface if and only if it is isomorphic to a quotient of a K3-surface by a fixed-pointfree involution. The latter property maKes the theory of Enriques surfaces a special case of the theory of K3-surfaces. Topologically. over the field of complex numbers an Enriques surface is a smooth 4-manifold whose universal cover is of degree 2 and diffeomorphic to a minimal resolution of the 16 nodes of a Kummer surface. The classification of algebraic surfaces was extended to the case of ground fields of positive characteristic in the worKS of E. Bombieri and D.
Mumford.
They gave a characteristic-free definition of
Enriques
surfaces
and undertooK the first study of these surfaces in characteristic 2. As far as we Know the first construction of Enriques surfaces in characteristic 2 was given by M. Reid (see IB-M 2)). We present these examples in Chapter 1. The case of characteristic 2 turned out to be the most special case for Enriques surfaces.
In this case only new pathological phenomena arise. as for
example the absence of a K3-cover of an Enriques surfaces. The paper of E. Bombieri and D. Mumford IB-M 1 J was the first in which these new phenomena were discovered and studied. They showed that the extension of the classification of algebraic surfaces
to
the case of positive characteristic
3 opens up an absolutely new world, where the familiar objects metamorphose into quite unrecognizable objects with equally rich and beautiful
geometrical
structure. The inclusion of this case in our treatment of Enriques surfaces is one of the main reasons for the unexpectedly large size of this bOOK As is mentioned in the Preface we are trying to study Enriques surfaces in all characteristics. We share the opinion that the geometry ends with
the application of
transcendental
methods and
the
true understanding
of the geometrical
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