Fabrizio Catanese
Singular bidouble covers and the construction of interesting algebraic surfaces
Preprint series: Mathematica Gottingensis
14J29 Surfaces of general type
14J10 Families, moduli, classification: algebraic theory
Abstract: \begin{abstract}
A bidouble cover is a finite flat Galois morphism with
Galois group ${\Bbb (Z/2)}^2 - $.
The structure theorem for smooth Galois ${\Bbb (Z/2)}^2 - $ covers was
given in
\cite{cat2} [pag. 491-493] where bidouble covers of
${\Bbb P}^1 \times{\Bbb P}^1$ were introduced in order to find interesting
properties of
the moduli spaces of surfaces of general type.
In this paper we develop general formulae for the case of
resolutions of singular bidouble covers.
P. Burniat used singular bidouble covers in order to fill out sectors of
surface geography. In this paper instead, the main application is for
the construction of surfaces with birational canonical map (so called
simple canonical surfaces) and high $K^2$, for instance we construct
such surfaces with $p_g = 4, 11 \leq K^2 \leq 28 $, against a prediction
of F. Enriques that $24$ should be the maximum allowed.

Moreover, we find, among several new examples of surfaces,
some surfaces with $p_g=q=1$, $K^2=4, 5$,
and also some infinite series of surfaces whose canonical map is
composed of a pencil of curves of genus $2$ or $3$,
with non costant moduli.

Keywords: Surfaces, covers, singularities