**MSC:**- 14J29 Surfaces of general type
- 14J10 Families, moduli, classification: algebraic theory

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.

\end{abstract}