Singular Bidouble Covers

A bidouble cover is a nite at Galois morphism with Galois group (Z=2) 2 ?. The structure theorem for smooth Galois (Z=2) 2 ? covers was given in Cat2] pag. 491-493] where bidouble covers of P 1 P 1 were introduced in order to nd 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 ll 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 pg = 4; 11 K 2 28, against a prediction of F. Enriques that 24 should be the maximum allowed. Moreover, we nd, among several new examples of surfaces, some surfaces with pg = q = 1, K 2 = 4;5, and also some innnite series of surfaces whose canonical map is composed of a pencil of curves of genus 2 or 3, with non costant moduli.