On surfaces with $p_g=q=2$ and non-birational bicanonical map

Abstract

This paper is devoted to the classification of irregular surfaces of general type with $p_g=q=2$ and non birational bicanonical map. The main result is that, if $S$ is such a surface and if $S$ is minimal with no pencil of curves of genus 2, then $S$ is a double cover of a principally polarized abelian surface $(A,\Theta )$, with $\Theta$ irreducible. The double cover $S\to A$ is branched along a divisor $B\in |2\Theta |$, having at most double points and so $K_S^2=4$.

To appear in Advances in Geometry, Latex 2e, 26 pages, title and introduction changed, minor corrections