On surfaces of general type with $p_g=q=1,K^2=3$

Abstract

The moduli space $\mathcal{M}$ of surfaces of general type with $p_g=q=1,K^2=g=3$ (where $g$ is the genus of the Albanese fibration) was constructed by Catanese and Ciliberto in \cite{CaCi93}. In this paper we characterize the subvariety ${\mathcal{M}}_2\subset \mathcal{M}$ corresponding to surfaces containing a genus 2 pencil, and moreover we show that there exists a non-empty, dense subset ${\mathcal{M}}^0\subset \mathcal{M}$ which parametrizes isomorphism classes of surfaces with birational bicanonical map.

35 pages. To appear in Collectanea Mathematica