The moduli space $\mathscr{ M}$ 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 $\mathscr{ M}_2 \subset \mathscr{ M}$ corresponding to surfaces containing a genus 2 pencil, and moreover we show that there exists a non-empty, dense subset $\mathscr{ M}^0 \subset \mathscr{ M}$ which parametrizes isomorphism classes of surfaces with birational bicanonical map.
Categories: math.AG