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

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 [14]. 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.