On surfaces of general type with p = q = 1, K = 3.
The moduli space M of surfaces of general type with p = q = 1, K = 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 M ⊂ M corresponding to surfaces containing a genus 2 pencil, and moreover we show that there exists a non-empty, dense subset M ⊂ M which parametrizes isomorphism classes of surfaces with birational bicanonical map.