Algebraic surfaces of general type with pg=q=1 and genus 2 Albanese fibrations

In this paper, we study algebraic surfaces of general type with pg=q=1 and genus 2 Albanese fibrations. We first study the examples of surfaces with pg=q=1,K2=5 and genus 2 Albanese fibrations constructed by Catanese using singular bidouble covers of P2. We prove that these surfaces give an irreducible and connected component of M1,15,2, the Gieseker moduli space of surfaces of general type with pg=q=1,K2=5 and genus 2 Albanese fibrations. Then by constructing surfaces with pg=q=1,K2=3 and a genus 2 Albanese fibration such that the number of the summands of the direct image of the bicanonical sheaf (under the Albanese map) is 2, we give a negative answer to a question of Pignatelli.