Algebraic Surfaces of General Type with $p_g=q=1$ and Genus 2 Albanese Fibrations

Abstract

In this paper, we study algebraic surfaces of general type with $p_g=q=1$ and genus 2 Albanese fibrations. We first study the examples of surfaces with $p_g=q=1,K^2=5$ and genus 2 Albanese fibrations constructed by Catanese using singular bidouble covers of ${\mathbb{P}}^2$. We prove that these surfaces give an irreducible and connected component of ${\mathcal{M}}_{1,1}^{5,2}$, the Gieseker moduli space of surfaces of general type with $p_g=q=1,K^2=5$ and genus 2 Albanese fibrations. Then by constructing surfaces with $p_g=q=1,K^2=3$ 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.

20 pages, minor changes

References in corpus (1)

• Standard isotrivial fibrations with p_g=q=1