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

TL;DR

This paper investigates algebraic surfaces of general type with specific invariants, constructing examples and analyzing their moduli, revealing new properties and answering existing questions in the classification theory.

Contribution

It provides new examples of surfaces with $p_g=q=1$ and genus 2 Albanese fibrations, and characterizes their moduli space components, also addressing a question by Pignatelli.

Findings

Surfaces with $p_g=q=1, K^2=5$ form an irreducible component of the moduli space.

Constructed surfaces with $p_g=q=1, K^2=3$ show the direct image of the bicanonical sheaf can have two summands.

Answer to Pignatelli's question is negative, demonstrating new behavior in the structure of these surfaces.

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.