Algebraic Surfaces with $p_g=q=1, K^2=4$ and Genus 3 Albanese Fibration

TL;DR

This paper investigates the moduli space of minimal algebraic surfaces with specific invariants, identifying two main components under a certain decomposability assumption, thereby advancing understanding of their geometric classification.

Contribution

It identifies two irreducible components of the moduli space of surfaces with given invariants, under a decomposability assumption of the canonical sheaf's direct image.

Findings

Two irreducible components of the moduli space are found.

One component has dimension 5, the other dimension 4.

The study relies on the assumption of decomposability of the direct image sheaf.

Abstract

In this paper, we study the Gieseker moduli space $\mathcal{M}_{1,1}^{4,3}$ of minimal surfaces with $p_g=q=1, K^2=4$ and genus 3 Albanese fibration. Under the assumption that direct image of the canonical sheaf under the Albanese map is decomposable, we find two irreducible components of $\mathcal{M}_{1,1}^{4,3}$, one of dimension 5 and the other of dimension 4.