On surfaces of general type with $p_g=q=1, K^2=3$

TL;DR

This paper studies the moduli space of surfaces of general type with specific invariants, characterizing a special subvariety and identifying a dense subset with particular geometric properties.

Contribution

It characterizes the subvariety of surfaces with a genus 2 pencil and shows the existence of a dense subset with birational bicanonical maps.

Findings

Characterization of the subvariety of surfaces with genus 2 pencils.

Existence of a dense subset parametrizing surfaces with birational bicanonical maps.

Abstract

The moduli space $\mathscr{M}$ of surfaces of general type with $p_g=q=1, K^2=g=3$ (where $g$ is the genus of the Albanese fibration) was constructed by Catanese and Ciliberto in \cite{CaCi93}. In this paper we characterize the subvariety $\mathscr{M}_2 \subset \mathscr{M}$ corresponding to surfaces containing a genus 2 pencil, and moreover we show that there exists a non-empty, dense subset $\mathscr{M}^0 \subset \mathscr{M}$ which parametrizes isomorphism classes of surfaces with birational bicanonical map.