Standard isotrivial fibrations with p_g=q=1. II

TL;DR

This paper classifies standard isotrivial fibrations with p_g=q=1, extending previous results, and provides the first examples of minimal surfaces with specific invariants, including cases where the surface is not minimal.

Contribution

It extends the classification of standard isotrivial fibrations with p_g=q=1 to include non-minimal cases and constructs explicit examples of minimal surfaces with particular invariants.

Findings

Classified all standard isotrivial fibrations with p_g=q=1 where the surface is minimal.

Constructed the first examples of minimal surfaces with p_g=q=1, K^2=5, and genus 3 Albanese fibration.

Provided explicit examples of non-minimal surfaces in this class.

Abstract

A smooth, projective surface $S$ is called a $\emph{standard isotrivial fibration}$ if there exists a finite group $G$ which acts faithfully on two smooth projective curves $C$ and $F$ so that $S$ is isomorphic to the minimal desingularization of $T:=(C \times F)/G$. Standard isotrivial fibrations of general type with $p_g=q=1$ have been classified in \cite{Pol07} under the assumption that $T$ has only Rational Double Points as singularities. In the present paper we extend this result, classifying all cases where $S$ is a minimal model. As a by-product, we provide the first examples of minimal surfaces of general type with $p_g=q=1$, $K_S^2=5$ and Albanese fibration of genus 3. Finally, we show with explicit examples that the case where $S$ is not minimal actually occurs.