Standard isotrivial fibrations with p_g=q=1. II

Details Standard isotrivial fibrations with p_g=q=1. II Standard isotrivial fibrations with p_g=q=1. II

2008-05-09

arxiv.org/abs/0805.1424v2

Journal of Pure and Applied Algebra 214 (2010), 344-369

Mathematics

Algebraic Geometry

32 pages. Final version, to appear in the Journal of Pure and Applied Algebra

Scientific paper

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.

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