On surfaces of general type with p_g=q=1 isogenous to a product of curves

Abstract

A smooth algebraic surface $S$ is said to be \emph{isogenous to a product of unmixed type} if there exist two smooth curves $C,F$ and a finite group $G$, acting faithfully on both $C$ and $F$ and freely on their product, so that $S=(C\times F)/G$. In this paper we classify the surfaces of general type with $p_g=q=1$ which are isogenous to an unmixed product, assuming that the group $G$ is abelian. It turns out that they belong to four families, that we call surfaces of type $I,II,III,IV$. The moduli spaces ${\mathfrak{M}}_I,{\mathfrak{M}}_{II},{\mathfrak{M}}_{IV}$ are irreducible, whereas ${\mathfrak{M}}_{III}$ is the disjoint union of two irreducible components. In the last section we start the analysis of the case where $G$ is not abelian, by constructing several examples.

23 pages. To appear in Communications in Algebra

References in corpus (2)

• Some new surfaces with $p_g=q=0$
• The classification of surfaces with p_g = q = 0 isogenous to a product of curves

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