Beauville surfaces with abelian Beauville group

Abstract

A Beauville surface is a rigid surface of general type arising as a quotient of a product of curves $C_1$, $C_2$ of genera $g_1,g_2\ge 2$ by the free action of a finite group $G$. In this paper we study those Beauville surfaces for which $G$ is abelian (so that $G\cong {\mathbb{Z}}_n^2$ with $\gcd (n,6)=1$ by a result of Catanese). For each such $n$ we are able to describe all such surfaces, give a formula for the number of their isomorphism classes and identify their possible automorphism groups. This explicit description also allows us to observe that such surfaces are all defined over $\mathbb{Q}$.

arXiv comment14 pages. This version contains amended references to related work of Garion and Penegini, who have recently replaced their paper arXiv:0910.5402v1 with arXiv:0910.5402v3 and arXiv:1107.5534, the latter resolving certain discrepancies between their results and ours