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Beauville Surfaces with Abelian Beauville Group

G. González-Diez, G. A. Jones, D. Torres-Teigell

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 $G\cong \mathsf{Z}_{n}^{2}$ 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 $\mathsf{Q}$.

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DOI: http://dx.doi.org/10.7146/math.scand.a-17106

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ISSN 0025-5521 (print) ISSN 1903-1807 (online)

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