On Beauville surfaces

Abstract

We prove that if a finite group $G$ acts freely on a product of two curves $C_1\times C_2$ so that the quotient $S=C_1\times C_2/G$ is a Beauville surface then $C_1$ and $C_2$ are both non hyperelliptic curves of genus $\ge 6$; the lowest bound being achieved when $C_1=C_2$ is the Fermat curve of genus $6$ and $G={\left(\mathbb{Z}/5\mathbb{Z}\right)}^2$. We also determine the possible values of the genera of $C_1$ and $C_2$ when $G$ equals $S_5$, ${\mathrm{PSL}}_2({\mathbb{F}}_7)$ or any abelian group. Finally, we produce examples of Beauville surfaces in which $G$ is a $p$-group with $p=2,3$.