Groups, Geometry, and Dynamics


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Volume 5, Issue 1, 2011, pp. 107–119
DOI: 10.4171/GGD/117

Published online: 2011-01-19

On Beauville surfaces

Yolanda Fuertes[1], Gabino González-Diez[2] and Andrei Jaikin-Zapirain[3]

(1) Universidad Autónoma de Madrid, Spain
(2) Universidad Autónoma de Madrid, Spain
(3) Universidad Autónoma de Madrid, Spain

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 $\geq 6$; 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 $p=2,3$.

Keywords: Beauville surfaces, Riemann surfaces, finite $p$-groups

Fuertes Yolanda, González-Diez Gabino, Jaikin-Zapirain Andrei: On Beauville surfaces. Groups Geom. Dyn. 5 (2011), 107-119. doi: 10.4171/GGD/117