On Beauville surfaces (Q646377)

The authors prove the following. (1) If a finite group \(G\) acts freely on a product of two curves \(X=C_1\times C_2\) so that \(X/G\) is a Beauville surface, then neither \(C_1\) nor \(C_2\) can be a hyperelliptic curve. (2) The minimum genera (in the lexicographic order) that can occur in the construction of a Beauville surface is \((g_1, g_2) = (6,6)\). The minimum is achieved by Beauville's example. (3) \(S_5\) and \(S_6\), hence \(S_n\) for all \(n \geq 5\), admit a Beauville structure. The authors determine which pair of genera \(g_1, g_2\) occur when \(G\) is \(S_5\), \(\text{PSL}_2({\mathbb F}_7)\), or an abelian group, and they produce examples of Beauville surfaces in which \(G\) is a \(p\)-group with \(p = 2, 3\). These appear to be the first known examples of \(2\)- and \(3\)-groups admitting Beauville structure.