Beauville surfaces and finite groups

TL;DR

This paper demonstrates that Beauville surfaces of unmixed type can be constructed from specific finite groups such as L_2(q), SL_2(q), Suzuki, and Ree groups, and characterizes when these surfaces are real.

Contribution

It extends previous results by showing new classes of finite groups that produce Beauville surfaces and characterizes conditions for their reality.

Findings

Beauville surfaces of unmixed type can be obtained from L_2(q), SL_2(q), Suzuki, and Ree groups.

L_2(q) and SL_2(q) admit strongly real Beauville structures if and only if q>5.

Constructs explicit examples of Beauville surfaces from these groups.

Abstract

Extending results of Bauer, Catanese and Grunewald, and of Fuertes and Gonz\'alez-Diez, we show that Beauville surfaces of unmixed type can be obtained from the groups L_2(q) and SL_2(q) for all prime powers q>5, and the Suzuki groups Sz(2^e) and the Ree groups R(3^e) for all odd e>1. We also show that L_2(q) and SL_2(q) admit strongly real Beauville structures, yielding real Beauville surfaces, if and only if q>5.