Characteristically simple Beauville groups, I: cartesian powers of alternating groups

TL;DR

This paper characterizes when cartesian powers of finite simple alternating groups are Beauville groups, showing they are so if and only if they are 2-generated and not isomorphic to A_5.

Contribution

It provides a complete characterization of Beauville groups among cartesian powers of alternating groups, extending understanding of Beauville structures in algebraic geometry.

Findings

Cartesian powers of alternating groups are Beauville if and only if 2-generated and not A_5.

A characterization criterion for Beauville groups among simple group powers.

Identification of the special case of A_5 as not forming a Beauville group.

Abstract

A Beauville surface (of unmixed type) is a complex algebraic surface which is the quotient of the product of two curves of genus at least 2 by a finite group G acting freely on the product, where G preserves the two curves and their quotients by G are isomorphic to the projective line, ramified over three points. Such a group G is called a Beauville group. We show that if a characteristically simple group G is a cartesian power of a finite simple alternating group, then G is a Beauville group if and only if it has two generators and is not isomorphic to A_5.