Quotients of Severi-Brauer surfaces

TL;DR

This paper characterizes when quotients of Severi-Brauer surfaces by finite groups are rational or birationally equivalent to the original surface, based on the divisibility of the group order by 3.

Contribution

It provides a complete criterion for the rationality of quotients of Severi-Brauer surfaces under finite group actions.

Findings

Quotients are rational iff group order divisible by 3.

Otherwise, quotients are birationally equivalent to the original surface.

The result holds over arbitrary fields of characteristic zero.

Abstract

We show that a quotient of a non-trivial Severi-Brauer surface $S$ over arbitrary field $\Bbbk$ of characteristic $0$ by a finite group $G \subset \operatorname{Aut}(S)$ is $\Bbbk$-rational, if and only if $|G|$ is divisible by $3$. Otherwise, the quotient is birationally equivalent to $S$.