Finite subgroups of automorphism groups of Severi--Brauer varieties of prime degree

TL;DR

This paper classifies finite subgroups of automorphism groups of non-trivial Severi--Brauer varieties of prime dimension and constructs examples illustrating group actions, also analyzing their birational rigidity.

Contribution

It provides a classification of finite automorphism subgroups and constructs explicit examples for any finite group set acting on such varieties.

Findings

Finite subgroups are classified for Severi--Brauer varieties of prime dimension.

Constructed examples for every finite group set acting on these varieties.

Showed non-rigidity of these varieties under certain conditions.

Abstract

We classify finite subgroups of automorphism groups of non-trivial Severi--Brauer varieties of dimension $q-1$, where $q \geqslant 3$ is a prime number, over an arbitrary field. We also construct families of examples, namely, for every consistent set of finite groups, we construct a field together with a non-trivial Severi--Brauer variety over that field such that every group in the set acts on the constructed variety. Additionally, we show that non-trivial Severi--Brauer varieties of dimension $q-1$, where $q \geqslant 3$ is a prime number, over a field of characteristic not equal to $q$ are not $G$-birationally rigid.