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

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.