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

TL;DR

The work classifies finite subgroups of automorphism groups of non-trivial Severi–Brauer varieties of dimension $p-1$ (with $p$ prime) over fields of characteristic not dividing $p$, via a reduction to central simple algebras and balanced semidirect products. It establishes a universal example in characteristic $0$ whose automorphism group contains all admissible finite subgroups, and provides positive-characteristic results showing that a single universal example does not exist there, though $(q,k)$-th universal examples can be constructed for any fixed $k$. The methods hinge on the structure of $ ext{Aut}(X) \, ext{(}X ext{ Severi–Brauer)} \\cong A^*( ext{K})/ ext{K}^*$ and the interplay with cyclic algebras and Galois actions. These results clarify how automorphism symmetries of Severi–Brauer varieties depend on characteristic and connect deeply with the arithmetic of central simple algebras, with implications for birational automorphisms in low dimension.

Abstract

We classify finite subgroups of automorphism groups of non-trivial Severi--Brauer varieties of dimension $p-1$, where $p \ge 3$ is prime, over an arbitrary field whose characteristic is coprime to $p$. In addition, we construct a universal example in the case when the base field has characteristic 0; that is, we exhibit a field of characteristic 0 together with a non-trivial Severi--Brauer variety such that every finite subgroup allowed by our classification acts on it. For positive characteristic, we provide universal examples in a weaker sense.

Theorems & Definitions (44)

• Theorem 1.1: ASav
• Theorem 1.2: Sh20
• Theorem 1.3: Sh20
• Theorem 1.4: Sh20B
• Theorem 1.5
• Theorem 1.6
• Definition 1.7
• Theorem 1.8
• Example 1.9
• Lemma 2.1: see e.g. Chat or ShV