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Finite subgroups of automorphism groups of Severi--Brauer varieties

Anna Savelyeva

TL;DR

The paper addresses the problem of classifying finite subgroups of automorphism groups of Severi–Brauer varieties, focusing on minimal SB varieties associated with division algebras. It develops a lifting framework that associates each finite subgroup $G\subset \mathrm{Aut}(A)\cong A^*/k^*$ with a normal subgroup $N_G$ such that $G/N_G$ is abelian and $|G/N_G|$ divides $n^2$, using the reduced norm and Amitsur’s classification to constrain possibilities. In the prime-degree case $p\ge3$, the results refine to a description of $G$ as a subgroup of a balanced semidirect product, implying that the only possible non-abelian simple action is via $\mathfrak{A}_5$ and leading to a structured, finite set of symmetry types for minimal SB varieties. The methods combine group-theoretic lifting, field-extension techniques, and division-algebra structure to yield sharp bounds and a clean classification in prime-degree scenarios, with potential applications to the study of automorphism groups of Severi–Brauer varieties. Overall, the work narrows the landscape of finite symmetry groups acting on these varieties and provides a concrete, computational framework for identifying admissible group actions.

Abstract

We discuss the structure of finite subgroups acting on minimal Severi--Brauer varieties and provide a complete description of such groups for the varieties of dimension $p-1$, where $p \geqslant 3$ is prime.

Finite subgroups of automorphism groups of Severi--Brauer varieties

TL;DR

The paper addresses the problem of classifying finite subgroups of automorphism groups of Severi–Brauer varieties, focusing on minimal SB varieties associated with division algebras. It develops a lifting framework that associates each finite subgroup with a normal subgroup such that is abelian and divides , using the reduced norm and Amitsur’s classification to constrain possibilities. In the prime-degree case , the results refine to a description of as a subgroup of a balanced semidirect product, implying that the only possible non-abelian simple action is via and leading to a structured, finite set of symmetry types for minimal SB varieties. The methods combine group-theoretic lifting, field-extension techniques, and division-algebra structure to yield sharp bounds and a clean classification in prime-degree scenarios, with potential applications to the study of automorphism groups of Severi–Brauer varieties. Overall, the work narrows the landscape of finite symmetry groups acting on these varieties and provides a concrete, computational framework for identifying admissible group actions.

Abstract

We discuss the structure of finite subgroups acting on minimal Severi--Brauer varieties and provide a complete description of such groups for the varieties of dimension , where is prime.

Paper Structure

This paper contains 7 sections, 23 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Lifting construction
  3. p-subgroups of automorphisms of division algebras
  4. Field extensions
  5. The case of trivial N$_G$
  6. General case
  7. Division algebras of prime degree

Key Result

Theorem 1.1

Let $A$ be a division algebra of degree $n$ over a field $k$ of characteristic coprime with $n$. Then for every finite subgroup there exists a finite subgroup $\widetilde{N}_G \subset A^*$ such that its image $N_G \subset A^*/k^*$ under the canonical projection is a normal subgroup of $G$ and the quotient $G/N_G$ is an abelian group of order dividing $n^2$.

Theorems & Definitions (49)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Proposition 2.3
  • ...and 39 more