Smoothness of stabilisers in generic characteristic

Martin, Ben; Stewart, David I.; Topley, Lewis

doi:10.2140/ant.2026.20.1159

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Smoothness of stabilisers in generic characteristic

Ben Martin, David I. Stewart and Lewis Topley

Vol. 20 (2026), No. 6, 1159–1184

DOI: 10.2140/ant.2026.20.1159

Abstract

Let $R$ be a commutative unital ring. Given a finitely presented affine $R$ -group scheme $G$ acting on a finitely presented separated scheme $X$ over $R$ , we show that there is a prime $p_{0}$ such that for any $R$ -algebra $k$ that is a field of characteristic $p \geq p_{0}$ , the centraliser in $G_{k}$ of any closed subscheme of $X_{k}$ is smooth. When $X$ is not necessarily separated we show similarly that for any closed finitely presented subscheme $Y \subseteq X$ there is a $p_{1}$ depending on $Y$ such that when $k$ has characteristic $p \geq p_{1}$ , the normaliser of $Y_{k}$ in $G_{k}$ is smooth. For the proof, we may assume $k$ is algebraically closed, whence we prove these results using the Lefschetz principle together with careful application of Gröbner basis techniques, and using a suitable notion of the complexity of an action.

We apply our results to demonstrate that the Kostant–Kirillov–Souriau theorem holds for Lie algebras of algebraic groups in large positive characteristics: the coadjoint module of every such Lie algebra decomposes as a disjoint union of symplectic varieties, each of which is a coadjoint orbit.

Keywords

algebraic group, smooth normaliser, smooth centraliser, smoothness, Gröbner basis, Lefschetz principle

Mathematical Subject Classification

Primary: 03C60

Secondary: 20G07

Milestones

Received: 3 November 2023

Revised: 6 December 2024

Accepted: 27 June 2025

Published: 26 May 2026

Authors

Ben Martin
Department of Mathematics University of Aberdeen Aberdeen United Kingdom

David I. Stewart
Department of Mathematics University of Manchester Manchester United Kingdom

Lewis Topley
Department of Mathematical Sciences University of Bath Bath United Kingdom

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Vol. 20, No. 6, 2026