On the orbits of plane automorphisms and their stabilizers

TL;DR

This paper characterizes the stabilizer subgroups of points under plane automorphisms over perfect fields, especially when the orbit closure is irreducible or a general curve, advancing understanding of automorphism group actions.

Contribution

It provides a detailed description of stabilizer subgroups for points with irreducible orbit closures and extends results to cases where the orbit closure is a general curve.

Findings

Explicit description of stabilizers for points with irreducible orbit closures.

Extension of results to cyclic groups and arbitrary orbit-closure curves.

Abstract

Let $\Bbbk$ be a perfect field with algebraic closure $\overline{\Bbbk}$. If $H$ is a subgroup of plane automorphisms over $\Bbbk$ and $p\in\overline{\Bbbk}^2$ is a point, we describe the subgroup consisting of plane automorphisms which stabilize the orbit of $p$ under $H$, when this orbit has irreducible closure in $\overline{\Bbbk}^2$. As an application, we treat the case where $H$ is cyclic and the closure of the orbit of $p$ is an arbitrary (non-necessarily irreducible) curve.