On the orbits of plane automorphisms and their stabilizers

Abstract

Let 𝕜 be a perfect field with algebraic closure 𝕜. If 𝐻 is a subgroup of plane automorphisms over 𝕜 and 𝑝 ∈ 𝕜 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 𝕜 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.