Automorphisms of the plane preserving a curve

TL;DR

This paper investigates the automorphism groups of affine planes that preserve specific curves, establishing their algebraic nature generally, and providing classifications for certain cases over perfect fields.

Contribution

It proves the algebraicity of automorphism groups preserving a curve, except for parallel lines, and classifies positive-dimensional groups for irreducible curves over perfect fields.

Findings

Automorphism groups are algebraic except for parallel lines

Classification of positive-dimensional groups for irreducible curves

Results hold over any field, with specific cases for perfect fields

Abstract

We study the group of automorphisms of the affine plane preserving some given curve, over any field. The group is proven to be algebraic, except in the case where the curve is a bunch of parallel lines. Moreover, a classification of the groups of positive dimension occuring is also given in the case where the curve is geometrically irreducible and the field is perfect.