Acyclic curves and group actions on affine toric surfaces

TL;DR

This paper classifies simply connected curves on affine toric surfaces as orbit closures of group actions, showing finiteness of embeddings of the affine line and extending classical theorems to toric surfaces.

Contribution

It provides a new classification of curves on affine toric surfaces, describing their relation to group actions and extending the Jung-van der Kulk theorem.

Findings

Irreducible, simply connected curves are orbit closures of group actions.

Finitely many non-equivalent affine line embeddings up to automorphisms.

Extension of classical theorems to affine toric surfaces.

Abstract

We show that every irreducible, simply connected curve on a toric affine surface X over the field of complex numbers is an orbit closure of a multiplicative group action on X. It follows that up to the action of the automorphism group Aut(X) there are only finitely many non-equivalent embeddings of the affine line in X. A similar description is given for simply connected curves in the quotients of the affine plane by small finite linear groups. We provide also an analog of the Jung-van der Kulk theorem for affine toric surfaces, and apply this to study actions of algebraic groups on such surfaces.