Acyclic curves and group actions on affine toric surfaces

We show that every irreducible, simply connected curve on a toric affine surface X over ℂ is an orbit closure of a 𝔾_m-action on X. It follows that up to the action of the automorphism group Aut(X) there are only finitely many nonequivalent embeddings of the affine line 𝔸¹ 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.