Acyclic curves and group actions on affine toric surfaces
Résumé
We show that every irreducible, simply connected curve on a toric affine surface $X$ over $\CC$ 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 non-equivalent embeddings of the affine line $Å^1$ 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.
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