Acyclic curves and group actions on affine toric surfaces

Ivan Arzhantsev (1) , Mikhail Zaidenberg (2)

1 MGU - Department of Algebra, Faculty of Mechanics and Mathematics
2 IF - Institut Fourier

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.

Mots clés: automorphism group; acyclic curve; Affine surface; torus action; quotient

Domaines: Géométrie algébrique [math.AG]; Théorie des groupes [math.GR]

Ivan Arzhantsev, Mikhail Zaidenberg. Acyclic curves and group actions on affine toric surfaces. Kayo Masuda, Hideo Kojima, Takashi Kishimoto (eds.). Affine Algebraic Geometry, World Scientific Publiching Co. pp.1-41, 2013, Proceedings of the Conference Osaka, Japan, 3 – 6 March 2011, 978-981-4436-71-7 ⟨10.1142/9789814436700_0001⟩. ⟨hal-00632263⟩