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Acyclic curves and group actions on affine toric surfaces

Ivan Arzhantsev 1, Mikhail Zaidenberg 2

(2011-10-14)

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.

1:	Department of Algebra, Faculty of Mechanics and Mathematics (MGU)
	Lomonosov Moscow State University>
2:	Institut Fourier (IF)
	CNRS : UMR5582 – Université Joseph Fourier - Grenoble I

Subject

:

Mathematics/Algebraic Geometry Mathematics/Group Theory

Keyword(s): Affine surface – acyclic curve – automorphism group – torus action – quotient

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From: Mikhail Zaidenberg <>

Submitted on: Thursday, 13 October 2011 23:01:21

Updated on: Tuesday, 19 June 2012 14:02:58