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Chapitre D'ouvrage Année : 2005

On the uniqueness of ${\bf C}^*$-actions on affine surfaces

M Zaidenberg
Hubert Flenner
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Résumé

We prove that a normal affine surface $V$ over $\bf C$ admits an effective action of a maximal torus ${\bf T}={\bf C}^{*n}$ ($n\le 2$) such that any other effective ${\bf C}^*$-action is conjugate to a subtorus of $\bf T$ in Aut $(V)$, in the following particular cases: (a) the Makar-Limanov invariant ML$(V)$ is nontrivial, (b) $V$ is a toric surface, (c) $V={\bf P}^1\times {\bf P}^1\backslash \Delta$, where $\Delta$ is the diagonal, and (d) $V={\bf P}^2\backslash Q$, where $Q$ is a nonsingular quadric. In case (a) this generalizes a result of Bertin for smooth surfaces, whereas (b) was previously known for the case of the affine plane (Gutwirth) and (d) is a result of Danilov-Gizatullin and Doebeli.

Dates et versions

hal-01623040 , version 1 (25-10-2017)

Identifiants

Citer

M Zaidenberg, Hubert Flenner. On the uniqueness of ${\bf C}^*$-actions on affine surfaces. Affine Algebraic Geometry: Special Session on Affine Algebraic Geometry at the First Joint AMS-RSME Meeting, Seville, Spain, June 18-21, 2003, 2005. ⟨hal-01623040⟩

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