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

Hubert Flenner; Mikhail Zaidenberg

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

Hubert Flenner·Mikhail Zaidenberg

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Abstract

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) $V={\bf P}^1\times {\bf P}^1\backslash Δ$, where $Δ$ 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.

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Hubert Flenner, Mikhail Zaidenberg. 2004-06-14. On the uniqueness of ${\bf C}^*$-actions on affine surfaces. https://arxiv.org/abs/math/0406239

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