On the uniqueness of ${\mathbf{C}}^{\ast }$-actions on affine surfaces

Abstract

We prove that a normal affine surface $V$ over $\mathbf{C}$ admits an effective action of a maximal torus $\mathbf{T}={\mathbf{C}}^{\ast n}$ ($n\le 2$) such that any other effective ${\mathbf{C}}^{\ast }$-action is conjugate to a subtorus of $\mathbf{T}$ in Aut $(V)$, in the following particular cases: (a) the Makar-Limanov invariant ML$(V)$ is nontrivial, (b) $V$ is a toric surface, (c) $V={\mathbf{P}}^1\times {\mathbf{P}}^1\backslash \Delta$, where $\Delta$ is the diagonal, and (d) $V={\mathbf{P}}^2\backslash Q$, 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.

11/06/2004 2 version 14/06/2004