Smooth affine surfaces with non-unique ℂ*-actions

Abstract

In this paper we complete the classification of effective C ∗ \mathbb {C}^* -actions on smooth affine surfaces up to conjugation in the full automorphism group and up to inversion λ ↦ λ − 1 \lambda \mapsto \lambda ^{-1} of C ∗ \mathbb {C}^* . If a smooth affine surface V V admits more than one C ∗ \mathbb {C}^* -action, then it is known to be Gizatullin i.e., it can be completed by a linear chain of smooth rational curves. In [Transformation Groups 13:2, 2008, pp. 305–354] we gave a sufficient condition, in terms of the Dolgachev-Pinkham-Demazure (or DPD) presentation, for the uniqueness of a C ∗ \mathbb {C}^* -action on a Gizatullin surface. In the present paper we show that this condition is also necessary, at least in the smooth case. In fact, if the uniqueness fails for a smooth Gizatullin surface V V which is neither toric nor Danilov-Gizatullin, then V V admits a continuous family of pairwise non-conjugated C ∗ \mathbb {C}^* -actions depending on one or two parameters. We give an explicit description of all such surfaces and their C ∗ \mathbb {C}^* -actions in terms of DPD presentations. We also show that for every k > 0 k>0 one can find a Danilov-Gizatullin surface V ( n ) V(n) of index n = n ( k ) n=n(k) with a family of pairwise non-conjugate C + \mathbb {C}_+ -actions depending on k k parameters.