ar X iv : 0 80 9 . 06 51 v 2 [ m at h . A G ] 6 S ep 2 00 8 SMOOTH AFFINE SURFACES WITH NON - UNIQUE C ∗ - ACTIONS

In this paper we complete the classification of effective C-actions on smooth affine surfaces up to conjugation in the full automorphism group and up to inversion λ 7→ λ of C. If a smooth affine surface V admits more than one C-action then it is known to be Gizatullin i.e., it can be completed by a linear chain of smooth rational curves. In [FKZ3] we gave a sufficient condition, in terms of the DolgachevPinkham-Demazure (or DPD) presentation, for the uniqueness of a 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 which is neither toric nor Danilov-Gizatullin, then V admits a continuous family of pairwise non-conjugated C-actions depending on one or two parameters. We give an explicit description of all such surfaces and their C-actions in terms of DPD presentations. We also show that for every k > 0 one can find a DanilovGizatullin surface V (n) of index n = n(k) with a family of pairwise non-conjugate C+-actions depending on k parameters.