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Abstract: In this paper we complete the classification of effective C*-actions onsmooth affine surfaces up to conjugation in the full automorphism group and upto inversion of C*. If a smooth affine surface V admits more than one C*-actionthen it is known to be Gizatullin i.e., it can be completed by a linear chainof smooth rational curves. In our previous paper we gave a sufficientcondition, in terms of the Dolgachev- Pinkham-Demazure or DPD presentation,for the uniqueness of a C*-action on a Gizatullin surface. In the present paperwe show that this condition is also necessary, at least in the smooth case. Infact, if the uniqueness fails for a smooth Gizatullin surface V which isneither toric nor Danilov-Gizatullin, then V admits a continuous family ofpairwise non-conjugated C*-actions depending on one or two parameters. We givean explicit description of all such surfaces and their C*-actions in terms ofDPD presentations. We also show that for every k > 0 one can find a Danilov-Gizatullin surface V n of index n = nk with a family of pairwisenon-conjugate C+-actions depending on k parameters.



Author: Hubert Flenner Fakultät für Mathematik, Shulim Kaliman, Mikhail Zaidenberg IF

Source: https://arxiv.org/







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