SMOOTH AFFINE SURFACES WITH NON-UNIQUE C*-ACTIONS

Hubert Flenner (1) , Shulim Kaliman (2) , Mikhail Zaidenberg (3)

1 Fakultät für Mathematik - Fakultät für Mathematik
2 Department of Mathematics [Miami]
3 IF - Institut Fourier

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 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 our previous paper we gave a sufficient condition, in terms of the Dolgachev- Pinkham-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 Danilov- Gizatullin surface V (n) of index n = n(k) with a family of pairwise non-conjugate C+-actions depending on k parameters.

Mots clés: C*-action; C+-action; affine surface

Domaines: Géométrie algébrique [math.AG]

Hubert Flenner, Shulim Kaliman, Mikhail Zaidenberg. SMOOTH AFFINE SURFACES WITH NON-UNIQUE C*-ACTIONS. Journal of Algebraic Geometry, 2011, Journal of Algebraic Geometry, 20, pp.329-398. ⟨10.1090/S1056-3911-2010-00533-4⟩. ⟨hal-00317965v2⟩