TY - JOUR

T1 - Smooth affine surfaces with non-unique ℂ*-actions

AU - Flenner, Hubert

AU - Kaliman, Shulim

AU - Zaidenberg, Mikhail

PY - 2011

Y1 - 2011

N2 - 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 → λ-1 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 [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 *-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.

AB - 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 → λ-1 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 [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 *-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.

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U2 - 10.1090/S1056-3911-2010-00533-4

DO - 10.1090/S1056-3911-2010-00533-4

M3 - Article

AN - SCOPUS:79751491466

VL - 20

SP - 329

EP - 398

JO - Journal of Algebraic Geometry

JF - Journal of Algebraic Geometry

SN - 1056-3911

IS - 2

ER -