Smooth affine surfaces with non-unique ℂ*-actions

Hubert Flenner, Shulim Kaliman, Mikhail Zaidenberg

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)329-398
Number of pages70
JournalJournal of Algebraic Geometry
Issue number2
StatePublished - 2011

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology


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