Smooth affine surfaces with non-unique ℂ*-actions

Hubert Flenner, Shulim Kaliman, Mikhail Zaidenberg

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

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
Volume20
Issue number2
StatePublished - 2011

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Pairwise
Uniqueness
Transformation group
Rational Curves
Effective Action
Smooth surface
Conjugation
Automorphism Group
Two Parameters
Inversion
Necessary
Sufficient Conditions
Family
Presentation

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

Cite this

Smooth affine surfaces with non-unique ℂ*-actions. / Flenner, Hubert; Kaliman, Shulim; Zaidenberg, Mikhail.

In: Journal of Algebraic Geometry, Vol. 20, No. 2, 2011, p. 329-398.

Research output: Contribution to journalArticle

Flenner, H, Kaliman, S & Zaidenberg, M 2011, 'Smooth affine surfaces with non-unique ℂ*-actions', Journal of Algebraic Geometry, vol. 20, no. 2, pp. 329-398.
Flenner, Hubert ; Kaliman, Shulim ; Zaidenberg, Mikhail. / Smooth affine surfaces with non-unique ℂ*-actions. In: Journal of Algebraic Geometry. 2011 ; Vol. 20, No. 2. pp. 329-398.
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