Uniqueness of ℂ* - and ℂ+ -actions on Gizatullin surfaces

Hubert Flenner, Shulim Kaliman, Mikhail Zaidenberg

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


A Gizatullin surface is a normal affine surface V over ℂ, which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of ℂ* -actions and double-struck A1 -fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with ℂ+ -actions on V considered up to a "speed change". Non-Gizatullin surfaces are known to admit at most one double-struck A 1 -fibration V → S up to an isomorphism of the base S. Moreover, an effective ℂ* -action on them, if it does exist, is unique up to conjugation and inversion t → -1 of ℂ*. Obviously, uniqueness of ℂ* -actions fails for affine toric surfaces. There is a further interesting family of nontoric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of ℂ* -actions and double-struck A1 -fibrations, see, e.g., [FKZ1]. In the present paper we obtain a criterion as to when double-struck A1 -fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base S ≅ double-struck A 1. We exhibit as well large subclasses of Gizatullin ℂ* -surfaces for which a ℂ* -action is essentially unique and for which there are at most two conjugacy classes of double-struck A1 -fibrations over double-struck A1.

Original languageEnglish (US)
Pages (from-to)305-354
Number of pages50
JournalTransformation Groups
Issue number2
StatePublished - Jun 1 2008

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology


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