## Abstract

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 A^{1} -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 A^{1} -fibrations, see, e.g., [FKZ_{1}]. In the present paper we obtain a criterion as to when double-struck A^{1} -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 A^{1} -fibrations over double-struck A^{1}.

Original language | English (US) |
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Pages (from-to) | 305-354 |

Number of pages | 50 |

Journal | Transformation Groups |

Volume | 13 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 2008 |

## ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology