### 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) |
---|---|

Pages (from-to) | 305-354 |

Number of pages | 50 |

Journal | Transformation Groups |

Volume | 13 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2008 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

_{+}-actions on Gizatullin surfaces.

*Transformation Groups*,

*13*(2), 305-354. https://doi.org/10.1007/s00031-008-9014-0

**Uniqueness of ℂ* - and ℂ _{+} -actions on Gizatullin surfaces.** / Flenner, Hubert; Kaliman, Shulim; Zaidenberg, Mikhail.

Research output: Contribution to journal › Article

_{+}-actions on Gizatullin surfaces',

*Transformation Groups*, vol. 13, no. 2, pp. 305-354. https://doi.org/10.1007/s00031-008-9014-0

_{+}-actions on Gizatullin surfaces. Transformation Groups. 2008 Jun;13(2):305-354. https://doi.org/10.1007/s00031-008-9014-0

}

TY - JOUR

T1 - Uniqueness of ℂ* - and ℂ+ -actions on Gizatullin surfaces

AU - Flenner, Hubert

AU - Kaliman, Shulim

AU - Zaidenberg, Mikhail

PY - 2008/6

Y1 - 2008/6

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=49049095886&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=49049095886&partnerID=8YFLogxK

U2 - 10.1007/s00031-008-9014-0

DO - 10.1007/s00031-008-9014-0

M3 - Article

VL - 13

SP - 305

EP - 354

JO - Transformation Groups

JF - Transformation Groups

SN - 1083-4362

IS - 2

ER -