### 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 language | English (US) |
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Pages (from-to) | 329-398 |

Number of pages | 70 |

Journal | Journal of Algebraic Geometry |

Volume | 20 |

Issue number | 2 |

DOIs | |

State | Published - 2011 |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology

### Cite this

*Journal of Algebraic Geometry*,

*20*(2), 329-398. https://doi.org/10.1090/S1056-3911-2010-00533-4