Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Smooth affine surfaces with non-unique $ \mathbb{C}^*$-actions


Authors: Hubert Flenner, Shulim Kaliman and Mikhail Zaidenberg
Journal: J. Algebraic Geom. 20 (2011), 329-398
DOI: https://doi.org/10.1090/S1056-3911-2010-00533-4
Published electronically: November 29, 2010
MathSciNet review: 2762994
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Abstract: In this paper we complete the classification of effective $ \mathbb{C}^*$-actions on smooth affine surfaces up to conjugation in the full automorphism group and up to inversion $ \lambda\mapsto \lambda^{-1}$ of $ \mathbb{C}^*$. If a smooth affine surface $ V$ admits more than one $ \mathbb{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 $ \mathbb{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 $ \mathbb{C}^*$-actions depending on one or two parameters. We give an explicit description of all such surfaces and their $ \mathbb{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 $ \mathbb{C}_+$-actions depending on $ k$ parameters.


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Hubert Flenner
Affiliation: Fakultät für Mathematik, Ruhr Universität Bochum, Geb. NA 2/72, Universitätsstr. 150, 44780 Bochum, Germany
Email: Hubert.Flenner@rub.de

Shulim Kaliman
Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33124
Email: kaliman@math.miami.edu

Mikhail Zaidenberg
Affiliation: Université Grenoble I, Institut Fourier, UMR 5582 CNRS-UJF, BP 74, 38402 St. Martin d’Hères cédex, France
Email: zaidenbe@ujf-grenoble.fr

DOI: https://doi.org/10.1090/S1056-3911-2010-00533-4
Received by editor(s): October 6, 2008
Received by editor(s) in revised form: March 20, 2009
Published electronically: November 29, 2010

American Mathematical Society