Kraft, Hanspeter and Russell, Peter.
(2014)
* Families of Group Actions, Generic Isotriviality, and Linearization.*
Transformation groups, 19.
pp. 779-792.

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Official URL: http://edoc.unibas.ch/43324/

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## Abstract

We study families of reductive group actions on A2 parametrized by curves and show that every faithful action of a non-finite reductive group on A3 is linearizable, i.e. G-isomorphic to a representation of G. The difficulties arise for non-connected groups G. We prove a Generic Equivalence Theorem which says that two affine mor- phisms p: S → Y and q: T → Y of varieties with isomorphic (closed) fibers become isomorphic under a dominant ́etale base change φ : U → Y . A special case is the following result. Call a morphism φ: X → Y a fibration with fiber F if φ is flat and all fibers are (reduced and) isomorphic to F. Then an affine fibration with fiber F admits an ́etale dominant morphism μ: U → Y such that the pull-back is a trivial fiber bundle: U ×Y X ≃ U × F . As an application we give short proofs of the following two (known) results: (a) Every affine A1-fibration over a normal variety is locally trivial in the Zariski-topology; (b) Every affine A2-fibration over a smooth curve is locally trivial in the Zariski-topology.

Faculties and Departments: | 05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Algebra (Kraft) |
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UniBasel Contributors: | Kraft, Hanspeter |

Item Type: | Article, refereed |

Article Subtype: | Research Article |

ISSN: | 1083-4362 |

Note: | Publication type according to Uni Basel Research Database: Journal article |

Language: | English |

Identification Number: | |

Last Modified: | 18 Oct 2016 14:49 |

Deposited On: | 18 Oct 2016 14:49 |

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