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Theses

Sur une classe de schémas avec actions de fibrés en droites

Abstract : On a affine algebraic variety S defined over a field k of characteristic zer o, there exists a well-known correspondence between algebraic actions of the add itive group k+=(k,+) and locally nilpotent derivations of the algebra of regular functions on S. Here we extend this equivalence between actions and derivations to the following relative situation : π:S → X is a scheme over a fixed base scheme X, which comes equipped with an algebraic action of a l ine bundle p:L→X over X. We study a special sub-class of schemes S as above, with the additional property that π:S → X factors through the structural morphism π':S→Y of a principal homogeneous bundle under a line bundle p':L'→Y over an X-scheme δ:Y→X, in such a way that the action of δ*L on S factors through the one of L'. We call them Danielewski-Fieseler schemes. We give different procedures to construct these schemes. In particular, in case that the base scheme X is affine, we give an algorithm which produces explicit embeddings of these schemes in relatives affine spaces over X. Then we study the case that the base scheme X is isomorphic to the affine line over a field k of characteristic zero. In this case, we establish that a Danielewski-Fieseler scheme is uniquely determined by a combinatorial data consisting of a weighted rooted tree. We classify these schemes through their associated trees. Finally, we give a combinatorial characterization of those schemes which admit many actions of the additive group k+ with distinct gener al orbits.
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https://tel.archives-ouvertes.fr/tel-00007733
Contributor : Arlette Guttin-Lombard <>
Submitted on : Monday, December 13, 2004 - 3:55:58 PM
Last modification on : Wednesday, November 4, 2020 - 2:06:55 PM
Long-term archiving on: : Thursday, September 13, 2012 - 1:40:43 PM

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  • HAL Id : tel-00007733, version 1

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Adrien Dubouloz. Sur une classe de schémas avec actions de fibrés en droites. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2004. Français. ⟨tel-00007733⟩

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