Sur quelques aspects des champs de revêtements de courbes algébriques

Abstract : The aim of this thesis is to study algebraic stacks of covers of algebraic curves, with special attention paid to positive characteristic. In the beginning we establish some results concerning group scheme actions on stacks: existence and algebraicity of stacks of fixed points and quotients; link with the classifying stack of the group. From now on, consider finite groups $G,G'$ of orders $n,n'$. Using Hurwitz theory of tame covers of curves, we first provide a smooth compactification of the stack ${\cal M}_g(G')$ of genus $g$ curves with level structure $G'$. This is also a desingularization of the proper stack given by Deligne-Mumford by normalization of the stack of stable curves of genus $g$ with respect to ${\cal M}_g(G')$. Then, letting certain groups act on the stack constructed above, we give a compactification for the stack of curves of genus $g$ with $G$ action, the base including this time all characteristics that divide $n$. This compactification is smooth a priori only over the prime-to-$n$ characteristics. At last, we focus on the local aspect of wild ramification. Assume that $G$ acts on a scheme $X$ over a discrete valuation ring of unequal characteristics (with the residue characteristic dividing $n$), and that this action is faithful on the generic fibre. We wish to find a model for $G$ which acts faithfully also on the special fibre, unique with this property. If $X$ is proper this ie not too hard. For $X$ affine, we provide a method, based on Néron blow-ups, that leads at least conjecturally to an effective approach to this model. In the case of the cyclic group of order $p$, this method yields the precise structure of covers of a germ of smooth curve. In the end, we conclude with an example illustrating the questions treated in the thesis.
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https://tel.archives-ouvertes.fr/tel-00002122
Contributor : Arlette Guttin-Lombard <>
Submitted on : Friday, December 13, 2002 - 10:07:41 AM
Last modification on : Wednesday, November 4, 2020 - 1:58:05 PM
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• HAL Id : tel-00002122, version 1

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Matthieu Romagny. Sur quelques aspects des champs de revêtements de courbes algébriques. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2002. Français. ⟨tel-00002122⟩

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