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Anneau de Cox de variétés avec action de groupe

Abstract : The Cox ring of an algebraic variety (satisfying some natural conditions) is a very rich invariant. It was introduced by Cox in 1995 for the study of toric varieties, and then generalized to normal varieties by Arzhantsev, Berchtold and Hausen. Later, Hu and Keel discovered that the normal varieties with a finitely generated Cox ring define a class of varieties whose birational geometry is particularly well understood. They called them the Mori Dream Spaces (MDS) by virtue of their good behaviour with respect to the minimal model program of Mori. A first problem is to find natural conditions for a normal variety to be an MDS. A second one is to describe the Cox ring of a given MDS: find a presentation by generators and relations, give the nature of its singularities, etc...Among algebraic varieties equipped with an action of an (affine) algebraic group, a particularly well understood class consists of normal varieties of complexity at most one: a connected reductive group is acting in such a way that the minimal codimension of an orbit of a Borel subgroup is at most one. The normal varieties of complexity zero are the spherical varieties (e.g. a toric variety is spherical). In 2007, Brion proved that spherical varieties are MDS, and gave a description of their Cox ring by generators and relations. A normal variety of complexity one is an MDS if and only if it is a rational variety (e.g. a normal rational surface with an effective action of the multiplicative group or a normal SL2-threefold with a dense orbit). This provides a natural class of MDS with group action for which the second problem has only been solved in very particular cases.In this thesis we begin by defining the equivariant Cox ring of a normal variety equipped with an action of an algebraic group. We study its general properties and relate it to the ordinary Cox ring. We then build on this preliminary work to study various aspects of the Cox ring of a normal rational variety of complexity one. Notably, we obtain a finiteness result for the iteration of Cox rings and we characterize the log terminality of singularities. Then, we study in detail a class of examples: the Cox rings of the SL2-threefolds which are almost homogeneous (i.e. normal and having a dense orbit). Extending the results obtained on these examples, we give a presentation by generators and relations of the Cox ring of an almost homogeneous variety of complexity one. To obtain this last result, we introduce and study an interesting class of equivariant completions of certain complexity one homogeneous spaces.
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Submitted on : Wednesday, July 20, 2022 - 4:03:11 PM
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Antoine Vezier. Anneau de Cox de variétés avec action de groupe. Géométrie algébrique [math.AG]. Université Grenoble Alpes [2020-..], 2021. Français. ⟨NNT : 2021GRALM081⟩. ⟨tel-03729499⟩

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