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DTSTART;TZID=Europe/Paris:20210406T113000
DTEND;TZID=Europe/Paris:20210406T123000
DTSTAMP:20220422T124140Z
CREATED:20210315T134033Z
LAST-MODIFIED:20220422T124140Z
UID:9068-1617708600-1617712200@lmv.math.cnrs.fr
SUMMARY:AG : Antoine Vezier (Institut Fourier) : "The Cox ring of a complexity one almost homogeneous variety"
DESCRIPTION: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… \nAmong 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 $\mathbb{G}_m$-action or a normal SL$_2$-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. \nIn this talk\, we will detail the construction of the Cox ring of a normal variety\, and explain its importance in algebraic geometry. Then\, we will describe the Cox ring of a complexity one almost homogeneous variety  (i.e. it is normal with a dense orbit)\, together with the methods developed to obtain this description. \n[L’exposé aura lieu en ligne\, sur Zoom. Pour avoir le lien de connexion\, contacter l’organisateur LP.] \nVidéo et slides de l’exposé.
URL:https://lmv.math.cnrs.fr/evenenement/expose-dantoine-vezier-institut-fourier-the-cox-ring-of-a-complexity-one-almost-homogeneous-variety/
CATEGORIES:Séminaire AG
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DTSTART;TZID=Europe/Paris:20210420T113000
DTEND;TZID=Europe/Paris:20210420T123000
DTSTAMP:20220422T124246Z
CREATED:20210412T085247Z
LAST-MODIFIED:20220422T124246Z
UID:9151-1618918200-1618921800@lmv.math.cnrs.fr
SUMMARY:AG : Mikhail Gorsky (Univ. de Stuttgart) :  « Exact structures\, Hall algebras\, and quantum groups »
DESCRIPTION:Abstract : Hall algebras and related notions play a prominent role in the modern representation theory. In their present form\, they first appeared in a series of papers by Ringel on quantum groups. After giving all the necessary definitions\, I will explain the interplay between different exact structures on an additive category and degenerations of the associated Hall algebras. For the categories of representations of Dynkin quivers\, this recovers degenerations of the nilpotent part of the corresponding quantum group. \nTo realize an entire quantum group as a version of a Hall algebra\, one has to consider a more complicated category. I will explain how to recover the comultiplication of the quantum group by taking a certain unexpected exact structure on this category. If time permits\, we will discuss several related conjectures. \nBased on joint work and an ongoing project with Xin Fang. \n  \n[ L’exposé sera virtuel et se déroulera sur Zoom. Contacter l’organisateur LP pour obtenir les codes de connexion ]. \nVidéo et slides de l’exposé.
URL:https://lmv.math.cnrs.fr/evenenement/mikhail-gorsky-univ-de-stuttgart-extriangulated-structures-and-degeneration-of-hall-algebras/
CATEGORIES:Séminaire AG
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