Lien ZOOM de la conférence: ID de réunion 842 5343 5365

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Monday 10 mai

Tuesday 11 mai

Wednesday 12 mai

10h

Introduction par J. d’Almeida







10h30-11h30

D. Markushevich

A case study in complex crystallographic groups: point group SL(2,7).



10h00-11h00

M. Brion

Variétés homogènes primitives.



10h-11h

O. Piltant

Old and new questions around the Local Uniformization problem.



11h45-12h45

A. Sarti

K3 surfaces with maximal finite automorphism groups.



11h30-12h30

K. Ranestad

Quartic forms and Gorenstein rings of codimension and regularity 4.



11h15-12h15

A. Fanelli

Rational simple connectedness and Fano threefolds.









14h15-15h15

L. Manivel

Sur les automorphismes des variétés de Mukai/On the automorphisms of Mukai varieties.



14h00-15h00

B. Remy

Géométrie non archimédienne, descente et théorie spectrale.



13h30-14h30

J. Weyman

Finite free resolutions and root systems.



15h45-16h45

P. Ellia

Negative instantons.



15h00-16h30


Discussions



14h45-15h45

A.M. Castravet

Blown-up toric surfaces with non-polyhedral effective cone.



17h00-18h00

R. Lazarsfeld

Saturation bounds for smooth varieties.



16h30-17h30

C. Peskine

Elliptic birational transformations of even complex projective spaces.



16h00-17h00

J. Kollár

Deformations of varieties of general type.





17h45-18h45

D. Eisenbud

Layered Free Resolutions of Cohen-Macaulay modules over complete intersections.



Résumés (Abstracts)

Michel Brion: Variétés homogènes primitives
Résumé : Le point de départ de l'exposé est la classification des actions birationnelles d'un groupe algébrique G en termes des couples (K,X) où K est un corps de fonctions et X une K-variété homogène sous G_K ; ceci amène à considérer des variétés homogènes sur un corps arbitraire. On présentera un résultat de structure pour les morphismes entre de telles variétés, qui amène à la notion de variété homogène primitive. Sur un corps algébriquement clos, on montrera que les variétés homogènes primitives sont en nombre fini à isomorphisme près.

Ana-Maria Castravet: Blown-up toric surfaces with non-polyhedral effective cone
Abstract: I will report on recent joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia. We construct examples of projective toric surfaces whose blow-up at a general point has a non-polyhedral effective cone, both in characteristic 0 and in prime characteristic. As a consequence, we prove that the effective cone of the Grothendieck-Knudsen moduli space of stable, n-pointed, rational stable curves, is not polyhedral if n>=10 in characteristic 0 and in positive characteristic for an infinite set of primes of positive density.

David Eisenbud: Layered Free Resolutions of Cohen-Macaulay modules over complete intersections
Abstract: Maximal Cohen-Macaulay modules over hypersurfaces correspond in a simple way to matrix factorizations, which provide minimal free resolutions of these modules. An extension of this corre spondence to complete intersections has been known for a long time, but does not lead to minimal resolutions. I will describe some of the history, and also recent joint work with Irena Peeva on a different construction, the ``layered free resolution, that is minimal in a large range of cases.

Philippe Ellia: Negative instantons
Abstract: This is a joint (unfinished) work with Laurent Gruson. A negative instanton is a stable rank two vector bundle, E, on P^3, with c_1(E) = −1 and h^1(E(−2)) = 0. We give some general properties and an application.

Andrea Fanelli: Rational simple connectedness and Fano threefolds.
Abstract: The notion of rational simple connectedness can be seen as an algebro-geometric analogue of simple connectedness in topology. The work of de Jong, He and Starr has already produced several recent studies to understand this notion. In this talk I will discuss the joint project with Laurent Gruson and Nicolas Perrin to study rational simple connectedness for Fano threefolds via explicit methods from birational geometry.

János Kollár: Deformations of varieties of general type
Abstract: We prove that small deformations of a projective variety of general type are also projective varieties of general type, with the same plurigenera.

Robert Lazarsfeld: Title: Saturation bounds for smooth varieties
Abstract: I will discuss joint work with Lawrence Ein and Tai Ha giving saturation bounds for ideals defining smooth projective varieties and their powers. The arguments revisit ideas I learned many years ago while collaborating with Gruson and Peskine.

Laurent Manivel: Sur les automorphismes des variétés de Mukai/On the automorphisms of Mukai varieties.
Résumé: Les variétés de Mukai de genre 7 à 10 sont des sections linéaires d'espaces homogènes. Pour certaines dimensions critiques, leurs groupes d'automorphismes génériques sont des groupes finis non triviaux, formés d'involutions ou d'automorphismes d'ordre trois. L'existence de ces automorphismes inattendus est liée à la théorie des theta-groupes explorée notamment, dans un contexte géométrique, par Laurent Gruson, Steven Sam et Jerzy Weyman. C'est le fruit d'une collaboration avec Thomas Dedieu. Version anglaise: Title: On the automorphisms of Mukai varieties
Abstract: Mukai varieties of genus 7 to 10 are linear sections of homogeneous varieties. For certain critical dimensions, their generic automorphisms groups are non trivial finite groups made of involutions and order three elements. The existence of these unexpected automorphisms is related to the theory of theta-groups, already used in a geometric context by Laurent Gruson, Steven Sam and Jerzy Weyman (joint work with Thomas Dedieu).

Dimitri Markouchevitch: A case study in complex crystallographic groups: point group SL(2,7).

Christian Peskine: Elliptic birational transformations of even complex projective spaces
Abstract

Olivier Piltant:Old and new questions around the Local Uniformization problem.
Résumé: Laurent Gruson's first works at the end of the 1960's belong to the field of $p$-adic analytic geometry. The stability or defectless assumption on a valued field $(K,v)$ plays an important role there because it allows to study finitely generated modules over the Tate algebra or rings of overconvergent functions on a polydisk by passing to the associated graded module. The defect in the ramification theory of valuations is also a central difficulty for the Local Uniformization problem. In this talk, I will review some related issues which are specific to residue characteristic $p>=0$ Resolution: defect, structure of Hironaka groups, local reducibility of arc spaces of varieties, and explain how they can be overcome for schemes of dimension three.

Kristian Ranestad: Quartic forms and Gorenstein rings of codimension and regularity 4.
Abstract: The dual socle generator of an Artinian Gorenstein ring A of regularity four is a quartic form F in four variables. We classify the forms F in terms of powersum decompositions according to the Betti table of A, and investigate how A may lift to the homogeneous coordinate ring of a Calabi-Yau threefold. I report on work in progress in collaboration with G. Kapustka, M. Kapustka, H. Schenck, M. Stillman, B. Yuan.

Bertrand Rémy: Géométrie non archimédienne, descente et théorie spectrale.
Résumé: Cet exposé fera un petit panorama des compactifications d’immeubles de groupes réductifs sur les corps locaux non archimédiens (avec A. Thuillier et A. Werner). V. Berkovich avait décrit la possibilité, dans le cas des groupes déployés, de faire de la théorie de Bruhat-Tits du point de vue de sa conception de la géométrie analytique. Pour passer au cas des groupes non nécessairement déployés, il a fallu développer l’approche. En particulier des procédés de descente ont dû être mis au point. Pour ce faire, un papier de Laurent Gruson sur la théorie de Fredholm p-adique a été déterminant.

Alessandra Sarti: K3 surfaces with maximal finite automorphism groups
Resumé: It was shown by Mukai that the maximum order of a finite group acting symplectically on a K3 surface is 960 and that the group is isomorphic to the Mathieu group $M_{20}$. Then Kondo showed that the maximum order of a finite group acting on a K3 surface is 3840 and this group contains the Mathieu group with index four. Kondo showed also that there is a unique K3 surface on which this group acts, which is a Kummer surface. I will present recent results on finite groups acting on K3 surfaces, that contain strictly the Mathieu group and I will classify them. I will show that there are exactly three groups and three K3 surfaces with this property. The contruction of these K3 surfaces is related to the action of some finite complex reflection groups on the 3-dimensional complex projective space and inspired by a previous work with W. Barth. If time permits I will explain this relation. This is a joint work with C. Bonnafé.

Jerzy Weyman: Title: Finite free resolutions and root systems.
Abstract: I will talk about the link between the finite free resolutions of length 3 and Kac-Moody Lie algebras related to the T-shaped graph T_{p,q,r}. For small formats related to the cases where T_{p,q,r} is a Dynkin graph one has precise conjectures on the structure of perfect ideals of codimension 3 with resolutions of such formats. The case of Gorenstein ideals of codimension 3 corresponds to half of D_n cases. The conjectures turn out to be related to the old question of Peskine-Szpiro. In the second part of the talk I will outline similar link between Gorenstein ideals of codimension 4 with n generators and the root system of type E_n.