Séminaire de géométrie algébrique

Dans cet exposé, nous allons étudier la réalisation motivique (et l-adique) de la catégorie de singularités attachées à un modèle de Landau-Ginzburg généralisé. En suite, on reliera cette réalisation à un formalisme du type "cycles évanescents". Enfin, on utilisera cette comparaison pour calculer la réalisation l-adique de la catégorie de singularités de la fibre spécial d'un schéma régulier, propre e plat sur un anneau régulier de dimension n.

(Joint with Piotr Pstragowski) We extend the construction of the normal cone of a closed embedding of schemes to any locally of finite type morphism of higher Artin stacks and show that in the Deligne-Mumford case our construction recovers the relative intrinsic normal cone of Behrend and Fantechi. We characterize our extension as the unique one satisfying a short list of axioms, and use it to construct the deformation to the normal cone. As an application of our methods, we associate to any morphism of Artin stacks equipped with a choice of a global perfect obstruction theory a relative virtual fundamental class in the Chow group of Kresch.

In a joint work with Longting Wu and Fenglong You, we defined relative quantum cohomology rings by introducing relative Gromov-Witten invariants with negative contact orders. I will compare the definitions of absolute and relative quantum rings, try to explain how the WDVV equation (associativity) motivates our definition of negative contacts, and perhaps show some examples if time permits.

Being the basis of Gromov-Witten theory, Kontsevich's moduli spaces of stable maps are important in different areas of mathematics and theoretical physics. For this reason, it is interesting to study their geometry, for example analyzing their (co)homology. In fact, determining their Betti/Hodge numbers is already a nontrivial problem. In this talk, I will present a method for computing the Betti/Hodge numbers of moduli spaces of stable maps from genus 0 curves to a Grassmann variety G(r,V). First, by using the combinatorial properties of these spaces, I will show that the problem can be reduced to the computation of the Hodge numbers of the open locus parametrizing stable maps from smooth curves. Then, I will show how the latter can be explicitly calculated, by means of a suitable Quot scheme compactification of the space of morphism of fixed degree from the projective line to G(r,V).

Je vais présenter une correspondance entre un problème essentiellement combinatoire, à savoir le comptage raffiné de courbes tropicales, et un problème de géométrie énumérative complexe, à savoir une variante de la théorie de Gromov-Witten des surfaces toriques. Ce résultat a des applications à la symétrie miroir en genre supérieur et à des questions a priori purement algébriques de construction d'algèbres non-commutatives (quantification par déformation avec base canonique des algèbres amassées de rang 2).

The plane Cremona group Cr2 is the group of birational automorphisms of P^2. From the half of the XIXth century the plane Cremona group has been studied under many aspects. We say that a curve is contractible if it can be contracted to a finite set of points. Starting from Castelnuovo and Enriques untill Calabri and Ciliberto, the problem has been investigated and related to adjoint linear systems. In this talk I will present the state of the art and give some general results about the contractibility of a curve. Then we will focus on the study of contractible configurations of lines.