Séminaire de géométrie algébrique

Le jeudi à 14h.
septembre-décembre ENS, janvier-mars Jussieu, avril-juin Sophie Germain

45 rue d'Ulm, Paris 5è (salle W) ou 4 place Jussieu, Paris 5e ou Bat Sophie Germain, av de France
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Janvier 2020 Affiche

23/01/2020 14h00 (5-02) Jussieu 15-25 Alastair Craw, Bath
Birational geometry of symplectic quotient singularities
Résumé: For a finite subgroup G of SL(2,C) and for n \geq 1, the Hilbert scheme X=Hilb^[n](S) of n points on the minimal resolution S of the Kleinian singularity C^2/G provides a crepant resolution of the symplectic quotient C^{2n}/G_n, where G_n is the wreath product of G with S_n. I'll explain why every projective, crepant resolution of C^{2n}/G_n is a quiver variety, and why the movable cone of X can be described in terms of an extended Catalan hyperplane arrangement of the root system associated to G by John McKay. These results extend the algebro-geometric aspects of Kronheimer's hyperkahler description of S to higher dimensions. This is recent joint work with Gwyn Bellamy.
23/01/2020 15h30 (5-02) Jussieu 15-25 Enrico Arbarello, Rome
(groupe de travail, suite de l'exposé du 17/1)

30/01/2020 14h(40+30) (5-02) Jussieu 15-25 Kieran O'Grady, Rome
Modular sheaves on HK varieties.
Résumé : Since moduli of sheaves on K3 surfaces play a key role in Algebraic Geometry, and since K3's are the two dimensional hyperkähler (HK) manifolds, it is natural to investigate moduli of sheaves on higher dimensional HK's. We propose to focus attention on (coherent) torsion free sheaves on a HK variety X whose discriminant in H^4(X) satisfies a certain condition. These are the modular sheaves of the title. For example a sheaf whose discriminant is a multiple of c_2(X) is modular. For HK's which are deformations of the Hilbert square of a K3 we prove an existence and uniqueness result for slope-stable vector bundles with certain ranks, c_1 and c_2. As a consequence we get uniqueness up to isomorphism of the tautological quotient rank 4 vector bundle on the variety of lines on a generic cubic 4-dimensional hypersurface, and on the Debarre-Voisin variety associated to a generic skew-symmetric 3-tensor in 10 variables. The last result implies that the period map from the moduli space of Debarre-Voisin varieties to the relevant period space is birational.

Fevrier 2020 Affiche

06/02/2020 14h(40+30) (5-02) Jussieu 15-25 Francesco Russo, Catania
Rationality of cubic fourfolds via Trisecant Flops and via (non minimal) associated K3 surfaces.
According to Kuznetsov Conjecture, in the moduli space of cubic fourfolds there exist infinitely many irreducible divisors (cubics of “admissible discriminant d” in the sense of Hassett), whose union should be  the locus of rational cubic fourfolds.
Via the construction of the Trisecant Flops and via the theory of the congruences of $3e-1$-secant curves of degree $e$ to surfaces in P5, we shall explain the role played by “associated" (non minimal) K3 surfaces in various rationality questions regarding cubic and Gushel-Mukai fourfolds. As an application we shall present uniform proofs of the cases d=14, 26, 38 and 42 of the Conjecture, classically known only for d=14 (Fano, 1943), and discuss further possible developments of this circle of ideas.
This is joint work with Giovanni Staglianò.

27/02/2020 14h00 (5-02) Jussieu 15-25 Enrico Fatighenti, Loughborough Univ.
(à préciser)

Mars 2020 Affiche

05/03/2020 14h00 (5-02) Jussieu 15-25 Ulrike Riess, ETH Zurich
(à préciser)