Program

Lectures will take place at Sala di Rappresentanza on the ground floor of the Department of Mathematics and online on zoom (write to the organizers to get the link).

If you want to participate we kindly ask you to write to the organizers in order to arrange the room to host the correct number of participants.

10:00-11:00

Giovanni Mongardi: Irrational Gushel Mukai threefolds.

Abstract: We construct an explicit complex smooth Fano threefold with Picard numer 1, index 1 and degree 10 (a Gushel-Mukai threefold) and prove that it is not rational by showing that its intermediate jacobian has a faithful PSL(2,F_11) action. The construction is based on a very special double EPW sextic. This is joint work with O. Debarre.

11:30-12:30

Maria Gioia Cifani: Reconstructing curves from their Hodge classes.

Abstract: A classical result of Griffiths establishes a connection between homogeneous polynomials in the Jacobian ring of a smooth projective hypersurface X and primitive cohomology classes in X. In view of this, recently Movasati and Sertöz pose several interesting questions about the reconstruction of subvarieties of X from their Hodge classes.

I will report on a joint work with Gian Pietro Pirola and Enrico Schlesinger, in which we give an answer to some of these questions, in the case of surfaces in $P^3$: in particular, we give sharp numerical sufficient conditions to reconstruct a curve C by forms of low degree in the ideal of the curve. Moreover, they give the notion of a perfect class: the Hodge class of a curve C is perfect if its annihilator is a sum of ideals of curves whose Hodge class is a nonzero rational multiple of that of C. We show that the Hodge class of an arithmetically Cohen-Macaulay curve is always perfect but not all the Hodge classes are perfect.

Lunch break

14:00-15:00

Andreas Hochenegger: Formality of P-Objects.

Abstract: Daniel Huybrechts and Richard Thomas introduced P-objects as Calabi-Yau objects whose derived endomorphism algebra is isomorphic to $k[t]/t^n$. The interest in P-objects comes from the fact that they induce autoequivalences of triangulated categories, the so-called P-twists, and that they appear naturally in the derived category of hyperkähler varieties. But P-objects appear also, for example, in symplectic geometry. Therefore one can ask to what extend the properties of P-objects and the associated P-twists depend on the geometry. In this talk, I will give a precise meaning to this question, which leads to the notion of formality. This talk is based on joint work with Andreas Krug.

15:30-16:30

Elisa Postinghel: Weyl cycles on blow-ups of projective 4-spaces.

Abstract: I will introduce the definition of Weyl cycles on the blow-up X of P^n in a collection of points in general position. These are irreducible components of the intersection of pairwise orthogonal effective divisors on X that live in the Weyl orbit of exceptional divisors, where the orthogonality is taken with respect to a Dolgachev-Mukai pairing on the Picard group of X. In the case where X is a Mori dream space of dimension four or less, we can describe the geometry of and classify these objects. I will also explain how this relates to certain polynomial interpolation problems.

This is joint work with M. C. Brambilla and O. Dumitrescu.

Coffee break

17:00-18:00

Ernesto Mistretta: An elementary problem on semiampleness of vector bundles.

Abstract: We will describe some recent results on various characterizations of parallelizable compact complex manifolds, and relate these to an apparently simple problem on semiampleness of vector bundles, allowing to characterize quotients of parallelizable compact complex manifolds, similarly to what can be done to obtain a characterization of quotients of abelian varieties (this is a work in progress).

There will be a social dinner, please write to the organizers if you plan to attend.