Program

Lectures will take place at Sala di Rappresentanza on the ground floor of the Department of Mathematics.

June 21

13:30-14:20

Howard Nuer: Unirationality of moduli spaces of special cubic fourfolds and K3 surfaces

Abstract: We provide explicit descriptions of the generic members of Hassett’s divisors C_d for relevant 18<=d<=38 and d=44. As a corollary, we prove the unirationality of these C_d. We obtain as an additional consequence that the moduli space N_d of polarized K3 surfaces of degree d is unirational for d=14,26,38. The case d=26 is entirely new, while the other two have been previously proven by Mukai. We also explain the construction of a new 19 dimensional family of hyperkahler manifolds not birational to any moduli space of (twisted) sheaves on a K3 surface. Time permitting, we discuss how some of these results have been used by Russo and Stagliano to prove the rationality of the generic cubic fourfold in C_26 and C_38.

14:30-15:20

Frank Gounelas: Positivity of the cotangent bundle of a K3 surface

Abstract: Kobayashi proved that symmetric powers of the cotangent bundle of a Calabi-Yau manifold have no global sections, and more recently Nakayama strengthened this to the statement that the tautological line bundle on the projectivised cotangent bundle is not even pseudoeffective. This raises the question: what are the cones of divisors of this projective bundle? I will discuss results in this direction (joint work with J.C. Ottem), ultimately related to the cones of divisors on the Hilbert scheme of two points of the K3.

15:20-16:00 Coffee break

16:00-16:50

Evgeny Shinder: Specialization of stable rationality

Abstract: The specialization question for rationality is the following one: assume that very general fibers of a flat proper morphism are rational, does it imply that all fibers are rational? I will talk about recent solution of this question in characteristic zero due to myself and Nicaise, and Kontsevich-Tschinkel. The proof relies on a construction of specialization morphisms for the Grothendieck ring of varieties and its variants. The method also applies more generally to deal with specialization and variation of stable birational types.

June 22

9:30-10:20

Emanuele Macrì: Derived categories of cubic fourfolds and non-commutative K3 surfaces, 1

Abstract: The derived category of coherent sheaves on a cubic fourfold has a subcategory which can be thought as the derived category of a non-commutative K3 surface. This subcategory was studied recently in the work of Kuznetsov and Addington-Thomas, among others. In this talk, I will present joint work in progress with Bayer, Lahoz, Nuer, Perry, Stellari, on how to construct Bridgeland stability conditions on this subcategory. This proves a conjecture by Huybrechts, and it allows to start developing the moduli theory of semistable objects in these categories, in an analogue way as for the classical Mukai theory for (commutative) K3 surfaces. I will also discuss a few applications of these results.

10:20-11:00 Coffee break

11:00-11:50

Martí Lahoz: Derived categories of cubic fourfolds and non-commutative K3 surfaces, 2

Abstract: In this talk, I will present joint work in progress with Bayer, Macrì, Nuer, Perry, Stellari. Following the previous talk by Emanuele Macrì, I will concentrate in the construction of Bridgeland stability conditions on the Kuznetsov component of a cubic fourfold and the definition of stability conditions in family.

12:00-12:50

Paola Comparin: Quasismooth hypersurfaces in toric varieties

Abstract: According to Cox's construction, any normal projective toric variety can be described as U//G, where U is an open subset in an affine space A^r and G an abelian group. Let Y be a hypersurface in X, defined as the zero set of a homogeneous polynomial in the Cox ring of X, with general coefficients. We call Y quasismooth if the intersection of its singular locus with U is empty, or equivalently, if the singular points of Y are in the irrelevant locus of X. When X is P^n, this definition gives exactly smooth hypersurfaces, while this is not true in general. In a joint work with M. Artebani and R. Guilbot, we study quasismoothness for hypersurfaces in toric variety and we give a characterization of it using combinatorial properties of the Newton polytope of the hypersurface.