Schedule & Programme

Senior talks

Ignacio Barros: Effective and Noether-Lefschetz divisors on moduli of K3 surfaces

Maulik—Pandharipande conjectured that the rational Picard group of moduli spaces of quasi-polarized K3 surfaces is generated by Noether-Lefschetz divisors. This is now a theorem by Bergeron—Li—Millson—Moeglin (2017) and a natural follow-up question is whether the cone of effective divisors is also generated by irreducible components of NL divisors. The question is: is every effective divisor linearly equivalent to an effective divisor supported on NL loci? I will report on joint work (in progress) with Emma Brakkee, Pietro Beri, and Laure Flapan, where we give a negative answer to this question by providing examples in infinitely many degrees of effective divisors that are not of NL type. 

Pieter Belmans:  Deformations, symmetries, and derived categories of quiver moduli

Vector bundles on curves and quiver representations have moduli theories that are very similar. Inspired by this connection, I will recall the canonical 4-term sequence on quiver moduli and its relationship to the Kodaira-Spencer morphism. This sequence can be used to prove a conjecture of Schofield, establish the rigidity of quiver moduli, describe their infinitesimal symmetries, and show that the universal representation furnishes a fully faithful Fourier-Mukai functor. These are all statements that have already been established for moduli of vector bundles on curves. The stacky perspective on these moduli spaces and Teleman quantization will play an important role in the proof. This is joint work with Ana-Maria Brecan, Hans Franzen, Gianni Petrella, and Markus Reineke.

Vladimiro Benedetti: The Coble quadric

A very classical result by Coble states that the Kummer threefold of a general genus 3 curve C is singular along a unique quartic hypersurface in P^7, named thus the Coble quartic. By results of Beauville and Narasimhan-Ramanan the Coble quartic can be identified, via the theta map, with the moduli space SU_C(2) of semi-stable rank two vector bundles on C with trivial determinant. Recent results by Gruson-Sam-Weyman (GSW) showed that this quartic (and the Kummer) can be constructed from a general skew-symmetric four-form in eight variables. In fact, GSW's construction also shows that there exists a close relationship between some special representations of graded Lie algebras and moduli spaces of vector bundles on curves of small genus. In this talk we will see how the same skew-symmetric four-form also allows to explicitly construct, inside the Grassmannian G(2,8), the moduli space SU_C(C,O(p)) of semi-stable rank two vector bundles on C with determinant equal to O(p). Then we will extend Coble's result in this situation: we will see that, for generic p in C, there exists a unique quadratic section of G(2,8) which is singular exactly along SU_C(2, O(p)), and thus deserves to be coined the Coble quadric of the pointed curve (C,p). This is a joint work with Michele Bolognesi, Daniele Faenzi and Laurent Manivel.

Marcello Bernardara: Conic Bundles of K3 type and Hyperkähler manifolds

Cubic and Gushel-Mukai fourfolds carry (1 and 2 respectively) conic bundle structures, whose discriminants are nodal surfaces whose double covers are of general type. The anti-invariant part of the intermediate cohomology of the latter surfaces carries the K3 structure corresponding to the one in the intermediate cohomology of the fourfolds.

In a work in collaboration with Fatighenti, G. Kapustka, M. Kapustka,  Manivel, Mongardi and Tanturri, we prove that in the case of Gushel-Mukai fourfolds, the discriminant double covers can be described as sections of HK manifolds with an anti-symplectic involution.

Morevoer, we analyze 3 other families of Fano fourfolds of K3 type with conic bundles degenerating along the same discriminants as above and we relate each family to one of the previous via hyperbolic splitting.

Valeria Bertini: Deformation families of IHS varieties: classification problem and new examples

A fruitful way to produce examples of IHS varieties is to consider terminalizations of symplectic quotients of symplectic varieties. In a work in collaboration with A. Grossi, M. Mauri and E. Mazzon we classify all terminalizations of quotients of Hilbert schemes of K3 surfaces and generalized Kummer varieties by the action of symplectic automorphisms induced by the underlying surface. Furthermore, we determine their second Betti number, the fundamental group of their singular locus and, in the Kummer case, we determine the singularities of their universal quasi-étale cover. Finally, we compare our deformation types with the examples known in literature, placing our work in the classification program proposed by Menet.

Cinzia Casagrande: Fano 4-folds with rational fibrations onto threefolds

Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. 

We will first discuss the following result: if rho(X)>12, then X is a product of del Pezzo surfaces; if rho(X)=12, then X has a rational contraction X-->Y where Y has a dimension 3. A rational contraction is a rational map that factors as a sequence of flips followed by a surjective morphism with connected fibers; we will see an explicit example of such setting.  

Then we will discuss the geometric properties of Fano 4-folds having a rational contraction onto a threefold. One goal is to find a sharp bound on the Picard number of such X, and possibly to classify the cases with large Picard number.

Another goal is to use the geometric description to construct new examples with large Picard number; this is a joint project with Saverio Secci. 

Olivier Debarre:  Subvarieties of abelian varieties

On a very general complex polarized abelian variety, the group of Hodge classes is, in each degree, of rank 1. In this survey talk, I will discuss the problem of whether these classes are represented by subvarieties, smooth or not, cycles, or Chern classes of vector bundles.

Francesco Denisi: On volumes and polygons of Newton-Okounkov type on hyper-Kähler manifolds

Among the asymptotic invariants of line bundles on projective varieties, the volume is one of the most studied. This asymptotic invariant defines a continuous function on the Néron-Severi space of the considered variety: the volume function. Also, the volume of line bundles can be interpreted in terms of convex geometry, by associating with any big line bundle L a convex body, known as the Newton-Okounkov body of L. This body contains a lot of information about the positivity of L. The purpose of this talk is twofold. Firstly, given a projective hyper-Kähler manifold X, we describe the behaviour of the volume function on X. Secondly, we show how to associate with any big divisor D on X a convex polygon, behaving like the Newton-Okounkov body of a divisor on a smooth complex projective surface.

Yajnaseni Dutta: Twists of intermediate Jacobian fibrations of OG10

Associated to a general cubic 4 fold X, Laza, Saccà and Voisin considered the universal hyperplane family of cubic 3-folds over B, the projective space of dimension 5. They constructed a compact hyperK manifold together with a Lagrangian fibration that compactifies the family of relative intermediate Jacobian of the smooth fibres. Arinkin and Fedorov's result then ensures that the smooth locus of the Lagrangian fibration forms an abelian group scheme over the base B. I will report on an on-going joint work with Dominique Mattei and Evgeny Shinder where we identify this group scheme in terms of the family of cubic 3-fold. This enables us to describe the torsors of this group scheme in terms of a finite group of order 3 and the Brauer-type group associated to the cubic 4-fold. In short, this puts the intermediate Jacobian Lagrangian fibration of OG10 in the very same footing as the Beauville--Mukai system for curves on K3 surfaces, opening the door for further constructions of Lagrangian fibrations.

Daniele Faenzi: Ulrich and instanton bundles on cubic fourfolds

We will recall some of the main properties and problems about Ulrich bundles on hypersurfaces, notably about the existence and the classification of Ulrich bundles of rank r on a given smooth cubic fourfold X. I will sketch a construction carried out with Yeongrak Kim for arbitrary X and r=6 based on deformations of pairs of sheaves arising from twisted cubics. Then I will review work in progress with G. Casnati and F. Galluzzi about fourfolds X lying on Hasset divisors C_d for low values of r, notably for (d,r)=(18,3) and (20,4), in connection with instanton bundles and pfaffian representations of X.

Alexander Kuznetsov: One-nodal Fano threefolds

Our recent joint work with Evgeny Shinder showed that 1-nodal nonfactorial Fano threefolds can be used as connections between smooth Fano threefolds of different types. This motivated our joint work with Yuri Prokhorov aimed at classification of all 1-nodal Fano threefolds. 

In my talk I will explain how the classification looks like in the case of Picard rank 1, emphasizing the relation to Mori-Mukai classification of smooth Fano threefolds of higher Picard rank and to complete intersections.

Alice Garbagnati: Singular irreducible symplectic surfaces

To extend the the Beauville-Bogomolov decomposition theorem, one has to define the singular version of the irreducible holomorphic symplectic manifolds. These are the so called "singular irreducible symplectic varities", i. e. compact, connected complex varieties with canonical singularities that have a holomorphic symplectic form \sigma on the smooth locus, and for which every finite quasi-étale covering has the algebra of reflexive forms spanned by the reflexive pull-back of \sigma. 

We classify all singular irreducible symplectic surfaces, which turn out to be some contractions of ADE configurations of curves on K3 surfaces. We describe the families of the surfaces obtained; we observe that these surfaces are generically non projective, and we distinguish them in two classes: the ones which do not admit any quasi-étale covering and the others, which admit quasi-étale coverings, necessarily with a K3 surface.

We briefely discuss the Hilber scheme  of two points on these surfaces.   

This is a joint work with Matteo Penegni and Arvid Perego.

Annalisa Grossi: Automorphisms of OG10 towards Enriques manifolds

Automorphisms of HK manifolds have been studied for many different reasons: construct symplectic quotients or study fixed loci in order to find examples of irreducible symplectic varieties, define maps among different deformation families of HK manifolds, find new examples of Enriques manifolds, that are higher dimensional analogue of Enriques surfaces, and for which Pacienza and Sarti recently proved the Morrison-Kawamata cone conjecture. In the first part of the talk I will show a recent result about symplectic rigidity of HK manifolds of OG10 type. Then I will show how to construct examples of Enriques manifolds considering nonysmplectic automorphisms of a Laza—Saccà—Voisin manifold that are induced by a nonysmplectic automorphism of the underlying cubic fourfold. The talk is based on a joint work with L. Giovenzana, Onorati and Veniani and on a joint work in progress with Billi, F and L. Giovenzana.

Emanuele Macrì: Deformations of stability conditions

Bridgeland stability conditions have been introduced about 20 years ago, with motivations both from algebraic geometry, representation theory and physics. One of the fundamental problem is that we currently lack methods to construct and study such stability conditions in full generality.

In this talk I would present a new technique to construct stability conditions by deformations, based on joint works with Li, Perry, Stellari and Zhao. As application, we can construct stability conditions on very general abelian varieties and deformations of Hilbert schemes of points on K3 surfaces.

Laurent Manivel: Coble type hypersurfaces from representations of simple Lie groups

There exists a surprising relationship between certain representations of simple Lie groups and certain families of curves, which was observed and studied systematically by Bhargava and his school. I will explain how to push this study one step further, so as to involve some moduli spaces of vector bundles on the curves. This will allow to define a few interesting hypersurfaces of Coble type, in the sense that they are singular exactly on these moduli spaces, and completely characterized by this property.

(Joint work with V. Benedetti, M. Bolognesi and D. Faenzi).

Giovanni Mongardi: Regenerations and applications

Chen-Gounelas-Liedtke recently introduced a powerful regeneration technique, a process opposite to specialization, to prove existence results for rational curves on projective K3 surfaces. We show that, for projective irreducible holomorphic symplectic manifolds, an analogous regeneration principle holds and provides a very flexible tool to prove existence of uniruled divisors, significantly improving known results.This is joint work with G. Pacienza.

Kieran Gregory O'Grady: Moduli spaces of stable sheaves on general polarized HK varieties  of type K3^[2].

I will report on work in progress whose goal is to prove that, for suitable choices  of rank, c_1, c_2 on general polarized HK varieties  of type K3^[2], the corresponding moduli space of stable sheaves contains a connected component which is a general polarized HK variety  of type K3^[n] (for n=a^2+1, a=1,2,3,etc).

Claudio Onorati: Deformations of moduli spaces of sheaves on K3 surfaces

We study some aspects of the geometry of moduli spaces of sheaves on K3 surfaces. More precisely, if v=mw is a non-primitive Mukai vector, the smooth moduli space M_w is included in the singular moduli space M_v as its most singular locus. We will show how this inclusion relates the geometry of M_v to the geometry of M_w by proving that it induces an isomorphism between the monodromy groups of these two varieties. Time permitting, we will discuss some future directions. This is a joint work with Arvid Perego and Antonio Rapagnetta.

Miles Reid: GRDB and generalised unprojections

The Graded Ring Database [GRDB] lists around 60,000 candidates for Hilbert series of Q-Fano 3-folds. A few hundred of these cases lead to convincing explicit constructions, and a few dozen to impossibility proofs. There are also a number of general structural results (for example concerning extreme cases). By contrast, the unsolved problems are numerous, and represent a diversity of research problems for future generations. I will discuss some recent aspects, including a partly new technique of quasi Gorenstein unprojection that used Eisenbud's theory ofmatrix factorisation.

[GRDB] Gavin Brown, Al Kasprzyk and others. The Graded Ring Database, + Fano 3-folds.

Alessandra Sarti: On the cone conjecture for Enriques Manifolds 

Enriques manifolds are non simply connected manifolds whose universal cover is irreducible holomorphic symplectic, and as such they are natural  generalizations of Enriques surfaces. The goal of the talk is to prove the Morrison-Kawamata cone  conjecture for such manifolds when the degree of the cover is prime using the analogous result (established by Amerik-Verbitsky) for their universal cover. If time permits I will also show the cone conjecture for the known examples having non-prime degree. This is a joint work with Gianluca Pacienza.

Claire Voisin: Complete intersections of two quadrics and Lagrangian fibrations

We show that, on  the cotangent bundle of a  n-dimensional smooth complete intersection X  of two quadrics, there are n quadratic functions (which are sections of Sym^2T_X)  providing a Lagrangian fibration whose fibers can be described as quotients of Jacobians of hyperelliptic curves by a group of translations of order 2. For n=3, X is a moduli space of rank 2 bundles and the fibration is the Hitchin fibration.

This is joint work with Beauville, Etesse, Höring and Liu.

Junior talks

Junior talks are addressed only to young participants and are meant to introduce the main topics of senior talks, in a more informal environment.

Ludovica Buelli: A gentle crash course on IHS manifolds

In this talk I will give a very general introduction to the theory of irreducible holomorphic symplectic manifolds (IHS). After recalling the definition and the main properties of this kind of manifolds, we will study some features of their second integral cohomology group, which encodes a lot of information about the geometry of the manifold itself and plays a central role in classification results such as Torelli Theorems. Subsequently, I will provide some useful tools for the study of divisors on IHS manifolds, such as cones of Cartier R-divisors in the Néron-Severi space. Finally, we will see the four known families of deformation classes of this kind of manifolds, and some of the several ways to construct them. 

Hannah Dell: Stability and moduli on curves and quivers

We will explore two classification problems that can give rise to Fano and hyperkähler varieties. In the first half, I will introduce Bridgeland stability, and explain how this generalises slope stability of sheaves on curves. We will need to make the sacrifice of working with derived categories, but the payoff will be some good deformation properties. In the second half, I will introduce quivers and their representations. Through the example of Kronecker quivers, we will see how their moduli have many nice properties.

Hanfei Guo:  Introduction of modular sheaves on hyperkähler varieties

In this talk, we introduce modular sheaves on hyperkähler (HK) varieties, as proposed by O'Grady. Initially, analogous to sheaves on surfaces, we show that modular sheaves exhibit well-behaved variations of stabilities, such as wall-chamber decomposition and stability on a Lagrangian HK variety. We then focus on the specific example of the Hilbert scheme of two points on a K3 surface S. Commencing with a vector bundle F on S subject to certain cohomology constraints, we construct a modular sheaf  F[2]^+ on S^[2].

Lucas Li Bassi: IHS manifolds: automorphisms and generalizations

In this talk we will introduce the terminology related to the theory of automorphisms on IHS manifolds. We will include also some properties and examples. Moreover, we will look at some possible generalizations of the definition of IHS manifold to the singular case. These will include irreducible symplectic orbifolds and irreducible symplectic varieties.

Saverio Andrea Secci: Fano varieties and Lefschetz defect

The Lefschetz defect is a numerical invariant associated to a Fano variety and is defined from the Picard rank of its prime divisors. After a brief introduction on Fano manifolds and their classification, we will see how the Lefschetz defect fits into this framework: I will show classification results (and examples) of Fano manifolds with Lefschetz defect > 2 and with maximal Lefschetz defect. At the end I will mention a partial result for Fano manifolds with Lefschetz defect 2.

Federico Tufo: Degeneracy loci of morphisms between vector bundles

Degeneracy loci of morphisms between vector bundles can be seen as a generalization of zero loci of sections of vector bundles, and they arise naturally, for example, when we study Grassmannians. In this talk, we want to introduce the basics of degeneracy loci with some examples and we want to give some tools that help the computations of some invariants of zero loci and degeneracy loci.