Programme

The Workshop will start on Monday 11th June 2018 after lunch (about 2.30 pm) and it will end on Friday 15th June 2018 lunch time (about 1pm).

More Information about talks, posters, abstracts and timetable will come as soon as possible.

Poster Programme.pdf Schedule.pdf

Talks:

"Negative" uniruled loci on holomorphic symplectic manifolds

Abstract. This is a joint work with Misha Verbitsky. Let X be an irreducible holomorphic symplectic manifold and z a Beauville-Bogomolov negative (1,1)-class. Using the ergodicity of the monodromy action, we prove some deformation-invariance statements for the locus covered by rational curves of class z in X.

Nef cones of Hilbert schemes of points via Bridgeland stability

Abstract. Carrying out the Minimal Model Program for moduli spaces is a classical and extremely challenging problem. In this talk, we will deal with a particular moduli space, namely the Hilbert scheme of points on a surface with irregularity zero. After explaining the connection between the birational models of a variety and the combinatorics of its Nef cone, we will show how Bridgeland stability conditions are a powerful machinery to produce extremal rays in the Nef cone of the Hilbert scheme. Time permitting, we will give a complete description of the Nef cone in some examples of low Picard rank. This is joint work with J. Huizenga, Y. Lin, E.Riedl, B. Schmidt, M. Woolf and X. Zhao.

Moduli spaces of cubic threefolds and automorphisms of irreducible holomorphic symplectic manifolds

Abstract. In this talk, I will describe an isomorphism between the moduli space of smooth cubic threefolds, as described by Allcock, Carlson and Toledo, and the moduli space of fourfolds of K3^[2]-type with a special non-symplectic automorphism of order three; then, I will show some consequences of this isomorphism concerning degenerations of non-symplectic automorphisms. This is a joint work in progress with S. Boissière and A. Sarti.

Curves on Enriques surfaces

Abstract. In this talk I will discuss some moduli problems for curves on Enriques surfaces. This is related with the existence of so called Enriques-Fano 3-folds, i.e., 3-folds with general hyperplane section an Enriques surface. This is work in progress in collaboration with Th. Dedieu, C. Galati and A. Knutsen.

On a Mori fibre space of dimension 4 and its automorphisms group

Abstract. A Mori fibre space with an action of a group G is G-birationally superrigid if every Mori fibre space that is birationally equivalent to it by a map compatible with the action of G is isomorphic to it and the isomorphism is compatible with the action of G. The notion of birational rigidity was originally introduced by Iskovskikh and Manin and is used as a measure of the non-rationality of a variety. Moreover, the study of birationally G-superrigid rational varieties is connected to the study of maximal subgroups of the Cremona group. In this talk we will present an example of a birationally G-superrigid Mori fibre space of dimension 4 fibred onto the projective line. This is a work in progress joint with Jérémy Blanc.

Families of singular curves on Enriques surfaces

Abstract. We will talk about a work in progress with C. Ciliberto, A.L. Knutsen and T. Dedieu on families of singular curves on Enriques surfaces. In particular we will focus on the

existence problem and the regularity problem.

Group actions on quiver moduli spaces and applications

Abstract. In joint work with Florent Schaffhauser, we study two types of actions on King's moduli spaces of quiver representations over a field k, and we decompose their fixed loci using group cohomology in order to give modular interpretations of the components. The first type of action arises by considering finite groups of quiver automorphisms. The second is the absolute Galois group of a perfect field k acting on the points of this quiver moduli space valued in an algebraic closure of k; the fixed locus is the set of k-rational points, which we decompose using the Brauer group of k, and we describe the rational points as quiver representations over central division algebras over k. Over the field of complex numbers, we describe the symplectic and holomorphic geometry of these fixed loci in hyperkaehler quiver varieties using the language of branes.

3-fold Calabi-Yau pairs

Abstract. A Calabi-Yau (CY) pair (X, D) consists of a normal projective variety and a reduced anti-canonical integral divisor D on it. Such pairs come up in a variety of contexts; for example, cluster varieties are obtained by glueing CY pairs by crepant birational maps. Recent developments in mirror symmetry suggest that cluster varieties are “natural" mirror partners of Fano varieties with a toric degeneration, and that understanding the birational geometry of CY pairs is an important step in the study of mirror symmetry and moduli of Fano varieties. In this talk, I will discuss the birational geometry of CY pairs and some 3-fold examples.

Cubic fourfolds and non-commutative K3 surfaces

Abstract. The derived category of coherent sheaves on a smooth cubic fourfold has a subcategory, recently studied by Kuznetsov, Addington-Thomas and Huybrechts among others, that can be thought as the derived category of a non-commutative K3 surface. In this talk, I will present this category and joint work in progress together with Bayer, Macrì, Nuer, Perry and Stellari about the construction of Bridgeland stability conditions on this category and some applications, like the construction of hyperkaehler manifolds as moduli spaces of objects in it.

Generalising abundance

Abstract. I will discuss how to generalise known nonvanishing and semiampleness conjectures from various contexts. This is joint work with Thomas Peternell.

Nikulin surfaces and moduli of Prym curves

Abstract. The relevance of K3 surface in the study of the moduli space of curves is well-established. Nikulin surfaces, that is, K3 surfaces endowed with a nontrivial double cover branched along eight disjoint rational curves, play a similar role at the level of the moduli space of Prym curves. I will report on a work in this direction joint with Knutsen and Verra. In particular, I will prove that a general Nikulin section of fixed genus lies exactly on one Nikulin surface with only a few exceptions occurring in low genus.

Deformations of algebraic schemes via Reedy cofibrant resolutions.

Abstract. In 1976 V. Palamodov (Deformations of complex spaces) introduced the tangent complex L of a complex space X as the differential graded Lie algebra of derivations of a resolvent, and proved that the first and second cohomology group of L give a tangent-obstruction pair for the deformation theory of X. The analogous construction can be easily done in the algebraic setting, for every separated scheme over a field of characteristic 0. In a joint work with Francesco Meazzini we prove that, up to a slight and harmless additional condition in the definition of the resolvent, the tangent complex controls the deformations of a separated scheme via the general principle of Maurer-Cartan equation modulus gauge equivalence

Birational geometry of moduli spaces of complete collineations

Abstract. The moduli space of complete collineations is roughly speaking a compactification of the space of linear maps between two fixed vector spaces, in which the boundary divisor is simple normal crossing. The space of complete collineations is a spherical wonderful variety. Exploiting its spherical nature we will investigate its birational geometry. More precisely we will compute the effective and nef cones, the Mori and moving cones of curves and the generators of the Cox ring. Finally, we will determine the Mori chamber decomposition of the space of complete collineations of the 3-dimensional projective space, and as a consequence we will recover a description of the Mori chamber decomposition of the space of complete quadric surfaces due to C. L. Huerta.

The tropicalization of the moduli space of stable spin curves.

Abstract. The theory of linear series on tropical curves, since its introduction by Baker and Norine about 10 years ago, has seen spectacular developments in recent years. In fact, the combinatorial systematic treatment of degenerations of classical linear series that the theory has led to the proof of many important results on algebraic curves. On the other hand, the introduction and study of a number of tropical moduli spaces of curves along with its realization as skeletons of their classical (compactified) counterparts allows for a deeper understanding of combinatorial aspects of moduli spaces and in particular of their compactifications. In this talk, which is based on ongoing joint work with Lucia Caporaso and Marco Pacini, I will explore this principle for certain moduli spaces of bundles on curves, as Cornalba's moduli space of spin structures and their compactified/tropical versions.

  • Arvid Perego (Università di Genova)

Moduli spaces of sheaves on K3 surfaces and irreducible symplectic varieties

Abstract: Irreducible symplectic manifolds are one of the three building blocks of compact K\"ahler manifolds with numerically trivial canonical bundle (together with abelian varieties and Calabi-Yau manifolds), thanks to the Beauville-Bogomolov decomposition theorem. A recent result of A. H\"oring and T. Peternell has completed the extension of this decomposition theorem to singular projective varieties: irreducible symplectic varieties are the singular analogue of irreducible symplectic manifolds, and they are one of the building blocks of normal, projective varieties having canonical singularities and numerically trivial canonical bundle. In a recent joint work with A. Rapagnetta we prove that all moduli spaces of semistable sheaves over projective K3 surfaces (with respect to a generic polarization) are irreducible symplectic varieties, with the only exception of those isomorphic to symmetric products of K3 surfaces, and compute their Beauville form and Fujiki constant. Similar results are shown to hold for the Albanese fiber of moduli spaces of sheaves over Abelian surfaces.

On the minimal model program of the moduli space of curves

Abstract. In this talk I will report on a work in progress with G. Codogni and F. Viviani in which we investigate the first possible steps of the minimal model program for the moduli space of stable pointed curves M. In particular, we show that such steps have a modular interpretation and we relate them to the so Hassett-Keel program, which predicts that the log canonical models of M (with natural boundaries) have also a modular interpretation.

On the stable cohomology for toroidal compactifications of A_g and its structure

Abstract. Principally polarized abelian varieties of dimension g are basic objects in algebraic geometry, but the cohomology of their moduli space A_g is largely unknown. However, by a classical result of Borel, the cohomology of A_g in degree k<g is is freely generated by the odd Chern classes of the Hodge bundle. Work of Charney and Lee provides an analogous result for the stable cohomology of the minimal compactification of A_g, the Satake compactification. For most geometric applications, it is more natural to consider toroidal compactifications of A_g instead. In this case, we have some stability results for the perfect cone compactification and the matroidal partial compactification. The features of stable cohomology classes are very different from those in the classical examples, although they have some combinatorial aspects in common with them.(This is joint work with Sam Grushevsky and Klaus Hulek).

Posters:

  • Mauro Fortuna (Leibniz Universitaet Hannover)

Cohomology of the moduli space of non-hyperelliptic genus four curves

Abstract. In this poster, I will present the intersection Betti numbers of the moduli space of non-hyperelliptic Petri-general genus four curves. This space has a canonical compactification as GIT quotient, which was proven to be the final step in the Hassett-Keel log MMP for stable genus four curves. The strategy of the cohomological computation relies on a general method developed by F. Kirwan to calculate the cohomology of GIT quotients of projective varieties, based on stratifications, a partial desingularisation and the decomposition theorem.

  • Inder Kaur (IMPA)

A Torelli-type theorem for moduli space of stable sheaves over a singular curve

Abstract. The classical Torelli theorem states that a complex, smooth, projective curve is uniquely determined by its principally polarized Jacobian variety. Mumford and Newstead further prove that the Jacobian of the curve is uniquely determined by the (second) intermediate Jacobian of the moduli space of stable rank 2 vector bundles with determinant of odd degree over the curve. However their techniques fail when the curve is singular. We use degeneration techniques from Hodge theory to prove an analogous result when the underlying curve is irreducible nodal with a single node. This is joint work with A. Dan and S. Basu.

  • Federico Lo Bianco (Institut de Mathématiques of Marseille)

The dynamics of an automorphism preserving a fibration

Abstract. Consider an automorphism (or more generally a birational transformation) f of a projective variety X, and suppose that f permutes the fibres of a non-trivial fibration p : X —> B. In this case, when studying the dynamical properties of f one can can first focus on the analysis of the induced automorphism of the base B; the easiest situation is when such automorphism has finite order (i.e. some iterate of f fixes each fibre of p). I will present a criterion for the finiteness of the action on the base, which can be applied for example to the case of a birational transformation of a hyperkaehler manifold preserving a Lagrangian fibration. The proof of the result is inspired to the original demonstration of Tits to the Tits alternative for finitely generated linear groups.

  • Diletta Martinelli (University of Edinburgh)

Birational geometry and stability conditions

Abstract. Recent advances have made possible to apply the machinery of wall-crossing and stability conditions on derived categories introduced by Bridgeland in 2007 in order to understand the birational geometry of moduli space of shaves M of a K3 surface X. In particular, Bayer and Macri’ classify the possible type of birational contractions of M in terms of wall-crossing in the space of stability conditions Stab(X). In will describe how to refine this analysis and describe the geometry of the exceptional locus of the contraction in terms of wall-crossing.

  • Luigi Lunardon (Imperial College)

The Motivic Monodromy Conjecture for K3 surfaces admitting a triple-point-free model

Abstract. Degenerations of Calabi-Yau varieties carry a natural monodromy action in cohomology. If the variety satisfies the monodromy property, a relation between this action and the degeneration is given by means of the motivic zeta function. Namely, each pole of this function determines an eigenvalue of the monodromy action. In this poster, we focus on K3 surfaces allowing a triple-point free model, a class of surfaces for which the monodromy property holds. Furthermore, these surfaces are the first examples for which the motivic zeta function may have more than a single pole. This is a joint work with L.H. Halle, A. Jaspers, and J. Nicaise.

  • Andrea Petracci (University of Nottingham)

Mirror Symmetry and deformations of Fano toric varieties

Abstract. Fano varieties are an important class of algebraic varieties and their study began long ago. Smooth Fano varieties of dimension 1, 2 and 3 were classified by del Pezzo, Fano, Iskovskikh, Mori and Mukai. It is known that in every dimension the number of deformation families of smooth Fano varieties is finite, but the classification of Fano varieties of dimension at least 4 is completely open. Ideas coming from Mirror Symmetry constitute a new way to tackle this problem. In this context it is crucial to study toric degenerations of smooth Fano varieties, or conversely study deformations/smoothings of (possibly singular) Fano toric varieties. Altmann has deeply studied homogeneous deformations of affine toric varieties, by noting that Minkowski decompositions of polyhedra induce deformations. In this poster, which is based an ongoing project with Alessio Corti and Paul Hacking, I will present an approach to construct non-homogeneous deformations of Fano toric varieties of dimension 3, with Gorenstein singularities. This approach lies in the context of the Gross-Siebert programme.

  • Luca Schaffer (University of Massachusetts Amherst)

Compactifications of moduli spaces of points and lines

Abstract. Projective duality identifies the moduli space B_n parametrizing configurations of n points on P^2 in general linear position with the moduli space X(3,n) parametrizing configurations of n lines on P^2 in general linear position. When considering degenerations of such objects, it is interesting to compare the resulting compactifications. In this work we consider Gerritzen-Piwek's compactification \barB_n and Kapranov's Chow quotient compactification \barX(3,n) and we show they are isomorphic. The ultimate goal is to construct and study an alternative compactification of B_n which parametrizes all possible marked degenerate central fibers of the Mustafin joins of stable lattices associated to one-parameter degenerations of n points on P^2, correcting some of the results in Gerritzen-Piwek's paper. This is joint work in progress with Jenia Tevelev.