Daniele Agostini (Max Planck Institute in Leipzig)

On the irrationality of moduli spaces of K3 and abelian surfaces

Abstract. In this talk, we consider quantitative measures of irrationality for moduli spaces of polarized K3 and abelian surfaces. We show that the degree of irrationality is bounded polynomially in terms of the degree of the polarization, so that these spaces become more irrational, but not too fast. The key insight is that the irrationality is bounded by the coefficients of a certain modular form.

This is joint work with Ignacio Barros and Kuan-Wen Lai.

Paolo Cascini (Imperial College London)

On the Minimal Model Program for algebraically integrable foliations

Abstract. Every fibration, or more in general, every dominant rational map between normal varieties, defines a natural foliation, which is called algebraically integrable. The canonical sheaf of such a foliation behaves, in many aspects, as the canonical sheaf of a normal variety. I will describe some recent results in this direction, such as a cone theorem, and some applications on the canonical bundle formula. In particular, this provides a proof of a conjecture by Shokurov.

This is joint work with Ambro, Shokurov and Spicer.

Sandra Di Rocco (KTH Stockholm)

Polar Geometry of Algebraic Data

Abstract. Classical projective algebraic geometry, like polar geometry, has proven very effective in the study of data arising from algebraic models.

I will try to introduce the (numerical and theoretical) problem of sampling algebraic varieties and illustrate some recent results based on joint work with D. Eklund, M. Weinstein and O. Gäfvert.

Enrica Floris (Université de Poitiers)

Connected algebraic groups acting on Fano fibrations over IP¹

Abstract. Let G be a connected algebraic group and X a variety endowed with a regular action of G and a Mori fibre space X/IP¹ whose fibre is a Fano variety of Picard rank at least 2.

In this talk I will explain why there is a proper horizontal subvariety of X which is invariant under the action of G, alongside with some applications of this result to the classification of connected algebraic subgroups of the Cremona group in dimension 4.

This is a joint work with Jérémy Blanc.

Paola Frediani (Università degli Studi di Pavia)

A canonical Hodge theoretic projective structure on compact Riemann surfaces

Abstract. In this talk we will show the existence of a canonical projective structure on every compact Riemann surface, coming from Hodge theory. We will show that it differs from the canonical projective structure produced by the uniformisation theorem. In fact the (0,1)-component of the differential of the corresponding sections of the moduli space of projective structures over the moduli space of curves are different. The one corresponding to the projective structure coming from uniformisation was computed by Zograf and Takhtadzhyan as the Weil-Petersson Kaehler form on the moduli space of curves. Ours is the pullback via the Torelli map of a nonzero constant scalar multiple of the Siegel form on the moduli space of principally polarised abelian varieties.

These are results obtained in collaboration with I. Biswas, E. Colombo and G.P. Pirola.

Margherita Lelli Chiesa (Università degli Studi Roma Tre)

Irreducibility of Severi varieties on K3 surfaces

Abstract. Let (S,L) be a general K3 surface of genus g. I will prove that the Severi variety parametrizing curves in |L| of geometric genus h is connected for h≥1 and irreducible for h≥4.

This is joint work with Andrea Bruno.

Emanuele Macrì (Université Paris-Saclay)

Surfaces on Cubic Fourfolds via Stability Conditions

Abstract. I will report on joint work in progress with Arend Bayer, Aaron Bertram, and Alex Perry on a possible characterization of Hassett divisors on the moduli space of cubic fourfolds by the property of containing special surfaces. I will sketch the construction of such special surfaces for infinitely many divisors and the relation with the work of Russo and Staglianò on rationality of such cubics in low discriminant.

Alex Massarenti (Università degli Studi di Ferrara)

Calabi-Yau structures on conic bundles

Abstract. In 1992 F. Campana, and J. Kollár, Y. Miyaoka, and S. Mori proved that smooth Fano varieties are rationally connected. Thanks to the work of V. G. Sarkisov (1980) and A. Corti (1995) we know that there are rationally connected varieties that are not birational to a Fano variety.

Furthermore, answering to a question of P. Cascini and Y. Gongyo (2013), in 2018 I. Krylov extended this result to varieties of Fano type.

Further weakening the positivity hypotheses on the anti-canonical divisor in 2017 J. Kollár addressed the problem of whether or not all rationally connected varieties are birational to a variety with pseudo-effective anti-canonical divisor, and proved that a general conic bundle over the projective plane with discriminant of degree at least 19 is not birational to a variety with pseudo-effective anti-canonical divisor.

Let \pi: Z -> IP^n-1 be a general minimal n-fold conic bundle branched over a hypersurface B_Z in IP^n-1 of degree d. We will prove that if d≥ 4n+1 then -K_Z is not pseudo-effective. Furthermore, if n = 3, d≥13 and \pi: Z -> IP² is a general minimal 3-fold conic bundle then Z is not birational to a normal projective variety Y such that -K_Y is pseudo-effective.

This is joint work with Massimiliano Mella.

Margarida Melo (Università degli Studi Roma Tre)

On the top weight cohomology of the moduli space of abelian varieties

Abstract. In the last few years, tropical methods have been applied quite successfully in understanding several aspects of the geometry of classical algebro-geometric moduli spaces. In particular, in several situations the combinatorics behind compactifications of moduli spaces have been given a tropical modular interpretation. Consequently, one can study different properties of these (compactified) spaces by studying their tropical counterparts.

In this talk, which is based in joint work with Madeleine Brandt, Juliette Bruce, Melody Chan, Gwyneth Moreland and Corey Wolfe, I will illustrate this phenomena for the moduli space A_g of abelian varieties of dimension g. In particular, I will show how to apply the tropical understanding of the classical toroidal compactifications of A_g to compute, for small values of g, the top weight cohomology of A_g.

The techniques we use follow the breakthrough results and techniques recently developed by Chan-Galatius-Payne in understanding the topology of the moduli space of curves via tropical geometry.

John Christian Ottem (University of Oslo)

Specialization of birational types and irrationality of complete intersections

Abstract. I will explain how tropical degenerations and birational specialization techniques can be used in rationality problems. In particular, I will apply these techniques to study quartic fivefolds and complete intersections of a quadric and a cubic in IP⁶.

This is joint work with Johannes Nicaise.

Gianluca Pacienza (Université de Lorraine)

Deformations of rational curves on primitive symplectic varieties and applications

Abstract. In the talk I will start by explaining why it is worth studying «singular» holomorphic symplectic varieties and rational curves on them. Then I will talk about a joint work with Ch. Lehn and G. Mongardi in which we study the deformation theory of rational curves on a (possibly singular) primitive symplectic variety and show that if the rational curves cover a divisor, then, as in the smooth case, they deform along their Hodge locus. As applications of our technique, I will present the extension of Markman's deformation invariance of prime exceptional divisors to this singular framework and provide existence results for uniruled ample divisors on primitive symplectic varieties which are locally trivial deformation of any moduli space of sheaves on a projective K3 surface or fibers of the Albanese map of those on an abelian surface.

Elisa Postinghel (Università degli Studi di Trento)

Weyl cycles on blow-ups of projective 4-spaces

Abstract. I will introduce the definition of Weyl cycles on the blow-up X of IPⁿ 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 classify these objects and describe the geometry. I will also explain how this relates to certain polynomial interpolation problems.

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

Joaquim Roé (Universitat Autonòma de Barcelona)

Singularities infinitely near to the branch locus

Abstract. Given a ramified morphism between smooth surfaces, we are interested in the behaviour of singularities of curves under pullback and pushforward. In this context, we rely on clusters of infinitely near points introduced by Enriques, and properties of valuations on 2-dimensional regular local rings elucidated by Zariski. Building on groundwork due to Casas-Alvero (2007) we give an explicit algorithm to determine the Enriques diagram of a pullback curve in terms of the original curve, and we will discuss consequences and more general cases.

Alessandra Sarti (Université de Poitiers)

Complex Reflection Groups and K3 surfaces

Abstract. In this talk I will classify all K3 surfaces that one can obtain as quotient of surfaces by certain subgroups of finite complex reflection groups of rank 4. Most of the K3 surfaces are singular with A-D-E singularities. The proof of these facts avoids as much as possible a case-by-case analysis and involves the theory of finite complex reflection groups. Moreover we show that each family contains K3 surfaces with maximal Picard number which is 20. This construction generalizes a previous result by W. Barth and myself.

This is a joint work with C. Bonnafé.

Stefan Schreieder (Universität Hannover)

The diagonal of quartic fivefolds

Abstract. We show that a very general quartic hypersurface in IP⁶ over a field of characteristic different from 2 does not admit a decomposition of the diagonal, hence is not retract rational. This generalizes a result of Nicaise–Ottem, who showed stable irrationality over fields of characteristic zero. To prove our result, we introduce a new cycle-theoretic obstruction that may be seen as an analogue of the motivic obstruction for rationality in characteristic zero, introduced by Nicaise–Shinder and Kontsevich–Tschinkel.

This is joint work with Nebojsa Pavic.

Federico Bongiorno (Imperial College London)

Quotients by Algebraic Foliations

Abstract. Given a variety defined over a field of characteristic zero and a foliation of corank less than or equal to two, we show the existence of a categorical quotient, defined on semistable points, through which every invariant morphism factors uniquely.

Stefano Canino (Politecnico di Torino)

Analytical classification of triple points

Abstract. Is there a classification up to analytical equivalence of the reduced algebraic plane curves singularities? A partial answer is given by the so-called ADE singularities, that classify completely the double points and partially the triple points. In order to analyze the latter, we use the Newton-Puiseux algorithm, aiming to find “representative” curves for triple points.

Hilbert Functions of subvarieties and c.i. subvarieties of Veronese surfaces

Abstract. If X is a subvariety of a Veronese surface, what is its Hilbert function? We give a theorem that characterizes the Hilbert functions of reduced points on Veronese surfaces. Finally, we use this to show that, except for the quadratic Veronese surface, the only complete intersections on Veronese surfaces are the trivial ones.

Francesco Denisi (Université de Lorraine)

Boucksom-Zariski chambers on irreducible holomorphic symplectic manifolds

Abstract. We provide for the big cone of a projective irreducible holomorphic symplectic manifold a decomposition into chambers, called Boucksom-Zariski chambers, in each of which the support of the negative part of the divisorial Zariski decomposition is constant. We show how the obtained decomposition allows to describe the volume function and we determine when the Boucksom-Zariski chambers are “numerically determined”.

Luca Schaffler (KTH Stockholm)

Families of pointed toric varieties and degenerations

Abstract. The Losev-Manin moduli space parametrizes pointed chains of projective lines. In this poster we study a possible generalization to families of pointed degenerate toric varieties. Geometric properties of these families, such as flatness and reducedness of the fibers, are explored via a combinatorial characterization. We show that such families are described by a specific type of polytope fibrations which generalize the twisted Cayley sums, introduced by Casagrande and Di Rocco to characterize elementary extremal contractions of fiber type associated to projective $\mathbb{Q}$-factorial toric varieties with positive dual defect. The case of a one dimensional simplex can be viewed as an alternative construction of the permutohedra. This is joint work with Sandra Di Rocco.