Schedule

The workshop will feature 3 mini-courses held by senior speakers, 12 talks held by junior speakers (selected among participants) and one poster session. There will be a social dinner.

Location of the talks: Collegezaal (lecture room) 1, Gorlaeus Lecture Hall (except Monday and Tuesday morning in lecture room 3)

Location of the poster session: Atrium DM1, on the 1st floor of the Faculty of Science building (situated just behind the Lecture Hall building)

Location and menu of the dinner buffet can be found here.

Schedule:

Senior Speakers:

• Luca Schaffler - Roma Tre University

  • Title: Explicit geometry of moduli spaces of stable pairs

  • Abstract: Moduli spaces of smooth algebraic varieties are often non-compact: in families, varieties can degenerate, acquiring singularities or breaking into components. A proper (in fact projective) moduli space can be obtained by enlarging the moduli problem to stable pairs (X,D) allowing semi-log canonical singularities and requiring KX+ D to be ample. This is the Kollár–Shepherd-Barron–Alexeev (KSBA) compactification, the higher-dimensional analogue of the Deligne–Mumford–Knudsen compactification of the moduli space of pointed curves. In this series of lectures we introduce the basics of the KSBA framework, with a focus on surfaces. A guiding theme will be the explicit description of boundary strata in KSBA moduli spaces by computing stable limits of degenerating families, especially in the context of (weighted) line arrangements in P2 and stable surfaces (the case D=0).

• Steven Groen & Valentijn Karemaker - University of Amsterdam 

  • Title: Dieudonné theory

  • Abstract: The main result of Dieudonné theory provides an anti-equivalence between the category of p-divisible groups over a perfect field k of characteristic p, and the category of Dieudonné modules that are free of finite rank over the Witt vectors W = W(k). This implies that Dieudonné modules play an important role in understanding varieties, in particular abelian varieties, in characteristic p through their p-divisible groups. In addition, Oda proved that the Dieudonné module of the p-torsion group scheme of an abelian variety can be described in terms of its de Rham cohomology.

In this mini-course we will introduce Dieudonné modules and p-divisible groups, discuss their rich theories, explain the anti-equivalence between them, and explore applications to abelian varieties in characteristic p.

Junior Speakers:

• Sophie Friesen - Leibniz University Hannover

  • Title: Finite groups acting on K3 surfaces

  • Abstract: Finite subgroups of automorphism groups of K3 surfaces have been an active area of study since foundational work of Nikulin in 1979. We give an overview of classical results over the complex numbers and report on recent progress in positive characteristic. 

For this, we will discuss the role of symplectic and non-symplectic automorphisms and highlight the differences between the complex and positive characteristic setting. We will introduce the notion of supersingular K3 surfaces (more precisely superspecial K3 surfaces) and their importance in the classification of finite symplectic actions in positive characteristic. Lastly, we show recent results on non-symplectic group actions with maximal symplectic subgroups on superspecial K3 surfaces.

• Gianluca Grassi - University of Ferrara

  • Title: Cremona quadratic transformations of the 4-dimensional projective space

  • Abstract: Cremona transformations of projective spaces P^n are extremely relevant in birational geometry. While Cremona transformations have been extensively studied for P^2, in every degree, going in higher dimensions gives a proper challenge, especially in the realm of classification. Quadratic Cremona transformations are also the easiest example and, for P^3, they have been classified back in 2001 by Pan, Ronga and Vust. 

In this talk we'll present the approach used to try a generalization in higher dimensions, focusing on the case study of P^4, showing off interesting properties of all components of the set of quadratic birational maps of P^4, understanding how, given a specific multidegree, we can build those maps. 

• Daniel Holmes - Institute of Science and Technology Austria

  • Title: From Gromov-Witten theory to GKM theory and back

  • Abstract: At the intersection of geometry, combinatorics, and algebra lies a fruitful two-way interaction of Gromov-Witten theory and GKM theory that is established by equivariant localization. In one direction, GKM theory provides a setting where Gromov-Witten invariants are explicitly computable (which we have implemented in joint work with Giosuè Muratore). In the other direction, the axiomatic behavior of Gromov-Witten invariants is a powerful constraint yielding structural results on algebraic and, more generally, Hamiltonian GKM spaces. I will present recent results in both directions.

• Lucas Lagarde - Sorbonne Paris Nord University

  • Title: On the Grunwald Problem

  • Abstract: The Grunwald problem for a finite group G over a number field k asks whether a compatible family of local Galois extensions can be realised as the completions of a single global extension with Galois group G. I will give an overview of this problem and explain how it can be reframed as a question about weak approximation for rational points on an associated k-variety X. This brings the problem into the realm of the Brauer-Manin obstruction, which provides a necessary and a priori algorithmically checkable condition for weak approximation controlled by Br(X), and conjectured to be the only one. The relevant part of Br(X) is known to be finite, but computing it explicitly is in general a subtle task. I will sketch some ideas to tackle this issue, and explain how this yields the decidability of Grunwald's problem for large families of groups G. If time permits, I will discuss recent connections with asymptotics for counting number fields with bounded discriminant.

• Isaac Martin - The University of Texas at Austin

  • Title: Logarithmic GW theory and localization

  • Abstract: We'll start with a brief account of traditional Gromov-Witten invariants and discuss one of the main computational tools in enumerative geometry: Atiyah-Bott Localization. I'll then motivate the generalization to Logarithmic GW invariants and highlight why on earth you'd ever think to define these in the first place. In whatever time remains, I'll explain why localization theorems fail in the logarithmic setting and showcase some preliminary results demonstrating how their failure can be rectified. 

• Enzo Pasquereau - Nantes University

  • Title: Combinatorial patchworking in higher codimension

  • Abstract: Combinatorial patchworking is a powerful tool used for constructing real algebraic hypersurfaces with controlled topology. It has led to signifcant advances including on the problem of the existence of maximal variety (the ones with the higher Betti numbers with respect to the Smith-Thom inequality).

I will present the combinatorial patchworking method originally due to Viro and then discuss generalizations to higher codimension. In particular, we give explicit patchworking rules similar to the formulation for hypersurfaces that allow to construct real complete intersections. As an application, we construct a families of pair of maximal surface and maximal curve in the real projective 3-space.

• Anna-Maria Raukh - University of Stavanger

  • Title: The topology of Tom and Jerry log singularities

  • Abstract: A conjecture inspired by mirror symmetry predicts that every Q-Gorenstein Fano threefold can be obtained by smoothing a singular space constructed from a reflexive polytope and a central subdivision. Such a singular space is a Gorenstein toroidal crossing and can be equipped with a compatible log structure. If the log structure is smooth and the underlying toroidal crossing space admits a smoothing, then the general fiber is homeomorphic to the relative Kato–Nakayama space. However, in this setting, log structures can be singular, and it can be difficult to construct the smoothing explicitly.

In this talk, we will look at two examples of singular log structures, known as Tom and Jerry. For each of them, we can construct a log-crepant resolution. An interesting question one can entertain oneself with is the topology of a nearby fiber. We compute the Euler characteristics of the relative Kato-Nakayama space of Tom and Jerry projective log-schemes and play with some curves in weighted projective spaces along the way.

• Davide Ricci - UVSQ Versailles University

  • Title: The Mori cone of certain moduli spaces of weighted pointed curves 

  • Abstract: The talk will introduce the F-conjecture, which states that the Mori cone of the moduli space of genus g stable curves ‾Mg,n is generated by 1-dimensional strata. We will see that the question can be formulated for the moduli spaces of weighted pointed curves (Hassett spaces) and has a positive answer when the universal family is a P1-bundle. In particular, we discuss how these particular Hassett spaces are linked to certain GIT quotients (P1)n // PGL2 and to the birational contractions of Bln-1Pn-3. 

• Francesca Rizzo - Paris Cité University

  • Title: On Hodge correspondences between Fano and Hyper-Kähler manifolds.

  • Abstract: Hyper-Kähler manifolds are among the fundamental building blocks in the Beauville–Bogomolov decomposition of compact Kähler manifolds with trivial first Chern class. Interestingly, all known examples of locally complete families of hyper-Kähler manifolds are related to Fano geometry. The first such example was studied by Beauville and Donagi, who showed that the Hilbert scheme parametrizing lines on a smooth cubic fourfold is a hyper-Kähler fourfold. Moreover, the associated incidence correspondence induces a Hodge isometry between the primitive middle cohomology of the cubic fourfold and the primitive second integral cohomology of its variety of lines.

We will discuss this example and explain how a similar phenomenon occurs for the other four known examples of projective families of hyper-Kähler manifolds. In particular, we will present some recent results on EPW cubes (hyper-Kähler sixfolds) and Gushel–Mukai sixfolds (Fano varieties).

• Sara Stephens - Cornell University

  • Title: Strictly semistable quasimaps on the wall

  • Abstract: Moduli spaces of ϵ-stable quasimaps exhibit a wall-and-chamber structure, interpolating between the moduli of stable quasimaps and Kontsevich’s moduli spaces of stable maps. In the case where the target is projective space, we develop an intrinsic framework for the wall-crossing phenomenon by constructing an algebraic stack where strictly semistable objects appear at a wall in the space of ϵ-stability conditions. We show this algebraic stack admits a proper good moduli space and analyze the variation of Θ-stratification on this stack. This yields a new approach to developing a K-theoretic wall-crossing formula for ϵ-stable quasimap invariants via non-abelian localization.

• Stijn Velstra - Sorbonne Paris - Paul Sabatier Toulouse

  • Title: A spin-off of spin volumes

  • Abstract: The Masur-Veech volumes of the strata of the Hogde bundle over the moduli space of curves can be computed intersection theoreticaly by Chen-Möller-Sauvaget-Zagier, starting from the volume of the minimal stratum, computed as such by Sauvaget.

Strata of even orders admit spin-parity decompositions, refining the story above. The missing intersection theoretic spin minimal stratum computation was recently solved in joint work with Andrei Bud and Georgios Politopoulos, resting on the moduli geometry of spin sections in Holmes-Politopoulos-Sauvaget and cancellations in spin Chiodo classes as in Giacchetto-Kramer-Lewański.

Chiodo classes appear in spin enumerative contexts in topological recursion and integrable hierarchies about which I shall say nothing. We will however observe a closed formula for certain Chiodo integrals by applying our method in reverse to the classical case.

• Alice Villa - University of Bologna

  • Title: Deformations of K-stable Fano varieties

  • Abstract: I present a toric construction of a Fano threefold and describe its Q-Gorenstein deformation space. By choosing a suitable point in the miniversal base, I obtain a K-stable Fano variety with obstructed Q-Gorenstein deformations. The talk will briefly outline the construction and the role of the torus action.