impanga 2023/24

We discuss recent progress in the program of classification of singular del Pezzo surfaces of Picard rank one over an algebraically closed field of arbitrary characteristic. We introduce a new invariant, which guides our uniform approach, the height. By definition, it is the minimal intersection number of a fiber of any P¹-fibration of the minimal resolution of singularities with the exceptional divisor. The geometry of del Pezzo surfaces gets more constrained as the height grows. Moreover, it turns out that the height is at most 4, with minor exceptions in case the characteristic of the field is positive and small, which allows for a complete classification. This is a joint project with Tomasz Pełka.

Let X be an n-dimensional smooth projective Fano hypersurface, where n is at least 3. Suppose X contains Z, a k-dimensional smooth projective hypersurface in P^k+1, where k is at least 1. We give a constructive proof that Y - the blowup of X along Z - is a Mori dream space. In particular, we describe its Mori chamber decomposition and the associated birational models of Y. This is joint work with L. Campo and E. Paemurru.

A smooth projective variety X is said to be Calabi-Yau if its canonical bundle is trivial. I will discuss joint work with Lukas Brantner, in which we use derived algebraic geometry to study deformations of Calabi-Yau varieties in characteristic p.
We prove a positive characteristic analogue of the Bogomolov-Tian-Todorov theorem (which states that deformations of Calabi-Yau varieties in characteristic 0 are unobstructed), and show that 'ordinary' Calabi-Yau varieties admit canonical lifts to characteristic zero (generalising earlier results of Serre-Tate, Deligne and Nygaard, and Achinger-Zdanowicz).

Double EPW sextics are one of the few complete families of hyper-Kähler fourfolds for which we have an explicit geometric model. Their six dimensional analogue, double EPW cubes, are the only such family in the case of hyper-Kähler sixfolds.
In the talk we will review the results on symmetries of double EPW sextics, then we will present analogous results from the joint work in progress with S. Billi and S. Mueller for EPW cubes. Later we will show the classification of the most symmetric manifolds in both cases and construct explicit examples.

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.

Moreover, 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.

Let X be a nonsingular complex projective toric variety of dimension n, equipped with an action of the n-dimensional complex torus T. A coherent torsion-free sheaf E on X is said to be T-equivariant if it admits a lift of the T-action on X, which is linear on the stalks of E. It is known that the tangent bundle T_X of a toric variety X is T-equivariant. In this talk, we will study the problem of μ-stability of T_X for a toric variety X using combinatorial techniques.
This is a joint work with Idranil Biswas, Arijit Dey and Mainak Poddar.

The Behrend function associates every point of a scheme X an integer which is supposed to measure the "singularity" at x. It is usually impossible to compute, yet it appears in Donaldson-Thomas theory of counting on threefolds: Behrend shows that the virtual Euler characteristic is the usual Euler characteristic weighted by the Behrend function.
It is also useful for the theory of moduli spaces themselves: if the Behrend function was constant on Hilb_d(X) for X a threefold, then Hilb_d(X) is generically reduced. Thus it was conjectured that it is constant.
In the first half of the talk I will explain a bit of the beautiful background on DT theory and motivation for the Behrend function (no special prerequisites necessary, not for experts). In the second part I will explain that the function is *not* constant (still no prerequisites).
This is joint work with M. Kool and R. Schmiermann.

It was proven by Manivel, Michalek, Monin, Seynnaeve, and Vodicka that the maximum likelihood degree for linear concentration models is a polynomial function. We calculate initial values of this polynomial and provide some conjectures on the negative values.

Some important conjectures involve the tool of quantum cohomology. Interest in this appeared at first on the side of the mathematical version of mirror symmetry. It has become apparent that it is also closely associated with other problems such as in number theory and algebraic geometry. I will discuss some aspects of these problems.

A classical work of Akyildiz, Brion, Carrell, Liebermann, Sommese from 80s shows how to see cohomology ring as a ring of functions on a thick point. 30 years later Brion and Carrell showed how to find the spectrum of the torus-equivariant cohomology as a geometrically defined scheme, provided that the Borel of SL₂ acts with a single fixed point of the regular unipotent. In a joint work with Tamas Hausel we show how to see equivariant cohomology as a ring of functions in more general setups. I would like to present those results together with ongoing work concerning K-theory and other cohomology theories, as well as singular spaces. An important class of examples are spherical varieties, which pose interesting computational challenges.

We describe when two hyper-Kahler fourfolds of K3^[2]-type of Picard rank 1 with isomorphic transcendental lattices are derived equivalent. Then we present new constructions of pairs of twisted derived equivalent hyper-Kaehler manifolds of Picard rank >1. This is a joint work with Michal Kapustka.

The Grothendieck-Riemann-Roch theorem fails to hold in the case of Deligne-Mumford stacks due to the presence of stabilisers. Several modified constructions, using the inertia stack and related objects, have been proposed by Edidin, Graham, Toën, and others.

In this talk, we will analyse Toën's formulation of the Riemann-Roch theorem from a motivic perspective. More specifically, we will construct an object (associated to every Deligne-Mumford stack) in Voevodsky's triangulated category of motives which we will then use to reformulate (and perhaps even generalise) Toën's Riemann-Roch isomorphism in the language of motivic cohomology. This is joint work with Utsav Choudhury and Amit Hogadi.

(Joint work with T. Keller (Groningen) and Y. Qin (Berkeley))
We consider versions for smooth varieties X over finitely generated fields K in positive characteristic p of several conjectures that can be traced back to Tate, and study their interdependence. In particular, let A/K be an abelian variety. Assuming resolutions of singularities in positive characteristic, I will explain how to relate the BSD-rank conjecture for A to the finiteness of the p-primary part of the Tate-Shafarevich group of A using rigid cohomology. Furthermore, I will discuss what is needed for a generalisation.

A theorem of Esnault states that smooth Fano varieties over finite fields have rational points. What happens if we relax the conditions related to the positivity properties of the anti-canonical class? In this seminar, I will discuss the case of 3-folds with nef anti-canonical class. Specifically, we show that in the case of negative Kodaira dimension, the existence of rational points is established if the cardinality is greater than 19. In the K-trivial case, we prove a similar result, provided that the Albanese morphism is non-trivial. This is joint work with S. Filipazzi.

In this talk I will describe a general philosophy, due to Patimo, for constructing positive combinatorial formulas for Kostka-Foulkes polynomials beyond type A. This amounts to constructing atomic decompositions for crystals as well as swapping functions which allow defining charge statistics. Then I will explain such a construction for crystals of type C₂ and point out future directions to follow. This is joint work with Leonardo Patimo.

The Hilbert scheme of points in the affine complex plane is a smooth variety. Several descriptions of its cohomology groups are known. One may use Białynicki-Birula decomposition, Nakajima creation operators, or the Kirwan map. In the talk I will present a relation between the last two of the mentioned methods. I will describe the action of Nakajima's creation operators on the characteristic classes of the tautological bundle. This is joint work with Magdalena Zielenkiewicz.

The title of the talk recalls some main keywords of much work that Piotr Pragacz did by himself and/or in collaboration with many authors, like e.g. Alain Lascoux, Jan Ratajski, Andrzej Weber, to mention a few. Inspired by the many enlightening conversations I had with Piotr, I will attempt to draw an elementary path connecting the four keywords in the title.

Classical theorems of Kulikov, Persson and Pinkham describe degeneration of one parameter families of K3 surfaces. In the first case it asserts that of a degeneration of K3 surfaces with finite monodromy, there is a semi-stable degeneration with a smooth central fiber. If the central fiber of degeneration admits A-D-E singularities then the theorem follows from the simultanous resolution.
It is known that this does not generalize to higher dimensional Calabi-Yau varieties: local monodromy of odd dimensional A₁ singularity has infinite order. There exists an example of Calabi-Yau threefold degeneration with A₂ singularity, it has local monodromy of order 6 but it has no smooth filling (in algebraic category). On the other case the A₂ singularity has no crepant resolution. I will present an example of a different type: a semi-stable one-parameter family of Calabi-Yau threefolds with trivial local monodromy and central fibers consisting of two components - a smooth, rigid Calabi-Yau manifold and a quadric bundle. I shall briefly discuss geometry of this degeneration.

Joint work with Duco van Straten (Johannes Gutenberg University Mainz).