impanga 2025/26

It has been known since the work of Tadeusz Mostowski in 1985 that the set of germs of complex surfaces up to bilipschitz equivalence is countable. Building on the work of Lev Birbrair, Walter Neumann and Anne Pichon on the bilipschitz classification of complex surface germs, we will describe how to explicitly construct an infinite number of germs of complex surfaces with isolated singularity that are pairwise homeomorphic (and in fact with same normalization up to isomorphism) but that are not pairwise bilipschitz equivalent.

The Brauer group of an algebraic variety is a group with many applications, in particular to the study of rational points. For a K3 surface over a number field, the transcendental part of its Brauer group is finite. It was shown by Cadoret-Charles that the size of its $p$-primary torsion is uniformly bounded for K3 surfaces in one-dimensional families. We give a new proof of this result for one-dimensional families of K3 surfaces with a polarization by a fixed lattice. To be precise, we construct moduli spaces of K3 surfaces with a lattice polarization and a Brauer class, and use the geometry of their complex points to prove boundedness of Brauer groups for the K3 surfaces they parametrize. This is joint work with D. Bragg and A. Várilly-Alvarado.

Given a hypersurface $X$ of the projective space, an Ulrich bundle is a sheaf on $X$ presented by a matrix of linear forms. In general it is hard to understand the rank of Ulrich sheaves on $X$, especially when the dimension $n$ and the degree $d$ of $X$ are high. The case of cubic fourfolds $(n,d)=(4,3)$ is the focus of this talk. I will mention a few results on Ulrich bundles and their moduli spaces, how to construct them using instanton bundles and generalized pfaffian representation, depending on whether the cubic lies on some Hasset divisor or is very general. Along the way we will discuss the Kuznetsov category of a cubic fourfold and sections of the Coble cubic.

Let $L$ be a generic linear space of symmetric matrices over the complex numbers. By inverting all invertible matrices in this space, we obtain an algebraic variety. Computing the degree of this variety is a natural geometric question in its own right, but is also interesting from the point of view of algebraic statistics: the number we obtain is the so-called maximum likelihood degree (ML-degree) of the generic linear concentration model. In 2010, Sturmfels and Uhler conjectured that if we fix the dimension of $L$, this ML-degree is a polynomial in the size of the matrices. Using Schubert calculus and intersection theory on the space of complete quadrics, we were able to prove this polynomiality conjecture, and to write an algorithm that can compute these polynomials efficiently. This talk is based on joint works with Rodica Dinu, Laurent Manivel, Mateusz Michałek, Leonid Monin, and Martin Vodička.

Instanton bundles originally arose in mathematical physics as vector bundles on projective space encoding solutions to the Yang–Mills equations. From an algebro-geometric perspective, they form a rich and intriguing class of $\mu$-stable bundles characterized by cohomological vanishing conditions. Over the past decades, the notion of instanton bundles has been extended beyond $\mathbb{P}^3$ to higher dimensional projective spaces, other Fano threefolds, and finally arbitrary projective varieties. In this talk, I will introduce the classical definition of instanton bundles on $\mathbb{P}^3$, discuss their key properties and moduli, and then explore how these ideas generalize to other varieties. Along the way, I will highlight both the challenges and the geometric insights that arise when adapting the instanton condition to broader settings.

Variations of GIT by $\mathrm{GL}(n)$ provide a rich landscape of birational transformations, inspired by the physics of gauged linear sigma models. I will present the general tools used to study such constructions, both at the geometric level and in the setting of derived categories, and then focus on a ninefold flop that admits a particularly simple geometric description. This is a joint work with Will Donovan, Wahei Hara, and Michał Kapustka.

Hyperkähler manifolds are natural generalizations of K3 surfaces in higher dimensions. Geometric descriptions for locally complete families of projective hyperkähler manifolds are only known in very few cases. In this talk, we first describe the projective geometry of the Hilbert square of a K3 surface of genus $7$ or $8$, by making use of the Mukai model: in both cases, it can be realized as a degeneracy locus on an ambient homogeneous space. From this, we deduce a geometric description for the two locally complete families of K3$^{[2]}$-type (square $4$ and square $6$ with divisibility $1$), in terms of Coble type hypersurfaces. Based on a joint work with Ángel Ríos Ortiz and Andrés Rojas, and an ongoing work also with Benedetta Piroddi.

A K3 surface $S$ is called a Nikulin surface if there exists another K3 surface $K$ with a symplectic involution $i$ such that $S$ is the minimal resolution of singularities of the quotient $K/i$. Equivalently, $S$ can be viewed as a component of the fixed locus of the natural involution induced by $i$ on the Hilbert scheme $K^{[2]}$, which is a hyperkähler manifold. This latter perspective admits a natural generalisation: we call a K3 surface a generalised Nikulin surface if it appears as a component of the fixed locus of a symplectic involution on some hyperkähler fourfold of K3$^{[2]}$-type. In this talk, we present a classification of generalised Nikulin surfaces in terms of their associated lattices, describe their connections with the corresponding hyperkähler varieties, and provide constructions of some of their projective models. This is joint work with C. Camere, A. Garbagnati, and G. Kapustka.

Although in recent years several authors have studied Cox rings of various varieties, such as log Fano varieties and moduli spaces of rational curves with marked points, little is known about Cox rings of varieties of general type. Consequently, only a few examples of Mori Dream spaces of general type are known. Moreover, in dimension two, a classification of Mori Dream surfaces of general type with geometric genus zero still seems far away.

A particular class of surfaces that appears promising to study is the so-called product-quotient surfaces. Furthermore, minimal product-quotient surfaces with geometric genus zero are already classified, so the main goal is to understand which among them are Mori Dream and which are not. In a recent paper, Keum and Lee showed, among other things, that product-quotient surfaces belonging to two specific families in the classification list are Mori Dream.

During the talk, we will briefly discuss the theory of product-quotient surfaces and explain the technique adopted by Keum and Lee to study their effective, nef, and semiample cones. We will then illustrate the main difficulties in studying the remaining product-quotient surfaces in the list and present some partial results in this direction, which are part of a joint work in progress with F. Polizzi.

I will address the question of adding closed points to a normal separated surface $X$, i.e. considering open embeddings of $X$ with complements of dimension zero. I will argue that to a surface $X$ one can add only finitely many points to get its saturated model, i.e. a surface to which one cannot add any points. I will show that the saturated model is unique and functorial in $X$. I will also describe how to construct it explicitly from the additive category of reflexive sheaves on $X$. I will say that a surface is saturated if it is isomorphic to its saturated model. I will discuss saturated surfaces and, more generally, saturated algebraic spaces of dimension two. The talk is based on joint work with A. Bondal, T. Pełka and D. Weissmann.