impanga 2025/26

A longstanding conjecture, recently solved positively by Chen, Gounelas and Liedtke, states that every projective $K3$ surface, as well as every Enriques surface, contains infinitely many rational curves. In this talk, we investigate the geometric properties of the rational curves lying on Enriques surfaces. We detect the algebraic class of many of them and in particular show that, for every natural number $k$ congruent to $1$ modulo $4$, the general Enriques surface admits irreducible rational curves of arithmetic genus $k$. We discuss the implications of our results for the geometry of rational curves on some elliptic $K3$ surfaces.

The Iarrobino scheme is a novel moduli space introduced by Jelisiejew, by combining the the Hilbert scheme of points with the variety of completed quadrics. Motivated by the rich enumerative geometry of the Hilbert scheme, one can try to also compute invariants of the Iarrobino scheme. In this talk, we will consider the motive in the Grothendieck ring of varieties of the Iarrobino scheme on a curve and get an explicit formula for the generating series. This is joint work with Joachim Jelisiejew and Andrea Ricolfi.

We sketch a proof of the generalised pro-2 real Section Conjecture for what we call equivariantly triangulable varieties over the reals.
Examples include all smooth varieties as well as all (possibly singular) affine/projective varieties. Building on this, we derive the generalised real Section Conjecture in the geometrically étale simply connected case.

The proétale topology on schemes, introduced by Bhatt and Scholze, provides a convenient framework for capturing the topological behavior of coefficient rings such as $\mathbb{Z}_\ell$ and $\mathbb{Q}_\ell$ in étale cohomology. While being sufficient for many purposes, general proétale sheaves on schemes can be quite wild, and there is no well-behaved formalism of six operations.

In this talk, we will introduce categories of solid proétale sheaves on schemes that aim to address this issue by adapting ideas of Fargues and Scholze. We will show that these categories admit a full six-functor formalism while remaining large enough to include all objects typically encountered in practice. We achieve this by comparing them to a suitable theory of proétale motives, which may be of independent interest. This work is joint with Raphaël Ruimy and Swann Tubach.

We consider the notion of eigenvalues for multilinear maps with values in a finite-dimensional vector space, generalizing the classical concept of matrix eigenvalues and giving some results for algebraic and geometric multiplicities.

The set of nonzero complex numbers can be decomposed into a copy of the positive real numbers and the unit circle using polar coordinates. In this talk, I will construct a topological analogue of this decomposition for complex varieties. For curves, this decomposition can be interpreted as a pair-of-pants decomposition of the corresponding Riemann surface, as I will explain during the talk. Our pairs of pants for higher-dimensional varieties are essential projective hyperplane complements $Z$. If $Z$ is for instance the projective line minus three points, then this recovers the classical notion of a pair of pants from hyperbolic geometry. Other important examples are the complements arising from braid and reflection arrangements. In the first main theorem of arXiv:2601.14116, we show that the angle map $$Z\to (S^{1})^{n}$$ associated to an essential projective hyperplane complement is a homotopy equivalence. This theorem generalizes results by Deligne, Salvetti, and Björner and Ziegler. I will explain the proof of this theorem, and I will show various examples of these angle sets. To construct a polar decomposition for complex varieties, we introduce the notion of a torically hyperbolic variety. For these, we can find a degeneration to a singular variety whose local components are the complements we studied above. This generalizes the usual notion of a hyperbolic variety, as any hyperbolic variety is automatically torically hyperbolic. I will show how tropical geometry can be used to construct these degenerations, and how this leads to an explicit van Kampen formula for the fundamental group. I will also show how the notion is broad enough to capture the topology of Calabi-Yau varieties such as $K3$ surfaces. Here, we find that the $K3$ surface is a homotopy colimit of $2$-tori over a polyhedral subdivision of the $2$-sphere with $24$ vertices.

In analogy with recent works on $K3$ surfaces, we study the existence of infinitely many ruled divisors on projective irreducible holomorphic symplectic (IHS) manifolds. We prove such an existence result for any projective IHS manifold of $K3^{[n]}$ or generalized Kummer type which is not a variety defined over $\overline{\mathbb{Q}}$ with Picard number one. The result is obtained as a combination of the regeneration principle and of a generalization to higher dimensions of a controlled degeneration technique, invented by Chen, Gounelas and Liedtke in dimension $2$. This is joint work with P. Beri and G. Pacienza.

The Smith inequality estimates the total Betti numbers with coefficients $\mathbb{F}_2$ of the fixed locus of an involution. This gives for example topological constraints for the real locus of a complex manifold with a real structure, which is given by an anti-holomorphic involution. Of particular interest are the involutions for which the Smith inequality is in fact an equality, these are called maximal. We give a criterion for when an (anti-)holomorphic involution on a complex surface $S$ with $h^1(S,\mathbb{F}_2)=0$ induces a maximal involution on the Hilbert scheme of $n$ points $S^{[n]}$. We moreover prove that if $S$ is a K3 surface, then there are no maximal (anti-)holomorphic involutions on any deformation of $S^{[n]}$. In other words, there are no maximal brane involutions on hyper-Kähler manifolds of $K3^{[n]}$ type.

We show that the homogeneous ideals of secant varieties of smooth projective curves and surfaces in sufficiently ample embeddings are determinantally presented. The same result holds for the first secant varieties of arbitrary smooth projective varieties in sufficiently ample embeddings. This completely settles a conjecture of Eisenbud-Koh-Stillman for curves and partially resolves a conjecture of Sidman-Smith in higher dimensions. To establish the results, we employ the geometry of Hilbert schemes of points. Based on our method, we also prove that the homogeneous ideals of arbitrary projective schemes in sufficiently ample embeddings are generated by quadrics of rank three, confirming a conjecture of Han-Lee-Moon-Park. This is joint work with Daniele Agostini.

Let $X$ be a proper smooth variety over a finite extension of $\mathbb{Q}_p$ with good reduction. Under a weak assumption on the Hodge numbers of the reduction of $X$, we show that the existence of a non-zero global $2$-form on $X$ implies the existence, over a finite extension, of Brauer classes with non-constant evaluation map. In particular, a smooth proper variety over a number field satisfying weak approximation over all finite extensions has no non-zero global $2$-forms. The proof is based on a prismatic interpretation of Brauer classes with geometrically constant evaluation map, and on a Newton-above-Hodge result for the modulo $p$ reduction of prismatic cohomology. This is joint work with Rachel Newton and Margherita Pagano.

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.