Oberseminar Algebraische Geometrie

Benedetta Piroddi (UdS)
A new symplectic fourfold

Irreducible symplectic varieties are singular analogues of hyperkähler manifolds. One way to construct them is to take terminalizations of quotients of hyperkähler manifolds by symplectic group actions. In this talk I'll present a new example of this construction, stanrting from an action of the group L_3(4) on a double EPW sextic. The resulting ISV has b_2=4 and non-quotient terminal singularities.

This is a joint work with M. Donten-Bury, G. Kapustka and T. Wawak.

15.04.2026

Simone Billi (UdS)
On the maximality of involutions for Hilbert schemes of points on surfaces

The Smith inequality estimates the total Betti numbers with coefficients 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 a (anti-)holomorphic involution on a complex surface S with h^1(S,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.

Riccardo Moschetti (Pavia)
The motive of the Hilbert scheme of points in all dimensions

Modern enumerative geometry is still driven by two ideas that Grothendieck introduced more than sixty years ago. In 1961, he defined the Hilbert scheme Hilb^d(A^n), whose points are length-d subschemes of affine n-space. Three years later, in a 1964 letter to Serre he sketched the Grothendieck ring of varieties K_0(Var/C), a framework that records every "additive and multiplicative" invariant of a variety in a single class. I will talk about a joint work with M. Graffeo, S. Monavari, and A. Ricolfi, where both of these ideas are present. Fixing d>0, we describe the generating function for the classes of the punctual Hilbert schemes [Hilb^d(A^n)_0]. We prove that this series is always rational and completely determined by just d-8 initial coefficients, therefore giving explicit formulas for all the cases d ≤ 8. This yields a formula for [Hilb^d(X)] for any smooth variety X. I will conclude by describing some conjectures that connect these results to the counting of higher-dimensional partitions.

Lucas Li Bassi

Automorphisms of Irreducible Symplectic Varieties via Kuznetsov components (of cubic fourfolds)

The study of automorphisms of Irreducible Symplectic Manifolds (ISMs) and their singular generalizations, namely Irreducible Symplectic Varieties (ISVs), is an active area of research in algebraic geometry. A modern approach to constructing examples of such varieties is via moduli spaces of (Bridgeland) stable objects in a subcategory of the derived category, known as the Kuznetsov component. Thanks to the works of Bayer--Lahoz--Macrì--Nuer--Perry--Stellari, Li--Pertusi--Zhao, and Saccà, these varieties can be either smooth of type $K3^{[n]}$, of OG10 type, or singular ISVs. A natural question is whether one can induce automorphisms on these varieties starting from automorphisms of the cubic fourfold.
In this talk, I will present some recent results obtained in collaboration with H. Dell and A. Jacovskis, which describe under which hypotheses such automorphisms can be induced, as well as the properties they inherit (for instance, their fixed loci).

Moritz Hartlieb (Bonn)

Twisted Arinkin transforms and derived categories of moduli spaces on Kuznetsov components

By a classical theorem of Mukai, the Poincaré sheaf induces a derived equivalence between an abelian variety and its dual. In this talk, I will discuss a generalization of this construction to a twisted derived equivalence between torsors under dual abelian schemes. Relying on results of Arinkin, this equivalence can be extended to twisted compactified Jacobians associated to linear systems on K3 surfaces. As an application, I will explain how the above can be used to relate the (twisted) derived category of Lagrangian fibered moduli spaces of stable objects in the Kuznetsov component of a cubic fourfold to the symmetric powers of the Kuznetsov component. This is joint work with Saket Shah.

Alexandre Zotine (UdS)

Dynamics of Projectivized Vector Bundles

Understanding the dynamics (self-maps) of a variety can often reveal information about its geometry. For example, the only curves with non-trivial surjective endomorphisms are of genus zero or one. In this talk, I will discuss the dynamics of projectivized vector bundles, where a similar classifying phenomenon seems to arise: the existence of non-trivial surjective endomorphisms suggests that the bundle splits into a direct sum of line bundles. This is joint work with Javier González Anaya and Brett Nasserden.

Enrico Fatighenti (Bologna)

Fano 4-folds of type c5 and even nodal surfaces

A classical construction in projective geometry realizes the blow-up of a cubic fourfold along a line as a conic bundle whose discriminant locus is a quintic surface with 16 nodes. A similar construction can be carried out for Gushel–Mukai fourfolds, as shown in joint work with M. Bernardara, G. Kapustka, M. Kapustka, L. Manivel, G. Mongardi, and G. Tanturri. In this talk, we explain how to obtain an analogous result via a degeneration of certain Küchle fourfolds of type c5, which are of K3 type. This is joint work with Federico Tufo.
 

Ernesto Mistretta (Padova)

Semiaampleness of vector bundles and topology of compact complex manifolds

We give two (non-equivalent) definitions of semi-ample vector bundles, and analyze their geometric meaning by considering some compact complex manifolds with semi-ample cotangent bundle. In particular, we conjecture that all compact complex manifolds of non-general type and with semi-ample cotangent bundle have an infinite fundamental group. This is a joint work with F. Esposito.

Sophie Friesen (Hannover)

Finite groups acting on K3 surfaces in positive characteristic

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.