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

Abstract: 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)

Abstract: 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).