Marian Aprodu
Chen ranks formulae
I report on a joint work in progress with G. Farkas, C. Raicu and A. Suciu. Koszul modules are finitely generated graded modules over polynomial algebras that arise in geometric group theory. Their main role is to compute the Chen ranks, which are essential numerical invariants of finitely-generated groups. The set-theoretic supports of Koszul modules, called resonance loci, are obtained from linear sections of Grassmannians via the incidence varieties. In our work, we find effective formulae for Chen ranks under certain geometric assumptions on the corresponding resonance loci.
Carolina Araujo
Blowups, Gale duality and moduli spaces
In this talk, we discuss the birational geometry of blowups of projective spaces at points in general position. For that, we explore Gale duality, a correspondence between sets of $n=r+s+2$ points in projective spaces $\mathbb{P}^s$ and $\mathbb{P}^r$. For small values of $s$, this duality has a remarkable geometric manifestation: the blowup of $\mathbb{P}^r$ at $n$ points can be realized as a moduli space of vector bundles on the blowup of $\mathbb{P}^s$ at the Gale dual points.
Valentina Beorchia
Variations of hypersurface hyperplane sections and syzygy bundles
In 1998 Harris, Mazur and Pandharipande posed the following question:
Does the family of all smooth hyperplane sections of a smooth irreducible projective hypersurface vary maximally in moduli? As observed by Beauville in 2025, the question is equivalent to the Weak Lefschetz Property of the Jacobian ideal in the hypersurface degree. We translate the problem into the vanishing of the cohomology of a suitable stable syzygy bundle. We give a positive answer in sufficiently high degree, by using Grauert-Mülich Theorem and Flenner Restriction Theorem. Joint work with R. M. Mirò-Roig.
Cinzia Casagrande
Classifying Fano 4-folds with large Picard number
We will present some classification results for (smooth and complex) Fano 4-folds X with Picard number rho(X)>6. First, if rho(X)>9, then X is a product of del Pezzo surfaces; this is sharp, since we know one family of Fano 4-folds with rho(X)=9 that is not a product of surfaces. Then, in the range rho(X)=7,8,9, we will explain some partial classification results. During the talk, we will also give an overview of the techniques, based on birational geometry in the framework of the MMP. A key ingredient is the properties of the Lefschetz defect, an invariant that relates the Picard number of X to that of its prime divisors.
Carel Faber
Tautological classes on the moduli space of bridgeless curves
A stable curve of genus at least 2 is called bridgeless if its dual
graph has no bridges, i.e., disconnecting edges. The moduli space Y_g
of bridgeless curves of genus g is the complement in Mgbar of the
boundary divisors Delta_i with i positive.
In a 2020 Zoom talk, Aaron Pixton stated and discussed several results
and conjectures concerning the tautological ring of Y_g. This included
some computational results. After recalling this work, I will discuss
joint work with Carolina Tamborini in which we prove Pixton's
conjectures for classes of dimension at most 3.
Gabi Farkas
Hurwitz-Brill-Noether theory via stability conditions.
We develop a novel approach to the Brill–Noether theory of curves endowed with a degree k cover of the projective line, via Bridgeland stability conditions on elliptic K3 surfaces.
We first develop the Brill–Noether theory on elliptic K3 surfaces via the notion of
Bridgeland stability type for objects in their derived category. As a main application,
we show that curves on elliptic K3 surfaces serve as the first known examples of smooth
k-gonal curves which are general from the viewpoint of Hurwitz–Brill–Noether theory.
In particular, we provide new proofs of the main non-existence and existence results in
Hurwitz–Brill–Noether theory. Finally, we construct explicit examples of curves defined over number fields which are general from the perspective of Hurwitz–Brill–Noether theory. Joint work with Soheyla Feyzbakhsh and Andres Rojas.
Camilla Felisetti
Parabolic bundles and intersection cohomology of moduli of vector bundles
Intersection cohomology is a topological notion adapted to the description of singular topological spaces, and the Decomposition Theorem for algebraic maps is a key tool in the subject. Motivated by the work of Mozgovoy and Reineke, in joint work with Andras Szenes and Olga Trapeznikova, we give a complete description of the intersection cohomology of the moduli space of vector bundles of any rank via a detailed analysis of the Decomposition Theorem applied to a certain map from parabolic bundles. We also give a new formula for the intersection Betti numbers of these moduli spaces, which has a clear geometric meaning.
Ajay Gautam
Cohomology of Tautological bundles on Quot schemes on P^1
Tautological bundles on Quot schemes on curves arise naturally from
bundles on curves. Recently, the study of cohomology of various wedge and
symmetric powers of these tautological bundles have yielded interesting
results in the works of Oprea-Sinha, Marian-Oprea-Sam, Krug and
Marian-Negut. I will discuss some results from joint work with Shubham
Sinha and Feiyang Lin in which we extend the study to the setting of to
the tautological bundles on Quot schemes of higher rank quotients on P^1.
Archi Kaushik
Hyperquot Schemes on Curves and Shifted Yangians.
Quot schemes parameterising rank 0 quotients of a fixed vector bundle on a smooth projective complex curve are interesting moduli spaces which in particular generalise the notion of symmetric products. Marian and Neguţ showed that the cohomology of Quot schemes on a curve admits an action of a shifted Yangian of sl_2 that can be used to recover the Betti numbers of these moduli spaces à la Nakajima and Grojnowski. I will present some results of my upcoming work where I generalise the aforementioned results to the case of Hyperquot schemes on curves, that is the moduli space of flags of rank 0 quotients of a fixed vector bundle on a smooth projective curve.
Danilo Lewanski
Invariants of nested Hilbert and Quot schemes on surfaces
We characterise the generating series of Euler characteristics of nested Hilbert and Quot schemes for a smooth pointed surface. This is achieved via a novel technique involving differential operators modelled on the enumerative problem, which we introduce. From this analysis, we deduce that in the Hilbert scheme case the generating series is the product of a rational function by the celebrated Euler's product formula counting integer partitions. In higher rank, we derive functional equations relating the nested Quot scheme generating series to the rank one series, corresponding to nested Hilbert schemes.
Joint work w/ Nadir Fasola, Michele Graffeo, Andrea T. Ricolfi.
Cristina Manolache
Higher genus reduced Gromov--Witten invariants
Gromov--Witten (GW) invariants ideally give counts of curves of genus g in a given variety. However, GW invariants with g greater than one, have a more subtle enumerative meaning: curves of lower genus also contribute to GW invariants. In genus one this problem was corrected by Vakil and Zinger, who defined more enumerative numbers called "reduced GW invariants". More recently Hu, Li and Niu gave a construction of reduced GW invariants in genus two. I will use a flattification result for sheaves to define reduced Gromov--Witten invariants in all genera. This is work with A. Cobos Rabano, E. Mann and R. Picciotto.
Anton Mellit
Local cohomology of Hilbert schemes and Higgs moduli spaces
In this talk I would like to draw your attention to the problem of computing the local cohomology of vector bundles on the Hilbert scheme of C^2 with respect to a certain Lagrangian subvariety. I will prove a general vanishing result and compute the local cohomology in a special case. This local cohomology is interesting because conjecturally, by some version of the so-called 3d mirror symmetry, it should be isomorphic to the Borel-Moore homology of certain Higgs moduli stacks. I will try to explain this conjecture and the motivation behind it.
Solomiya Mizyuk
Virtual invariants of Quot schemes of points on threefolds
We construct an almost perfect obstruction theory on the Quot scheme of a threefold parametrizing zero-dimensional quotients of a locally free sheaf. This yields a virtual class in degree zero and therefore allows one to define the virtual invariants. We then explain how these virtual invariants are computed.
Siao Chi Mok
Logarithmic Fulton—MacPherson configuration spaces
The Fulton—MacPherson configuration space is a well-known compactification of the ordered configuration space of a projective variety. We describe a construction of its logarithmic analogue: it is a simple normal crossings compactification of the configuration space of points on X\D, where X is a smooth projective variety and D is a simple normal crossings divisor, and is constructed via logarithmic and tropical geometry. Moreover, given a semistable degeneration of X, we construct a logarithmically smooth degeneration of the Fulton—MacPherson space of X. Both constructions parametrise point configurations on certain target degenerations, arising from both logarithmic geometry and the original Fulton–MacPherson construction. The degeneration satisfies a “degeneration formula” — each irreducible component of its special fibre can be described as a proper birational modification of a product of logarithmic Fulton–MacPherson configuration spaces. Time permitting, we explore some potential applications to enumerative geometry.
Riccardo Ontani
Virtual Jeffrey-Kirwan localisation
The localisation formula of Atiyah-Bott-Berline-Vergne expresses integrals over a space X acted on by a group G in terms of integrals over the fixed locus X^G. Jeffrey-Kirwan localisation expresses integrals over a quotient X/G in terms of integrals over (certain components of) the fixed locus X^G. I will discuss a virtual version of the Jeffrey-Kirwan localisation formula for the virtual classes of Behrend-Fantechi and Oh-Thomas. This is based on an ongoing project with Richard Thomas.
Andrea Petracci
On deformations of toric Fano varieties
In this talk I will explain certain properties of the deformation theory of toric Fano varieties. In particular I will describe some combinatorial inputs on polytopes which induce deformations. This talk is based on joint works with Anne-Sophie Kaloghiros, and with Alessio Corti and Paul Hacking.
Paolo Rossi
Cohomological field theories, their generalizations, and integrability.
The topology of the moduli spaces of stable curves has been known to be intimately related to integrability (of evolutionary PDEs) at least since the 90s, with the work of Witten and Kontsevich. Nowadays we have a much deeper understanding of this phenomenon and the mechanisms at its base. Cohomological field theories, are, in this sense pivotal: they are the cohomological counterpart of these integrable systems of PDEs and a subtle probe into the topology of the moduli spaces, as demonstrated by the role they play in the construction of Pixton's tautological relations. CohFTs are natural from the viewpoint of the Deligne-Mumford compactification and through the double ramification cycle construction of integrable systems from CohFTs, it's easy to control how their axioms correspond to properties of the integrable system. There are very natural generalizations of CohFTs either coming from considering partial compactifications or from wanting to include wider classes of integrable systems. These two approaches to generalizing the notion of CohFTs are suggestively related by a series of partial results and conjectures that I am going to present. This is a joint work with A. Buryak.
Shubham Sinha
Intersection Theory of Hyperquot Schemes on Curves
Hyperquot schemes parameterize chains of successive quotients of
a fixed vector bundle V on a curve C. When V is trivial, Hyperquot
schemes provide a compactification of the space of morphisms from C to a
flag variety. I will present a formula that computes top virtual
intersection numbers on Hyperquot schemes. Our result generalizes the
Vafa-Intriligator formula, proved by Bertram and Marian-Oprea, for
computing intersection numbers on Quot schemes. I will also discuss
conditions under which these virtual intersections yield enumerative
counts of maps to flag varieties. This talk is based on a joint work with
R. Ontani and W. Xu.
Balazs Szendroi
Moduli spaces around the McKay correspondence and Nakajima quiver varieties
This talk will review some results that identify singular Nakajima quiver varieties on finite and affine ADE quivers with Quot schemes of certain modules over cornered preprojective algebras and related moduli spaces. We will also point out some intricate representation theoretic questions that arise from this circle of ideas. Based on joint works with Lukas Bertsch, Alastair Craw, Søren Gammelgaard and Ádám Gyenge, and work of Craw-Yamagishi and Bertsch-Gammelgaard.
Orsola Tommasi
Geometry of fine compactified Jacobians
The universal Jacobian on the moduli space of curves parametrizes pairs consisting of a smooth curve of a fixed genus g and a line bundle of fixed degree d. Constructing a fine universal compactified Jacobian requires to extend this construction to a proper family over the moduli space of stable curves. In this talk, we introduce a simple definition of a fine compactified universal Jacobian, both for a single nodal curve and for families. We discuss their combinatorial characterization and explain cases in which this leads to new examples.
This is joint work with Nicola Pagani (Bologna/Liverpool).