Abstract:
Solomon-Terao polynomials of hyperplane arrangements are two variable x and t polynomials defined algebraically, and they have nice specializations x=1 to the topological Poincare polynomial of the complements of arrangements. Recently, for the t=-1-specialization, it was found that it coincides with the topological Poincare polynomial of the regular nilpotent Hessenberg variety when the arrangement comes from the lower ideal of the root poset. We call this specialization the Solomon-Terao polynomial, and it is a natural question to ask that is a general meaning of this polynomial.
However, almost nothing is known on it, in fact, we do not know the degrees of that polynomial. In this talk, we consider this problem by using the Castelnouvo-Mumford regularity.
Abstract:
The secants of an algebraic variety have a straightforward geometric definition, but their algebraic aspects, such as their equations are in general unknown. There are some natural equations, called catalecticant, that are conjectured to generate the ideal of the secants if the embedding is sufficiently positive. We confirm this conjecture for arbitrary secants of smooth curves and surfaces and for the first secant of arbitrary smooth varieties. Our main tool is the Hilbert scheme of points. This is joint work in progress with Jinhyung Park.
Abstract:
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.
Abstract:
In 2002 P. Cassou-Noguès, Luengo, Melle and myself studied the topological nature of Denef-Loeser topological zeta function, which involved the construction of formulas for the topological zeta functions of suspensions by 2 points. These formulas involved topological zeta functions twisted by a character and Loeser suggested a kind of general formula for suspensions by N points. The general idea of Loeser was right but the precise formula requires a correction. In this joint work with P. González-Pérez, M. González-Villa and E. León-Cardenal we provide correct formulas for arbitrary suspensions. These formulas involve arithmetic functions, and there is an affine recursion combining the topological zeta functions twisted by all the characters whose order divides N. This generalization goes further since we provide the formulas for the motivic level, by using a stratification principle and classical techniques of generating functions in toric geometry. The same strategy is used to obtain formulas for the motivic zeta functions of some families of non-isolated singularities related to superisolated, Lê-Yomdin, and weighted Lê-Yomdin singularities.
Abstract:
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.
Abstract:
The striking description found by Grojnowski and Nakajima of the homology of the Hilbert scheme of points on smooth surfaces is based on Heisenberg algebras associated with lattices. Khovanov initiated the categorification of these algebras and gave a graphically defined category from which, after decategorification, one can reproduce the Heisenberg algebra associated with the free boson or, equivalently, a rank 1 lattice.
In a sequence of papers, we extend Khovanov's graphical construction to derived categories of smooth and projective varieties or, more generally, to dg categories having a Serre functor. Additionally, we construct a 2-representation of our Heisenberg category on a categorical analogue of the Fock space. One advantage of our approach is that it has at least two natural decategorifications: the (numerical) Grothendieck group, which is a lattice, and the Hochschild homology, which is only a vector space without a canonical lattice generating it. To circumvent this issue, we introduce Heisenberg algebras modeled on vector spaces with a possibly non-symmetric form. Our results verify a conjecture of Grojnowski. Joint work with Timothy Logvinenko.
Abstract:
The series will aim to draw a picture of what is known and open for the Hilbert schemes of points, its singularities and topology. Many interesting results and directions emerged in this subfield in recent years. Questions or even requests for specific subtopics are welcome, but no guarantees.
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We discuss some geometric/numerical implications of the freeness property on curves having lines or smooth conics among the irreducible components.
Abstract:
I will explain how to evaluate the virtual number of maps from a fixed domain curve to a general hypersurface in a Grassmannian, subject to incidence conditions with special Schubert subvarieties. I will also discuss the question of whether this count is enumerative for sufficiently large degree of the maps. The talk is based on joint work with Shubham Sinha.
Abstract:
For a divisor representing a function and another divisor representing a differential form in a normal surface singularity, there is a notion of motivic and topological zeta function. In this talk we will discuss relations between the set of poles of these zeta functions whenthey are related by finite morphisms.
Abstract:
In this joint work with Alina Marian, we study the geometric representation theory of Quot schemes of 0-dimensional coherent sheaves on curves: the overall structure we obtain is a categorification of quantum loop sl_2 acting on the derived categories of Quot schemes. We have two uses for this machinery: we produce an explicit semiorthogonal decomposition of the derived categories in question, and we prove conjectures of Krug and Oprea-Sinha on the cohomology of tautological bundles on Quot schemes.
Abstract:
I will present several recent results regarding the tautological rings of the moduli spaces of abelian varieties and K3 surfaces. For the moduli of abelian varieties, I will introduce the decomposition of cycles into tautological and non-tautological components and give examples of non tautological classes. I will also discuss the complete computation of the Chow ring of the moduli space of degree 2 K3 surfaces. This is based on joint work with Samir Canning, Sam Molcho, and Rahul Pandharipande
Abstract:
I will present results old and new about the quantum cohomology and the higher genus enumerative geometry of the Hilbert scheme of points in the plane. I will highlight the remarkable number of connections to different streams of mathematics.
Abstract:
I will present a new approach to classify log canonical del Pezzo surfaces of rank one, over an algebraically closed field of arbitrary characteristic. It is based on studying P^1-fibrations of the minimal resolution which minimize the height, i.e. the intersection number of the fiber with the exceptional divisor. We prove that the height is at most 4 (with minor exceptions in characteristic 2 and 3) and the geometry becomes more rigid as the height grows. To illustrate this I will show a natural construction of some equisingular families of pairwise non-isomorphic del Pezzo surfaces of rank one and height 1 or 2, and explain why such families do not occur if the height is at least 3 (again, with minor exceptions).
This is a joint work with Karol Palka.
Abstract:
The space of binary forms of degree d admits a natural stratification determined by factorization patterns, indexed by the partitions of d. For example, the locus of binary forms that are d-th powers of a linear form is the rational normal curve, while those factoring as l^(d−1)l', with l, l' linear forms, describe its tangent developable. An important open problem is to determine the defining equations of the closures of these factorization strata, as well as their higher syzygy modules. In this talk, I will present various geometric incarnations of classical Hermite reciprocity for SL_2-representations, and explore some applications to the study of binary forms.
Abstract:
A finite collection of commutative local algebras will be called a multi-singularity. In singularity theory, just like in Hilbert schemes, multi-singularities can collide and degenerate into one another. In this lecture we will present some equivariant classes (and their h-deformed versions) that characterize the hierarchy of multi-singularities induced by these degenerations. Joint work with J. Koncki.
Abstract:
TBA
Abstract:
The Orlik-Solomon algebra of a simple matroid M is the quotient A = E/I, where E is the exterior algebra over the ground set of M and some field k and I is a homogeneous ideal defined in terms of the circuits in M. From this graded algebra, one may construct several other algebraic invariants, such as the holonomy Lie algebra h = h(A), the holonomy Chen Lie algebra h ′/h ′′, and the Koszul modules W_i(A) for i ≥ 1. One then may seek to compute the Hilbert series of these graded objects and determine the support loci and the resonance varieties (or schemes) associated with the modules W_i(A). In this talk, I will explain some of the known—or conjectured— relationships between these various objects, and how to interpret them topologically in the case when the lattice of flats of M can be realized as the intersection lattice of a complex hyperplane arrangement. I will also present some structural results and intriguing examples for “decomposable” matroids.