Hilbert Scheme Research Articles

On generators and relations of the rational cohomology of Hilbert schemes

We consider for $$d\ge 1$$ the graded commutative $${\mathbb {Q}}$$ -algebra $${\mathcal {A}}(d):=H^*({\text {Hilb}}^d({\mathbb {C}}^2);{\mathbb {Q}})$$ , which is also connected to the study of generalised Hurwitz spaces by work of the first author. These Hurwitz spaces are in turn related to the moduli spaces of Riemann surfaces with boundary. We determine two distinct, minimal sets of $${\lfloor }d/2{\rfloor }$$ multiplicative generators of $${\mathcal {A}}(d)$$ . Additionally, we prove when the lowest degree generating relations occur. For small values of d, we also determine a minimal set of generating relations, which leads to several conjectures about the necessary generating relations for $${\mathcal {A}}(d)$$ .

New elementary components of the Gorenstein locus of the Hilbert scheme of points

We construct new explicit examples of nonsmoothable Gorenstein algebras with Hilbert function . This gives a new infinite family of elementary components in the Gorenstein locus of the Hilbert scheme of points and solves the cubic case of Iarrobino’s conjecture.

Residual categories of quadric surface bundles

Abstract We show that the residual categories of quadric surface bundles are equivalent to the (twisted) derived categories of some scheme under the following hypotheses. • Case 1: The quadric surface bundle has a smooth section. • Case 2: The total space of the quadric surface bundle is smooth and the base is a smooth surface. We provide two proofs in Case 1 describing the scheme as the hyperbolic reduction and as a subscheme of the relative Hilbert scheme of lines, respectively. In Case 2, the twisted scheme is obtained by performing birational transformations to the relative Hilbert scheme of lines. Finally, we apply the results to certain complete intersections of quadrics.

The scheme of monogenic generators I: representability

This is the first in a series of two papers that study monogenicity of number rings from a moduli-theoretic perspective. Given an extension of algebras B/A, when is B generated by a single element $$\theta \in B$$ over A? In this paper, we show there is a scheme $${\mathcal {M}}_{B/A}$$ parameterizing the choice of a generator $$\theta \in B$$ , a “moduli space” of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ample examples.

Derived Categories of (Nested) Hilbert Schemes

In this paper, we provide several results regarding the structure of derived categories of (nested) Hilbert schemes of points. We show that the criteria of Krug–Sosna and Addington for the universal ideal sheaf functor to be fully faithful resp. a P-functor are sharp. Then we show how to embed multiple copies of the derived category of the surface using these fully faithful functors. We also give a semiorthogonal decomposition for the nested Hilbert scheme of points on a surface, and finally we give an alternative proof of a semiorthogonal decomposition due to Toda for the symmetric product of a curve.

Attribute-Aware RBFs: Interactive Visualization of Time Series Particle Volumes Using RT Core Range Queries.

Smoothed-particle hydrodynamics (SPH) is a mesh-free method used to simulate volumetric media in fluids, astrophysics, and solid mechanics. Visualizing these simulations is problematic because these datasets often contain millions, if not billions of particles carrying physical attributes and moving over time. Radial basis functions (RBFs) are used to model particles, and overlapping particles are interpolated to reconstruct a high-quality volumetric field; however, this interpolation process is expensive and makes interactive visualization difficult. Existing RBF interpolation schemes do not account for color-mapped attributes and are instead constrained to visualizing just the density field. To address these challenges, we exploit ray tracing cores in modern GPU architectures to accelerate scalar field reconstruction. We use a novel RBF interpolation scheme to integrate per-particle colors and densities, and leverage GPU-parallel tree construction and refitting to quickly update the tree as the simulation animates over time or when the user manipulates particle radii. We also propose a Hilbert reordering scheme to cluster particles together at the leaves of the tree to reduce tree memory consumption. Finally, we reduce the noise of volumetric shadows by adopting a spatially temporal blue noise sampling scheme. Our method can provide a more detailed and interactive view of these large, volumetric, time-series particle datasets than traditional methods, leading to new insights into these physics simulations.

Small elementary components of Hilbert schemes of points

AbstractWe answer an open problem posed by Iarrobino,Hilbert scheme of points: Overview of last ten years. Proceedings of Symposia in Pure Mathematics, 46 (American Mathematical Society, Providence, RI, 1987), 297–320: Is there a component of the punctual Hilbert scheme [Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert',in Séminaire Bourbaki, 6 (Societe Mathematique de France, Paris, 1995), 221, 249–276]$\operatorname {\mathrm {Hilb}}^d({\mathscr {O}}_{\mathbb {A}^n,p})$with dimension less than$(n-1)(d-1)$? For each$n\geq 4$, we construct an infinite class of elementary components in$\operatorname {\mathrm {Hilb}}^d(\mathbb {A}^n)$producing such examples. Our techniques also allow us to construct an explicit example of a local Artinian ring [Iarrobino and Kanev,Power sums, Gorenstein algebras, and determinantal loci(Springer-Verlag, Berlin, 1999), 221–226] of the formwith trivial negative tangents, vanishing nonnegative obstruction space, and socle-dimension$2$.

Hilbert schemes of points on smooth projective surfaces and generalized Kummer varieties with finite group actions

Hilbert schemes of points on smooth projective surfaces and generalized Kummer varieties with finite group actions

Derived categories of hyper-Kähler manifolds: extended Mukai vector and integral structure

We introduce a linearised form of the square root of the Todd class inside the Verbitsky component of a hyper-Kähler manifold using the extended Mukai lattice. This enables us to define a Mukai vector for certain objects in the derived category taking values inside the extended Mukai lattice which is functorial for derived equivalences. As applications, we obtain a structure theorem for derived equivalences between hyper-Kähler manifolds as well as an integral lattice associated to the derived category of hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of a K3 surface mimicking the surface case.

Extremal Gromov-Witten invariants of the Hilbert scheme of Points

AbstractWe determine all the extremal Gromov-Witten invariants of the Hilbert scheme of$3$points on a smooth projective complex surface. Our result for the genus-$1$case verifies a conjecture that we propose for the genus-$1$extremal Gromov-Witten invariant of the Hilbert scheme ofnpoints withnbeing arbitrary. The main ideas in the proofs are to use geometric arguments involving the cosection localization theory of Kiem and J. Li [17, 23], algebraic manipulations related to the Heisenberg operators of Grojnowski [13] and Nakajima [34], and the virtual localization formulas of Gromov-Witten theory [12, 20, 30].

Categorical and K-theoretic Donaldson–Thomas theory of (part II)

AbstractQuasi-BPS categories appear as summands in semiorthogonal decompositions of DT categories for Hilbert schemes of points in the three-dimensional affine space and in the categorical Hall algebra of the two-dimensional affine space. In this paper, we prove several properties of quasi-BPS categories analogous to BPS sheaves in cohomological DT theory.We first prove a categorical analogue of Davison’s support lemma, namely that complexes in the quasi-BPS categories for coprime length and weight are supported over the small diagonal in the symmetric product of the three-dimensional affine space. The categorical support lemma is used to determine the torsion-free generator of the torus equivariant K-theory of the quasi-BPS category of coprime length and weight.We next construct a bialgebra structure on the torsion free equivariant K-theory of quasi-BPS categories for a fixed ratio of length and weight. We define the K-theoretic BPS space as the space of primitive elements with respect to the coproduct. We show that all localized equivariant K-theoretic BPS spaces are one-dimensional, which is a K-theoretic analogue of the computation of (numerical) BPS invariants of the three-dimensional affine space.

Virasoro constraints for moduli spaces of sheaves on surfaces

AbstractWe introduce a conjecture on Virasoro constraints for the moduli space of stable sheaves on a smooth projective surface. These generalise the Virasoro constraints on the Hilbert scheme of a surface found by Moreira and Moreira, Oblomkov, Okounkov and Pandharipande. We verify the conjecture in many nontrivial cases by using a combinatorial description of equivariant sheaves found by Klyachko.

Syzygies in Hilbert schemes of complete intersections

Let e1,…,ec be positive integers and let Y⊆Pn be the monomial complete intersection defined by the vanishing of x1e1,…,xcec. In this paper, we study sharp upper bounds on the number of equations and syzygies of subschemes parametrized by the Hilbert scheme of points Hilbd(Y), and discuss applications to the Hilbert scheme of points Hilbd(X) of arbitrary complete intersections X⊆Pn.

Algebra and geometry of link homology: Lecture notes from the IHES 2021 Summer School

AbstractThese notes cover the lectures of the first named author at 2021 IHES Summer School on “Enumerative Geometry, Physics and Representation Theory” with additional details and references. They cover the definition of Khovanov‐Rozansky triply graded homology, its basic properties and recent advances, as well as three algebro‐geometric models for link homology: braid varieties, Hilbert schemes of singular curves and affine Springer fibers, and Hilbert schemes of points on the plane.

Cosection localization and the Quot scheme QuotSl(E)

Let E be a locally free sheaf of rank r on a smooth projective surface S . The Quot scheme Quot S l ( E ) of length l coherent sheaf quotients of E is a natural higher-rank generalization of the Hilbert scheme of l points of S . We study the virtual intersection theory of this scheme. If C ⊂ S is a smooth canonical curve, we use cosection localization to show that the virtual fundamental class of Quot S l ( E ) is ( − 1 ) l times the fundamental class of the smooth subscheme Quot C l ( E | C ) ⊂ Quot S l ( E ) . We then prove a structure theorem for virtual tautological integrals over Quot S l ( E ) . From this, we deduce, among other things, the equality of virtual Euler characteristics χ vir ( Quot S l ( E ) ) = χ vir ( Quot S l ( O ⊕ r ) ) .

Numerical solutions and geometric attractors of a fractional model of the cancer-immune based on the Atangana-Baleanu-Caputo derivative and the reproducing kernel scheme

• This paper discusses a mathematical model that aims to investigate the interaction between the IS and CCs by incorporating ℐℒ − 12 cytokine together with an anti- P D − L 1 inhibitor. • The utilized fractional differential problem is a modern mathematical formation model evaluated by the reproducing Hilbert scheme used with the fractional Atangana-Baleanu-Caputo derivative. • The required Hilbert spaces were included, and the representation theory of the solutions was presented together with several related results. • Some mathematical analyses like error convergence analysis, attitude, and its order degree were also included. • Novel steps are fitted for the covering fractional model to gain its numeric-analytic solutions. • Several plots and tables were presented and discussed to illustrate the mathematical and physiological behavior of this disease, in addition to the effect of the degree of fractional derivatives used in it. • The summary of the presented work and various scientific recommendations in addition to the future work that complements this work has been utilized in the last part Recently, it has become critical to attempt to recognize diseases with high death rates around the world like infectious diseases and cancer. As a result, modeling in mathematics may be utilized to give feedback on diseases that enabled influence every one of us. This paper discusses a mathematical model that aims to investigate the interaction between the IS and CCs by incorporating I L − 12 cytokine together with an anti- P D − L 1 inhibitor. The utilized fractional differential problem is a modern mathematical formation model evaluated by the reproducing Hilbert scheme used with the fractional Atangana-Baleanu-Caputo derivative. The required Hilbert spaces were included, and the representation theory of the solutions was presented together with several related results. Some mathematical analyses like error attitude and its order degree were also included. To illustrate the mathematical and physiological behavior of this disease, several plots and tables were presented and discussed, in addition to the effect of the degree of fractional derivatives used in it. The summary of the presented work and various scientific recommendations in addition to the future work that complements this work has been utilized in the last part.

Smooth Hilbert schemes: Their classification and geometry

Abstract Closed subschemes in projective space with a fixed Hilbert polynomial are parametrized by a Hilbert scheme. We classify the smooth ones. We identify numerical conditions on a polynomial that completely determine when the Hilbert scheme is smooth. We also reinterpret these smooth Hilbert schemes as generalized partial flag varieties and describe the subschemes being parametrized.

Some families of big and stable bundles on K3 surfaces and on their Hilbert schemes of points

Here we investigate meaningful families of vector bundles on a very general polarized K3 surface (X, H) and on the corresponding Hyper–Kähler variety given by the Hilbert scheme of points $$X^{[k]}:= {\textrm{Hilb}}^k(X)$$ , for any integer $$k \geqslant 2$$ . In particular, we prove results concerning bigness and stability of such bundles. First, we give conditions on integers n such that the twist of the tangent bundle of X by the line bundle nH is big and stable on X; we then prove a similar result for a natural twist of the tangent bundle of $$X^{[k]}$$ . Next, we prove global generation, bigness and stability results for tautological bundles on $$X^{[k]}$$ arising either from line bundles or from Mukai–Lazarsfeld bundles, as well as from Ulrich bundles on X, using a careful analysis on Segre classes and numerical computations for $$k = 2, 3$$ .

Hilbert schemes with two Borel-fixed points

We characterize Hilbert polynomials that give rise to Hilbert schemes with two Borel-fixed points and determine when the associated Hilbert schemes or its irreducible components are smooth. In particular, we show that the Hilbert scheme is reduced and has at most two irreducible components. By describing the singularities in a neighbourhood of the Borel-fixed points, we prove that the irreducible components are Cohen-Macaulay and normal. We end by giving many examples of Hilbert schemes with three Borel-fixed points.

Hermitian K-theory via oriented Gorenstein algebras

AbstractWe show that the hermitian K-theory space of a commutative ringRcan be identified, up to𝐀1{\mathbf{A}^{1}}-homotopy, with the group completion of the groupoid of oriented finite GorensteinR-algebras, i.e., finite locally freeR-algebras with trivialized dualizing sheaf. We deduce that hermitian K-theory is universal among generalized motivic cohomology theories with transfers along oriented finite Gorenstein morphisms. As an application, we obtain a Hilbert scheme model for hermitian K-theory as a motivic space. We also give an application to computational complexity: we prove that 1-generic minimal border rank tensors degenerate to the big Coppersmith–Winograd tensor.