Hilbert Scheme Research Articles

Logarithmic Pandharipande–Thomas spaces and the secondary polytope

Maulik and Ranganathan have recently introduced moduli spaces of logarithmic stable pairs. In the case of toric surfaces we recast this theory using three ingredients: Gelfand, Kapranov and Zelevinsky secondary polytopes, Hilbert schemes of points, and tautological vector bundles. In particular, logarithmic stable pairs spaces are expressed as the zero set of an explicit section of a vector bundle on a logarithmically smooth space, thus providing an explicit description of their virtual fundamental class. A key feature of our construction is that moduli spaces are completely canonical, unlike the existing construction, which is only well-defined up to logarithmic modifications. We calculate the Euler–Satake characteristics of our moduli spaces in a number of basic examples. These computations indicate the complexity of the spaces we construct.

AdS4 holography and the Hilbert scheme

We elucidate a holographic relationship between the enumerative geometry of the Hilbert scheme of N points in the plane ℂ2, with N large, and the entropy of certain magnetically charged black holes with AdS4 asymptotics. Specifically, we demonstrate how the entropy functional arises from the asymptotics of ’t Hooft and Wilson line operators in a 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 gauge theory. The gauge-Bethe correspondence allows us to interpret this calculation in terms of the enumerative geometry of the Hilbert scheme and thereby conjecture that the entropy is saturated by expectation values of certain natural operators in the quantum K-theory ring acting on the localised K-theory of the Hilbert scheme. We give numerical evidence that the large N limit is saturated by contributions from a certain vacuum/fixed point on the Hilbert scheme, associated to a particular triangular-shaped Young diagram, by evolving solutions to the Bethe equations numerically at finite (but large) N towards the classical limit. We thus conjecture a concrete geometric holographic dual of the so-called gravitational/Cardy block.

Ample cones of Hilbert schemes of points on hypersurfaces in ℙ³

Let X X be a very general degree d ≥ 5 d\geq 5 hypersurface in P 3 \mathbb {P}^3 . We compute the ample cone of the Hilbert scheme X [ n ] X^{[n]} of n n points on X X for various small values of n n (the answer is already known for large n n ). We obtain complete answers in some cases and find lower bounds in certain others. We also observe that in the case of X [ 2 ] X^{[2]} for quintic hypersurfaces X X , the existence (or absence) of hyperplane sections with points of high multiplicity also plays a role in the answer to the question at hand, in contrast with cases known earlier. Finally, in the case that a degree d ≥ 3 d\geq 3 smooth hypersurface X X contains a line, we compute the nef cone of X [ n ] X^{[n]} in a slice of the Néron-Severi space.

Birational geometry of Beauville–Mukai systems III: Asymptotic behavior

AbstractSuppose that a Hilbert scheme of points on a K3 surface of Picard rank one admits a rational Lagrangian fibration. We show that if the degree of the surface is sufficiently large compared to the number of points, then the Hilbert scheme is the unique hyperkähler manifold in its birational class. In particular, the Hilbert scheme is a Lagrangian fibration itself, which we realize as coming from a (twisted) Beauville–Mukai system on a Fourier–Mukai partner of . We also show that when the degree of the surface is small our method can be used to find all birational models of the Hilbert scheme.

Conjectural Criteria for the Most Singular Points of the Hilbert Schemes of Points

We provide conjectural necessary and (separately) sufficient conditions for the Hilbert scheme of points of a given length to have the maximum dimension tangent space at a point. The sufficient condition is claimed for 3D and reduces the original problem to a problem in convex geometry. Proving either of the two conjectural statements will in particular resolve a long-standing conjecture by Briançon and Iarrobino back in the ’70s for the case of the powers of the maximal ideal. Furthermore, for specific classes of lengths, we conjecturally classify points satisfying the conjectural sufficient conditions. This in particular (conjecturally) provides many new explicit families of examples of maximum dimension tangent space at a point of the Hilbert schemes of points of lengths strictly between two consecutive tetrahedral numbers ( 3 + k 3 ) .

Link splitting deformation of colored Khovanov–Rozansky homology

AbstractWe introduce a multiparameter deformation of the triply‐graded Khovanov–Rozansky homology of links colored by one‐column Young diagrams, generalizing the “‐ified” link homology of Gorsky–Hogancamp and work of Cautis–Lauda–Sussan. For each link component, the natural set of deformation parameters corresponds to interpolation coordinates on the Hilbert scheme of the plane. We extend our deformed link homology theory to braids by introducing a monoidal dg 2‐category of curved complexes of type A singular Soergel bimodules. Using this framework, we promote to the curved setting the categorical colored skein relation from our recent joint work and also the notion of splitting map for the colored full twists on two strands. As applications, we compute the invariants of colored Hopf links in terms of ideals generated by Haiman determinants and use these results to establish general link splitting properties for our deformed, colored, triply‐graded link homology. Informed by this, we formulate several conjectures that have implications for the relation between (colored) Khovanov–Rozansky homology and Hilbert schemes.

Rank Zero Segre Integrals on Hilbert Schemes of Points on Surfaces

Abstract The generating function of the Segre integrals on Hilbert schemes of points on a surface $X$ can be determined by five universal series $A_{0}(z)$, $A_{1}(z)$, $A_{2}(z)$, $A_{3}(z)$, $A_{4}(z)$, due to the result of Ellingsrud–Göttsche–Lehn. These five series do not depend on the surface $X$ and depend on the element $\alpha \in K(X)$, to which the Segre integrals are associated, only through the rank. Marian–Oprea–Pandharipande have determined $A_{0}(z),A_{1}(z),A_{2}(z)$ for all ranks. For rank 0, it is easy to see $A_{4}(z)=1$. Marian–Oprea–Pandharipande also conjectured that $A_{3}(z)=A_{0}(z)A_{1}(z)$ for rank 0. We prove this conjecture by showing that when $X$ is the projective plan, the Segre integrals associated to the structure sheaf of a curve in the anti-canoncial class are all zero. Hence, the rank zero Segre integrals on the Hilbert schemes of points for all surfaces are determined.

Calibrated representations of the double Dyck path algebra

AbstractThe double Dyck path algebra $$\mathbb {A}_{q,t}$$ A q , t and its polynomial representation first arose as a key figure in the proof of the celebrated Shuffle Theorem of Carlsson and Mellit. A geometric formulation for an equivalent algebra $$\mathbb {B}_{q,t}$$ B q , t was then given by the second author and Carlsson and Mellit using the K-theory of parabolic flag Hilbert schemes. In this article, we initiate the systematic study of the representation theory of the double Dyck path algebra $$\mathbb {B}_{q,t}$$ B q , t . We define a natural extension of this algebra and study its calibrated representations. We show that the polynomial representation is calibrated, and place it into a large family of calibrated representations constructed from posets satisfying certain conditions. We also define tensor products and duals of these representations, thus proving (under suitable conditions) the category of calibrated representations is generically monoidal. As an application, we prove that tensor powers of the polynomial representation can be constructed from the equivariant K-theory of parabolic Gieseker moduli spaces.

On the Chow theory of Quot schemes of locally free quotients

We prove a formula for the Chow groups of Quot schemes that resolve degeneracy loci of a map between vector bundles, under expected dimension conditions. This formula simultaneously generalizes the formulas for projective bundles, Grassmannian bundles, blowups, Cayley's trick, projectivizations, flops from Springer-type resolutions, and Grassmannian-type flips and flops. We also apply the formula to study the Chow groups of (i) blowups of determinantal ideals; (ii) moduli spaces of linear series on curves; and (iii) (nested) Hilbert schemes of points on surfaces.

Crepant Resolutions of Stratified Varieties via Gluing

Abstract Let $X$ be a variety with a stratification ${\mathcal{S}}$ into smooth locally closed subvarieties such that $X$ is locally a product along each stratum (e.g., a symplectic singularity). We prove that assigning to each open subset $U \subset X$ the set of isomorphism classes of locally projective crepant resolutions of $U$ defines an ${\mathcal{S}}$-constructible sheaf of sets. Thus, for each stratum $S$ and basepoint $s \in S$, the fundamental group acts on the set of germs of projective crepant resolutions at $s$, leaving invariant the germs extending to the entire stratum. Global locally projective crepant resolutions correspond to compatible such choices for all strata. For example, if the local projective crepant resolutions are unique, they automatically glue uniquely. We give criteria for a locally projective crepant resolution $\rho : \tilde X \to X$ to be globally projective. We show that the sheafification of the presheaf $U \mapsto \operatorname{Pic}(\rho ^{-1}(U)/U)$ of relative Picard classes is also constructible. The resolution is globally projective only if there exist local relatively ample bundles whose classes glue to a global section of this sheaf. The obstruction to lifting this section to a global ample line bundle is encoded by a gerbe on the singularity $X$. We show the gerbes are automatically trivial if $X$ is a symplectic quotient singularity. Our main results hold in the more general setting of partial crepant resolutions, that need not have smooth source. We apply the theory to symmetric powers and Hilbert schemes of surfaces with du Val singularities, finite quotients of tori, multiplicative, and Nakajima quiver varieties, as well as to canonical three-fold singularities.

Motivic zeta functions of the Hilbert schemes of points on a surface

Let K be a discretely-valued field. Let X\rightarrow\mathrm{Spec}K be a surface with trivial canonical bundle. In this paper we construct a weak Néron model of the schemes \mathrm{Hilb}^{n}(X) over the ring of integers R\subseteq K . We exploit this construction in order to compute the motivic zeta function of \mathrm{Hilb}^{n}(X) in terms of the motivic zeta functions of X and of its base-changes with respect to the totally ramified extensions of K . We determine the poles of Z_{\mathrm{Hilb}^n(X)} and study its monodromy property, showing that if the monodromy conjecture holds for X then it holds for \mathrm{Hilb}^{n}(X) too.

The spine of the \(T\)-graph of the Hilbert scheme of points in the plane

The spine of the \(T\)-graph of the Hilbert scheme of points in the plane

Components of the Hilbert scheme of smooth projective curves using ruled surfaces II: existence of non-reduced components

Components of the Hilbert scheme of smooth projective curves using ruled surfaces II: existence of non-reduced components

Construction and deformations of Calabi–Yau 3-folds in codimension 4

We construct polarized Calabi–Yau 3-folds with at worst isolated canonical orbifold points in codimension 4 that can be described in terms of the equations of the Segre embedding of P2×P2 in P8. We investigate the existence of other deformation families in their Hilbert scheme by either studying Tom and Jerry degenerations or by comparing their Hilbert series with those of existing low codimension Calabi–Yau 3-folds. Among other interesting results, we find a family of Calabi–Yau 3-fold with five distinct Tom and Jerry deformation families, a phenomenon not seen for Q-Fano 3-folds. We compute the Hodge numbers of P2×P2 Calabi–Yau 3-folds and corresponding manifolds obtained by performing crepant resolutions. We obtain a manifold with a pair of Hodge numbers that does not appear in the famously known list of 30108 distinct Hodge pairs of Kruzer–Skarke, in the list of 7890 distinct Hodge pairs corresponding to complete intersections in the product of projective spaces and in Hodge paris obtained from Calabi–Yau 3-folds having low codimension embeddings in weighted projective spaces.

Non-reduced Components of the Hilbert Scheme of Curves Using Triple Covers

Non-reduced Components of the Hilbert Scheme of Curves Using Triple Covers

Hilbert schemes of points on Calabi–Yau 4-folds via wall-crossing

Gross–Joyce–Tanaka [43] proposed a wall-crossing conjecture for Calabi–Yau fourfolds. Assuming it, we prove the conjecture of Cao–Kool [14] for 0-dimensional sheaf-counting invariants on projective Calabi–Yau 4-folds. From it, we extract the full topological information contained in the virtual fundamental classes of Hilbert schemes of points which turns out to be equivalent to the data of all descendent integrals. As a consequence, we can express many generating series of invariants in terms of explicit universal power series.i)On C4, Nekrasov proposed invariants with a conjectured closed form [72]. We show that an analog of his formula holds for compact Calabi–Yau 4-folds satisfying the wall-crossing conjecture.ii)We notice a relationship to corresponding generating series for Quot schemes on elliptic surfaces which are also governed by a wall-crossing formula. This leads to a Segre–Verlinde correspondence for Calabi–Yau fourfolds.

Rational Hodge isometries of hyper-Kähler varieties of type are algebraic

Let $X$ and $Y$ be compact hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of length $n$ subschemes of a $K3$ surface. A class in $H^{p,p}(X\times Y,{\mathbb {Q}})$ is an analytic correspondence, if it belongs to the subring generated by Chern classes of coherent analytic sheaves. Let $f:H^2(X,{\mathbb {Q}})\rightarrow H^2(Y,{\mathbb {Q}})$ be a rational Hodge isometry with respect to the Beauville–Bogomolov–Fujiki pairings. We prove that $f$ is induced by an analytic correspondence. We furthermore lift $f$ to an analytic correspondence $\tilde {f}: H^*(X,{\mathbb {Q}})[2n]\rightarrow H^*(Y,{\mathbb {Q}})[2n]$ , which is a Hodge isometry with respect to the Mukai pairings and which preserves the gradings up to sign. When $X$ and $Y$ are projective, the correspondences $f$ and $\tilde {f}$ are algebraic.

Positroid Catalan numbers

Given a (bounded affine) permutation f f , we study the positroid Catalan number C f C_f defined to be the torus-equivariant Euler characteristic of the associated open positroid variety. We introduce a class of repetition-free permutations and show that the corresponding positroid Catalan numbers count Dyck paths avoiding a convex subset of the rectangle. We show that any convex subset appears in this way. Conjecturally, the associated q , t q,t -polynomials coincide with the generalized q , t q,t -Catalan numbers that recently appeared in relation to the shuffle conjecture, flag Hilbert schemes, and Khovanov–Rozansky homology of Coxeter links.

The operator rings of topological symmetric orbifolds and their large N limit

We compute the structure constants of topological symmetric orbifold theories up to third order in the large N expansion. The leading order structure constants are dominated by topological metric contractions. The first order interactions are single cycles joining while at second order we can have double joining as well as splitting. At third order, single cycle joining obtains genus one contributions. We also compute illustrative small N structure constants. Our analysis applies to all second quantized Frobenius algebras, a large class of algebras that includes the cohomology ring of the Hilbert scheme of points on K3 among many others. We point out interesting open questions that our results raise.

Stratified bundles on the Hilbert Scheme of n points

Stratified bundles on the Hilbert Scheme of n points