Hilbert Scheme Research Articles

K-Theoretic Descendent Series for Hilbert Schemes of Points on Surfaces

We study the holomorphic Euler characteristics of tautological sheaves on Hilbert schemes of points on surfaces. In particular, we establish the rationality of K-theoretic descendent series. Our approach is to control equivariant holomorphic Euler characteristics over the Hilbert scheme of points on the affine plane. To do so, we slightly modify a Macdonald polynomial identity of Mellit.

Universality of Descendent Integrals over Moduli Spaces of Stable Sheaves on K3 Surfaces

We interprete results of Markman on monodromy operators as a universality statement for descendent integrals over moduli spaces of stable sheaves on K3 surfaces. This yields effective methods to reduce these descendent integrals to integrals over the punctual Hilbert scheme of the K3 surface. As an application we establish the higher rank Segre-Verlinde correspondence for K3 surfaces as conjectured by Göttsche and Kool.

Compact hyper-Kähler categories

We define and study the notion of hyper-Kahler category. On the theoretical side, we focus on construction techniques and deformation theory of such categories. We also study in details some examples : non-commutative Hilbert schemes of points on a K3 surface and a categorical resolution of a relative compactified Prymian constructed by Markushevich and Tikhomirov. *

Gromov–Witten invariants of Hilbert schemes of two points on elliptic surfaces

In this paper, we study the Gromov–Witten theory of the Hilbert scheme [Formula: see text] of two points on an elliptic surface [Formula: see text]. Assume that [Formula: see text] contains an element supported on the smooth fibers of [Formula: see text]. By analyzing the degeneracy locus and localized virtual cycle arising from the cosection localization theory of Kiem and Li [Y. Kiem and J. Li, Gromov–Witten invariants of varieties with holomorphic 2-forms, preprint; Y. Kiem and J. Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26 (2013) 1025–1050], we determine the [Formula: see text]-point genus-[Formula: see text] Gromov–Witten invariant [Formula: see text] up to some rational number [Formula: see text] depending only on [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] is a smooth fiber of [Formula: see text], [Formula: see text] with [Formula: see text] being a fixed point, and [Formula: see text]. Moreover, we propose a conjecture regarding [Formula: see text], and prove that the conjecture is true for [Formula: see text] where [Formula: see text] is an elliptic curve and [Formula: see text] is a smooth curve.

Automorphisms of some K3 surfaces and Beauville involutions of their Hilbert schemes

Automorphisms of some K3 surfaces and Beauville involutions of their Hilbert schemes

Double nested Hilbert schemes and the local stable pairs theory of curves

We propose a variation of the classical Hilbert scheme of points, thedouble nested Hilbert scheme of points, which parametrizes flags of zero-dimensional subschemes whose nesting is dictated by a Young diagram. Over a smooth quasi-projective curve, we compute the generating series of topological Euler characteristic of these spaces, by exploiting the combinatorics of reversed plane partitions. Moreover, we realize this moduli space as the zero locus of a section of a vector bundle over a smooth ambient space, which therefore admits a virtual fundamental class. We apply this construction to the stable pair theory of a local curve, that is the total space of the direct sum of two line bundles over a curve. We show that the invariants localize to virtual intersection numbers on double nested Hilbert scheme of points on the curve, and that the localized contributions to the invariants are controlled by three universal series for every Young diagram, which can be explicitly determined after the anti-diagonal restriction of the equivariant parameters. Under the anti-diagonal restriction, the invariants are matched with the Gromov–Witten invariants of local curves of Bryan–Pandharipande, as predicted by the Maulik–Nekrasov–Okounkov–Pandharipande (MNOP) correspondence. Finally, we discuss$K$-theoretic refinements à la Nekrasov–Okounkov.

Wreath Macdonald polynomials at 𝑞=𝑡 as characters of rational Cherednik algebras

Using the theory of Macdonald [Symmetric functions and Hall polynomials, The Clarendon Press, Oxford University Press, New York, 1995], Gordon [Bull. London Math. Soc. 35 (2003), pp. 321–336] showed that the graded characters of the simple modules for the restricted rational Cherednik algebra by Etingof and Ginzburg [Invent. Math. 147 (2002), pp. 243–348] associated to the symmetric group S n \mathfrak {S}_n are given by plethystically transformed Macdonald polynomials specialized at q = t q=t . We generalize this to restricted rational Cherednik algebras of wreath product groups C ℓ ≀ S n C_\ell \wr \mathfrak {S}_n and prove that the corresponding characters are given by a specialization of the wreath Macdonald polynomials defined by Haiman in [Combinatorics, symmetric functions, and Hilbert schemes, Int. Press, Somerville, MA, 2003, pp. 39–111].

ON THE NUMBER OF 2-HOOKS AND 3-HOOKS OF INTEGER PARTITIONS

AbstractLet $p_t(a,b;n)$ denote the number of partitions of n such that the number of t-hooks is congruent to $a \bmod {b}$ . For $t\in \{2, 3\}$ , arithmetic progressions $r_1 \bmod {m_1}$ and $r_2 \bmod {m_2}$ on which $p_t(r_1,m_1; m_2 n + r_2)$ vanishes were established in recent work by Bringmann, Craig, Males and Ono [‘Distributions on partitions arising from Hilbert schemes and hook lengths’, Forum Math. Sigma10 (2022), Article no. e49] using the theory of modular forms. Here we offer a direct combinatorial proof of this result using abaci and the theory of t-cores and t-quotients.

Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points

We study tautological vector bundles over the Hilbert scheme of points on surfaces. For each K-trivial surface, we write down a simple criterion ensuring that the tautological bundles are big and nef, and illustrate it by examples. In the K3 case, we extend recent constructions and results of Bini, Boissi\`ere and Flamini from the Hilbert scheme of 2 and 3 points to an arbitrary number of points. Among the K-trivial surfaces, the case of Enriques surfaces is the most involved. Our techniques apply to other smooth projective surfaces, including blowups of K3s and minimal surfaces of general type, as well as to the punctual Quot schemes of curves.

A certified iterative method for isolated singular roots

In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root and we compute its multiplicity structure. More precisely, given a polynomial system f=(f1,…,fN)∈C[x1,…,xn]N, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of f such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so-called inverse system that describes the multiplicity structure at the root. We use α-theory test to certify the quadratic convergence, and to give bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria.

K-Theoretic DT/PT Correspondence for Toric Calabi–Yau 4-Folds

Recently, Nekrasov discovered a new “genus” for Hilbert schemes of points on {mathbb {C}}^4. We extend its definition to Hilbert schemes of curves and moduli spaces of stable pairs, and conjecture a K-theoretic DT/PT correspondence for toric Calabi–Yau 4-folds. We develop a K-theoretic vertex formalism, which allows us to verify our conjecture in several cases. Taking a certain limit of the equivariant parameters, we recover the cohomological DT/PT correspondence for toric Calabi–Yau 4-folds recently conjectured by the first two authors. Another limit gives a dimensional reduction to the K-theoretic DT/PT correspondence for toric 3-folds conjectured by Nekrasov–Okounkov. As an application of our techniques, we find a conjectural formula for the generating series of K-theoretic stable pair invariants of text {Tot}_{{mathbb {P}}^1}({mathcal {O}}(-1) oplus {mathcal {O}}(-1) oplus {mathcal {O}}). Upon dimensional reduction to the resolved conifold, we recover a formula which was recently proved by Kononov–Okounkov–Osinenko.

Elliptic Stable Envelopes of Affine Type A Quiver Varieties

Abstract We generalize Smirnov’s formula for the elliptic stable envelopes of the Hilbert scheme of points in $\mathbb {C}^2$ to the case of affine type $A$ Nakajima quiver varieties constructed with positive stability condition. We allow for arbitrary choices of polarization and a fairly general choice of chamber. This paper is a companion to the Maple code developed by the author, which implements the formulas described in this paper.

Elliptic quintics on cubic fourfolds, O'Grady 10, and Lagrangian fibrations

For a smooth cubic fourfold Y, we study the moduli space M of semistable objects of Mukai vector 2λ1+2λ2 in the Kuznetsov component of Y. We show that with a certain choice of stability conditions, M admits a symplectic resolution M˜, which is a smooth projective hyperkähler manifold, deformation equivalent to the 10-dimensional examples constructed by O'Grady. As applications, we show that a birational model of M˜ provides a hyperkähler compactification of the twisted family of intermediate Jacobians associated to Y. This generalizes the previous result of Voisin [58] in the very general case. We also prove that M˜ is the MRC quotient of the main component of the Hilbert scheme of quintic elliptic curves in Y, confirming a conjecture of Castravet.

Pursuing Quantum Difference Equations II: 3D mirror symmetry

Abstract Let $\textsf {X}$ and $\textsf {X}^{!}$ be a pair of symplectic varieties dual with respect to 3D mirror symmetry. The $K$-theoretic limit of the elliptic duality interface is an equivariant $K$-theory class $\mathfrak {m} \in K(\textsf {X}\times \textsf {X}^{!})$. We show that this class provides correspondences $$ \begin{align*} & \Phi_{\mathfrak{m}}: K(\textsf{X}) \leftrightarrows K(\textsf{X}^{!}) \end{align*}$$mapping the $K$-theoretic stable envelopes to the $K$-theoretic stable envelopes. This construction allows us to relate various representation theoretic objects of $K(\textsf {X})$, such as action of quantum groups, quantum dynamical Weyl groups, $R$-matrices, etc., to those for $K(\textsf {X}^{!})$. In particular, we relate the wall $R$-matrices of $\textsf {X}$ to the $R$-matrices of the dual variety $\textsf {X}^{!}$. As an example, we apply our results to $\textsf {X}=\textrm {Hilb}^{n}({{\mathbb {C}}}^2)$—the Hilbert scheme of $n$ points in the complex plane. In this case, we arrive at the conjectures of Gorsky and Negut from [10].

On the tangent space to the Hilbert scheme of points in 𝐏³

In this paper we study the tangent space to the Hilbert scheme H i l b d P 3 \mathrm {Hilb}^d \mathbf {P}^3 , motivated by Haiman’s work on H i l b d P 2 \mathrm {Hilb}^d \mathbf {P}^2 and by a long-standing conjecture of Briançon and Iarrobino [J. Algebra 55 (1978), pp. 536–544] on the most singular point in H i l b d P n \mathrm {Hilb}^d \mathbf {P}^n . For points parametrizing monomial subschemes, we consider a decomposition of the tangent space into six distinguished subspaces, and show that a fat point exhibits an extremal behavior in this respect. This decomposition is also used to characterize smooth monomial points on the Hilbert scheme. We prove the first Briançon-Iarrobino conjecture up to a factor of 4 3 \frac {4}{3} , and improve the known asymptotic bound on the dimension of H i l b d P 3 \mathrm {Hilb}^d \mathbf {P}^3 . Furthermore, we construct infinitely many counterexamples to the second Briançon-Iarrobino conjecture, and we also settle a weaker conjecture of Sturmfels in the negative.

Irreducibility and singularities of some nested Hilbert schemes

Let S be a smooth projective surface over C. We study the local and global geometry of the nested Hilbert scheme of points S[n,n+1,n+2]. In particular, we show that S[n,n+1,n+2] has klt singularities and compute its Picard group when h1(S,OS)=0. We also make some improvements on the codimension bounds of some incidence loci in S×S[n,n+1], which are used to establish irreducibility of S[n,n+1,n+2]. From the irreducibility of S[n,n+1,n+2], we deduce irreducibility for four other infinite families of nested Hilbert schemes. We give the first effective example of a reducible nested Hilbert scheme, which allows us to show that S[n1,…,nk] is reducible for k>22.

From Representations of the Rational Cherednik Algebra to Parabolic Hilbert Schemes via the Dunkl-Opdam Subalgebra

From Representations of the Rational Cherednik Algebra to Parabolic Hilbert Schemes via the Dunkl-Opdam Subalgebra

Pursuing quantum difference equations I: stable envelopes of subvarieties

Let X be a symplectic variety equipped with an action of a torus \({\mathsf {A}}\). Let \({\varvec{\nu }}_{b}\subset {\mathsf {A}}\) be a finite cyclic subgroup. We show that K-theoretic stable envelope of the fixed point set \(X^{{\varvec{\nu }}_{b}}\subset X\) can be obtained via a limit of the elliptic stable envelopes of X. An example of X given by the Hilbert scheme of points in the complex plane is considered in detail.

Virtual Segre and Verlinde numbers of projective surfaces

Recently, Marian–Oprea–Pandharipande established (a generalization of) Lehn's conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. Extending the work of Johnson, they provided a conjectural correspondence between Segre and Verlinde numbers. For surfaces with holomorphic 2-form, we propose conjectural generalizations of their results to moduli spaces of stable sheaves of any rank. Using Mochizuki's formula, we derive a universal function which expresses virtual Segre and Verlinde numbers of surfaces with holomorphic 2-form in terms of Seiberg–Witten invariants and intersection numbers on products of Hilbert schemes of points. We prove that certain canonical virtual Segre and Verlinde numbers of general type surfaces are topological invariants and we verify our conjectures in examples. The power series in our conjectures are algebraic functions, for which we find expressions in several cases and which are permuted under certain Galois actions. Our conjectures imply an algebraic analog of the Mariño–Moore conjecture for higher rank Donaldson invariants. For ranks 3 and 4, we obtain explicit expressions for Donaldson invariants in terms of Seiberg–Witten invariants.

Classifying the irreducible components of moduli stacks of torsion free sheaves on K3 surfaces and an application to Brill-Noether theory

In this article, we classify the irreducible components of moduli stacks of torsion free sheaves of rank 2 on K3 surfaces of Picard number 1. For ruled surfaces, the components of moduli stacks of torsion free sheaves were classified by Walter ([18]). Moreover, by virtue of our result, we classify the irreducible components of Brill-Noether loci of Hilbert schemes of points on K3 surfaces.