Hilbert Scheme Research Articles

3D Mirror Symmetry for Instanton Moduli Spaces

We prove that the Hilbert scheme of k points on {mathbb {C}}^2 (hbox {Hilb}^k[{mathbb {C}}^2]) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the {mathbb {C}}^times _hbar -action. First, we find a two-parameter family X_{k,l} of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of hbox {Hilb}^k[{mathbb {C}}^2] is obtained via direct limit llongrightarrow infty and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted hbar -opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-N sheaves on {mathbb {P}}^2 with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.

A Combinatorial Proof of Buryak-Feigin-Nakajima

Buryak, Feigin and Nakajima computed a generating function for a family of partition statistics by using the geometry of the $\mathbb{Z}/c\mathbb{Z}$ fixed point sets in the Hilbert scheme of points on $\mathbb{C}^2$. Loehr and Warrington had already shown how a similar observation by Haiman using the geometry of the Hilbert scheme of points on $\mathbb{C}^2$ can be made purely combinatorial. We extend Loehr and Warrington's techniques to also account for cores and quotients. As a consequence, we obtain a purely combinatorial proof of Buryak, Feigin, and Nakajima's result.More precisely, we define a family of partition statistics $\{h_{x,c}^+, x\in (0,\infty]\}$ and give a combinatorial proof that for all $x$ and all positive integers $c$,$$\sum q^{|\lambda|}t^{h_{x,c}^+(\lambda)}=q^{|\mu|}\prod_{i\geq 1}\frac{1}{(1-q^{ic})^{c-1}}\prod_{j\geq 1}\frac{1}{1-q^{jc}t},$$where the sum ranges over all partitions $\lambda$ with $c$-core $\mu$.

A local study of the fiber-full scheme

We study some of the local properties of the fiber-full scheme, which is a fine moduli space that generalizes the Hilbert scheme by parametrizing closed subschemes with prescribed cohomological data. As a consequence, we provide sufficient conditions for cohomology to remain constant under Gröbner degenerations. We also describe a tangent-obstruction theory for the fiber-full scheme in analogy with the one for the Hilbert scheme.

Perverse-Hodge complexes for Lagrangian fibrations

Perverse-Hodge complexes are objects in the derived category of coherent sheaves obtained from Hodge modules associated with Saito's decomposition theorem. We study perverse-Hodge complexes for Lagrangian fibrations and propose a symmetry between them. This conjectural symmetry categorifies the "Perverse = Hodge" identity of the authors and specializes to Matsushita's theorem on the higher direct images of the structure sheaf. We verify our conjecture in several cases by making connections with variations of Hodge structures, Hilbert schemes, and Looijenga-Lunts-Verbitsky Lie algebras.

Superconformal quantum mechanics and growth of sheaf cohomology

We give a geometric interpretation for superconformal quantum mechanics defined on a hyper-Kähler cone which has an equivariant symplectic resolution. BPS states are identified with certain twisted Dolbeault cohomology classes on the resolved space and their index degeneracies can also be related to the Euler characteristic computed in equivariant sheaf cohomology. In the special case of the Hilbert scheme of K points on ℂ2, we obtain a rigorous estimate for the exponential growth of the index degeneracies of BPS states as K → ∞. This growth serves as a toy model for our recently proposed duality between a seven dimensional black hole and superconformal quantum mechanics.

Higher Gaussian Maps on K3 Surfaces

Abstract We give sufficient conditions for the surjectivity of higher Gaussian maps on a polarized K3 surface. As an application, we show that the $k$-th Gaussian map for a general curve of genus $g$ (that depends quadratically with $k$) is surjective. Along the proof, we also exhibit an ampleness criterion for divisors in the Hilbert scheme of two points of a K3 surface.

Deformations of half-canonical Gorenstein curves in codimension four

Recent work of Ablett (2021) and Kapustka, Kapustka, Ranestad, Schenck, Stillman and Yuan (2021) outlines a number of constructions for singular Gorenstein codimension four varieties. Earlier work of Coughlan, Gołȩbiowski, Kapustka and Kapustka (2016) details a series of nonsingular Gorenstein codimension four constructions with different Betti tables. In this paper we exhibit a number of flat deformations between Gorenstein codimension four varieties in the same Hilbert scheme, realising many of the singular varieties as specialisations of the earlier nonsingular varieties.

The Hilbert scheme of infinite affine space and algebraic K-theory

We study the Hilbert scheme \mathrm{Hilb}_{d}(\mathbf{A}^{\infty}) from an \mathbf{A}^{1} -homotopical viewpoint and obtain applications to algebraic K-theory. We show that the Hilbert scheme \mathrm{Hilb}_{d}(\mathbf{A}^{\infty}) is \mathbf{A}^{1} -equivalent to the Grassmannian of (d-1) -planes in \mathbf{A}^{\infty} . We then describe the \mathbf{A}^{1} -homotopy type of \mathrm{Hilb}_{d}(\mathbf{A}^{n}) in a certain range, for n large compared to d . For example, we compute the integral cohomology of \mathrm{Hilb}_{d}(\mathbf{A}^{n})(\mathbf{C}) in a range. We also deduce that the forgetful map \mathcal{FF}\mathrm{lat}\to\mathcal{V}\mathrm{ect} from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an \mathbf{A}^{1} -equivalence after group completion. This implies that the moduli stack \mathcal{FF}\mathrm{lat} , viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum \mathrm{kgl} representing algebraic K-theory. Combining our techniques with the recent work of Bachmann, we obtain Hilbert scheme models for the \mathrm{kgl} -homology of smooth proper schemes over a perfect field.

Effective divisors and Newton-Okounkov bodies of Hilbert schemes of points on toric surfaces

We compute the (unbounded) Newton-Okounkov body of the Hilbert scheme of points on C2. We obtain an upper bound for the Newton-Okounkov body of the Hilbert scheme of points on any smooth toric surface. We conjecture that this upper bound coincides with the exact Newton-Okounkov body for the Hilbert schemes of points on P2,P1×P1, and Hirzebruch surfaces. These results imply upper bounds for the effective cones of these Hilbert schemes, which are also conjecturally sharp in the above cases.

On integral points of some Fano threefolds and their Hilbert schemes of lines and conics

We prove some density results for integral points on affine open sets of Fano threefolds. For instance, let X^o=mathbb P^3{setminus } D where D is the union of two quadrics such that their intersection contains a smooth conic, or the union of a smooth quadric surface and two planes, or the union of a smooth cubic surface V and a plane Pi such that the intersection Vcap Pi contains a line. In all these cases we show that the set of integral points of X^o is potentially dense. We apply the above results to prove that integral points are potentially dense in some log-Fano or in some log-Calabi-Yau threefold.

Curve counting and S-duality

We work on a projective threefold $X$ which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda, such as $\mathbb P^3$ or the quintic threefold. We prove certain moduli spaces of 2-dimensional torsion sheaves on $X$ are smooth bundles over Hilbert schemes of ideal sheaves of curves and points in $X$. When $X$ is Calabi-Yau this gives a simple wall crossing formula expressing curve counts (and so ultimately Gromov-Witten invariants) in terms of counts of D4-D2-D0 branes. These latter invariants are predicted to have modular properties which we discuss from the point of view of S-duality and Noether-Lefschetz theory.

A CATEGORICAL QUANTUM TOROIDAL ACTION ON THE HILBERT SCHEMES

AbstractWe categorify the commutation of Nakajima’s Heisenberg operators $P_{\pm 1}$ and their infinitely many counterparts in the quantum toroidal algebra $U_{q_1,q_2}(\ddot {gl_1})$ acting on the Grothendieck groups of Hilbert schemes from [10, 24, 26, 32]. By combining our result with [26], one obtains a geometric categorical $U_{q_1,q_2}(\ddot {gl_1})$ action on the derived category of Hilbert schemes. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semidivisorial log terminal singularities.

Relative VGIT and an Application to Degenerations of Hilbert Schemes

We generalize the classical semicontinuity theorem for GIT (semi)stable loci under variations of linearizations to a relative situation of an equivariant projective morphism X→S over an affine base S. As an application to moduli problems, we consider degenerations of Hilbert schemes and give a conceptual interpretation of the (semi)stable loci of the degeneration families constructed in [GHH19].

Hilbert schemes with two Borel-fixed points in arbitrary characteristic

We extend the recent classification of Hilbert schemes with two Borel-fixed points to positive characteristic. We accomplish this by synthesizing Reeves’ algorithm for generating strongly stable ideals with the characteristic free properties of Borel-fixed ideals and our previous work classifying Hilbert schemes with unique Borel-fixed points in any characteristic.

Rational Singularities of Nested Hilbert Schemes

Abstract The Hilbert scheme of points $\textrm {Hilb}^{n}(S)$ of a smooth surface $S$ is a well-studied parameter space, lying at the interface of algebraic geometry, commutative algebra, representation theory, combinatorics, and mathematical physics. The foundational result is a classical theorem of Fogarty, stating that $\textrm {Hilb}^{n}(S)$ is a smooth variety of dimension $2n$. In recent years there has been growing interest in a natural generalization of $\textrm {Hilb}^{n}(S)$, the nested Hilbert scheme$\textrm {Hilb}^{(n_{1}, n_{2})}(S)$, which parametrizes nested pairs of zero-dimensional subschemes $Z_{1} \supseteq Z_{2}$ of $S$ with $\deg Z_{i}=n_{i}$. In contrast to Fogarty’s theorem, $\textrm {Hilb}^{(n_{1}, n_{2})}(S)$ is almost always singular, and very little is known about its singularities. In this paper, we aim to advance the knowledge of the geometry of these nested Hilbert schemes. Work by Fogarty in the 70’s shows that $\textrm {Hilb}^{(n,1)}(S)$ is a normal Cohen–Macaulay variety, and Song more recently proved that it has rational singularities. In our main result, we prove that the nested Hilbert scheme $\textrm {Hilb}^{(n,2)}(S)$ has rational singularities. We employ an array of tools from commutative algebra to prove this theorem. Using Gröbner bases, we establish a connection between $\textrm {Hilb}^{(n,2)}(S)$ and a certain variety of matrices with an action of the general linear group. This variety of matrices plays a central role in our work, and we analyze it by various algebraic techniques, including the Kempf–Lascoux–Weyman technique of calculating syzygies, square-free Gröbner degenerations, and the Stanley–Reisner correspondence. Along the way, we also obtain results on classes of irreducible and reducible nested Hilbert schemes, dimension of singular loci, and $F$-singularities in positive characteristic.

On the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces

AbstractThe aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let X be a nonsingular irreducible complex surface, and let E be a vector bundle of rank n on X. We use the m-elementary transformation of E at a point $x \in X$ to show that there exists an embedding from the Grassmannian variety $\mathbb{G}(E_x,m)$ into the moduli space of torsion-free sheaves $\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)$ which induces an injective morphism from $X \times M_{X,H}(n;\,c_1,c_2)$ to $Hilb_{\, \mathfrak{M}_{X,H}(n;\,c_1,c_2+m)}$ .

Existence and reducibility of the Hilbert scheme of smooth and linearly normal curves in Pr of relatively high degrees

Let H d , g , r be the Hilbert scheme parametrizing smooth irreducible and non-degenerate curves of degree d and genus g in P r . We denote by H d , g , r L the union of those components of H d , g , r whose general element is linearly normal. In this article we show that H d , g , r L ( d ≥ g + r − 3 ) is non-empty in a certain optimal range of triples ( d , g , r ) and is empty outside the range. This settles the existence (or non-emptiness if one prefers) of the Hilbert scheme H d , g , r L of linearly normal curves of degree d and genus g in P r for g + r − 3 ≤ d ≤ g + r , r ≥ 3 . We also determine all the triples ( d , g , r ) with g + r − 3 ≤ d ≤ g + r for which H d , g , r L is reducible (or irreducible).

Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes

These notes aim at providing a complete and systematic account of some foundational aspects of algebraic supergeometry, namely, the extension to the geometry of superschemes of many classical notions, techniques and results that make up the general backbone of algebraic geometry, most of them originating from Grothendieck's work. In particular, we extend to algebraic supergeometry such notions as projective and proper morphisms, finiteness of the cohomology, vector and projective bundles, cohomology base change, semicontinuity theorems, relative duality, Castelnuovo-Mumford regularity, flattening, Hilbert and Quot schemes, faithfully flat descent, quotient étale relations (notably, Picard schemes), among others. Some results may be found elsewhere, and, in particular, there is some overlap with [51]. However, many techniques and constructions are presented here for the first time, notably, a first development of Grothendieck relative duality for proper morphisms of superschemes, the construction of the Hilbert superscheme in a more general situation than the one already known (which in particular allows one to treat the case of sub-superschemes of supergrassmannians), and a rigorous construction of the Picard superscheme for a locally superprojective morphism of noetherian superschemes with geometrically integral fibres. Moreover, some of the proofs given here are new as well, even when restricted to ordinary schemes. In a final section we construct a period map from an open substack of the moduli of proper and smooth supercurves to the moduli stack of principally polarized abelian schemes.

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.