Punctual Hilbert Scheme Research Articles

Q,t$q,t$‐Catalan measures

AbstractWe introduce the ‐Catalan measures, a sequence of piece‐wise polynomial measures on . These measures are defined in terms of suitable area, dinv, and bounce statistics on continuous families of paths in the plane, and have many combinatorial similarities to the ‐Catalan numbers. Our main result realizes the ‐Catalan measures as a limit of higher ‐Catalan numbers as . We also give a geometric interpretation of the ‐Catalan measures. They are the Duistermaat–Heckman measures of the punctual Hilbert schemes parametrizing subschemes of supported at the origin.

Brill–Noether Theory of Hilbert Schemes of Points on Surfaces

Abstract We show that Brill–Noether loci in Hilbert scheme of points on a smooth connected surface $S$ are non-empty whenever their expected dimension is positive and that they are irreducible and have expected dimensions. More precisely, we consider the loci of pairs $(I, s)$, where $I$ is an ideal that locally at the point $s$ of $S$ needs a given number of generators. We give two proofs. The first uses Iarrobino’s description [9] of the Hilbert–Samuel stratification of local punctual Hilbert schemes, and the second is based on induction via birational relationships between different Brill–Noether loci given by nested Hilbert schemes.

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.

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.

Gröbner cells of punctual Hilbert schemes in dimension two

We begin with a comprehensive discussion of the punctual Hilbert scheme of the regular two-dimensional local ring in terms of the Gröbner cells. These schemes are the most degenerate fibers of the Grothendieck-Deligne norm map (the Hilbert-Chow morphism), playing an important role in the study of Hilbert schemes of smooth surfaces. They are generally singular, but their Gröbner cells are affine spaces; they admit an explicit parametrization due to Conca and Valla. We use this to obtain the Gröbner decomposition of compactified Jacobians of plane curve singularities, which is non-trivial even for the generalized Jacobians (principal ideals only). One of the applications is the topological invariance of certain variants of compactified Jacobians and the corresponding motivic superpolynomials for analytic deformations of quasi-homogeneous plane curve singularities and some similar families.

On the Equations Defining Some Hilbert Schemes

We work out details of the extrinsic geometry for two Hilbert schemes of some contemporary interest: the Hilbert scheme \(\text {Hilb}^{2} \mathbb {P}^{2}\) of two points on \(\mathbb {P}^{2}\) and the dense open set parametrizing non-planar clusters in the punctual Hilbert scheme \(\text {Hilb}^{4}_{0}(\mathbb {A}^{3})\) of clusters of length four on \(\mathbb {A}^{3}\) with support at the origin. We find explicit equations in projective, respectively affine, embeddings for these spaces. In particular, we answer a question of Bernd Sturmfels who asked for a description of the latter space that is amenable to further computations. While the explicit equations we find are controlled in a precise way by the representation theory of SL3, our arguments also rely on computer algebra.

Donaldson–Thomas invariants, linear systems and punctual Hilbert schemes

We study certain DT invariants arising from stable coherent sheaves in a nonsingular projective threefold supported on the members of a linear system of a fixed line bundle. When the canonical bundle of the threefold satisfies certain positivity conditions, we relate the DT invariants to Carlsson-Okounkov formulas for the twisted Euler's number of the punctual Hilbert schemes of nonsingular surfaces, and conclude they have a modular property.

Hilbert schemes of K3 surfaces, generalized Kummer, and cobordism classes of hyper-Kähler manifolds

We prove that the complex cobordism class of any hyper-K\{a}hler manifold of dimension $2n$ is a unique combination with rational coefficients of classes of products of punctual Hilbert schemes of $K3$ surfaces. We also prove a similar result using the generalized Kummer varieties instead of punctual Hilbert schemes. As a key step, we establish a closed formula for the top Chern character of their tangent bundles.

Generalized punctual Hilbert schemes and 𝔤-complex structures

We define and analyze various generalizations of the punctual Hilbert scheme of the plane, associated to complex or real Lie algebras. Out of these, we construct new geometric structures on surfaces whose moduli spaces share multiple properties with Hitchin components, and which are conjecturally homeomorphic to them. For simple complex Lie algebras, this generalizes the higher complex structure. For real Lie algebras, this should give an alternative description of the Hitchin–Kostant–Rallis section.

Punctual Hilbert schemes for Kleinian singularities as quiver varieties

For a finite subgroup $\Gamma\subset \mathrm{SL}(2,\mathbb{C})$ and $n\geq 1$, we construct the (reduced scheme underlying the) Hilbert scheme of $n$ points on the Kleinian singularity $\mathbb{C}^2/\Gamma$ as a Nakajima quiver variety for the framed McKay quiver of $\Gamma$, taken at a specific non-generic stability parameter. We deduce that this Hilbert scheme is irreducible (a result previously due to Zheng), normal, and admits a unique symplectic resolution. More generally, we introduce a class of algebras obtained from the preprojective algebra of the framed McKay quiver by a process called cornering, and we show that fine moduli spaces of cyclic modules over these new algebras are isomorphic to quiver varieties for the framed McKay quiver and certain non-generic choices of stability parameter.

Primary Ideals and Their Differential Equations

An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in a polynomial ring. We characterize primary ideals in terms of PDE, punctual Hilbert schemes, relative Weyl algebras, and the join construction. Solving the PDE described by a primary ideal amounts to computing Noetherian operators in the sense of Ehrenpreis and Palamodov. We develop new algorithms for this task, and we present efficient implementations.

Fourier–Mukai transformation and logarithmic Higgs bundles on punctual Hilbert schemes

Given a vector bundle E on a smooth projective curve or surface X carrying the structure of a V-twisted Hitchin pair for some vector bundle V, we observe that the associated tautological bundle E[n] on the punctual Hilbert scheme of points X[n] has an induced structure of a ((V∨)[n])∨-twisted Hitchin pair, where (V∨)[n] is a vector bundle on X[n] constructed using the dual V∨ of V. In particular, a Higgs bundle on X induces a logarithmic Higgs bundle on the Hilbert scheme X[n]. We then show that the known results on stability of tautological bundles and reconstruction from tautological bundles generalize to tautological Hitchin pairs.

Higher Complex Structures

Abstract We introduce and analyze a new geometric structure on topological surfaces generalizing the complex structure. To define this so-called higher complex structure, we use the punctual Hilbert scheme of the plane. The moduli space of higher complex structures is defined and is shown to be a generalization of the classical Teichmüller space. We give arguments for the conjectural isomorphism between the moduli space of higher complex structures and Hitchin’s component.

Smoothable Gorenstein Points Via Marked Schemes and Double-generic Initial Ideals

Over an infinite field K with we investigate smoothable Gorenstein K-points in a punctual Hilbert scheme and obtain the following results: (i) every K-point defined by local Gorenstein K-algebras with Hilbert function is smoothable (this is the only case non treated in the range considered by Iarrobino and Kanev in 1999; (ii) the Hilbert scheme has at least five irreducible components. As a byproduct of our study about we also find a new elementary component in We face the problem from a new point of view, that is based on properties of double-generic initial ideals and of marked schemes. The properties of marked schemes give us a simple method to compute the Zariski tangent space to a Hilbert scheme at a given K-point, which is very useful in this context. We also test our tools to find the already known result that K-points defined by local Gorenstein K-algebras with Hilbert function are smoothable. The problem that we consider is strictly related to the study of the irreducibility of the Gorenstein locus in a Hilbert scheme and, more generally, of the irreducibility of a Hilbert scheme, which is a very open question.

Conjugacy classes of commuting nilpotents

We consider the space M q , n \mathcal {M}_{q,n} of regular q q -tuples of commuting nilpotent endomorphisms of k n k^n modulo simultaneous conjugation. We show that M q , n \mathcal {M}_{q,n} admits a natural homogeneous space structure, and that it is an affine space bundle over P q − 1 {\mathbb {P}}^{q-1} . A closer look at the homogeneous structure reveals that, over C {\mathbb {C}} and with respect to the complex*1pt topology, M q , n \mathcal {M}_{q,n} is a smooth vector bundle over P q − 1 {\mathbb {P}}^{q-1} . We prove that, in this case, M q , n \mathcal {M}_{q,n} is diffeomorphic to a direct sum of twisted tangent bundles. We also prove that M q , n \mathcal {M}_{q,n} possesses a universal property and represents a functor of ideals, and we use it to identify M q , n \mathcal {M}_{q,n} with an open subscheme of a punctual Hilbert scheme. Using a result of A. Iarrobino’s, we show that M q , n → P q − 1 \mathcal {M}_{q,n} \to {\mathbb {P}}^{q-1} is not a vector bundle, hence giving a family of affine space bundles that are not vector bundles.

Constructions of $k$-regular maps using finite local schemes

A continuous map \mathbb R^m \to \mathbb R^N or \mathbb C^m \to \mathbb C^N is called k -regular if the images of any k points are linearly independent. Given integers m and k a problem going back to Chebyshev and Borsuk is to determine the minimal value of N for which such maps exist. The methods of algebraic topology provide lower bounds for N , but there are very few results on the existence of such maps for particular values m and k . Using methods of algebraic geometry we construct k -regular maps. We relate the upper bounds on N with the dimension of the locus of certain Gorenstein schemes in the punctual Hilbert scheme. The computations of the dimension of this family is explicit for k \leq 9 , and we provide explicit examples for k \leq 5 . We also provide upper bounds for arbitrary m and k .

PARABOLIC CONJUGATION AND COMMUTING VARIETIES

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of the general linear group on the variety of nilpotent matrices in its Lie algebra. Lie-theoretically, it is natural to wonder about the number of orbits of this action. We translate the setup to a representation-theoretic one and obtain a finiteness criterion which classifies all actions with only a finite number of orbits over an arbitrary infinite field. These results are applied to commuting varieties and nested punctual Hilbert schemes.

The tangent space of the punctual Hilbert scheme

The purpose of this paper is to study the Zariski tangent space of the punctual Hilbert scheme parametrizing subschemes of a smooth surface which are supported at a single point. We give a lower bound on the dimension of the tangent space in general and show the bound is sharp for subschemes of the affine plane cut out by monomials. Furthermore for monomial subschemes we give an explicit combinatorial formula for the dimension of the tangent space.

Mori cones of holomorphic symplectic varieties of K3 type

We determine the Mori cone of holomorphic symplectic varieties deformation equivalent to the punctual Hilbert scheme on a K3 surface. Our description is given in terms of Markman's extended Hodge lattice.

The punctual Hilbert schemes for the curve singularities of type A_{2d}

The punctual Hilbert schemes for the curve singularities of type A_{2d}