This is an expository account of Grothendieck's construction of Hilbert and Quot Schemes, following his talk `Techniques de construction et theoremes d'existence en geometrie algebriques IV : les schemas de Hilbert', Seminaire Bourbaki 221 (1960/61), together with further developments by Mumford and by Altman and Kleiman. Hilbert and Quot schemes are fundamental to modern Algebraic Geometry, in particular, for deformation theory and moduli constructions. These notes are based on a series of six lectures in the summer school `Advanced Basic Algebraic Geometry', held at the Abdus Salam International Centre for Theoretical Physics, Trieste, in July 2003.
We prove the algebraicity of the Hilbert functor, the Hilbert stack, the Quot functor and the stack of coherent sheaves on an algebraic stack X with (quasi-)finite diagonal without any finiteness assumptions on X. We also give similar results for Hom stacks and Weil restrictions.
Quot schemes of quotients of a trivial bundle of arbitrary rank on a nonsingular projective surface X carry perfect obstruction theories and virtual fundamental classes whenever the quotient sheaf has at most 1-dimensional support. The associated generating series of virtual Euler characteristics was conjectured to be a rational function when X is simply connected. We conjecture here the rationality of more general descendent series with insertions obtained from the Chern characters of the tautological sheaf. We prove the rationality of descendent series in Hilbert scheme cases for all curve c…
The aim of this note is to announce some results on the GIT problem for the Hilbert and Chow scheme of curves of degree d and genus g in P^{d-g}, whose full details will appear in a subsequent paper. In particular, we extend the previous results of L. Caporaso up to d>4(2g-2) and we observe that this is sharp. In the range 2(2g-2)<d<7/2(2g-2), we get a complete new description of the GIT quotient. As a corollary, we get a new compactification of the universal Jacobian over the moduli space of pseudo-stable curves.
In this note we investigate the generating series of the Euler characteristics of Hilbert scheme of points on cyclic quotient singularities of type (p,1). We link the appearing combinatorics to p-fountains, a generalization of the notion of fountain of coins. We obtain a representation of the generating series as coefficient of a two variable generating series.
We take a new look at the curvilinear Hilbert scheme of points on a smooth projective variety $X$ as a projective completion of the non-reductive quotient of holomorphic map germs from the complex line into $X$ by polynomial reparametrisations. Using an algebraic model of this quotient coming from global singularity theory we develop an iterated residue formula for tautological integrals over curvilinear Hilbert schemes.
In this paper we characterise the smoothness of the nested Quot scheme of points of a smooth variety, namely the moduli space parametrising flags of $0$-dimensional quotients of a fixed locally free sheaf. Our results extend Cheah's classification of smooth nested Hilbert schemes. -- Dans cet article on caractérise la lissité du schéma Quot ponctuel emboîté d'une variété lisse, c'est-à-dire l'espace de modules paramétrant les drapeaux de quotients de dimension $0$ d'un faisceau localement libre fixé. Nos résultats étendent la classification de Cheah concernant les schémas de Hilbert ponctuels …
We investigate some aspects of the geometry of two classical generalisations of the Hilbert schemes of points. Precisely, we show that parity conjecture for $\text{Quot}_r^d\mathbb{A}^3$ already fails for $d=8$ and $r=2$ and that lots of the elementary components of the nested Hilbert schemes of points on smooth quasi-projective varieties of dimension at least 4 are generically non-reduced. We also deduce that nested Hilbert schemes of points on smooth surfaces have generically non-reduced components. Finally, we give an infinite family of elementary components of the classical Hilbert schemes…
Let F(X,n):= X^n-Δbe the complementary of the union Δof the diagonals of X^n and let U be a quotient of F(X,n) (possibly trivial) by a subgroup of the symmetric group S_n. We construct compactifications of U in products of Hilbert schemes. Our approach generalizes and unifies classical constructions by Schubert-Semple, Le Barz-Keel, Kleiman and Cheah. An extensive study is done in the case n<4. This includes in particular a complete classification and a description of the quotients by the natural actions.
We prove Qingyuan Jiang's conjecture on semiorthogonal decompositions of derived categories of Quot schemes of locally free quotients. The author's result on categorified Hall products for Grassmannian flips is applied to prove the conjecture.
Let $X$ be a smooth projective surface over an algebraically closed field $k$ such that $char(k) \neq 2$. Let $X^{[d]}$ denote the punctual Hilbert scheme of zero dimensional quotients of degree $d$ and $X^{(d)}$ denote the symmetric product of $X$. For $\ell \neq 2$, we give a formula for the $\ell$-primary part of the Brauer group of $X^{[2]}$. We show that the Hilbert to Chow morphism induces an isomorphism of cohomological Brauer groups for $d=2$ and a similar result for $d \geq 3$. Let $Q(r,d)$ denote the punctual Quot-scheme parametrising zero dimensional quotients of $\mathcal{O}_X^{ \o…
We study Le Potier's strange duality on del Pezzo surfaces using quot schemes to construct independent sections of theta line bundles on moduli spaces of sheaves, one of which is the Hilbert scheme of n points. For n at most 7, we use multiple point formulas to count the length of the quot scheme, which agrees with the dimension of the space of sections on the Hilbert scheme. When the surface is the projective plane and n is arbitrary, we use nice resolutions of general stable sheaves to show that the quot schemes that arise are finite and reduced. Combining our results, we obtain a lower boun…
In this paper we construct $G$-Hilbert schemes for finite group schemes $G$. We find a construction of $G$-Hilbert schemes as relative $G$-Hilbert schemes over the quotient that does not need the Hilbert scheme of $n$ points, works under more natural assumptions and gives additional information about the morphism from the $G$-Hilbert scheme to the quotient.
We describe the closed strata that defines certain Quot schemes as closed subschemes in Grassmannians. The Quot schemes we consider are those parametrizing finite length $n$ quotient sheaves of the free, rank $p$ sheaf on projective $r$-space.
We compute the Euler characteristics of tautological vector bundles and their exterior powers over the Quot schemes of curves. We give closed-form expressions over punctual Quot schemes in all genera. For higher rank quotients of a trivial vector bundle, we obtain answers in genus zero. We also study the Euler characteristics of the symmetric powers of the tautological bundles, for rank zero quotients.
Given a locally free coherent sheaf on a smooth algebraic surface, we consider the Quot-scheme parametrizing zero-dimensional quotients of the sheaf and find the corresponding motivic class in the Grothendieck ring of algebraic varieties.
We construct a symplectic analog of the Quot scheme that parametrizes the torsion quotients of a trivial vector bundle over a compact Riemann surface. Some of its properties are investigated.
Let $k$ be an algebraically closed field. Let $C$ be an irreducible smooth projective curve over $k$. Let $E$ be a locally free sheaf on $C$ of rank $\geq 2$. Fix an integer $d \geq 2$. Let $\mathcal{Q}$ denote the Quot scheme parameterizing torsion quotients of $E$ of degree $d$. In this article we compute the $S$-fundamental group scheme of $\mathcal{Q}$.
We prove the existence of semiorthogonal decompositions of derived categories of Quot schemes of zero-dimensional quotients on curves in terms of derived categories of symmetric products of curves. The above result is a categorical analogue of a similar formula for the class of Quot schemes in the Grothendieck ring of varieties by Bagnarol-Fantechi-Perroni. It is a special case of a more general Quot formula of relative dimension one, which is regarded as a Bosonic counterpart of the Quot formula conjectured by Jiang and proved by the author. The proof involves categorical wall-crossing formul…
Let $\mathcal{E}$ be a locally free sheaf of rank $r$ on a smooth projective surface $S$. The Quot scheme $\mathrm{Quot}^{l}_{S}(\mathcal{E})$ of length $l$ coherent sheaf quotients of $\mathcal{E}$ 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 $C\subset S$ is a smooth canonical curve, we use cosection localization to show that the virtual fundamental class of $\mathrm{Quot}^{l}_{S}(\mathcal{E})$ is $(-1)^{l}$ times the fundamental class of the smooth subscheme $\mathrm{Quot}^{l}_{C}(\mathcal{E}\v…
We construct a virtual fundamental class on the Quot scheme parametrizing quotients of a trivial bundle on a curve. We use the virtual localization formula to calculate virtual intersection numbers on Quot. As a consequence, we reprove the Vafa-Intriligator formula; our answer is valid even when the Quot scheme is badly behaved. More intersections are computed by the same method. Finally, we present an application to the non-vanishing of the Pontrjagin ring of the moduli space of bundles.
We study Quot schemes of vector bundles on algebraic curves. Marian and Oprea gave a description of a topological quantum field theory (TQFT) studied by Witten in terms of intersection numbers on Quot schemes of trivial bundles. Since these Quot schemes can have the wrong dimension, virtual classes are required. But Quot schemes of general vector bundles always have the right dimension. Using the degree of the general vector bundle as an additional parameter, we construct a weighted TQFT containing both Witten's TQFT and the small quantum cohomology TQFT of the Grassmannian. This weighted TQFT…
We study virtual invariants of Quot schemes parametrizing quotients of dimension at most 1 of the trivial sheaf of rank N on nonsingular projective surfaces. We conjecture that the generating series of virtual K-theoretic invariants are given by rational functions. We prove rationality for several geometries including punctual quotients for all smooth projective surfaces and dimension 1 quotients for surfaces X with p_g>0. We also show that the generating series of virtual cobordism classes can be irrational. Given a K-theory class on X of rank r, we associate natural series of virtual Segre a…
We express the product of the cohomology ring of the Hilbert scheme in terms of the center of the algebra of the symmetric group. We give a conjecture for the case of crepant resolutions of symplectic quotient singularities.
Invariants of nested Hilbert and Quot schemes on surfaces
SISSA（斯issa国际高等研究学院）；Politecnico di Milano（米兰理工大学）；Università di Trieste（的里雅斯特大学）；Istituto Nazionale di Fisica Nucleare (INFN), sezione di Trieste（意大利国家核物理研究所（INFN）的里雅斯特分部）
Isotropic Quot schemes parameterize rank $r$ isotropic subsheaves of a vector bundle equipped with symplectic or symmetric quadratic form. We define a virtual fundamental class for isotropic Quot schemes over smooth projective curves. Using torus localization, we prescribe a way to calculate top intersection numbers of tautological classes, and obtain explicit formulas when $r=2$. These include and generalize the Vafa-Intriligator formula. In this setting, we compare the Quot scheme invariants with the invariants obtained via the stable map compactification.
We prove a formula for Chow groups of $Quot$-schemes which resolve degeneracy loci of a map between vector bundles, under expected dimension conditions. This result provides a unified way to understand known formulae for various geometric situations such as blowups, Cayley's trick, projectivizations, Grassmannian bundles, flops from Springer type resolutions, as well as provide new phenomena such as formulae for Grassmannain type flips/flops and virtual flips. We also give applications to blowups of determinantal ideals, moduli spaces of linear series on curves, and Hilbert schemes of points o…
We prove a closed formula for the integrals of the top Segre classes of tautological bundles over the Hilbert schemes of points of a K3 surface X. We derive relations among the Segre classes via equivariant localization of the virtual fundamental classes of Quot schemes on X. The resulting recursions are then solved explicitly. The formula proves the K-trivial case of a conjecture of M. Lehn from 1999. The relations determining the Segre classes fit into a much wider theory. By localizing the virtual classes of certain relative Quot schemes on surfaces, we obtain new systems of relations among…
We compute tautological integrals over Quot schemes on curves and surfaces. After obtaining several explicit formulas over Quot schemes of dimension 0 quotients on curves (and finding a new symmetry), we apply the results to tautological integrals against the virtual fundamental classes of Quot schemes of dimension 0 and 1 quotients on surfaces (using also universality, torus localization, and cosection localization). The virtual Euler characteristics of Quot schemes of surfaces, a new theory parallel to the Vafa-Witten Euler characteristics of the moduli of bundles, is defined and studied. Co…
For any locally free coherent sheaf on a fixed smooth projective curve, we study the class, in the Grothendieck ring of varieties, of the Quot scheme that parametrizes zero-dimensional quotients of the sheaf. We prove that this class depends only on the rank of the sheaf and on the length of the quotients. As an application, we obtain an explicit formula that expresses it in terms of the symmetric products of the curve.
Let $X$ be a smooth variety, $E$ a locally free sheaf on $X$. We express the generating function of the motives $[\textrm{Quot}_X(E,n)]$ in terms of the power structure on the Grothendieck ring of varieties. This extends a recent result of Bagnarol, Fantechi and Perroni for curves, and a result of Gusein-Zade, Luengo and Melle-Hernández for Hilbert schemes. We compute this generating function for curves and we express the relative motive $[\textrm{Quot}_{\mathbb A^d}(\mathscr{O}^{\oplus r}) \to \textrm{Sym}\, \mathbb A^d]$ as a plethystic exponential.
We construct good degenerations of Quot-schemes and coherent systems using the stack of expanded degenerations. We show that these good degenerations are separated and proper DM stacks of finite type. Applying to the projective threefolds, we derive degeneration formulas for DT-invariants of ideal sheaves and PT stable pair invariants.
Let $E$ be a vector bundle of rank $n$ on $\mathbb{P}^1$. Fix a positive integer $d$. Let $\mathcal{Q}(E,d)$ denote the Quot scheme of torsion quotients of $E$ of degree $d$ and let $Gr(E,d)$ denote the Grassmann bundle that parametrizes the $d$-dimensional quotients of the fibers of $E$. We compute Seshadri constants of ample line bundles on $\mathcal{Q}(E,d)$ and $Gr(E,d)$.
We prove that, given integers $m\geq 3$, $r\geq 1$ and $n\geq 0$, the moduli space of torsion free sheaves on $\mathbb P^m$ with Chern character $(r,0,\ldots,0,-n)$ that are trivial along a hyperplane $D \subset \mathbb P^m$ is isomorphic to the Quot scheme $\mathrm{Quot}_{\mathbb A^m}(\mathscr O^{\oplus r},n)$ of $0$-dimensional length $n$ quotients of the free sheaf $\mathscr O^{\oplus r}$ on $\mathbb A^m$.
We give an ADHM description of the Quot scheme of points ${\rm Quot}_{\mathbb{C}^{n}}(c,r),$ of length $c$ and rank $r$ on affine spaces $\mathbb{C}^{n}$ which naturally extends both Baranovsky's representation of the punctual Quot scheme on a smooth surface and the Hilbert scheme of points on affine spaces $\mathbb{C}^{n},$ described by the first author and M. Jardim. Using results on the variety of commuting matrices, and combining them with our construction, we prove new properties concerning irreducibility and reducedness of ${\rm Quot}_{\mathbb{C}^{n}}(c,r)$ and its punctual version ${\rm…
For a simple, rigid vector bundle $F$ on a Calabi-Yau $3$-fold $Y$, we construct a symmetric obstruction theory on the Quot scheme $\textrm{Quot}_Y(F,n)$, and we solve the associated enumerative theory. We discuss the case of other $3$-folds. Exploiting the critical structure on $\textrm{Quot}_{\mathbb A^3}(\mathscr O^r,n)$, we construct a virtual motive (in the sense of Behrend-Bryan-Szendrői) for $\textrm{Quot}_Y(F,n)$ for an arbitrary vector bundle $F$ on a smooth $3$-fold $Y$. We compute the associated motivic partition function. We obtain new examples of higher rank (motivic) Donaldson-Th…
We examine a sequence of examples of pairs of moduli spaces of sheaves on $\mathbb P^2$ where Le Potier's strange duality is expected to hold. One of the moduli spaces in these pairs is the Hilbert scheme of two points. We compute the sections of the relevant theta bundle as a representation of $SL(X)$, where $\mathbb P^2=\mathbb P(X)$. For the higher rank space, we construct a moduli space using the resolution of exceptional bundles from Coskun, Huizenga, and Woolf. We compute a subspace of the sections of the theta bundle which is dual to the sections on the Hilbert scheme. In the second par…
We consider a class of tautological top intersection products on the moduli space of stable pairs consisting of semistable vector bundles together with N sections on a smooth complex projective curve C. We show that when N is large, these intersection numbers can equally be computed on the Grothendieck Quot scheme of coherent sheaf quotients of the rank N trivial sheaf on C. The result has applications to the calculation of the intersection theory of the moduli space of semistable bundles on C.
In this paper, we show the full flag Hilbert scheme of points of nodal curves has only complete intersection singularities. We study the Hilbert scheme of points of the cusp curve and show the punctual Hilbert scheme is isomorphic to \mathbb{P}^1. Furthermore Hilbert scheme has only one singularty along the punctual one.
We define a derived enhancement of the classical quot functor of quotients associated to a coherent sheaf on a nonsingular quasiprojective variety. We prove its representability and show that it has the expected tangent complex. The derived quot scheme of points can be covered by affine charts obtained as spectra of commutative graded differential algebras (cdgas) and we compute an example. As a demonstration of the usefulness of this presentation, we write down an explicit shifted form on this example and conjecture that it agrees with the shifted symplectic structure on the derived stack of …
We consider the quot scheme $\mathrm{Quot}^d_{\mathcal F^r/ \mathbb P^1/ k}$ of locally free quotients of $\mathcal F^r:= \bigoplus ^{ r} \mathcal O_{\mathbb P^1 }$ with Hilbert polynomial $p(t)=d$. We prove that it is a smooth variety of dimension $dr$, locally isomorphic to $\mathbb A^{dr}$. We introduce a new notion of support for modules in $\mathrm{Quot}^d_{\mathcal F^r/ \mathbb P^1/ k}$, called Hilb-support that allows us to define a natural surjective morphism of schemes $ξ:\mathrm{Quot}^d_{\mathcal F^r/ \mathbb P^1/ k} \to \mathrm{Hilb}^d_{\mathcal O_{\mathbb P^1}} $ associating to eac…
Let $C$ be a complex projective smooth curve and $W$ a symplectic vector bundle of rank $2n$ over $C$. The Lagrangian Quot scheme $LQ_{-e}(W)$ parameterizes subsheaves of rank $n$ and degree $-e$ which are isotropic with respect to the symplectic form. We prove that $LQ_{-e}(W)$ is irreducible and generically smooth of the expected dimension for all large $e$, and that a generic element is saturated and stable.
Realizing a part of the Derived Deformation Theory program, we construct a "derived" analog of the Grothendieck's Quot scheme parametrizing subsheaves in a given coherent sheaf F on a smooth projective variety X. This analog is a differential graded manifold RQuot_h(F) (so it is always smooth in an appropriate sense) whose tangent space at a point represented by a subsheaf K in F, is a cochain complex quasiisomorphic to RHom(K, F/K).
