Quot Scheme Research Articles

Counting maximal isotropic subbundles of orthogonal bundles over a curve

Let C be a smooth projective curve and V an orthogonal bundle over C. Let IQe(V) be the isotropic Quot scheme parameterizing degree e isotropic subsheaves of maximal rank in V. We give a closed formula for intersection numbers on components of IQe(V) whose generic element is saturated. As a special case, for g≥2, we compute the number of isotropic subbundles of maximal rank and degree of a general stable orthogonal bundle in most cases when this is finite. This is an orthogonal analogue of Holla's enumeration of maximal subbundles in [20], and of the symplectic case studied in [8].

Infinitesimal deformations of some quot schemes, II

Infinitesimal deformations of some quot schemes, II

Quot scheme and deformation quantization

Quot scheme and deformation quantization

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.

Extension groups of tautological bundles on punctual Quot schemes of curves

We prove formulas for the cohomology and the extension groups of tautological bundles on punctual Quot schemes over complex smooth projective curves. As a corollary, we show that the tautological bundle determines the isomorphism class of the original vector bundle on the curve. We also give a vanishing result for the push-forward along the Quot–Chow morphism of tensor and wedge products of duals of tautological bundles.

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.

Infinitesimal Deformations of Some Quot Schemes

Abstract Let $E$ be a vector bundle on a smooth complex projective curve $C$ of genus at least two. Let $\mathcal{Q}(E,d)$ be the Quot scheme parameterizing the torsion quotients of $E$ of degree $d$. We compute the cohomologies of the tangent bundle $T_{\mathcal{Q}(E,d)}$. In particular, the space of infinitesimal deformations of $\mathcal{Q}(E,d)$ is computed. Kempf and Fantechi computed the space of infinitesimal deformations of $\mathcal{Q}({\mathcal O}_{C},d)\,=\, C^{(d)}$ [ 11, 19].

The virtual intersection theory of isotropic Quot schemes

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.

Application of Lagrange inversion to wall-crossing for Quot schemes on surfaces

Motivated by my work on enumerative invariants for Quot schemes, I related two power series obtained by two different means. One of them was computed using geometric arguments via virtual localization methods and the other one came from working with representation theoretic objects called vertex algebras. In this note, I give proof of the equality of the two power series by relying only on techniques related to Lagrange inversion. This makes my work on Quot schemes independent of the previous results in the literature and proves a new combinatorial identity.

The d-critical structure on the Quot scheme of points of a Calabi–Yau 3-fold

The Artin stack [Formula: see text] of zero-dimensional sheaves of length [Formula: see text] on [Formula: see text] carries two natural d-critical structures in the sense of Joyce. One comes from its description as a quotient stack [Formula: see text], another comes from derived deformation theory of sheaves. We show that these d-critical structures agree. We use this result to prove the analogous statement for the Quot scheme of points [Formula: see text], which is a global critical locus for every [Formula: see text], and also carries a derived-in-flavor d-critical structure besides the one induced by the potential [Formula: see text]. Again, we show these two d-critical structures agree. Moreover, we prove that they locally model the d-critical structure on [Formula: see text], where [Formula: see text] is a locally free sheaf of rank [Formula: see text] on a projective Calabi–Yau [Formula: see text]-fold [Formula: see text]. Finally, we prove that the perfect obstruction theory on [Formula: see text] induced by the Atiyah class of the universal ideal agrees with the critical obstruction theory induced by the Hessian of the potential [Formula: see text].

Quot schemes and Fourier-Mukai transformation

Abstract We consider several related examples of Fourier-Mukai transformations involving the quot scheme. A method of showing conservativity of these Fourier-Mukai transformations is described.

Euler characteristics of tautological bundles over Quot schemes of curves

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 0. We also study the Euler characteristics of the symmetric powers of the tautological bundles, for rank 0 quotients.

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.

Derived categories of Quot schemes of locally free quotients via categorified Hall products

Derived categories of Quot schemes of locally free quotients via categorified Hall products

Cosection localization and the Quot scheme QuotSl(E)

Let E be a locally free sheaf of rank r on a smooth projective surface S . The Quot scheme Quot S l ( E ) of length l coherent sheaf quotients of 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 ⊂ S is a smooth canonical curve, we use cosection localization to show that the virtual fundamental class of Quot S l ( E ) is ( − 1 ) l times the fundamental class of the smooth subscheme Quot C l ( E | C ) ⊂ Quot S l ( E ) . We then prove a structure theorem for virtual tautological integrals over Quot S l ( E ) . From this, we deduce, among other things, the equality of virtual Euler characteristics χ vir ( Quot S l ( E ) ) = χ vir ( Quot S l ( O ⊕ r ) ) .

Primary decomposition of modules: A computational differential approach

We study primary submodules and primary decompositions from a differential and computational point of view. Our main theoretical contribution is a general structure theory and a representation theorem for primary submodules of an arbitrary finitely generated module over a polynomial ring. We characterize primary submodules in terms of differential operators and punctual Quot schemes. Moreover, we introduce and implement an algorithm that computes a minimal differential primary decomposition for a module.

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.

On the motive of the nested Quot scheme of points on a curve

Let C be a smooth curve over an algebraically closed field k, and let E be a locally free sheaf of rank r. We compute, for every d>0, the generating function of the motives [QuotC(E,n)]∈K0(Vark), varying n=(0≤n1≤⋯≤nd), where QuotC(E,n) is the nested Quot scheme of points, parametrising 0-dimensional subsequent quotients E↠Td↠⋯↠T1 of fixed length ni=χ(Ti). The resulting series, obtained by exploiting the Białynicki-Birula decomposition, factors into a product of shifted motivic zeta functions of C. In particular, it is a rational function.

Components and singularities of Quot schemes and varieties of commuting matrices

Abstract We investigate the variety of commuting matrices. We classify its components for any number of matrices of size at most 7. We prove that starting from quadruples of 8 × 8 {8\times 8} matrices, this scheme has generically nonreduced components, while up to degree 7 it is generically reduced. Our approach is to recast the problem as deformations of modules and generalize an array of methods: apolarity, duality and Białynicki–Birula decompositions to this setup. We include a thorough review of our methods to make the paper self-contained and accessible to both algebraic and linear-algebraic communities. Our results give the corresponding statements for the Quot schemes of points, in particular we classify the components of Quot d ⁡ ( 𝒪 𝔸 n ⊕ r ) {\operatorname{Quot}_{d}(\mathcal{O}_{\mathbb{A}^{n}}^{\oplus r})} for d ≤ 7 {d\leq 7} and all r, n.

Higher rank motivic Donaldson–Thomas invariants of via wall-crossing, and asymptotics

AbstractWe compute, via motivic wall-crossing, the generating function of virtual motives of the Quot scheme of points on ${\mathbb{A}}^3$ , generalising to higher rank a result of Behrend–Bryan–Szendrői. We show that this motivic partition function converges to a Gaussian distribution, extending a result of Morrison.