Stability of some vector bundles on Hilbert schemes of points on K3 surfaces
Fabian Reede, Ziyu Zhang
备注：
21 pages. Major revision after referee report. Improved the presentation and highlighted the use of the spherical twist and its inverse. Comments welcome!
Let $X$ be a projective K3 surfaces. In two examples where there exists a fine moduli space $M$ of stable vector bundles on $X$, isomorphic to a Hilbert scheme of points, we prove that the universal family $\mathcal{E}$ on $X\times M$ can be understood as a complete flat family of stable vector bundles on $M$ parametrized by $X$, which identifies $X$ with a smooth connected component of some moduli space of stable sheaves on $M$.
We prove stability of rank two tautological bundles on the Hilbert square of a surface (under a mild positivity condition) and compute their Chern classes.
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 $((V^\vee)^{[n]})^\vee$-twisted Hitchin pair, where $(V^\vee)^{[n]}$ is a vector bundle on $X^{[n]}$ constructed using the dual $V^\vee$ 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 tauto…
We obtain effective results for the global generation of pluritheta line bundles on moduli spaces of vector bundles on curves. The main ingredient is an independent result giving an upper bound on the dimension of the Hilbert scheme of coherent quotients of a given vector bundle.
We construct five families of two-dimensional moduli spaces of parabolic Higgs bundles (respectively local systems) by taking the equivariant Hilbert scheme of a certain finite group acting on the cotangent bundle of an elliptic curve. We show that the Hilbert scheme of m points of these surfaces is again a moduli space of parabolic Higgs bundles (respectively local systems), confirming a conjecture of Boalch in these cases and extending a result of Gorsky--Nekrasov--Rubtsov. Using the McKay correspondence we establish the autoduality conjecture for the derived categories of the moduli spaces …
We construct the Hilbert compactification of the universal moduli space of semistable vector bundles over smooth curves. The Hilbert compactification is the GIT quotient of some open part of an appropriate Hilbert scheme of curves in a Grassmannian. It has all the properties asked for by Teixidor.
The purpose of this paper is to explore the geometry and establish the slope stability of tautological vector bundles on Hilbert schemes of points on smooth surfaces. By establishing stability in general we complete a series of results of Schlickewei and Wandel who proved the slope stability of these vector bundles for Hilbert schemes of 2 points or 3 points on K3 or abelian surfaces with Picard group restrictions. In exploring the geometry we show that every sufficiently positive semistable vector bundle on a smooth curve arises as the restriction of a tautological vector bundle on the Hilber…
In this paper, we prove that the notions of Hilbert stability and Mumford stability agree for vector bundles of arbitrary rank over smooth curves. The notion of Hilbert stability was introduced by Gieseker and Morrison in 1984, and they showed that for smooth curves and vector bundles of rank two it agrees with Mumford stability. A different proof for the rank two case was given by M. Teixidor i Bigas. Our proof uses a new approach and avoids complicated computations. Our results might serve as a first step in the construction of the Hilbert stable compactification of the universal moduli spac…
The cohomology of the Hilbert schemes of points on smooth projective surfaces can be approached both with vertex algebra tools and equivariant tools. Using the first tool, we study the existence and the structure of universal formulas for the Chern classes of the tangent bundle over the Hilbert scheme of points on a projective surface. The second tool leads then to nice generating formulas in the particular case of the Hilbert scheme of points on the affine plane.
The Hilbert scheme of n points in the projective plane parameterizes degree n zero-dimensional subschemes of the projective plane. We examine the dual cones of effective divisors and moving curves on the Hilbert scheme. By studying interpolation, restriction, and stability properties of certain vector bundles on the plane we fully determine these cones for just over three fourths of all values of n. A general Steiner bundle on P^N is a vector bundle admitting a linear resolution of a particular form. We complete the classification of slopes of semistable Steiner bundles on P^N by showing every…
Let $k$ be an algebraically closed field of characteristic $p > 3$ and $S$ be a smooth projective surface over $k$ with $k$-rational point $x$. For $n \geq 2$, let $S^{[n]}$ denote the Hilbert scheme of $n$ points on $S$. In this note, we compute the fundamental group scheme $π^{\textrm{alg}}(S^{[n]}, \tilde{nx})$ defined by the Tannakian category of stratified bundles on $S^{[n]}$.
We study the images of tautological bundles on Hilbert schemes of points on surfaces and their wedge powers under the derived McKay correspondence. The main observation of the paper is that using a derived equivalence differing slightly from the standard one considerably simplifies both the results and their proofs. As an application, we obtain shorter proofs for known results as well as new formulae for homological invariants of tautological sheaves. In particular, we compute the extension groups between wedge powers of tautological bundles associated to line bundles on the surface.
We prove de Cataldo-Hausel-Migliorini's P=W conjecture in arbitrary rank for parabolic Higgs bundles labeled by the affine Dynkin diagrams $\tilde{A}_0$, $\tilde{D}_4$, $\tilde{E}_6$, $\tilde{E}_7$, and $\tilde{E}_8$. Our proof relies on the study of the tautological classes on the Hilbert scheme of points on an elliptic surface with respect to the perverse filtration.
We compute the cone of effective divisors on the Hilbert scheme of points in the projective plane. We show the sections of many stable vector bundles satisfy a natural interpolation condition, and that these bundles always give rise to the edge of the effective cone of the Hilbert scheme. To do this, we give a generalization of Gaeta's theorem on the resolution of the ideal sheaf of a general collection of points in the plane. This resolution has a natural interpretation in terms of Bridgeland stability, and we observe that ideal sheaves of collections of points are destabilized by exceptional…
Several families of rank-two vector bundles on Hirzebruch surfaces are shown to consist of all very ample, uniform bundles. Under suitable numerical assumptions, the projectivization of these bundles, embedded by their tautological line bundles as linear scrolls, are shown to correspond to smooth points of components of their Hilbert scheme, the latter having the expected dimension. If e=0,1 the scrolls fill up the entire component of the Hilbert scheme, while for e=2 the scrolls exhaust a subvariety of codimension 1.
We develop a new general method for computing the decomposition type of the normal bundle to a projective rational curve. This method is then used to detect and explain an example of a Hilbert scheme that parametrizes all the rational curves in $\mathbb{P}^s$ with a given decomposition type of the normal bundle and that has exactly two irreducible components. This gives a negative answer to the very old question whether such Hilbert schemes are always irreducible. We also characterize smooth non-degenerate rational curves contained in rational normal scroll surfaces in terms of the splitting t…
Let $S$ be an irreducible smooth projective surface defined over an algebraically closed field $k$. For a positive integer $d$, let ${\rm Hilb}^d(S)$ be the Hilbert scheme parametrizing the zero-dimensional subschemes of $S$ of length $d$. For a vector bundle $E$ on $S$, let ${\mathcal H}(E)\, \longrightarrow\, {\rm Hilb}^d(S)$ be its Fourier--Mukai transform constructed using the structure sheaf of the universal subscheme of $S\times {\rm Hilb}^d(S)$ as the kernel. We prove that two vector bundles $E$ and $F$ on $S$ are isomorphic if the vector bundles ${\mathcal H}(E)$ and ${\mathcal H}(F)$ …
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, Boissière 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,…
In this paper we describe the Frobenius pull-backs of the syzygy bundles $Syz_C(X^a, Y^a, Z^a)$, $a \geq 1$, on the projective Fermat curve C of degree n in characteristics coprime to n, either by giving their strong Harder-Narasimhan Filtration if $Syz_C(X^a, Y^a, Z^a)$ is not strongly semistable or in the strongly semistable case by their periodicity behavior. Moreover, we apply these results to Hilbert-Kunz functions, to find Frobenius periodicities of the restricted cotangent bundle $Ω_{P^2}|_C$ of arbitrary length and a problem of Brenner regarding primes with strongly semistable reductio…
Here we investigate meaningful families of vector bundles on a very general polarized $K3$ surface $(X,H)$ and on the corresponding Hyper--Kaehler variety given by the Hilbert scheme of points $X^{[k]}:= {\rm Hilb}^k(X)$, for any integer $k \geqslant 2$. In particular, we prove results concerning bigness and stability of such bundles. First, we give conditions on integers $n$ such that the twist of the tangent bundle of $X$ by the line bundle $nH$ is big and stable on~$X$; we then prove a similar result for a natural twist of the tangent bundle of $X^{[k]}$. Next, we prove global generation, b…
Let $X$ be a Hilbert modular variety and $\mathbb{V}$ a non-trivial local system over $X$ with infinite monodromy. In this paper we study Saito's mixed Hodge structure (MHS) on the cohomology group $H^k(X,\mathbb{V})$ using the method of Higgs bundles. Among other results we prove the Eichler-Shimura isomorphism, give a dimension formula for the Hodge numbers and show that the mixed Hodge structure is split over $\mathbb{R}$. These results are analogous to Matsushima-Shimura [Annals of Mathematics 78, 1963] in the cocompact case and complement the results in Freitag [Book: Hilbert modular form…
We first give an alternative proof, based on a simple geometric argument, of a result of Marian, Oprea and Pandharipande on top Segre classes of the tautological bundles on Hilbert schemes of $K3$ surfaces equipped with a line bundle. We then turn to the blow-up of $K3$ surface at one point and establish vanishing results for the corresponding top Segre classes in a certain range. This determines, at least theoretically, all top Segre classes of tautological bundles for any pair $(Σ,H),\,H\in {\rm Pic}\,Σ$.
We compute the cohomology spaces for the tautological bundle tensor the determinant bundle on the punctual Hilbert scheme H of subschemes of length n of a smooth projective surface X. We show that for L and A invertible vector bundles on X, and w the canonical bundle of X, if $w^{-1}\otimes L$, $w^{-1}\otimes A$ and A are ample vector bundles, then the higher cohomology spaces on H of the tautological bundle associated to L tensor the determinant bundle associated to A vanish, and the space of global sections is computed in terms of $H^0(A)$ and $H^0(L\otimes A)$. This result is motivated by t…
Let (X,H) be a polarized smooth projective surface satisfying H^1(X,O_X)=0 and let F be either a rank one torsion-free sheaf or a rank two μH-stable vector bundle on X. Assume that c_1(F)/=0. In this article it is shown that the rank two, respectively rank four tautological sheaf F^{[2]} associated with F on the Hilbert square X^{[2]} is μ-stable with respect to a certain polarization.
We propose a conjecture on the generating series of Chern numbers of tautological bundles on Hilbert schemes of points on curves and establish the rank 1 and rank -1 case of this conjecture. Thus we compute explicitly the generating series of integrals of the Segre classes of tautological bundles of line bundles on curves, which has a similar structure as Lehn's conjecture for surfaces.
Let S be a nonsingular projective surface. Each vector bundle V on S of rank s induces a tautological vector bundle over the Hilbert scheme of n points of S. When s=1, the top Segre classes of the tautological bundles are given by a recently proven formula conjectured in 1999 by M. Lehn. We calculate here the Segre classes of tautological bundles for all ranks s over all K-trivial surfaces. Furthermore, in rank s=2, the Segre integrals are determined for all surfaces, thus establishing a full analogue of Lehn's formula. We also give conjectural formulas for certain series of Verlinde Euler cha…
We compute some cohomology spaces for the symmetric power of the tautological bundle tensor the determinant bundle on the punctual Hilbert scheme of a complex smooth projective surface.
We give an algorithmic description of the action of the Chern classes of tautological bundles on the cohomology of Hilbert schemes of points on surfaces within the framework of Nakajima's oscillator algebra. This leads to an identification of the cohomology ring of Hilbert schemes of the affine plane with a ring of differential operators on a Fock space. We end with the computation of the top Segre classes of tautological bundles associated to line bundles on Hilb^n up to n=7, and give a conjecture for the generating series.
In this paper we prove that, over a Hilbertian ground field, surfaces with two conic fibrations whose fibres have non-zero intersection product have the Hilbert property. We then give an application of this result, namely the verification of the Hilbert property for certain del Pezzo varieties.
The aim of this paper is two--fold. We first strongly improve our previous main result Theorem 3.1 in Arxiv 1702.00918v3 12Feb2018 ("Brill-Noether loci of rank two vector bundles on a general $ν$-gonal curve"), concerning classification of irreducible components of the Brill--Noether locus parametrizing rank 2 semistable vector bundles of suitable degrees $d$, with at least $d-2g+4$ independent global sections, on a general $ν$--gonal curve $C$ of genus $g$. We then uses this classification to study several properties of the Hilbert scheme of suitable surface scrolls in projective space, which…
Let $M(n,ξ)$ be the moduli space of stable vector bundles of rank $n\geq 3$ and fixed determinant $ξ$ over a smooth projective algebraic curve $X$ over $\mathbb{C}$ of genus $g\geq 4.$ We use the gonality of the curve and $r$-Hecke morphisms to describe a smooth open set and to compute the dimension of a component of the Hilbert scheme $Hilb_{M(n,ξ)}$, of the scheme of morphisms $Mor(\mathbb{G},M(n,ξ))$ and of the moduli space $ M_{X \times \mathbb{G}}$ of stable bundles over $X\times \mathbb{G},$ where $\mathbb{G}$ is the Grassmannian $\mathbb{G}(n-r,\mathbb{C}^n)$. In particular, we prove th…
We give a description of the image of tensor products of tautological bundles on Hilbert schemes of points on surfaces under the Bridgeland-King-Reid-Haiman equivalence. Using this, some new formulas for cohomological invariants of these bundles are obtained. In particular, we give formulas for the Euler characteristic of arbitrary tensor products on the Hilbert scheme of two points and of triple tensor products in general.
For a smooth projective variety $P$, we construct a Cartier divisor supported on the incidence locus in $\mathscr{C}_a (P) \times \mathscr{C}_{\dim(P)-a-1}(P)$. There is a natural definition of the corresponding line bundle on a product of Hilbert schemes, and we show this bundle descends to the Chow varieties. This answers a question posed by Mazur.
We describe the possible restrictions of the cotangent bundle Ω_{\PP^N} to an elliptic curve C \subset \PP^N. We apply this in positive characteristic to the computation of the Hilbert-Kunz function of a homogeneous R_+-primary ideal I \subset R in the graded section ring R = \bigoplus_{n \in \NN} Γ(C, Ø(n)).
We study general symmetric powers $S^k L^{[n]}$ of a tautological bundle $L^{[n]}$ on the Hilbert scheme $X^{[n]}$ of $n$ points over a smooth quasi-projective surface $X$, associated to a line bundle $L$ on $X$. Let $V_L$ be the $\mathfrak{S}_n$-vector bundle on $X^n$ defined as the exterior direct sum $L \boxplus \cdots \boxplus L$. We prove that the Bridgeland-King-Reid transform $\mathbfΦ(S^k L^{[n]})$ of symmetric powers $S^k L^{[n]}$ is quasi isomorphic to the last term of a finite decreasing filtration on the natural vector bundle $S^k V_L$, defined by kernels of operators $D^l_L$, whic…
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…
By using a Fourier-Mukai transform for sheaves on K3 surfaces we show that for a wide class of K3 surfaces $X$ the punctual Hilbert schemes $\Hilb^n(X)$ can be identified, for all $n\geq 1$, with moduli spaces of Gieseker stable vector bundles on $X$ of rank $1+2n$. We also introduce a new Fourier-Mukai type transform for such surfaces.
Let $(X, H)$ be a polarized smooth projective algebraic surface and $E$ is globally generated, stable vector bundle on $X$. Then the Syzygy bundle $M_E$ associated to it is defined as the kernel bundle corresponding to the evaluation map. In this article we will study the stability property of $M_E$ with respect to $H$.
In this paper, we give three bases for the cohomology groups of the Hilbert scheme of two points on projective space. Then, we use these bases to compute all effective and nef cones of higher codimensional cycles on the Hilbert scheme. Next, we compute the class in one of these bases of the Chern classes of tautological bundles coming from line bundles. Finally, we provide an application of these results to the degrees of secant varieties of complete intersections.
We determine the stability/instability of the tangent bundles of the Fano varieties in a certain class of two orbit varieties, which are classified by Pasquier in 2009. As a consequence, we show that some of these varieties admit unstable tangent bundles, which disproves a conjecture on stability of tangent bundles of Fano manifolds.
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.
We define Hecke transformation for orthogonal bundles over a compact Riemann surface. Using the cycles on a moduli space of orthogonal bundles given by Hecke transformations, we prove that the projectivized Picard bundle on the moduli space is stable.
Let $S$ be the first degeneracy locus of a morphism of vector bundles corresponding to a general matrix of linear forms in $\mathbb{P}^s$. We prove that, under certain positivity conditions, its Hilbert square $\mathrm{Hilb}^2(S)$ is isomorphic to the zero locus of a global section of an irreducible homogeneous vector bundle on a product of Grassmannians. Our construction involves a naturally associated Fano variety, and an explicit description of the isomorphism.
Roughly speaking, a conic bundle is a surface, fibered over a curve, such that the fibers are conics (not necessarily smooth). We define stability for conic bundles and construct a moduli space. We prove that (after fixing some invariants) these moduli spaces are irreducible (under some conditions). Conic bundles can be thought of as generalizations of orthogonal bundles on curves. We show that in this particular case our definition of stability agrees with the definition of stability for orthogonal bundles. Finally, in an appendix by I. Mundet i Riera, a Hitchin-Kobayashi correspondence is st…
In this paper, we prove the Bogomolov-Gieseker type inequality conjecture for threefolds with nef tangent bundles. As a corollary, there exist Bridgeland stability conditions on these threefolds.
We prove a closed formula expressing any multiplicative characteristic class evaluated on the tangent bundle of the Hilbert schemes of points on a non-compact simply-connected surface. As a corollary, we deduce a closed formula for the Chern character of the tangent bundles of these Hilbert schemes. We also give a closed formula for the multiplicative characteristic classes of the tautological bundles associated to a line bundle on the surface. We finally remark which implications the results here have for the Hilbert schemes of points of an arbitrary surface.
Given a rank $r$ stable bundle over a smooth irreducible projective curve $C,$ there is an associated rank $2r$ bundle over $S^2(C),$ the second symmetric power of $C.$ In this article we study the stability of this bundle. As a consequence we get an immersion from the moduli space of stable bundles over $C$ to the associated moduli space of stable bundles over $S^2(C).$
Using Bridgeland stability conditions we give sufficient criteria for a stable vector bundle on a surface to remain stable when restricted to a curve. We give a stronger criterion when the vector bundle is a general vector bundle on the plane. As an application, we compute the cohomology of such bundles for curves that lie in the plane or on Hirzebruch surfaces.
Let $L$ be a globally generated line bundle over a smooth irreducible complex projective surface $X$. The syzygy bundle $M_{L}$ is the kernel of the evaluation map $H^0(L)\otimes\mathcal O_X\to L$. We prove the $L$-stability of $M_L$ for Hirzebruch surfaces, del Pezzo surfaces and Enriques surfaces. The $(-K_X)$-stability of syzygy bundles $M_L$ over del Pezzo surfaces is also obtained.
We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-Kühn. Our theorem is thus suitable for application to some non-compact Shimura varieties with their bundles of cusp forms. As an application, we treat the case of Hilbert modular surfaces, establishing an arithmetic analogue of the classical result expressing the dimensions of spaces of cusp forms in terms of special values of Dedekind zeta functions.
We study the stable hyperelliptic locus, i.e. the closure, in the Deligne- Mumford moduli space of stable curves, of the locus of smooth hyperelliptic curves. Working on a suitable blowup of the relative Hilbert scheme (of degree 2) associated to a family of stable curves, we construct a bundle map ('degree-2 Brill-Noether') from a modification of the Hodge bundle to a tautological bundle, whose degeneracy locus is the natural lift of the stable hyperelliptic locus plus a simple residual scheme. Using intersection theory on Hilbert schemes and Fulton-Macpherson residual intersection theory, th…
We show that the residual categories of quadric surface bundles are equivalent to the (twisted) derived categories of some scheme under the following hypotheses. Case 1: The quadric surface bundle has a smooth section. Case 2: The total space of the quadric surface bundle is smooth and the base is a smooth surface. We provide two proofs in Case 1 describing the scheme as the hyperbolic reduction and as a subscheme of the relative Hilbert scheme of lines, respectively. In Case 2, the twisted scheme is obtained by performing birational transformations to the relative Hilbert scheme of lines. Fin…
