Moduli Spaces of Semistable Sheaves on Singular Genus One Curves
Daniel Hernández Ruipérez, Ana Cristina López Martín, Darío Sánchez Gómez, Carlos Tejero Prieto
备注：
26 pages, 4 figures. Added the structure of the biggest component of the moduli space of sheaves of degree 0 on a cycle of projective lines. Final version; to appear en IMRS (International Mathematics Research Notices 2009)
We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure dimensional sheaves. Using them we establish new identifications between certain Simpson moduli spaces of semistable sheaves on the curve. For rank zero, the moduli spaces are symmetric powers of the curve whilst for a fixed positive rank there are only a finite number of non-isomorphic spaces. We prove similar results for the relative semistable moduli spaces on an arbitrary genus one fibration with no conditions either on the base or on the total space. For a cycle $E_N$ of projective lines, we show that the unique degree 0 stable sheaves are the line bundles having degree 0 on every irreducible component and the sheaves $\mathcal{O}(-1)$ supported on one irreducible component. We also prove that the connected component of the moduli space that contains vector bundles of rank $r$ is isomorphic to the $r$-th symmetric product of the rational curve with one node.
This paper is devoted to generalizing the Mehta-Ramanathan restriction theorem to the case of ε-semistable and k-semistable decorated sheaves. After recalling the definition of decorated sheaves and their usual semistability we define the ε and k-(semi)stablility. We first prove the existence of a (unique) ε-maximal destabilizing subsheaf for decorated sheaves (Section 3.1). After some others preliminar results (such as the opennes condition for families of ε-semistable decorated sheaves) we finally prove, in Section 3.7, a restriction theorem for slope ε-semistable decorated sheaves. In Secti…
In this expository article, we follow the work of Langer to prove the boundedness of the moduli space of semistable torsion-free sheaves over a projective variety, in any characteristic.
In this paper we prove restriction theorems for torsion-free sheaves that are (semi)stable with respect to the truncated Hilbert polynomial over a smooth projective variety. Our results apply in particular to Gieseker-semistable sheaves and generalize the well-known restriction theorems of Mehta and Ramanathan. As an application we construct a moduli space of sheaves in higher dimensions.
We construct a moduli space of slope-semistable pure sheaves, building upon previous work of Le Potier and Jun Li on torsion-free sheaves over smooth surfaces. In particular, our construction provides a compactification of the Simpson moduli space of slope-stable reflexive sheaves. We also prove an effective restriction theorem for slope-(semi)stable pure sheaves following an approach due to Langer.
We decompose each moduli space of semistable sheaves on the complex projective plane with support of dimension one and degree four into locally closed subvarieties, each subvariety being the good or geometric quotient of a set of morphisms of locally free sheaves modulo a reductive or a nonreductive group. We find locally free resolutions of length one of all these sheaves and describe them.
In the 1980s Drézet and Le Potier realized moduli spaces of Gieseker-semistable sheaves on $\mathbb{P}^2$ as what are now called quiver moduli spaces. We discuss how this construction can be understood using t-structures and exceptional collections on derived categories, and how it can be extended to a similar result on $\mathbb{P}^1\times\mathbb{P}^1$.
We give a new simple proof of boundedness of the family of semistable sheaves with fixed numerical invariants on a fixed smooth projective variety. In characteristic zero our method gives a quick proof of Bogomolov's inequality for semistable sheaves on a smooth projective variety of any dimension $\ge 2$ without using any restriction theorems.
We describe some one-dimensional moduli spaces of rank 2 Gieseker semistable sheaves on an Enriques surface improving earlier results of H. Kim. In case of a nodal Enriques surface the obtained moduli spaces are reducible for general polarizations. For unnodal Enriques surfaces we show how to reduce the study of moduli spaces of high even rank Gieseker semistable sheaves to low ranks. To prove this we use the method of K. Yoshioka who showed that in the odd rank one can reduce to rank 1.
We construct a new infinite series of irreducible components of the Gieseker-Maruyama moduli scheme $\mathcal{M}(k), ~ k \geq 3$ of coherent semistable rank 2 sheaves with Chern classes $c_1=0,~ c_2=k,~ c_3=0$ on $\mathbb{P}^3$ whose general points are sheaves with singularities of mixed dimension. These sheaves are constructed by elementary transformations of stable and properly $μ$-semistable reflexive sheaves along disjoint union of collections of points and smooth irreducible curves which are rational or complete intersection curves. As a special member of this series we obtain a new compo…
This paper extends a number of known results on slope-semistable sheaves from the classical case to the setting where polarisations are given by movable curve classes. As applications, we obtain new flatness results for reflexive sheaves on singular varieties, as well as a characterisation of finite quotients of Abelian varieties via a Chern class condition.
Let $H$ and $H'$ be two ample line bundles over a smooth projective surface $X$, and $M(H)$ (resp. $M(H')$) the coarse moduli scheme of $H$-semistable (resp. $H'$-semistable) sheaves of fixed type $(r,c_1,c_2)$. We construct a sequence of blowing-ups which describes how $M(H)$ differs from $M(H')$ not only when $r=2$ but also when $r$ is arbitrary. Means we here utilize are elementary transforms and the notion of a sheaf with flag.
We classify (semi)stable sheaves on a rational curve with one node. The results are based on the classification of indecomposable torsion-free sheaves due to Drozd and Greuel "Tame and wild projective curves and classification of vector bundles", where the sheaves are described in terms of certain combinatorial data. We translate the condition of (semi)stability into this combinatorial language and solve the so obtained problem.
We show that the moduli space M(r,c) of semistable sheaves on n-dimensional projective space with support of dimension one, with multiplicity r and with Euler characteristic c is isomorphic to M(r,-c).
We describe new irreducible components of the Gieseker-Maruyama moduli scheme $\mathcal{M}(3)$ of semistable rank 2 coherent sheaves with Chern classes $c_1=0,\ c_2=3,\ c_3=0$ on $\mathbb{P}^3$, general points of which correspond to sheaves whose singular loci contain components of dimensions both 0 and 1. These sheaves are produced by elementary transformations of stable reflexive rank 2 sheaves with $c_1=0,\ c_2=2,\ c_3=2$ or 4 along a disjoint union of a projective line and a collection of points in $\mathbb{P}^3$. The constructed families of sheaves provide first examples of irreducible co…
In the present paper we describe new component of the Gieseker-Maruyama moduli space $\mathcal{M}(14)$ of coherent semistable rank-2 sheaves with Chern classes $c_1=0, \ c_2=14, \ c_3=0$ on $\mathbb{P}^{3}$ which is generically non-reduced. The construction of this component is based on the technique of elementary transformations of sheaves and famous Mumford's example of a non-reduced component of the Hilbert scheme of smooth space curves of degree 14 and genus 24.
Let $X$ be a del Pezzo surface. When the degree of $X$ is at least 4, we compute the cohomology of a general sheaf in the moduli space of Gieseker semistable sheaves. We also classify the Chern characters for which the general sheaf in the moduli space is non-special, i.e. has at most one nonzero cohomology group. Our results hold for arbitrary polarizations, slope semistability, and semi-exceptional moduli spaces. When the degree of $X$ is at least 3, we further show our construction of certain vector bundles implies the existence of stable and semistable sheaves with respect to the anti-cano…
For a smooth proper scheme over a local field of mixed characteristics which has semistable reduction we define the category of its semistable etale sheaves and under certain hypothesis we prove the appropriate semistable comparison isomorphisms.
We show that the moduli stacks of semistable sheaves on smooth projective varieties are analytic locally on their coarse moduli spaces described in terms of representations of the associated Ext-quivers with convergent relations. When the underlying variety is a Calabi-Yau 3-fold, our result describes the above moduli stacks as critical locus analytic locally on the coarse moduli spaces. The results in this paper will be applied to the wall-crossing formula of Gopakumar-Vafa invariants defined by Maulik and the author.
We give a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. The interest of this proof is that it provides simple effective bounds (depending on the degree of the canonical line bundle) for the degree of the isotrivial finite cove…
This paper has been withdrawn by the author due to a crucial error in the proofs. The error has been corrected and the paper has been expanded in arXiv:0910.5327
We compute the Picard group of the moduli space $U'$ of semistable vector bundles of rank $n$ and degree $d$ on an irreducible nodal curve $Y$ and show that $U'$ is locally factorial. We determine the canonical line bundles of $U'$ and $U'_L$, the subvariety consisting of vector bundles with a fixed determinant. For rank 2, we compute the Picard group of other strata in the compactification of $U'$.
Let X be a smooth projective complex variety, and let G be an algebraic reductive complex group. We define the notion of principal G-sheaf, that generalises the notion of principal G-bundle. Then we define a notion of semistability, and construct the projective moduli space of semistable principal G-sheaves on X. This is a natural compactification of the moduli space of principal G-bundles. This is the announcement note presented by the second author in the conference held at Catania (11-13 April 2001), dedicated to the 60th birthday of Silvio Greco. Detailed proofs will appear elsewhere.
Let $X$ be a Hirzebruch surface, and let $H$ be any ample divisor. In this paper, we algorithmically determine when the moduli space of semistable sheaves $M_{X,H}(r,c_1,c_2)$ is nonempty. Our algorithm relies on certain stacks of prioritary sheaves. We first solve the existence problem for these stacks and then algorithmically determine the Harder-Narasimhan filtration of the general sheaf in the stack. In particular, semistable sheaves exist if and only if the Harder-Narasimhan filtration has length one. We then study sharp Bogomolov inequalities $Δ\geq δ_H(c_1/r)$ for the discriminants of s…
We find the sharp bounds on $h^0(F)$ for one-dimensional semistable sheaves $F$ on a projective variety $X$ by using the spectrum of semistable sheaves. The result generalizes the Clifford theorem. When $X$ is the projective plane $\mathbb{P}^2$, we study the stratification of the moduli space by the spectrum of sheaves. We show that the deepest stratum is isomorphic to a subscheme of a relative Hilbert scheme. This provides an example of a family of semistable sheaves having the biggest dimensional global section space.
We show that certain semistable sheaves on the projective plane with linear Hilbert polynomial are cokernels of semistable morphisms of decomposable sheaves.We exhibit certain locally closed subvarieties of moduli spaces of semistable sheaves as quotients modulo nonreductive groups.
We study the vanishing cycles of a one-parameter smoothing of a complex analytic space and show that the weight filtration on its perverse cohomology sheaf of the highest degree is quite close to the monodromy filtration so that its graded pieces have a modified Lefschetz decomposition. We describe its primitive part using the weight filtration on the perverse cohomology sheaves of the constant sheaves. As a corollary we show in the local complete intersection case that 1 is not an eigenvalue of the monodromy on the reduced Milnor cohomology at any points if and only if the total space and the…
Let $M(d,χ)$ be the moduli space of semistable sheaves of rank 0, Euler characteristic $χ$ and first Chern class $dH (d>0)$, with $H$ the hyperplane class in $\mathbb{P}^2$. We give a description of $M(d,χ)$, viewing each sheaf as a class of matrices with entries in $\bigoplus_{i\geq0}H^0(\mathcal{O}_{\mathbb{P}^2}(i))$. We show that there is a big open subset of $M(d,1)$ isomorphic to a projective bundle over an open subset of a Hilbert scheme of points on $\mathbb{P}^2.$ Finally we compute the classes of M(4,1), M(5,1) and M(5,2) in the Grothendieck group of varieties, especially we conclude…
This paper is a sequel to arXiv:1109.4986, where we proved that a general smooth curve of odd genus, canonically or bicanonically embedded, has semistable finite Hilbert points. Here, we prove that a generic canonically embedded curve of even genus has semistable finite Hilbert points. More precisely, we prove that a generic canonically embedded trigonal curve of even genus has semistable finite Hilbert points. Furthermore, we show that the analogous result fails for bielliptic curves. Namely, the Hilbert points of bielliptic curves are asymptotically semistable but become non-semistable below…
We view the moduli space of semistable sheaves on a K3 surface as a global quotient stack, and compute its cotangent complex in terms of the universal sheaf on the Quot scheme. Relevant facts on the classical and reduced Atiyah classes are reviewed. We also define the notion of a symplectic stack, and show that it includes all moduli stacks of semistable sheaves on K3 surfaces.
Using an existence criterion for good moduli spaces of Artin stacks by Alper-Fedorchuk-Smyth we construct a proper moduli space of rank two sheaves with fixed Chern classes on a given complex projective manifold that are Gieseker-Maruyama-semistable with respect to a fixed Kähler class.
In this paper, we study the Picard group of the moduli space of semistable sheaves on a smooth quadric surface. We polarize the surface by an ample divisor close to the anticanonical class. We focus especially on moduli spaces of sheaves of small discriminant, where we observe new and interesting behavior. Our method relies on constructing certain resolutions for semistable sheaves and applying techniques of geometric invariant theory to the resulting families of sheaves.
On a Weierstraß elliptic surface $X$, we define a `limit' of Bridgeland stability conditions, denoted as $Z^l$-stability, by moving the polarisation towards the fiber direction in the ample cone while keeping the volume of the polarisation fixed. We describe conditions under which a slope stable torsion-free sheaf is taken by a Fourier-Mukai transform to a $Z^l$-stable object, and describe a modification upon which a $Z^l$-semistable object is taken by the inverse Fourier-Mukai transform to a slope semistable torsion-free sheaf. We also study wall-crossing for Bridgeland stability, and show th…
We investigate properties and describe examples of tilt-stable objects on a smooth complex projective threefold. We give a structure theorem on slope semistable sheaves of vanishing discriminant, and describe certain Chern classes for which every slope semistable sheaf yields a Bridgeland semistable object of maximal phase. Then, we study tilt stability as the polarisation $ω$ gets large, and give sufficient conditions for tilt-stability of sheaves of the following two forms: 1) twists of ideal sheaves or 2) torsion-free sheaves whose first Chern class is twice a minimum possible value.
We prove a new restriction theorem for semistable sheaves on varieties in all characteristics strengthening previous results. We also prove restriction theorem for strong semistability for varieties with some non-negativity constrains on the cotangent bundle (e.g., most of Fano and Calabi-Yau varieties).
We investigate the moduli spaces of one- and two-dimensional sheaves on projective K3 and abelian surfaces that are semistable with respect to a nongeneral ample divisor with regard to the symplectic resolvability. We can exclude the existence of new examples of projective irreducible symplectic manifolds lying birationally over components of the moduli spaces of one-dimensional semistable sheaves on K3 surfaces, and over components of many of the moduli spaces of two-dimensional sheaves on K3 surfaces, in particular, of those for rank two sheaves.
We introduce a notion of Gieseker stability for coherent sheaves on tame Deligne-Mumford stacks with projective moduli scheme and some chosen generating sheaf on the stack in the sense of Olsson and Starr \cite{MR2007396}. We prove that this stability condition is open, and pure dimensional semistable sheaves form a bounded family. We explicitly construct the moduli stack of semistable sheaves as a finite type global quotient, and study the moduli scheme of stable sheaves and its natural compactification in the same spirit as the seminal paper of Simpson \cite{MR1307297}. With this general mac…
We resolve pathological wall-crossing phenomena for moduli spaces of sheaves on higher-dimensional base manifolds. This is achieved by considering slope-semistability with respect to movable curves rather than divisors. Moreover, given a projective n-fold and a curve C that arises as the complete intersection of n-1 very ample divisors, we construct a modular compactification of the moduli space of vector bundles that are slope-stable with respect to C. Our construction generalises the algebro-geometric construction of the Donaldson-Uhlenbeck compactification by Joseph Le Potier and Jun Li. Fu…
We study a class of torsion-free sheaves on complex projective spaces which generalize the much studied mathematical instanton bundles. Instanton sheaves can be obtained as cohomologies of linear monads and are shown to be semistable if its rank is not too large, while semistable torsion-free sheaves satisfying certain cohomological conditions are instanton. We also study a few examples of moduli spaces of instanton sheaves.
Let S be a smooth projective surface, K be the canonical class of S and H be an ample divisor such that H.K<0 . In this paper we prove that for any rigid (Ext^1(F,F)=0) semistable sheaf F in the sense of Mumford--Takemoto stability w.r.t. H there exists an exceptional collection (E_1,...,E_n) of sheaves on S such that F can be constructed from {E_i} by a finite number of extensions.
Let $F$ be a torsionfree semistable coherent sheaf on a polarized normal projective variety. We prove that $F$ has a unique maximal locally free subsheaf $V$ such that $F/V$ is torsionfree and $V$ admits a filtration of subbundles for which each successive quotient is stable whose slope is $μ(F)$. We also prove that $F$ has a unique maximal reflexive subsheaf $W$ such that $F/W$ is torsionfree and $W$ admits a filtration of subsheaves for which each successive quotient is a stable reflexive sheaf whose slope is $μ(F)$. We show that these canonical subsheaves behave well with respect to the pul…
We study Le Potier's strange duality conjecture on $\mathbb{P}^2$. We show the conjecture is true for the pair ($W(2,0,2),~M(d,0)$) with $d>0$, where $W(2,0,2)$ is the moduli space of semistable sheaves of rank 2, zero first Chern class and second Chern class 2, and $M(d,0)$ is the moduli space of 1-dimensional semistable sheaves of first Chern class $dH$ and Euler characteristic 0.
We study the equivariant sheaf counting theory on K3 surfaces with finite group actions. Let $\sS=[S/G]$ be a global quotient stack, where $S$ is a K3 surface and $G$ is a finite group acting as symplectic homomorphisms on $S$. We show that the Joyce invariants counting Gieseker semistable sheaves on $\sS$ are independent on the Bridgeland stability conditions. As an application we prove the multiple cover formula of Y. Toda for the counting invariants for semistable sheaves on local K3 surfaces with a symplectic finite group action.
For the derived category of bounded complexes of sheaves on a smooth projective surface, Bridgeland and Arcara-Bertram constructed Bridgeland stability conditions $(Z_m, \mathcal P_m)$ parametrized by $m \in (0, +\infty)$. In this paper, we show that the set of mini-walls in $(0, +\infty)$ of a fixed numerical type is locally finite. In addition, we strengthen a result of Bayer by proving that the moduli of polynomial Bridgeland semistable objects of a fixed numerical type coincides with the moduli of $(Z_m, \mathcal P_m)$-semistable objects whenever $m$ is larger than a universal constant dep…
In this paper we deal with semistable sheaves which can be represented as the cokernel of an injective (or as the kernel of a surjective) morphism $E_1\otimes\CC^m\longrightarrow E_2\otimes\CC^n$ , where $E_1$ and $E_2$ are exceptional bundles. To each such a sheaf we assign the linear map $\CC^m\otimes\hom^*(E_1,E_2)\longrightarrow \CC^n$. We obtain sufficient conditions on the topological invariants of the sheaves for the moduli space of the sheaves to be isomorphic to the quotient of an open subset in $\PP{\cal L}(\CC^m\otimes\hom^*(E_1,E_2),\CC^n)$ under the action of SL$(\CC^m)\times$SL$(…
We describe new irreducible components of the moduli space of rank $2$ semistable torsion free sheaves on the three-dimensional projective space whose generic point corresponds to non-locally free sheaves whose singular locus is either 0-dimensional or consists of a line plus disjoint points. In particular, we prove that the moduli spaces of semistable sheaves with Chern classes $(c_1,c_2,c_3)=(-1,2n,0)$ and $(c_1,c_2,c_3)=(0,n,0)$ always contain at least one rational irreducible component. As an application, we prove that the number of such components grows as the second Chern class grows, an…
Sheaves on non-reduced curves can appear in moduli space of 1-dimensional semistable sheaves over a surface, and moduli space of Higgs bundles as well. We estimate the dimension of the stack $\mathbf{M}_{X}(nC,χ)$ of pure sheaves supported at the non-reduced curve $nC~(n\geq 2)$ with $C$ an integral curve on $X$. We prove that the Hilbert-Chow morphism $h_{L,χ}:\mathcal{M}^H_X(L,χ)\rightarrow |L|$ sending each semistable 1-dimensional sheaf to its support have all its fibers of the same dimension for $X$ Fano or with trivial canonical line bundle and $|L|$ contains integral curves.
Given a family of nodal curves, a semistable modification of it is another family made up of curves obtained by inserting chains of rational curves of any given length at certain nodes of certain curves of the original family. We give comparison theorems between torsion-free, rank-1 sheaves in the former family and invertible sheaves in the latter. We apply them to show that there are functorial isomorphisms between the compactifications of relative Jacobians of families of nodal curves constructed through Caporaso's approach and those constructed through Pandharipande's approach.
We study semistable sheaves of rank $2$ with Chern classes $c_1=0$, $c_2=2$ and $c_3=0$ on the Fano 3-fold $V_5$ of Picard number $1$, degree $5$ and index $2$. We show that the moduli space of such sheaves has a component that is isomorphic to $\mathbb{P}^5$ by identifying it with the moduli space of semistable quiver representations. This provides a natural smooth compactification of the moduli space of minimal instantons, as well as Ulrich bundles of rank $2$ on $V_5$.
We prove Bogomolov's inequality for semistable sheaves on product type varieties in arbitrary characteristic. This gives the first examples of varieties with positive Kodaira dimension in positive characteristic on which Bogomolov's inequality holds for semistable sheaves of any rank. The key ingredient in the proof is a high rank generalization of the slope inequality established by Xiao and Cornalba-Harris. This Bogomolov's inequality is applied to study the positivity of linear systems and semistable sheaves and construct Bridgeland stability conditions on product type surfaces in positive …
We study birational maps among 1) the moduli space of semistable torsion sheaves of Hilbert polynomial $4m+2$ on a smooth quadric surface, 2) the moduli space of semistable torsion sheaves of Hilbert polynomial $m^{2}+3m+2$ on $\mathbb{P}^{3}$, 3) Kontsevich's moduli space of genus zero stable maps of degree 2 to Grassmannian $Gr(2, 4)$. A regular birational morphism from 1) to 2) is described in terms of Fourier-Mukai transform. The map from 3) to 2) is Kirwan's partial desingularization. Also we investigate several geometric properties of 1) by using the variation of moduli spaces of stable …
We give another proof of Le Potier's result and some variants on moduli spaces of semistable sheaves on the projective plane, using the Bridgeland stability conditions. As an application we study the wall-crossing phenomena of the Hilbert schemes of points on the projective plane.
