Let X be a K3 surface of degree 8 in P^5 with hyperplane section H. We associate to it another K3 surface M which is a double cover of P^2 ramified on a sextic curve C. In the generic case when X is smooth and a complete intersection of three quadrics, there is a natural correspondence between M and the moduli space M' of rank two vector bundles on X with Chern classes c_1=H and c_2=4. We build on previous work of Mukai and others, giving conditions and examples where M' is fine, compact, non-empty; and birational or isomorphic to M. We also present an explicit calculation of the Fourier-Mukai transform when X contains a line and has Picard number two.
Generalizing the definitions originally presented by Kuznetsov and Faenzi, we study (possibly non locally free) instanton sheaves of arbitrary rank on Fano threefolds. We classify rank 1 instanton sheaves and describe all curves whose structure sheaves are rank 0 instanton sheaves. In addition, we show that every rank 2 instanton sheaf is an elementary transformation of a locally free instanton sheaf along a rank 0 instanton sheaf. To complete the paper, we describe the moduli space of rank 2 instanton sheaves of charge 2 on a quadric threefold $X$, and show that the full moduli space of rank …
We prove sharp bounds on the discriminants of stable rank two sheaves on surfaces in three-dimensional projective space. The key technical ingredient is to study them as torsion sheaves in projective space via tilt stability in the derived category. We then proceed to describe the surface itself as a moduli space of rank two vector bundles on it. Lastly, we give a proof of the Bogomolov inequality for semistable rank two sheaves on integral surfaces in three-dimensional projective space in all characteristics.
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…
Let A be a del Pezzo order on the projective plane over the field of complex numbers. We prove that every torsion-free A-module of rank one can be deformed into a locally free A-module of rank one.
The aim of this paper is to prove the existence of large complete subvarieties in moduli spaces of rank two stable sheaves with arbitrary $c_1$ and sufficiently large $c_2$ on algebraic surfaces. Then we study the restriction of these sheaves to curves of high degree embedded in the surface. In the final section we gives a relation with the spin strata defined by Pidstrigach and Tyurin.
We study the irreducible components of the moduli space of instanton sheaves on $\mathbb{P}^3$, that is rank 2 torsion free sheaves $E$ with $c_1(E)=c_3(E)=0$ satisfying $h^1(E(-2))=h^2(E(-2))=0$. In particular, we classify all instanton sheaves with $c_2(E)\le4$, describing all the irreducible components of their moduli space. A key ingredient for our argument is the study of the moduli space ${\mathcal T}(d)$ of stable sheaves on $\mathbb{P}^3$ with Hilbert polynomial $P(t)=d\cdot t$, which contains, as an open subset, the moduli space of rank 0 instanton sheaves of multiplicity $d$; we desc…
We give a complete classification of semistable rank two sheaves on three-dimensional projective space with maximal third Chern character. This implies an explicit description of their moduli spaces. As an open subset they contain rank two reflexive sheaves with maximal number of singularities. These spaces are irreducible, and apart from a single special case, they are also smooth. This extends a result by Okonek and Spindler to all missing cases and gives a new proof of their result. The key technical ingredient is variation of stability in the derived category.
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…
We prove that the singular locus of a rank 2 instanton sheaf $E$ on $\mathbb{P}^3$ which is not locally free has pure dimension 1. Moreover, we also show that the dual and double dual of $E$ are isomorphic locally free instanton sheaves, and that the sheaves $\mathcal{E}xt^1(E,\mathcal{O}_{\mathbb{P}^3)$ and $E^{\vee\vee}/E$ are rank $0$ instantons. We also provide explicit examples of instanton sheaves of rank $3$ and $4$ illustrating that all of these claims are false for higher rank instanton sheaves.
In this paper, we define $m$-tail reflexive sheaves as reflexive sheaves on projective spaces with the simplest possible cohomology. We prove that the rank of any $m$-tail reflexive sheaf $\mathcal{E}$ on $\mathcal{P}^n$ is greater or equal to $ nm-m$. We completely describe $m$-tail reflexive sheaves on $\mathcal{P}^n$ of minimal rank and we construct huge families of $m$-tail reflexive sheaves of higher rank.
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.
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.
We describe new components of the Gieseker--Maruyama moduli scheme $\mathcal{M}(n)$ of semistable rank 2 sheaves $E$ on $\mathbb{P}^3$ with $c_1(E)=0$, $c_2(E)=n$ and $c_3(E)=0$ whose generic point corresponds to non locally free sheaves. We show that such components grow in number as $n$ grows, and discuss how they intersect the instanton component. As an application, we prove that $\mathcal{M}(2)$ is connected, and identify a connected subscheme of $\mathcal{M}(3)$ consisting of 7 irreducible components.
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…
On a smooth projective threefold, we construct an essentially surjective functor $\mathcal{F}$ from a category of two-term complexes to a category of quotients of coherent sheaves, and describe the fibers of this functor. Under a coprime assumption on rank and degree, the domain of $\mathcal{F}$ coincides with the category of higher-rank PT stable objects, which appear on one side of Toda's higher-rank DT/PT correspondence formula. The codomain of $\mathcal{F}$ is the category of objects that appear on one side of another correspondence formula by Gholampour-Kool, between the generating series…
Let X be an Enriques surface over the field of complex numbers. We prove that there exists a nontrivial quaternion algebra A on X. Then we study the moduli scheme of torsion free A-modules of rank one. Finally we prove that this moduli scheme is an étale double cover of a Lagrangian subscheme in the corresponding moduli scheme on the associated covering K3 surface.
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 compute Betti numbers of the moduli spaces of arbitrary rank stable sheaves on ruled surfaces. Our result generalizes the formula of Goettsche for rank one sheaves and the formula of Yoshioka for rank two sheaves. It also confirms the conjecture of Manschot for arbitrary rank sheaves on the Hirzebruch surfaces.
By the description due to Mukai and Iliev, a smooth prime Fano threefold X of genus 9 is associated to a surface P(V), ruled over a smooth plane quartic Gamma. We use Kuznetsov's integral functor to study rank-2 stable sheaves on X with odd determinant. For each c_2 \geq 7, we prove that a component of their moduli space M_X(2,1,c_2) is birational to a Brill-Noether locus of bundles on Gamma having enough sections when twisted by V. Moreover we prove that M_X(2,1,7) is isomorphic to the blowing-up of the Picard variety Pic^2(Gamma) along the curve parametrizing lines contained in X.
We study the spectrum of rank $2$ torsion free sheaves on $\mathbb{P}^3$ with aim of producing examples of distinct irreducible components of the moduli space with the same spetrcum answering the question presented by Rao for the case of torsion free sheaves. In order to do so, we provide a full description of the spectrum of the sheaves in the moduli space of semistable rank $2$ torsion free sheaves on $\mathbb{P}^3$ with Chern classes $(c_1, c_2,c_3)$ equals to $(-1,2,0)$ and $(0,3,0)$.
We study twisted vector bundles of infinite rank on gerbes, giving a new spin on Grothendieck's famous problem on the equality of the Brauer group and cohomological Brauer group. We show that the relaxed version of the question has an affirmative answer in many, but not all, cases, including for any algebraic space with the resolution property and any algebraic space obtained by pinching two closed subschemes of a projective scheme. We also discuss some possible theories of infinite rank Azumaya algebras, consider a new class of "very positive" infinite rank vector bundles on projective variet…
We study the moduli space of rank 2 instanton sheaves on $\p3$ in terms of representations of a quiver consisting of 3 vertices and 4 arrows between two pairs of vertices. Aiming at an alternative compactification for the moduli space of instanton sheaves, we show that for each rank 2 instanton sheaf, there is a stability parameter $θ$ for which the corresponding quiver representation is $θ$-stable (in the sense of King), and that the space of stability parameters has a non trivial wall-and-chamber decomposition. Looking more closely at instantons of low charge, we prove that there are stabili…
We prove a theorem classifying the equivariant $K$-theoretic pushforwards of the product of arbitrary Schur functors applied to the tautological bundle on the moduli space of framed rank $r$ torsion-free sheaves on $\mathbb{P}^2$, and its dual. This is done by deriving a formula for similar coefficients on Grassmannian varieties, and by thinking of the moduli space as a class in the $K$-theory of the Grassmannian, in analogy with the construction of the Hilbert scheme when the rank is one. Our motivations stem from some vertex operator calculus studied recently by Nekrasov, Okounkov, and the a…
The Grothendieck-Ogg-Shafarevich formula is generalized to any dimensional scheme by Abbes-Kato-Saito. In this paper, we introduce two methods of localization of the characteristic classes for sheaves of rank 1 and compare them. As a corollary of this comparison, we obtain a refinement of a formula proved by Abbes-Saito without denominator for a smooth sheaf of rank 1 which is clean with respect to the boundary.
Fix a Chern character of a stable sheaf on the plane. Assume either the rank is at most 6 or the rank and first Chern class are coprime and the discriminant is sufficiently large. We use recent results of Bayer and Macri on Bridgeland stability to compute the ample cone of the moduli space of Gieseker semistable sheaves with the given Chern character. We recover earlier results, such as those by Stromme and Yoshioka, as special cases.
We compute the expected dimension of the moduli space of torsion-free rank 2 sheaves at a point corresponding to a stable reflexive sheaf and give conditions for the existence of a perfect tangent-obstruction complex on a class of smooth projective threefolds; this class includes Fano and Calabi-Yau threefolds. We also explore both local and global relationships between moduli spaces of reflexive rank 2 sheaves and the Hilbert scheme of curves.
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 shall study the existence condition of slope stable sheaves on Enriques surfaces. We also gives a different proof of the irreducibility of the moduli spaces of rank 2 stable sheaves.
Université de Pau et des Pays de l’Adour（波城大学）；CNRS, LMAP UMR 5142（法国国家科学研究中心，拉马实验室）；Université Bourgogne Europe（勃艮第欧洲大学）；CNRS, IMB UMR 5584（法国国家科学研究中心，IMB实验室）
We study rank-one sheaves and stable pairs on a smooth projective complex surface. We obtain an embedding of the moduli space of limit stable pairs into a smooth space. The embedding induces a perfect obstruction theory, which, over a surface with irregularity 0, agrees with the usual deformation-obstruction theory. The perfect obstruction theory defines a virtual fundamental class on the moduli space. Using the embedding, we show that the virtual class equals the Euler class of a vector bundle on the smooth ambient space. As an application, we show that on $\mathbb{P}^2$, the expected count o…
We classify a special type of arithmetically Cohen-Macaulay sheaves of rank two on reducible and reduced quadric hypersurfaces. As a consequence we show that a reducible and reduced quadric surface is of wild type.
We prove formulae for the motives of stacks of coherent sheaves of fixed rank and degree over a smooth projective curve in Voevodsky's triangulated category of mixed motives with rational coefficients.
Let S be a ruled surface without sections of negative self-intersection. We classify the irreducible components of the moduli stack of torsion-free sheaves of rank 2 sheaves on S. We also classify the irreducible components of the Brill-Noether loci in Hilb^N(P1xP1) given by W_N^0(D)={[X] | h^1(I_X(D)) >= 1 } for D an effective divisor class. Our methods are also applicable to P2 giving new proofs of theorems of Stromme (slightly extended) and Coppo.
We construct geometric realizations of the r-colored bosonic and fermionic Fock space on the equivariant cohomology of the moduli space of framed rank r torsion-free sheaves on CP^2. Using these constructions, we realize geometrically all level one irreducible representations of the affine Lie algebra \hat{gl(r)}. The cyclic group Z_k acts naturally on the moduli space of sheaves, and the fixed-point components of this action are cyclic Nakajima quiver varieties . We realize level k irreducible representations of \hat{gl(r)} on the equivariant cohomology of these quiver varieties.
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 sp…
We show that instanton bundles of rank $r\le 2n-1$, defined as the cohomology of certain linear monads, on an $n$-dimensional projective variety with cyclic Picard group are semistable in the sense of Mumford-Takemoto. Furthermore, we show that rank $r\le n$ linear bundles with nonzero first Chern class over such varieties are stable. We also show that these bounds are sharp.
Let U^{'s}_{L}(n,d) be the moduli space of stable vector bundles of rank $n$ and fixed determinant L of degree d on a nodal curve Y. The moduli space of semistable vector bundles of rank n and degree $d$ will be denoted by U'_Y(n,d). We calculate the Brauer groups of U^{'s}_{L}(n,d)$. We study the question of rationality of $U^{'s}_{L}(n,d)$ and $U'_Y(n,d)$.
Let (X,H) be a polarized, smooth, complex projective surface, and let v be a Chern character on X with positive rank and sufficiently large discriminant. In this paper, we compute the Gieseker wall for v in a slice of the stability manifold of X. We construct explicit curves parameterizing non-isomorphic Gieseker stable sheaves that become S-equivalent along the wall. As a corollary, we conclude that if there are no strictly semistable sheaves of character v, the Bayer-Macri divisor associated to the wall is a boundary nef divisor on the moduli space of sheaves M_H(v). We recover previous resu…
We show that codimension one distributions with at most isolated singularities on certain smooth projective threefolds with Picard rank one have stable tangent sheaves. The ideas in the proof of this fact are then applied to the characterization of certain irreducible components of the moduli space of stable rank 2 reflexive sheaves on $\mathbb{P}^3$, and to the construction of stable rank 2 reflexive sheaves with prescribed Chern classes on general threefolds. We also prove that if $\mathscr{G}$ is a subfoliation of a codimension one distribution $\mathscr{F}$ with isolated singularities, the…
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 $U'$.
We provide a definition of Tanaka-Thomas's Vafa-Witten invariants for étale gerbes over smooth projective surfaces using the moduli spaces of $μ_r$-gerbe twisted sheaves and Higgs sheaves. Twisted sheaves and their moduli are naturally used to study the period-index theorem for the corresponding $μ_r$-gerbe in the Brauer group of the surfaces. Deformation and obstruction theory of the twisted sheaves and Higgs sheaves behave like general sheaves and Higgs sheaves. We define virtual fundamental classes on the moduli spaces and define the twisted Vafa-Witten invariants using virtual localization…
Given an isolated, quasi-homogeneous singularity $X$ we prove that there is a group isomorphism between the group of rank one reflexive sheaves on $X$ and the free abelian group generated by $\mathbb{C}^*$-divisors, modulo linear equivalence. When $\dim(X)=2$ we reduce the problem of finding matrix factorizations of arbitrary reflexive $\mathcal{O}_X$-modules to the same question on rank one reflexive sheaves. We then enumerate the matrix factorizations of all rank one reflexive sheaves. As a consequence, we prove a conjecture of Etingof and Ginzburg on point modules.
We study the moduli space of rank 0 semistable sheaves on some rational surfaces. We show the irreducibility and stable rationality of them under some conditions. We also compute several (virtual) Betti numbers of those moduli spaces by computing their motivic measures.
It is well known that the Chern classes $c_i$ of a rank $n$ vector bundle on $\PP^N$, generated by global sections, are non-negative if $i\leq n$ and vanish otherwise. This paper deals with the following question: does the above result hold for the wider class of reflexive sheaves? We show that the Chern numbers $c_i$ with $i\geq 4$ can be arbitrarily negative for reflexive sheaves of any rank; on the contrary for $i\leq 3$ we show positivity of the $c_i$ with weaker hypothesis. We obtain lower bounds for $c_1$, $c_2$ and $c_3$ for every reflexive sheaf $\FF$ which is generated by $H^0\FF$ on …
We give an existence result on (H,A)-stable sheaves on a K3 or abelian surface X with primitive triple of invariants (rank,first Chern class,Euler characteristics) in the integral cohomology lattice. Such a result yields the existence of singular projective Q-factorial symplectic terminalisations of certain moduli spaces of sheaves on X that are Gieseker semistable with respect to a nongeneral ample divisor.
