We study moduli spaces $M_X(r,c_1,c_2)$ parametrizing slope semistable vector bundles of rank $r$ and fixed Chern classes $c_1, c_2$ on a ruled surface whose base is a rational nodal curve. We show that under certain conditions, these moduli spaces are irreducible, smooth and rational (when non-empty). We also prove that they are non-empty in some cases.
We show that for a rational ruled surface defined over real numbers, the moduli space $M_X(r,c_1,c_2)$ is rational as a variety defined over $\mathbb R$.
In this paper, we compare the moduli spaces of rank-3 vector bundles stable with respect to different ample divisors over rational ruled surfaces. We also discuss the irreducibility, unirationality, and rationality of these moduli spaces.
Very ampleness criteria for rank 2 vector bundles over smooth, ruled surfaces over rational and elliptic curves are given. The criteria are then used to settle open existence questions for some special threefolds of low degree.
Fix a ruled surface S obtained as the projective completion of a line bundle L on a complex elliptic curve; we study the moduli problem of parametrizing certain pairs consisting of a sheaf E on S and a map of E to a fixed reference sheaf on S. We prove that the full moduli stack for this problem is representable by a scheme in some cases. Moreover, the moduli stack admits an action by the group ${\mathbf C}^*$, and we determine its fixed-point set, which leads in some special cases to explicit formulas for the rational homology of the moduli space.
The present paper concerns the invariants of generically nef vector bundles on ruled surfaces. By Mehta - Ramanathan Restriction Theorem and by Miyaoka characterization of semistable vector bundles on a curve, the generic nefness can be considered as a weak form of semistability. We establish a Bogomolov type inequality for generically nef vector bundles with nef general fiber restriction on ruled surfaces with no negative section. This gives an affermative answer in this case to a problem posed by Th. Peternell. Concerning ruled surfaces with a negative section, we prove a a similar result fo…
We consider the following question: for which invariants $g$ and $e$ is there a geometrically ruled surface $S \rightarrow C$ over a curve $C$ of genus $g$ with invariant $e$ such that $S$ is the support of an Ulrich line bundle with respect to a very ample line bundle? A surprising relation between the existence of certain proper Theta divisors on some moduli spaces of vector bundles on $C$ with the existence of Ulrich line bundles on $S$ will be the key to completely solve the above question. The relation is realized by translating the vanishing conditions characterizing Ulrich line bundles …
Let $S$ be a ruled surface inside a smooth threefold $W$ and let $E$ be a vector bundle on a formal neighborhood of $S.$ We find minimal conditions under which the local moduli space of $E$ is finite dimensional and smooth. Moreover, we show that $E$ is a flat limit of a flat family of vector bundles whose general element we describe explicitly.
Let $π: X = \mathbb{P}_C(E) \longrightarrow C$ be a ruled surface over an algebraically closed field $k$ of characteristic 0, with a fixed polarization $L$ on $X$. In this paper, we show that pullback of a (semi)stable Higgs bundle on $C$ under $π$ is a $L$-(semi)stable Higgs bundle. Conversely, if $(V,θ)$ is a $L$-(semi)stable Higgs bundle on $X$ with $c_1(V)= π^*(\bf d \rm)$ for some divisor $\bf d \rm $ of degree $d$ on $C$ and $c_2(V)=0$, then there exists a (semi)stable Higgs bundle $(W,ψ)$ of degree $d$ on $C$ whose pullback under $π$ is isomorphic to $(V,θ)$. As a consequence, we get an…
In this paper we shall generalize the chamber structure of polarizations defined by Qin, and as an application we shall compute the Picard groups of moduli spaces of stable sheaves on a non-rational ruled surface.
Atiyah classifies vector bundles on elliptic curves $E$ over an algebraically closed field of any characteristic. On the other hand, a rank $2$ vector bundle on $E$ defines a surface $S$ with a $\mathbb{P}^1$-bundle structure on $E$. We study when $S$ has an elliptic fibration according to the Atiyah's classification, and what kinds of singular fibers appear.
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.
Let $S$ be a birationally ruled surface. We show that the moduli schemes $M_S(r,c_1,c_2)$ of semistable sheaves on $S$ of rank $r$ and Chern classes $c_1$ and $c_2$ are irreducible for all $(r,c_1,c_2)$ provided the polarization of $S$ used satisfies a simple numerical condition. This is accomplished by proving that the stacks of prioritary sheaves on $S$ of fixed rank and Chern classes are smooth and irreducible.
This is an expository article on the recent developments of Hodge theory on moduli spaces of smooth projective varieties with semi-ample canonical line bundles.
In this paper, it is proved that certain stable rank-3 vector bundles can be written as extensions of line bundles and stable rank-2 bundles. As an application, we show the rationality of certain moduli spaces of stable rank-3 bundles over the projective plane P^2.
We prove that the normalized Poincaré bundle on the moduli space of stable rank $r$ vector bundles with a fixed determinant on a smooth projective curve $X$ induces a family of nef vector bundles on the moduli space. Two applications follow. We show that when the genus of $X$ is large, the derived category of $X$ is embedded into the derived category of the moduli space for arbitrary rank and coprime degree, which extends the results of Narasimhan, Fonarev-Kuznetsov, and Belmans-Mukhopadhyay. As the second application, we construct a family of ACM bundles on the moduli space. A key ingredient …
Recent work on the log minimal model program for the moduli space of curves, as well as past results of Caporaso, Pandharipande, and Simpson motivate an investigation of compactifications of the universal moduli space of slope semi-stable vector bundles over moduli spaces of curves arising in the Hassett-Keel program. Our main result is the construction of a universal moduli space of slope semi-stable sheaves which compactifies the moduli space of vector bundles over the moduli space of pseudo-stable curves.
We construct a compactification of the universal moduli space of semistable principal $G$-bundles over $\overline{\textrm{M}}_{g}$, the fibers of which over singular curves are the moduli spaces of $δ$-semistable singular principal G-bundles.
We discuss some relations of moduli of sheaves on rational surfaces by using universal extensions. These are a generalization of Maruyama's method to construct Uhlenbeck compactification of moduli of vector bundles.
We examine the moduli of framed holomorphic bundles over the blowup of a complex surface, by studying a filtration induced by the behavior of the bundles on a neighborhood of the exceptional divisor.
In this paper, we study rank 2 (quasi) parabolic bundles over the Riemann sphere with an effective divisor and these moduli spaces. First we consider a criterium when a parabolic bundle admits a unramified irregular singular parabolic connection. Second, to give a good compactification of the moduli space of semistable parabolic bundles, we introduce a generalization of parabolic bundles, which is called refined parabolic bundles. Third, we discuss a stability condition of refined parabolic bundles and define elementary transformations of the refined parabolic bundles. Finally, we describe the…
We study moduli of holomorphic vector bundles on non-compact varieties. We discuss filtrability and algebraicity of bundles and calculate dimensions of local moduli. As particularly interesting examples, we describe numerical invariants of bundles on some local Calabi-Yau threefolds.
We consider the moduli space of stable vector bundles on curves embedded in P_2 with Hilbert polynomial 3m+1 and construct a compactification of this space by vector bundles. The result is a blow up of the Simpson moduli space M_{3m+1}(P_2).
We generalize the construction of M. Lieblich for the compactification of the moduli stack of $\PGL_r$-bundles on algebraic spaces to the moduli stack of Tanaka-Thomas $\PGL_r$-Higgs bundles on algebraic schemes. The method we use is the moduli stack of Higgs version of Azumaya algebras. In the case of smooth surfaces, we obtain a virtual fundamental class on the moduli stack of $\PGL_r$-Higgs bundles. An application to the Vafa-Witten invariants is discussed.
We present a systematic study of involutions on the moduli space of $G$-Higgs bundles over an elliptic curve $X$, where $G$ is complex reductive affine algebraic group. The fixed point loci in the moduli space of $G$-Higgs bundles on $X$, and in the moduli space of representations of the fundamental group of $X$ into $G$, are described. This leads to an explicit description of the moduli spaces of pseudo-real $G$-Higgs bundles over $X$.
In this paper we construct the moduli of semistable principal bundles with structure group a semisimple group, over nonsingular curves in positive characteristic, with good bounds on the prime.
Let X be a geometrically irreducible smooth projective curve over a field k. We describe the algebra of endomorphisms of indecomposable unstable vector bundles over X of rank 2 and degree d. Fixing some numerical invariants, namely the Harder-Narasimhan type and the dimension of the algebra of endomorphisms, we construct algebraic stacks and moduli schemes for such bundles.
We compute the Brauer group of the moduli stack of stable PGL(r)-bundles on a curve $X$ over an algebraically closed field of characteristic zero. We also show that the Brauer group of such a moduli stack coincides with the Brauer group of the smooth locus of the corresponding coarse moduli space of stable PGL(r)-bundles on $X$.
We use hypersurfaces containing unexpected linear spaces to construct interesting vector bundles on complete intersection surfaces in projective space. We discover examples of moduli spaces of rank 2 stable bundles on surfaces of Picard rank one with arbitrarily many connected components.
In this article we extend the proof given by Biswas and Gomez of a Torelli theorem for the moduli space of Higgs bundles with fixed determinant, to the parabolic situation.
A morphism of the moduli functor of admissible semistable pairs to the Gieseker -- Maruyama moduli functor (of semistable coherent torsion-free sheaves) with the same Hilbert polynomial on the surface, is constructed. It is shown that these functors are isomorphic, and main components of moduli scheme for semistable admissible pairs are isomorphic to main components of the Gieseker -- Maruyama moduli scheme
We prove that the moduli spaces of parabolic symplectic/orthogonal bundles on a smooth curve are globally F regular type. As a consequence, all higher cohomology of theta line bundle vanish. During the proof, we develop a method to estimate codimension, and consider the infinite grassmannians for parabolic $G$ bundles.
Let $X$ be a compact Riemann surface of genus $g\geq 3$ and let $\mathcal{M}_{\mathrm{par}}$ be the moduli space of semistable parabolic bundles over $X$. Let $\mathcal{M}_{\mathrm{parH}}$ denote the moduli space of semistable parabolic Higgs bundles over $X$. In this article, we study the Picard group of $\mathcal{M}_{\mathrm{par}}$ and $\mathcal{M}_{\mathrm{parH}}$.
We study noncompact Calabi-Yau threefolds, their moduli spaces of vector bundles and deformation theory. We present Calabi-Yau threefolds that have infinitely many distinct deformations, constructing them explicitily, and describe the effect that such deformations produce on moduli spaces of vector bundles.
Let $C$ be a smooth irreducible complex projective curve of genus $g$ and let $B^k(2,K_C)$ be the Brill-Noether loci parametrizing classes of (semi)-stable vector bundles $E$ of rank two with canonical determinant over $C$ with $h^0(C,E)\geq k$. We show that $B^4(2,K_C)$ it has an irreducible component $\mathcal B$ of dimension $3g-13$ on a general curve $C$ of genus $g\geq 8$. Moreover, we show that for the general element $[E]$ of $\mathcal B$, $E$ fits in a exact sequence $0\to\mathcal O_C(D)\to E\to K_C(-D)\to 0$, with $D$ a general effective divisor of degree three, and the corresponding …
Let M(j) denote the moduli space of bundles on the blown-up plane which restrict to the exceptional divisor as O(j)+O(-j). We show that there is a topological embedding of M(j) into M(j+1).
We study the rational Chow motives of certain moduli spaces of vector bundles on a smooth projective curve with additional structure (such as a parabolic structure or Higgs field). In the parabolic case, these moduli spaces depend on a choice of stability condition given by weights; our approach is to use explicit descriptions of variation of this stability condition in terms of simple birational transformations (standard flips/flops and Mukai flops) for which we understand the variation of the Chow motives. For moduli spaces of parabolic vector bundles, we describe the change in motive under …
Mathematical instanton bundles of rank 4 and $c_2=2$ on ${\mathbb P}^4$ have a smoothquasiprojective moduli space, which is shown via a direct GIT construction. A complete classification of jumping lines of these vector bundles is obtained. The duals of these bundles are described. Dimension computations and irreducibility proofs for some "interesting" subsets of the moduli space are presented. Restrictions of vector bundles from ${\mathbb P}^4$ to ${\mathbb P}^3$ are shown to produce mathematical instanton bundles on ${\mathbb P}^3$.
We study certain moduli spaces of stable vector bundles of rank two on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing intermediate cohomology. In one case, all except one component of the moduli space has such vector bundles.
We prove a canonical identifications of the spaces of generalized theta functions on the moduli spaces of anti-invariant vector bundles in the ramified case and the conformal blocks associated to twisted Kac-Moody affine algebras.
We study the geometry of the moduli stack of vector bundles of fixed rank and degree over an algebraic curve by introducing a filtration made of open substacks build from $(k, l)$-stable vector bundles. $(k, l)$-stability was introduced by Narasimhan and Ramanan to study the geometry of the coarse moduli space of stable bundles. We will exhibit the stacky picture and analyse the geometric and cohomological properties of the moduli stacks of $(k, l)$-stable vector bundles. For particular pairs $(k, l) $ of integers we also show that these moduli stacks admit coarse moduli spaces and we discuss …
We prove that the moduli spaces of framed bundles over a smooth projective curve are rational. We compute the Brauer group of these moduli spaces to be zero under some assumption on the stability parameter.
We define a new class of holomorphic (n-1)-bundles on the n-dimensional complex projective space, that contains the Tango bundles and their stable Horrock's pull-backs; we show that these bundles are invariant under small deformations and that their isomorphism classes correspond to smooth points of moduli spaces.
Let $X$ be a smooth irreducible projective curve with an involution $σ$. A vector bundle $E$ over $X$ is called anti-invariant if there exists an isomorphism $σ^*E\rightarrow E^*$. In this paper, we give a construction of the moduli spaces of anti-invariant vector bundles over $X$.
In this paper we proof the existence of a linearization for singular principal G-bundles not depending on the base curve. This allow us to construct the relative compact moduli space of δ-(semi)stable singular principal G-bundles over families of reduced projective and connected nodal curves, and to reduce the construction of the universal moduli space over $\overline{M}_{g}$ to the construction of the universal moduli space of swamps.
