Moduli Space Of Vector Bundles Research Articles

$A_{\infty}$-structures, Brill–Noether loci and the Fourier–Mukai transform

We use the -formalism to study variation in cohomology spaces under formal deformations of coherent sheaves on projective varieties. As an application we describe formal neighborhoods of some singular points in twisted Brill–Noether loci in the moduli spaces of vector bundles on a curve. Another application is a computation of the Fourier–Mukai transform of some natural line bundles on symmetric powers of a curve.

Elementary transformations and the rationality of the Moduli Spaces of Vector Bundles on ?2

In this work, using elementary transformations and prioritary sheaves, we establish birational maps between certain moduli spaces of stable vector bundles over ℙ2 with the same rank and different Chern classes. As an application we give a simple proof of the rationality of the moduli spaces M(r; c 1, c 2) of rank r stable vector bundles over ℙ2 with given Chern classes for a huge families of the triples (r; c 1, c 2).

Stable maps and Quot schemes

In this paper we study the relationship between two different compactifications of the space of vector bundle quotients of an arbitrary vector bundle on a curve. One is Grothendieck's Quot scheme, while the other is a moduli space of stable maps to the relative Grassmannian. We establish an essentially optimal upper bound on the dimension of the two compactifications. Based on that, we prove that for an arbitrary vector bundle, the Quot schemes of quotients of large degree are irreducible and generically smooth. We precisely describe all the vector bundles for which the same thing holds in the case of the moduli spaces of stable maps. We show that there are in general no natural morphisms between the two compactifications. Finally, as an application, we obtain new cases of a conjecture on effective base point freeness for pluritheta linear series on moduli spaces of vector bundles.

Theta Functions and Szegö Kernels

We study relations between two fundamental constructions associated to vector bundles on a smooth complex projective curve; the theta function (a section of a line bundle on the moduli space of vector bundles) and the Szego kernel (a section of a vector bundle on the square of the curve). Two types of relations are demonstrated. First, we establish a higher-rank version of the prime form, describing the pullback of determinant line bundles by difference maps, and show that the theta function pulls back to the determinant of the Szego kernel. Next, we prove that the expansion of the Szego kernel at the diagonal gives the logarithmic derivative of the theta function over the moduli space of bundles for a fixed, or moving, curve. In particular, we recover the identification of the space of connections on the theta line bundle with moduli space of flat vector bundles, when the curve is fixed. When the curve varies, we identify this space of connections with the moduli space of extended connections, which we introduce.

Rational surfaces and moduli spaces of vector bundles on rational surfaces

Let X be a smooth algebraic surface, \( L \in \textrm{Pic}(X) \) and H an ample divisor on X. Set M X,H (2; L, c 2) the moduli space of rank 2, H-stable vector bundles F on X with det(F) = L and c 2(F) = c 2. In this paper, we show that the geometry of X and of M X,H (2; L, c 2) are closely related. More precisely, we prove that for any ample divisor H on X and any \( L \in \textrm{Pic}(X) \), there exists \( n_0 \in \mathbb{Z} \) such that for all \( n_0 \leqq c_2 \in \mathbb{Z} \), M X,H (2; L, c 2) is rational if and only if X is rational.

Rationality of moduli spaces of vector bundles on rational surfaces

Let X be a smooth rational surface. In this paper, we prove the rationality of the moduli space MX,L(2; c1; c2) of rank two L-stable vector bundles E on X with det (E) = c1 ∈ Pic(X) and c2(E) = c2 ≫ 0.

THE COMPACTIFIED JACOBIAN OF A REDUCIBLE HYPERELLIPTIC CURVE

There is a one-to-one correspondence between nonsingular pencils of quadrics in an odd-dimensional projective space and nonsingular hyperelliptic curves. There is also a close relation between the moduli spaces of vector bundles of ranks 1, 2 on the hyperelliptic curve and spaces associated to the intersection of quadrics of the corresponding pencil. This has been studied extensively by many authors [1, 5, 8, 9, 11]. An interesting application of this was in the proof of the Verlinde formulae by Szenes [12]. This relationship was generalised to a relation between spaces related to singular pencils of quadrics associated to irreducible nodal curves and moduli of torsion-free sheaves on the nodal curves [2, 4]. The projective models of the moduli of vector bundles on smooth curves (given by this relationship) led to explicit representations of integrable systems of [6, 7]. Similar results on singular curves would provide important classes of special solutions. The pencils of quadrics considered in previous works had associated Segre symbols of type [1 … 1] and [2 … 2 1 … 1]. Kleiman (and Emma Previato) had raised the question: What happens in the case of Segre symbol [2 2 2]? In this case the associated curve Y has two rational components intersecting transversely at three points. Unlike in the previous cases, the compactified Jacobian has more than one component. Moreover the stability of line bundles and the compactification of the Jacobian depends on the choice of a polarisation on Y, making the problem more interesting and difficult.

Moduli spaces of vector bundles on higher dimensional varieties

Moduli spaces of vector bundles on higher dimensional varieties

Dimension estimates for Hilbert schemes and effective base point freeness on moduli spaces of vector bundles on curves

We give an upper bound on the dimension of the Hilbert scheme of quotients of an arbitrary vector bundle on a smooth projective curve, depending on the minimal degree of such a quotient. The bound is used for deriving effective base point freeness statements for generalized theta linear series on moduli spaces of vector bundles.

Birational classification of moduli spaces of vector bundles over

The depth of a vector bundle E over ▪ is the largest integer h such that [ E]/ h is in the Grothendieck group of coherent sheaves on ▪ where [ E] is the class of E in this Grothendieck group. We show that a moduli space of vector bundles is birational to a suitable number of h by h matrices up to simultaneous conjugacy where h is the depth of the vector bundles classified by the moduli space. In particular, such a moduli space is a rational variety if h ≤ 4 and is stably rational when h divides 420.

The hodge numbers of the moduli spaces of vector bundles over a Riemann surface

Inductive formulas for the Betti numbers of the moduli spaces of stable holomorphic vector bundles of coprime rank and degree over a fixed Riemann surface of genus at least two were obtained by Harder, Narasimhan, Desale and Ramanan using number theoretic methods and the Weil conjectures and were rederived by Atiyah and Bott using gauge theory. In this note we observe that there are similar inductive formulas for determining the Hodge numbers of these moduli spaces.

Poisson Structures on Moduli Spaces of Framed Vector Bundles on Surfaces

In this paper we prove that the moduli spaces of framed vector bundles over a surface X, satisfying certain conditions, admit a family of Poisson structures parametrized by the global sections of a certain line bundle on X. This generalizes to the case of framed vector bundles previous results obtained in [B2] for the moduli space of vector bundles over a Poisson surface. As a corollary of this result we prove that the moduli spaces of framed SU(r) – instantons on S4 = ℝ4 ∪ {∞} admit a natural holomorphic symplectic structure.

On the Connectedness of the Real Part of Moduli Spaces of Vector Bundles on Real Algebraic Surfaces

AbstractLet X be a smooth projective surface with q(X) = 0 defined over R and M(X;r;c1;c2;H) the moduli space of H-stable rank r vector bundles on X with Chern classes c1 and c2. Assume either r = 3 and X(R) connected or r = 3 and X(R) =ø or r=2 and X(R) = ø. We prove that quite often M is connected.

Intermediate Jacobians and Hodge structures of moduli spaces

The mixed Hodge structure on the low degree cohomology of the moduli space of vector bundles on a curve is studied. Analysis of the third cohomology yields a new proof of a Torelli theorem.

A modular compactification of the general linear group

We define a certain compactifiction of the general linear group and give a modular description for its points with values in arbitrary schemes. This is a first step in the construction of a higher rank generalization of Gieseker's degeneration of moduli spaces of vector bundles over a curve. We show that our compactification has similar properties as the “wonderful compactification” of algebraic groups of adjoint type as studied by de Concini and Procesi. As a byproduct we obtain a modular description of the points of the wonderful compactification of \operatorname{PGl}_n .

A remark on the $2$-dimensional moduli spaces of vector bundles on K$3$ surfaces

A remark on the $2$-dimensional moduli spaces of vector bundles on K$3$ surfaces

Smoothness in Algebraic Geography

Let $V$ be a vector space and let $\{ e_1,\hdots,e_r \}$ be a basis of $V$. An algebra structure on $V$ is given by $r^3$ structure constants $c_{ij}^h$ where $e_i\cdot e_j = \sum_h c_{ij}^h e_h$. We require this algebra structure to be associative with unit element $e_1$. This limits the sets of structure constants $(c_{ij}^h)$ to a subvariety of $k^{r^3}$, which we denote by $\mbox{Alg}_r$. Base changes in $V$ (leaving $e_1$ fixed) give rise to the natural transport of structure action on $\mbox{Alg}_r$; isomorphism classes of $r$-dimensional algebras are in one-to-one correspondence with the orbits under this action. In this paper we classify the smooth closed subvarieties of $\mbox{Alg}_r$ which are invariant under the transport of structure action and study the singularities which may occur. In particular, we show that if $r=n^2$ then the closure of the locus corresponding to the matrix algebra $M_n(k)$ is not smooth for $n \geq 3$. This gives a negative answer to a question of Seshadri on the desingularization of moduli spaces of vector bundles over curves.

Picard groups of the moduli spaces of vector bundles

Picard groups of the moduli spaces of vector bundles

On the rationality of moduli spaces of vector bundles on Fano surfaces

In this paper we prove the rationality of the moduli space of rank two, H-stable vector bundles on Fano surfaces.

Rationality of Moduli Spaces of Parabolic Bundles

The moduli space of parabolic bundles with fixed determinant over a smooth curve of genus greater than one is proved to be rational whenever one of the multiplicities associated to the quasi-parabolic structure is equal to one. It follows that if rank and degree are coprime, the moduli space of vector bundles is stably rational, and the bound obtained on the level is strong enough to conclude rationality in many cases.