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The moduli space of Gieseker vector bundles is a compactification of moduli of vector bundles on a nodal curve. This moduli space has only normal crossing singularity and it provides a flat degeneration. We prove a Torelli type theorem for…

We discuss topological rigidity of vector bundles with asymptotically conical (AC) total spaces of rank greater than 1 with a sufficiently connected link; our focus will mainly be on ALE (asymptotically locally Euclidean) bundles. Within…

Consider the moduli space $M_C(r; K_C)$ of stable rank r vector bundles on a curve $C$ with canonical determinant, and let $h$ be the maximum number of linearly independent global sections of these bundles. If $C$ embeds in a K3 surface $X$…

We show that the category of affine bundles over a smooth manifold M is equivalent to the category of affine spaces modelled on projective finitely generated C^\infty(M)-modules. Using this equivalence of categories, we are able to give an…

We generalize Bertram's work on rank two vector bundles to an irreducible projective nodal curve C. We use extensions of a line bundle L by O_C and the associated `forgetful' map to study a compactification of the moduli space of…

Let X be a smooth complex projective curve of genus g bigger or equal to 1. If g is bigger than 1 assume further that X is either bielliptic or with general moduli. Under a natural condition on slopes, we prove that there exists a short…

Let $\text{M}_C( 2, \mathcal{O}_C) \cong \mathbb{P}^3$ denote the coarse moduli space of semistable vector bundles of rank $2$ with trivial determinant over a smooth projective curve $C$ of genus $2$ over $\mathbb{C}$. Let $\beta_C$ denote…

We study different notions of slope of a vector bundle over a smooth projective curve with respect to ampleness and affineness in order to apply this to tight closure problems. This method gives new degree estimates from above and from…

In this paper we consider a manifold with a dynamical vector field and inquire about the possible tangent bundle structures which would turn the starting vector field into a second order one. The analysis is restricted to manifolds which…

Let $C$ be a hyperelliptic curve of genus $g \geq 3$. We give a new description of the theta map for moduli spaces of rank 2 semistable vector bundles with trivial determinant. In orther to do this, we describe a fibration of (a birational…

This is a continuation of "Rational families of vector bundles on curves, I". Let C be a smooth projective curve of genus at least 2 and let M be the moduli space of rank 2, stable vector bundles on C, with fixed determinant of degree 1.…

Let $C$ be an irreducible smooth complex projective curve, and let $E$ be an algebraic vector bundle of rank $r$ on $C$. Associated to $E$, there are vector bundles ${\mathcal F}_n(E)$ of rank $nr$ on $S^n(C)$, where $S^n(C)$ is $ $n$-th…

We determine the splitting type of the Verlinde vector bundles in higher genus in terms of simple semihomogeneous factors. In agreement with strange duality, the simple factors are interchanged by the Fourier-Mukai transform, and their…

Given two vector bundles E and F on a variety X and a morphism from Sym^2(E) to F, we compute the cohomology class of the locus in X where the kernel of this morphism contains a quadric of prescribed rank. Our formulas have many…

Let C be a smooth projective curve with genus g>1 and Clifford index c(C) and let L be a line bundle on C generated by its global sections. The morphism i:C -->P(H^0(L))=P is well-defined and i*T is the restriction to C of the tangent…

Inside the moduli space of curves of genus 2 with 2 marked points we consider the loci of curves admitting a map to P^1 of degree d totally ramified over the two marked points, for d>= 2. Such loci have codimension two. We compute the class…

In this note we give an easy proof of the existence of generically smooth components of the expected dimension of certain Brill--Noether loci of stable rank 2 vector bundles on a curve with general moduli, with related applications to…

We study the moduli stacks of real vector bundles of fixed rank and degree on a type I real algebraic curve and determine its mod 2 cohomology algebra in terms of characteristic classes.

We use the theory of p-curvature of connections to analyze stable vector bundles of rank 2 on curves of genus 2 which pull back to unstable bundles under the Frobenius morphism. We take two approaches, first using explicit formulas for…

Given an orthogonal bundle $E$ over a smooth projective curve $X$ we define a Hecke transformation in the moduli space of orthogonal bundles by performing an elementary transformation with respect to a Lagrangian submodule $L \subset…