Related papers: Splitting criteria for vector bundles on higher di…

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.…

Consider $E$ a vector bundle over a smooth curve $C$. We compute the $\delta$-invariant of all ample ($\mathbb{Q}$-) line bundles on $\mathbb{P}(E)$ when $E$ is strictly Mumford semistable. We also investigate the case when one assumes that…

We provide a classification of globally generated vector bundles with $c_1 = 5$ on the projective 3-space. The classification is complete (except for one case) but not as detailed as the corresponding classification in the case $c_1 = 4$…

Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of…

Given a vector bundle $E$ on a tree of smooth rational curves $C$, we give necessary and sufficient conditions for a vector bundle $E'$ on $\mathbb{P}^1$ to specialize to $E$ on $C$, generalizing the rank 2 case, due to Coskun.

Given a smooth projective toric variety $X$ of Picard rank 2, we resolve the diagonal sheaf on $X \times X$ by a linear complex of length $\dim{X}$ consisting of finite direct sums of line bundles. As applications, we prove a new case of a…

Let $k$ be an algebraically closed field of any characteristic. Let $X$ be a polarized irreducible smooth projective algebraic variety over $k$. We give criterion for semistability and stability of system of Hodge bundles on $X$. We define…

We obtain criteria for detecting complete intersections in projective varieties. Motivated by a conjecture of Hartshorne concerning subvarieties of projective spaces, we investigate situations when two-codimensional smooth subvarieties of…

A symplectic or orthogonal bundle $V$ of rank $2n$ over a curve has an invariant $t(V)$ which measures the maximal degree of its isotropic subbundles of rank $n$. This invariant $t$ defines stratifications on moduli spaces of symplectic and…

For a vector bundle V over a curve X of rank n and for each integer r in the range 1 \le r \le n-1, the Segre invariant s_r is defined by generalizing the minimal self-intersection number of the sections on a ruled surface. In this paper we…

In this paper we show that semistable vector bundles on a Castelnuovo curve of genus g >= 2 have theta divisors. As a corollary, we deduce that semistable vector bundles on a smooth, general curve of genus g >= 2 which extend to semistable…

In Butler, J.Differential Geom. 39 (1):1--34,1994, the author gives a sufficient condition for a line bundle associated with a divisor D to be normally generated on $X=P(E)$ where E is a vector bundle over a smooth curve C. A line bundle…

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…

The paper studies a rank 2 vector bundle on P1 x P3. Similarly to the Horrocks - Mumford bundle on P4 this vector bundle encodes a lot of geometric information. It is defined via the Serre construction by an abelian surface in P1 x P3. The…

Suppose $X$ is a smooth affine real variety and $\mathscr{E}$ is a vector bundle over $X$. We analyze the problem of splitting off a free rank one summand from $\mathscr{E}$ in corank $0$ and $1$. The problem in corank $0$ can be viewed as…

We exhaustively classify topological equivariant complex vector bundles over two-torus under a compact Lie group (not necessarily effective) action. It is shown that inequivariant Chern classes and isotropy representations at (at most) six…

We show that the idea used by Kempf (1990) in order to obtain a splitting criterion for vector bundles on projective spaces leads to an elementary proof of the Babylonian tower theorem for this class of bundles, a result due to Barth--Van…

In this paper, complex vector bundles of rank $r$ over $8$-dimensional spin$^{c}$ manifolds are classified in terms of the Chern classes of the complex vector bundles and the cohomology ring of the manifolds, where $r = 3$ or $4$. As an…

We show that the semistable locus is the unique maximal open substack of the moduli stack of principal bundles over a curve that admits a schematic moduli space. For rank $2$ vector bundles it coincides with the unique maximal open substack…

Let E and F be vector bundles over a complex projective smooth curve X, and suppose that 0 -> E -> W -> F -> 0 is a nontrivial extension. Let G be a subbundle of F, and D an effective divisor on X. We give a criterion for the subsheaf G(-D)…