Related papers: Holomorphic Rank Two Vector Bundles on Blow-ups

The existence problem for vector bundles on a smooth compact complex surface consists in determining which topological complex vector bundles admit holomorphic structures. For projective surfaces, Schwarzenberger proved that a topological…

Let E be a stable rank two normalized vector bundle on P3. If H is a general plane we show that h^0(E_H(1)) \leq 2+c_1. It follows that h^0(E(1)) \leq 2+c_1.

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…

Several families of rank-two vector bundles on Hirzebruch surfaces are shown to consist of all very ample, uniform bundles. Under suitable numerical assumptions, the projectivization of these bundles, embedded by their tautological line…

In this paper we look at Grothendieck's work on classifying holomorphic bundles over the complex projective line. The paper is divided into $4$ parts. The first and second part we build up the necessary background to talk about vector…

We prove that a blowing-up formula for the intersection cohomology of the moduli space of rank 2 Higgs bundles over a curve with trivial determinant holds. As an application, we derive the Poincar\'{e} polynomial of the intersection…

Since Schwarzenberger and his celebrated paper called "Vector bundles on the projective plane" we know that any rank two vector bundle on $\P^2$ is a direct image of a line bundle on a double covering of the plane. This theorem suggests to…

We study the moduli space $\mathfrak M_k^r(\tilde{\mathbb P}^2_{\!q})$ of rank $r$ holomorphic bundles with trivial determinant and second Chern class $c_2=k$, over the blowup $\tilde{\mathbb P}^2_{\!q}$ of the projective plane at $q$…

In this work we study the topology of holomorphic rank two bundles over complex surfaces. We consider bundles that are constructed by glueing and show that under certain conditions the topology of the bundle does not depend on the glueing.…

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…

The goal of this paper is to start a study of aCM and Ulrich sheaves on non-integral projective varieties. We show that any aCM vector bundle of rank two on the double plane is a direct sum of line bundles. As a by-product, any aCM vector…

The depth of a vector bundle E over the projective plane P^2 is the largest integer h such that [E]/h is in the Grothendieck group of coherent sheaves on P^2 where [E] is the class of E in this Grothendieck group. We show that a moduli…

The purpose of this paper is to study minimal monads associated to a rank two vector bundle $\mathcal E$ on $\mathbb P^n$. In particular, we study situations where $\mathcal E$ has $H^i_*(\mathcal E) =0$ for $1<i<n-1$, except for one pair…

Let E be the restriction of the null-correlation bundle on $\mathbb{P}^{3}$ to a hyperplane. In this article, we show that the projective bundle $\mathbb{P}(E)$ is isomorphic to a blow-up of a non-singular quadric in $\mathbb{P}^{4}$ along…

Grothendieck-Birkhoff Theorem states that every finite dimensional vector bundle over the projective line P1 splits as the sum of one dimensional vector bundles. This can be rephrased, in terms of orders, as stating that all maximal orders…

Grothendieck-Birkhoff Theorem states that every finite dimensional vector bundle over the projective line splits as the sum of one dimensional vector bundles. In this work we study simultaneous splittings of two dimensional vector bundles…

For a reduced curve $C:f=0$ in the complex projective plane $\mathbb{P}^2$, we study the set of jumping lines for the rank two vector bundle $T\langle C \rangle $ on $\mathbb{P}^2$, whose sections are the logarithmic vector fields along…

Let $X$ be a smooth variety defined over an algebraically closed field of arbitrary characteristic and $\O_X(H)$ be a very ample line bundle on $X$. We show that for a semistable $X$-bundle $E$ of rank two, there exists an integer $m$…

I prove that a vector bundle on a minuscule homogeneous variety splits into a direct sum of line bundles if and only if its restriction to the union of two-dimensional Schubert subvarieties splits. A case-by-case analysis is done.

We prove that any unstable holomorphic 2-bundle over the complex projective space of complex dimension n at least 6 must split into a direct sum of two holomorphic line bundles. The statement with the weaker dimension condition of n at…