In this work we introduce the definition of Schwarzenberger bundle on a Grassmannian. Recalling the notion of Steiner bundle, we generalize the concept of jumping pair for a Steiner bundle on a Grassmannian. After studying the jumping locus variety and bounding its dimension, we give a complete classification of Steiner bundle with jumping locus of maximal dimension, which all are Schwarzenberger bundles.
Verlinde bundles are vector bundles $V_k$ arising as the direct image $π_*(\mathcal L^{\otimes k})$ of polarizations of a proper family of schemes $π\colon \mathfrak X \to S$. We study the splitting behavior of Verlinde bundles in the case where $π$ is the universal family $\mathfrak X \to |\mathcal O(d)|$ of hypersurfaces of degree $d$ in $\mathbb P^n$ and calculate the cohomology class of the locus of jumping lines of the Verlinde bundles $V_{d+1}$ in the cases $n=2,3$.
In this work we study $k$-type uniform Steiner bundles, being $k$ the lowest degree of the splitting. We prove sharp upper and lower bounds for the rank in the case $k=1$ and moreover we give families of examples for every allowed possible rank and explain which relation exists between the families. After dealing with the case $k$ in general, we conjecture that every $k$-type uniform Steiner bundle is obtained through the proposed construction technique.
A Steiner bundle is a vector bundle on projective space arising as the cokernel of the map defined by a matrix of linear forms. These come up in various geometric settings, and by now they are the subject of a considerable literature. Starting with work of Dolgachev and Kapranov from 1993, several authors have considered the question of whether one can recover from the bundle the geometric data used to construct it. Here we prove such Torelli-type statements for the tautological bundles associated to sufficiently positive divisors on any very ample linear series. In an appendix, we give a new …
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 $C$. We point out the relations of these jumping lines with the Lefschetz type properties of the Jacobian module of $f$ and with the Bourbaki ideal of the module of Jacobian syzygies of $f$. In particular, when the vector bundle $T\langle C \rangle $ is unstable, a line is a jumping line if and only if it meets the 0-dimensional subscheme defined by this Bourba…
In this paper, we study the cohomology of vector bundles on projective space defined as kernels or cokernels of general maps $V_1 \to V_2$, where the $V_i$ are direct sums of line bundles or certain exceptional bundles. We prove an asymptotic cohomology vanishing theorem. We characterize the stability of general Steiner bundles on projective space. We also give a criterion for a Steiner bundle to be ample.
Let $X$ be a compact complex manifold and $E$ be a holomorphic vector bundle on $X$. Given a deformation $(\mathcal{X},\mathcal{E})$ of the pair $(X,E)$ over a small polydisk $B$ centered at the origin, we study the jumping phenomenon of the cohomology groups $\dim_{\mathbb{C}}H^q(\mathcal{X}_t,\mathcal{E}_t)$ near $t = 0$. Generalizing previous results of X. Ye for the tangent bundle $E = T_{\mathcal{X}_t}$ and exterior powers of the cotangent bundle $E = Ω^p_{\mathcal{X}_t}$, we show that there are precisely two cohomological obstructions to the stability of $\dim_{\mathbb{C}}H^q(\mathcal{X}…
