Instanton bundles on two Fano threefolds of index 11

We deal with instanton bundles on the product ${\mathbb P}^1\times{\mathbb P}^2$ and the blow up of ${\mathbb P}^3$ along a line. We give an explicit construction leading to instanton bundles. Moreover, we also show that they correspond to smooth points of a unique irreducible component of their moduli space.Comment: 25 pages. The final version will appear in Forum Mathematicum. arXiv admin note: text overlap with arXiv:1909.1028

Even and odd instanton bundles on Fano threefolds

We define non-ordinary instanton bundles on Fano threefolds $X$ extending the notion of (ordinary) instanton bundles. We determine a lower bound for the quantum number of a non-ordinary instanton bundle, i.e. the degree of its second Chern class, showing the existence of such bundles for each admissible value of the quantum number when $i_X\ge 2$ or $i_X=1$, $\mathrm{Pic}(X)$ is cyclic and $X$ is ordinary. In these cases we deal with the component inside the moduli spaces of simple bundles containing the vector bundles we construct and we study their restriction to lines. Finally we give a monadic description of non-ordinary instanton bundles on $\mathbb{P}^3$ and the smooth quadric studying their loci of jumping lines, when of the expected codimension.Comment: 34 pages. Minor changes. The final version will appear in The Asian Journal of Mathematic

On stability of tangent bundle of toric varieties

Let $X$ be a nonsingular complex projective toric variety. We address the question of semi-stability as well as stability for the tangent bundle $T{X}$. In particular, a complete answer is given when $X$ is a Fano toric variety of dimension four with Picard number at most two, complementing earlier work of Nakagawa. We also give an infinite set of examples of Fano toric varieties for which $TX$ is unstable; the dimensions of this collection of varieties are unbounded. Our method is based on the equivariant approach initiated by Klyachko and developed further by Perling and Kool.Comment: Revised version. To appear in Proc. Indian Acad. Sci. Math. Sc

Ulrich bundles on Veronese surfaces

We prove that every Ulrich bundle on the Veronese surface has a resolution in terms of twists of the trivial bundle over $\mathbb{P}^{2}$. Using this classification, we prove existence results for stable Ulrich bundles over $\mathbb{P}^{k}$ with respect to an arbitrary polarization $dH$.Comment: to appear in the Proceedings of the AM

l-away ACM Bundles on DelPezzo Surfaces

We propose the definition of $l$-away ACM bundles on varieties. Then we give constructions of $l$-away ACM bundles on \p2, \p1 \times \p1 and blow-up of \p2 up to three points. Also, we give complete classification of special $l$-away ACM bundles of rank 2 for small values of $l$ on \p2 and \p1 \times \p1. Moreover, we prove that \mathrm{H}_*^1 (\p1 \times \p1, \cE) is connected for any special rank 2 bundle \cE, which is already known for \p2.Comment: 22 page

Instanton bundles on P1×F1

In this paper we deal with a particular class of rank two vector bundles (emph{instanton} bundles) on the Fano threefold of index one $F:=mathbb{F}_1 imes mathbb{P}^1$. We show that every instanton bundle on $F$ can be described as the cohomology of a monad whose terms are free sheaves. Furthermore we prove the existence of instanton bundles for any admissible second Chern class and we construct a nice component of the moduli space where they sit. Finally we show that minimal instanton bundles (i.e. with the least possible degree of the second Chern class) are aCM and we describe their moduli space