arXiv subjects

Ozhan Genc

Publications and source records attributed to Ozhan Genc.

11 recordsLinked to original sources

arXiv2026

H-Instanton Bundles on Three-Dimensional Smooth Toric Varieties with Picard Number Two

We study $H$-instanton bundles on the infinite family of smooth three-dimensional varieties $X_e=\mathbb{P}(\mathcal{O}_{\mathbb{P}^2} \oplus \mathcal{O}_{\mathbb{P}^2}(e))$, for $e \geq 0$. We provide two distinct monadic descriptions of $H$-instanton bundles on $X_e$, generalizing the classical monads on $\mathbb P^3$. We then characterize $H$-instanton bundles with second Chern class supported in a single degree, and investigate their existence and moduli spaces. Finally, for $e\leq 3$, we prove the existence of $H$-instanton bundles for all admissible second Chern classes. These results extend previous constructions on specific cases and contribute to the study of instanton bundles on threefolds with higher Picard number.

math.AG↗

arXiv2024

Instanton Sheaves on Ruled Fano 3-folds of Picard Rank 2 and Index 1

We study rank 2 $h$-instanton sheaves on projective threefolds. We demonstrate that any orientable rank 2, non-locally free $h$-instanton sheaf with defect 0 on a threefold can be obtained as an elementary transformation of a locally free $h$-instanton sheaf. Our focus then shifts to ruled Fano threefolds of Picard rank 2 and index 1, of which there are five deformation classes. We establish the existence of orientable rank 2 $h$-instanton bundles on such varieties. Additionally, we prove the existence of Ulrich bundles on such varieties, which correspond to $h$-instanton sheaves of minimum charge.

math.AG↗

arXiv2023

$\ell$-away ACM Bundles on Fano Surfaces

We propose the definition of $\ell$-away ACM bundle on a polarized variety $(X, \mathcal{O}_{X}(h))$. Then we give constructions of $\ell$-away ACM bundles on $(\mathbb{P}^2 , \mathcal{O}_{\mathbb{P}^2}(1))$, $(\mathbb{P}^1 \times \mathbb{P}^1, \mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(1,1))$ and the anticanonically polarized blow up of $\mathbb{P}^2$ up to three non collinear points. Also, we give the complete classification of $\ell$-away ACM bundles $\mathcal{E}$ of rank 2 for values $1 \leq \ell \leq 2$ on $(\mathbb{P}^2 , \mathcal{O}_{\mathbb{P}^2}(1))$. Similarly, on $(\mathbb{P}^1 \times \mathbb{P}^1, \mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(1,1))$, we give such a classification if $\mathrm{det}(\mathcal{E}) = \mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(a,a)$ for some $a \in \mathbb{Z}$. Moreover, we prove that the corresponding graded module $\mathrm{H}_*^1 ( \mathcal{E}) = \underset{{t \in \mathbb{Z} }}{\bigoplus} \mathrm{H}^1 (\mathcal{E} (th))$ is connected, extending the similar result for bundles on $\mathbb{P}^2$.

math.AG↗

arXiv2021

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.

math.AG↗

arXiv2021

Instanton bundles on $\mathbb{P}^1\times\mathbb{F}_1$

In this paper we deal with a particular class of rank two vector bundles (\emph{instanton} bundles) on the Fano threefold of index one $F:=\mathbb{F}_1 \times \mathbb{P}^1$. 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.

math.AG↗

arXiv2020

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.

math.AG↗

arXiv2020

Instanton bundles on two Fano threefolds of index $1$

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.

math.AG↗

arXiv2020

Ulrich Trichotomy on del Pezzo surfaces

In this article, we use a correspondence between Ulrich bundles on a projective variety and quiver representations to prove that certain del Pezzo surfaces satisfy the Ulrich trichotomy, for any given polarization.

math.AG↗

arXiv2019

Instanton bundles on the blow up of the projective $3$-space at a point

We propose a general definition of mathematical instanton bundle with given charge on any Fano threefold extending the classical definitions on $\mathbb P^3$ and on Fano threefold with cyclic Picard group. Then we deal with the case of the blow up of $\mathbb P^3$ at a point, giving an explicit construction of instanton bundles satisfying some important extra properties: moreover, we also show that they correspond to smooth points of a component of the moduli space.

math.AG↗

arXiv2017

Stable Ulrich Bundles on Fano Threefolds with Picard Number 2

In this paper, we consider the existence problem of rank one and two stable Ulrich bundles on imprimitive Fano 3-folds obtained by blowing-up one of $\mathbb{P}^{3}$, $Q$ (smooth quadric in $\mathbb{P}^{4}$), $V_{3}$ (smooth cubic in $\mathbb{P}^{4}$) or $V_{4}$ (complete intersection of two quadrics in $\mathbb{P}^{5}$) along a smooth irreducible curve. We prove that the only class which admits Ulrich line bundles is the one obtained by blowing up a genus 3, degree 6 curve in $\mathbb{P}^{3}$. Also, we prove that there exist stable rank two Ulrich bundles with $c_{1}=3H$ on a generic member of this deformation class.

math.AG↗

arXiv2017

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 $dH$.

math.AG↗