Related papers: Nef cone of flag bundles over a curve

In this article, we give a description of the closed cone of curves of the projective bundle $\mathbb{P}(E)$ over a smooth projective variety $X$. Using duality, we then calculate the nef cone of divisors in $\mathbb{P}(E)$ over some…

Let $C$ be a smooth projective curve over $\mathbb C$. Let $n,d\geq 1$. Let $\mathcal Q$ be the Quot scheme parameterizing torsion quotients of the vector bundle $\mathcal O^n_C$ of degree $d$. In this article we study the nef cone of…

Let $X = \mathbb{P}(E_1) \times_C \mathbb{P}(E_2)$ where $C$ is a smooth curve and let $E_1$, $E_2$ be vector bundles over $C$. In this paper, we extend the results in \cite{K-M-R} by computing the nef cone of $X$ without restriction on the…

Let $X$ be a smooth complex projective curve and let $E$ be a vector bundle on $X$ which is not semistable. We consider a flag bundle $\pi: \text{Fl}(E) \to X$ parametrizing certain flags of fibers of $E$. The dimensions of the successive…

We classify nef vector bundles on a smooth hyperquadric of dimension $\geq 4$ with first Chern class two over an algebraically closed field of characteristic zero.

We classify nef vector bundles on a smooth hyperquadric of dimension three with first Chern class two over an algebraically closed field of characteristic zero. In particular, we see that they are globally generated.

Let $E$ be a vector bundle of rank $r$ on a smooth complex projective variety $X$. In this article, we compute the nef and pseudoeffective cones of divisors in the Grassmann bundle $Gr_X(k,E)$ parametrizing $k$-dimensional subspaces of the…

We classify nef vector bundles on a smooth quadric surface with first Chern class $(2,2)$ over an algebraically closed field of characteristic zero.

We study the positivity properties of the projectivization of a parabolic bundle over a smooth complex projective curve. The generators of its N\'eron--Severi group are computed, and the positive cone is determined. In particular, we…

Using recent results of Bayer-Macr\`i, we compute in many cases the pseudoeffective and nef cones of the projectivised cotangent bundle of a smooth projective K3 surface. We then use these results to construct explicit families of smooth…

Following ideas of V. Batyrev, we prove an analogue of the Cone Theorem for the closed cone of nef curves: an enlargement of the cone of nef curves is the closure of the sum of a K_X-non-negative portion and countably many K_X-negative…

Let $L$ be a line bundle on a scheme $X$, proper over a field. The property of $L$ being nef can sometimes be "thickened", allowing reductions to positive characteristic. We call such line bundles arithmetically nef. It is known that a line…

A Fano manifold $X$ with nef tangent bundle is of flag-type if it has the same type of elementary contractions as a complete flag manifold. In this paper we present a method to associate a Dynkin diagram $\mathcal{D}(X)$ with any such $X$,…

Let $X$ be a projective variety and let $C$ be a rational normal curve on $X$. We compute the normal bundle of $C$ in a general complete intersection of hypersurfaces of sufficiently large degree in $X$. As a result, we establish the…

We classify nef vector bundles on a projective space with first Chern class three over an algebraically closed field of characteristic zero; we see, in particular, that these nef vector bundles are globally generated if the second Chern…

Let $C$ be a smooth projective curve over $\mathbb{C}$ of genus $g(C)\geqslant 3$ (respectively, $g(C)=2$). Fix integers $r,k$ such that $2\leqslant k\leqslant r-2$, (respectively, $3\leqslant k\leqslant r-2$). Let $\mathcal{Q}:={\rm…

We describe nef vector bundles on a projective space with first Chern class three and second Chern class eight over an algebraically closed field of characteristic zero by giving them a minimal resolution in terms of a full strong…

In this short article, we determine the bigness of the tangent bundle $T_X$ of the projective bundle $X=\mathbb{P}_C(E)$ associated to a vector bundle $E$ on a smooth projective curve $C$.

Let $X$ be a smooth projective variety over the complex numbers. One knows by the Cone Theorem that the closed cone of curves of $X$ is rational polyhedral whenever $c_1(X)$ is ample. For varieties $X$ such that $c_1(X)$ is not ample,…

On a projective surface it is well-known that the set of curves orthogonal to a nef line bundle is either finite or uncountable. We show that this dichotomy fails in higher dimension by constructing a nef line bundle on a threefold which is…