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We give a criterion for a projectivized toric vector bundle to be a Mori dream space and describe its Cox ring using generators and relations. Both of these results are in terms of the matroids of all symmetric powers of the bundle. We also…

We study projectivizations of a special class of toric vector bundles that includes cotangent bundles, whose associated Klyachko filtrations are particularly simple. For these projectivized bundles, we give generators for the cone of…

We construct a presentation for the Cox ring of the projectivization $\mathbb{P}\mathcal{E}$ of any rank $n$ irreducible toric vector bundle on $\mathbb{P}^n$. We use this presentation to show that $\mathbb{P}\mathcal{E}$ always satisfies…

In this note we prove that any toric Fano manifold with nef tangent bundle is a product of projective spaces. In particular, it implies that Campana-Peternell conjecture hold for toric manifolds.

In this article, we study Newton-Okounkov bodies on projective vector bundles over curves. Inspired by Wolfe's estimates used to compute the volume function on these varieties, we compute all Newton-Okounkov bodies with respect to linear…

Extending classical algebro-geometric constructions to arbitrary matroids, we construct a $K$-class $T_M\in K(M)$ for every loopless matroid $M$. When $M$ is realizable by a linear subspace $L$, $T_M$ recovers the $K$-class of the tangent…

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…

In an unpublished preprint, A. King conjectured that there are tilting bundles over projective varieties which are obtained as invariant quotients of affine spaces for linear actions of reductive groups. The goal of this paper is to give…

Work of Gonz\'alez, Hering, Payne, and S\"uss shows that it is possible to find both examples and non-examples of Mori dream spaces among projectivized toric vector bundles. This result, and the combinatorial nature of the data of…

In this paper, using Klyachko's classification theorem we study positivity and semi-stability of toric vector bundles on a class of nonsingular projective toric varieties, known as Bott towers. In particular, we give a criterion of $s$-jet…

We show that a strong version of the geometric Merkurjev-Panin conjecture holds for the Cox category of a projective toric variety. That is, we prove that the full strong exceptional collection of Bondal-Thomsen line bundles is invariant…

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…

In continuation of our work in Comm. in Algebra, vol. 28 (2000), we study ramified coverings of projective manifolds, in particular over Fano manifolds and investigate positivity properties of the associated vector bundle. Moreover we study…

For a variety with a finitely generated total coordinate ring, we describe basic geometric properties in terms of certain combinatorial structures living in its divisor class group. For example, we describe the singularities, we calculate…

Let $X$ be a smooth complex projective variety. A recent conjecture of S. Kov\'acs states that if t\ he $p^{\text{th}}$-exterior power of the tangent bundle $T_X$ contains the $p^{\text{th}}$-exterior power of an ample vector bundle, then…

We study the positivity of exterior powers of the tangent sheaf on toric varieties in order to generalize results by Campana and Peternell about 3-folds with nef second exterior power of the tangent bundle. Using the theory of equivariant…

We consider some conditions under which a smooth projective variety X is actually the projective space. We also extend to the case of positive characteristic some results in the theory of vector bundle adjunction. We use methods and…

We give criteria for determining the positivity of line bundles coming from vertex operator algebras (VOAs) on the moduli space $\overline{\mathrm{M}}_{0,n}$ of rational curves with $n$ marked points. The criteria use the multiplicative…

We give a classification of rank $r$ torus equivariant vector bundles $\mathcal{E}$ on a toric scheme $\mathfrak{X}$ over a discrete valuation ring $\mathcal{O}$, in terms of graded piecewise linear maps $\Phi$ from the fan of…

In this note we study properties of partially ample line bundles on simplicial projective toric varieties. We prove that the cone of q-ample line bundles is a union of rational polyhedral cones, and calculate these cones in examples. We…