Positivity properties of divisors on Toric Vector Bundles

TL;DR

This paper investigates the positivity properties of divisors on toric vector bundles by computing geometric cones and analyzing Fano conditions, using Cox rings and matroid theory.

Contribution

It introduces methods to compute Newton-Okounkov bodies and cones for projectivized toric vector bundles, and verifies Fujita's conjectures for certain classes.

Findings

Computed Newton-Okounkov bodies, effective, and nef cones for these bundles.

Established Fano property for specific classes of bundles.

Confirmed Fujita's freeness and ampleness conjectures in new cases.

Abstract

We use presentations of the Cox rings of projectivized toric vector bundles and elements of matroid theory to compute Newton-Okounkov bodies, effective cones, and nef cones of these spaces. As an application we analyze the Fano property and establish Fujita's freeness and ampleness conjectures for several classes of projectivized toric vector bundles.