Some positivity results for toric vector bundles

TL;DR

This paper establishes criteria for when projectivized toric vector bundles are Mori dream spaces and big, using matroid-based conditions, and explores the complexity of positivity properties beyond line bundles.

Contribution

It introduces new criteria for positivity and Mori dream space status of toric vector bundles based on matroids of symmetric powers, expanding understanding beyond line bundles.

Findings

Criteria for projectivized toric vector bundles to be Mori dream spaces.

Description of Cox rings via generators and relations.

Examples illustrating the complexity of positivity properties.

Abstract

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 give a criterion for a toric vector bundle to be big and describe several interesting examples of toric vector bundles which highlights how positivity properties for toric vector bundles are more complicated than for toric line bundles.