Algebra and Geometry of Irreducible toric vector bundles of rank $n$ on $\mathbb{P}^n$

TL;DR

This paper constructs a Cox ring presentation for irreducible toric vector bundles on projective space and demonstrates that their projectivizations satisfy key positivity conjectures, advancing understanding of their geometric properties.

Contribution

It provides a explicit Cox ring presentation for these bundles and proves they meet Fujita's freeness and ampleness conjectures.

Findings

Cox ring presentation for projectivizations of irreducible toric vector bundles

Verification that these bundles satisfy Fujita's conjectures

Enhancement of geometric understanding of toric vector bundles

Abstract

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 Fujita's freeness and ampleness conjectures.