Projectivized rank two toric vector bundles are Mori dream spaces

TL;DR

This paper proves that the projectivization of rank two toric vector bundles over certain toric varieties has a finitely generated Cox ring, making it a Mori dream space under specific conditions.

Contribution

It establishes the finite generation of the Cox ring for projectivized rank two toric vector bundles, extending the class of Mori dream spaces.

Findings

Cox ring of P(E) is finitely generated

P(E) is a Mori dream space when X is projective and simplicial

Provides new examples of Mori dream spaces

Abstract

We prove that the Cox ring of the projectivization P(E) of a rank two toric vector bundle E, over a toric variety X, is a finitely generated k-algebra. As a consequence, P(E) is a Mori dream space if the toric variety X is projective and simplicial.