Cox rings and pseudoeffective cones of projectivized toric vector bundles

TL;DR

This paper investigates the structure of Cox rings and pseudoeffective cones for projectivized toric vector bundles, including cotangent bundles, revealing their algebraic properties and limitations as Mori dream spaces.

Contribution

It provides explicit generators for effective cones and presents Cox rings as polynomial algebras over blowup Cox rings, highlighting new algebraic descriptions for these bundles.

Findings

Cox rings are polynomial algebras over blowup Cox rings.

Some projectivized cotangent bundles are not Mori dream spaces.

Examples of Cox rings isomorphic to that of M_{0,n} are constructed.

Abstract

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 effective divisors and a presentation of the Cox ring as a polynomial algebra over the Cox ring of a blowup of a projective space along a sequence of linear subspaces. As applications, we show that the projectivized cotangent bundles of some toric varieties are not Mori dream spaces and give examples of projectivized toric vector bundles whose Cox rings are isomorphic to that of M_{0,n}.