  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}.

Algebraic Geometry