Tangent bundles on toric varieties their quotient and their Cox rings

Hering, Mustata and Payne raised the question if a projectivized equivariant vector bundle on a toric variety is a Mori dream space. For rank two bundles and the cotangent bundle this is indeed the case, but even on surfaces there are counterexamples of rank three and in dimension three one can construct a toric variety having a tangent bundle which is not a Mori dream space. To show this we calculate the torus quotients of such bundles.