Two kinds of bundles with torus action (joint work with J. Hausen / J. L. Gonzales, M. Hering, S. Payne)

In this talk we consider bundles admitting a “horizontal” or “vertical” torus action, respectively:
1) equivariant vector bundles on toric varieties,
2) (affine) fibre bundles with toric fibres over projective varieties.
For both classes of objects we shortly present a combinatorial description, one introduced by A. A. Klyachko the other one by K. Altmann and Hausen. But our main interest concerns the question under which conditions these objects are Mori dream spaces. It turns out, that the answer to this question is quite different for both types of bundles.