When is a (projectivized) toric vector bundle a Mori dream space?

Like toric varieties, toric vector bundles are a rich class of varieties which admit a combinatorial description. Following the classification due to Klyachko, a toric vector bundle is captured by a subspace arrangement decorated by toric data. This makes toric vector bundles an accessible testbed for concepts from algebraic geometry. Along these lines, Hering, Payne, and Mustata asked if the projectivization of a toric vector bundle is always a Mori dream space. Süß and Hausen, and Gonzales showed that the answer is "yes" for tangent bundles of smooth, projective toric varieties, and rank 2 vector bundles, respectively. Then Hering, Payne, Gonzales, and Süß showed the answer in general must be "no" by constructing an elegant relationship between toric vector bundles and various blow-ups of projective space, in particular the blow-ups of general arrangements of points studied by Castravet and Tevelev. In this talk I'll review some of these results, and then show a new description of toric vector bundles by tropical information. This description allows us to characterize the Mori dream space property in terms of tropical and algebraic data. I'll describe new examples and pose some questions. This is joint work with Kiumars Kaveh.