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

Abstract: 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 test-bed 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. Suess 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 Suess showed the answer in general must be "no" by constructing an elegant relationship between toric vector bundles and various blow-ups of projective spaces, in particular the blow-ups of general arrangements of points studied by Castravet, Tevelev and Mukai. In this talk I'll review some of these results, and then give 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, and produce new families of Mori dream spaces indexed by the integral points in a locally closed polyhedral complex. Along the way I'll discuss plenty of examples and some questions. This is joint work with Kiumars Kaveh.

algebraic geometrydifferential geometryrepresentation theory

Audience: researchers in the topic

Toric Degenerations

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