Algebraic Geometry Seminar: Jose Gonzalez (UBC)

Abstract:

Projectivized toric vector bundles are a large class of rational varieties that share some of the pleasant properties of toric varieties and other Mori dream spaces. Hering, Mustata and Payne proved that the Mori cones of these varieties are polyhedral and asked if their Cox rings are indeed finitely generated. We present the complete answer to this question. There are several proofs of a positive answer in the rank two case [Hausen-Suss, Gonzalez]. One of these proofs relies on the simple structure of the Okounkov body of these varieties with respect to a special flag of subvarieties. For higher ranks 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 [Gonzalez-Hering-Payne-Suss]. 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}.