Toric Projective Bundles.

Abstract

In this dissertation we study some invariants of projectivized toric vector bundles such as their global Okounkov bodies, Cox rings and cones of pseudoeffective divisors. Projectivized toric vector bundles need not be toric varieties in general, however, they have a well-understood combinatorial description and they enjoy some of the finiteness properties of Mori dream spaces, such as finite generation of their nef and Mori cones. In one of our main results we associate a flag of torus invariant subvarieties to the projectivization of any given rank two toric vector bundle, and we describe the corresponding global Okounkov body in terms of the combinatorial data of the toric variety on the base and the data in the Klyachko filtrations of the toric vector bundle. Later on, we provide two proofs of the finite generation of the Cox rings of projectivizations of rank two toric vector bundles. In one of these proofs we obtain the finite generation using our description of the global Okounkov body in this setting, and in the other we follow a direct combinatorial approach and describe this Cox ring in terms of the Klyachko filtrations of the toric vector bundle. Both of our approaches differ from the ones of J. Hausen and H. Suess in their solutions to this finite generation problem. In the final part, I present my joint work with M. Hering, S. Payne and H. Suess, in which we give negative answers to the finite generation of the Cox rings and pseudoeffective cones of projectivizations of higher rank toric vector bundles. Our counterexamples have two flavors, some are very general in their moduli spaces, and some are determined by the combinatorial data of the toric varieties such as their cotangent bundles.