Toric flat families, valuations, and applications to projectivized toric vector bundles

Abstract

Using the notion of a valuation into the semifield of piecewise linear functions, we give a classification of torus equivariant flat families of finite type over a toric variety base, by certain piecewise linear maps between fans. As a consequence we derive a classification of toric vector bundles phrased in terms of tropicalized linear spaces. We use these tools to give a characterization of the Mori dream space property for a projectivized toric vector bundle.

arXiv commentTitle changed to emphasize applications to the study of projectivized toric vector bundles. Revised in many places and exposition improved. Some material rearranged and new sections introduced. References updated and a list of notation added. 44 pages