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

## TL;DR

This paper introduces a valuation-based framework to classify torus equivariant flat families and toric vector bundles, linking tropical geometry with algebraic geometry to analyze Mori dream space properties.

## Contribution

It develops a valuation approach using piecewise linear functions to classify flat families and toric vector bundles, connecting tropical geometry with classical algebraic geometry.

## Key findings

- Classification of torus equivariant flat families via piecewise linear maps
- Tropicalized linear spaces characterize toric vector bundles
- Criteria for Mori dream space property in projectivized toric 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.