# Global Spherical Tropicalization via Toric Embeddings

## TL;DR

This paper introduces a new approach to spherical tropicalization by embedding spherical varieties into toric varieties, enabling a simplified description and proving that closure operations commute with tropicalization.

## Contribution

It provides an equivalent description of spherical tropicalization using toric embeddings and demonstrates that taking closures commutes with the tropicalization process.

## Key findings

- New description of spherical tropicalization via toric embeddings
- Proof that closure operations commute with tropicalization
- Simplification of the spherical tropicalization process

## Abstract

The first steps in defining tropicalization for spherical varieties have been taken in the last few years. There are two parts to this theory: tropicalizing subvarieties of homogeneous spaces and tropicalizing their closures in spherical embeddings. In this paper, we obtain a new description of spherical tropicalization that is equivalent to the other theories. This works by embedding in a toric variety, tropicalizing there, and then applying a particular piecewise projection map. We use this theory to prove that taking closures commutes with the spherical tropicalization operation.