Faithful tropicalization of the Grassmannian of planes

TL;DR

This paper establishes a homeomorphism between the tropical projective Grassmannian of planes and a subset of the Berkovich analytic Grassmannian, providing explicit descriptions and confirming uniform tropical multiplicities.

Contribution

It constructs a continuous section to the tropicalization map for the Grassmannian of planes and describes the algebraic and polyhedral structures involved.

Findings

The tropical projective Grassmannian of planes is homeomorphic to a subset of the Berkovich Grassmannian.

All tropical multiplicities in this setting are equal to one.

A piecewise linear structure on the image of the section matches the polyhedral structure of the tropical Grassmannian.

Abstract

We show that the tropical projective Grassmannian of planes is homeomorphic to a closed subset of the analytic Grassmannian in Berkovich's sense by constructing a continuous section to the tropicalization map. Our main tool is an explicit description of the algebraic coordinate rings of the toric strata of the Grassmannian. We determine the fibers of the tropicalization map and compute the initial degenerations of all the toric strata. As a consequence, we prove that the tropical multiplicities of all points in the tropical projective Grassmannian are equal to one. Finally, we determine a piecewise linear structure on the image of our section that corresponds to the polyhedral structure on the tropical projective Grassmannian.