Faithful tropicalization of hypertoric varieties

TL;DR

This paper explicitly describes the polyhedral structure of the tropicalization of hypertoric varieties and proves the existence of a continuous section of the tropicalization map, advancing understanding of their tropical geometry.

Contribution

It provides an explicit description of the tropicalization's polyhedral structure and establishes a continuous section of the tropicalization map for hypertoric varieties.

Findings

Explicit polyhedral description of hypertoric variety tropicalization

Existence of a continuous section of the tropicalization map

Advancement in tropical geometry of hypertoric varieties

Abstract

The hypertoric variety $\mathfrak{M}_{\mathcal{A}}$ defined by an affine arrangement $\mathcal{A}$ admits a natural tropicalization, induced by its embedding in a Lawrence toric variety. We explicitly describe the polyhedral structure of this tropicalization. Using a recent result of Gubler, Rabinoff, and Werner, we prove that there is a continuous section of the tropicalization map.