An introduction to local tropicalization

TL;DR

This paper introduces four perspectives on local tropicalization of formal subgerms of toric germs, extending the concept from plane curve singularities to higher dimensions and discussing various extensions.

Contribution

It provides a comprehensive overview of multiple viewpoints on local tropicalization and introduces new variants and extensions of the concept.

Findings

Four viewpoints on local tropicalization are explained.

Extension of local tropicalization from plane curves to higher dimensions.

Descriptions of variants and extensions of local tropicalization.

Abstract

In this paper we explain four viewpoints on the local tropicalization of formal subgerms of toric germs, which is a local analog of the global tropicalization of subvarieties of algebraic tori. We start by illustrating some of those viewpoints for plane curve singularities, then we pass to arbitrary dimensions. We conclude by describing several variants and extensions of the notion of local tropicalization presented in this paper.