Tropicalization of toric prevarieties

TL;DR

This paper develops a tropicalization procedure for toric prevarieties, extending existing tropical limit theorems to a broader class of algebraic structures, facilitating their study in tropical geometry.

Contribution

It introduces a tropicalization framework for toric prevarieties and generalizes key tropical limit theorems to divisorial schemes.

Findings

Established a tropicalization procedure for toric prevarieties.

Proved a generalized tropical limit theorem for divisorial schemes.

Enhanced understanding of tropical geometry for non-separated toric structures.

Abstract

The homogeneous spectrum of a multigraded finitely generated algebra (in the sense of Brenner-Schr\"oer) always admits an embedding into a toric variety that is not necessarily separated, a so-called toric prevariety. In order to have a convenient framework to study the tropicalization of homogeneous spectra we propose a tropicalization procedure for toric prevarieties and study its basic properties. With these tools at hand, we prove a generalization of Payne's and Foster--Gross--Payne's tropical limit theorem for divisorial schemes.