On Bands and Limit Theorems in Tropical Geometry

TL;DR

This paper explores the algebraic framework of bands and band schemes for tropicalization and analytification, establishing a scheme-theoretic limit theorem that enhances Payne's tropical limit results.

Contribution

It introduces a scheme-theoretic approach to tropicalization using band schemes, providing a new perspective and generalization of Payne's limit theorem.

Findings

Proves that an affine scheme is the limit of its associated band schemes.

Recovers Payne's tropicalization limit theorem via band scheme theory.

Extends the limit theorem to real tropical settings.

Abstract

We review the basic theory of bands and band schemes introduced by Baker-Jin-Lorscheid, which is an algebraic framework for tropicalization, analytification, and $\mathbb{F}_1$-geometry. For an affine scheme $X$ over a non-Archimedean valued field $k$, one can associate to every affine embedding $\iota$ of $X$ a naturally defined affine band scheme $Y_\iota$ whose rational points over the tropical band $\mathbb{T}$ recover the tropicalization $Trop(X,\iota)$. We prove that $X$ is the limit of the $Y_\iota$ in the category of band schemes, thereby obtaining a scheme-theoretic enhancement of Payne's limit theorem. By taking $\mathbb{T}$-rational points, this recovers Payne's theorem for affine tropicalizations from the perspective of band scheme theory and the same method provides an analogous result in the real tropical setting.