Introduction to adic tropicalization

TL;DR

This paper introduces adic tropicalization, a new structure combining tropical and algebraic data on varieties, and discusses its relation to Huber analytification and curve complexes.

Contribution

It presents the concept of adic tropicalization, blending polyhedral and algebraic structures, and establishes a limit theorem relating it to Huber analytification.

Findings

Adic tropicalization combines tropical and algebraic data.

Huber analytification is the limit of all adic tropicalizations.

Connections to curve complexes are explored.

Abstract

This is an expository article on the adic tropicalization of algebraic varieties. We outline joint work with Sam Payne in which we put a topology and structure sheaf of local topological rings on the exploded tropicalization. The resulting object, which blends polyhedral data of the tropicalization with algebraic data of the associated initial degenerations, is called the adic tropicalization. It satisfies a theorem of the form "Huber analytification is the limit of all adic tropicalizations." We explain this limit theorem in the present article, and illustrate connections between adic tropicalization and the curve complexes of O. Amini and M. Baker.