Constructing smooth and fully faithful tropicalizations for Mumford curves

TL;DR

This paper develops methods to construct smooth and fully faithful tropicalizations for Mumford curves, enhancing the understanding of their algebraic and combinatorial structures through new embedding techniques.

Contribution

It introduces constructions of fully faithful and smooth tropicalizations for Mumford curves, and establishes that smooth tropicalizations characterize Mumford curves.

Findings

Constructed fully faithful tropicalizations with sections to Berkovich spaces.

Proved that smooth tropicalizations imply the curve is a Mumford curve.

Developed a lifting theorem for rational functions on metric graphs.

Abstract

The tropicalization of an algebraic variety X is a combinatorial shadow of X, which is sensitive to a closed embedding of X into a toric variety. Given a good embedding, the tropicalization can provide a lot of information about X. We construct two types of these good embeddings for Mumford curves: Fully faithful tropicalizations, which are embeddings such that the tropicalization admits a section to the associated Berkovich space $X^{an}$ of X, and smooth tropicalizations. We also show that a smooth curve that admits a smooth tropicalization is necessarily a Mumford curve. Our key tool is a variant of a lifting theorem for rational functions on metric graphs.