The moduli space of the tropicalizations of Riemann surfaces

TL;DR

This paper explores the structure of the moduli space of tropicalized Riemann surfaces, constructing a compact, Hausdorff space and relating it to classical moduli spaces through stratification comparisons.

Contribution

It introduces a new compact moduli space for tropicalizations of Riemann surfaces and establishes a stratification correspondence with classical moduli spaces.

Findings

The moduli space of tropicalizations is compact and Hausdorff.

A stratification-preserving correspondence exists between tropical and classical moduli spaces.

The construction uses hyperbolic pair of pants decompositions and weighted contractions.

Abstract

In this paper we study the moduli space of the tropicalizations of Riemann surfaces. We first tropicalize a smooth pointed Riemann surface by a graph defined by its (hyperbolic) pair of pants decomposition. Then we can construct the moduli space of tropicalizations based on a fixed regular tropicalization, and compactify it by adding strata parametrizing weighted contractions. We show that this compact moduli space is also Hausdorff. In the end, we compare this moduli space with the moduli space of Riemann surfaces, establishing a partial order-preserving correspondence between the stratifications of these two moduli spaces.