Algebraic and tropical curves: comparing their moduli spaces

TL;DR

This paper constructs and analyzes the moduli space of tropical curves, compares it with algebraic curve moduli spaces, and explores their topological and combinatorial relationships, providing a comprehensive expository overview.

Contribution

It introduces a new construction of the tropical curves moduli space and compares it to algebraic counterparts from multiple mathematical perspectives.

Findings

Established basic topological properties of the tropical moduli space

Compared tropical and algebraic moduli spaces from combinatorial and topological viewpoints

Generalized previous results on tropical curves' moduli spaces

Abstract

We construct the moduli space for equivalence classes of n-pointed tropical curves of genus g, together with its compactification given by weighted tropical curves, and establish some of its basic topological properties. We compare it to the moduli spaces of smooth and stable algebraic curves, from the combinatorial, the topological, and the Teichm\"uller point of view. The paper is written in an expository style, and it generalizes some results contained in sections 4-6 of arXiv:1001.2815v3.