# A moduli stack of tropical curves

## TL;DR

This paper develops a foundational framework for tropical geometry by constructing a moduli stack of tropical curves, enabling a rigorous approach to tropical moduli problems and their relation to algebraic curves.

## Contribution

It introduces a moduli functor for tropical curves, proves its representability by a geometric stack, and establishes a smooth tropicalization morphism compatible with tautological structures.

## Key findings

- Constructed a geometric stack representing tropical moduli of curves.
- Defined a tropicalization morphism from algebraic to tropical moduli spaces.
- Extended tropical moduli problems to logarithmic schemes.

## Abstract

We contribute to the foundations of tropical geometry with a view towards formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves. Our approach to tropical geometry permits tropical moduli problems---moduli of curves or otherwise---to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms.