The tropicalization of the moduli space of curves

TL;DR

This paper establishes a natural correspondence between the skeleton of the moduli stack of stable curves and the moduli space of extended tropical curves, bridging algebraic geometry and tropical geometry.

Contribution

It provides a canonical identification of the moduli space skeleton with tropical moduli space and constructs compatible tautological maps in the tropical setting.

Findings

Skeleton of the moduli stack matches the tropical moduli space

Compatibility of tropicalization with algebraic tautological maps

Structural results on skeletons of toroidal stacks

Abstract

We show that the skeleton of the Deligne-Mumford-Knudsen moduli stack of stable curves is naturally identified with the moduli space of extended tropical curves, and that this is compatible with the "naive" set-theoretic tropicalization map. The proof passes through general structure results on the skeleton of a toroidal Deligne-Mumford stack. Furthermore, we construct tautological forgetful, clutching, and gluing maps between moduli spaces of extended tropical curves and show that they are compatible with the analogous tautological maps in the algebraic setting.