The tropicalization of the moduli space of curves

TL;DR

This work forges a precise bridge between non-archimedean analytic geometry and tropical geometry for moduli of curves by identifying the skeleton of the toroidal Deligne–Mumford stack $\overline{\mathcal{M}}_{g,n}$ with the extended tropical moduli space $\overline{M}_{g,n}^{trop}$, via an isomorphism $\Phi_{g,n}$ and its extension $\overline{\Phi}_{g,n}$. It extends Thuillier's skeleton construction to toroidal DM stacks, producing a functorial skeleton $\overline{\Sigma}(\mathcal{X})$ embedded in $X^{an}$ and realized as an extended generalized cone complex, with monodromy given by automorphisms of stable graphs. The tropicalization map Trop from $\overline{M}_{g,n}^{an}$ to $\overline{M}_{g,n}^{trop}$ is shown to be the projection to the skeleton and is compatible with tropicalization of forgetful, clutching, and gluing maps, yielding commutative diagrams that mirror the algebraic world. Overall, the paper provides a rigorous, functorial dictionary between algebraic moduli of curves and their tropical counterparts, enabling canonical degeneration analyses and tropical computations within a unified framework.

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.

Figures (6)

• Figure 1: Barycentric subdivisions
• Figure 2: The barycentric subdivision of an extended generalized cone complex is not the extended cone complex of the barycentric subdivision. The dashed line on the left indicates folding.
• Figure 3: A four-legged weighted graph of genus 6
• Figure 4: Fiber of $\pi^{{\operatorname{trop}}}$ over the smallest stratum of ${\overline{M}}_{1,1}^{\operatorname{trop}}$
• Figure 5: Fiber of $\pi^{{\operatorname{trop}}}$ over $[{\boldsymbol{\Gamma}}_d]\in {\overline{M}}_{1.1}^{\operatorname{trop}}$ with $d>0$.

Theorems & Definitions (59)

• Definition 1.1.1
• Theorem 1.2.1
• Theorem 1.2.2
• Remark 2.2.1
• Proposition 2.2.2
• Remark 2.4.1
• Example 2.4.2
• Remark 2.4.3
• Remark 2.6.1
• Proposition 2.6.2