The tropicalization of the moduli space of curves
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.
Keywords
Cite
@article{arxiv.1212.0373,
title = {The tropicalization of the moduli space of curves},
author = {Dan Abramovich and Lucia Caporaso and Sam Payne},
journal= {arXiv preprint arXiv:1212.0373},
year = {2025}
}
Comments
v2: 55 pages. Expanded Section 2 with improved treatment of the category of generalized cone complexes. Clarified the role of the coarse moduli space and its analytification in the construction of the skeleton for a toroidal DM stack