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.

Abramovich, D., Caporaso, L., & Payne, S. (2015). The tropicalization of the moduli space of curves. ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE, 48(4), 765-809 [10.24033/asens.2258].

The tropicalization of the moduli space of curves.

CAPORASO, Lucia;
2015

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.
On d'{e}montre que la squelette du champ des modules des courbes stables de Deligne-Mumford-Knudsen est naturellement identifi'{e} avec l'espace des modules des courbes tropicales de fac{c}on compatible avec l'application de tropicalisation ``na"ive" d'ensembles. La d'{e}monstration emploi des r'{e}sultats g'{e}n'{e}raux de structure sur la squelette des champs toro"{i}daux de Deligne-Mum-ford. En outre, on construit les morphismes tautologiques entre les espaces de modules des courbes tropicales '{e}tendues, et l'on d'{e}montre qu'ils sont compatibles avec leurs analogues dans le cadre alg'{e}brique.
Abramovich, D., Caporaso, L., & Payne, S. (2015). The tropicalization of the moduli space of curves. ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE, 48(4), 765-809 [10.24033/asens.2258].
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11590/136842
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