Geodesic
The tropicalization of the moduli space of curves
[La tropicalisation de l'espace des modules des courbes]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 4, pp. 765-809.

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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 démontre que le squelette du champ des modules des courbes stables de Deligne-Mumford-Knudsen est naturellement identifié avec l'espace des modules des courbes tropicales de façon compatible avec l'application de tropicalisation « naïve » d'ensembles. La démonstration emploie des résultats généraux de structure sur le squelette des champs toroïdaux de Deligne-Mumford. En outre, on construit les morphismes tautologiques entre les espaces de modules des courbes tropicales étendues, et on démontre qu'ils sont compatibles avec leurs analogues dans le cadre algébrique.

Publié le :
DOI : 10.24033/asens.2258
Classification : 14T05, 14H10, 14G22.
Keywords: Tropicalization, moduli of curves, Berkovich spaces, skeletons, toroidal Deligne-Mumford stacks.
Mots-clés : Tropicalisation, espaces de modules des courbes, espaces de Berkovich, champs toroïdaux de Deligne-Mumford 
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Abramovich, Dan; Caporaso, Lucia; Payne, Sam. The tropicalization  of the moduli space of curves. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 4, pp. 765-809. doi : 10.24033/asens.2258. http://geodesic.mathdoc.fr/articles/10.24033/asens.2258/

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