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Annales scientifiques de l'ENS - Parutions - série 4, 48 (2015)

Parutions < série 4, 48

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 48, fascicule 4 (2015)

Dan Abramovich, Lucia Caporaso, Sam Payne
The tropicalization of the moduli space of curves
Annales scientifiques de l'ENS 48, fascicule 4 (2015), 765-809

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Résumé :
La tropicalisation de l'espace des modules des courbes
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.

Mots-clefs : Tropicalisation, espaces de modules des courbes, espaces de Berkovich, champs toroïdaux de Deligne-Mumford

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: Tropicalization, moduli of curves, Berkovich spaces, skeletons, toroidal Deligne-Mumford stacks.

Class. math. : 14T05, 14H10, 14G22.


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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