The tropicalization of the moduli space of curves
Dan Abramovich, Lucia Caporaso, Sam Payne

TL;DR
This paper establishes a natural correspondence between the skeleton of the moduli stack of stable curves and the moduli space of extended tropical curves, bridging algebraic geometry and tropical geometry.
Contribution
It provides a canonical identification of the moduli space skeleton with tropical moduli space and constructs compatible tautological maps in the tropical setting.
Findings
Skeleton of the moduli stack matches the tropical moduli space
Compatibility of tropicalization with algebraic tautological maps
Structural results on skeletons of toroidal stacks
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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Polynomial and algebraic computation · Geometric and Algebraic Topology
