Tropical geometry of moduli spaces of weighted stable curves

TL;DR

This paper develops a tropical geometric framework for Hassett's moduli spaces of weighted stable curves, linking algebraic and tropical geometry through deformation retraction and skeletons.

Contribution

It introduces a tropical analogue of Hassett's moduli spaces and establishes a natural deformation retraction onto the non-Archimedean skeleton, extending previous work on moduli space compactifications.

Findings

Tropical moduli spaces are identified with deformation retractions onto skeletons.

Tautological maps have tropical analogues that relate to weight data.

The dependence of tropical moduli spaces on weights is analyzed, with examples like Losev-Manin spaces.

Abstract

Hassett's moduli spaces of weighted stable curves form an important class of alternate modular compactifications of the moduli space of smooth curves with marked points. In this article we define a tropical analogue of these moduli spaces and show that the naive set-theoretic tropicalization map can be identified with a natural deformation retraction onto the non-Archimedean skeleton. This result generalizes work of Abramovich, Caporaso, and Payne treating the Deligne-Knudsen-Mumford compactification of the moduli space of smooth curves with marked points. We also study tropical analogues of the tautological maps, investigate the dependence of the tropical moduli spaces on the weight data, and consider the example of Losev-Manin spaces.