Moduli spaces of rational weighted stable curves and tropical geometry

TL;DR

This paper explores the structure of moduli spaces of rational weighted stable tropical curves, connecting them with classical Hassett spaces, and demonstrates their realization as Bergman fans, tropicalizations, and Berkovich skeletons.

Contribution

It characterizes when tropical moduli spaces form balanced fans and expresses them as Bergman fans of graphic matroids, linking tropical and algebraic moduli spaces.

Findings

Tropical moduli spaces are balanced fans if weights are heavy/light.

These spaces can be expressed as Bergman fans of graphic matroids.

The tropical moduli space is realized as a tropicalization and Berkovich skeleton of the classical space.

Abstract

We study moduli spaces of rational weighted stable tropical curves, and their connections with the classical Hassett spaces. Given a vector w of weights, the moduli space of tropical w-stable curves can be given the structure of a balanced fan if and only if w has only heavy and light entries. In this case, we can express the moduli space as the Bergman fan of a graphic matroid. Furthermore, we realize the tropical moduli space as a geometric tropicalization, and as a Berkovich skeleton, of the classical moduli space. This builds on previous work of Tevelev, Gibney--Maclagan, and Abramovich--Caporaso--Payne. Finally, we construct the moduli spaces of heavy/light weighted tropical curves as fiber products of unweighted spaces, and explore parallels with the algebraic world.