Topology of tropical moduli of weighted stable curves

TL;DR

This paper studies the topology of tropical moduli spaces of weighted stable curves, revealing their homotopy types, Betti numbers, and fundamental groups, and contrasting these with known cases, with implications for genus 0 and 1 spaces.

Contribution

It provides new topological descriptions and explicit formulas for weighted tropical moduli spaces, including their homotopy types, Betti numbers, and fundamental groups, especially in genus 0 and 1.

Findings

elta_{0,w} is homotopic to a wedge sum of spheres.

Explicit Betti number formulas for certain weight configurations.

Existence of disconnected spaces and torsion in fundamental groups.

Abstract

The moduli space $\Delta_{g,w}$ of tropical $w$-weighted stable curves of volume $1$ is naturally identified with the dual complex of the divisor of singular curves in Hassett's spaces of $w$-weighted stable curves. If at least two of the weights are $1$, we prove that $\Delta_{0,w}$ is homotopic to a wedge sum of spheres, possibly of varying dimensions. Under additional natural hypotheses on the weight vector, we establish explicit formulas for the Betti numbers of the spaces. We exhibit infinite families of weights for which the space $\Delta_{0,w}$ is disconnected and for which the fundamental group of $\Delta_{0,w}$ has torsion. In the latter case, the universal cover is shown to have a natural modular interpretation. This places the weighted variant of the space in stark contrast to the heavy/light cases studied previously by Vogtmann and Cavalieri-Hampe-Markwig-Ranganathan. Finally, we prove a structural result relating the spaces of weighted stable curves in genus $0$ and $1$, and leverage this to extend several of our genus $0$ results to the spaces $\Delta_{1,w}$.