Topology of tropical moduli spaces of weighted stable curves in higher genus

TL;DR

This paper investigates the topology of tropical moduli spaces of weighted stable curves of higher genus, revealing their simple connectivity and providing formulas for their Euler characteristics based on combinatorial data.

Contribution

It establishes the simple connectivity of these tropical moduli spaces for genus at least one and derives a combinatorial formula for their Euler characteristic.

Findings

The space elta_{g,w} is simply connected for all weights when g .

Provides a formula for the Euler characteristic of elta_{g,w}.

Links the topology of tropical moduli spaces to combinatorial properties of weights.

Abstract

Given integers $g \geq 0$, $n \geq 1$, and a vector $w \in (\mathbb{Q} \cap (0, 1])^n$ such that ${2g - 2 + \sum w_i > 0}$, we study the topology of the moduli space $\Delta_{g, w}$ of $w$-stable tropical curves of genus $g$ with volume 1. The space $\Delta_{g, w}$ is the dual complex of the divisor of singular curves in Hassett's moduli space of $w$-stable genus $g$ curves $\overline{\mathcal{M}}_{g, w}$. When $g \geq 1$, we show that $\Delta_{g, w}$ is simply connected for all values of $w$. We also give a formula for the Euler characteristic of $\Delta_{g, w}$ in terms of the combinatorics of $w$.