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

Abstract

Given integers $g\ge 0$, $n\ge 1$, 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 $g\ge 1$, 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 $w$.

14 pages; 1 figure; final version accepted at Advances in Geometry