Filtrations of moduli spaces of tropical weighted stable curves

Abstract

We consider tropical versions of Hassett’s moduli spaces of weighted stable curves $M_{g,\mathcal{A}}^{\mathrm{trop}}$, ${\overline{M}}_{g,\mathcal{A}}^{\mathrm{trop}}$ and ${\Delta }_{g,\mathcal{A}}$ associated to a weight datum $\mathcal{A}=(a_1,\dots ,a_n)\in (\mathbb{Q}\cap (0,1]{)}^n$, their associated graph complexes $G^{(g,\mathcal{A})}$, and study the topology of these spaces as $\mathcal{A}$ changes. We show that for fixed $g$ and $n$, there are particular filtrations of these topological spaces and their graph complexes which may be used to compute the reduced rational homology of ${\Delta }_{g,\mathcal{A}}$ and the top weight cohomology of the moduli space ${\mathcal{M}}_{g,\mathcal{A}}$ of smooth $(g,\mathcal{A})$-stable algebraic curves.