Filtrations of moduli spaces of tropical weighted stable curves

Abstract

We study how changing the weight datum $\mathcal{A}=(a_1,...,a_n)\in (\mathbb{Q}\cap (0,1]{)}^n$ affects the topology of tropical moduli spaces $M_{g,\mathcal{A}}^{trop}$, ${\overline{M}}_{g,\mathcal{A}}^{trop}$ and ${\Delta }_{g,\mathcal{A}}$ and the homology of the latter one. We show that for fixed $g$ and $n$, there are particular filtrations of these topological spaces which we can use 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.

Final version, to appear in Portugaliae Mathematica