Filtrations of moduli spaces of tropical weighted stable curves

TL;DR

This paper investigates how varying weight data influences the topology of tropical moduli spaces of weighted stable curves, providing filtrations to compute their homology and cohomology.

Contribution

It introduces filtrations of tropical moduli spaces based on weight data, enabling the computation of their homology and cohomology.

Findings

Filtrations of tropical moduli spaces are constructed.

These filtrations facilitate homology and cohomology calculations.

Results connect tropical and algebraic moduli space invariants.

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.