$\mathbb{Z}_2$-homology of the orbit spaces $G_{n,2}/T^n$

We study the $\mathbb Z_2$-homology groups of the orbit space $X_n = G_{n,2}/T^n$ for the canonical action of the compact torus $T^n$ on a complex Grassmann manifold $G_{n,2}$. Our starting point is the model $(U_n, p_n)$ for $X_n$ constructed by Buchstaber and Terzić (2020), where $U_n = \Delta _{n,2}\times \mathcal{F}_{n}$ for a hypersimplex $\Delta_{n,2}$ and an universal space of parameters $\mathcal{F}_{n}$ defined in the works of Buchstaber and Terzić (2019), (2020). It is proved by Buchstaber and Terzić (2021) that $\mathcal{F}_{n}$ is diffeomorphic to the moduli space $\mathcal{M}_{0,n}$ of stable $n$-pointed genus zero curves. We exploit the results of Keel (1992) and Ceyhan (2009) on homology groups of $\mathcal{M}_{0,n}$ and express them in terms of thestratification of $\mathcal{F}_{n}$ which are incorporated in the model $(U_n, p_n)$. In the result we provide the description of cycles in $X_n$, inductively on $ n. $ We obtain as well explicit formulas for $\mathbb Z_2$-homology groups for $X_5$ and $X_6$. The results for $X_5$ recover by different method the results from Buchstaber and Terzić (2021) and Süss (2020). The results for $X_6$ we consider to be new.