Z₂ - homology of the orbit spaces G₍, ₂/ T^n

We study the Z₂-homology groups of the orbit space Xₙ = G₍, ₂/Tⁿ for the canonical action of the compact torus Tⁿ on a complex Grassmann manifold G₍, ₂. Our starting point is the model (Uₙ, pₙ) for Xₙ constructed by Buchstaber and Terzi\'c (2020), where Uₙ = ₍, ₂ F₍ for a hypersimplex ₍, ₂ and an universal space of parameters F₍ defined in Buchstaber and Terzi\'c (2019), (2020). It is proved by Buchstaber and Terzi\'c (2021) that F₍ is diffeomorphic to the moduli space M₀, ₍ of stable n-pointed genus zero curves. We exploit the results from Keel (1992) and Ceyhan (2009) on homology groups of M₀, ₍ and express them in terms of the stratification of F₍ which are incorporated in the model (Uₙ, pₙ). In the result we provide the description of cycles in Xₙ, inductively on n. We obtain as well explicit formulas for Z₂-homology groups for X₅ and X₆. The results for X₅ recover by different method the results from Buchstaber and Terzi\'c (2021) and S\"uss (2020). The results for X₆ we consider to be new.