Resolution of Singularities of the Orbit Spaces $G_{n,2}/T^n$

We study the orbit space $X_n = G_{n,2}/T^n$ of the standard action of the compact torus $T^n$ on the complex Grassmann manifold $G_{n,2}$. We describe the structure of the set of critical points $\operatorname {Crit}G_{n,2}$ of the generalized moment map $\mu _n: G_{n,2}\to \mathbb {R}^n$ whose image is a hypersimplex $\Delta _{n,2}$. The canonical projection $G_{n,2}\to X_n$ maps the set $\operatorname {Crit} G_{n,2}$ to the set $\operatorname {Crit}X_n$, which by definition consists of the orbits $x\in X_n$ with nontrivial stabilizer subgroup in $T^{n-1}=T^n/S^1$, where $S^1\subset T^n$ is the diagonal one-dimensional torus. Introducing the notion of a singular point $x\in \operatorname {Sing}X_n \subset X_n$ in terms of the parameter spaces of the orbits, we prove that the set $Y_n = X_n\setminus \operatorname {Sing}X_n$ is an open manifold and is dense in $X_n$. We show that $\operatorname {Crit}X_n \subset \operatorname {Sing}X_n$ for $n>4$, but $\operatorname {Sing}X_4\subset \operatorname {Crit}X_4$. Our central result is the construction of a projection $p_n: U_n= \mathcal {F}_n\times \Delta _{n,2}\to X_n$, $\dim U_n = \dim X_n$, where $\mathcal {F}_n$ is a universal parameter space. Earlier, we have proved that $\mathcal {F}_n$ is a closed smooth manifold diffeomorphic to a known manifold $\,\overline {\!\mathcal {M}}(0,n)$. We show that the map $p_n: Z_n = p_n^{-1}(Y_n)\to Y_n$ is a diffeomorphism, and describe the structure of the sets $p_n^{-1}(x)$ for $x\in \operatorname {Sing}X_n$.