Smooth manifolds in $G_{n,2}$ and $\mathbb{C}{P}^N$ defined by symplectic reductions of $T^n$-action

Plücker coordinates define the $T^n$-equivariant embedding $p:G_{n,2}\to \text{C}{P}^N$ of a complex Grassmann manifold $G_{n,2}$ into the complex projective space $\text{C}{P}^N$, $N=\left(\genfrac{}{}{0px}{}{n}{2}\right)-1$ for the canonical $T^n$-action on $G_{n,2}$ and the $T^n$-action on $\text{C}{P}^N$ given by the second exterior power representation $T^n\to T^N$ and the standard $T^N$-action. Let $\mu :G_{n,2}\to {\mathrm{\Delta }}_{n,2}\subset {\mathbb{R}}^n$ and $\tilde{\mu }:\text{C}{P}^N\to {\mathrm{\Delta }}_{n,2}\subset {\mathbb{R}}^n$ be the moment maps for the $T^n$-actions on $G_{n,2}$ and $\text{C}{P}^N$ respectively, such that $\tilde{\mu }\circ p=\mu$. The preimages ${\mu }^{-1}(\mathbf{x})$ and ${\tilde{\mu }}^{-1}(\mathbf{y})$ are smooth submanifolds in $G_{n,2}$ and $\text{C}{P}^N$, for any regular values $\mathbf{x},\mathbf{y}\in {\mathrm{\Delta }}_{n,2}$ for these maps, respectively. The orbit spaces ${\mu }^{-1}(\mathbf{x})\mathrm{/}{T}^n$ and ${\tilde{\mu }}^{-1}(\mathbf{y})\mathrm{/}{T}^n$ are symplectic manifolds, which are known as symplectic reduction. The regular values for $\mu$ and $\tilde{\mu }$ coincide for $n=4$ and we prove that ${\mu }^{-1}(\mathbf{x})$ and ${\tilde{\mu }}^{-1}(\mathbf{x})$ do not depend on a regular value $\mathbf{x}\in {\mathrm{\Delta }}_{4,2}$. We provide their explicit topological description, that is we prove ${\mu }^{-1}(\mathbf{x})\cong S^3\times T^2$ and ${\tilde{\mu }}^{-1}(\mathbf{x})\cong S^5\times T^2$. The Deligne - Mumford compactification ${\bar{\mathcal{M}}}_{0,n}$ is proved to be a symplectic reduction of $G_{n,2}$ by the canonical $T^n$-action if and only if $n=4,5$, while the Losev - Manin compactification is a such symplectic reduction if and only if $n=5$.