The orbit spaces $G_{n,2}/T^n$ and the Chow quotients $G_{n,2}\!/\!/(\mathbb{C} ^{\ast})^{n}$ of the Grassmann manifolds $G_{n,2}$

Abstract

The focus of our paper is on the complex Grassmann manifolds $G_{n,2}$ which appear as one of the fundamental objects in developing the interaction between algebraic geometry and algebraic topology. In his well-known paper Kapranov has proved that the Deligne-Mumford compactification $\overline{\mathcal{M}}(0,n)$ of $n$-pointed curves of genus zero can be realized as the Chow quotient $G_{n,2}\!/\!/(\mathbb{C} ^{\ast})^{n}$. In our recent papers, the constructive description of the orbit space $G_{n,2}/T^n$ has been obtained. In getting this result our notions of the CW-complex of the admissible polytopes and the universal space of parameters $\mathcal{F}_{n}$ for $T^n$-action on $G_{n,2}$ were of essential use. Using technique of the wonderful compactification, in this paper it is given an explicit construction of the space $\mathcal{F}_{n}$. Together with Keel's description of $\overline{\mathcal{M}}(0,n)$, this construction enabled us to obtain an explicit diffeomorphism between $\mathcal{F}_{n}$ and $\overline{\mathcal{M}}(0,n)$. Thus, we showed that the space $G_{n,2}\!/\!/(\mathbb{C} ^{\ast})^{n}$ can be realized as our universal space of parameters $\mathcal{F}_{n}$. In this way, we give description of the structure in $G_{n,2}\!/\!/(\mathbb{C} ^{\ast})^{n}$, that is $\overline{\mathcal{M}}(0,n)$ in terms of the CW-complex of the admissible polytopes for $G_{n,2}$ and their spaces of parameters.