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 wellknown paper Kapranov has proved that the Deligne-Mumford compactification M(0, n) of n-pointed curves of genus zero can be realized as the Chow quotient G n,2 //(C * ) 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 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 F n . Together with Keel's description of M(0, n), this construction enabled us to obtain an explicit diffeomorphism between F n and M(0, n). Thus, we showed that the space G n,2 //(C * ) n can be realized as our universal space of parameters F n . In this way, we give description of the structure in G n,2 //(C * ) n , that is M(0, n) in terms of the CW-complex of the admissible polytopes for G n,2 and their spaces of parameters. 1