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

Abstract

The complex Grassmann manifolds $G_{n,k}$ appear as one of the fundamental objects in developing an interaction between algebraic geometry and algebraic topology. The case $k=2$ is of special interest on its own as the manifolds $G_{n,2}$ have several remarkable properties which distinguish them from the $G_{n,k}$ for $k>2$. In our paper we obtain results which, essentially using the specifics of the Grassmann manifolds $G_{n,2}$, develop connections between algebraic geometry and equivariant topology. They are related to well-known problems of the canonical action of the algebraic torus $(\mathbb{C}^{\ast})^n$ on $G_{n,2}$ and the induced action of the compact torus $T^n\subset(\mathbb{C}^{\ast})^n$. Kapranov proved that the Deligne-Mumford-Grothendieck-Knudsen compactification $\overline{\mathcal{M}}(0,n)$ of the space of $n$-pointed rational stable curves can be realized as the Chow quotient $G_{n,2}//(\mathbb{C}^{\ast})^{n}$. In recent papers of the authors a constructive description of the orbit space $G_{n,2}/T^n$ was obtained. In deducing this result the notions of the complex of admissible polytopes and the universal space of parameters $\mathcal{F}_{n}$ for the $T^n$-action on $G_{n,2}$ were of essential use. Using the techniques of wonderful compactification, in this paper an explicit construction of the space $\mathcal{F}_{n}$ is presented. In combination with Keel's description of $\overline{\mathcal{M}}(0,n)$, this construction enabled one to obtain an explicit diffeomorphism between $\mathcal{F}_{n}$ and $\overline{\mathcal{M}}(0,n)$. In this way, we give a description of $G_{n,2}//(\mathbb{C}^{\ast})^{n}$ as the space $\mathcal{F}_{n}$ with a structure described in terms of admissible polytopes $P_\sigma$ and spaces $F_\sigma$. Bibliography: 32 titles.