Abstract:
The complex Grassmann manifolds $G_{n,2}$ are of special interest as they have several remarkable properties which distinguish them from $G_{n,k}$ for $k>2$. In this talk we present an explicit construction of the model $U_n= \Delta_{n,2}\times \mathcal{F}_{n}$ for the orbit space $G_{n,2}/T^n$ in a sense that there exists a continuous surjection $U_n\to G_{n,2}/T^n$, where $\Delta_{n,2}$ is a hypersimplex and $\mathcal{F}_{n}$ is a smooth, compact manifold. In addition, we provide an explicit description of $\mathcal{F}_{n}$ by the method of wonderful compactification and prove that it coincides with Grothendieck-Knudsen compactification $\overline{\mathcal{M}(0,n)}$ of $n$-pointed curves of genus zero, that is with the Chow quotient $G_{n,2}/\!/\!(\mathbb C ^{\ast})^{n}$. As a corollary we describe the build up points in this compactification in terms of the ingredients for the model $U_n$.
The talk is based on joint results with Victor M. Buchstaber.