Moduli space of weighted pointed stable curves and toric topology of Grassmann manifolds

Abstract

We relate the theory of moduli spaces $\overline{\mathcal{M}}_{0,\mathcal{A}}$ of stable weighted curves of genus $0$ to the equivariant topology of complex Grassmann manifolds $G_{n,2}$, with the canonical action of the compact torus $T^n$. We prove that all spaces $\overline{\mathcal{M}}_{0,\mathcal{A}}$ can be isomorphically or up to birational morphisms embedded in $G_{n,2}/T^n$ . The crucial role for proving this result play the chamber decomposition of the hypersimplex $\Delta _{n,2}$ which corresponds to $(\mathbb{C} ^{\ast})^{n}$-stratification of $G_{n,2}$ and the spaces of parameters over the chambers, which are subspaces in $G_{n,2}/T^n$. We show that the points of these moduli spaces $\overline{\mathcal{M}}_{0, \mathcal{A}}$ have the geometric realization as the points of the spaces of parameters over the chambers. We single out the characteristic categories among such moduli spaces. The morphisms in these categories correspond to the natural projections between the universal space of parameters and the spaces of parameters over the chambers. As a corollary, we obtain the realization of the orbit space $G_{n,2}/T^n$ as an universal object for the introduced categories. As one of our main results we describe the structure of the canonical projection from the Deligne-Mumford compactification to the Losev-Manin compactification of $\mathcal{M}_{0,n}$, using the embedding of $\mathcal{M}_{0, n}\subset \bar{L}_{0, n, 2}$ in $(\mathbb{C} P^{1})^{N}$, $N=\binom{n-2}{2}$, the action of the algebraic torus $(\mathbb{C} ^{\ast})^{n-3}$ on $(\mathbb{C} P^{1})^{N}$ for which $\bar{L}_{0, n, 2}$ is invariant, and the realization of the Losev-Manin compactification as the corresponding permutohedral toric variety.