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

In this paper we establish fundamental relations between the famous problem of compactifications of the moduli space $M_{0,n}$ of ordered n distinct points on $C{P}^1$ and toric topology of the complex Grassmann manifolds $G_{n,2}$. The best known is the Deligne-Mumford compactification ${\bar{M}}_{0,n}$. The Losev-Manin compactification ${\overline{L}}_{0,n,2}$ is also closely related to important questions in mathematical physics and toric geometry. These compactifications belong to the family of Hassett compactifications ${\bar{M}}_{0,n,A}$ of moduli spaces of weighted pointed stable curves. In this paper we show that the orbit space $G_{n,2}/T^n$ of complex Grassmann manifolds $G_{n,2}$ by the canonical action of the compact torus $T^n$ of the complexity $n-3$, $n\ge 3$ serves as a universal space in a sense that for any Hassett compactification ${\bar{M}}_{0,n,A}$ there exists a subspace $F_{\omega }$ in $G_{n,2}/T^n$ and birational morphism ${\bar{M}}_{0,n,A}\to F_{\omega }$. We describe large class of the set of weights $A$ for which this birational morphism gives rise to an isomorphism. This provides topological model for the Hassett category in which Deligne-Mumford compactification ${\bar{M}}_{0,n}$ is the initial object. Manin showed that compactification ${\overline{L}}_{0,n,2}$ is a smooth toric variety over permutohedron. We provide explicit geometric realization of the manifold ${\overline{L}}_{0,n,2}$ as a toric compactification of an orbit for algebraic torus ${\left(C^{⁎}\right)}^{n-3}$ - action on ${\left(C{P}^1\right)}^N$, $N=\left(\begin{array}{c}n-2\\ 2\end{array}\right)$ and describe the epimorphism ${\bar{M}}_{0,n}\to {\overline{L}}_{0,n,2}$ in terms of Fulton - MacPherson, De Concini-Procesi and Li wonderful compactification.¹