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

Abstract

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