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

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

**Online talk**

In this talk 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 is played by 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 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 a universal object for the introduced categories. We describe as well 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.

The talk is based on joint works with Victor M. Buchstaber