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