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Online talk

ÌýIn this talk we relate the theory of moduli spaces $\overline{\mathcal{M}}_{0,\mathcal{A}}$ of stable weighted Ìýcurves of genus $0$ Ìýto the equivariant topology of complex Grassmann manifolds $G_{n,2}$ Ìýwith Ìýthe canonical action Ìýof the compact torus $T^n$. ÌýWe prove that all spaces $\overline{\mathcal{M}}_{0,\mathcal{A}}$ Ìýcan be isomorphically or up to birational morphisms Ìýembedded in $G_{n,2}/T^n$. The crucial role for proving this result Ìýis played by Ìýthe 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 Ìýchambers, which are subspaces in $G_{n,2}/T^n$. We single out Ìýthe characteristic Ìýcategories among Ìýsuch moduli spaces. ÌýThe morphisms in Ìýthese Ìýcategories correspond to the natural projections between the universal space of parameters and Ìýthe spaces of parameters over the chambers.

ÌýAs Ìýa corollary, we obtain the Ìýrealization of the orbit space $G_{n,2}/T^n$ Ìýas a universal object for the introduced categories. We describe as well Ìýthe structure of the canonical projection from the Deligne-Mumford compactification to the Losev-Manin compactification of $\mathcal{M}_{0,n}$, Ìýusing the Ìýembedding 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, Ìýand the realization of Ìýthe Losev-Manin compactification as the corresponding permutohedral toric variety.

ÌýThe talk is based Ìýon joint works with Victor M. Buchstaber