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# Svjetlana Terzić

### Univerzitet Crne Gore, Montenegro

### The spaces of parameters for $T^n$-action on $G_{n,2}$ and the moduli spaces $\overline{\mathcal{M}}_{0, (t_1,\ldots , t_n)}$}

The canonical action of the compact torus $T^n$ on a complex Grassmann manifold $G_{n,2}$ of two-dimensional complex subspaces in $\C ^{n}$ is an important and widely known example which appears to many areas of mathematics.
The main stratum $W_{n}\subset G_{n,2}$ given by those points from $G_{n,2}$ whose all Pl\"ucker coordinates are non-zero, plays a crucial role in description of the space $G_{n,2}/T^n$, as it is an open dense set in $G_{n,2}$ and it belongs to any Pl\"ucker chart for $G_{n,2}$. In addition, we earlier proved that $W_n\cong \stackrel{\circ}{\Delta}_{n,2}\times F_{n}$, where $\Delta_{n,2}$ is the hypersimplex and $F_{n}=W_{n}/ (\C ^{\ast})^{n}\subset \C P^{N}, N={n-2\choose 2}$ is an open algebraic manifold. In order to construct a model for the orbit space $G_{n,2}/T^n$ it turns out to be important to look for a compactification $\mathcal{F}_{n}$ for $F_n$ such that any automorphism of $F_n$ induced by the transition maps between the Pl\"ucker charts extends to the automorphism of $\mathcal{F}_{n}$. Such compactification $\mathcal{F}_{n}$ we call the universal space of parameters. We obtain the space $\mathcal{F}_{n}$ by resolving singularities that arise, using the techniques of wonderful compactification from algebraic geometry.
On the other side, the moduli space $\overline{\mathcal{M}}_{0,n}$ of $n$-pointed stable genus zero curves is the Deligne-Mumford-Grothendieck-Knudsen compactification of the moduli space $\mathcal{M}_{0,n}$ of $n$-pointed genus zero curves. The space $\overline{\mathcal{M}}_{0,n}$ is proved by Kapranov to coincide with the Chow
quotient $G_{n,2}/\!/\! (\C ^{\ast})^{n}$. By proving that $\mathcal{F}_{n}$ coincides with $\overline{\mathcal{M}}_{0,n}$ we present another approach to the compactification of $\mathcal{M}_{0,n}$.
In addition, we discuss the relations between the spaces of parameters $F_{\omega}$ of the chambers $C_{\omega}\subset \Delta_{n,2}$, which arise from the matroidal decomposition of $\Delta _{n,2}$, and
the moduli space $\overline{\mathcal{M}}_{0, (t_1,\ldots , t_n)}$ of weighted $n$-pointed stable genus zero curves, where $(t_1,\ldots , t_n)\in C_{\omega}$.
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The talk is based on the results obtained jointly with Victor M. Buchstaber.