The moduli spaces of weighted $n$-pointed stable curves of genus $g$ together with reduction and forgetful morphisms were introduced by Hassett in [4]. Following his construction we introduce in [3] the category among such genus zero curves, which we call Hassett category. Losev and Manin introduced in [5] the spaces which parametrize the stable curves of genus $g$ endowed with smooth painted by black and white points. Manin proved in [6] that this space can be realized as a Hassett space with suitable weighted points. Analogously, we introduce the Losev-Manin category determined by the these genus zero curves.

The space $\overline{\mathcal{M}}_{0,n}$ is the well-known GKDM compactification of the moduli space of genus zero $n$-pointed curves. It is proved by Kapranov that $\overline{\mathcal{M}}_{0,n}$ can be identified with the Chow quotient of the complex Grassmann manifold $G_{n,2}$ by the action of the algebraic torus. In the paper [2] Buchstaber and Terzi\'c introduced the notion of the universal space of parameters $\mathcal{F}_{n}$ for the canonical compact torus action $T^n$ on $G_{n,2}$, which is a compactification of the space of parameters of the main stratum. For the description of the outgrows in this compactification we used the structure ingredients of the orbit space $G_{n,2}/T^n$ and proved that $\mathcal{F}_{n}$ can be identified with $\overline{\mathcal{M}}_{0,n}$, providing the description of $\overline{\mathcal{M}}_{0,n}$ in terms of the equivariant topology of $G_{n,2}$.

In this talk we show that the Hassett category as well as the Losev-Manin category can be modeled in terms of the ingredients of the topological model $(U_n, p_n)$ constructed by Buchstaber and Terzi\'c [1] for the description of the orbit space $G_{n,2}/T^n$, where $U_n = \Delta_{n,2}\times \mathcal{F}_{n}$ for the hypersimplex $\Delta_{n,2}$ and a smooth compact manifold $\mathcal{F}_{n}$ and $p_n : U_n \to G_{n,2}/T^n$ is a continuous surjection.

The talk is based on the joint work with Victor M. Buchstaber.

[1] Victor M.~Buchtaber, Svjetlana Terzi\'c, {\em Resolution of Singularities of the Orbit Spaces $G_{n,2}/T^n$}, Trudy Mat. Inst. Steklova, {\bf 317} (2022), 27--63.

[2] Victor M.~Buchstaber, Svjetlana Terzi\'c, {\em The orbit spaces $G_{n,2}/T^n$ and the Chow quotients $G_{n,2}/\!/\! (\mathbb{C}^{\ast})^{n}$} of the Grassmann manifolds $G_{n,2}$, Mat.~Sbornik, {\bf 214}, Iss.~12, (2023), 46--75.

[3] Victor M.~Buchtaber, Svjetlana Terzi\'c, {\em Moduli spaces of weighted pointed stable curves and toric topology of Grassmann manifolds}, preprint, 2024.

[4] Brendan Hassett, {\em Moduli spaces of weighted pointed stable curves}, Advan.~in Math.~{\bf 173}, Iss.~2, (2003), 316--352.

[5] Andrey Losev, Yuri Manin, {\em New moduli spaces of pointed curves and pencils of flat connections}, Michigan Journ.~of Math., {\bf 48} (Fulton’s Festschrift) (2000), 443--472.

[6] Yuri Manin, {\em Moduli stacks $\overline{L}_{g,S}$}, Mosc.~Math.~ Journal, {\bf 4}, Iss.~1, (2004), 181--198.