arXiv subjects

Svjetlana Terzić

Publications and source records attributed to Svjetlana Terzić.

5 recordsLinked to original sources

arXiv2026

Smooth manifolds in $G_{n,2}$ and $\mathbb{C} P^{N}$ defined by symplectic reductions of $T^n$-action

We study symplectic reductions arising from the canonical action of the maximal compact torus $T^n$ on the complex Grassmann manifold $G_{n,2}$ as well as those arising from the $T^n$-action on the complex projective space $\C P^{N}$, $N=\binom{n}{2}-1$ for which the Plücker embedding $G_{n,2}\to \C P^{N}$ is $T^n$-equivariant. We investigate the topology of regular level sets of the moment maps and the corresponding symplectic reductions. For $n=4$ we show that the regular level sets of the moment maps do not depend on a regular value. We prove this set to be homeomorphic to $S^3\times T^2$ in the case $G_{4,2}$, while in the case $\C P^5$ it is homeomorphic to $ S^5\times T^2$. We relate our constructions to moduli spaces of weighted pointed stable genus zero curves. The Deligne-Mumford compactification $\overline{\mathcal{M}}_{0, n}$ is proved to arise as a symplectic reduction of $G_{n,2}$ by the canonical $T^n$-action in precisely the cases $n=4,5$. In the case $n\geq 5$, there is well known the Losev-Manin compactification different from Deligne-Mumford and it appears to be this symplectic reduction only in the case $n=5$. In this way we show that for $n=5$, a symplectic reduction depends on a regular value of the moment map.

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arXiv2026

Cohomology of symplectic $T^n$ - reductions and compactifications of $\mathcal{M}_{0, n}$

A symplectic $T^n$ - reduction on a complex Grassmann manifold $G_{n,2}$ for the canonical action of the maximal compact torus depends on the $S_n$ - orbit of a maximal chamber in a hypersimplex $Δ_{n,2}$. The chamber decomposition of $Δ_{n,2}$ is defined by the admissible polytopes, which can be realized as matroids. In our previous work we described this chamber decomposition by means of the hyperplane arrangement. It is important to note that to any chamber it corresponds the compact space, which is a smooth compact manifold for a chamber of maximal dimension. In this paper we obtain explicit description of the cohomology rings of the symplectic $T^n$ - reductions on $G_{n,2}$ for the standard moment map in terms of the chamber decomposition of $Δ_{n,2}$ and the well known results of Kirwan and Goldin. We recently introduced the Hassett category whose objects are special compactifications of the space $\mathcal{M}_{0, n}$. The initial object in this category is the Deligne-Mumford compactification $\overline{\mathcal{M}}_{0, n}$. In this paper we describe the cohomology rings of various compactifications for $\mathcal{M}_{0, n}$, which correspond to the objects of the Hassett category. Our results show that even in the case $n=5$ there are objects in the Hassett category with different cohomology rings.

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arXiv2024

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

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 play the chamber decomposition of the hypersimplex $Δ_{n,2}$ 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 show that the points of these moduli spaces $\overline{\mathcal{M}}_{0, \mathcal{A}}$ have the geometric realization as the points of the spaces of parameters over the chambers. 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 an universal object for the introduced categories. As one of our main results we describe 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.

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arXiv2024

$ \mathbb{Z}_{2} $- homology of the orbit spaces $ G_{n,2}/ T^{n} $

We study the $\mathbb{Z}_2$-homology groups of the orbit space $X_n = G_{n,2}/T^n$ for the canonical action of the compact torus $T^n$ on a complex Grassmann manifold $G_{n,2}$. Our starting point is the model $(U_n, p_n)$ for $X_n$ constructed by Buchstaber and Terzić (2020), where $U_n = Δ_{n,2}\times \mathcal{F}_{n}$ for a hypersimplex $Δ_{n,2}$ and an universal space of parameters $\mathcal{F}_{n}$ defined in Buchstaber and Terzić (2019), (2020). It is proved by Buchstaber and Terzić (2021) that $\mathcal{F}_{n}$ is diffeomorphic to the moduli space $\mathcal{M}_{0,n}$ of stable $n$-pointed genus zero curves. We exploit the results from Keel (1992) and Ceyhan (2009) on homology groups of $\mathcal{M}_{0,n}$ and express them in terms of the stratification of $\mathcal{F}_{n}$ which are incorporated in the model $(U_n, p_n)$. In the result we provide the description of cycles in $X_n$, inductively on $ n. $ We obtain as well explicit formulas for $\mathbb{Z}_2$-homology groups for $X_5$ and $X_6$. The results for $X_5$ recover by different method the results from Buchstaber and Terzić (2021) and Süss (2020). The results for $X_6$ we consider to be new.

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arXiv2021

The orbit spaces $G_{n,2}/T^n$ and the Chow quotients $G_{n,2}\!/\!/(\mathbb{C} ^{\ast})^{n}$ of the Grassmann manifolds $G_{n,2}$

The focus of our paper is on the complex Grassmann manifolds $G_{n,2}$ which appear as one of the fundamental objects in developing the interaction between algebraic geometry and algebraic topology. In his well-known paper Kapranov has proved that the Deligne-Mumford compactification $\overline{\mathcal{M}}(0,n)$ of $n$-pointed curves of genus zero can be realized as the Chow quotient $G_{n,2}\!/\!/(\mathbb{C} ^{\ast})^{n}$. In our recent papers, the constructive description of the orbit space $G_{n,2}/T^n$ has been obtained. In getting this result our notions of the CW-complex of the admissible polytopes and the universal space of parameters $\mathcal{F}_{n}$ for $T^n$-action on $G_{n,2}$ were of essential use. Using technique of the wonderful compactification, in this paper it is given an explicit construction of the space $\mathcal{F}_{n}$. Together with Keel's description of $\overline{\mathcal{M}}(0,n)$, this construction enabled us to obtain an explicit diffeomorphism between $\mathcal{F}_{n}$ and $\overline{\mathcal{M}}(0,n)$. Thus, we showed that the space $G_{n,2}\!/\!/(\mathbb{C} ^{\ast})^{n}$ can be realized as our universal space of parameters $\mathcal{F}_{n}$. In this way, we give description of the structure in $G_{n,2}\!/\!/(\mathbb{C} ^{\ast})^{n}$, that is $\overline{\mathcal{M}}(0,n)$ in terms of the CW-complex of the admissible polytopes for $G_{n,2}$ and their spaces of parameters.

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