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The orbit spaces V. M. Buchstaberab, S. Terzićc a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia b National Research University Higher School of Economics, Moscow, Russia c Faculty of Science and Mathematics, University of Montenegro, Podgorica, Montenegro
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     § 1. IntroductionQuestions about the canonical action of the algebraic torus $(\mathbb{C}^{\ast})^n$ and the induced action of the compact torus $T^n$ on complex Grassmann manifolds arise naturally in many areas of mathematics (see [14]–[17] and [19]). In this paper we discuss problems which are at the crossroads of toric geometry (a part of algebraic geometry) and toric topology (a part of algebraic topology). The central object of toric geometry is algebraic manifolds endowed with an action of the algebraic torus. A toric variety is defined to be the closure of one orbit of the algebraic torus. The equivariant structure of a nonsingular toric manifold can effectively be described in terms of the combinatorial structure of the moment polytope, the image of the moment map. One important example of a toric manifold is the complex projective space $\mathbb{C} P^n=G_{n+1,1}$. The issue which arises naturally in this context is to study algebraic varieties with algebraic torus actions whose family of algebraic torus orbits defines, in a certain way, a good stratification of the variety. The first such nontrivial examples are the Grassmann manifolds $G_{n,2}$ which can also be interpreted as the spaces of all projective lines in $\mathbb{C} P^{n-1}$. It follows from results due to Gelfand and Serganova [15] that in the case of $G_{n,2}$ not only the strata given by orbits of the algebraic torus can be described explicitly, but also the structure of the stratification, that is, the gluing of these strata. However, this is no longer true for the Grassmannians $G_{n,k}$, where $n\geqslant 7$ and $k\geqslant 3$, as it was demonstrated in an example in [15]; see also [3]. Note that in [3] based on the gluing of strata in $G_{n,2}$ we introduced the notion of the complex of admissible polytopes. In this paper we establish a connection between the well-known constructions from algebraic geometry based on the notion of wonderful compactification and the results on the equivariant algebraic topology of the Grassmann manifolds $G_{n,2}$. This connection was discovered in the course of solving of the problem of the description of the universal space of parameters $\mathcal{F}_n$ of the canonical $T^n$-action on $G_{n,2}$. The general concept of universal space of parameters was introduced in the theory of $(2m, k)$-manifolds developed by these authors: see [3]. This theory focuses on smooth $2m$-dimensional manifolds with a smooth action of $k$-dimensional compact torus, which satisfy a certain set of axioms. An important class of such manifolds consists of the manifolds on which an action of the compact torus $T^k$ is induced by an action of the algebraic torus $(\mathbb{C}^{\ast})^{k}$. Among these manifolds a remarkable role is played by the manifolds $G_{n,2}$, which are $(2m,k)$-manifold for $m=2(n-2)$ and $k=n-1$. The universal space of parameters $\mathcal{F}$ for a $(2m,k)$-manifold $M^{2m}$ with an effective action of the compact torus $T^k$, $k\leqslant m$, is a compactification of the space of parameters $F$ of the main stratum. Its detailed definition was presented in [3]. The universal space of parameters $\mathcal{F}_{5}$ for the Grassmannian $G_{5,2}$ with regard to the canonical action of $T^5$ was explicitly described in [2], where it is proved that $\mathcal{F}_{5}$ is the blow-up of the cubic surface $c_1c_2'c_3=c_1'c_2c_{3}'$ in $(\mathbb{C} P^{1})^{3}$ at one point, which is the same as the blow-up of $\mathbb{C} P^{2}$ at four points in general position. Thus, for $G_{5,2}$ the universal space of parameters is the del Pezzo surface of degree $5$, which is well known in algebraic geometry; see also [30]. Klemyatin [25] presented a construction of a compactification of the space $F_n$. On the one hand the method in [25] appeals to the Gelfand-MacPherson correspondence [14] between the space $F_n$ and the configurations of $n$ pairwise distinct points in $(\mathbb{C} P^1)^{n}$ up to the action of $\mathrm{GL}(2,\mathbb{C})$ on $\mathbb{C} P^1$, and on the other hand it is based on the description of the Grothendieck-Knudsen compactification $\overline{\mathcal{M}}(0,n)$ of the moduli space $\mathcal{M}(0, n)$ in terms of the cross ratios of quadruples of points, as described by McDuff and Salamon in [29]. Unfortunately, there was a gap in the proof of the claim that this construction produces the universal space of parameters $\mathcal{F}_n$; see Remark 22. In [19] Kapranov considered the Chow quotient $G_{n,2}/\!/ (\mathbb{C}^{\ast})^{n}$ and proved that it coincides with the Grothendieck-Knudsen compactification $\overline{\mathcal{M}}(0,n)$ of the moduli space. It is a well known fact that $\overline{\mathcal{M}}(0,5)$ is the del Pezzo surface of degree $5$. The cohomology rings of the spaces $\overline{\mathcal{M}}(0,n)$ were described in [22] (see also [21]). Complex cobordism classes of the spaces $\overline{\mathcal{M}}(0,n)$ are important for applications of the Chern character to complex cobordism theory; see [6] and [4]. In this paper we provide an explicit construction of the universal space of parameters $\mathcal{F}_{n}$ for $G_{n,2}$. Using only the equivariant topology of the Grassmann manifolds $G_{n,2}$ we construct a smooth compact manifold $\mathcal{F}_{n}$ and prove that it is diffeomorphic to the moduli space $\overline{\mathcal{M}}(0,n)$, that is, to the Chow quotient $G_{n,2}/\!/ (\mathbb{C}^{\ast})^{n}$. We propose to formulate the problem of the compactification of an algebraic variety as follows: consider an algebraic variety $X$ in $(\mathbb{C} P^1)^{N}$ which is an open subset of its closure $\overline{X}$ in $(\mathbb{C} P^{1})^{N}$, where $\overline{X}$ is a smooth compact submanifold of $(\mathbb{C} P^1)^{N}$, and fix a family of its automorphisms $\mathcal{A}(X)$. Problem 1. Find a compactification $\mathcal{X}$ of $X$ for which there exists a projection $p\colon \mathcal{X}\to \overline{X}$ whose restriction to $X$ is the identity map, and such that each automorphism $f\in \mathcal{A}(X)$ extends to an automorphism of $\mathcal{X}$. In our case $X$ is the space of parameters $F_n$ of the main stratum $W_n$, which is expressed in coordinates in a fixed standard chart on $G_{n,2}$, explicitly defined in terms of the Plücker coordinates. Recall that $W_n$ lies in the intersection of all standard charts on $G_{n,2}$. As $\mathcal{A}(X)$ we take the group of automorphisms of $X$ induced by the coordinate changes between the fixed chart and all other charts. Our construction of the required compactification $\mathcal{X}$ of $F_n$ uses the construction of wonderful compactification, which is well known in algebraic geometry. Using the properties of this compactification we can show that the automorphisms of $F_n$ described above extends to automorphisms of the space $\mathcal{X}$. As a result, we prove that $\mathcal{X}$ coincides with the space $\mathcal{F}_n$ in question, which is diffeomorphic to the Chow quotient $G_{n,2}/\!/ (\mathbb{C}^{\ast})^{n}$, that is, to the Grothendieck-Knudsen compactification $\overline{\mathcal{M}}(0, n)$ of the space $\mathcal{M}(0,n)$. The wonderful compactification of a complex manifold $M$ is a compact smooth manifold $\widetilde{X}$ such that $D=\widetilde{X}\setminus M$ is a divisor with normal crossings in $\widetilde{X}$ whose irreducible components are smooth, and any number of connected components of $D$ intersect transversally. Such strong conditions on a compactification turn out to be of essential importance for many algebraic and geometric problems, such as problems of enumerative algebraic geometry and the Schubert enumerative problems, the description of the rational homotopy type of $M$, its mixed Hodge structure, the Chow ring, and so on. The notion of wonderful compactification appeared for the first time in the paper [8] by De Concini-Procesi, in the context of an equivariant compactification of the symmetric spaces $G/H$; see also [28] and [31] for a comprehensive overview of the subject. Then this idea was developed further and applied in many directions, such as Fulton-MacPherson compactification [13], De Concini-Procesi wonderful models [8], [9], the wonderful compactification of Li [27] and, more recently, the projective wonderful models of toric arrangements by De Concini-Gaiffi and others [10]–[12]. Furthermore, this paper reveals the advantage of the notion of wonderful compactification for the description of the equivariant topology of the Grassmannians $G_{n,2}$ for the canonical $T^n$-action. More explicitly, we show that the wonderful compactification of arrangements of subvarieties from [27] enables one to solve successfully of the space of parameters $F_n$ of the main stratum $W_n\subset G_{n,2}$. In this way we obtain the smooth manifold $\mathcal{F}_{n}$, which enables us to construct the model $U_n = \mathcal{F}_{n}\times \Delta_{n,2}$ for the orbit space $G_{n,2}/T^n$ which describes the structure of $G_{n,2}/T^n$ in terms of the continuous surjection $p_n\colon U_n \to G_{n,2}/T^n$; see [5]. The Chow quotient $G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$ is a compactification of the orbit space $W_n/(\mathbb{C}^{\ast})^{n}$ for the main stratum $W_n\subset G_{n,2}$ described in terms of the Chow variety for $G_{n,2}$; see [19] and [16]. It coincides with Grothendieck-Knudsen compactification of the space $\mathcal{M}(0,n)$ of $n$-pointed curves of genus zero. The space $\overline{\mathcal{M}}(0,n)$ has a number of remarkable realizations: (1) it is the moduli space of stable $n$-pointed rational curves; (2) it is the log canonical compactification [24]; (3) it provides the canonical description of limits of one-parameter compactifications [20]. In [23] the first two realizations were generalized to the complex Grassmann manifold $G_{n,k}$ for any $k\geqslant 2$, while the third, which holds for $G_{n,2}$, is proved to be false except conjecturally in few special cases. Moreover, it was proved by Kapranov that the Chow quotient $X/\!/\,G$ dominates all quotients in geometric invariant theory (GIT). In [18] various interpretations of the Chow quotient $X/\!/\,G$ of a projective variety $X$ by a reductive algebraic group $G$ were provided from the point of view of algebraic geometry, symplectic geometry and topology. The topological interpretation of the Chow quotient $X/\!/\,G$ in [18] is as follows: one considers the polar decomposition $G= K\cdot A$ of a connected group, where $K\subset G$ is a maximal compact subgroup and $A\subset G$ is a completely solvable subgroup. For example, $(\mathbb{C}^{\ast})^{n}$ has a unique polar decomposition, which is given by $(\mathbb{C}^{\ast})^{n}= T^n\cdot \mathbb{R}^{n}_{>0}$. The closures of the $A$-orbits of points in $X$ are called action manifolds and the moment map $\mu\colon X \to \mathbb{R}^{\dim K}$ gives a bijection between $\overline{x\cdot A}$ and $\mu (\overline{x\cdot A})$; action manifolds through generic points are called generic action manifolds, and the moduli space of generic action manifolds by the action of $K$ can be considered. It was proved that the compactification of this moduli space can be identified with the Chow quotient $X/\!/G$, and the outgrow points of this compactification are called stable action manifolds; stable action manifolds can be obtained as unions of the closures of $A$-orbits of maximum dimension. The union of the moment polytopes for orbits from a stable action manifold is the moment polytope for $X$. The construction described generalizes in fact the description due to Kapranov [19] of the construction of algebraic cycles which give the outgrows in the compactification for $W/(\mathbb{C}^{\ast})^{n}$ in the Chow quotient in $G_{n,k}/\!/(\mathbb{C}^{\ast})^{n}$ for the main stratum $W\subset G_{n,k}$. The focus of our paper, as already mentioned, is on the universal space of parameters $\mathcal{F}_{n}$ for the canonical $T^n$-action on $G_{n,2}$, for which we provide an explicit construction and prove that $\mathcal{F}_{n}$ can be identified with the Chow quotient $G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$. As a result, we describe this Chow quotient in terms of certain combinatorial and topological notions, which we introduced previously for the description of the orbit space $G_{n,2}/T^n$. The outgrow points for the Chow-quotient compactification were described in [19] in terms of maximal algebraic cycles in $G_{n,2}$. In this paper we push this further and show that outgrow components of $G_{n,2}/\!/ (\mathbb{C}^{\ast})^{n}$ can explicitly be described in terms of our complex of admissible polytopes and their spaces of parameters. In this description the key role is played by cortéges of $(n-1)$-dimensional admissible polytopes, which define polyhedral decompositions for $\Delta_{n,2}$. In the case of the Chow quotient $G_{5,2}/\!/(\mathbb{C}^{\ast})^{5}$ we compare thoroughly our description of $\mathcal{F}_5$ with Kapranov’s description of outgrow points in the Chow quotient. In this context our central result is Theorem 30, which describes outgrow components of this Chow quotient in terms of the chambers in the hypersimplex $\Delta_{n,2}$ and their spaces of parameters. § 2. Space of parameters $F_n$ of the main stratum for $G_{n,2}$2.1. The embedding of $F_n$ in $( {\mathbb{C}}P^{1})^{N}$First we recall from [2] and [3] the notions of the main stratum $W_n$ in $G_{n,2}$ and its space of parameters $F_n$ as a background introduction to the objects we are going to consider. The main stratum $W_n\subset G_{n,2}$ is characterized by the condition that all Plücker coordinates of its points are nonzero, which implies that $W_n$ belongs to any standard chart $M_{ij} = \{ L\in G_{n,2}\mid P^{ij}(L)\neq 0\}$, $1\leqslant i<j\leqslant n$, of a Grassmann manifold $G_{n,2}$. The main stratum $W_n$ is invariant under the canonical action of the algebraic torus $(\mathbb{C}^{\ast})^{n}$, and the orbit space $F_{n} = W_n/(\mathbb{C}^{\ast})^{n}$ is said to be the space of parameters for $W_n$. Without loss of generality we treat $W_n$ as a subspace of $ M_{12}$. Any point $L\in M_{12}$ can be represented by an $ n\times 2 $ matrix
$$
\begin{equation*}
A_{L}= \begin{pmatrix} 1 & 0\\ 0 & 1\\ z_3 & w_3\\ \vdots & \vdots \\ z_n & w_n \end{pmatrix},
\end{equation*}
\notag
$$
where $z_3,\dots,z_n,w_3,\dots,w_n$ are the coordinates of $L$ in the chart $M_{12}$. Points in the $(\mathbb{C}^{\ast})^n$-orbit of a point $L^{0}=(z_3^{0},\dots,z_n^{0},w_{3}^{0},\dots,w_n^{0})\in W_n$ satisfy the equalities
$$
\begin{equation}
c_{ij}'z_iw_{j}=c_{ij}z_jw_{i}, \qquad 3\leqslant i<j\leqslant n,
\end{equation}
\tag{1}
$$
where $c_{ij}=z_{j}^{0}w_{i}^{0}$ and $c_{ij}'=z_{i}^{0}w_{j}^{0}$. Note that $(c_{ij}:c_{ij}')\in \mathbb{C} P^1$, and we have $c_{ij}, c_{ij}'\neq 0$ and $c_{ij}\neq c_{ij}'$ for all $3\leqslant i<j\leqslant n$. Hence in the chart $M_{12}$ the main stratum $W_n$ is described by the system of equations (1) with parameters $(c_{ij}:c_{ij}')\in \mathbb{C} P^1$, where $c_{ij},c_{ij}'\neq 0$ and $c_{ij}\neq c_{ij}'$ for all $3\leqslant i<j\leqslant n$. The number of parameters $(c_{ij} : c_{ij}')$ is $N=\binom{n-2}{2}$, and it follows from (1) that these parameters satisfy the following equations:
$$
\begin{equation}
c_{ij}'c_{ik}c_{jk}'=c_{ij}c_{ik}'c_{jk}, \qquad 3\leqslant i<j<k\leqslant n.
\end{equation}
\tag{2}
$$
The number of these equations is $\binom{n-2}{3}$. We see from (2) that the parameters $(c_{ij}: c_{ij}')$ satisfy the relations
$$
\begin{equation}
(c_{ij}:c_{ij}')=(c_{3i}'c_{3j}: c_{3i}c_{3j}'), \qquad 4\leqslant i<j\leqslant n,
\end{equation}
\tag{3}
$$
and
$$
\begin{equation}
(c_{3i}:c_{3i}')\neq (c_{3j}:c_{3j}'), \qquad 4\leqslant i<j\leqslant n.
\end{equation}
\tag{4}
$$
The number of these relations is $M=\binom{n-3}{2}$. Therefore, $F_n$ is homeomorphic to the space
$$
\begin{equation}
F_n\cong (\mathbb{C} P^{1}_{A})^{n-3}\setminus \Delta,
\end{equation}
\tag{5}
$$
where $A=\{(0:1), (1:0), (1:1)\}$, $\mathbb{C} P^{1}_{A} = \mathbb{C} P^{1}\setminus {A}$ and $\Delta = \bigcup_{3\leqslant i<j\leqslant n}\Delta_{ij}$ for the diagonals $\Delta_{ij} = \{((c_{34}:c_{34}'), \dots, (c_{n-1,n}, c_{n-1,n}'))\in (\mathbb{C} P^{1}_{A})^{n-3}\mid (c_{3i}:c_{3i}') = (c_{3j}:c_{3j}')\}$. Relations (2) define an embedding of the space $F_n$ into $(\mathbb{C} P^{1})^{N}$, $N=\binom{n-2}{2}$, that is, we have the following result. Lemma 2. The space of parameters $F_n$ of the main stratum $W_n$ is defined by equations (2) as a subspace of $(\mathbb{C} P^{1})^{N}$, $N=\binom{n-2}{2}$. Moreover, we see that $F_n$ is an open algebraic manifold in $\overline{F}_{n}\subset (\mathbb{C} P^{1})^{N}$ given by the intersection of the cubic hypersurfaces (2) and the conditions that ${(c_{ij}':c_{ij})}\in \mathbb{C} P^{1}_{A}$. Note that the dimension of $F_n$ is $2(n-3)$, which is exactly equal to ${2(N-M)}$. Proposition 3. The compactification $\overline{F}_{n}$ of $F_{n}$ in $(\mathbb{C} P^{1})^{N}$ is the smooth algebraic variety given by Proof. We consider the gradients of the functions $f_{ijk} = c_{ij}c_{ik}'c_{jk} - c_{ij}'c_{ik}c_{jk}'$ defining $\overline{F}_{n}$. It can be verified directly that there are $M=\binom{n-3}{2}$ linearly independent vectors among these gradients at any point of $\overline{F}_{n}$. This implies that $\overline{F}_{n}$ is a smooth algebraic variety of real dimension $2(N-M)$ for $N=\binom{n-2}{2}$ and $M=\binom{n-3}{2}$. 2.2. The problem of compactification of $F_{n}$The motivation for our approach in finding a proper compactification of the space of parameters $F_{n}$ of the main stratum $W_{n}$ in $G_{n,2}$ comes from our works [2], [3] on the description of the topology of the orbit space $G_{n,2}/T^n$. Using the Plücker coordinates one can define a stratification of the Grassmannian $G_{n,2}$, that is, $G_{n,2}=\bigcup_{\sigma}W_{\sigma}$, where $\sigma \subset \{\{i,j\}\subset \{1, \dots,n\},\ i\neq j\}$ and
$$
\begin{equation*}
W_{\sigma}=\{ L\in G_{n,2} \mid P^{ij}(L)\neq 0 \ \text{only for } \{i,j\} \in \sigma\}\neq\varnothing.
\end{equation*}
\notag
$$
Strata are pairwise disjoint and $(\mathbb{C}^{\ast})^n$-invariant, so they are $T^n$-invariant. This induces a stratification of the orbit space: The main stratum $W_{n}$ is an open dense set in $G_{n,2}$, and the torus $T^{n-1}=T^{n}/S^1$ acts freely on it. In [2] we proved that where $\Delta_{n,2}$ is the hypersimplex. Definition 4. A polytope $P_{\sigma}\subset \Delta_{n,2}$ such that is called an admissible polytope of the stratum $W_{\sigma}$. For each stratum $W_{\sigma}$ we have proved the existence of a homeomorphism
$$
\begin{equation*}
W_{\sigma}/T^n\cong \mathring{P}_{\sigma}\times F_{\sigma},
\end{equation*}
\notag
$$
where $F_{\sigma}=W_{\sigma}/(\mathbb{C}^{\ast})^n$; see [2]. Note that the fact that $W_n$ is an open dense set in $G_{n,2}$, in combination with (1), implies that to any stratum $W_{\sigma}$ we can assign to it a subspace $\widetilde{F}_{\sigma}\subset \mathcal{F}_{n}$ for any compactification $\mathcal{F}_{n}$ of $F_{n}$. In our construction of the orbit space $G_{n,2}/T^n$ we look for such a compactification $\mathcal{F}_{n}$ of $F_n$ for which $\mathcal{F}_{n}= \bigcup_{\sigma}\widetilde{F}_{\sigma}$, where for any stratum $W_{\sigma}$ there exists a projection $p_{\sigma}\colon \widetilde{F}_{\sigma}\to F_{\sigma}$. In finding such a compactification we start by noting the following: if we consider a stratum $W_{\sigma}$ in a chart $M_{ij}$, then we assign to a space $\widetilde{F}_{\sigma, ij}\subset \mathcal{F}_{n}$ using the fact that $W_{\sigma}/T^{n}$ is on the boundary of $W_{n}/T^n$. We obtain the required compactification $\widehat{\mathcal{F}}_n$ if $\widetilde{F}_{\sigma, ij}$ does not depend on the chart $M_{ij}$ such that $W_{\sigma}\subset M_{ij}$. Thus, we start by considering the above smooth compact manifold $\overline{F}_{n}\subset (\mathbb{C} P^{1})^{N}$ and all possible subspaces $\widetilde{F}_{\sigma, ij}\subset \overline{F}_{n}$, and then we find modifications $\widetilde{F}_{\sigma}$ of $\widetilde{F}_{\sigma, ij}$ such that they do not depend on the chart $M_{ij}$. 2.3. Automorphisms of $F_n$ induced by changes of coordinatesThe main stratum $W_n$ lies in the intersection of the standard charts and is defined in these charts in terms of the Plücker coordinates. Equations (1) and (2) provide its definition in the coordinates of $M_{12}$. Identifying $W_n$ with its descriptions in the charts $M_{ij}$, $i<j$, we see that transition maps between the charts produce automorphisms of the space of parameters $F_n$ of the main stratum. We deduce explicitly the automorphisms of $F_n$ induced by the transition maps between the charts $M_{12}$ and $M_{ij}$, $i<j$, $\{1,2\}\neq \{i,j\}$. Denote the local coordinates in $M_{12}$ by and let
$$
\begin{equation*}
\begin{gathered} \, z_{1}', \dots, z_{i-1}', z_{i+1}', \dots, z_{j-1}', z_{j+1}', \dots, z_n', \\ w_{1}', \dots, w_{i-1}',w_{i+1}', \dots, w_{j-1}', w_{j+1}', \dots, w_n' \end{gathered}
\end{equation*}
\notag
$$
be the local coordinates in $M_{ij}$. Further let $(c_{pq}: c_{pq}')$, $3\leqslant p<q\leqslant n$, be the coordinate record of the space $F_n$ in the chart $M_{12}$ and $(d_{kl}: d_{kl}')$, $1\leqslant k<l\leqslant n$, $k,l\neq i,j$, be the coordinate record of $F_n$ in the chart $M_{ij}$. We differentiate between the following cases. 1) $i=1$ and $3\leqslant j\leqslant n$, that is, we consider the chart $M_{1j}$. In this case
$$
\begin{equation}
\begin{gathered} \, z_{2}'=-\frac{z_j}{w_j}, \qquad z_{k}'=\frac{z_{k}w_{j}-z_{j}w_{k}}{w_j}, \quad k\geqslant 3, \\ \notag w_{2}'=\frac{1}{w_{j}}, \qquad w_{k}'=\frac{w_{k}}{w_{j}}, \quad k\geqslant 3. \end{gathered}
\end{equation}
\tag{6}
$$
Lemma 5. The coordinates $(c_{pq}: c_{pq}')$ and $(d_{kl}:d_{kl}')$ of the space of parameters $F_n$ in the charts $M_{12}$ and $M_{1j}$, $3\leqslant j\leqslant n$, are related by The second expression can be written as Proof. In the chart $M_{1j}$ the main stratum writes as
Substituting (6) into these equations of the main stratum we obtain
$$
\begin{equation*}
- d_{2l}'\frac{z_j}{w_j}\,\frac{w_{l}}{w_{j}}=d_{2l}\frac{z_lw_j-z_jw_l}{w_j}\, \frac{1}{w_j},
\end{equation*}
\notag
$$
which implies that
$$
\begin{equation*}
d_{2l}'=-d_{2l} \biggl(\frac{z_lw_j}{z_jw_l}-1\biggr) =d_{2l}\biggl(1-\frac{c_{jl}'}{c_{jl}}\biggr),
\end{equation*}
\notag
$$
that is, For $k\geqslant 3$, in a similar way we obtain
$$
\begin{equation*}
d_{kl}'w_{l}(z_kw_j-z_jw_k)=d_{kl}w_{k}(z_lw_j - z_jw_l),
\end{equation*}
\notag
$$
which implies that that is, The lemma is proved. 2) $i=2$ and $3\leqslant j\leqslant n$, that is, we consider the chart $M_{2j}$. Then we have
$$
\begin{equation*}
\begin{gathered} \, z_{1}'=-\frac{w_j}{z_j}, \qquad z_{k}'=\frac{z_{j}w_{k}-z_{k}w_{j}}{z_j} \quad\text{for } k\geqslant 3, \\ w_{1}'=\frac{1}{z_{j}}\quad\text{and} \quad w_{k}'=\frac{z_{k}}{z_{j}} \quad\text{for } k\geqslant 3. \end{gathered}
\end{equation*}
\notag
$$
Substituting this into the equations of the main stratum written in the chart $M_{2j}$ we obtain the following result. Lemma 6. The coordinates $(c_{pq}: c_{pq}')$ and $(d_{kl}:d_{kl}')$ of the space of parameters $F_n$ in the charts $M_{12}$ and $M_{2j}$, $3\leqslant j\leqslant n$, are related by and 3) $3\leqslant i<j\leqslant n$, so that we are in the chart $M_{ij}$, and we have
$$
\begin{equation*}
z_{1}'=\frac{w_{j}}{z_{i}w_{j}-z_{j}w_{i}}, \qquad z_{2}'= -\frac{z_{j}}{z_{i}w_{j}-z_{j}w_{i}}\quad\text{and} \quad z_{k}'=\frac{z_{k}w_{j}-z_{j}w_{k}}{z_{i}w_{j}-z_{j}w_{i}} \quad\text{for } k\geqslant 3;
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
w_{1}'=\frac{w_{i}}{z_{j}w_{i}-z_{i}w_{j}}, \qquad w_{2}'= -\frac{z_{i}}{z_{j}w_{i}-z_{i}w_{j}}\quad\text{and} \quad w_{k}'= \frac{z_{k}w_{i}-z_{i}w_{k}}{z_{j}w_{i}-z_{i}w_{j}} \quad\text{for } k\geqslant 3.
\end{equation*}
\notag
$$
For the coordinates of the space of parameters $F_n$ we obtain in this chart the following result. Lemma 7. The coordinates $(c_{kl}: c_{kl}')$ and $(d_{kl}:d_{kl}')$ of the space of parameters $F_n$ in the charts $M_{12}$ and $M_{ij}$, $3\leqslant i<j\leqslant n$, are related by The proof of this lemma is analogous to the proof of Lemma 5. Remark 8. In this way we obtain the family $\{f_{12, ij}\}$, $i=1$, $j\geqslant 3$ or $2\leqslant i<j\leqslant n$, of homeomorphisms of the space $F_{n}$ induced by the transitions from $M_{12}$ to the charts $M_{ij}$. Any homeomorphism $f_{kl, pq}$ of $F_{n}$ induced by the transition map between the charts $M_{kl}$ and $M_{pq}$ can be represented in the form $f_{kl, pq} = f^{-1}_{12, kl} \circ f_{12, pq}$. § 3. Constructing the space $\widehat{\mathcal{F}}_{n}$ by wonderful compactificationWe want to find a compactification $\widehat{\mathcal{F}}_n$ of $F_{n}$ such that the homeomorphisms $\{f_{12, ij}\}$ of $F_{n}$, induced by the transition maps between the chart $M_{12}$ and the charts $M_{ij}$, extend the homeomorphisms of $\widehat{\mathcal{F}}_n$. According to Remark 8, it is enough to consider just the family $\{f_{12, ij}\}$, since then any homeomorphism $\{f_{kl, pq}\}$ of $F_n$ induced by the transition map between the two charts $M_{kl}$ and $M_{pq}$ can canonically be extended to a homeomorphism of $\widehat{\mathcal{F}}_n$. 3.1. The compactification $\overline{F}_{n}$ of $F_n$ in $( {\mathbb{C}} P^{1})^{N}$We begin with $\overline{F}_{n}\!\subset\! (\mathbb{C} P^{1})^{N}$, $N=\binom{n-2}{2}$, given by
$$
\begin{equation*}
\overline{F}_n=\{ ((c_{ij}:c_{ij}')),\ 3\leqslant i<j\leqslant n,\ c_{ik}c_{il}'c_{kl}=c_{ik}'c_{il}c_{kl}',\ 3\leqslant i<k<l\leqslant n\},
\end{equation*}
\notag
$$
which is, by Proposition 3, a smooth algebraic variety obtained as the compactification of $F_{n}$ in $(\mathbb{C} P^{1})^{N}$. The compactification $\overline{F}_{n}$ is not the one we are looking for, as it is obvious that the homeomorphisms of $F_n$ defined in previous lemmas for $n>4$ cannot be extended continuously to $\overline{F}_n$. Note that for $n=4$ these homeomorphisms extend to homeomorphisms of $\overline{F}_{4} = \mathbb{C} P^{1}$. In greater detail, the boundary of $F_n$ in $\overline{F}_{n}$ is $\overline{F}_{n}\setminus F_{n}$, and it consists of the points $(c_{ij}:c_{ij}')\in \overline{F}_{n}$ such that $c_{ij}=0$ or $c_{ij}'=0$ or $c_{ij}=c_{ij}'$ for some $ 3\leqslant i<j\leqslant n$. The homeomorphisms given by the previous lemmas do not continuously extend to the points on the boundary of $F_n$ that satisfy $c_{jk} = c_{jk}'$ and $c_{jl} = c_{jl}'$ for some $3\leqslant j<k<l\leqslant n$. Moreover, even if they do continuously extend to some subset of $\overline{F}_{n}\setminus F_{n}$, these extensions are not necessarily homeomorphisms. For example, the subvariety in $\overline{F}_{n}\setminus F_{n}$ given by $c_{34}' = c_{35}'=0$ is mapped by the continuous extension of the homeomorphism $f_{12, 13}$, which is defined by Lemma 5 for $j=3$, to the subvariety $d_{24} =d_{24}'$, $d_{25}=d_{25}'$, $d_{45}=d_{45}'$. In particular, for $n=5$ this means that the subvariety $(1:0), (1:0), (c_{45}:c_{45}')$ is mapped to the point $((1:1), (1:1), (1:1))$; see also [2]. Thus, this extension cannot be a homeomorphism. We call the subvariety of $\overline{F}_n$ consisting of the points to which the maps $\{f_{12, ij}\}$ do not extend as homeomorphisms the singular subvariety for $\{f_{12, ij}\}$. To overcome these problems the idea is to blow up the smooth, compact variety $\overline{F}_{n}$ along the singular subvarieties for the $\{f_{12, ij}\}$. In order to do this we use the technique from algebraic geometry known as the wonderful compactification of an arrangement of subvarieties. 3.2. Basic facts on wonderful compactificationFix a smooth, compact complex variety $X$ and a subvariety of $X$ which is considered singular in a certain sense. A wonderful compactification of $X$ is a kind of compactification aimed to resolve the singularities of $X$ along this subvariety. More precisely, a smooth compactification $\widetilde{X}$ of a complex manifold $M$ is wonderful in the case when $D=\widetilde{X}\setminus M$ is a divisor with normal crossings in $\widetilde{X}$ whose irreducible components are smooth and any number of connected components of $D$ intersect transversally. There are several compactifications in the literature which provide examples of wonderful compactification. We mention first the compactification of symmetric spaces proposed by De Concini and Procesi [7], in which $X$ is a symmetric space $G/H$ of an adjoint semisimple Lie group and $\widetilde{X}$ is a smooth, compact variety with an $G$-action such that $\widetilde{X}$ has an open orbit isomorphic to $G/H$ with finitely many $G$-orbits, all orbit closures are smooth, and any number of orbit closures intersect transversally. Another example is the compactification of configuration spaces defined by Fulton and MacPherson [13], in which $X$ is an open subset of a Cartesian product $M^n$ of a given nonsingular variety $M$ (that is, $X$ is defined as the complement to all diagonals), while $\widetilde{X}$ is defined by a sequence of blow-ups of $M^n$ along all nonsingular subvarieties corresponding to diagonals. A further example of wonderful compactification is the compactification of arrangements of complements to linear subspaces given also by De Concini and Procesi [8], in which $X$ is a finite-dimensional vector space and $\widetilde{X}$ is obtained by replacing any prescribed family of its subspaces by a divisor with normal crossings. In the recent paper [27] by Li a wonderful compactification is defined for a prescribed arrangement of subvarieties: in this case $X$ is a nonsingular variety and $\widetilde{X}$ is obtained by replacing any arrangement of given subvarieties by a divisor with normal crossings. It has been proved that any of these compactifications can be constructed by a sequence of blow-ups along appropriate subvarieties and some transformations of the results of these blow-ups. We follow here Li [27] to recall the basic facts on wonderful compactification for the setting we are going to use. Let $Y$ be a nonsingular variety. We say that the subvarieties $S_1,\dots,S_k$ of $Y$ intersect transversally if either $k=1$, or, for $k>1$, for each point $p\in \bigcap_{i=1}^{k}S_i$ we have In this case
$$
\begin{equation*}
\operatorname{codim}\biggl(\bigcap_{i=1}^{k} T(S_i), T(Y)\biggr) =\sum_{i=1}^{k}\operatorname{codim}(S_i, Y).
\end{equation*}
\notag
$$
Definition 9. A simple arrangement of subvarieties of $Y$ is a finite set $\mathcal{S}=\{S_{i}\}$ of nonsingular proper closed subvarieties $S_i$ properly contained in $Y$ and satisfying the following conditions. (1) Each intersection $S_{i}\cap S_{j}$ is either equal to some $S_k$ or empty. (2) Each nonempty intersection of two varieties $S_i$ and $S_j$ is clean, that is, it is nonsingular and the following relation holds for tangent bundles: Definition 10. Let $\mathcal{S}$ be an arrangement of subvarieties of $Y$. A subset $\mathcal{G} \subseteq \mathcal{S}$ is called a building set of $\mathcal{S}$ if, for all $S\in \mathcal{S}$, minimal elements of $\{G\in \mathcal{G}\colon S\subseteq G\}$ intersect transversally and their intersection is $S$. Such minimal elements are called $\mathcal{G}$-quotients of $S$. A finite set $\mathcal{G}$ of nonsingular subvarieties of $Y$ is called a building set if the set of all possible intersections of collections of subvarieties from $\mathcal{G}$ form an arrangement $\mathcal{S}$ and if $\mathcal{G}$ is a building set of $\mathcal{S}$. Then $\mathcal{S}$ is called the arrangement induced by $\mathcal{G}$. Definition 11. A subset $\mathcal{T}$ of $ \mathcal{G}$ is called a $\mathcal{G}$-nest if there exists a flag $S_1\subseteq S_2\subseteq \dots\subseteq S_\ell$ of elements of the configuration $\mathcal{S}$ such that one of the following equivalent conditions is satisfied. (1) The following equality holds:
$$
\begin{equation*}
\mathcal{T}=\bigcup_{i=1}^\ell \{ A\colon \text{$A$ ranges over the $\mathcal{G}$-quotients of $S_i$} \}.
\end{equation*}
\notag
$$
One says that $\mathcal{T}$ is induced by the flag $S_1\subseteq S_2\subseteq \dots\subseteq S_\ell$. (2) Let $A_1,\dots,A_k$ be the minimal elements of $\mathcal{T}$. Then each of them is a $\mathcal{G}$-quotient of some element of $\mathcal{S}$. For each $i$, $1 \leqslant i \leqslant k$, the set $\{ A\in \mathcal{T}\colon A\supsetneq A_i \}$ is also a $\mathcal{G}$-nest, defined inductively. Note that condition (2) implies that each set $\mathcal{T}\setminus A_i$ is also a $\mathcal{G}$-nest, in the sense of a repeated application of the above inductive definition. Definition 12. Let $Y$ be a smooth variety, $\mathcal{G}$ be a nonempty building set, and let $Y^{\circ}=Y\setminus \bigcup_{G\in \mathcal{G}}G$. Then the closure of the image of the closed diagonal embedding if called the wonderful compactification with building set $\mathcal{G}$ and denoted by $Y_{\mathcal{G}}$. We formulate two crucial theorems from [27], the first of which states that the wonderful compactification $Y_{\mathcal{G}}$ is a nonsingular variety and the second describes $Y_{\mathcal{G}}$ as a series of blow-ups determined by subvarieties of the building set $\mathcal{G}$. Theorem 13. Let $Y$ be a nonsingular variety, and let $\mathcal{G}$ be a nonempty building set of subvarieties of $Y$. Then the wonderful compactification $Y_{\mathcal{G}}$ is a nonsingular variety. Moreover, for any $G\in \mathcal{G}$ there is a nonsingular divisor $D_{G}\subset Y_{\mathcal{G}}$ such that Now recall a concept related to blow-ups and required for the statement of the theorem going next. Definition 14. Let $Z$ be a nonsingular subvariety of a nonsingular variety $Y$, and let $\operatorname{Bl}_{Z}Y$ be the blow-up of $Y$ along $Z$. Let $\pi\colon \operatorname{Bl}_{Z}Y \to Y$ denote the canonical projection. Then for each irreducible subvariety $V\subset Y$ its dominant transform $\widetilde{V}$ is described as follows. The reason for introducing the notion of dominant transform is to correct the fact that the strict transform of a subvariety contained in the centre of the blow-up is empty. In our applications we always have $V\not\subset Z$, so $\widetilde{V}$ is always a strict transform. Theorem 15. Let $Y$ be a nonsingular variety, and let $\mathcal{G}$ be a nonempty building set of subvarieties of $Y$. Let the building set $\mathcal{G} = \{G_1, \dots, G_{Q}\}$ be ordered so that for any $1\leqslant i\leqslant Q$ the subvarieties $\{G_1, \dots, G_i\}$ form a building set. Then iterated blow-ups produce the smooth variety where $\widetilde{G}_{i}$ is a nonsingular variety obtained as the iterated dominant transform of $G_{i}$ in $\operatorname{Bl}_{\widetilde{G}_{i-1}}\dotsb \operatorname{Bl}_{\widetilde {G}_{2}}\operatorname{Bl}_{G_1}Y$, $2\leqslant i\leqslant Q$. The smooth manifold $X_{\mathcal{G}}$ coincides with the wonderful compactification $Y_{\mathcal{G}}$. We point out the following observation, which will be useful in what follows. Lemma 16. Let the finite set of nonsingular subvarieties $\mathcal{G}$ of a nonsingular variety $Y$ satisfy the following: Then $\mathcal{G}$ is a building set. Proof. In this case the set $\mathcal{G}$ is a simple arrangement whose building set is $\mathcal{G}$, since any $S\in \mathcal{G}$ is a minimal element of the set $\{G\in \mathcal{G}\mid S\subseteq G\}$. The lemma is proved. We comment briefly on the proof of Theorem 15; see [27]. The proof relies essentially on the following result proved in [27]. Let $Y$ be a nonsingular and $F$ be a minimal element of the building set $\mathcal{G} = \{G_1, \dots, G_N\}$ induced by an arrangement $\mathcal{S}$, and let $E$ be the exceptional divisor in the blow-up $\operatorname{Bl}_{F}Y$. Then the collection of subvarieties $\widetilde{\mathcal{S}}$ in $\operatorname{Bl}_{F}Y$ defined by
$$
\begin{equation*}
\widetilde{\mathcal{S}}=\{\widetilde{S}\}_{S\in \mathcal{S}} \cup \{\widetilde{S}\cap E\}_{\varnothing \neq S\cap F\varsubsetneq S}
\end{equation*}
\notag
$$
is an arrangement of subvarieties in $\operatorname{Bl}_{F}Y$, and $\widetilde{\mathcal{G}}= \{\widetilde{G}\}_{G\in \mathcal{G}}$ is a building set in $\widetilde{\mathcal{S}}$. The scheme of the proof of Theorem 15 is as follows. The idea is to order (partially) the set $\mathcal{G}$ in accordance with the inclusion relation and to start with the blow-up of $Y$ along a minimal element $F\subset Y$. Then using the result just mentioned we repeat the procedure iteratively with the dominant transform $\widetilde{\mathcal{G}}$ of the set $\mathcal{G}$ in $\operatorname{Bl}_{F}Y$. 3.3. The space $\mathcal{F}_{n}$ as the wonderful compactification based on $\overline{F}_{n}$Let $\overline{F}_{n}\subset (\mathbb{C} P^{1})^{N}$ be as defined in Proposition 3, and let
$$
\begin{equation}
\widehat{F}_{I}=\overline{F}_n\cap \{(c_{ik}:c_{ik}')=(c_{il}:c_{il}')= (c_{kl}:c_{kl}')=(1:1)\}
\end{equation}
\tag{7}
$$
for $I=\{i, k, l\}\in \{I\subset \{1, \dots, n\},\ |I|=3\}$ and $n\geqslant 5$. We take $Y=\overline{F}_{n}$ and let the building set $\mathcal{G}_{n}$ be as follows:
We denote an element $G\in \mathcal{G}_n$ of the form $G = \widehat{F}_{I_1}\cap \dotsb \cap \widehat{F}_{I_k}$ by $\widehat{F}_{I_1, \dots, I_k}$. Proof. Any intersection of elements of $\mathcal{G}_n$ belongs to $\mathcal{G}_n$. The set $\mathcal{G}_n$ is a simple arrangement since the intersection of two elements of $\mathcal{G}_n$ is obviously either empty or belongs to $\mathcal{G}_n$, and every two elements intersect cleanly. We see this from the description of the subvarieties ${F}_{I_1, \dots, I_k}\subset (\mathbb{C} P^{1})^{N}$ given by (2) and from the observation that for $S_1=\widehat{F}_{I_{i_1}, \dots, I_{i_k}}$ and $S_2=\widehat{F}_{I_{j_1},\dots, I_{j_l}}\in \mathcal{S}$ we have
Then Lemma 16 implies that $\mathcal{G}_n$ is a building set. The lemma is proved. Next we prove that the building set $\mathcal{G}_n$ satisfies the condition in Theorem 15. Lemma 18. The above building set $\mathcal{G}_n$ can be ordered as $\mathcal{G}_n=\{G_1, \dots, G_{Q}\}$ so that for any $i$, $1\leqslant i\leqslant Q$, the subvarieties $\{G_1, \dots, G_i\}$ form a building set. Proof. To an element $G=\widehat{F}_{I_1, \dots, I_k}\in \mathcal{G}_n$ we assign the integer $\mathfrak{o}(G)$ equal to the number of coordinates of the points in $\overline{F}_n\subset (\mathbb{C} P^1)^{N}$ that are determined by the set $I_1\cup \dotsb \cup I_k$. In other words $\mathfrak{o}(G)$ is the number of coordinates of the form $(1:1)$ that are the same for all points in $G$.
Using equations (2) and formula (7) we see, for example, that if $k=2$, $I_1=\{3,4,5\}$ and $I_2=\{3,4,6\}$, then $\mathfrak{o}(G) =6$, while for $I_1=\{3,4,5\}$ and $I_2=\{3,6,7\}$ we have $\mathfrak{o}(G) = 10$, and in the case when $I_1\cap I_2=\varnothing$ we always have $\mathfrak{o}(G) = 6$. We define an equivalence relation on $\mathcal{G}$ as follows: $G_1$ and $G_2$ are equivalent if and only if $\mathfrak{o}(G_1)=\mathfrak{o}(G_2)$. Denote the corresponding equivalence classes by $\widetilde{\mathcal{G}}_{1}, \dots, \widetilde{\mathcal{G}}_{m}$. We put these equivalence classes in the reverse order to the corresponding numbers $\mathfrak{o}(\widetilde{\mathcal{G}}_{i})$, that is, $\widetilde{\mathcal{G}}_{i}<\widetilde{\mathcal{G}}_{j}$ if and only if $\mathfrak{0}(\widetilde{\mathcal{G}}_{i})>\mathfrak{o}(\widetilde{\mathcal{G}}_{j})$. This implies that $\widetilde{\mathcal{G}}_{1}$ contains only the point $S=(1:1)^{N}$, while $\widetilde{\mathcal{G}}_{m}$ consists of all elements $\widehat{F}_{I}$. Next we order the elements of $\mathcal{G}_n$ as follows: we set $G_1=(1:1)^{N}$, and then we put the elements of $\widetilde{\mathcal{G}}_{2}$ in an arbitrary order, after which we put the elements of $\widetilde{\mathcal{G}}_{3}$ in an arbitrary order and so on. At the end we put the elements of $\widetilde{\mathcal{G}}_{m}$, that is, $\widehat{F}_{I}$ in an arbitrary order. We denote this ordering by $\mathcal{G}_n=\{G_1, \dots,G_{Q}\}$. Since elements of $\mathcal{G}$ intersect cleanly, it follows that $\{G_1, \dots, G_{i}\}$ is a building set for any $1\leqslant i\leqslant Q$. The lemma is proved. We denote by $\widehat{\mathcal{F}}_n$ the smooth, compact manifold $Y_{\mathcal{G}}$ that is the wonderful compactification with building set $\mathcal{G} = \mathcal{G}_{n}$ and $Y= \overline{F}_{n}$. Remark 19. Note that for $n=5$ the building set $\mathcal{G}_{5}$ consists of one point $P=({(1:1)}, (1:1), (1:1))$, and therefore $\widehat{\mathcal{F}}_{5}=\operatorname{Bl}_{P}\overline{F}_{5}$; cf. [3]. For $n=6$ the description of $\widehat{\mathcal{F}}_{6}$ is more complicated and allows us to demonstrate the general approach; see Example 4.18 in [5] and § 4.2 below. We are in the situation that for each $n$ we are given a smooth manifold ${F_{n}\subset (\mathbb{C} P^{1})^{N}}$, which is an open subset of $\overline{F}_{n}\subset (\mathbb{C} P^{1})^{N}$, and a group of automorphisms $\mathcal{A}= \{f_{ij, kl}\}$ for the space of parameters $F_{n}$ of the main stratum $W_n$ that are induced by the transition maps between charts $M_{ij}$ and $M_{kl}$ for $G_{n,2}$. The manifold $\widehat{\mathcal{F}}_n$ that is a compactification of $F_{n}$ has the required property described in the introduction. Theorem 20. The homeomorphisms of $F_{n}$ from the set $\mathcal{A}$ extend to homeomorphisms of $\widehat{\mathcal{F}}_n$. Proof. Since $f_{ij,kl}=f_{12, kl} \circ f^{-1}_{12,ij}$, it is enough to prove the statement for the homeomorphisms $f_{12, ij}$. We present the proof for the homeomorphisms $f_{12, 1j}$ described in Lemma 5; the other cases are considered in the analogous way.
First we discuss the homeomorphic extensions of the $f_{12, ij}$ to the boundary $\overline{F}_{n}\setminus F_{n}$. This boundary is given by the conditions $(c_{kl}:c_{kl}') = (1:0)$ or $(0:1)$ or $(1:1)$ for some $3\leqslant k<l\leqslant n$. We analyze each of these cases using the equations $c_{kl}c_{kp}'c_{lp}=c_{kl}'c_{kp}c_{lp}'$, which define $\overline{F}_{n}$, and the expressions for $f_{12,1j}$ given in Lemma 5. $\bullet$ Assume that $k\neq j$. For $c_{kl}'=0$ we have $c_{kp}' = 0$ or $c_{lp}=0$, which implies that $d_{kl}' = 0$ and either $d_{kp}' = 0$ or $d_{lp}=0$. For $c_{kl} = 0$ we have $c_{kp}=0$ or $c_{lp}'=0$, which gives $d_{kl} =0$ and either $d_{kp}=0$ or $d_{lp}' =0$. For $c_{kl}=c_{kl}'$ we have $(c_{lp}:c_{lp}')=(c_{kp}:c_{kp}')$, which implies that $d_{kl}=d_{kl}'$ and $(d_{lp}:d_{lp}')=(d_{kp}:d_{kp}')$. $\bullet$ Assume that $k=j$. If $c_{jl} = c_{jp}=0$, then we have $d_{2l}=d_{2p}=0$, while for $c_{jl}=c_{lp}'=0$ we have $d_{2l}=0$ and $d_{lp}'=0$. If $c_{jl}'= c_{jp}'=0$, then $(c_{lp}:c_{lp}')$ can be an arbitrary point in $\mathbb{C} P^1$, because we have $(d_{2l}:d_{2l}')= (d_{2p}:d_{2p}') = (d_{lp}:d_{lp}') = (1:1)$. Note that in this case $f_{12, 1j}$ extends to such points on the boundary but it is not a homeomorphism. If $c_{jl}' = c_{lp}=0$, then we obtain $d_{2l}=d_{2l}'$ and $d_{lp} = 0$. For $c_{jl}=c_{jl}'$ we have $(c_{lp}:c_{lp}')=(c_{jp}:c_{jp}')$ which implies that $d_{2l}'=0$ and $(d_{lp}:d_{lp}')=(d_{jp}:d_{jp}')$. Altogether, we conclude that $f_{12, 1j}$ cannot be extended continuously to the subvarieties $\widehat{F}_{I} \subset \overline{F}_{n}\setminus F_{n}$, $I=\{j,l,p\}$, given by $(c_{jl}:c_{jl}') = (c_{jp}:c_{jp}') = (c_{lp}:c_{lp}')=(1:1)$ and that it can continuously, but not homeomorphically, be extended to the subvarieties $\breve{F}_{I}\subset \overline{F}_{n}\setminus F_{n}$, $I=\{j,l,p\}$, that is, to the family of subvarieties consisting of all possible nonempty intersections of the subvarieties defined by the equations $(c_{jl};c_{jl}')=(c_{jp}:c_{jp}')=(1:0)$. We denote by $\mathcal{G}(j)$ the family of subvarieties consisting of all possible nonempty intersections of subvarieties $\widehat{F}_{I}$ and by $\mathcal{H}(j)$ the family of all possible nonempty intersections of subvarieties $\breve{F}_{I}$. We see from the previous discussion that $f_{12, 1j}$ extends continuously and homeomorphically to the complement in $\overline{F}_{n}$ to the union of the subvarieties in $\mathcal{G}(j)$ and $\mathcal{H}(j)$, that is, to $\overline{F}_{n}\setminus (\mathcal{G}(j)\cup \mathcal{H}(j))$. Moreover, note that the preimage of a subvariety $\widehat{F}_{I}$ under this extension of $f_{12,1j}$ is equal to the subvariety $\breve{F}_{I}$. Now we extend $f_{12, 1j}$ to a homeomorphism $\widetilde{f}_{12, 1j}\colon \widehat{\mathcal{F}}_n\to \widehat{\mathcal{F}}_n$) as follows. $\bullet$ On the complement to the union of the subvarieties in $\mathcal{G}(j)$ and $\mathcal{H}(j)$ the map $\widetilde{f}_{12, 1j}$ is given by the natural homeomorphic extension of $f_{12, 1j}$. $\bullet$ Let $\widetilde{S}\in \mathcal{H}(j)$; then $\widetilde{S}=\widetilde{F}_{I_1}\cap\dotsb \cap \widetilde{F}_{I_k}$ for some $I_1, \dots, I_{k}\subset \{I\subset \{1,\dots, n\}, |I|=3,\ j\in I\}$. Let $\widehat{S}\in \mathcal{G}(j)$ have the form $\widehat{S}= \widehat{F}_{I_1}\cap\dotsb\cap \widehat{F}_{I_k}$. Then we define $\widetilde{f}_{12, 1j}$ as the map taking $\widetilde{S}$ homeomorphically to the exceptional divisor $E(\widehat{S})$ for $\widehat{S}$ in $\widehat{\mathcal{F}}_n$. This can be done in a natural way because of our previous observation on the behaviour of the extension of $f_{12, 1j}$ on the subvarieties $\breve{F}_{(j,l,p)}$. $\bullet$ Let $E(\widehat{S})\subset \widehat{\mathcal{F}}_n$ be the exceptional divisor for $\widehat{S}\in \mathcal{G}(j)$, where $\widehat{S}=\widehat{F}_{I_1}\cap\dotsb\cap \widehat{F}_{I_k}$, $I_1, \dots, I_{k}\in \{I\subset \{1,\dots, n\},\ |I|=3,\ j\in I\}$. We define $\widetilde{f}_{12, 1j}$ as the homeomorphism of $E(\widehat{S})$ onto $\widetilde{S}=\widetilde{F}_{I_1}\cap\dotsb \cap \widetilde{F}_{I_k}$ that is the inverse of the above extension $\widetilde{f}_{12, 1j}\colon \widetilde{S}\to E(\widehat{S})$. The theorem is proved. 3.4. The space $\widehat{\mathcal{F}}_n$ as the universal space of parameters $\mathcal{F}_n$The universal space of parameters $\mathcal{F}$ for $(2n,k)$-manifolds with effective $T^k$-action and its axioms were introduced in [3]. For a detailed description of the axioms and comments to them, see [3]. Here is a brief statements of the axioms defining $\mathcal{F}$. 1. The space $\mathcal{F}$ is a smooth manifold and a compactification of the space of parameters $F$ of the main stratum $W$. 2. $\mathcal{F}$ coincides with the union of the virtual spaces of parameters $\widetilde{F}_\sigma$ of all strata $W_\sigma$. 3. There exist continuous projections $p_{\sigma}\colon \widetilde{F}_{\sigma}\to F_{\sigma}$. 4. There exists a continuous projection $G \colon \bigcup_{\sigma} \mathring{P}_{\sigma} \times \widetilde{F}_{\sigma} \to M^{2n}/T^k$, where the topology in the disjoint union $\bigcup_{\sigma} \mathring{P}_{\sigma} \times \widetilde{F}_{\sigma}$ is determined by the embedding
$$
\begin{equation*}
\bigcup_{\sigma} \mathring{P}_{\sigma} \times \widetilde{F}_{\sigma}\to \operatorname{CQ}(M^{2n}, P^k)\times \mathcal{F},
\end{equation*}
\notag
$$
and $\operatorname{CQ}(M^{2n},P^k)$ is the complex of admissible polytopes with the relevant topology introduced in [3]. Theorem 21. The space $\widehat{\mathcal{F}}_n$ is the universal space of parameters of the $T^n$-action on $G_{n,2}$, that is, it satisfies conditions 1–4. An explicit description of the virtual spaces of parameters for all $n$ in accordance with our description of $\widehat{\mathcal{F}}_n$ was presented in [5]. The most difficult step in the proof of the above theorem is the construction of a continuous projection $G_n\colon \bigcup_{\sigma} \mathring{P}_{\sigma} \times \widetilde{F}_{\sigma} \to G_{n,2}/T^n$ for $n>4$. In [2] we presented a detailed proof of the existence if such a projection for $n=5$. It was pointed out in [5] that in the general case the construction of a continuous projection of $G_n$ is analogous to the case $n=5$. Remark 22. In the introduction we mentioned the paper [25] by Klemyatin, presenting a construction of a space satisfying conditions 1–3. However, it does not contain a proof that condition 4 holds. Thus, the main theorem of [25] was not proved there. In [5], using some specifics of $G_{n,2}$, we showed that in our case condition 4 can be replaced by the following one. $4^*$. There exists a continuous projection $H_n\colon \Delta_{n,2}\times \mathcal{F}_n \to G_{n,2}/T^n$ such that the composition $\widehat{\mu}\circ H_n=\mathrm{pr}_{1}$ is the projection onto the first factor. The construction of $\widehat{\mathcal{F}}_n$ by wonderful compactification enables us to produce the required map $H_n$ explicitly. Here we have essentially used the fact that the admissible polytopes $P_\sigma$ define a disjoint partition of the hypersimplex $\Delta_{n,2}$ into chambers, and we proved in [5] that for each chamber $C_{\omega} \subset \mathring{\Delta}_{n,2}$ the space $\widehat{\mathcal{F}}_n$ is subdivided into a disjoint union of the virtual spaces of parameters of admissible polytopes forming $C_{\omega}$. § 4. $\mathcal{F}_{n}$ and the moduli space $\overline{\mathcal{M}}(0,n)$4.1. The main resultWe denote by $\mathcal{M}(0,n)$, as usual, the moduli space of curves of genus $0$ with $n$ marked distinct points. The space $\mathcal{M}(0,n)$ parametrizes $n$-tuples of distinct points on the Riemann sphere $\mathbb{C} P^1$ up to biholomorphisms of the sphere, that is,
$$
\begin{equation*}
\mathcal{M}(0,n)=((\mathbb{C} P^{1})^n\setminus \Delta)/\mathrm{PGL}_{2}(\mathbb{C}),
\end{equation*}
\notag
$$
where $\Delta = \bigcup_{i\neq j}\{(x_1, \dots, x_n)\in (\mathbb{C} P^{1})^{n}\mid x_i=x_j\}$. Any triple of points in $\mathbb{C} P^{1}$ can be taken to $A=\{(0:1),(1:1),(1:0)\}$ by a unique projective transformation. It follows that $\mathcal{M}(0,n)$ can be identified with the space
$$
\begin{equation}
(\mathbb{C} P^{1}_{A})^{n-3}\setminus \Delta=\{ (x_1,\dots,x_{n-3})\in \mathbb{C}^{n-3}\mid x_i\neq 0,1;\, x_i\neq x_j \text{ if } i\neq j \},
\end{equation}
\tag{8}
$$
where $\mathbb{C} P^{1}_{A}=\mathbb{C} P^{1}\setminus A=\{x\in\mathbb{C}\mid x\neq 0,1 \}$. For example, $\mathcal{M}(0,3)$ is a point, while $\mathcal{M}(0,4)=\mathbb{C} P^{1}_A$. Note that the moduli space $\mathcal{M}(0,n)$ coincides with $F_n$, our space of parameters of the main stratum $W_n$; cf. (5). The moduli space $\overline{\mathcal{M}}(0,n)$ is the space of biholomorphism classes of stable curves of genus $0$ with $n$ marked distinct points. It is a compact, smooth complex manifold of dimension $n-3$, in which $\mathcal{M}_{0, n}$ is a Zariski-open subset. The moduli space $\overline{\mathcal{M}}(0,n)$ is a compactification of $\mathcal{M}(0,n)$, known as the Grothendieck-Knudsen compactification. Recall that for $g=0$ the Deligne-Mumford compactification $\overline{\mathcal{M}}(g,n)$ of the moduli space $\mathcal{M}(g,n)$ coincides with the Grothendieck-Knudsen compactification. In [22] Keel gave an alternative (to Grothendieck-Knudsen’s) construction of the smooth variety $\overline{\mathcal{M}}(0,n)$. Li noted in [27] that the application of Theorem 15 to this construction immediately shows that $\overline{\mathcal{M}}(0,n)$ is the wonderful compactification $Y_{\mathcal{G}}$ where $Y =(\mathbb{C} P^1)^{n-3}$ and the building set $\mathcal{G}$ consists of all diagonals and augmented diagonals. More precisely, $\mathcal{G}$ consists of
$$
\begin{equation*}
\Delta_{I}=\{(c_4,\dots, c_n)\in (\mathbb{C} P^{1})^{n-3} \mid c_i=c_j \text{ for all }i,j\in I\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\Delta_{I,a}=\{(c_4,\dots, c_n)\in (\mathbb{C} P^{1})^{n-3} \mid c_i=a\text{ for all } i\in I\},
\end{equation*}
\notag
$$
where $I\subseteq \{4, \dots, n\}$, $|I|\geqslant 2$ and $a\in \{0, 1, \infty\}$. The corresponding arrangement is the set of all intersections of elements of $\mathcal{G}$. Using these results we prove that our compactification $\mathcal{F}_{n}$ of $F_{n}$, which is obtained as the wonderful compactification with building set $\mathcal{G}_n$ consisting of the nonsingular subvarieties of $\overline{F}_{n}$ (see § 3.3), coincides with the Grothendieck-Knudsen compactification of $\mathcal{M}(0,n) = F_{n}$. Theorem 23. The manifold $\mathcal{F}_{n}$ is diffeomorphic to $\overline{\mathcal{M}}(0,n)$ for $n\geqslant 4$. Remark 24. Recall that for $n=4$ it is already known that $\mathcal{F}_{4}= \mathbb{C} P^1 = \overline{\mathcal{M}}(0,4)$. Also, for $n=5$ it was already noted in [2], Remark 7.13, that $\mathcal{F}_{5}$ is diffeomorphic to $\overline{\mathcal{M}}(0,5)$. Remark 25. The decomposition of $\overline{\mathcal{M}}(0,n)$, $n\geqslant 4$, into a disjoint union of subspaces given by a fixed chamber has found important applications to the problem of the algebraic topology of $\overline{\mathcal{M}}(0,n)$: see [21] and [22]. We discuss this question in a forthcoming paper. 4.2. The spaces $\mathcal{F}_{6}$ and $\overline{\mathcal{M}}(0,6)$First, for the sake of clarity we elaborate on Theorem 20 and prove Theorem 23 in the case of the Grassmannian $G_{6,2}$. Note that $G_{6,2}$ is of particular importance in algebraic geometry being one of the six Severi varieties; see [32] and [26]. For $n=6$ we begin with the manifold
$$
\begin{equation*}
\begin{aligned} \, \overline{F}_{6} &=\{((c_{34}:c_{34}'), (c_{35}:c_{35}'), (c_{36}:c_{36}'), (c_{45}:c_{45}'), (c_{46}:c_{46}'), (c_{56}:c_{56}'))\in (\mathbb{C} P^{1})^{6}, \\ &\qquad c_{34}'c_{35}c_{45}'=c_{34}c_{35}'c_{45}, \ c_{34}'c_{36}c_{46}'=c_{34}c_{36}'c_{46}, \\ &\qquad c_{35}'c_{36}c_{56}'=c_{35}c_{36}'c_{56}, \ c_{45}'c_{46}c_{56}'=c_{45}c_{46}'c_{56}\}. \end{aligned}
\end{equation*}
\notag
$$
The building set is given by the following subvarieties of $\overline{F}_{6}$:
$$
\begin{equation*}
\begin{aligned} \, \widehat{F}_{345} &=\{((1:1), (1:1), (c_{36}:c_{36}'), (1:1), (c_{46}: c_{46}'), (c_{56}:c_{56}')), \\ &\qquad c_{36}c_{46}'=c_{36}'c_{46},\ c_{36}c_{56}'=c_{36}'c_{56},\ c_{46}c_{56}'=c_{46}'c_{56}\}, \\ \widehat{F}_{346} &=\{((1:1), (c_{35}:c_{35}'), (1:1), (c_{45}:c_{45}'), (1: 1), (c_{56}:c_{56}')), \\ &\qquad c_{35}c_{45}'=c_{35}'c_{45},\ c_{35}c_{56}'=c_{35}'c_{56}, \ c_{45}c_{56}'=c_{45}'c_{56}\}, \\ \widehat{F}_{356} &=\{((c_{34}:c_{34}'), (1:1), (1:1), (c_{45}:c_{45}'), (c_{46}: c_{46}'), (1:1)), \\ &\qquad c_{34}c_{45}'=c_{34}'c_{45},\ c_{34}c_{46}'=c_{34}'c_{46},\ c_{45}c_{46}'=c_{45}'c_{46}\}, \\ \widehat{F}_{456} &=\{((c_{34}:c_{34}'), (c_{35}:c_{35}'), (c_{36}:c_{36}'), (1:1), (1:1), (1:1))\}, \\ &\qquad c_{34}c_{35}'=c_{34}'c_{35},\ c_{34}c_{36}'=c_{34}'c_{36},\ c_{35}'c_{36}=c_{35}c_{36}'\}, \end{aligned}
\end{equation*}
\notag
$$
together with the point $S= (1:1)^{6}$. Every two of the above subvarieties intersect at this point. The smooth, compact manifold $\mathcal{F}_{6}$ is a wonderful compactification with building set $\mathcal{G}_{6}=\{ S, \widehat{F}_{345}, \widehat{F}_{346}, \widehat{F}_{356},\widehat{F}_{456}\}$, that is,
$$
\begin{equation}
\mathcal{F}_{6}= \operatorname{Bl}_{\widetilde{F}_{456}}\operatorname{Bl}_{\widetilde{F}_{356}} \operatorname{Bl}_{\widetilde{F}_{346}}\operatorname{Bl}_{\widetilde{F}_{345}} \operatorname{Bl}_{S}\overline{F}_{6}.
\end{equation}
\tag{9}
$$
Note that the dominant transform $\widetilde{F}_{ijk}$ in $\operatorname{Bl}_{S}\overline{F}_{6}$ of any submanifold $\widehat{F}_{ijk}$, $3\leqslant i<j<k\leqslant 6$, intersects the exceptional divisor $\mathbb{C} P^2$ in one point. In addition, the four points obtained in this way are different. Hence the wonderful compactification given by (9) does not depend on the order of blow-ups along the subvarieties $\widetilde{F}_{ijk}$. We show that the manifold $\mathcal{F}_{6}$ coincides with the moduli space $\overline{\mathcal{M}}(0,6)$. As we have already mentioned, the construction in [22] describes $\overline{\mathcal{M}}(0,6)$ as a sequence of blow-ups. This result, due to Keel, was then used by Li [27] to note that $\overline{\mathcal{M}}(0,6)$ is the wonderful compactification $Y_{\mathcal{G}}$ where $Y =(\mathbb{C} P^1)^{3}$ and the building set $\mathcal{G}$ consists of all diagonals
$$
\begin{equation*}
\Delta_{I}=\{ (p_1,p_2, p_{3})\in (\mathbb{C} P^{1})^{3} \mid p_i=p_j\text{ for } i,j\in I\}
\end{equation*}
\notag
$$
and augmented diagonals
$$
\begin{equation*}
\Delta_{I, a}=\{ (p_1, p_2, p_{3})\in (\mathbb{C} P^{1})^{3} \mid p_i=a\text{ for all }i\in I\},
\end{equation*}
\notag
$$
where $I\subset \{1, 2, 3\}$, $|I|\geqslant 2$ and $a\in A=\{(0:1),(1:1),(1:0)\}$. Note that $\Delta_{I}$ is a complex two-dimensional submanifold of $(\mathbb{C} P^{1})^3$ for $|I|=2$, so the blow-up of $(\mathbb{C} P^{1})^3$ along the diagonals $\Delta_{I}$, $|I|=2$, leaves $(\mathbb{C} P^{1})^3$ unchanged. Thus, in the wonderful compactification that describes $\overline{\mathcal{M}}(0,6)$ it is sufficient to take the full diagonal $\Delta_{\{1,2,3\}}$ and augmented diagonals $\Delta_{I,a}$ as a building set. Theorem 26. The manifold $\mathcal{F}_{6}$ is diffeomorphic to the space $\overline{\mathcal{M}}(0,6)$. Proof. Consider the smooth map $f\colon (\mathbb{C} P^{1})^{3} \to (\mathbb{C} P^{1})^{6}$ given by
$$
\begin{equation*}
\begin{aligned} \, &f((c_{34}:c_{34}'), (c_{35}: c_{35}'), (c_{36}:c_{36}'))=((c_{34}:c_{34}'), (c_{35}: c_{35}'), (c_{36}:c_{36}'), \\ &\qquad\qquad\qquad\qquad\qquad (c_{34}'c_{35} : c_{34}c_{35}'), (c_{34}'c_{36}:c_{34}c_{36}'), (c_{35}'c_{36}:c_{35}c_{36}')). \end{aligned}
\end{equation*}
\notag
$$
Set $0=(0:1)$ and $\infty=(1:0)$. This map is not defined at points in the following submanifolds of $(\mathbb{C} P^{1})^{3}$:
$$
\begin{equation*}
\begin{gathered} \, \Delta_{\{1,2\}, \infty}=\{((1:0), (1:0), (c_{36}:c_{36}'))\}, \quad \Delta_{\{1,2\}, 0}=\{((0:1), (0:1), (c_{36}:c_{36}'))\}, \\ \Delta_{\{1,3\}, \infty}=\{((1:0), (c_{35}:c_{35}'), (1:0))\}, \quad \Delta_{\{1,3\}, 0}=\{((0:1), (c_{35}:c_{35}'), (0: 1))\}, \\ \Delta_{\{2,3\}, \infty}=\{((c_{34}:c_{34}'), (1:0), (1:0))\}, \quad \Delta_{\{2,3\}, 0}= \{((c_{34}:c_{34}'), (0:1), (0:1))\}. \end{gathered}
\end{equation*}
\notag
$$
Then for $P = ((1:0), (1:0), (1:0))$ and $Q= ((0:1), (0:1), (0:1))$ the set
$$
\begin{equation*}
\mathcal{G}'=\{\Delta_{\{i,j\}, a}, a=0, \infty\} \cup \{P, Q\}
\end{equation*}
\notag
$$
is a building set in $(\mathbb{C} P^{1})^{3}$, and we consider the wonderful compactification ${Z=(\mathbb{C} P^{1})^{3}_{\mathcal{G}'}}$. The map $f$ extends to a diffeomorphism $\overline{f}$ between $Z$ and $\overline{F}_6$. For example, at points of the exceptional divisor $\mathbb{C} P^1$ along the submanifold $\Delta_{\{1, 2\},0}$ the map $\overline{f}$ has the form
$$
\begin{equation*}
\begin{aligned} \, &\overline{f}( (1:0), (1:0), (c_{36}:c_{36}'), (x_1:x_2)) \\ &\qquad=((1:0), (1:0), (c_{36}:c_{36}'), (x_1:x_2), (0:1), (0:1)), \end{aligned}
\end{equation*}
\notag
$$
while at points $(x_1:x_2:x_3)$ in the exceptional divisor $\mathbb{C} P^2$ of the blow-up at the point $P$ the map $\overline{f}$ has the form Since in the neighbourhood $((1 : c_{34}'), (1 : c_{35}'), (1 : c_{36}'))$ of $P=((1:0), {(1:0)}, {(1:0)})$ we have the points $((1:0), (1:0), (1:0), (x_1:x_2), (x_1:x_3), (x_2:x_3))$ belong to $\overline{F}_6$.
In order to finish the proof it remains to note that the wonderful compactification of $\overline{F}_6$ with building set consisting of the $\overline{F}_{ijk}$ and $S$ corresponds to the wonderful compactification of $Z$ with building set where $1=(1:1)$ and $R=((1:1), (1:1), (1:1))$. Hence $\mathcal{F}_{6}$ and $\overline{\mathcal{M}}(0,6)$ are diffeomorphic. The theorem is proved.4.3. The proof of the main result Proof of Theorem 23. The proof proceeds in a way analogous to $n=6$. According to (3), (4) and (5) we can begin with the smooth map $f\colon (\mathbb{C} P^{1})^{n-3} \to (\mathbb{C} P^{1})^{N}$, $N= \binom{n-2}{2}$, given by
$$
\begin{equation*}
\begin{aligned} \, &f((c_{34}:c_{34}'),\dots, (c_{3n}:c_{3n}')) \\ &\qquad =((c_{34}:c_{34}'), \dots, (c_{3n}:c_{3n}'), (c_{34}'c_{35} : c_{34}c_{35}'),\dots, (c_{3n-1}'c_{3n}:c_{3n-1}c_{3n}')). \end{aligned}
\end{equation*}
\notag
$$
The map $f$ is not defined at points in the submanifolds $G_{pq}, G_{pg}'\subset (\mathbb{C} P^{1})^{n-3}$, $4\leqslant p<q\leqslant n$, defined by and
It obviously follows from (3) and (4) that the map $f$ defines a diffeomorphism $f\colon (\mathbb{C} P^{1}_{A})^{n-3}\setminus \Delta \to F_{n}$. It is easy to verify that the set $\mathcal{G}'$ of all possible intersections of subvarieties $G_{pq}$ and $G_{pq}'$ is a building set. Let $Z$ be a smooth manifold obtained as the wonderful compactification of $(\mathbb{C} P^{1})^{n-3}$ with building set $\mathcal{G}'$, that is, $Z= (\mathbb{C} P^{1})^{n-3}_{\mathcal{G}'}$. Then, as in the case $n=6$, we see that $f$ extends to a diffeomorphism $\overline{f}$ between $Z$ and $\overline{F}_{n}$. Now let
$$
\begin{equation*}
H_{pq}=\{((c_{3i}:c_{3j}))\in (\mathbb{C} P^{1})^{n-3} \mid (c_{3p}:c_{3p}')=(c_{3q}:c_{3q}')=(1:1)\},
\end{equation*}
\notag
$$
and let $\widetilde{H}_{pq}$ be a proper transform of $H_{pq}$ in $Z$. The set $\mathcal{G}''$ of all possible intersections of subvarieties $\widetilde{H}_{pq}$ is a building set. It was explained in [27], § 4.4, that the manifold $\overline{\mathcal{M}}(0, n)$ coincides with the wonderful compactification $Z_{\mathcal{G}''}$. It is left to note that the diffeomorphism $\overline{f}$ extends to a diffeomorphism between the wonderful compactification $Z_{\mathcal{G}^{''}}$ and the wonderful compactification $(\overline{F}_{n})_{\mathcal{G}}$, where the building set $\mathcal{G}$ is given by all possible intersections of the subvarieties (7). Thus, the smooth manifolds $\overline{\mathcal{M}}(0,n)$ and $\mathcal{F}_{n}$ are diffeomorphic. The theorem is proved. § 5. $\mathcal{F}_{n}$ and Chow quotient $G_{n,2}/\!/( {\mathbb{C}}^{\ast})^{n}$5.1. The basic facts on Chow varieties and Chow quotientWe follow the monograph [16], Ch. 4, to recall the basic facts on Chow varieties, while for the notion of Chow quotient $G_{n,k}/\!/(\mathbb{C}^{\ast})^{n}$ we follow Kapranov’s paper [19]. The idea behind the definition of the Chow quotient is the construction in algebraic geometry known as Chow varieties, that is, compact varieties whose points parametrize the algebraic cycles of fixed dimension and degree in a given variety. The Chow variety for $G_{n,k}$, which we need in the definition of the Chow quotient $G_{n,k}/\!/(\mathbb{C}^{\ast})^n$ can be defined as follows. Let $\delta \in H_{2(n-1)}(G_{n,k}, \mathbb{Z})$ be the homology class of the closure of a generic $(\mathbb{C}^{\ast})^{n}$-orbit in $G_{n,k}$, and let $C_{2(n-1)}(G_{n,k}, \delta)$ denote the set of all algebraic cycles in $G_{n,k}$ of dimension $2(n-1)$ whose homology class is $\delta$. The Grassmann manifolds $G_{n,k}$ embed into $\mathbb{C} P^{N}$, $N=\dbinom{n}{k}-1$, via the Plücker embedding, so let $d\in H_{2(n-1)}(\mathbb{C} P^{N}, \mathbb{Z})\cong \mathbb{Z}$ be the image of the homology class $\delta$ under this embedding. Now consider the set $G(N, d, 2(n-1))$ of algebraic cycles in $\mathbb{C} P^{N}$ of dimension $2(n-1)$ and degree $d$. In other words, one considers algebraic cycles whose multiplicity in $H_{2(n-1)}(\mathbb{C} P^{N}, \mathbb{Z})$ with respect to the canonical generator is $d$. Denote by $\mathcal{B}$ the coordinate ring of $G_{n,k}$ via the Plücker embedding, that is, the quotient of the polynomial ring $\mathbb{C} [z_{1}, \dots, z_{N+1}]$ by Plücker’s relations. Then $\mathcal{B}=\bigoplus_{k\geqslant 0}\mathcal{B}_k$, where $\mathcal{B}_k$ is the complex linear space spanned by the homogeneous polynomials of degree $k$. By the theorem of Chow and van der Waerden, the set $G(N, d, 2(n-1))$ becomes a closed (in particular, compact) projective algebraic variety under the Chow embedding $G(N, d, 2(n-1))\to P(\mathcal{B}_{d})$. The set $C_{2(n-1)}(G_{n,k}, \delta)$ endowed with the resulting structure of an algebraic variety induced by the embedding $C_{2(n-1)}(G_{n,k}, \delta)\subset G(N, d, 2(n-1))$ is the required Chow variety for $G_{n,k}$. The Chow embedding can be defined in greater detail as follows (see [16]). For any irreducible algebraic cycle $X\in G(N, d, 2(n-1))$ one can consider the set $\mathcal{Z}(X)$ of all $(N-2(n-1)-1)$-dimensional projective subspaces $L$ in $\mathbb{C} P^{N}$ which intersect $X$. The set $\mathcal{Z}(X)$ is a subvariety of the Grassmannian $G(N, {N-2(n-1)+1})$. It can be proved that $\mathcal{Z}(X)$ is defined by some element $R_{X}\in \mathcal{B}_{d}$, which is unique up to a constant factor and is called the Chow form of $X$. If $X$ is not an irreducible cycle, then $X=\sum a_iX_i$, where the $X_i$ are $2(n-1)$-dimensional closed irreducible varieties and the $a_i$ are nonnegative integer coefficients, and the Chow form for $X$ is defined by $R_X=\prod R_{X_i}^{a_i}\in \mathcal{B}_{d}$. The map $X\to R_{X}$ defines an embedding of $G(N, d, 2(n-1))$ into the projective space $P(\mathcal{B}_{d})$, which is called the Chow embedding. In order to define the Chow quotient we consider the natural map
$$
\begin{equation*}
W/(\mathbb{C}^{\ast})^n \to C_{2(n-1)}(G_{n,k}, \delta), \qquad x \to \overline{(\mathbb{C}^{\ast})^n\cdot x},
\end{equation*}
\notag
$$
where $W$ is the main stratum in $G_{n,k}$, which consists of the points all of whose Plücker coordinates are nonzero. By definition the Chow quotient $G_{n,k}/\!/(\mathbb{C}^{\ast})^{n}$ is the closure of the image of this map. We recall the following results from [19], Propositions (1.2.11) and (1.2.15), which provide a description of the outgrow components of $W/(\mathbb{C}^{\ast})^{n}$ in $G_{n,k}/\!/(\mathbb{C}^{\ast})^{n}$. Proposition 27. The algebraic cycles in the Chow quotient $G_{n,k}/\!/ (\mathbb{C}^{\ast})^{n}$ are of the form $Z= \sum_{i}Z_{i}$, where the $Z_i$ are the closures of $(\mathbb{C}^{\ast})^{n}$-orbits in $G_{n,k}$ such that the matroid polytopes $\mu (Z_i)$ form a polyhedral decomposition of $\Delta_{n,k}$. We point out that in the case of $G_{n,2}$ the matroid polytopes defined in [19] coincide with our admissible polytopes; see [5]. Thus, in our terminology the Chow quotient provides a special compactification of the space of parameters $F_n$ of the main stratum $W \subset G_{n,2}$. We want to emphasize that the Chow quotient $X/\!/H$ can be defined for any complex projective variety $X\subset \mathbb{C} P^{N}$ with an action of an algebraic group $H$; see [19]. Namely, the orbit closure $\overline{H\cdot x}$ is a compact subvariety of $X$ for any point $x\in X$, and for a small Zariski open $H$-invariant subset $U\subset X$ consisting of generic points, all varieties $\overline{H \cdot x}$ for $x\in U$ have the same dimension $m$ and represent the same homology class $\delta \in H_{2m}(X, \mathbb{Z})$. One can consider the Chow variety $C_{2m}(X, \delta)\subset G(N, d, 2m)$, where $d$ is the image of $\delta$ under the embedding $X\to \mathbb{C} P^{N}$, and the Chow quotient $X/\!/H$ is the closure of the image of the map $U/H \to C_{2m}(X, \delta)$ defined by $x\to \overline{x\cdot H}$. Using the Gelfand-MacPherson construction Kapranov proved in [19] that for $k=2$ the Chow quotient $G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$ is isomorphic to the Chow quotient $(\mathbb{C} P^1)^{n}/\!/\mathrm{GL}(2)$. Based on this, he constructed an isomorphism between the Chow quotient $G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$ and the Grothendieck-Knudsen compactification $\overline{\mathcal{M}}(0,n)$. On the other hand it follows from (8) that $\mathbb{C} P^{n-3}$, a compactification of $\mathbb{C}^{n-3}$, is also a compactification of $\mathcal{M}(0,n)$. Kapranov [19] proved that $\overline{\mathcal{M}}(0,n)$ is a finer compactification, which can be mapped to $\mathbb{C} P^{n-3}$ by a regular birational morphism. Moreover, he showed that for every $n-1$ generic points $q_1,\dots,q_{n-1}$ in $\mathbb{C} P^{n-3}$ the variety $\overline{\mathcal{M}}(0,n)$ can be obtained from $\mathbb{C} P^{n-3}$ by blowing up all projective spaces spanned by the $q_i$. In addition, using the isomorphism obtained Kapranov provided in [19] the description of the quotient $G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$ as a sequence of blow-ups along certain subvarieties of $\mathbb{C} P^{n-3}$. In conclusion, we want to point out that Theorem 23 implies that our construction of the space $\mathcal{F}_{n}$ provides a new, purely topological approach to the description of the Chow quotient $G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$, which is closely connected with our description of the orbit space $G_{n,2}/T^n$. 5.2. The structures of $G_{n,2}/\!/( {\mathbb{C}}^{\ast})^{n}$ and $G_{n,2}/T^n$In [5] we described the orbit space $G_{n,2}/T^n$ in terms of the complex of admissible polytopes and the universal space of parameters $\mathcal{F}_{n}$. In this description the chamber decomposition of $\Delta_{n,2}$ induced by the admissible polytopes plays a fundamental role. Using the results of [19], first we describe the Chow quotient $G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$ in terms of cortéges of admissible polytopes which define the polyhedral decomposition of $\Delta_{n,2}$ and the spaces of parameters of these polytopes. In addition, we describe $G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$ in terms of the virtual spaces of parameters for the admissible polytopes that form a $(n-1)$-dimensional chamber in $\Delta_{n,2}$. Let $\mathcal{P}$ denote the family of admissible polytopes of dimension $n-1$ for the standard $T^n$-action on $G_{n,2}$. Let the set $\{\mathcal{P}_{1}, \dots, \mathcal{P}_{l}\}$ consist of all subfamilies $\mathcal{P}_{i}=\{P_{i_1}, \dots, P_{i_s}\} \subset \mathcal{P}$ such that the polytopes $P_{i_1}, \dots, P_{i_s}$ form a polyhedral decomposition of $\Delta_{n,2}$, that is, $\bigcup_{j=1}^{s}P_{i_j}=\Delta_{n,2}$ and $\mathring{P}_{i_j}\cap \mathring{P}_{i_k} = \varnothing$ for any $1\leqslant j<k\leqslant s$. To a family $\mathcal{P}_{i} = \{P_{i_1}, \dots, P_{i_s}\}$ we assign the set $\mathcal{W}_{i}= \{W_{i_1},\dots, W_{i_s}\}$, where the $W_{i_j}$ are the strata in $G_{n,2}$ corresponding to the admissible polytopes $P_{i_j}$, that is, $\mu (W_{i_j}) = \mathring{P}_{i_j}$. Taking the quotient of these strata by the $(\mathbb{C}^{\ast})^{n}$-action, to any $\mathcal{P}_{i}$ we can assign the set $\{ F_{i_1}, \dots, F_{i_s}\}$. Next we introduce the space of parameters $\mathcal{F}_{i}$ of the family $\mathcal{P}_{i}$ as a topological space homeomorphic to the direct product $\mathcal{F}_{i} = F_{i_1} \times \dotsb \times F_{i_s}$. It follows from Proposition 27 that the Chow quotient $G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$ is the disjoint union of connected components $\mathcal{C}_{i}$, each consisting of algebraic cycles determined by the family $\mathcal{P}_{i}$, $1\leqslant i\leqslant l$. In particular, the complement to $F_{n}$ in $G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$ is the disjoint union of the components $\mathcal{C}_{i}$ for which $\mathcal{P}_{i} \neq \{\Delta_{n,2}\}$. Proposition 28. There is a bijection between $\mathcal{F}_{i}$ and $\mathcal{C}_{i}$ for any $1\leqslant i\leqslant l$. Proof. We define the bijection $g_i\colon \mathcal{F}_{i}\to \mathcal{C}_{i}$ as follows: where $Z_{i_j}(c_{i_j})$ is the $(\mathbb{C}^{\ast})^{n}$-orbit in the stratum $W_{i_j}$ that is determined by the parameter $c_{i_j}$. The proposition is proved. The Chow quotient $G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$ and the complement to $F_{n}$ in $G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$ can also be interpreted in the following way. Let $P_{\sigma}$ be an admissible polytope, and consider the set ${\mathcal{P}_{\sigma} \,{=}\, \{\mathcal{P}_{\sigma, 1}, \dots, \mathcal{P}_{\sigma, s}\}\,{\subset}\, \mathcal{P}}$ of all decompositions $\mathcal{P}_{\sigma, i}$ of $\Delta_{n,2}$ which contain $P_{\sigma}$, that is, $\mathcal{P}_{\sigma, i}\in \mathcal{P}$ if and only if $P_{\sigma} \in \mathcal{P}_{\sigma, i}$. Let $\widetilde{Z}_{\sigma, i}\subset G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$ be the family of algebraic cycles determined by the decomposition $\mathcal{P}_{\sigma, i} \,{=}\, \{P_{\sigma_{i_1}}, \dots, P_{\sigma_{i_q}}\}$. By Proposition 28 these cycles have the form
$$
\begin{equation*}
Z_{\sigma_{i_1}}(c_{\sigma_{i_1}})+\dots + Z_{\sigma,_{i_q}}(c_{\sigma_{i_q}})
\end{equation*}
\notag
$$
for $(c_{\sigma_{i_1}}, \dots, c_{\sigma_{i_q}}) \in F_{\sigma_{i_1}}\times \dotsb \times F_{\sigma_{i_q}}$, where $Z_{\sigma_{i_j}}(c_{\sigma_{i_j}})$ is the closure in $W_{\sigma_{i_j}}$ of an orbit of the algebraic torus, and this orbit is determined by $c_{\sigma_{i_j}}\in F_{\sigma_{i_j}} = W_{\sigma_{i,j}}/(\mathbb{C}^{\ast})^{n}$. Now let
$$
\begin{equation*}
\widetilde{Z}_{\sigma}=\bigcup_{i=1}^{s}\widetilde{Z}_{\sigma, i}.
\end{equation*}
\notag
$$
Proposition 29. For any admissible set $\sigma$ there exists a projection $p_{\sigma}\colon \widetilde{Z}_{\sigma} \to F_{\sigma}$, where $F_{\sigma}=W_{\sigma}/(\mathbb{C}^{\ast})^n$. Proof. For an algebraic cycle $Z\in \widetilde{Z}_{\sigma}$ there exists a $(\mathbb{C}^{\ast})^{n}$-orbit $Z_{\sigma}$ in the stratum $W_{\sigma}$ whose admissible polytopes is $P_{\sigma}$ and which is an irreducible summand of $Z$. Since $F_{\sigma} = W_{\sigma}/(\mathbb{C}^{\ast})^{n}$, there is a canonical $(\mathbb{C}^{\ast})^{n}$-invariant projection $q_{\sigma}\colon {W_{\sigma}\to F_{\sigma}}$. We define $p_{\sigma}(Z) = q_{\sigma}(Z_{\sigma})$. The proposition is proved. Recall (see [17] and [5]) that the admissible polytopes (see Definition 4) define the chamber decomposition for $\Delta_{n,2}$: for some subset $\omega$ of all admissible sets, $C_{\omega}$ is a chamber if
$$
\begin{equation*}
C_{\omega}=\bigcap_{\sigma \in \omega} \mathring{P}_{\sigma}\quad\text{and} \quad C_{\omega} \cap \mathring{P}_{\sigma}=\varnothing, \qquad \sigma\notin \omega.
\end{equation*}
\notag
$$
Using the diffeomorphism between $\mathcal{F}_{n}$ and $G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$ (see Theorem 23) we obtain the following result. Theorem 30. Let $C_{\omega}\subset \Delta_{n,2}$ be a chamber such that $\dim C_{\omega} = n-1$. Then $C_{\omega}$ defines a decomposition of $G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$ into a disjoint union and there are homeomorphisms where the topology on $\bigcup_{\sigma \in \omega} \widetilde{Z}_{\sigma}$ is defined by the first bijective map. Proof. Let $Z\in G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$. Then $Z$ is determined by some polyhedral decomposition $\mathcal{P}_{i} = \{ P_{\sigma_{i_1}}, \dots, P_{\sigma_{i_l}}\}$. Note that for any admissible polytope $P_{\sigma}$ we have either $C_{\omega}\subset \mathring{P}_{\sigma}$ or $C_{\omega}\cap \mathring{P}_{\sigma} = \varnothing$. Thus, there exists $P_{\sigma_{i_{j}}}\in \mathcal{P}_{i}$ such that $C_{\omega}\subset \mathring{P}_{\sigma_{i_{j}}}$, which means that $\sigma_{i_{j}}\in \omega$. Therefore, $Z\in \widetilde{Z}_{\sigma_{i_{j}}}$, which implies that $Z\in \bigcup_{\sigma \in \omega} \widetilde{Z}_{\sigma}$.
In order to prove that the union (10) is disjoint, we note that for $\sigma_1, \sigma_2\in \omega$ we have $C_{\omega}\subset \mathring{P}_{\sigma_1}, \mathring{P}_{\sigma_2}$, that is, $\mathring{P}_{\sigma_1}\cap \mathring{P}_{\sigma_2}\neq \varnothing$. This implies that there is no decomposition of $\Delta_{n,2}$ which contains both $P_{\sigma_1}$ and $P_{\sigma_2}$. Thus, there is no algebraic cycle $Z\in G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$ such that $Z\in \widetilde{Z}_{\sigma_1}$ and $Z\in \widetilde{Z}_{\sigma_2}$. The theorem is proved. Note that $\widetilde{Z}_{\sigma} = F_n$ for $P_{\sigma}= \Delta_{n,2}$, which implies that, apart from the Chow quotient, (10) describes the outgrow components to $F_{n}$ in $G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$, which are not disjoint in general. Remark 31. In our papers [2] and [3], for the description of the orbit space $G_{n,2}/T^n$ we introduced the notion of the virtual space of parameters $\widetilde{F}_{\sigma}$ for the stratum $W_{\sigma}$. Note that the properties of the spaces $\widetilde{Z}_{\sigma}$ formulated in Proposition 29 and Theorem 30 confirm that for $\dim P_{\sigma}=n-1$ the spaces $\widetilde{Z}_{\sigma}$ correspond to the spaces $\widetilde{F}_{\sigma}$. In particular, Theorem 7 in [5] is an analogue of Theorem 30. Theorem 23 and the result of [19] that the manifolds $G_{n,2}/\!/(\mathbb{C}^{\ast})^{n}$ and $\overline{\mathcal{M}}(0,n)$ are isomorphic imply that the universal space of parameters $\mathcal{F}_{n}$ describes the topology of the gluing of the outgrow components in $G_{n,2}/\!/ (\mathbb{C}^{\ast})^{n}$. Having the description of $\mathcal{F}_{n}$ as a wonderful compactification of $\overline{F}_{n}$, we present an explicit demonstration of this correspondence in the cases $n=4$ and $n=5$. 5.3. The spaces $G_{4,2}/\!/( {\mathbb{C}}^{\ast})^{4}$ and $G_{4,2}/T^4$For $n=4$ the outgrow components in $G_{4,2}/\!/ (\mathbb{C}^{\ast})^{4}$ consist of three points, and glued to $F_{4}\cong \mathbb{C} P^{1}_{A}$ in $G_{4,2}/\!/ (\mathbb{C}^{\ast})^{4}$ they produce $\mathbb{C} P^{1}$. This was observed in [19], but also follows independently from [1] and Theorem 23. Using our notation we describe this in the following way. There are exactly three decompositions of the octahedron $\Delta_{4,2}$, that is, $\mathcal{P}= \{\mathcal{P}_{1}, \mathcal{P}_{2}, \mathcal{P}_{3}\}$, and they are given by three pairs of four-sided complementary pyramids. The space of parameters of a stratum for any of these pyramids is a point. Then Proposition 28 implies that $(G_{4,2}/\!/ (\mathbb{C}^{\ast})^{4})\setminus F_{4}$ consists of three points, that is, $G_{4,2}/\!/ (\mathbb{C}^{\ast})^{4}\cong \mathbb{C} P^1$. More precisely, these three points correspond to the algebraic cycles formed by the $(\mathbb{C}^{\ast})^{4}$-orbits whose admissible polytopes are complementary pyramids in the octahedron $\Delta_{4,2}$. In addition, for any pyramid $P_{\sigma}$ the cycle $\widetilde{Z}_{\sigma}$ is given by one algebraic cycle, which implies that $F_{\sigma} = \widetilde{Z}_{\sigma}$. 5.4. The spaces $G_{5,2}/\!/( {\mathbb{C}}^{\ast})^{5}$ and $G_{5,2}/T^5$In the case $n=5$ we use the results from [2] to describe the correspondence between $\mathcal{F}_{5}$ and the Chow quotient $G_{5,2}/\!/ (\mathbb{C}^{\ast})^{5}$. Recall that the hypersimplex $\Delta_{5,2}$ has 10 vertices $\{\Lambda_{ij}, 1\leqslant i<j\leqslant 5 \}$. By [2] the following result holds. Lemma 32. There are $25$ decompositions of the hypersimplex $\Delta_{5,2}$ given by the admissible polytopes for the $T^5$-action on $G_{5,2}$. They are given by the pairs $\{K_{ij}, P_{ij}\}$, $1\leqslant i<j\leqslant 5$, and the triples $\{P_{ij}, K_{ij, kl}, P_{kl}\}$, $1\leqslant i<j\leqslant 5$, $1\leqslant k<l\leqslant5$, $\{i,j\}\cap \{k,l\}=\varnothing$. Here $K_{ij}$ is a polytope with nine vertices which is the convex hull of the nine vertices of $\Delta_{5,2}$ distinct from $\Lambda_{ij}$, $P_{ij}$ is the seven-sided pyramid with apex $\Lambda_{ij}$, while $K_{ij, kl}$ is the polytope with eight vertices that does not contain $\Lambda_{ij}$ and $\Lambda_{kl}$. The space of parameters for an admissible polytope $K_{ij}$ is $\mathbb{C} P^{1}_{A}$, while for the polytopes $P_{ij}$ and $K_{ij, kl}$ it is a point. Then Proposition 28 yields the following result. Corollary 33. The disjoint outgrow components for $F_5$ in $G_{5,2}/\!/ (\mathbb{C}^{\ast})^{5}$ are the spaces $\mathcal{C}_{ij} \cong \mathbb{C} P^{1}_{A}$ and the points $\mathcal{C}_{ij, kl}$ for $1\leqslant i<j\leqslant 5$, $1\leqslant k<l\leqslant 5$. A component $\mathcal{C}_{ij}$ consists of the cycles of the form $Z_{ij,9}(c)+Z_{ij,7}$, while a component $\mathcal{C}_{ij,kl}$ consists of the cycle $Z_{ij,7}+Z_{ij, kl} + Z_{kl,7}$, where $c\in \mathbb{C} P^{1}_{A}$. In this case the irreducible algebraic cycles are as follows: We proved in [2] that the universal space of parameters $\mathcal{F}_{5}$ is the blow-up of the surface $\overline{F}_{5} = \{(c_1:c_1'), (c_2:c_2'), (c_3:c_3'))\in (\mathbb{C} P^{1})^{3},\ c_1c_{2}'c_3=c_{1}'c_2c_{3}'\}$ at the point $((1:1), (1:1), (1:1))$. The identification of $\mathcal{F}_{5}$ with $G_{5,2}/\!/ (\mathbb{C}^{\ast})^{5}$ translates as follows into the gluing of the outgrow components in $G_{5,2}/\!/ (\mathbb{C}^{\ast})^{5}$. Corollary 34. The gluing of the outgrow components in $G_{5,2}/\!/ (\mathbb{C}^{\ast})^{5}$ corresponds to the compactification of $F_5\subset \mathcal{F}_{5}$ in accordance with the following pattern. We also describe the correspondence between the subspaces $\widetilde{Z}_{ij,9}$, $\widetilde{Z}_{ij, 7}$ and $\widetilde{Z}_{ij,kl}$ and subspaces of $\mathcal{F}_{5}$. Corollary 35. The spaces $\widetilde{Z}_{ij,9}$, $\widetilde{Z}_{ij, 7}, \widetilde{Z}_{ij,kl} \subset G_{5,2}/\!/ (\mathbb{C}^{\ast})^{5}$ are homeomorphic to $\mathbb{C} P^{1}_{A}$, $\mathbb{C} P^{1}$ and a point, respectively. The corresponding spaces in $\mathcal{F}_{5}$ are Note that the spaces $\widetilde{Z}_{ij,9}$, $\widetilde{Z}_{ij, 7}$ and $\widetilde{Z}_{ij, k}$ coincide with the virtual spaces of parameters $\widetilde{F}_{ij, 9}$, $\widetilde{F}_{ij, 7}$ and $\widetilde{F}_{ij, kl}$ for the corresponding strata in $G_{5,2}$: see [5]. AcknowledgementThe authors are grateful to A. A. Gaifullin for a useful discussion of the results of the paper.
Citation in format AMSBIB
\Bibitem{BucTer23} |
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