ea}}^{-1}\cap \{x_{i_j}=0\}$ for $j = 1,\cdots, n$. We denote by $\Delta_{p,\theta}$ such a hyperbolic polyhedron. Then the conditions for $\theta$ as in the definition of $\Theta_n$ ensures us that the hyperbolic polyhedron $\Delta_{p,\theta}$ has exactly $n$ facets.

Let us denote by $(i_1 i_2)i_3 \ldots i_n$ or simply $(i_1 i_2)$ the face of $\Delta_{p,\theta}$ represented by $\Delta_{p,\theta} \cap \{x_{i_1}=0\}$ since it corresponds to the degenerate configurations where the points marked by $i_1$ and $i_2$ collide. Similarly we use $(i_1i_2)(i_3 i_4)i_5 \ldots i_n$ or $(i_1 i_2 i_3) i_4 \ldots i_n$, etc. to represent the codimension two faces of $\Delta_{p,\theta}$.

Now gluing $(n-1)!/2$ hyperbolic polyhedra $\Delta_{p, \theta}$ for all labels $p$ along the faces which represent the same degenerate configurations, we obtain $\overline{X_{n,\theta}}$ in which $X_{n, \theta}$ lies as an open dense subset.

 The case $n=5$
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When $n=5$, $\Delta_{p, \theta}$ is a hyperbolic right pentagon where the edges are labeled by $(i_1i_2)$, $(i_3i_4)$, $(i_5i_1)$, $(i_2i_3)$, $(i_4i_5)$ cyclically. For the equal weight $\theta_0 =(2\pi/5, \cdots, 2\pi/5)$, $\Delta_{p, \theta_0}$ is a hyperbolic regular pentagon, i.e, with all edges having equal lengths.

For any $\theta \in \Theta_5$, the space $\overline{X_{5,\theta}}$ is a closed hyperbolic surface homeomorphic to ${\#}^5 {{\bold R}}{{\bold P}}^2$. The space of hyperbolic structures, which we called a Teichmüller space and denoted by ${\cal T}({\#}^5 {{\bold R}}{{\bold P}}^2)$, is parameterized by the lengths and twisting amounts for the 2-sided ones of a maximal family of mutually disjoint nonparallel simple closed curves on ${\#}^5 {{\bold R}}{{\bold P}}^2$. It is homeomorphic to ${{\bold R}}^9$. Thus we have a map $\Phi_5: \Theta_5 \to {\cal T}({\#}^5 {{\bold R}}{{\bold P}}^2)$ assigning to each $\theta$ the marked hyperbolic structure of $\overline{X_{5,\theta}}$.

On the other hand, our surface $\overline{X_{5,\theta}}$ has a geometric cell decomp