The moduli space of \(n\)+1 points in complex hyperbolic \(n\)-space (Q1397893)

They argue the \((n+1)\)-point configuration space in the complex hyperbolic \(n\)-space \(H^n_C\). They give a coordinate system on the moduli space \(\widetilde M\) with respect to the holomorphic isometry group \(PU (n,1)\). In the case \(n=2\), the result is well known result as shown by Brehm and by Cartan. The natural extension to the case \(n>2\) is obtained by natural suggestion and observation. The authors are interested in the moduli space \(M\) with respect to \(PU(n,1) \ltimes S_{n+1}\), where \(S_{n+1}\) is the symmetry group acting on \(n+1\) points. The results in this paper give good tools to investigate the topology type and singularity of \(M\).