Abstract
If \(n \ge 3\), then moduli space \({\mathcal M}_{0,[n+1]}\), of isomorphisms classes of \((n+1)\)-marked spheres, is a complex orbifold of dimension \(n-2\). Its branch locus \({\mathcal B}_{0,[n+1]}\) consists of the isomorphism classes of those \((n+1)\)-marked spheres with non-trivial group of conformal automorphisms. We prove that \({\mathcal B}_{0,[n+1]}\) is connected if either \(n \ge 4\) is even or if \(n \ge 6\) is divisible by 3, and that it has exactly two connected components otherwise. The orbifold \({\mathcal M}_{0,[n+1]}\) also admits a natural real structure, this being induced by the complex conjugation on the Riemann sphere. The locus \({\mathcal M}_{0,[n+1]}({\mathbb R})\) of its fixed points, the real points, consists of the isomorphism classes of those marked spheres admitting an anticonformal automorphism. Inside this locus is the real locus \({\mathcal M}_{0,[n+1]}^{\mathbb R}\), consisting of those classes of marked spheres admitting an anticonformal involution. We prove that \({\mathcal M}_{0,[n+1]}^{\mathbb R}\) is connected for \(n \ge 5\) odd, and that it is disconnected for \(n=2r\) with \(r \ge 5\) being odd.
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The authors would like to thanks to the anonymous referees for their valuable comments and suggestions which permitted to improve the presentation of this paper.
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Atarihuana, Y., Hidalgo, R.A. On the connectivity of the branch and real locus of \({\mathcal M}_{0,[n+1]}\). RACSAM 113, 2981–2998 (2019). https://doi.org/10.1007/s13398-019-00669-6
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DOI: https://doi.org/10.1007/s13398-019-00669-6