2019/04/03 by Yasmina Atarihuana, Atarihuana, Yasmina, Rubén A. Hidalgo +1
Mathematics · #30F10 #30F60 #32G15 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1904.01982
openalex publication_date 2019/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If n ≥ 3, then moduli space \mathcal M0,[n+1], of isomorphisms classes of (n+1)-marked spheres, is a complex orbifold of dimension n-2. Its branch locus \mathcal B0,[n+1] consists of the isomorphism classes of those (n+1)-marked spheres with non-trivial group of conformal automorphisms. We prove that \mathcal B0,[n+1] is connected if either n ≥ 4 is even or if n ≥ 6 is divisible by 3, and that it has exactly two connected components otherwise. The orbifold \mathcal M0,[n+1] also admits a natural real structure, this being induced by the complex conjugation on the Riemann sphere. The locus \mathcal M0,[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 M0,[n+1]\mathbb R, consisting of those classes of marked spheres admitting an anticonformal involution. We prove that \mathcal M0,[n+1]\mathbb R is connected for n ≥ 5 odd, and that it is disconnected for n=2r with r ≥ 5 is odd.