On the connectivity of the branch and real locus of $${\mathcal M}_{0,[n+1]}$$M0,[n+1

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.