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

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 $r\ge 5$ is odd.