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

Abstract: If $n \geq 3$, 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 $n \geq 4$ is even or if $n \geq 6$ 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 $n \geq 5$ odd, and that it is disconnected for $n=2r$ with $r \geq 5$ is odd.