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

## TL;DR

This paper investigates the topological structure of the branch and real loci of the moduli space of marked spheres, establishing their connectivity properties depending on the number of marked points and symmetries.

## Contribution

It proves new connectivity results for the branch locus and the real locus of the moduli space of marked spheres, clarifying their topological structure.

## Key findings

- The branch locus is connected if n ≥ 4 even or divisible by 3 for n ≥ 6.
- The branch locus has exactly two connected components otherwise.
- The real locus is connected for odd n ≥ 5 and disconnected for even n ≥ 10 with odd r.

## 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.