Connectivity of the real locus

Yasmina Atarihuana 
Universidad de La Frontera (Chile) 
Abstract: 
The moduli space \[{\mathcal M}_{0,[n+1]}\] of isomorphisms classes of \[(n+1)\] marked spheres, where \[n\geq 3\] , is a complex orbifold of dimension \[n2\] . Its admits a natural real structure, this induced by complex conjugation on the Riemann sphere. The locus \[{\mathcal B}_{0,[n+1]}({\mathbb R})\] of its fixed points consists of the isomorphism classes of those marked spheres admitting an anticonformal automorphism. Inside this locus is the real locus \[{\mathcal B}_{0,[n+1]}^{\mathbb R}\] , consisting of those classes of marked spheres admitting an anticonformal involution. We prove that \[{\mathcal B}_{0,[n+1]}^{\mathbb R}\] is connected for \[n\geq 5\] odd. It is disconnected for \[n=2r\] , where \[r\geq 5\] is odd. The connectivity of \[{\mathcal B}_{0,[n+1]}^{\mathbb R}\] is described by the intersection graph \[{\mathcal G}_{n}\] of \[{\mathcal B}_{0,[n+1]}^{\mathbb R}\] , whose vertices represent the irreducible components of the real locus and the edges their intersection. 
This is part of the results obtained for my Ph. D. Thesis under the supervision of R. A. Hidalgo and S. Quispe. 