Abstract

If n = 3, then moduli spaceM0,[n+ 1], of isomorphisms classes of ( n+ 1)-marked spheres, is a complex orbifold of dimension n-2. Its branch locus B0,[n+ 1] consists of the isomorphism classes of those ( n+ 1)-marked spheres with non-trivial group of conformal automorphisms. We prove that B0,[n+ 1] is connected if either n = 4 is even or if n = 6 is divisible by 3, and that it has exactly two connected components otherwise. The orbifoldM0,[n+ 1] also admits a natural real structure, this being induced by the complex conjugation on the Riemann sphere. The locusM0,[n+ 1]( 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 locusMR 0,[n+ 1], consisting of those classes ofmarked spheres admitting an anticonformal involution. We prove that MR 0,[n+ 1] is connected for n = 5 odd, and that it is disconnected for n = 2r with r = 5 being odd.