On the connectivity of the branch and real locus of $${\mathcal M}_{0,[n+1]}$$M0,[n+1]

Abstract

If n ≥ 3, then moduli space M 0,[n+1] , of isomorphisms classes of (n+1)-marked spheres, is a complex orbifold of dimension n − 2. Its branch locus 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 B 0,[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 orbifold M 0,[n+1] also admits a natural real structure, this being induced by the complex conjugation on the Riemann sphere. The locus M 0,[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 locus M R 0,[n+1] , consisting of those classes of marked spheres admitting an anticonformal involution. We prove that M R 0,[n+1] is connected for n ≥ 5 odd, and that it is disconnected for n = 2r with r ≥ 5 is odd.2010 Mathematics Subject Classification. 30F10, 30F60, 32G15.

Smooth quotients of generalized Fermat curves

“…. , n + 1)) = t. In particular, this allowd us to note that Θ : S n+1 → G n is a surjective homomorphism with G 3 S 3 and, for n ≥ 4, G n S n+1 (details can be found in [2]).…”

“…. , λ n−2 ) ⊂ P n is an embedding into a lower dimensional projective space for (p, n) (2,4). Nevertheless, in [14] it was noted that C p (λ 1 , .…”

“…Such types of complete intersections of (n − 1) quadrics in P n where studied by Edge in [9]. (3) Generalized Fermat curves of type (2,4), also called classical Humbert curves, define a 2-dimensional locus in the moduli space M 5 (of genus five Riemann surfaces). These were firstly considered by Humbert in [16] as degree seven curves in P 3 .…”

“…, λ n−2 . As noted in Remark 2, there is a subgroup K Z m 2 of H, acting freely on S = C (2) (λ 1 , . .…”