Connectivity of the branch and the real locus in Moduli space
\[ M_{0,[n+1]}\]
 
Yasmina Atarihuana
Universidad de La Frontera (Chile)
Resumen:
Let
\[{\mathcal M}_{0,[n+1]}\]
be the moduli space of isomorphisms classes of
\[(n+1)\]
-marked spheres, where
\[n\geq 3\]
. It is know that
\[{\mathcal M}_{0,[n+1]}\]
has a complex orbifold structure of dimension
\[n-2\]
. Moreover, the space
\[{\mathcal M}_{0,[n+1]}\]
admits a natural real structure
\[\hat{J}\]
, this being induced by the complex conjugation on the Riemann sphere. The fixed points of
\[\hat{J}\]
are called the real points and these points corresponds to the classes of isomorphisms of 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.
Let us denote by
\[{\mathcal B}_{0,[n+1]}\]
the branch locus of
\[{\mathcal M}_{0,[n+1]}\]
(the isomorphism classes of those
\[(n+1)\]
-marked spheres with non-trivial group of conformal automorphisms). It is known that
\[{\mathcal B}_{0,[4]}={\mathcal M}_{0,[4]}\]
(as any collection of four points in the Riemann sphere is invariant by a subgroup of Möbius transformations isomorphic to
\[{\mathbb Z}_{2}^{2}\]
) and that
\[{\mathcal B}_{0,[n+1]} \neq {\mathcal M}_{0,[n+1]}\]
for
\[n \geq 4\]
.
The main aim of this talk is to observe the following:
1) 
\[{\mathcal B}_{0,[n+1]}\]
is connected if either (i)
\[n \geq 4\]
is even or (ii) if
\[n \geq 6\]
is divisible by
\[3\]
. It has exactly two connected components otherwise.
2) 
\[{\mathcal M}_{0,[n+1]}^{\mathbb R}\]
is connected for
\[n \geq 5\]
odd. It is disconnected for
\[n=2r\]
with
\[r \geq 5\]
odd.
This is part of the results obtained for my Ph. D. Thesis under the supervision of R. A. Hidalgo and S. Quispe.