Fotos
Connectivity of the branch and the real locus in Moduli space
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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. |