Special Loci in Moduli Spaces of Curves

Summary

Let S be a topological surface of genus g with n marked points, and let be a finite-order element of the mapping class group of S. We study the special locus associated to in the moduli space M(S) of Riemann surfaces of topological type (g, n); this is the set of points in M(S) corresponding to Riemann surfaces admitting as an automorphism. Another definition of the special locus is that it is the image on M(S) of the points in the Teichmiiller space τ (S) fixed by under the natural action of the mapping class group on τ (S). We completely describe all special loci in the moduli spaces of small type (0,4), (0,5), (1,1) and (1,2), and also of the general genus zero spaces (0,n), including determining their fields of moduli. Then, based on results of Harvey et al., we show how the (normalization of the) special locus of in M(S) provides a finite covering of the moduli space of the topological quotient, and give conditions on for this covering to be as close as possible to an isomorphism. Finally, we translate these results in terms of the mapping class groups and show that when the conditions on are satisfied, we obtain a homomorphism between mapping class groups which has geometric and arithmetic significance, and that in genus zero, these two conditions are always satisfied. We end with two explicit examples of such homomorphisms, one in genus zero and one in genus one.