On the connectivity of branch loci of moduli spaces

Abstract

The moduli space M g of compact Riemann surfaces of genus g has orbifold structure and the set of singular points of the orbifold is the branch locus B g .In this article we show that B g is connected for genera three, four, thirteen, seventeen, nineteen and fiftynine, and disconnected for any other genus.In order to prove this we use Fuchsian groups, automorphisms of order 5 and 7 of Riemann surfaces, and calculations with GAP for some small genera.