Connecting p-gonal loci in the compactification of moduli space

Consider the moduli space M g of Riemann surfaces of genus g ≥ 2 and its Deligne-Mumford compactification M g. We are interested in the branch locus B g for g > 2, i.e., the subset of M g consisting of surfaces with automorphisms. It is well-known that the set of hyperelliptic surfaces (the hyperelliptic locus) is connected in M g but the set of (cyclic) trigonal surfaces is not. By contrast, the set of (cyclic) trigonal surfaces is connected in M g. We exhibit an explicit nodal surface that lies in the completion of every equisymmetric set of 3-gonal Riemann surfaces providing an alternative proof of a result of Achter and Pries in [AP]. For p > 3 the connectivity of the p-gonal loci becomes more involved. We show that for p ≥ 11 prime and genus g = p − 1 there are one-dimensional strata of cyclic p-gonal surfaces that are completely isolated in the completion B g of the branch locus in M g .