Connecting p-gonal loci in the compactification of moduli space

الأصل · EN

Consider the moduli space Mg of Riemann surfaces of genus g≥ 2 and its Deligne-Munford compactification Mg. We are interested in the branch locus Bg for g>2, i.e., the subset of Mg consisting of surfaces with automorphisms. It is well-known that the set of hyperelliptic surfaces (the hyperelliptic locus) is connected in Mg but the set of (cyclic) trigonal surfaces is not. By contrast, we show that for g≥ 5 the set of (cyclic) trigonal surfaces is connected in Mg. To do so we exhibit an explicit nodal surface that lies in the completion of every equisymmetric set of 3-gonal Riemann surfaces. 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 Bg of the branch locus in Mg.