2013/05/01 by Antonio F. Costa, Costa, Antonio F., Milagros Izquierdo +3
Mathematics · #14H15 #30F10 #30F60 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #math.AG #math.GT #msc:14H15 #msc:30F10 #msc:30F60
paper · pdf · doi:10.48550/arxiv.1305.0284
openalex publication_date 2013/05/01 · arxiv created 2013/05/22 · arxiv updated 2013/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.