Connecting $$p$$ p -gonal loci in the compactification of moduli space

Abstract

Consider the moduli space \(\mathcal {M}_{g}\) of Riemann surfaces of genus \(g\ge 2\) and its Deligne–Mumford compactification \(\overline{\mathcal {M}_{g}}\). We are interested in the branch locus \({\mathcal {B}_{g}}\) for \(g>2\), i.e., the subset of \(\mathcal {M}_{g}\) consisting of surfaces with automorphisms. It is well-known that the set of hyperelliptic surfaces (the hyperelliptic locus) is connected in \(\mathcal {M}_{g}\) but the set of (cyclic) trigonal surfaces is not. By contrast, the set of (cyclic) trigonal surfaces is connected in \(\overline{\mathcal {M}_{g}}\). 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 (Math Ann 338:187–206, 2007). For \(p>3\) the connectivity of the \(p\)-gonal loci becomes more involved. We show that for \(p\ge 11\) prime and genus \(g=p-1\) there are one-dimensional strata of cyclic \(p\)-gonal surfaces that are completely isolated in the completion \(\overline{\mathcal {B}_{g}}\) of the branch locus in \(\overline{\mathcal {M}_{g}}\).