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

TL;DR

This paper investigates the connectivity of loci of Riemann surfaces with cyclic automorphisms within the moduli space and its compactification, revealing new connectedness properties for trigonal surfaces and isolated strata for higher prime degrees.

Contribution

It proves that for genus g≥5, the set of cyclic trigonal surfaces is connected in the compactified moduli space, and identifies isolated strata for prime p≥11.

Findings

Trigonal locus is connected in the compactification for g≥5.

Existence of an explicit nodal surface connecting all trigonal surfaces.

Higher prime p≥11 loci can be completely isolated in the compactified space.

Abstract

Consider the moduli space $\mathcal{M}_{g}$ of Riemann surfaces of genus $g\geq 2$ and its Deligne-Munford compactification $\bar{\mathcal{M}_{g}}$. 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, we show that for $g\geq 5$ the set of (cyclic) trigonal surfaces is connected in $\bar{\mathcal{M}_{g}}$. 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 $p\geq 11$ prime and genus $g=p-1$ there are one-dimensional strata of cyclic $p$-gonal surfaces that are completely isolated in the completion $\bar{\mathcal{B}_{g}}$ of the branch locus in $\bar{\mathcal{M}_{g}}$.