On isolated strata of p-gonal Riemann surfaces in the branch locus of moduli spaces

TL;DR

This paper investigates isolated strata of p-gonal Riemann surfaces within the branch locus of moduli spaces, generalizing previous results for specific cases and identifying conditions for their existence.

Contribution

It extends prior work by characterizing isolated p-gonal strata in the branch locus for genera satisfying certain divisibility conditions, broadening understanding of the moduli space's structure.

Findings

Identifies isolated p-gonal strata in the branch locus for genera g ≥ 3(p-1)/2

Generalizes previous results from pentagonal to p-gonal cases

Provides conditions under which these isolated strata exist

Abstract

The moduli space $\mathcal{M}_{g}$ of compact Riemann surfaces of genus $g$ has orbifold structure, and the set of singular points of such orbifold is the \textit{branch locus} $\mathcal{B}_{g}$. Given a prime number $p \ge 7$, $\mathcal{B}_{g}$ contains isolated strata consisting of $p$-gonal Riemann surfaces for genera $g \ge {3(p-1)\over 2}$,that are multiple of ${p-1 \over 2}$. This is a generalization of the results obtained in \cite{[BCI1]} for pentagonal Riemann surfaces, and the results of \cite{[K]} and \cite{[CI3]} for zero- and one-dimensional isolated strata in the branch locus.