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

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 $g\ge \frac{3(p-1)}{2}$,that are multiple of $\frac{p-1}{2}$. 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.