On triangular (Dn)-actions on cyclic p-gonal Riemann surfaces

Abstract

A compact Riemann surface $X$ of genus $g\gt 1$ which has a conformal automorphism $\rho$ of prime order $p$ such that the orbit space $X/ \langle \rho \rangle$ is the Riemann sphere is called cyclic $p$-gonal. Exceptional points in the moduli space $\mathcal{M}_g$ of compact Riemann surfaces of genus $g$ are unique surface classes whose full group of conformal automorphisms acts with a triangular signature. We study symmetries of exceptional points in the cyclic $p$-gonal locus in $\mathcal{M}_g$ for which $\text{Aut}(X)/ \langle \rho \rangle$ is a dihedral group $D_n$.