Abstract:
Riemann surfaces (and algebraic curves) have been comprehensively studied when they are regular (Galois) coverings of the Riemann sphere, but barely addressed in the general case of being non-regular coverings. In this article we deal with this less known case for a special type of non-regular $p$-coverings ($p$ prime greater than 2), those with monodromy group isomorphic to the dihedral group $D_p$, which we call dihedral $p$-gonal coverings (the particular case $p=3$ has been already studied by A. F. Costa and M. Izquierdo). We have focused on real algebraic curves (those that have a special anticonformal involution) and we study real dihedral $p$-gonal Riemann surfaces. We found out the restrictions, besides Harnack's theorem and generalizations, that apply to the possible topological types of real dihedral $p$-gonal Riemann surfaces.
Key words and phrases:real Riemann surface, real algebraic curve, automorphism, anticonformal automorphism, $p$-gonal morphism, Klein surface.