Defining algebraic polynomials for cyclic prime covers of the Riemann sphere

Abstract

A compact Riemann surface X is said to be a cyclic p-gonal surface if it admits an automorphism φ of prime order p such that the quotient space X/(φ) has genus 0. It is said to be normal cyclic p-gonal if in addition, the group generated by φ is normal in the full automorphism group of X. In the following notes, we determine a method to find defining polynomial equations for any cyclic p-gonal surface X. If the surface X is assumed to be normal cyclic p-gonal, then all redundancies--equations which are equations for the same surface up to conformal equivalence--are also found.