Smooth quotients of generalized Fermat curves

Abstract

A closed Riemann surface S is called a generalized Fermat curve of type (p, n), where n, p ≥ 2 are integers such that (p − 1)(n − 1) > 2, if it admits a group H Z n p of conformal automorphisms with quotient orbifold S /H of genus zero with exactly n + 1 cone points, each one of order p; in this case H is called a generalized Fermat group of type (p, n). In this case, it is known that S is non-hyperelliptic and that H is its unique generalized Fermat group of type (p, n). Also, explicit equations for them, as a fiber product of classical Fermat curves of degree p, are known. For p a prime integer, we describe those subgroups K of H acting freely on S , together with algebraic equations for S /K, and determine those K such that S /K is hyperelliptic.