On generalized superelliptic Riemann surfaces

A closed Riemann surface X, of genus g ≥ 2, is called a generalized superelliptic curve of level n ≥ 2 if it admits an order n conformal automorphism τ so that X/hτihas genus zero and τ is central in Aut(X); the cyclic group H = hτiis called a generalized superelliptic group of level n for X. These Riemann surfaces are natural generalizations of hyperelliptic Riemann surfaces. We provide an algebraic curve description of these Riemann surfaces in terms of their groups of automorphisms. Also, we observe that the generalized superelliptic group H of level n is unique, with the exception of a very particular family of exceptional generalized superelliptic Riemann surfaces for n even. In particular, the uniqueness holds if either: (i) n is odd or (ii) the quotient X/H has all its cone points of order n. In the non-exceptional case, we use this uniqueness property to observe that the corresponding curves are definable over their fields of moduli if Aut(X)/H is neither trivial or cyclic.