Generalized Fermat curves

A closed Riemann surface S is a generalized Fermat curve of type (k,n) if it admits a group of automorphisms such that the quotient O=S/H is an orbifold with signature (0,n+1;k,…,k), that is, the Riemann sphere with (n+1) conical points, all of same order k. The group H is called a generalized Fermat group of type (k,n) and the pair (S,H) is called a generalized Fermat pair of type (k,n). We study some of the properties of generalized Fermat curves and, in particular, we provide simple algebraic curve realization of a generalized Fermat pair (S,H) in a lower-dimensional projective space than the usual canonical curve of S so that the normalizer of H in Aut(S) is still linear. We (partially) study the problem of the uniqueness of a generalized Fermat group on a fixed Riemann surface. It is noted that the moduli space of generalized Fermat curves of type (p,n), where p is a prime, is isomorphic to the moduli space of orbifolds of signature (0,n+1;p,…,p). Some applications are: (i) an example of a pencil consisting of only non-hyperelliptic Riemann surfaces of genus gk=1+k3−2k2, for every integer k⩾3, admitting exactly three singular fibers, (ii) an injective holomorphic map ψ:C−{0,1}→Mg, where Mg is the moduli space of genus g⩾2 (for infinitely many values of g), and (iii) a description of all complex surfaces isogenous to a product with invariants pg=q=0 and covering group equal to or .