Abstract

A closed Riemann surface S is a generalized Fermat curve of type (k, n) if it admits a group of automorphisms H ? Zk n 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 - 2 k2, for every integer k = 3, admitting exactly three singular fibers, (ii) an injective holomorphic map ? : 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 Z5 2 or Z2 4. © 2009 Elsevier Inc. All rights reserved.