Generalized Fermat curves

Abstract

A closed Riemann surface S is a generalized Fermat curve of type ( k , n ) if it admits a group of automorphisms H ≅ Z k 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 g k = 1 + k 3 − 2 k 2 , for every integer k ⩾ 3 , admitting exactly three singular fibers, (ii) an injective holomorphic map ψ : C − { 0 , 1 } → M g , where M g 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 p g = q = 0 and covering group equal to Z 5 2 or Z 2 4 .