2017/10/03 by Hidalgo, Ruben A.
#14H05 #14H30 #14H45 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1710.01349
A non-singular complete irreducible algebraic curve Fk,n, defined over an algebraically closed field K, is called a generalized Fermat curve of type (k,n), where n, k ≥ 2 are integers and k is relatively prime to the characteristic p of K, if it admits a group H ≅ \mathbb Zkn of automorphisms such that Fk,n/H is isomorphic to \mathbb PK1 and it has exactly (n+1) cone points, each one of order k. By the Riemann-Hurwitz-Hasse formula, Fk,n has genus at least one if and only if (k-1)(n-1) >1. In such a situation, we construct a basis, called an standard basis, of its space H1,0(Fk,n) of regular forms, containing a subset of cardinality n+1 that provides an embedding of Fk,n into \mathbb PKn whose image is the fiber product of (n-1) classical Fermat curves of degree k. For p=2, we obtain a lower bound (which is sharp for n=2,3) for the dimension of the space of the exact one-forms, that is, the kernel of the Cartier operator. Also, we done this for p=3, k=2 and n=4.