Holomorphic differentials of Generalized Fermat curves

TL;DR

This paper studies the structure of holomorphic differentials on generalized Fermat curves, constructing explicit bases and embeddings, and analyzing the dimension of exact forms in special characteristics.

Contribution

It provides a explicit basis for the space of holomorphic differentials on generalized Fermat curves and describes their embeddings into projective space, including bounds on exact forms in characteristic 2 and 3.

Findings

Constructed a standard basis for holomorphic differentials on $F_{k,n}$.

Embedded $F_{k,n}$ into projective space as a fiber product of Fermat curves.

Derived bounds for the dimension of exact forms in characteristic 2 and 3.

Abstract

A non-singular complete irreducible algebraic curve $F_{k,n}$, defined over an algebraically closed field $K$, is called a generalized Fermat curve of type $(k,n)$, where $n, k \geq 2$ are integers and $k$ is relatively prime to the characteristic $p$ of $K$, if it admits a group $H \cong {\mathbb Z}_{k}^{n}$ of automorphisms such that $F_{k,n}/H$ is isomorphic to ${\mathbb P}_{K}^{1}$ and it has exactly $(n+1)$ cone points, each one of order $k$. By the Riemann-Hurwitz-Hasse formula, $F_{k,n}$ 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 $H^{1,0}(F_{k,n})$ of regular forms, containing a subset of cardinality $n+1$ that provides an embedding of $F_{k,n}$ into ${\mathbb P}_{K}^{n}$ 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 $n=4$.