Uniqueness of Generalized Fermat Groups in positive characteristic

Abstract

Let $X\subset {\mathbb{P}}_K^m$ be a smooth irreducible projective algebraic variety of dimension $d$, defined over an algebraically closed field $K$ of characteristic $p>0$. We say that $X$ is a generalized Fermat variety of type $(d;k,n)$, where $n\ge d+1$ and $k\ge 2$ is relatively prime to $p$, if there is a Galois branched covering $\pi :X\to {\mathbb{P}}_K^d$, with deck group ${\mathbb{Z}}_k^n\cong H<\mathrm{Aut}(\mathrm{X})$, whose branch divisor consists of $n+1$ hyperplanes in general position (each one of branch order $k$). In this case, the group $H$ is called a generalized Fermat group of type $(d;k,n)$. We prove that, if $k-1$ is not a power of $p$ and either (i) $p=2$ or (ii) $p>2$ and $(d;k,n)\notin \{(2;2,5),(2;4,3)\}$, then a generalized Fermat variety of type $(d;k,n)$ has a unique generalized Fermat group of that type.