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 $n \geq d+1$ and $k \geq 2$ is relatively prime to $p$, if there is a Galois branched covering $\pi\colon X\to {\mathbb P}{K}^{d}$, with deck group ${\mathbb Z}_k^n\cong H<\rm{Aut}(X)$, 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.