Uniqueness of Generalized Fermat Groups in positive characteristic

TL;DR

This paper proves the uniqueness of generalized Fermat groups for certain algebraic varieties in positive characteristic, under specific conditions on the parameters and the characteristic of the field.

Contribution

It establishes conditions under which a generalized Fermat variety has a unique associated Fermat group, extending understanding of automorphism groups in positive characteristic.

Findings

Uniqueness holds if $k-1$ is not a power of $p$ and either $p=2$ or specific parameter exceptions.

The result applies to varieties with Galois coverings branched over hyperplanes in general position.

Provides criteria for the automorphism group structure of generalized Fermat varieties.

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.