On the triviality of a family of linear hyperplanes

TL;DR

This paper generalizes conditions under which certain higher-dimensional affine varieties defined by linear hyperplanes are isomorphic to affine space, providing new counterexamples to the Zariski Cancellation Problem in positive characteristic.

Contribution

It extends previous results to higher dimensions, characterizing when these varieties are affine spaces and analyzing their automorphisms and isomorphism classes.

Findings

Identifies conditions for higher-dimensional varieties to be affine spaces

Provides a family of counterexamples to the Zariski Cancellation Problem

Describes automorphisms and isomorphism classes of the constructed varieties

Abstract

Let $k$ be a field, $m$ a positive integer, $\mathbb{V}$ an affine subvariety of $\mathbb{A}^{m+3}$ defined by a linear relation of the form $x_{1}^{r_{1}}\cdots x_{m}^{r_{m}}y=F(x_{1}, \ldots , x_{m},z,t)$, $A$ the coordinate ring of $\mathbb{V}$ and $G= X_1^{r_1}\cdots X_m^{r_m}Y-F(X_1, \dots, X_m,Z,T)$. In \cite{com}, the second author had studied the case $m=1$ and had obtained several necessary and sufficient conditions for $\mathbb{V}$ to be isomorphic to the affine 3-space and $G$ to be a coordinate in $k[X_1, Y,Z,T]$. In this paper, we study the general higher-dimensional variety $\mathbb{V}$ for each $m \geqslant 1$ and obtain analogous conditions for $\mathbb{V}$ to be isomorphic to $\mathbb{A}^{m+2}$ and $G$ to be a coordinate in $k[X_1, \dots, X_m, Y,Z,T]$, under a certain hypothesis on $F$. Our main theorem immediately yields a family of higher-dimensional linear hyperplanes for which the Abhyankar-Sathaye Conjecture holds. We also describe the isomorphism classes and automorphisms of integral domains of the type $A$ under certain conditions. These results show that for each $d \geqslant 3$, there is a family of infinitely many pairwise non-isomorphic rings which are counterexamples to the Zariski Cancellation Problem for dimension $d$ in positive characteristic.