论文标题
关于线性超平面家族的微不足道
On the triviality of a family of linear hyperplanes
论文作者
论文摘要
令$ k $为一个字段，$ m $ a正整数，$ \ mathbb {v} $ $ \ mathbb {a}^{m+3} $的仿射子变量由$ x_ {1}^{r_ {1}}^r_ {1}} \ cdots的线性关系定义x_ {m}^{r_ {m}} y = f（x_ {1}，\ ldots，x_ {m}，z，t）$，$ a $ a $ coortion环$ \ mathbb {v} $和$ g = x_1} x_m，z，t）$。在\ cite {com}中，第二作者研究了$ m = 1 $的案例，并获得了$ \ mathbb {v} $的几个必要条件，以使其与offine 3空间和$ g $同构和$ g $是$ k [x_1，y，y，z，t] $。在本文中，我们研究了每个$ m \ geqslant 1 $的一般高维品种$ \ mathbb {v} $，并获得$ \ mathbb {v} $的类似条件，以等同于$ \ mathbb {mathbb {a} a}^a}^{m+2} $+2} $和$ g $ a coortion ym y，z，t] $，根据$ f $的一定假设。我们的主要定理立即产生一个高维线性超平面的家族，为Abhyankar-Sathaye猜想所构的。我们还描述了在某些条件下类型$ a $的整体域的同构类别和自动形态类别。这些结果表明，对于每个$ d \ geqslant 3 $，都有一个无限的成对非同构环的家族，这是对Zariski取消问题的反例，用于尺寸$ d $的积极特征。
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.
