Planes of the form $b(X,Y)Z^n-a(X,Y)$ over a DVR

Abstract

In this paper we extend an epimorphism theorem of D. Wright to the case of discrete valuation rings. We will show that if $(R,t)$ is a discrete valuation ring, $n\ge 2$ is an integer not divisible by the characteristic of the residue field $R/tR$, and $g\in R[X,Y,Z]$ is a polynomial of the form $g=b(X,Y)Z^n-a(X,Y)$ such that $R[X,Y,Z]/(g)$ is a polynomial algebra in two variables, then $g$ and $Z$ form a pair of variables in $R[X,Y,Z]$. We will also show that the result holds over any Noetherian domain containing $\mathbb{Q}$.

16 pages