On the triviality of a family of linear hyperplanes

Let k be a field, m a positive integer, V an affine subvariety of Am+3 defined by a linear relation of the form x1r1⋯xmrmy=F(x1,…,xm,z,t), A the coordinate ring of V and G=X1r1⋯XmrmY−F(X1,…,Xm,Z,T). In [13], the second author had studied the case m=1 and had obtained several necessary and sufficient conditions for V to be isomorphic to the affine 3-space and G to be a coordinate in k[X1,Y,Z,T]. In this paper, we study the general higher-dimensional variety V for each m⩾1 and obtain analogous conditions for V to be isomorphic to Am+2 and G to be a coordinate in k[X1,…,Xm,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⩾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.