On the family of affine threefolds a(x)y = F(x,z,t)

Abstract

In recent decades, linear affine threefolds have enabled researchers to solve some of the challenging problems on affine spaces. Koras-Russell threefolds, especially the Russell Cubic over and Asanuma threefolds over a field of positive characteristic, are striking examples of such linear threefolds. In this paper, we apply tools from K-theory and theory of -actions to linear threefolds of the form , over an arbitrary field k.We give some equivalent conditions for G to be a hyperplane (i.e., ) in the following cases: (i) k is a field of characteristic zero (ii) k is an arbitrary field and has only multiple roots. We also establish the Abhyankar-Sathaye Conjecture affirmatively in these cases.