On the family of affine threefolds x^my=F(x,z,t)

Abstract

Let k be a field and V the affine threefold in A4k defined by xmy=F(x,z,t), m⩾2. In this paper, we show that V≅A3k if and only if f(z,t):=F(0,z,t) is a coordinate of k[z,t]. In particular, when k is a field of positive characteristic and f defines a non-trivial line in the affine plane A2k (we shall call such a V as an Asanuma threefold), then V≆A3k although V×A1k≅A4k, thereby providing a family of counter-examples to Zariski’s cancellation conjecture for the affine 3-space in positive characteristic. Our main result also proves a special case of the embedding conjecture of Abhyankar–Sathaye in arbitrary characteristic.