On the family of affine threefolds

Let $k$ be a field and $\mathbb{V}$ the affine threefold in $\mathbb{A}^4_k$ defined by $x^m y=F(x, z, t)$ , $m \ge 2$ . In this paper, we show that $\mathbb{V} \cong \mathbb{A}^3_k$ 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 $\mathbb{A}^2_k$ (we shall call such a $\mathbb{V}$ as an Asanuma threefold), then $\mathbb{V}\ncong \mathbb{A}^3_k$ although $\mathbb{V} \times \mathbb{A}^1_k \cong \mathbb{A}^4_k$ , 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.