1 results
On the family of affine threefolds
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}x^m y= F(x, z, t)$
- Part of
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- Journal:
- Compositio Mathematica / Volume 150 / Issue 6 / June 2014
- Published online by Cambridge University Press:
- 09 June 2014, pp. 979-998
- Print publication:
- June 2014
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- Article
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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.
