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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: General commutative ring theory Affine geometry Ring extensions and related topics

Published online by Cambridge University Press:  09 June 2014

Neena Gupta
Neena Gupta*
Affiliation:
Stat-Math Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700108, India email neenag@isical.ac.in, rnanina@gmail.com
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Abstract

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.

Keywords

polynomial algebragraded ringGa-actionDerksen invariantMakar-Limanov invariantcancellation problemembedding problemA2-fibrationlocalization theorem of K-theory

MSC classification

Secondary: 14R10: Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) 14R25: Affine fibrations 13B25: Polynomials over commutative rings 13A50: Actions of groups on commutative rings; invariant theory

Information

Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 6 , June 2014 , pp. 979 - 998
Copyright
© The Author 2014 

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