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

Gupta, Neena (2014) On the family of affine threefolds xmy=F(x,z,t) Compositio Mathematica, 150 (6). pp. 979-998. ISSN 0010-437X

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Official URL: http://doi.org/10.1112/S0010437X13007793

Related URL: http://dx.doi.org/10.1112/S0010437X13007793


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.

Item Type:Article
Source:Copyright of this article belongs to London Mathematical Society.
Keywords:Polynomial Algebra; Graded Ring; Ga-action;Derksen Invariant; Makar-Limanov Invariant; Cancellation Problem; Embedding Problem; A2-Fibration, Localization Theorem Of K-Theory.
ID Code:121186
Deposited On:12 Jul 2021 12:37
Last Modified:12 Jul 2021 12:37

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