Affine-ruled varieties without the Laurent cancellation property
type: Item
instance of: scholarly article
title: Affine-ruled varieties without the Laurent cancellation property
zbMATH Open document ID: 1394.14039
author: Adrien Dubouloz
author: Pierre-Marie Poloni
published in: Bulletin of the London Mathematical Society
publication date: 2016-10-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/1509.07803
full work available at URL: DubPol-Torus-SemiInvariants.pdf
Mathematics Subject Classification ID: 14R05
Mathematics Subject Classification ID: 14R25
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 6645454
zbMATH Keywords: affine algebraic variety
zbMATH Keywords: cancellation conjecture
zbMATH Keywords: torus action
MaRDI profile type: Publication
OpenAlex ID: W2274631109
arXiv ID: 1509.07803
cites work: The structure of a Laurent polynomial fibration in \(n\) variables
cites work: Additive group actions on Danielewski varieties and the cancellation problem
cites work: On the cancellation problem for algebraic tori
cites work: The group of units on an affine variety
cites work: Laurent Cancellation for Rings of Transcendence Degree One Over a Field
cites work: On prime Galois coverings of the Riemann sphere
cites work: Q4123440
cites work: Open Algebraic Surfaces
cites work: Affine surfaces containing cylinderlike open sets
cites work: ON THE EQUIVALENCE PROBLEM OF CYCLIC BRANCHED COVERINGS OF THE RIEMANN SPHERE
cites work: A note on the stable equivalence problem
DOI: 10.1112/BLMS/BDW045
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review text: This nicely written paper deals with questions related to the so-called \textbf{Cancellation conjecture:}NEWLINENEWLINELet \(V\) be an affine variety. Suppose \(V\times {\mathbb C}\simeq {\mathbb C}^{n+1}\). Does it follows that \(V\simeq {\mathbb C}^{n}\)?NEWLINENEWLINEFor algebraic closed ground fields of positive characteristic, \textit{N. Gupta} [Invent. Math. 195, No. 1, 279--288 (2014; Zbl 1309.14050)] found a counterexample. The similar question for fields was solved by negative way by Beauville, Colliot-Thelene, Sansuc and Swinnerton-Dyer [\textit{A. Beauville} et al., Ann. Math. (2) 121, 283--318 (1985; Zbl 0589.14042)].NEWLINENEWLINEW. Danielewski constructed examples of non-isomorphical affine varieties \(V_1, V_2\) with isomorphical cylinders \(V_1\times {\mathbb C}\simeq V_1\times{\mathbb C}\). [\textit{W. Danielewski}, ``On a cancellation problem and automorphism groups of affine algebraic varieties'', Preprint (1989)] (Appendix by K. Fieseler).) For a more detailed review of this subject see [\textit{N. Gupta}, Indian J. Pure Appl. Math. 46, No. 6, 865--877 (2015; Zbl 1409.13039)].NEWLINENEWLINEIn the paper under review, the authors describe a method to construct hypersurfaces of the complex affine \(n\)-space with isomorphic \(\mathbb{C}^*\)-cylinders. Among these hypersurfaces, they find new explicit counterexamples to the Laurent cancellation problem, that is, hypersurfaces that are nonisomorphic, although their \(\mathbb{C}^*\)-cylinders are isomorphic as abstract algebraic varieties. They also provide examples of nonisomorphic varieties \(X\) and \(Y\) with isomorphic Cartesian squares \(X\times X\) and \(Y\times Y\).NEWLINENEWLINEThe method of morphism control is based on consideration of semiinvariant hypersurfaces with torus \({\mathbb G}_m\)-actions and construction varieties such that the global functions can be controlled (see paragraph 1.2).NEWLINENEWLINEOn the other hand, non-isomorphic properties are established via consideration of cove…
reviewed by: A. Ya Belov
description: scientific article; zbMATH DE number 6645454