Affine-ruled varieties without the Laurent cancellation property
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 covering properties and degenerate fibres.
scientific article; zbMATH DE number 6645454
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