Abstract
A special Danielewski surface is an affine surface which is the total space of a principal \(({{\mathbb {C}}},+)\)-bundle over an affine line with a multiple origin. Using a fiber product trick introduced by Danielewski, it is known that cylinders over two such surfaces are always isomorphic provided that both bases have the same number of origins. The goal of this note is to give an explicit method to find isomorphisms between cylinders over special Danielewski surfaces. The method is based on the construction of appropriate locally nilpotent derivations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alhajjar, B.: Locally Nilpotent Derivations of Integral Domains, Ph.D. Thesis, Université de Bourgogne (2015)
Danielewski, W.: On the cancellation problem and automorphism groups of affine algebraic varieties, preprint (1989)
Dubouloz, A.: Danielewski–Fieseler surfaces. Transform. Groups 10(2), 139–162 (2005)
Dubouloz, A., Poloni, P.-M.: On a class of Danielewski surfaces in affine 3-space. J. Algebra 321(7), 1797–1812 (2009)
Fieseler, K.-H.: On complex affine surfaces with \({ C}^+\)-action. Comment. Math. Helv. 69(1), 5–27 (1994)
Freudenburg, G.: Algebraic Theory of Locally Nilpotent Derivations, Encyclopaedia of Mathematical Sciences, vol. 136. Invariant Theory and Algebraic Transformation Groups, VII, 2nd edn. Spinger, Berlin (2017)
Gupta, N., Sen, S.: On double Danielewski surfaces and the cancellation problem. J. Algebra 533, 25–43 (2019)
Shpilrain, V., Yu, J.-T.: Affine varieties with equivalent cylinders. J. Algebra 251(1), 295–307 (2002)
Acknowledgements
Part of this work was done during the first joint meeting Brazil-France in Mathematics. The second-named author gratefully acknowledges financial support from the Réseau Franco-Brésilien de Mathématiques (RFBM).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Lucy Moser-Jauslin acknowledges support from the French “Investissements d’Avenir” program, project ISITE-BFC (contract ANR-lS-IDEX-OOOB) and from the EIPHI Graduate School (contract ANR-17-EURE-0002). Pierre-Marie Poloni acknowledges support from the Swiss National Science Foundation Grant “Curves in the spaces” 200021-169508.
Rights and permissions
About this article
Cite this article
Moser-Jauslin, L., Poloni, PM. Isomorphisms between cylinders over Danielewski surfaces. Beitr Algebra Geom 62, 755–771 (2021). https://doi.org/10.1007/s13366-020-00548-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13366-020-00548-x