Abstract
For an affine algebraic variety X, we study the subgroup Autalg(X) of the group of regular automorphisms Aut(X) of X generated by all the connected algebraic subgroups. We prove that Autalg(X) is nested, i.e., is a direct limit of algebraic subgroups of Aut(X), if and only if all the 𝔾a-actions on X commute. Moreover, we describe the structure of such a group Autalg(X).
Similar content being viewed by others
References
Arzhantsev, I., Gaifullin, S.: The automorphism group of a rigid affine variety. Math. Nachrichten. 290(5-6), 662–671 (2017)
J. Blanc, A. Dubouloz, Affine surfaces with a huge group of automorphisms, Int. Math. Res. Notices 2015 (2015), no. 2, 422-459.
Cohen, I.S., On the structure and ideal theory of complete local rings. Trans. Amer. Math. Soc. 59(1), 54–106 (1946)
H. Flenner, M. Zaidenberg, On the uniqueness of ℂ*-actions on affine surfaces, in: Affine Algebraic Geometry, Contemporary Math., Vol. 369, Amer. Math. Soc. Providence, RI, 2005, pp. 97-111.
J.-P. Furter, H. Kraft, On the geometry of the automorphism groups of affine varieties, ar**v:1809.04175 (2018).
G. Freudenburg, Algebraic Theory of Locally Nilpotent Derivations, Encyclopaedia of Mathematical Sciences, Vol. 136, Subseries Invariant Theory and Algebraic Transformation Groups, Vol. VII, Springer-Verlag, Berlin, 2006.
Gurjar, R.V., Masuda, K., Miyanishi, M.: 𝔸1-fibrations on affine threefolds. J. Pure Appl. Algebra. 216, 296–313 (2012)
Kishimoto, T., Prokhorov, Y., Zaidenberg, M.: Group actions on affine cones. Transform. Groups. 18, 1137–1153 (2013)
S. Kovalenko, A. Perepechko, M. Zaidenberg, On automorphism groups of affine surfaces, in: Algebraic Varieties and Automorphism Groups, Advanced Studies in Pure Mathematics, Vol. 75, Math. Soc. Japan, Tokyo, 2017, pp. 207-286.
Kraft, H.: Automorphism groups of affine varieties and a characterization of affine n-Space. Trans. Moscow Math. Soc. 78, 171–186 (2017)
S. Kumar, Kac-Moody Groups, Their Flag Varieties and Representation Theory, Progress in Mathematics, Vol. 204, Birkhäuser Boston, Boston, MA, 2002.
V. Lin, M. Zaidenberg, Configuration spaces of the affine line and their auto-morphism groups, in: Automorphisms in Birational and Affine Geometry, Levico Terme, Italy, October 2012, Springer Proceedings in Mathematics and Statistics, Vol. 79, Springer, Cham, 2014, 431-467.
Matsusaka, T.: Polarized varieties, fields of moduli and generalized Kummer varieties of polarized varieties. Amer. J. Math. 80, 45–82 (1958)
Miyanishi, M., Ga-actions and completions. Journal of Algebra. 319, 2845–2854 (2008)
A. Perepechko, M. Zaidenberg, Automorphism groups of affine ML2-surfaces: dual graphs and Thompson groups, in preparation.
I. R. Shafarevich, On some infinite-dimensional groups, Rend. Mat. Appl. (5) 25 (1966), no. 1-2, 208-212.
The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu (2016).
Winkelmann, J.: Invariant rings and quasiaffine quotients. Math. Z. 244, 163–174 (2003)
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.
Alexander Perepechko is supported by the Russian Foundation for Sciences (project no. 18-71-00153).
Andriy Regeta is partially supported by SNF (Schweizerischer Nationalfonds), project number P2BSP2 165390.
Rights and permissions
About this article
Cite this article
PEREPECHKO, A., REGETA, A. WHEN IS THE AUTOMORPHISM GROUP OF AN AFFINE VARIETY NESTED?. Transformation Groups 28, 401–412 (2023). https://doi.org/10.1007/s00031-022-09711-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-022-09711-1