Skip to main content
Log in

WHEN IS THE AUTOMORPHISM GROUP OF AN AFFINE VARIETY NESTED?

  • Published:
Transformation Groups Aims and scope Submit manuscript

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).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Arzhantsev, I., Gaifullin, S.: The automorphism group of a rigid affine variety. Math. Nachrichten. 290(5-6), 662–671 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Blanc, A. Dubouloz, Affine surfaces with a huge group of automorphisms, Int. Math. Res. Notices 2015 (2015), no. 2, 422-459.

  3. Cohen, I.S., On the structure and ideal theory of complete local rings. Trans. Amer. Math. Soc. 59(1), 54–106 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Flenner, M. Zaidenberg, On the uniqueness of*-actions on affine surfaces, in: Affine Algebraic Geometry, Contemporary Math., Vol. 369, Amer. Math. Soc. Providence, RI, 2005, pp. 97-111.

  5. J.-P. Furter, H. Kraft, On the geometry of the automorphism groups of affine varieties, arXiv:1809.04175 (2018).

  6. 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.

  7. Gurjar, R.V., Masuda, K., Miyanishi, M.: 𝔸1-fibrations on affine threefolds. J. Pure Appl. Algebra. 216, 296–313 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kishimoto, T., Prokhorov, Y., Zaidenberg, M.: Group actions on affine cones. Transform. Groups. 18, 1137–1153 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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.

  10. Kraft, H.: Automorphism groups of affine varieties and a characterization of affine n-Space. Trans. Moscow Math. Soc. 78, 171–186 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Kumar, Kac-Moody Groups, Their Flag Varieties and Representation Theory, Progress in Mathematics, Vol. 204, Birkhäuser Boston, Boston, MA, 2002.

  12. 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.

  13. Matsusaka, T.: Polarized varieties, fields of moduli and generalized Kummer varieties of polarized varieties. Amer. J. Math. 80, 45–82 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  14. Miyanishi, M., Ga-actions and completions. Journal of Algebra. 319, 2845–2854 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Perepechko, M. Zaidenberg, Automorphism groups of affine ML2-surfaces: dual graphs and Thompson groups, in preparation.

  16. I. R. Shafarevich, On some infinite-dimensional groups, Rend. Mat. Appl. (5) 25 (1966), no. 1-2, 208-212.

  17. The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu (2016).

  18. Winkelmann, J.: Invariant rings and quasiaffine quotients. Math. Z. 244, 163–174 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to ALEXANDER PEREPECHKO.

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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-022-09711-1

Navigation