Skip to main content
SpringerLink
Log in

Limit points and additive group actions

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

We show that an effective action of the one-dimensional torus \({\mathbb G}_m\) on a normal affine algebraic variety X can be extended to an effective action of a semi-direct product \({\mathbb G}_m\rightthreetimes {\mathbb G}_a\) with the same general orbit closures if and only if there is a divisor D on X that consists of \({\mathbb G}_m\)-fixed points. This result is applied to the study of orbits of the automorphism group \({{\,\mathrm{Aut}\,}}(X)\) on X.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Altmann, K., Hausen, J.: Polyhedral divisors and algebraic torus actions. Math. Ann. 334(3), 557–607 (2006)

    Article  MathSciNet  Google Scholar 

  2. Arzhantsev, I.: On actions of reductive groups with one-parameter family of spherical orbits. Sb. Math. 188(5), 639–655 (1997)

    Article  MathSciNet  Google Scholar 

  3. Arzhantsev, I.: On rigidity of factorial trinomial hypersurfaces. Int. J. Algebra Comput. 26(5), 1061–1070 (2016)

    Article  MathSciNet  Google Scholar 

  4. Arzhantsev, I., Bazhov, I.: On orbits of the automorphism group on an affine toric variety. Cent. Eur. J. Math. 11(10), 1713–1724 (2013)

    MathSciNet  Google Scholar 

  5. Arzhantsev, I., Flenner, H., Kaliman, S., Kutzschebauch, F., Zaidenberg, M.: Flexible varieties and automorphism groups. Duke Math. J. 162(4), 767–823 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Arzhantsev, I., Kuyumzhiyan, K., Zaidenberg, M.: Flag varieties, toric varieties, and suspensions: three instances of infinite transitivity. Sb. Math. 203(7), 923–949 (2012)

    Article  MathSciNet  Google Scholar 

  8. Bialynicki-Birula, A.: Some theorems on actions of algebraic groups. Ann. Math. 98, 480–497 (1973)

    Article  MathSciNet  Google Scholar 

  9. Boldyrev, I., Gaifullin, S.: Automorphisms of Nonnormal Toric Varieties. Math. Notes (2021), in print; arXiv:2012.03346

  10. Cox, D., Little, J., Schenck, H.: Toric Varieties. Graduate Studies in Mathematics, vol. 124. AMS, Providence (2011)

  11. Flenner, H., Zaidenberg, M.: On the uniqueness of \(\mathbb{C}^*\)-actions on affine surfaces. In: Affine Algebraic Geometry, Contemporary Mathematics, vol. 369, pp. 97–111. American Mathematical Society, Providence (2005)

  12. Flenner, H., Zaidenberg, M.: Locally nilpotent derivations on affine surfaces with a \({\mathbb{C}}^*\)-action. Osaka J. Math. 42(4), 931–974 (2005)

    MathSciNet  Google Scholar 

  13. Freudenburg, G.: Algebraic Theory of Locally Nilpotent Derivations, 2nd edn. Encyclopaedia of Mathematical Sciences, vol. 136. Springer, Berlin (2017)

  14. Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993)

  15. Gaifullin, S., Shafarevich, A.: Flexibility of normal affine horospherical varieties. Proc. Am. Math. Soc. 147(8), 3317–3330 (2019)

    Article  MathSciNet  Google Scholar 

  16. Hausen, J., Wrobel, M.: Non-complete rational T-varieties of complexity one. Math. Nachr. 290(5–6), 815–826 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Liendo, A.: \({\mathbb{G}}\)-actions of fiber type on affine \(\mathbb{T}\)-varieties. J. Algebra 324, 3653–3665 (2010)

    Article  MathSciNet  Google Scholar 

  19. Oda, T.: Convex Bodies and Algebraic Geometry: An Introduction to Toric Varieties. A Series of Modern Surveys in Mathematics, vol. 15. Springer, Berlin (1988)

  20. Popov, V., Vinberg, E.: Invariant Theory. Algebraic Geometry IV, Encyclopaedia of Mathematics Sciences, vol. 55, pp. 123–284. Springer, Berlin (1994)

Download references

Acknowledgements

The author is grateful to Alvaro Liendo for a fruitful discussion, and to Sergey Gaifullin, Michail Zaidenberg, and two anonymous reviewers for many useful comments, suggestions and references to related results.

Author information

Authors and Affiliations

  1. Faculty of Computer Science, HSE University, Pokrovsky Boulevard 11, Moscow, Russia, 109028

    Ivan Arzhantsev

Authors
  1. Ivan Arzhantsev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ivan Arzhantsev.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research was supported by Russian Science Foundation, Grant 19-11-00056.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arzhantsev, I. Limit points and additive group actions. Ricerche mat 73, 715–724 (2024). https://doi.org/10.1007/s11587-021-00630-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-021-00630-z

Keywords

Mathematics Subject Classification

Search

Navigation