Skip to main content

Quotients of Algebraic Group Actions

  • Chapter
Automorphisms of Affine Spaces

Abstract

The main goal of this paper is to investigate whether an algebraic group action on a variety has a nice algebraic quotient. A classical book in this respect is “Geometric Invariant Theory” of Mumford (see [15]). The major part of the book only concerns reductive groups. More recently some work has been done to do similar things for general algebraic groups (see [8], [5], [6], [7] and [4]).

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Borel, Linear Algebraic Groups, W.A. Benjamin, Inc., 1969.

    MATH  Google Scholar 

  2. M. Demazure and P. Gabriel, Groupes algébriques, Masson-North-Holland, Paris Amsterdam, 1970.

    MATH  Google Scholar 

  3. J.K. Deveney and D.R. Finston, A proper G a action on (5 which is not locally trivial,to appear in Proceedings of the American Mathematical Society.

    Google Scholar 

  4. J. Dixmier and M. Raynaud, Sur le quotient d’une variété algébrique par un groupe algébrique, Adv. Math. Suppl. Stud. 7A (1981), 327–344.

    MathSciNet  Google Scholar 

  5. A. Fauntleroy, Geometric Invariant Theory for General Algebraic Groups, Michigan Math. J. 39 (1983), 131–142.

    MathSciNet  Google Scholar 

  6. A. Fauntleroy, Geometric Invariant Theory for General Algebraic Groups II (char k arbitrary), Compositio Math. 68 (1988), 23–29.

    MathSciNet  MATH  Google Scholar 

  7. A. Fauntleroy, Unipotent group actions: Corrections, J. of Pure and Applied Algebra 50 (1988), 209–210.

    Article  MathSciNet  MATH  Google Scholar 

  8. G.-M. Greuel and G. Pfister, Geometric quotients of unipotent group actions, Proc. London Math. Soc. 67 (1993), 75–105.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977.

    Google Scholar 

  10. H. Hironaka, An example of a non-Kahlerian deformation, Annals of Mathematics 75 (1962), 190.

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Kraft, Geometrische Methoden in der Invariantentheorie, Vieweg-Verlag, 1984.

    Google Scholar 

  12. H. Kraft and C. Procesi, Classi coniugate in Gl(n, (U), Rend. Sem. Mat. Univ. Padova 59 (1978), 209–222.

    MathSciNet  MATH  Google Scholar 

  13. H. Kraft and C. Procesi, Closures of Conjugacy Classes of Matrices are Normal, Inventiones math. 53 (1979), 227–247.

    Article  MathSciNet  MATH  Google Scholar 

  14. D. Luna, Slices étales, Bull. Soc. Math. France Mémoire 33 (1973), 81–105.

    MATH  Google Scholar 

  15. D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, Springer-Verlag, 1994.

    Google Scholar 

  16. M. Nagata, Note on orbit spaces, J. Math. Kyoto Univ. 3 (1964), 369.

    MathSciNet  MATH  Google Scholar 

  17. M. Rosenlicht, A remark on Quotient Spaces, An. Acad. Brasil. Ciênc. 35 (1963), 487–489.

    MathSciNet  MATH  Google Scholar 

  18. C.S. Seshadri, Quotient Spaces modulo Reductive Algebraic Groups, Annals of Mathematics 95 (1972), 511–556.

    Article  MathSciNet  MATH  Google Scholar 

  19. P. Slodowy, Der Scheibensatz für algebraische Transformationsgruppen, Algebraische Transformationsgruppen und Invariantentheorie (H. Kraft, P. Slodowy, and T.A. Springer, eds. ), Birkhäuser Verlag, 1989.

    Google Scholar 

  20. T.A. Springer, Linear Algebraic Groups, Birkhäuser Verlag, 1981.

    Google Scholar 

  21. T.A. Springer, Aktionen Reduktiver Gruppen auf Varietäten, Algebraische Transformationsgruppen und Invariantentheorie (H. Kraft, P. Slodowy, and T.A. Springer, eds. ), Birkhäuser Verlag, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Derksen, H. (1995). Quotients of Algebraic Group Actions. In: van den Essen, A. (eds) Automorphisms of Affine Spaces. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8555-2_13

Download citation

  • DOI: https://doi.org/10.1007/978-94-015-8555-2_13

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4566-9

  • Online ISBN: 978-94-015-8555-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics