Skip to main content

Rationality of Moduli Spaces via Invariant Theory

  • Chapter

Part of the book series: Progress in Mathematics ((PM,volume 80))

Abstract

The aim of this paper is to convey some idea, by means of concrete examples, of what it means for a moduli space in algebraic geometry to be rational, and of what this question has to do with group theory.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Beauville, J.-L. Colliot-Thélène, J.-J. Sansuc and Sir Peter Swinner-ton-Dyer, Variétés stablement rationelles non-rationelles, Ann Math. 121 (1985), 283–318.

    Article  MATH  Google Scholar 

  2. F. A. Bogomolov, Holomorphic tensors and vector bundles on projective varieties, Math. USSR Izvestiya 13 (1979), 499–556.

    Article  Google Scholar 

  3. F. A. Bogomolov, Stable rationality of quotient varieties by simply connected groups, Mat. Sbornik 130 (1986), 3–17.

    MathSciNet  Google Scholar 

  4. F. A. Bogomolov and P. I. Katsylo, Rationality of some quotient varieties, Mat. Sbornik 126 (1985), 584–589.

    MathSciNet  Google Scholar 

  5. M. Brion, Sur l’image de l’application moment, Springer LNM 1296 (1986), 177–192.

    MathSciNet  Google Scholar 

  6. M. Chang and Z. Ran, Unirationality of the moduli spaces of curves of genus 11, 13 (and 12), Invent. Math. 76 (1984), 41–54.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. B. Coble, An application of Moore’s cross-ratio group to the solution of the sextic equation, Trans. A.M.S. 12 (1911), 311–325.

    MathSciNet  MATH  Google Scholar 

  8. I. V. Dolgachev, Rationality of fields of invariants, in “Algebraic Geometry,” Bowdoin, 1985. A.M.S., 1987

    Google Scholar 

  9. D. Eisenbud and J. Harris, The Kodaira dimension of the moduli space of urves of genus ≥ 23, Invent. Math. 90 (1987), 359–387.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. H. Grace and W. H. Young, “The algebra of invariants,” Chelsea Publ. Company, New York, 1903.

    MATH  Google Scholar 

  11. J. Harris, On the Kodaira dimension of the moduli space of curves, II. The even-genus case, Invent. Math. 75 (1984).

    Google Scholar 

  12. J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), 23–97.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Kempf, Instability in invariant theory, Ann. of Math. 108 (1978), 299–316.

    Article  MathSciNet  MATH  Google Scholar 

  14. F. Kirwan, “Cohomology of quotients in algebraic and symplectic geometry,” Mathematical Notes 31, Princeton Univ. Press, 1984.

    Google Scholar 

  15. D. Mumford and J. Fogarty, “Geometric Invariant Theory,” 2nd edition, Ergebnisse der Mathematik, Springer, 1985.

    Google Scholar 

  16. L. Ness, A stratification of the null-cone via the moment map, Am. J. Math. 106 (1984), 1281–1330.

    Article  MathSciNet  MATH  Google Scholar 

  17. G. Salmon, “Lessons introductory to the modern higher algebra,” Chelsea Publ. Company, New York.

    Google Scholar 

  18. D. Saltman, Noether’s problem over an algebarically closed field, Invent. Math. 77 (1984), 71–84.

    Article  MathSciNet  MATH  Google Scholar 

  19. D. Saltman, Multiplicative field invariants, J. Alg. 106 (1987), 221–238.

    Article  MathSciNet  MATH  Google Scholar 

  20. N. I. Shepherd-Barron, The rationality of certain spaces associated to trigonal curves, in “Algebraic Geometry,” Bowdoin, 1985. A.M.S. 1987

    Google Scholar 

  21. N. I. Shepherd-Barron, The rationality of some moduli spaces of plane curves, Comp. Math. 67 (1988), 51–88.

    MathSciNet  MATH  Google Scholar 

  22. N. I. Shepherd-Barron, Invariant theory for S 5 and the rationality of M 6, Comp. Math. (to appear).

    Google Scholar 

  23. N. I. Shepherd-Barron, Apolarity and its applications, unpublished.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Birkhäuser Boston, Inc.

About this chapter

Cite this chapter

Shepherd-Barron, N.I. (1989). Rationality of Moduli Spaces via Invariant Theory. In: Kraft, H., Petrie, T., Schwarz, G.W. (eds) Topological Methods in Algebraic Transformation Groups. Progress in Mathematics, vol 80. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3702-0_10

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-3702-0_10

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4612-8219-8

  • Online ISBN: 978-1-4612-3702-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics