Advertisement

Representations of SL2 and the distribution of points in ℙn

  • J. Kuttler
  • N. Wallach
Chapter
Part of the Progress in Mathematics book series (PM, volume 220)

Abstract

In 1958 Nagata [5] gave an ingenious argument that demonstrated the existence of counterexamples to Hilbert’s Fourteenth Problem. Recall that the original problem is the following: Let K = F(x 1, x 2,..., x n ) be the function field of affine n-space V = F n over an algebraically closed field F, and suppose LK is any subfield. Then the question is: Is A = LF[x 1 , x 2 ,..., xn] a finitely generated F-algebra. In most cases of interest, L is the field of invariants of an algebraic group G acting linearly on V, and A becomes the ring of invariant regular functions on V. Certainly, if G is reductive, the answer is yes, a result due to Hilbert himself for char(F) = 0, but much more subtle in positive characteristic (see []). More generally for G reductive O(X) G is a finitely generated F-algebra for any affine variety X over F on which G acts.

Keywords

Generic Point Algebraic Group Transformation Rule Formal Power Series Hilbert Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. Alexander, Singularités imposables en position générale à une hypersurface projective, Compositio Math. 68 (1988), 305–354.MathSciNetMATHGoogle Scholar
  2. [2]
    J. Alexander, A. Hirschowitz, La méthode d’Horace éclatée: application à l’interpolation en degré quatre, Invent. Math. 107 (1992), 585–602.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    C. Ciliberto, R. Miranda, Linear systems of plane curves with base points of equal multiplicity, Trans. Amer. Math. Soc. 352, no. 9 (2000), 4037–4050.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    D. Daigle, G. Freudenburg, A counterexample to Hilbert’s fourteenth problem in dimension 5, J. Algebra 221, no. 2 (1999), 528–535.MathSciNetMATHGoogle Scholar
  5. [5]
    M. Nagata, On the 14-th problem of Hilbert, American J. Math. 81 (1959), 766–772.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    M. Nagata, On rational surfaces. I. Irreducible curves of arithmetic genus 0 or 1, Mem. Coli. Sci. Univ. Kyoto Ser. A Math. 32 (1960), 351–370.MathSciNetMATHGoogle Scholar
  7. [7]
    M. Nagata, On rational surfaces. II, Mem. Coli. Sci. Univ. Kyoto Ser. A 33 (1960/1961), 271–293.MathSciNetMATHGoogle Scholar
  8. [8]
    R. Steinberg, Nagata’s example, Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., 9, Cambridge Univ. Press, Cambridge, 1997, pp. 375–384.Google Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • J. Kuttler
    • 1
  • N. Wallach
    • 2
  1. 1.Matematisches InstitutUniversität BaselBaselSwitzerland
  2. 2.Department of MathematicsUniversity of CaliforniaSan Diego La JollaUSA

Personalised recommendations