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Arkiv för Matematik

, Volume 40, Issue 1, pp 189–200 | Cite as

Resultants and the Hilbert scheme of points on the line

  • Roy Mikael Skjelnes
Article

Abstract

We present an elementary and concrete description of the Hilbert scheme of points on the spectrum of fraction ringsk[X] U of the one-variable polynomial ring over a commutative ringk. Our description is based on the computation of the resultant of polynomials ink[X]. The present paper generalizes the results of Laksov-Skjelnes [7], where the Hilbert scheme on spectrum of the local ring of a point was described.

Keywords

Local Ring Polynomial Ring Hilbert Scheme Concrete Description 
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References

  1. 1.
    Deligne, P., Cohomologie à supports propres, inThéorie des topos et cohomologie étale des schémas. Tome 3,Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Lecture Notes in Math.305, pp. 250–462, Springer-Verlag, Berlin-Heidelberg, 1973.Google Scholar
  2. 2.
    Ferrand, D., Un foncteur norme,Bull. Soc. Math. France 126 (1998), 1–49.MATHMathSciNetGoogle Scholar
  3. 3.
    Grothendieck, A., Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert, inSéminaire Bourbaki, Vol.6, Exp.221, pp. 249–276, Soc. Math. France, Paris, 1995.Google Scholar
  4. 4.
    Iversen, B.,Linear Determinants with Applications to the Picard Scheme of a Family of Algebraic Curves, Lecture Notes in Math.174, Springer-Verlag, Berlin-Heidelberg, 1970.Google Scholar
  5. 5.
    Kleiman, S. L., Multiple-point formulas. II. The Hilbert scheme, inEnumerative Geometry (Sitges, 1987) (Xambó-Descamps, S., ed.), Lecture Notes in Math.1436, pp. 101–138, Springer-Verlag, Berlin-Heidelberg, 1990.Google Scholar
  6. 6.
    Laksov, D., Pitteloud, Y. andSkjelnes, R. M., Notes on flatness and the Quot functor on rings,Comm. Algebra 28 (2000), 5613–5627.MathSciNetGoogle Scholar
  7. 7.
    Laksov, D. andSkjelnes, R. M., The Hilbert scheme parameterizing finite length subschemes of the line with support at the origin,Compositio Math. 126 (2001), 323–334.CrossRefMathSciNetGoogle Scholar
  8. 8.
    Laksov, D., Svensson, L. andThorup, A., The spectral mapping theorem, norms on rings, and resultants,Enseign. Math. 46 (2000), 349–358.MathSciNetGoogle Scholar
  9. 9.
    Roby, N., Lois polynômes et lois formelles en théorie des modules,Ann. Sci. École Norm. Sup. 80 (1963), 213–248.MATHMathSciNetGoogle Scholar
  10. 10.
    Skjelnes, R. M. andWalter, C., Infinite intersections of open subschemes and the Hilbert scheme of points, In preparation.Google Scholar

Copyright information

© Institut Mittag-Leffler 2002

Authors and Affiliations

  • Roy Mikael Skjelnes
    • 1
  1. 1.Department of MathematicsRoyal Institute of TechnologyStockholmSweden

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