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.
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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.
Ferrand, D., Un foncteur norme,Bull. Soc. Math. France 126 (1998), 1–49.
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.
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.
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.
Laksov, D., Pitteloud, Y. andSkjelnes, R. M., Notes on flatness and the Quot functor on rings,Comm. Algebra 28 (2000), 5613–5627.
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.
Laksov, D., Svensson, L. andThorup, A., The spectral mapping theorem, norms on rings, and resultants,Enseign. Math. 46 (2000), 349–358.
Roby, N., Lois polynômes et lois formelles en théorie des modules,Ann. Sci. École Norm. Sup. 80 (1963), 213–248.
Skjelnes, R. M. andWalter, C., Infinite intersections of open subschemes and the Hilbert scheme of points, In preparation.
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Skjelnes, R.M. Resultants and the Hilbert scheme of points on the line. Ark. Mat. 40, 189–200 (2002). https://doi.org/10.1007/BF02384509
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DOI: https://doi.org/10.1007/BF02384509