An intersection number for the

punctual Hilbert scheme of a surface

Authors:
Geir Ellingsrud and Stein Arild Strømme

Journal:
Trans. Amer. Math. Soc. **350** (1998), 2547-2552

MSC (1991):
Primary 14C17, 14C05

DOI:
https://doi.org/10.1090/S0002-9947-98-01972-2

MathSciNet review:
1432198

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Abstract | References | Similar Articles | Additional Information

Abstract: We compute the intersection number between two cycles and of complementary dimensions in the Hilbert scheme parameterizing subschemes of given finite length of a smooth projective surface . The -cycle corresponds to the set of finite closed subschemes the support of which has cardinality 1. The -cycle consists of the closed subschemes the support of which is one given point of the surface. Since is contained in , indirect methods are needed. The intersection number is , answering a question by H. Nakajima.

**1.**L. L. Avramov. Complete intersections and symmetric algebras.*J. of Algebra*, 73:248-263, 1981. MR**83e:13024****2.**J. Briançon. Description de .*Invent. Math.*, 41:45-89, 1977. MR**56:15637****3.**J. Cheah.*On the cohomology of Hilbert schemes of points*, J. Algebraic Geometry 5 (1996), 479-511. MR**97b:14005****4.**G. Ellingsrud. Irreducibility of the punctual Hilbert scheme of a surface. Unpublished.**5.**L. Göttsche. The Betti numbers of the Hilbert scheme of points on a smooth projective surface.*Math. Ann.*, 286:193-207, 1990. MR**91h:14007****6.**M. Nakajima. Heisenberg algebra and Hilbert schemes of points on a projective surface. Duke e-print alg-geom/950712.**7.**A. S. Tikhomirov. On Hilbert schemes and flag varieties of points on algebraic surfaces. Preprint (1992).

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Additional Information

**Geir Ellingsrud**

Affiliation:
Mathematical Institute, University of Oslo, P. O. Box 1053, N–0316 Oslo, Norway

Email:
ellingsr@math.uio.no

**Stein Arild Strømme**

Affiliation:
Mathematical Institute, University of Bergen, Johannes Brunsg. 12, N-5008 Bergen, Norway

Email:
stromme@mi.uib.no

DOI:
https://doi.org/10.1090/S0002-9947-98-01972-2

Keywords:
Punctual Hilbert scheme,
intersection numbers

Received by editor(s):
September 1, 1996

Article copyright:
© Copyright 1998
American Mathematical Society