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

MathSciNet review:
1432198

Full-text PDF Free Access

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.**Luchezar L. Avramov,*Complete intersections and symmetric algebras*, J. Algebra**73**(1981), no. 1, 248–263. MR**641643**, 10.1016/0021-8693(81)90357-4**2.**Joël Briançon,*Description de 𝐻𝑖𝑙𝑏ⁿ𝐶{𝑥,𝑦}*, Invent. Math.**41**(1977), no. 1, 45–89. MR**0457432****3.**Jan Cheah,*On the cohomology of Hilbert schemes of points*, J. Algebraic Geom.**5**(1996), no. 3, 479–511. MR**1382733****4.**G. Ellingsrud. Irreducibility of the punctual Hilbert scheme of a surface. Unpublished.**5.**Lothar Göttsche,*The Betti numbers of the Hilbert scheme of points on a smooth projective surface*, Math. Ann.**286**(1990), no. 1-3, 193–207. MR**1032930**, 10.1007/BF01453572**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).

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
14C17,
14C05

Retrieve articles in all journals with MSC (1991): 14C17, 14C05

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