Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

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Additional Information

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

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

Keywords: Punctual Hilbert scheme, intersection numbers
Received by editor(s): September 1, 1996
Article copyright: © Copyright 1998 American Mathematical Society