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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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An intersection number for the punctual Hilbert scheme of a surface
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by Geir Ellingsrud and Stein Arild Strømme PDF
Trans. Amer. Math. Soc. 350 (1998), 2547-2552 Request permission

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.
References
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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
  • Received by editor(s): September 1, 1996
  • © Copyright 1998 American Mathematical Society
  • 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