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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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Ideals of the cohomology rings of Hilbert schemes and their applications
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by Wei-Ping Li, Zhenbo Qin and Weiqiang Wang PDF
Trans. Amer. Math. Soc. 356 (2004), 245-265 Request permission

Abstract:

We study the ideals of the rational cohomology ring of the Hilbert scheme $X^{[n]}$ of $n$ points on a smooth projective surface $X$. As an application, for a large class of smooth quasi-projective surfaces $X$, we show that every cup product structure constant of $H^*(X^{[n]})$ is independent of $n$; moreover, we obtain two sets of ring generators for the cohomology ring $H^*(X^{[n]})$. Similar results are established for the Chen-Ruan orbifold cohomology ring of the symmetric product. In particular, we prove a ring isomorphism between $H^*(X^{[n]}; \mathbb {C})$ and $H^*_\textrm {orb}(X^n/S_n; \mathbb {C})$ for a large class of smooth quasi-projective surfaces with numerically trivial canonical class.
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Additional Information
  • Wei-Ping Li
  • Affiliation: Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
  • MR Author ID: 334959
  • Email: mawpli@ust.hk
  • Zhenbo Qin
  • Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
  • Email: zq@math.missouri.edu
  • Weiqiang Wang
  • Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
  • MR Author ID: 339426
  • Email: ww9c@virginia.edu
  • Received by editor(s): July 5, 2002
  • Published electronically: August 26, 2003
  • Additional Notes: The first author was partially supported by the grant HKUST6170/99P
    The second author was partially supported by an NSF grant
    The third author was partially supported by an NSF grant
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 245-265
  • MSC (2000): Primary 14C05; Secondary 14F25, 17B69
  • DOI: https://doi.org/10.1090/S0002-9947-03-03422-6
  • MathSciNet review: 2020031