## 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.## References

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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