Ideals of the cohomology rings of Hilbert schemes and their applications

Authors:
Wei-Ping Li, Zhenbo Qin and Weiqiang Wang

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

Published electronically:
August 26, 2003

MathSciNet review:
2020031

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the ideals of the rational cohomology ring of the Hilbert scheme of points on a smooth projective surface . As an application, for a large class of smooth quasi-projective surfaces , we show that every cup product structure constant of is independent of ; moreover, we obtain two sets of ring generators for the cohomology ring .

Similar results are established for the Chen-Ruan orbifold cohomology ring of the symmetric product. In particular, we prove a ring isomorphism between and 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

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

Email:
ww9c@virginia.edu

DOI:
https://doi.org/10.1090/S0002-9947-03-03422-6

Keywords:
Heisenberg algebra,
Hilbert scheme,
cohomology ring.

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

Article copyright:
© Copyright 2003
American Mathematical Society