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Ideals of the cohomology rings of Hilbert schemes and their applications

Wei-Ping Li·Zhenbo Qin·Weiqiang Wang
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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]}, C) and H^*_{orb}(X^{[n]}/S_n, C) for a large class of smooth quasi-projective surfaces with numerically trivial canonical class.

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Wei-Ping Li, Zhenbo Qin, Weiqiang Wang. 2003-10-23. Ideals of the cohomology rings of Hilbert schemes and their applications. https://arxiv.org/abs/math/0208070

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