Ideals of the cohomology rings of Hilbert schemes and their applications

Li, Wei-Ping; Qin, Zhenbo; Wang, Weiqiang

doi:10.1090/S0002-9947-03-03422-6

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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