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arxiv: math/0208070 · v2 · submitted 2002-08-09 · 🧮 math.AG · math.QA

Ideals of the cohomology rings of Hilbert schemes and their applications

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keywords ringcohomologyclasssmoothhilbertidealslargeproduct
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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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