Ideals of the cohomology rings of Hilbert schemes and their applications

Details Ideals of the cohomology rings of Hilbert schemes and their applications Ideals of the cohomology rings of Hilbert schemes and their applications

2002-08-09

arxiv.org/abs/math/0208070v2

Trans. AMS, Vol.356 No.1 (2004), p245-265

Mathematics

Algebraic Geometry

Final version to appear on Trans. AMS

Scientific paper

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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