Ideals of the cohomology rings of Hilbert schemes and their applications

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 /Sn; C) for a large class of smooth quasi-projective surfaces with numerically trivial canonical class.