Ideals of the cohomology rings of Hilbert schemes and their applications

Abstract

We study the ideals of the rational cohomology ring of the Hilbert scheme X [ n ] X^{[n]} of n n points on a smooth projective surface X X . As an application, for a large class of smooth quasi-projective surfaces X X , we show that every cup product structure constant of H ∗ ( X [ n ] ) H^*(X^{[n]}) is independent of n n ; moreover, we obtain two sets of ring generators for the cohomology ring H ∗ ( X [ n ] ) 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 ) H^*(X^{[n]}; \mathbb {C}) and H orb ∗ ( X n / S n ; C ) H^*_\textrm {orb}(X^n/S_n; \mathbb {C}) for a large class of smooth quasi-projective surfaces with numerically trivial canonical class.