Universality and stability of cohomology rings of Hilbert schemes of points on surfaces

We prove that the cohomology ring structure of the Hilbert scheme X [n] of n-points on a projective surface X is determined in a universal way by the cohomology ring of X. In particular, if there exists a ring isomorphism H ∗(X) → H∗(Y ) for two projective surfaces X and Y which matches the canonical classes, then the cohomology rings of the Hilbert schemes X [n] and Y [n] are isomorphic for every n. We further establish a remarkable stability of the cohomology rings of X [n] as n varies. This enables us to construct a Hilbert ring which depends on the ring H∗(X) only and which encodes all the structures of the cohomology rings of X [n] for each n. In addition, we determine the structure of the Hilbert ring.