The Chow of $S^{[n]}$ and the universal subscheme

Abstract

We prove that any element in the Chow ring of the Hilbert scheme ${\text{Hilb}}_n$ of $n$ points on a smooth surface $S$ is a universal class, i.e. the pushforward of a polynomial in the Chern classes of the universal subscheme on ${\text{Hilb}}_n\times S^k$ for some $k\in \mathbb{N}$, with coefficients pulled back from the Chow of $S^k$.