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

TL;DR

This paper proves that all elements in the Chow ring of the Hilbert scheme of points on a smooth surface are universal classes, expressible via Chern classes of the universal subscheme, revealing a fundamental structure.

Contribution

It establishes that every Chow class on the Hilbert scheme can be represented as a universal class derived from the universal subscheme, providing a new structural understanding.

Findings

All Chow ring elements are universal classes.

Universal classes are generated by Chern classes of the universal subscheme.

The result applies to Hilbert schemes of points on smooth surfaces.

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 $S^k$.