An algebro-geometric realization of the cohomology ring of Hilbert scheme of points in the affine plane

TL;DR

This paper establishes an algebraic geometric isomorphism between the cohomology ring of the Hilbert scheme of points in the affine plane and the coordinate ring of a fixed point scheme, providing a new perspective on its structure.

Contribution

It introduces a novel algebro-geometric realization of the cohomology ring of the Hilbert scheme of points in the affine plane, linking it to fixed point schemes of symmetric products.

Findings

Cohomology ring is isomorphic to a coordinate ring of a fixed point scheme.

Provides an analogue of a known theorem for Springer fibers.

Establishes a new geometric interpretation of the cohomology ring.

Abstract

We show that the cohomology ring of Hilbert scheme of $n$-points in the affine plane is isomorphic to the coordinate ring of $\mathbb{G}_{m}$-fixed point scheme of the $n$-th symmetric product of $\mathbb{C}^{2}$ for a natural $\mathbb{G}_{m}$-action on it. This result can be seen as an analogue of a theorem of DeConcini, Procesi and Tanisaki on a description of the cohomology ring of Springer fiber of type A.