On the equivariant cohomology of Hilbert schemes of points in the plane

TL;DR

This paper investigates the equivariant cohomology of Hilbert schemes of points on the affine plane, providing explicit formulas, basis transformations, and structural results for nested schemes.

Contribution

It introduces new computational formulas and relations in the equivariant Chow ring of Hilbert schemes, including basis change formulas and irreducibility of nested schemes.

Findings

Computed base change formulas between natural bases.

Derived equivariant commutation relations for operators.

Proved irreducibility of nested Hilbert schemes.

Abstract

Let $S$ be the affine plane regarded as a toric variety with an action of the 2-dimensional torus $T$. We study the equivariant Chow ring $A_{K}^*(Hilb^n(S))$ of the punctual Hilbert scheme $Hilb^n(S)$ with equivariant coefficients inverted. We compute base change formulas in $A_{K}^*(Hilb^n(S))$ between the natural bases introduced by Nakajima, Ellingsrud and Str{\o}mme, and the classical basis associated with the fixed points. We compute the equivariant commutation relations between creation/annihilation operators. We express the class of the small diagonal in $Hilb^n(S)$ in terms of the equivariant Chern classes of the tautological bundle. We prove that the nested Hilbert scheme $Hilb^[n,n+1](S)$ parametrizing nested punctual subschemes of degree $n$ and $n+1$ is irreducible.