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

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 $A_K^{\ast }(Hil{b}^n(S))$ of the punctual Hilbert scheme $Hil{b}^n(S)$ with equivariant coefficients inverted. We compute base change formulas in $A_K^{\ast }(Hil{b}^n(S))$ between the natural bases introduced by Nakajima, Ellingsrud and Strømme, 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 $Hil{b}^n(S)$ in terms of the equivariant Chern classes of the tautological bundle. We prove that the nested Hilbert scheme $Hil{b}^{[}n,n+1](S)$ parametrizing nested punctual subschemes of degree $n$ and $n+1$ is irreducible.

arXiv comment41 pages