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# 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 $A_{K}^*(Hilb^n(S))$ 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.
Document type :
Journal articles

https://hal.archives-ouvertes.fr/hal-00713204
Contributor : Laurent Evain Connect in order to contact the contributor
Submitted on : Friday, June 29, 2012 - 3:26:59 PM
Last modification on : Wednesday, April 27, 2022 - 4:16:36 AM

### Citation

Pierre-Emmanuel Chaput, Laurent Evain. On the equivariant cohomology of Hilbert schemes of points in the plane. Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2015, 65 (3), pp.1201-1250. ⟨10.5802/aif.2955⟩. ⟨hal-00713204⟩

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