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

2012/05/24 by Pierre-Emmanuel Chaput, Chaput, Pierre-Emmanuel, Laurent Evain +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics

paper · doi:10.48550/arxiv.1205.5470

openalex publication_date 2012/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^*(Hilbⁿ(S)) of the punctual Hilbert scheme Hilbⁿ(S) with equivariant coefficients inverted. We compute base change formulas in A_K^*(Hilbⁿ(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 Hilbⁿ(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.

Related

• Irreducible components of the equivariant punctual Hilbert schemes
• The Chow ring of punctual Hilbert schemes of toric surfaces
• Intersection theory on punctual Hilbert schemes and graded Hilbert schemes
• Equivariant cohomology, symmetric functions and the Hilbert schemes of points on the total space of the invertible sheaf O(-2) over the projective line
• On generating series of classes of equivariant Hilbert schemes of fat points