Irreducible components of the equivariant punctual Hilbert schemes

Abstract

Let H_{ab} be the equivariant Hilbert scheme parametrizing the 0-dimensional subschemes of the affine plane invariant under the natural action of the one-dimensional torus T_{ab}:={(t^{-b},t^a), t\in k^*}. We compute the irreducible components of H_{ab}: they are in one-one correspondence with a set of Hilbert functions. As a by-product of the proof, we give new proofs of results by Ellingsrud and Stromme, namely the main lemma of the computation of the Betti numbers of the Hilbert scheme H^l parametrizing the 0-dimensional subschemes of the affine plane of length l and a description of Bialynicki-Birula cells on H^l by means of explicit flat families. In particular, we precise conditions of applications of this last description.