Irreducible components of the equivariant punctual Hilbert schemes

Let Hab be the equivariant Hilbert scheme parametrizing the 0-dimensional subschemes of the affine plane invariant under the natural action of the one-dimensional torus Tab := {(t, t) t ∈ k}. We compute the irreducible components of Hab: 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 Strømme, namely the main lemma of the computation of the Betti numbers of the Hilbert scheme H parametrizing the 0-dimensional subschemes of the affine plane of length l [4], and a description of Bialynicki-Birula cells on H by means of explicit flat families [5]. In particular, we precise conditions of applications of this last description.