Irreducible components of the equivariant punctual Hilbert schemes

الأصل · EN

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⁻ᵇ,tᵃ), 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 Stromme, namely the main lemma of the computation of the Betti numbers of the Hilbert scheme Hˡ parametrizing the 0-dimensional subschemes of the affine plane of length l and a description of Bialynicki-Birula cells on Hˡ by means of explicit flat families. In particular, we precise conditions of applications of this last description.