Irreducible components of the equivariant punctual Hilbert schemes

Abstract

Let H ab be the equivariant Hilbert scheme parameterizing the zero-dimensional subschemes of the affine plane invariant under the natural action of the one-dimensional torus T ab ≔{(t −b ,t a )t∈k ∗ } . We compute the irreducible components of H ab : they are in one-to-one correspondence with the set of possible 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 l parametrizing the zero-dimensional subschemes of the affine plane of length l (Invent. Math. 87 (1988) 343), and a description of Bialynicki-Birula cells on H l by means of explicit flat families (Invent. Math. 91 (1988) 365). In particular, we give precise conditions when this last description applies.