Irreducible components of the equivariant punctual Hilbert schemes

Details Irreducible components of the equivariant punctual Hilbert schemes Irreducible components of the equivariant punctual Hilbert schemes

2001-06-26

arxiv.org/abs/math/0106218v1

Mathematics

Algebraic Geometry

13 pages

Scientific paper

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.

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