Irreducible components of the equivariant punctual Hilbert schemes

2001/06/26 by Laurent Evain, Evain, Laurent
Mathematics · Physics and Astronomy · #14C05 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic and Geometric Analysis #FOS: Mathematics #Nonlinear Waves and Solitons #math.AG #msc:14C05

paper · pdf · doi:10.48550/arxiv.math/0106218

13 pages

arxiv created 2001/06/26 · openalex publication_date 2001/06/26 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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∈ 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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