On a cell decomposition of the Hilbert scheme of points in the plane.
type: Item
instance of: scholarly article
title: On a cell decomposition of the Hilbert scheme of points in the plane.
zbMATH Open document ID: 1064.14500
DOI: 10.1007/BF01389372
author: Geir Ellingsrud
author: Stein Arild Strømme
published in: Inventiones Mathematicae
publication date: 2005-03-31T00:00:00Z
full work available at URL: https://eudml.org/doc/143544
review text: The authors compute a cell decomposition of the Hilbert scheme \(\text{Hilb}^d(\mathbb{P}^2)\) induced by the theorem of \textit{A. Bialynicki-Birula} [Ann. Math.~(2)~98, 480--497 (1973; Zbl 0275.14007)] when applied to the natural torus action. The existence of such a decomposition was already used by the authors to compute the Betti numbers of \(\text{Hilb}^d(\mathbb{P}^2)\) [Invent. Math. 87, 343--352 (1987; Zbl 0625.14002)] and can be explicitly given by a slight modification of the argument used there. Each cell is in some \(\text{Hilb}^d(\mathbb{A}^2)\) and is constructed via invariant ideals of colength \(d\) having a prescribed resolution (in this way, they are parametrized by the partitions of \(d\)). The authors construct to each partition a corresponding quasifinite and flat family of subschemes. It should be noted that this family is not always finite over their base (as used in the paper), however, the main results still hold due to the corrections of \textit{M. Huibregtse} [Invent. Math. 160, 165--172 (2005; Zbl 1064.14005)].
reviewed by: Olaf Teschke
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 14D22
zbMATH DE Number: 2150890
MaRDI profile type: Publication
cites work: Introduction to Grothendieck duality theory
cites work: Some theorems on actions of algebraic groups
cites work: Q4126461
cites work: On the homology of the Hilbert scheme of points in the plane
cites work: Q3328651
OpenAlex ID: W2004681780
description: scientific article; zbMATH DE number 2150890