×

On a cell decomposition of the Hilbert scheme of points in the plane. (English) Zbl 1064.14500

The authors compute a cell decomposition of the Hilbert scheme \(\text{Hilb}^d(\mathbb{P}^2)\) induced by the theorem of 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 M. Huibregtse [Invent. Math. 160, 165–172 (2005; Zbl 1064.14005)].

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
14C15 (Equivariant) Chow groups and rings; motives
14D22 Fine and coarse moduli spaces
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] [A-K] Altman, A., Kleiman, S.: Introduction to Grothendieck Duality Theory. Lect. Notes Math., vol. 146. Berlin Heidelberg New York: Springer 1970 · Zbl 0215.37201
[2] [B1] Bialynicki-Birula, A.: Some theorems on actions of algebraic groups. Ann. Math.98, 480-497 (1973) · Zbl 0275.14007
[3] [B2] Bialynicki-Birula, A.: Some properties of the decompositions of algebraic varieties determined by actions of a torus. Bull. Acad. Pol. Sci. Sér. Sci. Math. astron. Phys.24, (No. 9) 667-674 (1976) · Zbl 0355.14015
[4] [E-S] Ellingsrud, G., Strømme, S.A.: On the homology of the Hilbert scheme of points in the plane. Invent. Math.87, 343-352 (1987) · Zbl 0625.14002
[5] [F] Fulton, W.: Intersection Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. Berlin Heidelberg New York: Springer 1984 · Zbl 0541.14005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.