On a cell decomposition of the Hilbert scheme of points in the plane.
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)].
scientific article; zbMATH DE number 2150890