Combinatorial duality of Hilbert schemes of points in the affine plane

TL;DR

This paper explores the duality between two stratifications of the Hilbert scheme of points in the affine plane, revealing a combinatorial duality in their Białynicki-Birula decompositions.

Contribution

It establishes that the two Białynicki-Birula decompositions are stratifications with dual partial orderings on monomial ideals, uncovering a combinatorial duality.

Findings

Both decompositions are stratifications with closure relations.

The partial orderings on monomial ideals are dual to each other.

The duality links the open and closed subschemes of the Hilbert scheme.

Abstract

The Hilbert scheme of $n$ points in the affine plane contains the open subscheme parametrizing $n$ distinct points in the affine plane, and the closed subscheme parametrizing ideals of codimension $n$ supported at the origin of the affine plane. Both schemes admit Bia{\l}ynicki-Birula decompositions into moduli spaces of ideals with prescribed lexicographic Gr\"obner deformations. We show that both decompositions are stratifications in the sense that the closure of each stratum is a union of certain other strata. We show that the corresponding two partial orderings on the set of of monomial ideals are dual to each other.