Homology of the three flag Hilbert scheme

TL;DR

This paper proves the existence of an affine paving for the three-step flag Hilbert scheme at the origin of ^2, enabling explicit computation of its Poincare9 polynomials and revealing new combinatorial relations with related Hilbert schemes.

Contribution

It introduces a stratification of the three-step flag Hilbert scheme into smooth subvarieties with affine pavings, and derives explicit formulas for their Poincare9 polynomials.

Findings

Affine paving of the three-step flag Hilbert scheme at the origin.

Explicit formula for the generating function of Poincare9 polynomials.

Relation between the homology of these spaces and known subspaces of ^2.

Abstract

We prove the existence of an affine paving for the three-step flag Hilbert scheme $$ \text{Hilb}^{n, n+1, n+2}(0) := \left\{\mathbb{C}[[x,y]]\supset I_n\supset I_{n+1}\supset I_{n+2}: I_i \,\,\text{ ideals with } \text{dim}_{\mathbb{C}} {\mathbb{C}[x,y]}/{I_i} = i \right\} $$ of 0-dimensional subschemes that are supported at the origin of $\mathbb{C}^2$. This is done by showing that the space stratifies in smooth subvarieties, the Hilbert-Samuel's strata, each of which has an affine paving with cells of known dimension, indexed by marked Young diagrams. The affine pavings of the Hilbert-Samuel's strata allow us to prove that the Poincar\'{e} polynomials for $\text{Hilb}^{n,n+1, n+2}(0)$ satisfy:$$ \sum_{n\geq 0} P_q\left(\text{Hilb}^{n,n+1, n+2}(0)\right) z^n = \frac{q+1}{(1-zq)(1-z^2q^2)}\,\, \prod_{k\geq 1} \frac{1}{1-z^kq^{k-1}}. $$ In the process of proving this formula we relate combinatorially the homology of our spaces with that of known subspaces of $\text{Hilb}^{n+1, n+3}(0)$. As a corollary we find an affine paving and a formula for the generating function of the Poincar\'{e} polynomials of $\text{Hilb}^{n, n+2}(0)$ for all $n\in \mathbb{N}$.