Homology of the three flag Hilbert scheme

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 Poincaré polynomials for ${\text{Hilb}}^{n,n+1,n+2}(0)$ satisfy:$$\sum _{n\ge 0}{P}_q\left({\text{Hilb}}^{n,n+1,n+2}(0)\right)z^n=\frac{q+1}{(1-zq)(1-z^2{q}^2)}\prod _{k\ge 1}\frac{1}{1-z^k{q}^{k-1}}.$$ 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 Poincaré polynomials of ${\text{Hilb}}^{n,n+2}(0)$ for all $n\in \mathbb{N}$.