Homology of the three flag Hilbert scheme

Abstract

We prove the existence of an affine paving for the three-step flag Hilbert scheme Hilbn, n+1, n+2(0) := \C[[x,y]]⊃ In⊃ In+1⊃ In+2: Ii \,\, ideals with dimC C[x,y]/Ii = i \ of 0-dimensional subschemes that are supported at the origin of C2. 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 Hilbn,n+1, n+2(0) satisfy: Σn≥ 0 Pq(Hilbn,n+1, n+2(0)) zn = q+1(1-zq)(1-z2q2)\,\, Πk≥ 1 11-zkqk-1. In the process of proving this formula we relate combinatorially the homology of our spaces with that of known subspaces of Hilbn+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 Hilbn, n+2(0) for all n∈ N.