Homology of the three flag Hilbert Scheme

We prove the existence of an affine paving for the three-step flag Hilbert scheme that parametrizes flag of three 0-dimensional subschemes of length, respectively, n, n+1 and n+2 that are supported at the origin of the affine plane. 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 our spaces satisfy a generating function. In the process of proving the formula for the generating function we relate combinatorially the homology of our spaces with that of known smooth subspaces of another Hilbert scheme of flags, this time of length n and n+2. As a corollary we find an affine paving and a combinatorial formula for the Poincaré of these last ambient spaces.