dimensions of strata of Pfaffian varieties

Question

Let $V$ a complex vector space of dimension $2n$. Let us consider $W=\wedge^2V$ and the Pfaffian variety $Pf\subset \mathbb{P}W$ that parametrize degenerate skew-symmetric matrices. $Pf$ is naturally stratified by the (even) rank of the matrices. $Pf$ is the locus of rank at most $2n-2$, the singular locus $Sing(Pf)$ is the locus of rank at most $2n-4$, etc. down to the smallest stratum, which is the Grassmannian $Gr(2,2n)$, that parametrizes matrices of rank 2. While the dimensions of the intermediate strata in the case of generic determinantal varieties are known and easy to check, I would like to know if the dimensions of these strata have already been computed somewhere.

Answer 1

The codimension of forms of rank $2n-2k$ is $k(2k-1)$. In fact, it is easy to construct a resolution for each strata (a projective bundle over appropriate Grassmannian).