On the Hilbert scheme of degeneracy loci of twisted differential forms

We prove that, for 3 < m < n − 1, the Grassmannian of mdimensional subspaces of the space of skew-symmetric forms over a vector space of dimension n is birational to the Hilbert scheme of degeneracy loci of m global sections of Ωℙn−1 (2), the twisted cotangent bundle on ℙn−1. For 3 = m < n − 1 and n odd, this Grassmannian is proved to be birational to the set of Veronese surfaces parameterized by the Pfaffians of linear skewsymmetric matrices of order n.