Degeneracy loci of twisted differential forms and linear line complexes

We prove that the Hilbert scheme of degeneracy loci of pairs of global sections of , the twisted cotangent bundle on , is unirational and dominated by the Grassmannian of lines in the projective space of skew-symmetric forms over a vector space of dimension n. We provide a constructive method to find the fibers of the dominant map. In classical terminology, this amounts to giving a method to realize all the pencils of linear line complexes having a prescribed set of centers. In particular, we show that the previous map is birational when n = 4.