Families of distributions and Pfaff systems under duality

A singular distribution on a non-singular variety $X$ can be defined either by a subsheaf $D$ of the tangent sheaf, or by the zeros of a subsheaf $D^0$ of the sheaf of 1-forms. Although both definitions are equivalent under mild conditions on $D$, they give rise, in general, to non-equivalent notions of flat families. In this work we investigate conditions under which both notions of flat families are equivalent. In the last sections we focus on the case where the distribution is integrable, and we use our results to generalize a theorem of Cukierman and Pereira.