Moduli of distributions via singular schemes

Let X be a smooth projective variety. We show that the map that sends a codimension one distribution on X to its singular scheme is a morphism from the moduli space of distributions into a Hilbert scheme. We describe its fibers and, when X= Pn, compute them via syzygies. As an application, we describe the moduli spaces of degree 1 distributions on P3. We also give the minimal graded free resolution for the ideal of the singular scheme of a generic distribution on P3.