Moduli of distributions via singular schemes

Let $X$ be a smooth projective variety. We show that the map sends codimension one distribution on to its singular scheme is morphism from moduli space of distributions into Hilbert scheme. describe fibers and, when $X = \mathbb{P}^n$, compute them via syzygies. As an application, we spaces degree 1 $\mathbb{P}^3$. also give minimal graded free resolution for ideal generic