Moduli of distributions via singular schemes

Abstract

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 \(X = {\mathbb {P}^{n}}\), compute them via syzygies. As an application, we describe the moduli spaces of degree 1 distributions on \({\mathbb {P}^{3}}\). We also give the minimal graded free resolution for the ideal of the singular scheme of a generic distribution on \({\mathbb {P}^{3}}\).