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 ${\mathbb{P}}^3$.

21 pages. Some results and exposition were improved. Final version to appear in Math. Zeitschrift

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