Moduli of Distributions via Singular Schemes

TL;DR

This paper studies the moduli space of codimension one distributions on smooth projective varieties, linking it to singular schemes via a morphism, and explicitly describes fibers and resolutions, especially for projective spaces.

Contribution

It introduces a morphism from the moduli space of distributions to a Hilbert scheme and computes fibers using syzygies, providing new insights into the structure of these moduli spaces.

Findings

The map from distributions to singular schemes is a morphism.

Fibers of this map are described explicitly for projective spaces.

Minimal graded free resolutions for generic distributions on P^3 are provided.

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 $\mathbb{P}^3$.