Codimension one distributions of degree 3 on the three-dimensional projective space

TL;DR

This work analyzes codimension one distributions of degree $3$ on $\mathbb{P}^3$, focusing on the tangent-sheaf invariants and the structure of the singular scheme. It develops a forgetful morphism from the distribution moduli to tangent-sheaf moduli, uses Castelnuovo-Mumford regularity and stability criteria to constrain possible invariants, and constructs explicit moduli components. The authors identify a stable family with tangent-sheaf data $(-1,3,5)$, prove its moduli $\mathcal{D}^{st}(3,3,5)$ is irreducible of dimension $42$, and show it surjects onto a $19$-dimensional reflexive-sheaf moduli, yielding a coherent, nonempty picture of degree-$3$ codimension-one distributions on $\mathbb{P}^3$. These results advance the classification of higher-degree distributions, providing concrete invariant lists, existence results, and moduli-dimension computations that underpin the broader global moduli program for foliations and distributions on projective varieties.

Abstract

We make a classification of codimension one degree 3 distributions on the projective three space, giving possible Chern classes of the tangent sheaf and describing de zero and one dimensional components of the singular scheme of the distribution. Also, we show the existence and describe some moduli spaces of such distributions, using the concept of stability of the tangent sheaf.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 3.1
• Proposition 3.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof