Codimension one holomorphic distributions on the projective three-space

TL;DR

This paper classifies and analyzes codimension one holomorphic distributions on projective three-space, focusing on their singular schemes, tangent sheaves, and moduli spaces, revealing stability properties and geometric structures.

Contribution

It provides a classification of low-degree distributions with locally free tangent sheaves and describes the moduli space structure, including stability results and irreducibility.

Findings

Classified distributions of degree at most 2 with locally free tangent sheaves.

Showed distributions with isolated singularities have stable tangent sheaves.

Described the moduli space as an irreducible, nonsingular quasi-projective variety.

Abstract

We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves, and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck's Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. Finally, we prove that every rational foliation, and certain logarithmic foliations have stable tangent sheaves.