Foliations by curves on threefolds

TL;DR

This paper investigates the properties of 1-dimensional foliations on smooth threefolds, establishing stability conditions for conormal sheaves, and classifying certain low-degree foliations on a quadric threefold.

Contribution

It proves the stability of conormal sheaves under specific conditions and classifies local complete intersection foliations of degree 0 and 1 on a quadric threefold.

Findings

Conormal sheaves are μ-stable when the singular scheme is zero-dimensional and the tangent bundle is stable.

Characterization of irreducible components of moduli spaces of rank 2 reflexive sheaves.

Classification of low-degree local complete intersection foliations on Q_3.

Abstract

We study the conormal sheaves and singular schemes of 1-dimensional foliations on smooth projective varieties $X$ of dimension 3 and Picard rank 1. We prove that if the singular scheme has dimension 0, then the conormal sheaf is $\mu$-stable whenever the tangent bundle $TX$ is stable, and apply this fact to the characterization of certain irreducible components of the moduli space of rank 2 reflexive sheaves on $\mathbb{P}^3$ and on a smooth quadric hypersurface $Q_3\subset\mathbb{P}^4$. Finally, we give a classification of local complete intersection foliations, that is, foliations with locally free conormal sheaves, of degree 0 and 1 on $Q_3$.