We study the conormal sheaves and singular schemes of 1-dimensional foliations on smooth projective varieties X of dimension 3 of Picard rank 1. We prove that if the singular scheme has dimension 0, then the conormal sheaf is µ-stable whenever the tangent bundle T X is stable, and apply this fact to the characterization of certain irreducible components of the moduli space of rank 2 reflexive sheaves on P 3 and on a smooth quadric hypersurface Q 3 ⊂ 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 . Contents17 7.3. Foliations of odd degree 19 7.4. Local complete intersection foliations of degree 0 20 7.5. Local complete intersection foliations of degree 1 21 References 22

We study the conormal sheaves and singular schemes of one‐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 μ‐stable whenever the tangent bundle TX$TX$ is stable, and apply this fact to the characterization of certain irreducible components of the moduli space of rank 2 reflexive sheaves on double-struckP3$\mathbb {P}^3$ and on a smooth quadric hypersurface Q3⊂P4$Q_3\subset \mathbb {P}^4$. Finally, we give a classification of local complete intersection foliations, that is, foliations with locally free conormal sheaves, of degree 0 and 1 on Q3.

We show that codimension 1 distributions with at most isolated singularities on threefold hypersurfaces Xd ⊂ P4 of degree d provide interesting examples of stable rank 2 reflexive sheaves. When d ≤ 5, these sheaves can be regarded as smooth points within an irreducible component of the moduli space of stable reflexive sheaves. Our second goal goes in the reverse direction: we start from a well-known family of stable locally free sheaves and provide examples of codimension 1 distributions of local complete intersection type on Xd.

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