Foliations by curves on threefolds

Abstract

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.

On the Milnor number of non‐isolated singularities of holomorphic foliations and its topological invariance

“…For a study on the number of residual isolated singularities of foliations on complex projective spaces, see [8,11,16] and the interested reader may consult [7,9] for results about the classification of one-dimensional foliations of low degree on threefolds.…”

“…Let (𝑢 1𝑘 , 𝑢 2𝑘 ) = lim 𝑡→0 𝑢 𝑡 𝑘 = lim 𝑡→0 (𝑢 𝑡 1𝑘 , 𝑢 𝑡 2𝑘 ). We get (𝑢 2𝑘 ) 6 − (𝑢 2𝑘 ) 7 = 0, which implies that either 𝑢 2𝑘 = 0 or 𝑢 2𝑘 = 1. If 𝑢 2𝑘 = 0, then 𝑢 1𝑘 = 0 and if…”