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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 $Q_3subsetmathbb{P}^4$. Finally, we give a classification of local complete intersection foliations, that is, foliations with locally free conormal sheaves, of degree 0 and 1 on $Q_3$.
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We study foliations by curves on the three-dimensional projective space with no isolated singularities, which is equivalent to assuming that the conormal sheaf is locally free. We provide a classification of such foliations by curves up to degree 3, also describing the possible singular schemes. In particular, we prove that foliations by curves of degree 1 or 2 are either contained on a pencil of planes or legendrian, and are given by the complete intersection of two codimension one distributions. We prove that the conormal sheaf of a foliation by curves of degree 3 with reduced singular scheme either splits as a sum of line bundles or is an instanton bundle. For degree larger than 3, we focus on two classes of foliations by curves, namely legendrian foliations and those whose conormal sheaf is a twisted null correlation bundle. We give characterizations of such foliations, describe their singular schemes and their moduli spaces.
We develop a method to compute limits of dual plane curves in Zeuthen families of any kind. More precisely, we compute the limit 0-cycle of the ramification scheme of a general linear system on the generic fiber, only assumed geometrically reduced, of a Zeuthen family of any kind.
We give a criterion for a nef divisor $D$ to be semiample on a Calabi--Yau threefold $X$ when $D^3=0=c_2(X)cdot D$ and $c_3(X) eq 0$. As a direct consequence, we show that on such a variety $X$, if $D$ is strictly nef and $ u(D) eq 1$, then $D$ is ample; we also show that if there exists a nef non-ample divisor $D$ with $D otequiv 0$, then $X$ contains a rational curve when its topological Euler characteristic is not $0$.
This paper is devoted to the study of holomorphic distributions of dimension and codimension one on smooth weighted projective complete intersection Fano manifolds threedimensional, with Picard number equal to one. We study the relations between algebro-geometric properties of the singular set of singular holomorphic distributions and their associated sheaves. We characterize either distributions whose tangent sheaf or conormal sheaf are arithmetically Cohen Macaulay (aCM) on smooth weighted projective complete intersection Fano manifolds threefold. We also prove that a codimension one locally free distribution with trivial canonical bundle on any Fano threefold, with Picard number equal to one, has a tangent sheaf which either splits or it is stable.
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