On holomorphic distributions on Fano threefolds

Abstract

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.

Foliations by curves on threefolds

“…We will use the following relation between the conormal sheaf of a foliation by curves and its singular locus [7].…”

Foliations by curves on threefolds

“…Finally, we consider local complete intersection (LCI) foliations, which are defined as foliations by curves with locally free conormal sheaves; the nomenclature is motivated by the fact that they are given locally as the intersection of two codimension one distributions. When the conormal sheaf N ∨ F splits as a sum of line bundles, we say that F is a complete intersection (CI) foliation; CI foliations by curves on Fano thereefolds are studied in [7], where characterizations in terms of the singular scheme are provided. Here, motivated by the classification of LCI foliations by curves of low degree on P 3 given in [8], we give the first steps towards a classification of LCI foliations by curves on smooth quadric hypersurfaces in P 4 of degree 0 and 1.…”

“…A systematic study of for the case X = P 3 was initiated in [10] and continued in [8]. Furthermore, the authors of [7] consider foliations by curves on Fano threefolds, obtaining results regarding the connectedness of the singular scheme Sing(F ) and the stability of the conormal sheaf N ∨ F . In all of these papers the focus was on foliations whose singular scheme has pure dimension 1; we will consider arbitrary foliations by curves and a wider class of threefolds, generalizing many of the results obtained in [7,8,10].…”

“…If deg(C) = 1, then C must be a line, contradicting the equality χ(O C ) = 2. If deg(C) = 2, so that C is 2 disjoint lines or a double line, and hence C is a 1-Buchsbaum curve.The same happens when deg(C) = 3: a curve of degree 3 and genus −1 must be a conic plus a line which is again a 1-Buchsbaum curve.It follows from[7, Theorem 4.4], that N ∨ F is arithmetically Cohen-Macaulay, so N ∨ F is a direct sum of line bundles and twisted spinor bundles by[4, Theorem 2.7]. Since rk(N ∨ F ) = 2, we are left with two possibilities: eitherN ∨ F = O Q3 (a) ⊕ O Q3 (b) with a + b = −3, or N ∨ F = S(−1).…”

Codimension 1 distributions on three dimensional hypersurfaces

“…whose singular locus Z is a curve see [1,Lemma 2.1]. Some properties of these split distributions have been studied in [3].…”