On the Singular Scheme of Split Foliations

Abstract

We prove that the tangent sheaf of a codimension one locally free distribution splits as a sum of line bundles if and only if its singular scheme is arithmetically Cohen-Macaulay. In addition, we show that a foliation by curves is given by an intersection of generically transversal holomorphic distributions of codimension one if and only if its singular scheme is arithmetically Buchsbaum. Finally, we establish that these foliations are determined by their singular schemes, and deduce that the Hilbert scheme of certain arithmetically Buchsbaum schemes of codimension $2$ is birational to a Grassmannian.

21 pages

References in corpus (3)

• On globally generated vector bundles on projective spaces II
• On degeneracy schemes of maps of vector bundles and applications to holomorphic foliations
• On the singular scheme of codimension one holomorphic foliations in P^3

Cited by in corpus (8)

• Codimension one holomorphic distributions on the projective three-space
• On holomorphic distributions on Fano threefolds
• Codimension one distributions of degree 2 on the three-dimensional projective space
• Moduli of Distributions via Singular Schemes
• GSV-index for holomorphic Pfaff Systems
• On the geometry of the singular locus of a codimension one foliation in ${\mathbb{P}}^n$
• Rational Morita equivalence for holomorphic Poisson modules
• Classification of holomorphic Pfaff systems on Hopf manifolds