  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.
