On the Singular Scheme of Split Foliations

TL;DR

This paper characterizes when the tangent sheaf of certain foliations splits into line bundles based on the properties of their singular schemes, linking geometric structures to algebraic conditions.

Contribution

It establishes a precise equivalence between splitting of tangent sheaves and the singular scheme being arithmetically Cohen-Macaulay or Buchsbaum, and relates foliations to their singular schemes.

Findings

Tangent sheaf splits iff singular scheme is arithmetically Cohen-Macaulay.

Foliation by curves is an intersection of transversal distributions iff singular scheme is arithmetically Buchsbaum.

Hilbert scheme of certain Buchsbaum schemes is birational to a Grassmannian.

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.