On the singular scheme of codimension one holomorphic foliations in P^3

TL;DR

This paper investigates the properties of codimension one holomorphic foliations in P^3, establishing conditions under which their tangent sheaves are locally free or split, based on the nature of their singular schemes.

Contribution

It provides new criteria linking the structure of the tangent sheaf to the singular scheme, including conditions for local freeness and splitting, and explores when foliations are determined by their singular schemes.

Findings

Tangent sheaf is locally free iff the singular scheme is a curve.

The sheaf splits iff the curve is arithmetically Cohen-Macaulay.

Split foliations are determined by their singular schemes under certain conditions.

Abstract

In this work, we begin by showing that a holomorphic foliation with singularities is reduced if and only if its normal sheaf is torsion free. In addition, when the codimension of the singular locus is at least two, it is shown that being reduced is equivalent to the reflexivity of the tangent sheaf. Our main results state on one hand, that the tangent sheaf of a codimension one foliation in P^3 is locally free if and only the singular scheme is a curve, and that it splits if and only if that curve is arithmetically Cohen-Macaulay. On the other hand, we discuss when a split foliation in P^3 is determined by its singular scheme.