On the geometry of the singular locus of a codimension one foliation in $\mathbb{P}^n$

TL;DR

This paper investigates the structure of singular loci in codimension one holomorphic foliations on complex projective spaces, proving the Kupka set is arithmetically Cohen-Macaulay and establishing connectedness and splitting properties.

Contribution

It demonstrates that under certain conditions, the Kupka set is arithmetically Cohen-Macaulay and shows the connectedness and tangent sheaf splitting of foliations.

Findings

Kupka set is arithmetically Cohen-Macaulay

Kupka set is connected in projective space

Tangent sheaf splits if locally free

Abstract

We will work with codimension one holomorphic foliations over the complex projective space, represented by integrable forms $\omega\in H^0(\Omega^1_{\PP^n}(e))$. Our main result is that, under suitable hypotheses, the Kupka set of the singular locus of $\omega\in H^0(\Omega^1_{\PP^3}(e))$, defined algebraically as a scheme, turns out to be arithmetically Cohen-Macaulay. As a consequence, we prove the connectedness of the Kupka set in $\PP^n$, and the splitting of the tangent sheaf of the foliation, provided that it is locally free.