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

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.