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 $\omega \in H^0({\Omega }_{{\mathbb{P}}^n}^1(e))$. Our main result is that, under suitable hypotheses, the Kupka set of the singular locus of $\omega \in H^0({\Omega }_{{\mathbb{P}}^3}^1(e))$, defined algebraically as a scheme, turns out to be arithmetically Cohen–Macaulay. As a consequence, we prove the connectedness of the Kupka set in ${\mathbb{P}}^n$, and the splitting of the tangent sheaf of the foliation, provided that it is locally free.