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

Omegar Calvo-Andrade, Ariel Molinuevo, Federico Quallbrunn

Published 2016-11-11Version 1

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