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

### Omegar Calvo-Andrade

Centro de Investigación en Matemáticas, A.C., Guanajuato, Mexico### Ariel Molinuevo

Universidade Federal do Rio de Janeiro, Brazil### Federico Quallbrunn

Universidad de Buenos Aires, Argentina

## Abstract

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

## Cite this article

Omegar Calvo-Andrade, Ariel Molinuevo, Federico Quallbrunn, On the geometry of the singular locus of a codimension one foliation in $\mathbb P^n$. Rev. Mat. Iberoam. 35 (2019), no. 3, pp. 857–876

DOI 10.4171/RMI/1073