Revista Matemática Iberoamericana


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Volume 35, Issue 3, 2019, pp. 857–876
DOI: 10.4171/rmi/1073

Published online: 2019-04-08

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

Omegar Calvo-Andrade[1], Ariel Molinuevo[2] and Federico Quallbrunn[3]

(1) Centro de Investigación en Matemáticas, A.C., Guanajuato, Mexico
(2) Universidade Federal do Rio de Janeiro, Brazil
(3) Universidad de Buenos Aires, Argentina

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.

Keywords: Projective space, foliations, arithmetically Cohen–Macaulay

Calvo-Andrade Omegar, Molinuevo Ariel, Quallbrunn Federico: On the geometry of the singular locus of a codimension one foliation in $\mathbb P^n$. Rev. Mat. Iberoam. 35 (2019), 857-876. doi: 10.4171/rmi/1073