On the Camacho-Lins Neto regularity

TL;DR

This paper investigates the regularity of codimension one foliations in projective space, linking it to the triviality of first order unfoldings and characterizing regular foliations with reduced singularities.

Contribution

It establishes the equivalence between Camacho-Lins Neto regularity and trivial first order unfoldings, and characterizes regular foliations with reduced singular locus.

Findings

Regular projective foliations with reduced singular locus are of Kupka type.

Camacho-Lins Neto regularity is equivalent to trivial first order unfoldings.

Any dicritical foliation with regular initial form is isomorphic to it.

Abstract

We work with codimension one foliations in the projective space $\mathbb{P}^{n}$, given a differential one form $\omega\in H^0(\mathbb{P}^n,\Omega^1_{\mathbb{P}^n}(e))$, such differential form verifies the Frobenius integrability condition $\omega\wedge d\omega =0$. In this work we show that the Camacho-Lins Neto regularity, applied for $\omega$, is equivalent to the fact that every first order unfolding of $\omega$ is trivial up to isomorphism. We do this by computing the Castelnuovo-Mumford regularity of the ideal $I(\omega)$ of first order unfoldings. With this result, we are also showing that the only regular projective foliations, with reduced singular locus, are the ones that have singular locus only Kupka type singularities. At last we use these results to show that every foliation $\varpi\in \Omega^1_{\mathbb{C}^{n+1}}$, with initial form $\omega$ regular and dicritical, is isomorphic to $\omega$.