coincide when the singular locus it is radical.

([@mmq]\[Lemma 4.6, p.1034\])\[K=Kset\] Let $\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e)$ such that $J=\sqrt{J}$. Then $${\mathpzc{Kup}(\omega)}= \mathpzc{K}_{set}.$$

We have the following chain of inclusions, see Proposition \[JJcIIcKK\] and Proposition \[JJcIIcKK2\] for a generalization, in the codimension one and codimension $q$ case, respectively:

([@mmq]\[Proposition 4.7, p. 1035\])\[incl2\] Let $\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e)$. Then, we have the following relations $$J\subseteq I\subseteq K\ .$$

Let $\mathfrak{p}$ be a point in ${\mathbb{P}}^n$, *e.g.*, an homogeneous prime ideal in $S$ different from the *irrelevant ideal* $(x_0.\ldots,x_n)$, and let $\omega$ be an integrable differential 1-form. We will denote with a subscript $\mathfrak{p}$ the localization at the point $\mathfrak{p}$ and with $\widehat{S}_\mathfrak{p}$ the completion of the local ring $S_\mathfrak{p}$ with respect to the maximal ideal defined by $\mathfrak{p}$.

We say that $\mathfrak{p}\in{\mathbb{P}}^n$ is a *division point of $\omega$* if $1\in I(\omega)_\mathfrak{p}$.

We now define a subset of the moduli space of foliations on which we are going to state our next result.

\[generic\] We define the set ${\mathcal{U}}\subseteq{\mathcal{F}}^1({\mathbb{P}}^n,e)$ as $${\mathcal{U}}= \left\{\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e): \ \forall \mathfrak{p}\not\in{\mathpzc{Kup}(\omega)},\,\mathfrak{p}\text{ is a division point of }\omega\right\}.$$

See Theorem \[propP=K\] for a generalization of the following:

([@mmq]\[Theorem 4.12, p. 1036\])\[teo1\] Let $\omega\in\mathcal{U}\subseteq{\mathcal{F}}^1({\mathbb{P}}^n,e)$. Then, $$\sqrt{I}=\sqrt{K}.$$ Furthermore, if $\sqrt{I}=\sqrt{K}$ then $\omega\in{\mathcal{U}}$.

See Theorem \[teoKnotempty\] for a generalization 