not of the form $f.\omega'$, for some global section $f\in H^0({\mathbb{P}}^n,{\mathcal{O}}_{{\mathbb{P}}^n}(d))$ and a 1-form $\omega'\in H^0({\mathbb{P}}^n,\Omega^1_{{\mathbb{P}}^n}(e-d))$. Also, integrable differential 1-forms define the same foliation up to scalar multiplication. Then, we will denote the set of codimension 1 foliations of geometric degree $e-2$ as $$\label{torsion}
{\mathcal{F}}^1({\mathbb{P}}^n,e) := \left\{\omega\in{\mathbb{P}}\left(H^0({\mathbb{P}}^n,\Omega^1_{{\mathbb{P}}^n}(e))\right):\ \omega\wedge d\omega=0,\ {\mathrm{codim}}({\mathrm{Sing}(\omega)}_{set})\geq 2 \right\}.$$

\[IJ\] We define the graded ideals of $S$ associated to $\omega$ as $$\begin{aligned}
I(\omega) &:= \left\{ h\in S:\  h\ d\omega = \omega\wedge\eta\text{ for some } \eta\in\Omega^1_S\right\}\\
J(\omega) &:= \left\{ i_X(\omega)\in S:\ X\in T_S \right\}.\end{aligned}$$ We will name $I(\omega)$ the *ideal of persistent singularities* of $\omega$. We will also denote them $I=I(\omega)$ and $J=J(\omega)$ if no confusion arises.

\[1notinI\] Notice that $1\not\in I$, since the class of $d\omega$ in the Koszul complex of $\omega$, $H^2(\omega)$ is not zero, see Definition \[koszulcomplex\]. Also $J(\omega)$ equals the ideal defining the singular locus of $\omega$. This last thing, can be seen by contracting with the vector fields ${\partial}/{\partial x_i}$. The definition given for $J(\omega)$ is better suited for our schematic approach that we will develop next.

For $\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e)$, we define the *Kupka set* as the subset of the singular set $$\mathpzc{K}_{set} = \overline{\{p\in {\mathrm{Sing}(\omega)}_{set} :  d\omega(p)\neq 0 \}}\ .$$

Notice that the 