we will generalize later. The scheme ${\mathpzc{Kup}(\omega)}$ and the ideal $I$ were of central importance in those results. We refer the reader to [@mmq] for a full overview of this subjects.

With the exception of Theorem \[teodivision\] through this section we will restrict to the projective space ${\mathbb{P}}^n$. So let us denote $S={\mathbb{C}}[x_0,\dots,x_n]$ to the homogeneous coordinate ring of ${\mathbb{P}}^n$ and $\Omega^1_{{\mathbb{P}}^n}(e)$ the sheaf of twisted differential 1-forms in ${\mathbb{P}}^n$ of degree $e$. With ${\mathrm{Sing}(\omega)}_{set}$ we will denote the (set theoretic) singular set of $\omega\in H^0({\mathbb{P}}^n, \Omega^1_{{\mathbb{P}}^n}(e))$ in ${\mathbb{P}}^n$, $${\mathrm{Sing}(\omega)}_{set} = \{p\in {\mathbb{P}}^n: \omega(p) = 0 \}\ .$$

\[foliation\] Let ${\mathcal{L}}\simeq {\mathcal{O}}_{{\mathbb{P}}^n}(-e)$, $e\geq 2$, be a line bundle and $\omega:{\mathcal{L}}\to \Omega^1_{{\mathbb{P}}^n}$ be a morphism of sheaves, we will say that $\omega$ defines an *algebraic foliation of codimension 1* on ${\mathbb{P}}^n$, if $\Omega^1_{{\mathbb{P}}^n}/{\mathcal{L}}$ is torsion free and the morfism is generated by a non zero global section $\omega\in H^0({\mathbb{P}}^n,\Omega^1_{{\mathbb{P}}^n}(e))$ such that $\omega\wedge d\omega = 0$. We recall that such foliations have *geometric degree* $e-2$.

The condition of $\Omega^1_{{\mathbb{P}}^n}/{\mathcal{L}}$ to be torsion free in the definition of a foliation is equivalent to ask the singular set to have codimension greater than 2. Indeed, this is the same to ask that $\omega$ is 