of the following:

([@mmq]\[Theorem 4.24, p. 1041\])\[teo3\] Let $\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e)$ such that $J=\sqrt{J}$. Then $${\mathpzc{Kup}(\omega)}=\mathpzc{K}_{set}\neq \emptyset.$$

The following statement is valid in a non-singular variety $X$ and we will use it later. We will consider a $1$-form $\omega$ on $X$ with singular set of codimension equal or greater than 2. And we will denote with $\mathcal{J}$ the ideal sheaf of ${\mathrm{Sing}(\omega)}$.

([@mmq]\[Theorem 2.7, p. 1030\])\[teodivision\] Let $\omega$ be an integrable $1$-form in a non-singular variety $X$ and let ${\mathfrak{p}}\in{\mathrm{Sing}(\omega)}$ be such that ${\mathscr{J}}_{\mathfrak{p}}$ is radical and such that $d\omega_{\mathfrak{p}}\in {\mathscr{J}}_{\mathfrak{p}}\cdot \Omega^2_{X,{\mathfrak{p}}}$. Then there is a formal $1$-form $\eta$ such that $d\omega=\omega\wedge\eta$.

Unfoldings over schemes {#unf-schemes}
=======================

Along this section we will give the definition of *codimension $q$ foliation* on a smooth variety $X$. Then we will redefine the *singular locus* with a scheme theoretic approach. Finally we define an *unfolding* of a codimension $q$ foliation.

For the rest of the article, until Section \[section\_toric\], let us consider $X$ as a non-singular projective variety unless stated otherwise.

If $\Xi \in \Gamma(U, \bigwedge^p TX)$ is a multivector and $\varpi\in \Gamma(U,\Omega^q_X)$ a $q$-form we will denote by $i_\Xi \varpi \in \Gamma(U,\Omega^{q-p}_X)$ the contraction. Recall that the *Plücker relations* for $\varpi$ are given by $$i_\Xi 