Theorem \[teoConnectionE\]:

Let $X$ be a projective variety and ${\mathcal{L}}\xrightarrow{\omega}\Omega_X^q$ be an integrable $q$-form and ${\mathcal{E}}$ be the associated subsheaf of $1$-forms ${\mathcal{E}}\subseteq \Omega^1_X$. Let $\mathrm{Sym}^2({\mathcal{E}})$ denote the symmetric power of ${\mathcal{E}}$ and suppose ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))=0$. If ${\mathscr{I}}(\omega)={\mathcal{O}}_X$ then ${\mathcal{E}}$ admits a holomorphic connection, in particular is locally free and every Chern class of ${\mathcal{E}}$ vanishes.

In the case of toric varieties we can extend some of the results known for ${\mathbb{P}}^n$ by making use of the Cox ring, in this regard we obtain Theorem \[prop1notinI\]:

Let $X_\Sigma$ be a projective simplicial toric variety, $\omega\in H^0(X_\Sigma,\widehat{\Omega}^q(\beta))$ a codimension $q$ foliation and let $\mathcal{E}=\{\eta\in\widehat\Omega^1\,\colon\,\omega\wedge\eta=0\}$. Assume that $\mathcal{E}$ is locally free (*e.g.* $q=1$), $H^1(X_\Sigma,\mathcal{E})=0$ and $\beta$ not a torsion element. Then $\omega$ has persistent singularities.

Furthermore, if instead ${\mathcal{E}}$ locally free, we require $\widehat\Omega^1/{\mathcal{E}}$ reflexive, the same conclusion holds.

Kupka scheme in the Projective space for codimension 1 foliations
=================================================================

Along this section we will revisit some definitions that we used in [@mmq], among them we will define the Kupka variety as a projective scheme ${\mathpzc{Kup}(\omega)}$ over ${\mathbb{P}}^n$ and $I=I(\omega)$ the *ideal of persistent singularities* (*a.k.a.* *unfoldings ideal*) of $\omega$. Then we will recall some results that we proved in *loc. cit.* that 