modules over this ring. In codimension $1$ there is a direct relation between unfoldings and a certain type of singularities which we call persistent singularities. In this respect we prove Proposition \[JJcIIcKK\] relating Kupka and persistent singularities:

Let ${\mathscr{J}}$ be the ideal sheaf of the singular locus of $\omega$, ${\mathcal{K}}$ the ideal of the Kupka singularities of $\omega$ and ${\mathscr{I}}$ the ideal of persistent singularities. Then the following inclusions hold, $${\mathscr{J}}\subseteq {\mathscr{I}}\subseteq {\mathcal{K}}.$$

and Theorem \[teoKnotempty\] stating the existence of Kupka points under certain hypotheses:

Let $X$ be a projective variety and ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$ a foliation of codimension $1$ such that ${\mathscr{J}}(\omega)$ is a sheaf of radical ideals and such that $c_1({\mathcal{L}})\neq 0$ and $H^1(X,{\mathcal{L}})=0$. Then $\omega$ has Kupka singularities.

In higher codimension the relation of persistent and Kupka singularities is not so clear, specially in the case where the foliation is not given locally by a complete intersection of $1$-forms, as in Example \[exampleCodimension2inA3\]. However, under suitable cohomological conditions the absence of persistent singularities impose very strong consequences on the foliation. If ${\mathcal{E}}$ is the sheaf of $1$-forms defining the foliation, and if ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}}, \mathrm{Sym}^2{\mathcal{E}})=0$ then the absence of persistent singularities implies the existence of a connection on ${\mathcal{E}}$, see 