as ${\mathscr{J}}$ if no confusion arises.

This definition agrees with Remark \[1notinI\], where we said that the ideal $J(\omega)$ gives the ideal defining the singular locus of $\omega$.

From [@suwa-review]\[(4.6) Definition, p. 192\] we get the following definition for a codimension $q$ foliation:

Let $S$ be a scheme, $p\in S$ a closed point, and ${\mathcal{L}}\xrightarrow{\omega} \Omega^q_X$ a codimension $q$ foliation on $X$. An *unfolding* of $\omega$ is a codimension $q$ foliation $\widetilde{{\mathcal{L}}}\xrightarrow{\widetilde{\omega}}\Omega^q_{X\times S}$ on $X\times S$ such that $\widetilde{\omega}|_{X\times\{p\}}\cong \omega$. In the case $S=\mathrm{Spec}(k[x]/(x^2))$ we will call $\widetilde{\omega}$ a *first order infinitesimal unfolding*.

Kupka scheme in general for codimension 1 foliations
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Over this section we will restate the definition of *persistent singularities* and of the *Kupka scheme*, through its ideal sheaf, in a more general setting, see Definition \[II2\] and Definition \[KK2\], respectively. In [@mmq] we showed that persistent singularities are related to unfoldings in codimension one. We want to extend this relation to higher codimension.

First we prove Proposition \[JJcIIcKK\], generalizing Proposition \[incl2\] in the codimension one case. Then we define the Kupka scheme and we prove Theorem \[propP=K\] and Theorem \[teoKnotempty\], generalizing Theorems \[teo1\] and \[teo3\].

Given a line bundle ${\mathcal{L}}$ and a global section $\omega\in H^0\left(X,\Omega^1_X\otimes 