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.

This follows from Proposition \[propc1L=0\] and Theorem \[propP=K\], as a foliation with $c_1({\mathcal{L}})\neq 0$ and $H^1(X,{\mathcal{L}})=0$ has persistent singularities on one hand, and having radical singular ideal implies the reduced scheme defined by persistent singularities is equal to the reduced scheme of Kupka singularities, in particular this last scheme is not empty.

Infinitesimal unfoldings in codimension $q$
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Along this section we review the definition of unfolding of a codimension $q$ foliation on a variety $X$. We will also generalize the definitions of persistent singularities and of Kupka singularities for codimension $q$ foliations, see Definition \[defII\] and Definition \[defKK\], respectively. We classify which singular points of $\omega$ are such that they extend to singular points of every unfolding $\widetilde{\omega}$ (Proposition \[prop\]) and then, we generalize Proposition \[incl2\] and Proposition \[JJcIIcKK\] to the codimension $q$ case (Proposition \[JJcIIcKK2\]). Finally, with Theorem \[teoConnectionE\] we establish that the absence of persistent singularities implies the existence of a connection on ${\mathcal{E}}$, the sheaf of 1-forms defining the foliation.

Let $S=\mathrm{Spec}(k[\varepsilon]/(\varepsilon^2))$, $0\in S$ be its closed point, $p:X\times S\to S$ be the projection and $\iota:X\cong X\times\{0\}\hookrightarrow X\times S$ 