which proves the second inclusion.

With the following results we can generalize Theorem \[teo1\] and Theorem \[teo3\] giving conditions for the existence of Kupka singularities:

Let $X$ be a smooth projective variety and ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$ a foliation of codimension $1$, we are going to call ${\mathpzc{Per}(\omega)}\subseteq X$ the subschemes of persistent singularities.

\[propP=K\] Let $X$ be a smooth 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. Let ${\mathpzc{Per}(\omega)}\subseteq X$ and ${\mathpzc{Kup}(\omega)}\subseteq X$ be the subschemes of persistent and Kupka singularities respectively. Then ${\mathpzc{Per}(\omega)}_{\mbox{red}}={\mathpzc{Kup}(\omega)}_{\mbox{red}}$.

We are going to prove that $X\setminus{\mathpzc{Per}(\omega)}=X\setminus{\mathpzc{Kup}(\omega)}$. By Proposition \[JJcIIcKK\] we have ${\mathpzc{Kup}(\omega)}\subseteq{\mathpzc{Per}(\omega)}$, so $X\setminus {\mathpzc{Per}(\omega)}\subseteq X\setminus {\mathpzc{Kup}(\omega)}$. Now suppose ${\mathfrak{p}}$ is a point *not* in ${\mathpzc{Kup}(\omega)}$, by abuse of notation we will call $\omega$ a local generator of ${\mathcal{L}}_{\mathfrak{p}}$ viewed as a $1$-form. As ${\mathfrak{p}}$ is not in ${\mathpzc{Kup}(\omega)}$ then $d\omega \in {\mathscr{J}}_{\mathfrak{p}}\cdot\Omega^2_{X,{\mathfrak{p}}}$. By hypothesis ${\mathscr{J}}_{\mathfrak{p}}$ is radical and so by Theorem \[teodivision\] we have that $d\omega$ decomposes as $\omega\wedge\eta$ for some formal $1$-form $\eta$, this implies $1\in {\mathscr{I}}_{\mathfrak{p}}$, so ${\mathfrak{p}}$ is not in ${\mathpzc{Per}(\omega)}$ (see Remark \[formalremark\]).

\[teoKnotempty\] Let $X$ be a smooth projective variety and ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$ a foliation of codimension $1$ such that ${\mathscr{J}}(\omega)$ is 