unfolding $\widetilde{\omega}$ of $\omega$ in $X_{\mathfrak{p}}$, the point $({\mathfrak{p}}, 0)\in X_{\mathfrak{p}}\times \mathrm{Spec}(k[\varepsilon]/(\varepsilon^2))$ is a singular point of $\widetilde{\omega}$.

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 $\iota:X\cong X\times\{0\}\hookrightarrow X\times S$ be the inclusion. Then the sheaf $\Omega^1_{X\times S}$ can be decomposed as direct sum of $\iota_*({\mathcal{O}}_X)$-modules as $$\Omega^1_{X\times S}\cong \iota_*(\Omega^1_X)\oplus \epsilon\cdot \iota_*(\Omega^1_X)\oplus \iota_*({\mathcal{O}}_X)d\epsilon.$$ A point ${\mathfrak{p}}\in{\mathrm{Sing}(\omega)}$ is *not* a persistent singularity if and only if $1\in {\mathscr{I}}_{\mathfrak{p}}\subseteq {\mathcal{O}}_{X,{\mathfrak{p}}}$ which, by Remark \[remarkI\], means that there is an open neighborhood $U\subseteq X$ of ${\mathfrak{p}}$, a local generator $\varpi$ of the image of ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$, and a section $\eta\in\Gamma(U,\Omega^1_X/{\mathcal{L}})$ such that $d\omega=\omega\wedge \eta$. By shrinking $U$ if necessary we can take a lifting of $\eta$ in $\Omega^1_X$ which by abuse of notation we also call $\eta$ and define $$\widetilde{\omega}= \omega + \varepsilon \eta + d\varepsilon.$$ Thus $\widetilde{\omega}$ is a form in $\Omega^1_{X\times S}$ and $\widetilde{\omega}({\mathfrak{p}},0)=
d\varepsilon\neq 0$, so ${\mathfrak{p}}\times \{0\}$ is *not* a singular point of $\widetilde{\omega}$. Reciprocally, if there is an unfolding $\widetilde{\omega}$ of $\omega|_U$, then $$\widetilde{\omega}({\mathfrak{p}}, 0)=\omega({\mathfrak{p}}) + h(0) d\varepsilon.$$ As ${\mathfrak{p}}$ is a singular point of $\omega$, we have $\omega({\mathfrak{p}})$, so if $({\mathfrak{p}},0)$ is not a singular point of $\widetilde{\omega}$, then $h(0)\neq 0$, 