definition above it is not the standard definition of the Kupka set. Usually it is defined just as the set of points in ${\mathrm{Sing}(\omega)}_{set}$ such that $d\omega(p)\neq 0$. Instead, we consider the closure of that set.

\[KK\] For $\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e)$, we define the *Kupka scheme* ${\mathpzc{Kup}(\omega)}$ as the scheme theoretic support of $d\omega$ at $\Omega^2_{S}\otimes_S 
(S\big/J)$. Then, ${\mathpzc{Kup}(\omega)}={\mathrm{Proj}}(S/K(\omega))$ where $K(\omega)$ is the homogeneous ideal defined as $$K(\omega)={\mathrm{ann}}(\overline{d\omega})+J(\omega)\subseteq S,\quad \overline{d\omega}\in \Omega^2_{S}\otimes_S \left(S\big/J(\omega)\right).$$ We will denote $K=K(\omega)$ if no confusion arises.

We recall the notion of *ideal quotient* of two $S$-modules $M$ and $N$ as $$(N:M) := \left\{a\in S: a.M\subseteq N\right\},$$ then, one could also define $K(\omega)$ as $K(\omega)=(J\cdot \Omega^2_S: d\omega)$. Also, given that $\Omega^2_S$ is free, we can also write $$\label{Kbis}
K(\omega)=(J(\omega):J(d\omega)),$$ where $J(d\omega)$ denotes the ideal generated by the polynomial coefficients of $d\omega$.

From the properties of ideal quotient, it follows that if $J$ is radical, then $K$ is radical as well. With the Example 4.5 in [@mmq]\[p. 1034\] we showed that the algebraic geometric approach is indeed necessary, since the reduced structure associated to the Kupka scheme ${\mathcal{K}}$ differs from the reduced variety associated to $\mathpzc{K}_{set}$. With the following lemma we show that the Kupka scheme and the Kupka set 