The Kupka Scheme and Unfoldings

Let $ω$ be a differential 1-form defining an algebraic foliation of codimension 1 in projective space. In this article we use commutative algebra to study the singular locus of $ω$ through its ideal of definition. Then, we expose the relation between the ideal defining the Kupka components of the singular set of $ω$ and the first order unfoldings of $ω$. Exploiting this relation, we show that the set of Kupka points of $ω$ is generically not empty. As an application of this results, we can compute the ideal of first order unfoldings for some known components of the space of foliations.

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