The Kupka Scheme and Unfoldings

Abstract

Let $\omega$ 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 $\omega$ through its ideal of definition. Then, we expose the relation between the ideal defining the Kupka components of the singular set of $\omega$ and the first order unfoldings of $\omega$. Exploiting this relation, we show that the set of Kupka points of $\omega$ 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.