The Kupka Scheme and Unfoldings

Abstract

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 these results, we can compute the ideal of first order unfoldings for some known components of the space of foliations.