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.

24 pages. Corrected version. To appear in The Asian Journal of Mathematics

References in corpus (1)

• Families of distributions and Pfaff systems under duality

Cited by in corpus (1)

• Foliations with persistent singularities