The Kupka Scheme and Unfoldings

TL;DR

This paper investigates the structure of algebraic foliations in projective space, focusing on Kupka components and their relation to unfoldings, providing new insights into their generic properties and computational aspects.

Contribution

It establishes a connection between Kupka components and first order unfoldings, demonstrating that Kupka points are generically present and enabling computation of unfolding ideals.

Findings

Kupka points are generically non-empty in the foliation space.

The relation between Kupka components and unfoldings is explicitly characterized.

The method allows computation of unfolding ideals for known foliation components.

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.