The Kupka scheme and unfoldings

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

Keywords

Kupka singularities, foliation, unfoldings

2010 Mathematics Subject Classification

32S65, 37F75

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The authors were fully supported by CONICET, Argentina.

Received 15 December 2015

Accepted 16 February 2017

Published 6 February 2019