---
abstract: 'Let $\omega$ be a differential $q$-form defining a foliation of codimension $q$ in a projective variety. In this article we study the singular locus of $\omega$ in various settings. We relate a certain type of singularities, which we name *persistent*, with the unfoldings of $\omega$, generalizing previous work done on foliations of codimension $1$ in projective space. We also relate the absence of persistent singularities with the existence of a connection in the sheaf of $1$-forms defining the foliation. In the latter parts of the article we extend some of these results to toric varieties by making computations on the Cox ring and modules over this ring.'
author:
- 'César Massri[^1]'
- 'Ariel Molinuevo[^2]'
- 'Federico Quallbrunn$^*$'
title: Foliations with persistent singularities
---

Introduction {#introduction .unnumbered}
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Overview of the subject and existing work {#overview-of-the-subject-and-existing-work .unnumbered}
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Foliations of arbitrary codimension over algebraic varieties have been considered at least since the seminal works of Malgrange [@malgrange; @malgrange2] in the local case, and Jouanolou [@jou] in a more global approach. Aside from the main result of [@malgrange2] and general definitions, most of the early theorems about foliations on projective algebraic varieties have been formulated for codimension $1$ foliations on 