On the Resolutions of Non-Dicritical Foliations

TL;DR

This paper introduces jet schemes for holomorphic foliations to characterize simple singularities and establishes the existence of desingularizations for non-dicritical foliations at the germ level.

Contribution

It provides a new jet scheme-based characterization of simple singularities and proves the universal existence of desingularizations for non-dicritical foliations at the germ level.

Findings

Jet schemes characterize simple singularities independently of normal forms.

A criterion for the existence of desingularizations in non-dicritical cases.

Existence of desingularizations at the germ level for non-dicritical foliations.

Abstract

We introduce the jet schemes of a holomorphic foliation, and thereby prove an alternate characterisation of simple singularities of codimension-$1$ foliations, independent of any normal form. This leads to an equivalent condition for the existence of a desingularisation in the non-dicritical case. We then prove that such a desingularisation always exists, at least on the level of germs.