On the Resolutions of 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.

45 pages Addition of new theorems to Section 7 of the paper, replacing the conjectures in the previous version. Adjustment to the proof of Lemma 5.16; and expansion of Remark 6.14