A series of Nash resolutions of a singular foliation

TL;DR

This paper introduces a series of blowups of singular foliations using Nash modifications, transforming any singular foliation into a Debord foliation after one blowup, with connections to existing concepts.

Contribution

It constructs a new series of blowups of singular foliations via Nash modifications, unifying and extending previous notions by Sert"oz and Mohsen.

Findings

Any singular foliation becomes a Debord foliation after one blowup.

The series of blowups generalizes previous constructions by Sert"oz and Mohsen.

Examples illustrate the effectiveness of the method.

Abstract

We construct a series of blowups $(\widetilde M_i,\pi_i)_{i\in \mathbb N_0}$ of a singular foliation by applying to the universal Lie $\infty$-algebroid of a singular foliation the so-called Nash modification. For $i=0$, we recover a blowup introduced Sinan Sert\"oz, and for $i=1$, we recover a notion due to Omar Mohsen. One of the important features is that any singular foliation becomes a Debord foliation (= projective singular foliation) after one blowup. Examples are also given.