A series of Nash resolutions of a singular foliation

We construct a series of blowups $({\tilde{M}}_i,{\pi }_i{)}_{i\in {\mathbb{N}}_0}$ of a singular foliation by applying to the universal Lie $\mathrm{\infty }$-algebroid of a singular foliation the so-called Nash modification. For $i=0$, we recover a blowup introduced Sinan Sertöz, 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.