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Zbl 0786.32023
Cano, Felipe
Reduction of the singularities of non-dicritical singular foliations. Dimension three.
(English)
[J] Am. J. Math. 115, No.3, 509-588 (1993). ISSN 0002-9327; ISSN 1080-6377/e

Let ${\cal F}$ be a singular foliation of codimension one on $\bbfC\sp 3$, i.e. ${\cal F} \subseteq \Omega\sb{\bbfC\sp 3}$ is locally free of rank one, ${\cal F} \wedge d {\cal F}=0$ and $\Omega\sb{\bbfC\sp 3}/{\cal F}$ is torsion free. ${\cal F}$ is nondecritical if and only if all the irreducible components of the exceptional divisor are leaves of the strict transform $\tilde{\cal F}$ of ${\cal F}$ under any finite sequence of permissible blowing-ups.\par Using the notion of an adapted multiplicity of $\tilde{\cal F}$ (which coincides with the multiplicity in case of ${\cal F}=(df))$ it is proved that for nondecritical singular foliations there is a finite sequence of permissible blowing-ups such that each singularity of the strict transform has adapted multiplicity less or equal to one.
[G.Pfister (Berlin)]
MSC 2000:
*32S65 Singularities of holomorphic vector fields
32S45 Resolution of singularities, etc. (analytic spaces)

Keywords: singular foliation

Cited in: Zbl 0922.32021

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