Reduction of Singularities of Three-Dimensional Line Foliations

We give a birational reduction of singularities for one dimensional foliations in ambient spaces of dimension three. To do this, we first prove the existence of a Local Uniformization in the sense of Zariski. The reduction of singularities is then obtained by a gluing procedure for Local Uniformization similar to Zariski’s one, using an axiomatization of Zariski’s work by Olivier Piltant.