Vector fields tangent to foliations and blow-ups (Q2879061)

In the paper under review, the authors consider the germs of holomorphic vector fields at the origin of \((\mathbb C^3,0)\) having a formal invariant curve that is totally transcendental. Let \(\xi\) be such a germ of vector field. They prove that if there exists a germ of codimension one holomorphic foliation of \((\mathbb C^3,0)\) such that \(\xi\) is tangent to it, then the vector field can be desingularized by a sequence of blowing-ups with center a point or some invariant curve. Then it follows that any germ of vector field tangent to a codimension one foliation can be desingularized.