Characteristic Curves of Holomorphic Foliations

Let F be a germ of holomorphic foliation with an isolated singularity at 0 ∈ C2. A characteristic curve of F is a continuous one-dimensional curve tending to 0 ∈ C2, tangent to F and having some “tame” oscillating behavior, which is a kind of generalization of a separatrix. We define a notion of resolution of the set of characteristic curves of F and show that this process gives another way of obtaining the resolution of singularities of the foliation.