Ramifications, Reduction of Singularities, Separatrices

Abstract

We study the behaviour of sequences of blowing-ups under ramifications, and use the results to give a simple proof of Camacho-Sad's Theorem on the existence of Separatrices for singularities of plane holomorphic foliations. The main result we prove is that for any finite sequence $\pi$ of blowing-ups, there is a ramification morphism $\rho$ such that the elimination of indeterminations $\tilde{\pi}$ of $\pi^{-1}\circ \rho$ is a sequence of blowing-ups \emph{with centers at regular points of the exceptional divisors}; moreover, we show that if $\pi$ is the reduction of singularities of a foliation $\mathcal{F}$, then $\rho$ can be such that $\tilde{\pi}^{\star}\rho^{\star}\mathcal{F}$ has only simple singularities.