Ramifications, Reduction of Singularities, Separatrices
P. Fortuny Ayuso
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 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.
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\begin{document}
\title{Ramifications, Reduction of Singularities, Separatrices} 
\author{P. Fortuny Ayuso} 
\address{Department of Mathematical Sciences. School of Mathematics.
  Queen Mary College, University of London. Mile End Road. E1 4NS London, UK.}
\email{p.fortuny@maths.qmul.ac.uk}
%\ead{pf@maths.qmul.ac.uk} 
\thanks{Supported by the European Commission Marie
  Curie Fellowship Programme.}
\begin{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.
\end{abstract}
\maketitle
\section{Introduction}\label{sec:introduction}
In 1982, C. Camacho and P. Sad \cite{Camacho-Sad} proved that any holomorphic foliation
in $({\mathbb C}^{2},0)$ has at least one invariant anaylitc curve passing
through $(0,0)$. This result was searched for, at least, since XIX Century
(cf. \cite{Briot-Bouquet} in which it was proved for what are now called
\emph{simple} singularities). Afterwards, the same Camacho generalized it to
vector fields in singular surfaces whose dual resolution graph is a tree
\cite{Camacho}. In 1993, J. Cano \cite{CanoJ2} gave a
constructive proof, which he simplified in 1997 \cite{CanoJ4}. Camacho and Sad's
requires a detailed (and rather cumbersome) study of the combinatorial behaviour
of what has since been called the \emph{Camacho-Sad index} of an invariant
manifold. J. Cano's, on the other hand, although being much simpler, introduces
a somewhat artificial condition on that index.\par
Our work relies on the ---new, as far as the author knows--- study of the
behaviour of sequences of blowing-ups under ramification morphisms. These maps
were used successfully in \cite{Fortuny-Sanz} regarding Thom's Gradient
Conjecture (see \cite{Kurdyka-Mostowski-Parusinski}) and \cite{Corral} for
studying some properties of the polar curves of a foliation. In the present
paper we show ---roughly speaking--- how a sequence of blowing-ups can be modified
by a ramification to obtain a new sequence \emph{without combinatorial
  terms}: in the same way as one can modify a (singular) plane curve by a ramification in
order to get a union of regular curves, we ``modify'' a sequence of blowing-ups
in order to get another one which is ``regular''.
\input{reductions_and_ramific.ltx}
\input{residues.ltx}
\input{the_proof.ltx}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\begin{thebibliography}{10}

\bibitem{Briot-Bouquet}
C.~A. Briot and J.~C. Bouquet, \emph{Propri{\'e}t{\'e}s des fonctions
  d{\'e}finies par des equations diff{\'e}rentielles}, Journal de L'{\'E}cole
  Polytechnique (1856), no.~36, 133--198.

\bibitem{Camacho}
C.~Camacho, \emph{Quadratic forms and holomorphic foliations on singular
  surfaces}, Math. Ann. (1988), no.~282, 177--184.

\bibitem{Camacho-Sad}
C.~Camacho and P.~Sad, \emph{Invariant varieties through singularities of
  holomorphic vector fields}, Ann. of Math. \textbf{115} (1982), 579--595.

\bibitem{Camacho-Sad-Coloquio}
C.~Camacho and P.~Sad, \emph{Pontos singulares de equa{\c{c}}{\~o}es
  diferenciais analiticas}, 16 Col{\'o}quio Brasileiro de Matem{\'a}tica, IMPA,
  Rio de Janeiro, 1985.

\bibitem{CanoJ2}
J.~Cano, \emph{An extension of the {Newton-Puiseux} polygon construction to
  give solutions of pfaffian forms}, Ann. de L'Institut Fourier (1993), no.~43,
  125--142.

\bibitem{CanoJ4}
\bysame, \emph{Construction of invariant curves for singular holomorphic vector
  fields}, Proc. of the AMS \textbf{125} (1997), no.~9, 2649--2650.

\bibitem{Corral}
N.~Corral, \emph{Polar curves of singular foliations}, Ph.D. thesis, Univ. de
  Valladolid, 2001.

\bibitem{Fortuny-Sanz}
P.~Fortuny and F.~Sanz, \emph{Gradient vector fields do not generate twister
  dynamics}, Journal of Differential Equations \textbf{174} (2001), 91--100.

\bibitem{Hironaka}
H.~Hironaka, \emph{Resolution of singularities of an algebraic variety over a
  field of characteristic zero}, Ann. of Math. \textbf{79} (1964), 109--306.

\bibitem{Kurdyka-Mostowski-Parusinski}
K.~Kurdyka, T.~Mostowski, and A.~Parusinski, \emph{Proof of the {G}radient
  {C}onjecture of {R}. {T}hom}, To appear in \emph{Ann. of Math.}

\end{thebibliography}
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math0111244.extracted_bib
\begin{thebibliography}{10}

\bibitem{Briot-Bouquet}
C.~A. Briot and J.~C. Bouquet, \emph{Propri{\'e}t{\'e}s des fonctions
  d{\'e}finies par des equations diff{\'e}rentielles}, Journal de L'{\'E}cole
  Polytechnique (1856), no.~36, 133--198.

\bibitem{Camacho}
C.~Camacho, \emph{Quadratic forms and holomorphic foliations on singular
  surfaces}, Math. Ann. (1988), no.~282, 177--184.

\bibitem{Camacho-Sad}
C.~Camacho and P.~Sad, \emph{Invariant varieties through singularities of
  holomorphic vector fields}, Ann. of Math. \textbf{115} (1982), 579--595.

\bibitem{Camacho-Sad-Coloquio}
C.~Camacho and P.~Sad, \emph{Pontos singulares de equa{\c{c}}{\~o}es
  diferenciais analiticas}, 16 Col{\'o}quio Brasileiro de Matem{\'a}tica, IMPA,
  Rio de Janeiro, 1985.

\bibitem{CanoJ2}
J.~Cano, \emph{An extension of the {Newton-Puiseux} polygon construction to
  give solutions of pfaffian forms}, Ann. de L'Institut Fourier (1993), no.~43,
  125--142.

\bibitem{CanoJ4}
\bysame, \emph{Construction of invariant curves for singular holomorphic vector
  fields}, Proc. of the AMS \textbf{125} (1997), no.~9, 2649--2650.

\bibitem{Corral}
N.~Corral, \emph{Polar curves of singular foliations}, Ph.D. thesis, Univ. de
  Valladolid, 2001.

\bibitem{Fortuny-Sanz}
P.~Fortuny and F.~Sanz, \emph{Gradient vector fields do not generate twister
  dynamics}, Journal of Differential Equations \textbf{174} (2001), 91--100.

\bibitem{Hironaka}
H.~Hironaka, \emph{Resolution of singularities of an algebraic variety over a
  field of characteristic zero}, Ann. of Math. \textbf{79} (1964), 109--306.

\bibitem{Kurdyka-Mostowski-Parusinski}
K.~Kurdyka, T.~Mostowski, and A.~Parusinski, \emph{Proof of the {G}radient
  {C}onjecture of {R}. {T}hom}, To appear in \emph{Ann. of Math.}

\end{thebibliography}