\documentclass[graybox,envcountsect,envcountsame]{svmult}
\usepackage{newtxmath}% selects Times Roman as basic font
%\usepackage{newtxtext}       %  selects Times Roman as basic font
% see the list of further useful packages
% in the Reference Guide
\usepackage{amssymb,amsmath, pinlabel, mathabx, tikz, xcolor, lineno}% choose options for [] as required from the list
% in the Reference Guide

\usepackage{type1cm}        % activate if the above 3 fonts are
                            % not available on your system
%
\usepackage{makeidx}         % allows index generation
\usepackage{graphicx}        % standard LaTeX graphics tool
                             % when including figure files
\usepackage{multicol}        % used for the two-column index

\usepackage[bottom]{footmisc}% places footnotes at page bottom

%\usepackage{mathabx}
\usepackage{kantlipsum} %para subti?tulo
\usepackage{mathtools}



%\usepackage{mathrsfs}
%\usepackage{mathabx}
%\usepackage{hyperref}
% \usepackage{mathrsfs}
\usepackage{kantlipsum}
\usepackage[curve]{xypic}
\usepackage{xy}
\xyoption{all}
%\usepackage{tikz}
\usetikzlibrary{arrows,decorations,shapes,shadows}
\usetikzlibrary{decorations.pathreplacing}
\usetikzlibrary{shapes.geometric}
\usetikzlibrary{calc,shadows}
%\usepackage{verbatim}
%\usepackage[normalem]{ulem}
\usepackage{bookmark} % package to set bookmarks to the pdf file

%%%%  Macros for the package bookmark to set bookmark level to title and author
%\makeatletter
%\providecommand*{\toclevel@titlech}{0} 
%\edef\toclevel@authorch{\the\numexpr\toclevel@titlech+1} 
%\makeatother
\def\negthinspace{\kern-2.5em}



\newcommand\Q{{\mathbf Q}}
\newcommand\R{{\mathbf R}}
\newcommand\C{{\mathbf C}}
\newcommand\Z{{\mathbf Z}}
\newcommand\N{{\mathbf N}}
\newcommand\Nn{{\mathcal N}}
\newcommand{\norm}[1]{\lVert #1 \rVert}
\newcommand{\grassman}{\mathbf G}
\newcommand{\lcm}{\operatorname{lcm}}
\newcommand{\red}{\operatorname{red}}
\newcommand{\tot}{\operatorname{tot}}
\newcommand{\specan}{\operatorname{specan}}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\bl}{\operatorname{Bl}}
\newcommand{\proj}{\operatorname{Proj}}
\newcommand{\Oo}{\mathcal{O}}
\newcommand{\Pp}{\mathbf{P}}
\newcommand{\B}{\mathbf{B}}
\newcommand{\Dd}{\mathbf{D}}
\newcommand\Ll{{$\mathcal L$}}
\renewcommand\P{{$\cal P$}}
\newcommand\T{{$\cal T$}}
\newcommand\A{{$\cal A$}}
 
\makeindex           % used for the subject index
                       % please use the style svind.ist with
                       % your makeindex program

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\title{The biLipschitz geometry of complex curves: an algebraic approach}

\author{Arturo Giles Flores, Otoniel N. Silva, Bernard Teissier}
\titlerunning{biLipschitz geometry of complex curves: an algebraic approach}
\authorrunning{Arturo Giles Flores, Otoniel N. Silva, Bernard Teissier}

\begin{document}
\maketitle
\begin{abstract}\noindent
The purpose of these notes is to explain why a generic projection to the plane of a germ of reduced space curve is a biLipschitz homeomorphism for the outer metric. This is related to the fact that all topologically equivalent germs of plane curves are exactly the generic projections of a single space curve. The analytic algebra of this space curve is the algebra of Lipschitz meromorphic functions on any of its generic projections. An application to the geometry of polar curves is given.
\end{abstract}
%\tableofcontents
\section*{Introduction}
These are the lecture notes of the course given by Bernard Teissier during the second week of the `` International School on Singularities and Lipschitz Geometry" which took place in Cuernavaca, Mexico from june 11 to june 22, 2018. The aim of the course was to explore the concept of ``generic plane linear projection'' of a complex analytic germ of curve in $\C^N$. The objects of our study will therefore be germs of curves $(X,0) \subset (\C^N,0)$, linear map germs $\pi\colon(\C^N,0) \rightarrow (\C^2,0)$, and the images $(\pi(X),0) \subset (\C^2,0)$.\par Intuitively, a projection $\pi$ is generic for $(X,0)$ if a small variation of $\pi$ does not change the "equisingularity type" (or embedded topological type) of the image $(\pi(X),0)$ in $(\C^2,0)$. \\
The main objective was to provide algebraic criteria for a projection to be generic and to use them to prove two results related to Lipschitz geometry:\par\smallskip\noindent
(1) That all equisingular (topologically equivalent) germs of reduced plane curves are, up to analytic isomorphism,  images of a single space curve $(X,0)\subset (\C^N,0)$  by generic linear projections $\pi\colon\C^N\to\C^2$, and that the restriction $\pi\vert (X,0)\colon (X,0)\rightarrow (\pi(X),0)$ to $(X,0)$ of such a generic projection is a biLipschitz map for the metrics induced by the hermitian metrics of their respective ambient spaces. In particular, all topologically equivalent germs of plane curves are biLipschitz equivalent.\par\smallskip\noindent
(2) Given a reduced equidimensional germ of a complex space $(X,0)\hookrightarrow (\C^N,0)$, with dimension $d$, we consider a ``general'' projection $\pi:\C^N\rightarrow \C^2$ and the \textit{polar curve on X} associated to the projection $\pi$. It is the closure in $X$ of the critical locus of the restriction of $\pi$ to the smooth part of $X$. If it is not empty, it is a curve usually denoted by $P_{d-1}(X,\pi)$ which plays an important role in the study of the Lipschitz geometry of $X$. We can consider $\pi$ as defining a plane projection of the space curve $(P_{d-1}(X,\pi),0)$ which varies with $\pi$. The result is that if the projection $\pi$ is sufficiently general, then it is a generic plane projection for the curve $(P_{d-1}(X,\pi),0) \subset (\C^N,0)$.\par
The course assumed a certain familiarity with algebraic or complex analytic geometry, such as the definition of a complex analytic space $X$, the fact that its local algebras of functions are analytic algebras, that is, quotients of rings of convergent power series with complex coefficients, that the singular locus ${\rm Sing}X$ consisting of points where the local algebra is not isomorphic to a ring of convergent power series, is a closed analytic subspace, etc. The reader is also encouraged to consult the article \cite{Sn} of Jawad Snoussi in this volume.

\subsection{What is a germ of complex analytic curve?}\label{subsec1}

$\ \ \ \ $   \index{complex curve}A complex analytic curve\footnote{For more details on what follows in this section, we refer the reader to \cite{TeiBia}.} $X$ may be locally  regarded as a family of points in an open subset $U$ of the complex affine space
   $\C^N$ which is the union of finitely many sets of points depending analytically on one complex parameter. It can also be defined as the zero set of a finite number of holomorphic functions $f_1, \ldots,f_s$ on $U$ satisfying certain algebraic conditions:
   \[X =\{ z \in U \, | \, f_1(z)=\cdots=f_s(z)=0\}.\]
 
  \noindent \index{germ of curve}A germ of curve $(X,0)\subset (\C^N,0)$ at a point which we take to be the origin is an equivalence class of curves in open neighborhoods of the origin. Two such objects defined respectively in $U$ and $U'$ are equivalent if their restrictions to a third neighborhood of the origin $U''\subset U\cap U'$ coincide. Of course, when we speak of germs, we think of representatives in some "sufficiently small" neighborhood of the origin. Because of analyticity, to give a germ is equivalent to giving the convergent power series of $f_1,\ldots,f_s$ around the origin with respect to some coordinate system.\\
   
   This allows us to associate to the germ $(X,0) \subset (\C^N,0)$ the analytic algebra of germs of holomorphic functions on $(X,0)$:
   \[\Oo_{X,0}:= \C\{z_1,\ldots,z_N\} / \langle f_1,\ldots,f_s\rangle, \]
   where $\C\{z_1,\ldots,z_N\}$ denotes the ring of convergent power series.  In these notes we will only be interested in reduced germs, meaning that the ideal $J:=\langle f_1,\ldots,f_s\rangle$ is radical and 
   $\Oo_{X,0}$ is a reduced analytic algebra of pure dimension $1$.\\
   
   In the case of plane curves $(N=2)$ the ideal $I=\langle f\rangle\C\{x,y\}$ is principal and $f$ is square free, which means that $f$ has a factorization of the form $f=f_1\cdots f_r$, where each $f_i$ is irreducible in $\C\{x,y\}$ and they are all different. The point is that the $f_i$'s 
   correspond to germs $(X_i,0) \subset (\C,0)$ of analytically irreducible curves called the \textit{branches}\index{branches} 
   of the curve.  \[(X,0) = \bigcup_{i=1}^r (X_i,0).\]
   For arbitrary $N$, the branches $(X_i,0)$ correspond to the prime ideals appearing in the primary decomposition
   of the ideal $(0) $ in $\Oo_{X,0}$ \[ (0)= P_1\cap \ldots \cap P_r,\ {\rm where\ each}\ P_i\ {\rm is\ a\ minimal\ prime}\ {\rm in}\  \Oo_{X,0}\]
   
   A germ of curve $(X,0) \subset (\C^N,0)$ may also be described\index{parametrization} parametrically by $r$ sets 
   of power series 
   \[\varphi_1^i(t_i), \ldots, \varphi_N^i (t_i) \in \C\{t_i\},\ 1\leq i\leq r,\]
   where again $r$ is the number of branches. For each $i,\  z_k=\varphi_k^i (t_i),\ 1\leq k\leq N$ defines a germ of map $(\mathbf{D}_i,0) \longrightarrow (\C^N,0)$ where $\mathbf{D}_i$ is a disk in $\C$. Together these $r$ $n$-uples of series correspond to a multigerm of map 
   \begin{equation} \label{eq:1} \varphi: \bigsqcup_{i=1}^r (\mathbf{D}_i,0) \longrightarrow (\C^N,0);\  z_k=\varphi_k^i (t_i).\end{equation} The connexion between these two definitions goes back to Newton, who showed that an equation $f(x,y)=0,\ \ {\rm with}\ f(0,0)=0$ has solutions $y(x)$ which are power series in $x$ with \textit{rational exponents with bounded denominators} and coefficients in the algebraic closure of the smallest field containing the coefficients of $f(x,y)$. For Newton $f(x,y)$ is a polynomial with real coefficients, but the method works for series over any field $k$ of characteristic zero. Note that if  $\frac{\partial f(x,y)}{\partial y}$ does not vanish at $(0,0)$ the implicit function Theorem gives us a power series $y(x)$ with integers as exponents.  In the general case, such a series $y(x)=\sum_{i\in \N} a_ix^{\frac{i}{n}}$ gives rise to a parametrization \[x=t^n,\ \ y=\sum_{i\in \N} a_it^i.\] of one of the branches of the curve over an algebraic extension of $k$. 
 \subsection{Structuring a parametrization}\label{stru} Suppose that we have an irreducible and reduced germ of curve in $(\C^N,0)$, given by $z_k=\varphi_k(t)\in \C\{t\},\ k=1,\ldots ,N$. For simplicity we shall write $z_k=\varphi_k(t)=\sum_ia^{(i)}_kt^i$. We assume that the group generated by the exponents is $\Z$, which means that they are coprime. Let $n$ be the smallest exponent appearing in all the series $\varphi_k(t)$; up to reindexing the variables $z_i$ we may assume that it is the order of $\varphi_1(t)$, so that we may write $\varphi_1(t) =a^{(n)}_1t^n(1+\psi(t))$ with $\psi(0)=0$. By making a homothetic change on the variable $z_1$ we may assume that $a^{(n)}_1=1$. Since we are in characteristic zero, we may extract an $n$-th root of the unit $1+\psi(t)$ so that $1+\psi(t)=u(t)^n$ where $u(t)$ is again invertible in $\C\{t\}$. Now we make the change of parameter $t'=tu(t)$ so that $\varphi_1(t')=t'^n$. Now by making a linear change of the form $z_i-a_iz_1$ on the coordinates $z_2,\ldots ,z_N$ we may assume that $z_1$ is the only variable where the lowest exponent $n$ appears. Geometrically this means that  our curve is tangent to the $z_1$-axis at the origin: its set-theoretic tangent cone is the $z_1$-axis. Similarly, by making now a non linear change of coordinates of the form $z_i-\sum a^{(i)}_kz_1^k$ we may assume that the first exponent appearing in each $\varphi_k(t')$ is not divisible by $n$. This is geometrically more subtle and corresponds to Hironaka's \textit{maximal contact}. Since $t'$ is now our uniformizing parameter, we call it $t$ henceforth.\par\noindent
Let us now compare $z_1=t^n$ with one of the other coordinates, which we may write (up to a homothetic change of variables) $z_i=\varphi_i(t)=t^{b_i}+\cdots$. It may be that the exponents appearing in $\varphi_i(t)$ and $n$ are not coprime. As we shall see below it means that the projection of our curve to the $(z_1,z_i)$-plane is not reduced. If that is the case, we may begin by dividing all the exponents by their greatest common divisor. The interesting case is therefore that of two series expansions $t^n,\varphi(t)$ with coprime exponents: we are in the case $N=2$ of a plane branch to which we now turn.\par\medskip\noindent 
\textbf{The case of a plane branch.}
As we saw, after a change of coordinates and of uniformizing parameter, we can describe our plane branch by: $z_1=t^n,z_2=\varphi(t)\in\C\{t\}$ where the smallest exponent of $t$ in $\varphi(t)$ is not divisible by $n$. This smallest exponent is traditionally denoted by $\beta_1$. We take the g.c.d. of $n$ and $\beta_1$; set $e_1=(n,\beta_1)<n$. If $e_1=1$, the series $\varphi(t)$ is of the form $t^{\beta_1}+\sum_{k\geq 1}a_kt^{\beta_1+k}$. If $e_1>1$, since the exponents are coprime, there has to be a smallest exponent $\beta_2$ in the series $\varphi(t)$ which is not divisible by $e_1$. Then we set $e_2=(e_1,\beta_2)<e_1$, and we continue in this manner. Since $n>e_1>e_2>\cdots$ there exists an integer $g$ such that $e_g=(e_{g-1},\beta_g)=1$. Finally we have the following structure for $\varphi(t)$: its expansion is decomposed into segments corresponding to the divisibility properties of the exponents. 

 {\small $$z_2=t^{\beta_1}+\sum_{k=1}^{s_1}a_{\beta_1+ke_1}t^{\beta_1+ke_1}+a_{\beta_2}t^{\beta_2}+\sum_{k=1}^{s_2}a_{\beta_2+ke_2}t^{\beta_2+ke_2}+\cdots+a_{\beta_j}t^{\beta_j}+\sum_{k=1}^{s_j}a_{\beta_j+ke_j}t^{\beta_j+ke_j}+$$
$$\cdots+a_{\beta_g}t^{\beta_g}+\sum_{k=1}^\infty a_{\beta_g+k}t^{\beta_g+k},$$}
\noindent where all $a_{\beta_i}$ are $\neq 0$ and each sum has to stop before the g.c.d. of the exponents drops and only the last segment is possibly infinite. 
The set of integers $n,\beta_1,\beta_2,\ldots ,\beta_g$, which is often also denoted by $\beta_0,\beta_1,\beta_2,\ldots ,\beta_g$, is called the \textit{Puiseux characteristic}\index{Puiseux characteristic} of the branch and the $\beta_i$, or sometimes the $\frac{\beta_i}{n}$, are called the \textit{characteristic exponents}. It determines and is determined by the \textit{embedded topological type}\index{embedded topological type}  of the branch (see \cite[\S 7]{Za1}, \cite[Theorem 2.1, pg. 983]{Za3}, \cite{Lej73}). This means that if two germs of plane branches $(X_1,0)$ and $(X_2,0)$ have the same Puiseux characteristic there exists a homeomorphism $(U_1,0)\to (U_2,0)$ of neighborhoods of the origin mapping the representative $X_1\subset U_1$ to $X_2\subset U_2$, and conversely. The two germs are also said to be \index{equisingular}\textit{equisingular}. We shall meet this Puiseux characteristic again after Example \ref{op} below, where we shall see that it determines not only the topology but also the biLipschitz geometry of the branch. \par
After what we have seen, the expansion above can be reinterpreted as a Newton expansion in terms of $t=z_1^{\frac{1}{n}}$, but here we have to choose a $n$-th root of $z_1$. The algebraic interpretation is that $\varphi (z_1^{\frac{1}{n}})\in \C\{\{z_1\}\}\{z_1^{\frac{1}{n}}\}$ determines a cyclic extension of the field $\C\{\{z_1\}\}$ of meromorphic functions in $z_1$ with Galois group equal to the group $\mu_n$ of $n$-th roots of $1$. The $n$ series $\varphi(\omega z_1^{\frac{1}{n}}),\ \omega\in \mu_n$, are the roots of a unitary polynomial $\prod_{\omega\in\mu_n}\bigl(z_2- \varphi(\omega z_1^{\frac{1}{n}})\bigr)\in \C\{z_1\}[z_2]$ whose vanishing is an equation for our germ of curve in the sense we shall see in the next section.\par\noindent
The structure of the series gives rise to a filtration of the Galois group:
\[\mu_n\supset \mu_{e_1}\supset \mu_{e_2}\supset\cdots\supset  \mu_{e_g}=\{1\},\]
with the characteristic property that if we set $n=e_0$ and denote by $\nu_t$ the $t$-adic order of a series, then for $1\leq k\leq g$, we have that \[\omega\in \mu_{e_{k-1}}\setminus \mu_{e_k}\iff \nu_t(\varphi(\omega t)-\varphi(t))=\beta_k.\]
\indent Let us now refine the structure according to \cite[Chapters III, IV,V]{Za2}. The parametrization of a branch by $t^n,y(t)$ as above presents its analytic algebra $\Oo_{X,0}$ as a subalgebra of $\C\{t\}$. The $t$-adic orders of the series in $t$ which are in $\Oo_{X,0}$ form a numerical semigroup $\Gamma\subset \N$ since one can multiply them and stay in $\Oo_{X,0}$. Since the exponents are coprime the complement of $\Gamma$ in $\N$ is finite (Dickson's Lemma) and the semigroup $\Gamma$ is finitely generated. The smallest element $c$ of $\N$ such that all integers $\geq c$ are in $\Gamma$ is called the \index{conductor of a numerical semigroup}\textit{conductor} of the semigroup. It is not difficult to verify (see \cite[Chapter III, Lemma 1.1]{Za2}) that if the order of a series $\xi(t)\in \Oo_{X,0}$ is $>\beta_1$, then $\xi(t)\in \langle x,y\rangle^2$, and therefore if the order $s$ of $\xi(t)$ is in $\Gamma$ we can make a change of coordinates $x'=x, y'=y-\xi(t)$ to eliminate a term in $t^s$ from the expansion of $y(t)$. Using this, and the fact that by definition any element of $\Gamma$ is the order of a series in $\Oo_{X,0}$, Zariski proved in \cite[Chapter III, Proposition 1.2]{Za2}:
\begin{proposition}\label{short}{\rm (Zariski)} 1) Assume that $n>2$. Then one has $c\geq\beta_1+1$. Let $s_1,\ldots ,s_q$ be the integers of the set $\{\beta_1+1,\ldots, c\}$ which do not belong to $\Gamma$. The branch $(X,0)$ is analytically isomorphic to a branch given parametrically by:
\[x'(t)=t^n,\ \ \ y'(t)=t^{\beta_1}+\sum_{i=1}^qa'_{s_i}t^{s_i}.\]
2) If $n=2$ then $\beta_1$ is odd since our germ is irreducible and the conductor is $\beta_1$; our curve is analytically isomorphic to $x(t)=t^2, y(t)=t^{\beta_1}$.
\end{proposition}
Zariski calls this a \textit{short representation}. There are more simplifications of the expansion of $y(t)$ one can make without changing the analytic type. See \cite[Chapters III, IV,V]{Za2}. \par
The next thing we need to know is that the semigroup $\Gamma$ determines and is determined by the Puiseux characteristic of the branch: it is a complete invariant of the equisingularity class. See \cite[Chap. II, \S 3]{Za2}. In particular, in the short expansion, the coefficients of the $t^{\beta_i}$ are $\neq 0$.

With this description of branches, we are able to describe the \textit{contact} of two branches, which plays a key role in the characterization of the topological (and biLipschitz) type of a reduced germ of plane curve.

We shall see below how, conversely, the image of a parametrization can be defined by equations.\par\noindent The modern presentation of the parametrization of a curve goes through the normalization, which is the topic of the next section. 
   
   \newpage

\section{Normalization}

$\ \ \ \ $   The property of being normal has an algebraic aspect which has to do with \index{integral extension}integral extension of rings. 
   
  \begin{definition}  Let $R \subset S$ be rings. 
    \begin{itemize}
        \item The inclusion $R \subset S$ is called a finite extension if $S$ is a finitely generated $R-$module.
        \item An element $s \in S$ is called integral over $R$ if and only if it satisfies an equation
        \[s^h + a_1s^{h-1}+\cdots+a_{h-1}s + a_h =0\]
        with all $a_i \in R$. The extension is called integral if every element $s \in S$ is integral 
        over $R$. (Just as in field theory, if the extension $R \subset S$ is finite it is integral. See \cite[Lemma 1.5.2]{DeJo-Pfi})
        \item The ring $R$ is said to be integrally closed in $S$ if every element in $S$ which is integral over $R$ already
        belongs to R.
        \item The ring $R$ is called normal if it is reduced and integrally closed in its total quotient ring $Q(R)$.
    \end{itemize}
  \end{definition}
 
   Suppose that $R$ is a reduced ring. Recall that the set of non-zero divisors of a ring $R$ is a multiplicatively closed set and the corresponding ring of fractions $Q(R)$ is called the total ring of fractions. It has the property that the canonical morphism $R \to Q(R)$ is injective.\\
   
      The \index{normalization}\emph{normalization of $R$} is defined as  the set $\overline{R}$ of all elements of $Q(R)$ which are
   integral over $R$. It is a reduced ring, integrally closed in $Q(R)$ and whose total ring of fractions coincides with $Q(R)$. In particular, the normalization $\overline{R}$ is a normal ring. Moreover, for the rings appearing in analytic or algebraic geometry,  the 
   extension $R \subset \overline{R}$ is finite in the sense that $\overline{R}$ is a finitely generated $R$-module.\footnote{It is interesting to note that the term "integral" comes from algebraic number theory in the tradition of Dedekind and the definition of the ring of integers of an algebraic number field, while the term "normal" was used by Zariski (see \cite{Za}) in the course of his studies in birational geometry and resolution of singularities to designate an algebraic variety which could \textit{not} be presented as the image of a different one by a finite birational map. This is why the terms "integral closure in the total ring of fractions" and "normalization" are used in algebraic or analytic geometry as names for the algebraic and geometric aspects of the same operation.}\\
   
     So what about if we start with the analytic algebra $\Oo_{X,0}$ of a germ of analytic space $(X,0) \subset (\C^N,0)$? We will say that the germ $(X,0)$ is normal if $\Oo_{X,0}$ is a normal ring.\\
     \begin{itemize}
         \item Unique factorization domains are normal (\cite[Thm 1.5.5]{DeJo-Pfi}) so the ring of power series $\C\{z_1,\ldots,z_n\}$ and the corresponding smooth germ $(\C^N,0)$ are normal. 
         \item Noetherian normal local rings are integral domains (\cite[Thm 1.5.7]{DeJo-Pfi}), so a normal
               germ $(X,0)$ is irreducible.
                \item Suppose $(X,0)$ is irreducible. Since $\Oo_{X,0}$ and its normalization have the same 
          total ring of fractions, which in this case is a field, it follows from what we have just seen that $\overline{\Oo_{X,0}}$ is a local 
          noetherian domain. Moreover,  by \cite[Cor. 3.325]{DeJo-Pfi} it is an analytic algebra and so 
          we can associate to it a normal germ $(\overline{X},0)$. In particular we have:
          \[ \Oo_{\overline{X},0} = \overline{\Oo_{X,0}}.\]
 \item Splitting of normalization (\cite[Thm. 1.5.20]{DeJo-Pfi}) tells us that that if we
         have the irreducible decomposition
         \[ (X,0)= (X_1,0) \cup \ldots \cup (X_s,0),\]
         then the normalization $\overline{\Oo_{X,0}}$ is equal to a direct sum of analytic algebras which are the normalizations of 
         the analytic algebras $\Oo_{X_i,0}$ corresponding to the irreducible components $(X_i,0)$:
         \[ \overline{\Oo_{X,0}}= \bigoplus_{i=1}^s \overline{\Oo_{X_i,0}}. \]
                  Note that this implies that $(X,0)$ and $(\overline{X},0)$ have the same dimension. 
     \end{itemize}
     
      A \index{multi-germ}\emph{multi-germ} of analytic spaces $(X,x)$ is a finite disjoint union:
          \[ (X,x):= (X_1,x_1) \sqcup (X_2,x_2) \sqcup \ldots \sqcup (X_r,x_r)\]
          of germs of analytic spaces. The ring $\Oo_{X,x}$ by definition is equal to $\bigoplus_{i=1}^r \Oo_{X_i,x_i}$. The multigerm $(X,x)$ is called normal if $\Oo_{X,x}$ is a normal ring. 
          
      Let $(Y,y)=(Y_1,y_1) \sqcup  \ldots \sqcup (Y_s,y_s)$ be another multi-germ. A map  
      $\varphi: (X,x) \to (Y,y)$ of multi-germs is given by a system of maps
      \[ \varphi_i: (X_i,x_i) \to (Y_{\alpha(i)}, y_{\alpha(i)}), \hspace{0.2in} i\in 
      \{1,\ldots,r\}, \, \alpha(i) \in \{1,\ldots,s\} .\]
      Such a map $\varphi$ induces, and is induced by, a $\C-$algebra map $\varphi^*: \Oo_{Y,y} \to
      \Oo_{X,x}$.
        
     \begin{definition}\label{Normalization}
         Let $(X,x)$ be a germ of analytic space. A normalization of $(X,x)$ is a normal
      multi-germ $(\overline{X},\overline{x})$ together with a finite, generically 1-1 map
       \[n:(\overline{X},\overline{x}) \to (X,x).\]
     \end{definition}
     
        With this definition at hand, for any germ of analytic space $(X,0)$ with irreducible decomposition
    \[ (X,0)= (X_1,0) \cup \ldots \cup (X_s,0),\]
      we can now obtain a normal multigerm 
    \[ (\overline{X},\overline{x}) = (\overline{X_1},x_1) \sqcup \ldots \sqcup (\overline{X_s},x_s)\] 
    with associated normal ring
    \[\overline{\Oo_{X,0}}= \bigoplus_{i=1}^s \overline{\Oo_{X_i,0}} = \bigoplus_{i=1}^s \Oo_{\overline{X_i},x_i},\]
   and it is not hard to prove that the inclusion map $\Oo_{X,0} \hookrightarrow \overline{\Oo_{X,0}}$ induces 
   a finite and generically 1-1 map, proving thus the existence of normalization (\cite[Thm 4.4.8]{DeJo-Pfi}).
   Note that, geometrically, the normalization of a germ separates the irreducible components and normalizes each of them separately.
   
   \begin{example}
      Let $(X,0) \subset (\C^2,0)$ be the germ of plane curve defined by $f(x,y)=x^2-y^2$.
      It has two irreducible components $(X_1,0)$ and $(X_2,0)$ with associated analytic algebras
      \[\Oo_{X_1,0}= \C\{x,y\}/\langle x-y\rangle \hspace{0.3in} \Oo_{X_2,0}= \C\{x,y\}/\langle x+y\rangle .\]
      These two germs are smooth,  in particular they are normal and we have:
      \begin{align*}
              \Oo_{X,0} = \frac{\C\{x,y\}}{\langle x^2-y^2\rangle} & \longrightarrow  \frac{\C\{x,y\}}{\langle x-y\rangle}
              \bigoplus \frac{\C\{x,y\}}{\langle x+y\rangle}= \overline{\Oo_{X,0}} \\
              f & \longmapsto \left( f + \langle x-y\rangle , f + \langle x+y\rangle \right)
      \end{align*}
      Since the germs are smooth and of dimension 1, their analytic algebras are isomorphic to the ring of convergent power series $\C\{t\}$:
      \[ \C\{x,y\}/\langle x-y\rangle \to \C\{t\} \hspace{0.2in} x \mapsto t, \, y \mapsto t\]
       \[ \C\{x,y\}/\langle x+y\rangle \to \C\{u\} \hspace{0.2in} x \mapsto u, \, y \mapsto -u\]
     This means that the resulting normalization map 
     \[n:(\C,0) \sqcup (\C,0) \to (X,0) \]
     is the parametrization of each of the branches $t_1 \mapsto (t,t)$ and $t_2 \mapsto (u,-u)$.  
   \end{example}
   
    It is useful to consider a function-theoretic interpretation of normal spaces. A general result tells us that
    in a smooth germ $(\C^d,0)$  if you have a meromorphic function which is (locally) bounded then it is actually
    holomorphic (See for example \cite[IV.4]{Fr-Gr}).  The algebraic version is that a locally bounded meromorphic function $h$ satisfies an integral dependence relation of the form:
       \[h^m +c_1h^{m-1}+ \cdots+ c_m =0; \hspace{0.2in} c_j \in \Oo_n:=\C\{z_1,\ldots, z_n\},\]
    and since $\Oo_n$ is normal then $h \in \Oo_n$.\\
    Now there are many more analytic spaces for which $\Oo_{X,x}$ is normal than just the non singular ones.
    
    \begin{definition}\label{WeaklyHolDef}
       Given a reduced germ of analytic space $(X,x)$, we call a function $f: X \setminus \mathrm{Sing}\,X \to \C$ \index{weakly holomorphic}\emph{weakly
    holomorphic} at $x\in X$ if : \begin{itemize}
        \item $f$ is holomorphic on $X\setminus \mathrm{Sing}\,X$ in a neighborhood of $x$.
        \item $f$ is (locally) bounded near $x$.
    \end{itemize}
    A function is weakly holomorphic on $X$ if it is so at every point.
    \end{definition}
   
    The key point is proving that the germs at $x\in X$ of weakly holomorphic functions on $X$ form a ring which is canonically isomorphic to the normalization of $\Oo_{X,x}$. That is, $f$ is weakly holomorphic on $X$ if and only if it is 
    meromorphic and satisfies an integral dependence relation. This gives us the following characterization:
    
    \begin{theorem}\label{WealyHolThm} {\rm \cite[Thm 4.4.15]{DeJo-Pfi}} \\
    \begin{itemize}
        \item [1)] Let $(X,x)$ be a germ of reduced analytic space. Then a function $f$ is weakly
        holomorphic on $X$ if and only if $f$ is in the integral closure of $\Oo_{X,x}$ in its total ring of quotients.
        \item [2)] The integral closure of $\Oo_{X,x}$  in its total ring of quotients is a direct sum of analytic algebras.
        \item [3)] The reduced germ $(X,x)$ is normal if and only if every weakly holomorphic function germ can be extended to a holomorphic function.
    \end{itemize}
    \end{theorem}
    \begin{remark}Since this fact is fundamental for what follows, here is an idea of why boundedness and polynomial equation are related: The roots of a polynomial are bounded in terms of its coefficients, so a solution of a polynomial equation with holomorphic coefficients is bounded because holomorphic functions are. In the other direction, let $h=\frac{f}{g}$, with $f,g\in m_{(X,0)}$ be our meromorphic function, let $(Y,0)\subset (X,0)$ be the subset defined by the ideal $\langle f,g\rangle \Oo_{(X,0)}$, and consider the analytic subspace $X'$ of $X\times \Pp^1(\C)$ which is the closure of the graph of the map $X\setminus Y\to \Pp^1(\C)$ defined by $x\mapsto (f(x):g(x))\in \Pp^1(\C)$. It is contained in the hypersurface of $X\times \Pp^1(\C)$ defined by $T_2f(x)-T_1g(x)=0$ where $(T_1:T_2)$ are projective coordinates on $\Pp^1$. The first projection induces a holomorphic map $e\colon X'\to X$ (we are blowing-up the ideal $\langle f,g\rangle$). The fiber over $0$ is a complex analytic subspace of $\Pp^1(\C)$ and therefore is either $\Pp^1(\C)$ or a finite subset of it. If our meromorphic function is bounded, the point $(1:0)\in\Pp^1(\C)$ is not in the fiber, so that by the Weierstrass preparation Theorem (see Theorem \ref{W} below), for a small enough representative $X$ of the germ $(X,0)$ the map $X'\to X$ is finite and $X'$ has to be a hypersurface in $X\times \C$: its equation is our integral dependence relation.
    \end{remark}
    \begin{example}\label{WeaklyHolEx}
      For the germ $(X,0) \subset (\C^2,0)$ defined by $xy=0$ we have
       \[ \overline{\Oo_{X,0}} = \C\{x\} \oplus \C\{y\}.\]
       The function $f=(1,0)$, meaning it is the constant function $1$ on the $x$ axis and the 
       constant function $0$ on the $y$ axis, is holomorphic on $X\setminus \mathrm{Sing}\,X = X 
       \setminus \{0\}$ and is certainly bounded so it is weakly holomorphic. Note that it can not be continuously extended to $(X,0)$. As a meromorphic function it can be written as
        \[f(x,y) =\frac{x}{x+y}.\]
    \end{example}
    
    Let us wrap up this discussion on normal spaces and normalization by stating several important properties of which
    you can find detailed expositions in \cite{Loj}, \cite{GLS07} and \cite{Ka}. \begin{enumerate}
        \item If $X$ is reduced, the \textit{non normal locus} is the set of points $x\in X$ where the local algebra $\Oo_{X,x}$ is not normal; it is the complement of the \index{normal locus}\textit{normal locus} and is a closed analytic subspace contained in the singular locus ${\rm Sing}X$ of $X$. It is defined by the \textit{conductor sheaf} which is the annihilator of the coherent $\Oo_X$-module $\overline{\Oo_X}/\Oo_X$ and thus a coherent sheaf of ideals.
        \item If $T$ is a normal space and $X$ is reduced then any map $T \to X$ which does not map any irreducible
              component of $T$ to the non-normal locus of $X$ factors uniquely through the normalization 
              $n:\overline{X} \to X$.      
        \item If $X$ is normal then $\mathrm{dim}\,\mathrm{Sing}(X) \leq \mathrm{dim}\,X -2$ (Singular locus of 
              codimension at least 2). 
        \item If $X$ is normal, the polar locus of a meromorphic function is either of codimension $1$ or empty. 
    \end{enumerate}   
  \noindent    Going back to the curve case, a classical result of commutative algebra (\cite[Thm 4.4.9]{DeJo-Pfi}) states
    that a Noetherian local ring of dimension one is normal if and only if it is regular. This implies that if 
    $(X,0)=\bigcup_{i=1}^r (X_i,0)\subset (\C^N,0)$ is a germ of analytic curve with $r$ branches then the 
    normal ring $\overline{\Oo_{X,0}}$ is isomorphic to a direct sum of $r$ copies of $\C\{t\}$ and the corresponding
    normalization map is equal to the parametrization of each branch, thus recovering the description in (\ref{eq:1}).\par\noindent
        For plane curves, this result can also be seen using algebraic field extensions, but first we need a couple of definitions and
    the \index{Wierstrass preparation thoerem}Weierstrass preparation Theorem. A  convergent power series $f \in \C\{z_1,\ldots,z_N\}$ is called regular of order $b$ in $z_N$ if the power series $f(0,\ldots,0,z_N)$ in the variable $z_N$ has a zero of order $b$. A
    simple calculation shows that if $f$ is of order $b$ in the sense that $f\in \langle z_1,\ldots,z_N\rangle^b\setminus \langle z_1,\ldots,z_N\rangle^{b+1}$, then after a general linear change of coordinates, $f$ is regular of order $b$ in $z_N$ (see \cite[Lemma 3.2.2]{DeJo-Pfi}). Geometrically this means that if we consider the germ of hypersurface $(X,0) \subset (\C^{N-1} \times \C,0)$ defined by $f$ and the first projection 
    $p: X \to \C^{N-1}$, then for a small enough representative the fiber $p^{-1}(0)$ is the single point $0$.\par\medskip
       
    \begin{theorem}\label{W} \textbf{(Weierstrass Preparation Theorem)} {\rm (see \cite[Thm 3.2.4]{DeJo-Pfi})}\\
       Let $f \in \C\{z_1,\ldots,z_N\}$ be regular of order $b$ in $z_N$. Then there exists a  unique
       monic polynomial $P \in \C\{z_1,\ldots,z_{N-1}\}[z_N]$  
       \[P(z_1,\ldots,z_N)= z_N^b + a_1(z_1,\ldots,z_{N-1})z_N^{b-1}+ \cdots + a_N(z_1,\ldots,z_{N-1})\]
       with $a_i(0)=0$, and a unit $u \in \C\{z_1,\ldots,z_N\}$ such that we have the equality of convergent power
       series
       \[f=uP.\]
    \end{theorem}
    
     As a consequence of this result we deduce two important facts: if we choose adequate coordinates such
     that $f=uP$ then it is equivalent to seek solutions of $f(z_1,\ldots,z_N)=0$ and of $P(z_1,\ldots,z_N)=0$.
     As a geometric consequence of this we get that if we consider the first projection as before
     and $p^{-1}(0)=\{0\}$, then for any point $q=(q_1,\ldots,q_{N-1}) \in \C^{N-1}$ sufficiently close to 
     the origin the points of the fiber $p^{-1}(q)$ correspond to the roots of the polynomial of degree $b$
     \[P(q_1,\ldots,q_{N-1},z_N)= z_N^b + a_1(q_1,\ldots,q_{N-1})z_N^{b-1}+ \cdots + a_N(q_1,\ldots,q_{N-1}),\]
     and so all nearby fibers are also finite. More generally one uses this result to prove that if a complex analytic map $p\colon X'\to X$ is such that for some point $0\in X$ we have that $p^{-1}(0)$ is a finite set, then there exists a neighborhood $U$ of $0$ in $X$ such that the restricted map $p^{-1}(U)\to U$ is finite. See \cite[Thm 3.4.24]{DeJo-Pfi}.\par\medskip
      For curve singularities, there is a classical invariant which measures how far the singularity is from being normal, or non singular. It has several geometric interpretations, the classical one being "diminution of genus", and we shall see more about it below.
    \begin{definition}\label{delinv}
    Let $(X,0)$ be a reduced curve singularity. Its \index{$\delta$ invariant}$\delta$ invariant is $\delta={\rm dim}_\C\frac{\overline{\Oo_{X,0}}}{\Oo_{X,0}}$.
    \end{definition}
    This quotient is a finite dimensional vector space because it is the stalk of a coherent sheaf supported at the origin. For plane, and more generally Gorenstein, branches we have the equality $c=2\delta$, where $c$ is the conductor defined before Proposition \ref{short}. See \cite[Chap. II, \S 1]{Za2}.
\par\medskip

     
     Going back to the plane curve case, that is curves $(X,0) \subset (\C^2,0)$ defined by a convergent 
     power series $f \in \C\{x,y\}$, or according to the Weierstrass preparation Theorem and possibly after a linear change of coordinates,  by a polynomial 
     $P \in \C\{x\}[y]$. Now from an algebraic point of view, consider the field of fractions $\C\{\{x\}\}$ of 
     the integral domain $\C\{x\}$; the irreducible polynomial  $y^n-x \in \C\{\{x\}\}[y ]$ defines an algebraic extension of degree $n$ of $\C\{\{x\}\}$, denoted by $\C\{\{x^{\frac{1}{n}}\}\}$, which is a Galois extension with Galois group equal to the group $\mathbf{\mu}_n$ of $n$-th roots of unity in $\C$. The action of $\mathbf{\mu}_n$ is exactly the change in
     determination of $x^{\frac{1}{n}}$ determined by $x^{\frac{1}{n}} \mapsto \omega x^{\frac{1}{n}}$ for $\omega \in \mathbf{\mu}_n $. A series of
     the form $y=\sum a_{i} x^ {\frac{i}{n}}$ such that the greatest common divisor of $n$ and all the exponents $i$ which effectively appear is $1$ gives $n$ different series as $\omega$ runs through $\mathbf{ \mu}_n$.\\
     
   Suppose now that our polynomial $P$ is an irreducible element of $\C\{x\}[y]$ of degree $n$. 
    Then the Newton polygon method (see for example \cite{TeiBia}, \cite{Chen}, or \cite[Section 8.3]{Brie})  provides a series $y(x^{1/m})\in \C\{x^{\frac{1}{n}}\}$ such that $P(x,y(x^{\frac{1}{n}})) = 0$ 
   and we have the equality:
   \[P(x,y)= \prod_{\omega \in \mathbf{\mu}_{n}}\left(y-y(\omega x^{\frac{1}{n}})\right).\]
   In particular we have that
   \[ \C\{\{x\}\}^*:= \bigcup_{n \in \N} \C\{\{x^{\frac{1}{n}}\}\}  \]
   is an algebraically closed field (See \cite[IV.3]{Wal} or \cite[Thm 8.2.1]{Chen}), and so 
   every polynomial $P \in \C\{x\}[y]$ has all its roots in $\C\{\{x\}\}^*$. Finally, the relation with 
   the parametrizations given by the normalization is the following, if \[y(x^{\frac{1}{n}})\in \C\{x^{\frac{1}{n}}\} \subset
   \C\{\{x\}\}^*\] is a root of $P(x,y)$, then by taking $x=t^n$ we get the parametrization
    \[ t \mapsto (t^n,y(t)).\]
        
    Let us finish this section by looking at plane projections from an algebraic perspective.  For simplicity suppose $(X,0) \subset (\C^N,0)$ is a reduced and irreducible germ of complex analytic curve with $N\geq 3$. Let us write the associated analytic algebra 
    \[\Oo_{X,0}= \frac{\C\{z_1,\ldots,z_N\}}{I}, \]
    where $I$ is a prime ideal, and so $\Oo_{X,0}$ is an integral domain. If we choose a sufficiently general coordinate system (or 
    if you prefer after a general linear coordinate change) the Noether normalization Theorem (\cite[Corollary 3.3.19]{DeJo-Pfi})
    tells us that we have a finite ring extension $\C\{z_1\} \hookrightarrow \Oo_{X,0}$. This implies that the we have an algebraic
    field extension \[\C\{\{z_1\}\} \subset \mathrm{Quot}\left(\Oo_{X,0}\right),\]
    and by the primitive element Theorem there exists an element $f \in \Oo_{X,0}$ such that $\mathrm{Quot}\left(\Oo_{X,0}\right) =
    \C\{\{z_1\}\}[f]$. 
    
    So if we denote by $\C\{z_1,f\}$ the analytic algebra obtained as the quotient of $\C\{x,y\}$ by the kernel of the 
    map $\C\{x,y\} \to \Oo_{X,0}$ defined by $x \mapsto z_1 + J$, $y \mapsto f$ then we have finite ring extensions with the 
    same field of fractions
    \[\C\{z_1,f\} \hookrightarrow \Oo_{X,0} \hookrightarrow \C\{t\}.\]
    Now $\C\{z_1,f\}$ is the analytic algebra of a plane curve $(X_1,0) \subset (\C^2,0)$ and it has the same normalization as $\Oo_{X,0}$. We have used the primitive element Theorem as a substitute for the proof of the existence of a projection 
    $\C^N \to \C^2$ sufficiently general for it to induce a `` bimeromorphic" map $(X,0) \to (X_1,0)$. However the primitive
    element Theorem does not tell us much about the geometric nature of the projection. That is the object of the following sections.
     
  
   
\newpage  
   
\section{Fitting Ideals - A good structure for the image of a finite map}   
     
$\ \ \ \ $ In this section, following \cite[\S 3]{Te2} and \cite [Definition 5.6]{Rim}, \cite[\S 5]{hunting}, we will give the definitions of Fitting ideals, which we will use later to give a definition of the image, as a \textit{complex analytic space}, of a finite map between complex analytic spaces.

Let $A$ be a ring, and let $M$ be an $A$-module of finite presentation, that is, there is an exact sequence, called a presentation of $M$:

\begin{center}
$A^q \overset{\Psi}{\longrightarrow} A^p \longrightarrow M \longrightarrow 0 , $   
\end{center}

\noindent where $p,q \in \N$. For each integer $j$ we associate to $M$ the ideal $F_j(M)$ of $A$ generated by the $(p-j)\times (p-j)$ minors of the matrix (with entries in $A$) representing $\Psi$. Here we need the convention that if there are no $(p-j)\times (p-j)$ minors because $j$ is too large, i.e., $j \geq p$, then $F_j(M)=A$ (the empty  determinant is equal to $1$) and if, at the other extreme, $p-j > q$, set $F_j(M)=0$ (the ideal generated by the empty set is $0$).

A Theorem of Fitting (see \cite[Chap. I, \S 2]{To72}, \cite[Chap. 20, \S 2]{Eis95}) asserts that the ideals $F_j(M)$ depend only on the $A$-module $M$ and not on the choice of a presentation. We call $F_j(M)$ the $j$-th Fitting ideal\index{Fitting ideal} of $M$.

More generally, if $(X,\Oo_X)$ is a ringed space, and $\mathcal{M}$ a coherent sheaf of $\Oo_X$-modules, we can define a sheaf of ideals $\mathcal{F}_i(\mathcal{M})$ of $\Oo_X$, by defining $\mathcal{F}_i(\mathcal{M})$ locally as above, and then by uniqueness the ideals found locally patch up into a sheaf of ideals. Remark also that since $\mathcal{F}_i(\mathcal{M})$ is locally finitely generated, $\mathcal{F}_i(\mathcal{M})$ will be a coherent sheaf of ideals as soon as $\Oo_X$ is coherent, which is the case for a complex analytic space by Oka's Theorem (see \cite[Chap. VI, 1.3]{Loj}).

Let now $f:(X,\Oo_X) \rightarrow (Y,\Oo_Y)$ be a map of complex analytic spaces. We would like to define the image of $f$ as a complex analytic \textit{subspace} of $(Y,\Oo_Y)$. This is not always possible, and in particular if one hopes to get a closed complex subspace of $Y$ it is better to assume $f$ is proper, and here we will consider only the case where $f$ is \textit{finite} (that is, proper with finite fibres).

The first sheaf of ideals that comes to mind as a candidate to define $f(X)$ is the sheaf of functions $g$ on $Y$ such that $g \circ f=0$ on $X$, i.e., the annihilator sheaf of the sheaf of $\Oo_Y$-modules $f_{\ast}\Oo_X$, which is coherent by a theorem of Grauert-Remmert:

\begin{center}
    ${\rm Ann}_{\Oo_Y}(f_{\ast}(\Oo_X))= \  {\rm sheaf}   \lbrace {\rm functions}\  g \ {\rm on}\  Y$ such that $g\cdot f_{\ast}\Oo_X=0 \ \rbrace . $
\end{center}

This is \textbf{not} a good choice because its formation does not commute with base extension, as we will show by an example below (Example \ref{exfitting1}).

The second option is the $0$th Fitting ideal of $f_{\ast}\Oo_X$, which set theoretically also defines the image of $f$, since as a set the subspace of $Y$ defined by it is $\lbrace y \in Y \ | \ {\rm dim}_{\C}(f_{\ast}\Oo_X) > 0 \rbrace = \lbrace y \in Y \ | \ (f_{\ast}\Oo_{X})_y \neq 0 \rbrace $. Indeed, since the $\Oo_Y$-module $f_{\ast}\Oo_X$ is coherent, it has locally on suitable open sets $U$ of $Y$ a presentation by an exact equence of $\Oo_Y(U)$-modules:
$$ \Oo_Y(U)^q \overset{\Psi}{\longrightarrow}  \Oo_Y(U)^p\longrightarrow f_{\ast}\Oo_X(U)\longrightarrow (0).$$
The sheaf of ideals $\mathcal{F}_0(f_{\ast}\Oo_X)$ is then generated on $U$ by the $p\times p$ minors of a matrix representing $\Psi$.

Since both the formation of direct images and the formation of Fitting ideals commute with base change (see proposition \ref{fit} below), this definition of the image will also have this property. So we set:

\begin{definition} Let $f:X \rightarrow Y$ be a finite morphism of complex analytic spaces. The image ${\rm im}(f)$ of $f$ is the subspace of $Y$ defined by the coherent sheaf of ideals $\mathcal{F}_0(f_{\ast}\Oo_X)$. It is sometimes called the \textit{Fitting image}\index{Fitting image} of $f$ to distinguish it from the one defined by the annihilator.
\end{definition}

\begin{proposition}\label{fit}\begin{enumerate} \item The formation of ${\rm im}(f)$ commutes with base change: Given a complex analytic map $\phi\colon T\to Y$, consider the map $f_T\colon X\times_YT\to T$ obtained by base extension, where $X\times_YT$ is the fiber product. Then ${\rm im}(f_T)=\phi^{-1}({\rm im}(f))$ as analytic spaces.
 \item We have the inclusion  $ \mathcal{F}_{0}(f_{\ast}\Oo_X)\subset{\rm Ann}(f_{\ast}\Oo_X)$ and the equality $ \sqrt{\mathcal{F}_{0}(f_{\ast}\Oo_X)}= \sqrt{{\rm Ann}(f_{\ast}\Oo_X)}$. 
\end{enumerate}
\end{proposition} 
\begin{proof}1) Since $\Oo_X$ is a finitely generated $\Oo_Y$-module the $\Oo_T$-module $\Oo_{X\times_YT}$ is equal to $\Oo_X\otimes_{\Oo_Y}\Oo_T$ and if $M$ is a finitely presented $A$-module as above and $A\to B$ is a map of algebras, then \begin{center}$B^q \overset{\Psi\otimes_A1}{\longrightarrow} B^p \longrightarrow M \otimes_AB\longrightarrow 0$\end{center}
is a presentation of $M \otimes_AB$ as a $B$-module and the matrix of $\Psi\otimes_A1$ is the matrix of $\Psi$ so that $F_j(M \otimes_AB)=F_j(M).B$.\par\noindent
2) The inclusion follows directly from Cramer's rule and the equality from the definition of the Fitting ideal as defining the set of points where the cokernel of the second arrow is not zero.
\end{proof}
\begin{example}\label{exfitting1} Let $f:(\C,0) \rightarrow (\C^2,0)$ be given by $x=t^{2k}$, $y=t^{3k}$ for some integer $k$. The set-theoretic image of $f$ is the curve $y^2-x^3=0$. However, we wish to obtain an ideal defining a space supported on that curve, but possibly with nilpotent functions. Let us compute $\mathcal{F}_0(f_{\ast}(\Oo_{\C})_0$ as the $0$-th Fitting ideal of $\C \lbrace t \rbrace $ considered as $\C \lbrace x,y \rbrace $-module via the map of rings $\C\lbrace x,y \rbrace \rightarrow \C \lbrace t \rbrace $ sending $x$ to $t^{2k}$ and $y$ to $t^{3k}$. We must write a presentation of $\C \lbrace t \rbrace $ as $\C \lbrace x,y \rbrace$-module. Let $e_0=1$, $e_1=t,\ldots, e_{2k-1}=t^{2k-1}$. It is easily seen that they form a system of generators of $\C\lbrace t \rbrace$ as $\C\lbrace x,y\rbrace$-module, and that between them we have the following $2k$ relations:

 \begin{center}$
    \begin{array}{cccc}
    xe_k-ye_0 & =0, & x^2e_0-ye_k & =0    \\
    xe_{k+1}-ye_1 & =0, & x^2e_1-ye_{k+1} & =0    \\ 
    \vdots  &     &   \vdots  &     \\
    xe_{2k-1}-ye_{k-1} & =0, & x^2e_{k-1}-ye_{2k-1} & =0    
    \end{array}$
\end{center}

\noindent which are independent. Hence we have a sequence of $\C\lbrace x,y\rbrace$-modules:


\[0 \longrightarrow \displaystyle \bigoplus_{i=0}^{2k-1}\C\lbrace x,y \rbrace e_i  \xrightarrow[\text{}]{\text{$\psi$}} \displaystyle \bigoplus_{i=0}^{2k-1}\C\lbrace x,y \rbrace e_i \xrightarrow[\text{}]{\text{$\varphi$}} \C\lbrace t \rbrace \longrightarrow 0\]   


\noindent with $\varphi(e_i)=t^i$, and $\psi$ is given by the $2k \times 2k$ matrix
\begin{center}
    

$\psi = \left[\begin{array}{ccccccccc}
-y & 0  & \cdots & 0 & x & 0 & \cdots & 0 \\

0 & -y  & \cdots & 0 & 0 & x & \cdots & 0 \\

\vdots &  & \ddots & \vdots &  &  & \ddots & 0 \\ 
 
0 & 0  & \cdots & -y & 0 & 0 & \cdots & x \\ 

x^2 & 0  & \cdots & 0 & -y & 0 & \cdots & 0 \\

0 & x^2  & \cdots & 0 & 0 & -y & \cdots & 0 \\

\vdots &  & \ddots & \vdots &  &  & \ddots & 0 \\ 
 
0 & 0  & \cdots & x^2 & 0 & 0 & \cdots & -y \\  

\end{array}\right]$

\end{center}
\noindent
It is not hard to see that the sequence is exact, which means that the independent relations we have found must generate all relations between the $e_i$. Indeed, there is a general reason why $\C\lbrace t \rbrace$ must have a resolution of length $1$ as $\C\lbrace x,y \rbrace$-module: the $\C\lbrace x,y \rbrace$-module  $ \C \lbrace t \rbrace $ is of homological dimension one  (see \cite{Mo-Pe}) and therefore the module of relations between the $e_i$ is a free submodule of $\bigoplus_{i=0}^{2k-1}\C\lbrace x,y \rbrace$ and thus of rank $\leq 2k-1$.

By permuting rows and columns of $\psi$ one checks that ${\rm det}(\psi)=(y^2-x^3)^k$ i.e., we have shown that

\begin{center}
   $F_0(f_{\ast}\Oo_{\C})_0=(y^2-x^3)^k \C\lbrace x,y \rbrace$
\end{center}

Let us now calculate  ${\rm Ann}_{\C\lbrace x,y \rbrace}\C\lbrace t \rbrace$; the annihilator is just the kernel of the map $\C\lbrace x,y \rbrace \rightarrow \C\lbrace t \rbrace$, which is the ideal generated by $(y^2-x^3)$, certainly different from our Fitting ideal if $k>1$.

Let us now make a base change by restricting our map over the $x$-axis, i.e., by the inclusion $\lbrace y=0 \rbrace \subset (\C^2,0)$ or algebraically by $\C\lbrace x,y \rbrace \rightarrow \C\lbrace x \rbrace$ sending $y$ to $0$. Then the annihilator of $\C\lbrace t \rbrace \bigotimes_{\C\lbrace x,y \rbrace}\C\lbrace x \rbrace=\C\lbrace t\rbrace/(t^{3k})$ viewed as $\C\lbrace x \rbrace$-module is $(x^2)\C\lbrace x \rbrace$ while the image in $\C\lbrace x \rbrace$ of $(y^2-x^3)\C\lbrace x,y \rbrace$ is $(x^3)\C\lbrace x \rbrace$. This shows that the formation of the annihilator does not commute with base change.

\end{example}

\subsection{Equations versus Parametrizations}

$\ \ \ \ $ As we said in subsection \ref{subsec1}, a germ of curve $(X_0,0)$, abstractly, is a germ of a purely $1$-dimensional analytic space, hence it is described by an analytic algebra $\Oo_{X_0,0}$ of pure dimension $1$. Geometrically, $(X_0,0)$ can be effectively given in two ways:\\

\noindent \textit{By equations:} By giving an ideal $I=\langle f_1,\ldots,f_m \rangle$ in $ \C\lbrace x_1,\ldots,x_N \rbrace$ such that $\Oo_{X_0,0}\simeq \C\lbrace x_1,\ldots,x_N \rbrace / I$. Saying that $\Oo_{X_0,0}$ is purely one-dimensional means that the ideal $\langle 0 \rangle$ has a primary decomposition $\langle 0 \rangle = Q_1 \cap \ldots \cap Q_r$ where $\sqrt{Q_i}=P_i$ is a minimal prime ideal in $\Oo_{X_0,0}$, and ${\rm dim} (\Oo_{X_0,0}/P_i)=1$ for $i=1,\ldots, r$.\\

\noindent \textit{By a parametrization:} By giving ourselves a germ of finite map $p:\bigsqcup_{i=1}^r (\C,0)\rightarrow (\C^N,0)$.

Here one has to be very careful: except when $n=2$, it is not true, even if $r=1$ and $p$ is generically $1$-to-$1$ so that the image (given by the Fitting structure) of this mapping is a reduced curve: it will have ``embedded components'' concentrated at the singular points, as will be shown in Example \ref{exfitting}. The analysis of this phenomenon is beyond the scope of these notes. The case where $n=2$ is explained in Proposition \ref{plane} in the next section.

\begin{example}\label{exfitting} Consider the curve $(X_0,0)$ parametrized by $n(t)=(t^4,t^6,t^7)$ which is a complete intersection (with the reduced structure) with ideal 

\begin{center}
$\langle y^2-x^3,z^2-x^2y \rangle \C \lbrace x,y,z \rbrace $.
\end{center}

We have that $\C\lbrace t \rbrace $ is generated as a $\C\lbrace x,y,z \rbrace$-module by $e_0=1$, $e_1=t$, $e_2=t^2$ and $e_3=t^3$ and it is not difficult to see that the relations are described by the following matrix

   {  $$ \Psi = \left[ \begin{array}{cccc}
 
y &  0 &  -x & 0 \\
0 &  y &  0 & -x \\
-x^2 & 0 &  y & 0 \\
0 &  -x^2 & 0 & y \\
z &  0 &  0 &  -x \\
-x^2 & z &  0 &  0 \\
0 &  -x^2 & z &  0 \\
0 &  0 &  -x^2 & z

\end{array} \right]$$}


\noindent that is, $\Psi$ is the matrix of a presentation

\begin{center}
$\C\lbrace x,y,z \rbrace^8 \overset{\Psi}{\longrightarrow} \C\lbrace x,y,z \rbrace^4 \longrightarrow \C\lbrace t \rbrace \longrightarrow 0. $    
\end{center}

\noindent of the $\C\lbrace x,y,z \rbrace$-module $ \C\lbrace t \rbrace$. Computing the $4\times 4$ minors of $\Psi$ we find that:

\begin{center}
$F_0(\C\lbrace t \rbrace) = \langle y^2-x^3,z^2-x^2y \rangle \cap \langle z^2, xy^3 ,y^4, xy^2z-x^4z, y^3z, x^4y, x^3y^2, x^3yz, x^6, x^5z \rangle \C \lbrace x,y,z \rbrace ,$   
\end{center}

\noindent where $\sqrt{\langle z^2, xy^3 ,y^4, xy^2z-x^4z, y^3z, x^4y, x^3y^2, x^3yz, x^6, x^5z \rangle}= \langle x,y,z \rangle \C \lbrace x,y,z \rbrace $.\par\noindent
The ring $\C\lbrace x,y,z \rbrace/F_0(\C\lbrace t \rbrace)$ is not purely $1$-dimensional: it has an \textit{embedded component}: an ideal of the primary decomposition of the ideal $(0)$ which defines a subspace of strictly lower dimension, in this case dimension zero.
\end{example}

\subsection{Deformations of equations vs. deformations of parametrizations}
In this subsection we consider deformations of a curve. We will follow the presentation given in \cite{buch}. The results in this subsection are due to B. Teissier (see \cite{hunting}).

Let $(X_0,0)\subset (\C^N,0)$ be a germ of a reduced curve and $X_0 \subset \B_0$ a representative, where $\B_0 \subset \C^N$ is a small open ball with center $0$. Let

\begin{center}
$\varphi_0:\overline{X}_0=\displaystyle\bigsqcup_{j=1}^r \Dd_j \rightarrow X_0 \subset \B_0$,\\

$\varphi_0(\underline t)=(\varphi_0(t_1),\ldots,\varphi_0(t_r))$\ ,\ \ $\underline t=(t_1,\ldots ,t_r)$,
\end{center}
 
\noindent be a representative of the normalization of $X_0$, where $\displaystyle \bigsqcup_{j=1}^r \Dd_j$ is the disjoint union of $r$ open discs centered at the origin in $\C$, $\varphi_0$ is a $r$-uple of distinct maps $\varphi_0\vert \Dd_j\colon (\Dd_j,0)\to \B_0$, and for each $j$ the restriction $\varphi_0\vert \Dd_j$ is a homeomorphism $(\Dd_j,0)\to (X_0^j,0)$, where $(X_0^j,0)$ is the $j$-th branch of $(X_0,0)$. It induces an analytic isomorphism $(\Dd_j,0)\setminus\{0\}\to (X_0^j,0)\setminus\{0\}$ because an analytic map $\C\to\C$ which is a homeomorphism is an isomorphism.

\begin{definition} Let $\Dd \subset \C^q$ be a small disc with center $0$. A deformation of the normalization of $X_0$ is a holomorphic mapping

\begin{center}
$\varphi: \overline{X}_0\times \Dd = \displaystyle\bigsqcup_{j=1}^r (\Dd_j \times \Dd) \rightarrow \B_0$,
\end{center}

\noindent such that $\varphi(\underline t,v)=\varphi_0(\underline t)+v\psi(\underline t,v)$, $\underline t \in \overline{X}_0$, $v\in \Dd$ and $\psi(\underline t,v)=(\psi(t_1,v),\ldots ,\psi( t_r,v))$ with $\psi(t_j,v)\colon (\Dd_j\times\Dd,0)\to \B_0$. Note that we are dealing with representatives of germs of deformations of germs.

\end{definition}

Then for sufficiently small $\Dd_j$ and $\Dd$ we have that $\phi=(\varphi,v):\overline{X}_0 \times \Dd \rightarrow \B_0 \times \Dd$ is a finite mapping and therefore

\begin{center}
$X=\phi(\overline{X}_0 \times \Dd) \subset \B_0 \times \Dd$
\end{center}

\noindent is a $q+1$-dimensional analytic subset. 
\begin{proposition}\label{plane} Given a germ of finite map $n:\bigsqcup_{i=1}^r (\Dd_j\times \Dd,0)\rightarrow (\C^2,0)\times \Dd$ with $\Dd \subset \C^q$ as above, corresponding to a map of analytic $\C$-algebras ${\mathcal O}_{\C^q,0}\{x,y\}\rightarrow \bigoplus_{i=1}^r{\mathcal O}_{\C^q,0}\{t_i\}$ which is the identity on ${\mathcal O}_{\C^q,0}$ and makes the second algebra a finite module over the first, the Fitting ideal $F_0(\bigoplus_{i=1}^r{\mathcal O}_{\C^q,0}\{t_i\})$ is a non zero principal ideal of ${\mathcal O}_{\C^q,0}\{x,y\}$.
\end{proposition}
\begin{proof} The argument goes back to \cite[Chap.III, 3.4]{Te2} (see also \cite[\S 5]{hunting}, \cite[Exercise 1.6.4]{GLS07}, \cite[Proposition 3.1]{Mo-Pe}): the depth of the ${\mathcal O}_{\C^q,0}\{x,y\}$-module $\bigoplus_{i=1}^r{\mathcal O}_{\C^q,0}\{t_i\}$ is $q+1$ because it is Cohen-Macaulay (see \cite[Theorem B.8.11]{GLS07}) of dimension $q+1$, so by the Auslander-Buchsbaum formula its homological dimension is one (see  (see \cite[Theorem B.9.3]{GLS07}), which implies that its minimal presentations are exact sequences of ${\mathcal O}_{\C^q,0}\{x,y\}$-modules
\[(0)\rightarrow {\mathcal O}_{\C^q,0}\{x,y\}^p\rightarrow {\mathcal O}_{\C^q,0}\{x,y\}^p\rightarrow \bigoplus_{i=1}^r{\mathcal O}_{\C^q,0}\{t_i\}\rightarrow (0).\]
Therefore the $0$-th Fitting ideal is generated by the determinant of a $p\times p$-matrix.\end{proof}
See also \cite[Lemma 2.1]{Mo-Pe}.\hfill$\square$\par\noindent
Applying Proposition \ref{plane} with $q=0$, we see that none of the plane images we consider has embedded components. The Fitting image of a parametrization is reduced if and only if for $i=1,\ldots ,r$ the set of all exponents appearing in the series $x(t_i)$ or in $ y(t_i)$ is coprime. For given $i$, the gcd of those exponents is the degree of the map $(\Dd,0)\to (\C^2,0)$  defined by $x(t_i),y(t_i)$ and it is also the degree at which the equation $f_i(x,y)=0$ of the \textit{reduced} image curve is raised to give a generator of the Fitting ideal of the $\C\{x,y\}$-module $\C\{t_i\})$. Compare with Example \ref{exfitting1} and see \cite[Proposition 3.1]{Mo-Pe} for a more general result.\par
Going back to the case $q>0$, we see that if the parametrization for $v=0$ has a reduced image $X_0$, then the hypersurface $X$ in $\Dd\times \C^2$ which is the image of the parametrization is reduced. Otherwise, since it is the Fitting image, applying the compatibility with base change to $0\in \Dd$, we find that the special fiber $X_0$ would not be reduced.\par\noindent
But then each fiber is reduced and its parametrization is an isomorphism outside of the singularities; it is a bimeromorphic map. \par\noindent This implies that the map $p:\bigsqcup_{i=1}^r (\Dd_j\times \Dd,0\times 0)\rightarrow (X,0)$ is the normalization of the hypersurface $X$ because the source is normal, and the map $p$ is finite and bimeromorphic. It is a non singular normalization of the hypersurface $X$ which induces the normalization of each fiber of the projection map $X\to \Dd$. So we have proved:
\begin{corollary}\label{simnor} The parametrization of the total space of the family of plane curves $n\colon (X,0)\rightarrow (\Dd,0)$ obtained by deforming the parametrization of the germ of reduced plane curve $(X_0,0)$ is a simultaneous normalization.
\end{corollary}
Moreover, we can observe that the ${\mathcal O}_{\C^q,0}$-modules ${\mathcal O}_{X,0}$ and $\overline {{\mathcal O}_{X,0}}=\bigoplus_{i=1}^r{\mathcal O}_{\C^q,0}\{t_i\}$ are flat. The first one because a hypersurface is Cohen-Macaulay (see \cite[Theorem B.8.11]{GLS07}) and the second one because it is a sum of flat modules (see \cite[Corollary I 1.88]{GLS07}).\par\noindent
We have the exact sequence of ${\mathcal O}_{\C^q,0}$-modules:
$$(0)\rightarrow {\mathcal O}_{X,0}\rightarrow \overline {{\mathcal O}_{X,0}}\rightarrow\frac{ \overline {{\mathcal O}_{X,0}}}{{\mathcal O}_{X,0}}\rightarrow (0),$$
and the flatness of the first two modules implies that of the third.\par\noindent
Now by the Weierstrass Preparation Theorem, the singular locus of $X$ is finite over $\Dd,0)$ and is the support of the ${\mathcal O}_{\C^q,0}$-module $\frac{ \overline {{\mathcal O}_{X,0}}}{{\mathcal O}_{X,0}}$, which is flat and thus locally free. So the dimensions of its fibers over points $v\in \Dd$ is constant. Provided $\Dd$ and $\B_0$ are sufficiently small, this dimension is the sum of the $\delta$ invariants (see Definition \ref{delinv}) of the finitely many singularities of the curve $X_v=n^{-1}(v)$, which all tend to ) as $v\to 0$. So we have:
\begin{corollary}\label{deldel}In a small enough representative of family of reduced plane curves obtained by deformation of the parametrization of the special fiber, the sum of the $\delta$ invariants of the singularities of the fibers is constant.
\end{corollary}
We shall use this in the proof of Proposition \ref{gengen} below.\par\medskip
We have just seen that the space $X$ defined by deforming a parametrization of a plane curve can also be described as the space defined by deforming an equation for the reduced plane curve $X_0$, since the equation for $X$ has to reduce to the equation for $X_0$ when setting $v=0$. The situation is more delicate for parametrized curves in $\C^N$ with $N>2$, not only because of the behavior of the Fitting ideal, but also because in general deforming equations does not produce a flat family. The general definition of a deformation of a germ $(X_0,0)\subset (\C^N,0)$ is a germ $(X,0)\subset (\C^N,0)\times (S,0)$, \textit{flat over $S$} and defined by equations which, when restricted over $0\in S$ give a set of equations of $X_0$, up to isomorphism. It is therefore a reasonable question to ask whether any such deformation can be obtained by a deformation of the parametrization of $X_0$ in the sense that $X$ is the reduced image of such a deformation.\par

 The answer is a converse to Corollary \ref{deldel}, as follows:
\begin{proposition}\label{constant} {\rm (See \cite[Theor\`eme 1, page 80]{Tei80}, and \cite[\S 2.6]{GLS07} for plane curves)}  Let $p\colon (X,0)\rightarrow (S,0)$ be a flat morphism where all the fibers are reduced curves and $S$ is non singular. Then, for suitable representatives,  the following conditions are equivalent:
\begin{itemize} 
\item The normalization $\overline X$ is non singular, the composed map $\overline X\stackrel{n}\rightarrow X\stackrel{p}\rightarrow S$ is flat and for each $s\in S$ the map of fibers $(\overline X)_s\to X_s$ is the normalization.
\item The sum $\delta_s$ of the $\delta$ invariants of the singular points of the fibers $X_s$ is independent of $s\in S$.
\end{itemize}
\end{proposition}
Since the map $p\circ n$ is flat with non singular fiber, at every point of $\overline X$ lying above $0\in X$, the space $\overline X$ is locally isomorphic over $S$ to a product of a disk by $S$. This shows that the map $n$ is a deformation of the parametrization of $X_0$.
The assumption that the fibers are reduced is necessary, as evidenced by the following example. 

\begin{example} Consider the of curve $X_0$ in $\C^3$ given by the equations $x=0, \ z^2-y^3=0$. The normalization of $X_0$ is given by

\begin{center}
 $\varphi(t)=(0,t^2,t^3)$. 
\end{center}
 
\noindent Consider the deformation $\Phi(v,t)=(vt,t^2,t^3,v)$. So, the reduced image $X_{red}$ of $\Phi$ is given by the following equations:

\begin{center}
$x^2-v^2y=0, \ \  xy-vz=0, \ \  xz-vy^2=0, \ \ z^2-y^3=0$.
\end{center}   

\noindent Now, when we consider the projection $f:X_{red}\rightarrow \Dd$, the fiber $f^{-1}(0)=X_0$ is given by

\begin{center}
$x^2=0, \ \  xy=0, \ \  xz=0, \ \ z^2-y^3=0$.
\end{center}
\end{example}
\noindent Note that it is not reduced at the origin, hence there is no deformation of the equations $x=z^2-y^3=0$ which defines $X_{red}$. One can understand this as follows: while the special fiber of our family of curves has embedding dimension two, the general fiber has embedding dimension three. In an analytic family the embedding dimension of the fibers can only increase by specialization so that in our analytic family $f:X_{red}\rightarrow \Dd$ the ideal defining the special fiber has in its primary decomposition an infinitesimal embedded component with ideal $\langle x^2,y,z\rangle$ sticking out of the $x=0$ plane, which makes the embedding dimension of $f^{-1}(0)$ equal to three as it must be. This fact was stressed also in \cite[\S 3, section 3.5]{hunting}.\par\medskip
More material on the plane curve case is found in \cite[Chap. II, \S 2]{GLS07}. There are generalizations of these results to the cases where the fibers may be non reduced and have embedded components. There are definitions of the invariant $\delta$ which apply to this more general situation. We refer the reader to \cite{L15} and the references therein.


\begin{remark} We note that one can use Mond-Pellikaan's algorithm in \cite{Mo-Pe} to find a presentation matrix of a finite analytic map germ $g:(X,0)\rightarrow(\C^{d+1},0)$, where $(X,x)$ is a germ of Cohen-Macaulay analytic space of dimension $d$. For the computations one can use also the software Singular \cite{ref21} and the implementation of Mond-Pellikaan's algorithm given by Hernandes, Miranda, and Pe\~{n}afort-Sanchis in \cite{HMP}. At the web page of Miranda \cite{ref10} one can find a Singular library to compute presentation matrices based on the results of \cite{HMP}.
\end{remark} \newpage

\section{General projections}

    For a reduced and equidimensional germ of complex analytic variety $(X,0) \subset (\C^N,0)$ Whitney 
  gave 6 possible definitions of tangent vectors (\cite{Whi}), the sets of which constitute \index{tangent cones}tangent cones:
    \[C_1(X,0) \subset C_2(X,0) \subset C_3(X,0) \subset C_4(X,0) \subset C_5(X,0) \subset C_6(X,0), \]
   and when the germ $(X,0)$ is smooth they all coincide with the tangent space $T_0X$.  \\
   
   What is usually known as the tangent cone $C_{X,0}$ is what Whitney defined as the cone $C_3(X,0)$ and is constructed by taking
  limits of secants through the origin. This means that if we take a representative $(X,0) \subset (\C^N,0)$ then
  a vector $\Vec{v} \in \C^N$ is in $C_3(X,0)$ if there exists a sequence of points $\{p_i\} \subset X\setminus \{0\}$ tending to $0$ and a sequence of complex numbers $\{\lambda_i\} \subset \C^*$ such that
  \[ \lambda_ip_i \to \Vec{v}.\]
  Algebraically it is constructed by blowing up the point 
     \[e_0: {\rm Bl}_0X \to X\]
  and the fiber over the origin is the projectivized tangent cone $e_0^{-1}(0) = \mathbf{P} C_3(X,0)$. 
  In particular it is a pure d-dimensional algebraic cone where $d$ is the dimension of $(X,0)$. \\
  
  If $(X,0) $ is a curve then the cone $C_3(X,0)$ is a finite number of lines, one for
  each branch of $X$. By abuse of language they are called the tangents to $X$ at $0$. Of course different branches may have the same tangent. 
  
  \begin{definition}\label{C3trans}
    A linear projection $\pi:(\C^N,0) \to (\C^M,0)$ with kernel $D$ is called \index{$C_3$-general}$C_3$-general (with respect to X) if 
    it is transversal to the tangent cone. That is 
    \[D \cap C_3(X,0)=\{0\}.\]  
  \end{definition}
  
   Note that  by the Weierstrass Preparation Theorem (see \ref{W} and  \cite[Thm 3.4.24]{DeJo-Pfi}) the condition is \textit{equivalent} to the fact that the map \[\pi\vert C_3(X,0)\colon C_3(X,0)\to C_3(\C^M,0)=T_{\C^M,0}\] is finite (proper with finite fibers), which implies $M\geq d$. The restriction of a $C_3$-general projection to $X$
      \[ \pi\vert X: (X,0) \to (\C^M,0)\]
   satisfies $\pi^{-1}(0)=\{0\}$ since otherwise the tangent cone to $\pi^{-1}(0)$, which is contained in $C_3(X,0)$, would be contained in $D$. Again by the Weierstrass Preparation Theorem, this is equivalent to $\pi\vert X$ being a finite map. However the finiteness of $\pi\vert X$ does not imply that the projection is $C_3$-general; consider the projection of $y^3-x^2=0$ to the $x$-axis.\par Since $C_3(X,0)$ is of dimension $d$, almost all (an open dense set of) linear projections $\pi: (\C^N,0) \to (\C^{d+1},0)$ are $C_3$-general for $(X,0)$. Since we assume $X$ to be equidimensional, this tells us that $\pi(X) \subset \C^{d+1}$ 
   is a hypersurface. In the curve case (d=1) this guarantees the existence of linear projections with image a plane curve. \par
By \cite[Cor. 8.2]{Chi} we have that $C_3(\pi(X),0) = \pi\left( C_3(X,0)\right)$ in $\C^{d+1}$. We leave it as an exercise for the reader to verify that this last equality is an equality of Fitting images. Hint: use the specialization spaces $\mathcal X$ and $\mathcal Y$ to the tangent cones for $X$ and $\C^{d+1}$ respectively (see \cite[\S 2, 2.4]{GiTei})  and the fact that the natural map $\mathcal X\to\mathcal Y$ is finite by the Weierstrass Preparation Theorem because the genericity assumption is equivalent to the finiteness of the map $C_3(X,0)\to T_{\C^{d+1},0}$, and apply \cite[Prop. 1.6]{Mo-Pe}. \par\noindent 
Finally, a projection $\pi:(\C^N,0) \to (\C^d,0)$ is $C_3$-general for $(X,0)$ if and only if the map $C_3(X,0)\to C_3(\C^d,0)=T_{\C^d,0}$ which it induces is finite and surjective, and thus a ramified covering. These $C_3$-general maps all induce on $(X,0)$ ramified analytic coverings  $(X,0)\to (\C^d,0)$ of degree equal to the multiplicity of $(X,0)$. \\
   
   The cone $C_4(X,0)$ is constructed by taking limits of tangent vectors at smooth points. One can 
   prove that it is equivalent to taking limits at $0$ in the appropriate 
   Grasmannian of tangent spaces at non singular points of $X$ and so it is determined by the fiber over $0$ of the Semple-Nash modification of a representative $X$ of $(X,0)$. Of course there is an analogous definition of a $C_4$-general linear projection and they do have interesting 
   equisingularity properties.  However, since in the curve case the cones $C_3$ and $C_4$ coincide we will skip this part and ask the interested reader to look at \cite{Chi}, \cite{Stu1} and \cite{Stu2}.\\
   
   The cone $C_5(X,0)$ is constructed by taking limits of secants. This means that if we take a representative $X \subset \C^N$ then
  a vector $\Vec{v} \in \C^N$ is in $C_5(X,0)$ if there exist  sequences of pairs of distinct points $\{p_i\}, \{q_i\} \subset X\setminus \{0\}$ tending to $0$ as $i\to\infty$ and a sequence of complex numbers $\{\lambda_i\} \subset \C^*$ such that
  \[ \lambda_i(p_i-q_i) \to \Vec{v}.\]
  To prove that $C_5(X,0)$ is an algebraic cone and have a bound for its dimension, take a small representative $X \subset \C^n$
  consider the (closed) diagonal embedding $\delta: X \hookrightarrow X \times X$ and blow up its image $\Delta$: 
 
     \[e_\Delta: {\rm Bl}_\Delta (X \times X)  \to X\times X.\] 
     If we choose coordinates $(z_1,\ldots,z_N,w_1,\ldots,w_N)$ of the ambient space $\C^{2N}$, then we can obtain the space ${\rm Bl}_\Delta (X \times X)$ as the closure of the graph of the secant map defined away from the diagonal $\Delta$ by:
     \begin{align*} X \times X \setminus \Delta &\longrightarrow \mathbf{P}^{N-1} \\
                            (z,w) & \longmapsto [z_1-w_1: \cdots: z_N-w_N].  \end{align*}
     So we have ${\rm Bl}_\Delta (X \times X) $ as a closed subspace of the product $X \times X \times \mathbf{P}^{N-1}$,  the map $e_\Delta$ is induced by the projection to $X \times X$, and the exceptional fiber is the divisor $ D:=e_\Delta^{-1}(\Delta) \subset \Delta \times \mathbf{P}^{N-1}$  which comes with a map $D \to \Delta$ such that for every point 
     $(q,q) \in \Delta$ the fiber is the projective subvariety  corresponding to the projectivization of the $C_5$-cone of $X$ at $q$, that is $\mathbf{P}C_5(X,q)$.     
     This is roughly the way Whitney proved that the $C_5$-cone is an algebraic variety  in \cite[Th. 5.1]{Whi}.  Now  $C_5(X)$ is the analytic space obtained by  deprojectivization of the (fibers of) the divisor $D$ and $\psi$ corresponds to the pullback of $e_\Delta$ by $\delta$:
      
            \[ \xymatrix{X\times \C^N \supset \negthinspace &C_5(X) \ar@{.>}[r] \ar[d]_\psi  & {\rm Bl}_\Delta (X \times X) \ar[d]^{e_\Delta} \\
                          &X \ar[r]_\delta   &  X \times X}\]   
  where the upper arrow is defined only outside of $X\times \{0\}$.  Note that the dimension of $C_5(X)$ is $2d$, and the dimension of $\psi^{-1}(p) = C_5(X,p)$ for a smooth point $ p \in X$ is equal to $d$ since in this case we have $C_5(X,p)=T_pX$. By the semicontinuity of the dimensions of the fibers of an analytic morphism, this implies that:
     \[d \leq \mathrm{dim} \, C_5(X,0) \leq 2d \]
    
    \begin{definition}
    A linear projection $\pi:(\C^N,0) \to (\C^M,0)$ with kernel $D$ is called \index{generic projection}\emph{generic} (or $C_5$-general) with respect to X if it is transversal to the cone $C_5(X,0)$. That is 
    \[D \cap C_5(X,0)=\{0\}.\]  
  \end{definition}
 \noindent  In other words, no limit at $0$ of secants to $X$ is contained in $D$.\par\noindent
    Note that a generic projection is in particular $C_3$-general and $C_4$-general. 
    
 \begin{proposition}\label{C5smooth} Let $(X,0) \subset (\C^N,0)$ be a reduced equidimensional germ of complex
    analytic variety of dimension $d$ and $\pi:(\C^N,0) \to (\C^M,0)$ a linear projection.
 \begin{itemize}
        \item [a)] If $\pi$ is generic then the restriction to $X$ induces a homeomorphism with its image.
        \item [b)] $(X,0)$ is smooth if and only if dim $C_5(X,0)=d$
  \end{itemize}
    \end{proposition}
    \begin{proof}
      First of all note that the transversality to the cone $C_5(X,0)$ implies that the restriction
      $\pi\vert X$ is injective for a small enough representative of $X$. But then the induced map
      $\pi \vert X\colon X \rightarrow \C^M$ is injective, continous and the map $X\rightarrow \pi(X)$ is open since $\pi$ is and so it should be a homeomorphism of $X$ with its image $\pi(X)$.\par\noindent
      Now for $b)$: sufficiency is clear since $(X,0)$ smooth implies $C_5(X,0)=T_0X$ and so it is of dimension
      d. Conversely, if the dimension of $C_5(X,0)$ is $d$ there exist generic linear projections of $X$ to $\C^d$.
      By $a)$ this gives us a homeomorphism between $(X,0)$ and $(\C^d,0)$. Note that $\pi$ is also $C_3$-general so it induces a ramified covering of degree equal to the multiplicity of $(X,0)$, 
      but the injectivity gives us multiplicity $1$ and so $(X,0)$ is smooth. 
    \end{proof}
For more on this and more general results see  \cite{Stu1},\cite{Stu3} and \cite[Section 9.4]{Chi}\\
     
        An important thing to notice is that in the reducible case the cone $C_5(X,0)$ contains but is \textit{not} equal 
        to the union of the $C_5$-cones of its irreducible components. For instance if $(X_,0)$ is a curve consisting 
        of two smooth branches $X_1$ and $X_2$ then both cones $C_5(X_i,0)$ are one-dimensional but since $(X,0)$ 
        is singular then by the previous result $C_5(X,0)$ can not have dimension 1.\\
     
        So now we have that if $(X,0)$ is singular then $d+1 \leq \mathrm{dim}\, C_5(X,0) \leq 2d$, and 
       for curves this gives dim $C_5(X,0)=2$. This guarantees the existence of generic projections of curves
        to $\C^2$.\\
        
    \begin{corollary} \label{homeo}  Let $(X,0) \subset (\C^N,0)$ be a germ of reduced analytic curve. Then 
    almost all (an open dense set of) linear projections $\pi: (\C^N,0) \to (\C^2,0)$ are generic and
    their Fitting images $\pi(X) \subset \C^2$ are reduced plane curves homeomorphic to $X$. Moreover, $\pi$ induces an analytic isomorphism $X\setminus\{0\}\rightarrow \pi(X)\setminus\{0\}$.
    \end{corollary} 
    \begin{proof}This follows from Proposition \ref{plane} and the fact that an analytic map $\C\to\C$ which is a homeomorphism is an isomorphism.
    \end{proof}

\subsection{The case of dimension 1.}

   In the case of curves we have the following important results:
   
   \begin{proposition}\label{C5curve}{\rm (see \cite[Prop IV.1]{BGG})}\\
       Let $(X,0) \subset (\C^N,0)$ be a germ of reduced analytic curve. If $(X,0)$ is singular then the cone $C_5(X,0)$ is a finite union of 2-planes each one of them containing at least one tangent line to $(X,0)$. 
   \end{proposition}
   \begin{proof}
    We will only give an idea of the proof. \\
     By Proposition \ref{C5smooth} the cone $C_5(X,0)$ is two dimensional and by the blowup construction it has a finite number of irreducible components. So what one has to prove is that all the irreducible components are 2-planes.  Again, by this blowup construction, any (direction of) line contained in $C_5(X,0)$ can be picked off by lifting an arc 
     \[(\psi_1,\psi_2): (\C,0) \to (X \times X, (0,0))\]
     to ${\rm Bl}_\Delta(X \times X)$ like $\left( \psi_1(t), \psi_2(t), \left[ \psi_1(t)-\psi_2(t)\right] \right)$.
     Now each $\psi_i(t)$ is an arc $(\C,0) \to (X,0)$ and can be obtained using the parametrization of 
     one of the branches of $(X,0)$. Once you see this, what you have to do is consider the different cases and work out the calculations.
     
     The first case is when $(X,0) \subset (\C^N,0)$ is irreducible of multiplicity $n$ so according to subsection \ref{stru}, in suitable coordinates we have a parametrization of the form: 
     \[ \varphi(t)= \left(t^n, \sum_{i>n} a_{2,i} t^i, \ldots, \sum_{i>n} a_{N,i} t^i\right) \]
     with the tangent line being the $z_1$-axis $[1:0:\cdots:0]$. For every $n$-th root of unity 
     $\omega\neq 1$ the lifted arc   
     \[t \mapsto \left(\varphi(t), \varphi(\omega t), \left[\varphi(t) - \varphi(\omega t) \right] \right)\in X\times X\times {\mathbf P}^{N-1}\]
     will define a limit  line $\ell_\omega\in {\mathbf P}^{N-1}$ as $t\to 0$ which is distinct from the $z_1$ axis and if you define $H_\omega$ as the 2-plane generated by the $z_1$-axis
     and the line in $\C^N$ corresponding to $\ell_\omega$, then you can prove that
     \[C_5(X,0) = H_{\omega_1} \cup \ldots \cup H_{\omega_{n-1}},\]
     by verifying that any line obtained by lifting an arc is contained in one of these 2-planes. We note that they are not necessarily all different.
     
     For the reducible case it is enough to consider two branches $(X,0) = (X_1,0) \cup (X_2,0)$.
     In this case you have that the $C_5$-cone of each irreducible component $(X_i,0)$ will be contained
     in $C_5(X,0)$ but you will have additional components that come from the configuration of these two branches. For instance if they have different tangent lines $\ell_1$ and $\ell_2$ then all you have to 
     add is the plane $H_{12}$ generated by these two lines.i.e.,
     \[C_5(X,0) = C_5(X_1,0) \cup C_5(X_2,0) \cup H_{12}.\]
     When the two branches are tangent (have the same tangent line) then you have to play a game very similar
     to the irreducible case by reparametrizing your branches in such a way as to travel through them at 
     the same ``speed'' and using roots of unity to find lines $\ell_\omega$ in the $C_5(X,0)$ that are
     different from the tangent line and these will give you the additional 2-planes. i.e.,:
     \[C_5(X,0) = C_5(X_1,0) \cup C_5(X_2,0) \cup H_{\omega_1} \cup \ldots \cup H_{\omega_k}.
   \eqno\square\]
   \end{proof}
   
   \begin{proposition}\label{gengen}{\rm (see \cite[Prop IV.2]{BGG})}\\
      Let $(X,0) \subset (\C^N,0)$ be a germ of reduced analytic curve, and let $\Omega \subset G(N-2,N)$ be the non-empty
      Zariski open set of the Grassmannian of $(N-2)$-planes of $\C^N$ which are transversal to $C_5(X,0)$. Then:
      \begin{itemize}
          \item [a)] For $H \in \Omega$ the plane curve $\left(\pi_H(X),0\right)$ is reduced and of constant  
          topological (equisingularity) type with Milnor number $\mu_0$.
          \item[b)] If $H \notin \Omega$ then one of the following statements is verified:
                 \begin{itemize}
                     \item 0 is not an isolated point of $H \cap X$.
                     \item 0 is an isolated point of $H \cap X$ but the curve $\left(\pi_H(X),0\right)$ is not reduced.
                     \item 0 is an isolated point of $H \cap X$, the curve $\left(\pi_H(X),0\right)$ is reduced
                      but its Milnor number is greater than $\mu_0$.
                 \end{itemize}
      \end{itemize}
   \end{proposition}
   \begin{proof}
       Let $W' \subset G(N-2,N)$ be the open subset of the the Grassmannian of $(N-2)$-planes of $\C^N$
    defined by the condition that $H \in W'$ if and only if $0 \in \C^N$ is an isolated point of $H \cap X$. Let 
   $W \subset \C^{2N}$ with coordinate system $(a_1,\ldots,a_N, b_1,\ldots,b_N)$ be the associated
   open subset, where $d=(\underline a,\underline b) \in W$ if and only if the linear forms
    \[a_1z_1 + \cdots + a_Nz_N \hspace{0.2in} \mathrm{and} \hspace{0.2in} b_1z_1+ \cdots +b_Nz_N\]
    are linearly independent and the $N-2$ plane $H_d \subset \C^N$ they define is in $W'$. Let $\pi_d$ be the 
    linear projection
    \begin{align*}
         \pi_d: \C^N &\longrightarrow \C^2 \\
              (z_1, \ldots,z_N) &\mapsto (a_1z_1 + \cdots + a_Nz_N, b_1z_1+ \cdots +b_Nz_N )
    \end{align*}
    Note that for $d \in W$ the germ $\pi_d: (X,0) \to (\C^2,0)$ is finite, and if we denote 
    by $(\pi_d(X),0)\subset (\C^2,0)$ the image germ with the Fitting structure then by \cite[Lemma 2.1]{Mo-Pe}
    it is a (not necessarily reduced, but without embedded component, by Proposition \ref{plane}) plane curve.
    
    We put all these projections in an analytic family by considering the map
     \begin{align*}
         \Pi: \C^N \times W &\longrightarrow \C^2 \times W \\
              (z_1,\ldots,z_N,d) &\mapsto \left( \pi_d(z_1,\ldots,z_N),d\right)
     \end{align*}
     Note that for every $d \in W$ the map germ
     \[\Pi: \left(X \times W, (0,d)\right)\to \left( \C^2 \times W, (0,d)\right) \]
     is finite. And since the analytic algebra $\Oo_{X \times W, (0,d)}$ is Cohen-Macaulay 
     again by \cite[Lemma 2.1]{Mo-Pe} we have a germ of hypersurface $\left(\Pi(X \times,W), (0,d)
     \right) \subset \left(\C^2 \times W, (0,d)\right) $.
     By projecting to $W \subset \C^{2N}$ we obtain (by \cite[Thm B.8.11]{GLS07}) a flat map:
     \[ G:\left(\Pi(X \times W), (0,d)\right) \to (W,d). \]
     Since the Fitting structure commutes with base change we have that the germ $\left(G^{-1}(d),(0,d)\right)$ is isomorphic to $(\pi_d(X),0)$, and so we have a flat deformation of $(\pi_d(X),0)$ where all the fibers are plane curves. 
      
     Note that if $\varphi: (\C,0) \to (\C^N,0)$, $t \mapsto \left( \varphi_1(t), \ldots, \varphi_N(t)\right)$ is the normalization of a branch of $(X,0)$ then the plane curve
     $(\pi_d(X),0)$ is parametrized by:
     \[t \mapsto \left(a_1\varphi_1(t) + \cdots + a_N\varphi_N(t), b_1\varphi_1(t) + \cdots + b_N\varphi_N(t)\right), \]
     and by varying $d$ we get that the deformation space of $G$ admits a parametrization in family.\\ 
     \textbf{Proof of a):} When $H_d$ is transversal to $C_5(X,0)$ then for every $d'$ in a small neighborhood of $d$ the $(n-2)$-plane $H_{d'}$ is also transversal to $C_5(X,0)$ and all the corresponding projections $\pi_{d'}$ are therefore generic. By Corollary \ref{homeo} this tells
     us that $\pi_{d'}: X \setminus \{0\} \to G^{-1}(d') \setminus \{0\}$ is an analytic isomorphism
     for every $d'$ sufficiently close to $d$. This implies: 
     \begin{itemize}
         \item All the curves in the family $G^{-1}(d')$ have the same number of branches as $X$.
         \item The parametrization in family is actually a normalization in family and by Corollary \ref{deldel}, or \cite[\S 3]{hunting}, see also \cite [II, Thm 2.56]{GLS07}
         the family is $\delta$ constant. 
     \end{itemize}
     By the Milnor formula $\mu = 2\delta - r +1 $ the family $G:\left(\Pi(X \times,W), (0,d)\right) \to (W,d)$ is $\mu$-constant and thus equisingular by \cite{L-R76} or \cite[Thm II.4]{BGG}.\\
     
     \textbf{Proof of b):} For $H_d \in W \setminus \Omega$ we have that the map 
       \[\pi_d: (X,0) \to (\C^2,0)\] 
       is finite but if it is generically $k$ to $1$ then by \cite[Prop. 3.1]{Mo-Pe} the 
       Fitting structure of ($\pi_d(X),0)$ is not reduced. \\
       When $\pi_d$ is generically $1-1$ then $(\pi_d(X),0)$ is reduced but by assumption 
       there is a line $\ell \subset H_d \cap C_5(X,0)$. Take a sequence of secants $\ell_k$ going through the points $x_k,y_k \in X \setminus \{0\}$ such that $\ell_k$ converges (in direction) to $\ell$, since $\Omega$ is Zariski open we can find a sequence $d_k$ tending to $d$ such that $H_{d_k} \in \Omega$ and it
       contains (the direction of) $\ell_k$.  Note that $\pi_{d_k} (\ell_k)= q_k \neq 0$ and so the plane curve $(G^{-1}(d_k),q_k)$ is singular which implies that $\mu\left( (\pi_d(X),0) \right) > \mu\left( (\pi_{d_k}(X),0)\right)$.
   \end{proof}
   
   \begin{example}
     Let $(X,0) \subset (\C^3,0)$ the germ of irreducible curve parametrized by 
     \[ t\mapsto (t^4,t^5,t^7)\]
     then the tangent cone $C_3(X,0)$ is the $z_1$-axis.\\
     By taking other arcs $t \mapsto (t^4,\omega t^5,\omega^3t^7)$ were $\omega\in\mu_4\setminus\{1\}$ and taking the limit as $t\to 0$ of the difference $(0:(1-\omega)t^5:(1-\omega^3)t^7)$ we get the $z_2-$ axis as a limit of secants and we can deduce that the cone $C_5(X,0)$ is the 
     $z_1z_2$-plane.\\
     For $d=(1,0,0,0,1,0)$ the corresponding projection
     \[\pi_d (z_1,z_2,z_3) = (z_1,z_2)\] 
     is $C_5$-general and its image $\pi_d(X,0) \subset (\C^2,0)$ is 
     the reduced plane curve $y^4-x^5=0$ with Milnor number $\mu=12$.\\
     On the other hand For $d_0=(1,0,0,0,0,1)$ the corresponding projection
     \[\pi_{d_0} (z_1,z_2,z_3) = (z_1,z_3)\] 
     is not $C_5$-general and its image $\pi_{d_0}(X,0) \subset (\C^2,0)$ is 
     the reduced plane curve $y^4-x^7=0$ with Milnor number $\mu=18$.\\
     By taking $d_\alpha=(1,0,0,0,-\alpha^2,1)$ we get a sequence of $C_5$-general
     projections $\pi_{d_\alpha}$ converging to $\pi_{d_0}$
     \[\pi_{d_\alpha} (z_1,z_2,z_3) = (z_1,z_3-\alpha^2z_2)\]
     Note that the plane curve  $X_\alpha:= \pi_{d_\alpha}(X)$ has a singular point
     in $(\alpha^4,0)$ coming from the image of the secant going through the points
     $(\alpha^4, \alpha^5, \alpha^7)$ and $(\alpha^4, -\alpha^5, -\alpha^7)$ in 
     $X$. Moreover as $\alpha$ tends to $0$ these secants $d_\alpha=[0:1:\alpha^2]$ converge to the $z_2$-axis
     $[0:1:0]$ in $\mathbf{P}^2$ which is precisely the intersection $H_{d_0}\cap C_5(X,0)$.
   \end{example}

 \newpage  

\section{Main result}

  We have just seen that all ($C_5$-)generic plane projections of a reduced analytic curve are equisingular. Now our objective is to prove that all equisingular germs of reduced plane curves are generic projections of a single space curve. As we shall see, given a reduced plane curve $(X,0)\subset (\C^2,0)$ this space curve corresponds to the one dimensional analytic algebra which is the Lipschitz saturation $\Oo_{X,0}^s$ of $\Oo_{X,0}$ in the sense of \cite{P-T69}. In doing so we will also give another reason why a) of Proposition \ref{gengen} is true, since we shall see that a projection $\pi$ is generic for a space curve $(X,0)\subset (\C^N,0)$  if and only if it induces an isomorphism of the saturated algebras $\Oo_{X,0}^s$ and  $\Oo_{\pi(X),0}^s$. In particular, two germs of reduced plane curves are equisingular (topologically equivalent) if and only if their saturations are \textit{analytically} isomorphic.\par\noindent In order to define these saturations we need the theory of integral closure of ideals.
  
  \subsection{Integral closure of ideals}\label{intcl}
  
      Our main references for this subsection are, \cite{Lej-Te}, \cite{Lip1}, \cite{Te2} and \cite{SwaHu}.
 \begin{definition}
     Let $I$ be an ideal in a ring $R$. An element $r \in R$ is said to be integral over $I$ if there
     exists an integer $h$ and elements $a_j \in I^j$, $j=1, \ldots,h$, such that
  \[r^h+a_{1} r^{h-1}+a_{2} r^{h-2}+\cdots+a_{h-1} r+a_h=0.\]
    The set of all elements of $R$ that are integral over $I$ is an ideal called the \index{Integral closure of an ideal}\textbf{integral closure of} $I$
    and denoted by $\overline{I}$. We say that $I$ is integrally closed if $I=\overline{I}$.  If $I \subset J$ are ideals we say that $J$ is integral over $I$
    if $J \subset \overline{I}$.
 \end{definition}   
    
 \begin{remark} \label{RemIntDepId}The following properties are easily verified:
    \begin{enumerate}
        \item $I\subset \overline{I}$. For each $r \in I$ choose $n=1$ and $a_1=-r$.
        \item If $I \subset J$ are ideals then $\overline{I} \subset \overline{J}$ since
        an integral dependence equation for $r$ over $I$ is also an an integral dependence equation 
        for $r$ over $J$.
        \item $\overline{I} \subset \sqrt{I}$ since the integral dependence equation implies
        $r^n \in \langle a_1, \ldots, a_n\rangle \subset I$.
        \item Radical ideals are integrally closed. 
        \item If $\varphi:R \to S$ is a ring morphism and $I \subset S$ is an 
        integrally closed ideal of S then $\varphi^{-1}(I)$ is an integrally 
        closed ideal of $R$.
    \end{enumerate}
 \end{remark}
 
      A related concept is that of \index{reduction of an ideal}\textit{reduction}: For ideals $J \subset I \subset R$ we say
       that $J$ is a \textbf{reduction of $I$} if there exists a non-negative integer $n$
       such that $I^{n+1}=JI^n$. This implies that $\overline{I}=\overline{J}$. We can express integral dependence using equalities of ideals and modules. 
    
   \begin{proposition} {\rm (see \cite[Chapter 1]{Lej-Te}, \cite[Prop 1.1.7, Cor. 1.1.8 $\&$ Cor. 1.2.2]{SwaHu})}
       For any element $r \in R$ and ideal $I \subset R$. The following are equivalent: 
       \begin{itemize}
           \item [a)] $r \in \overline{I}$.
           \item[b)]  There exists an integer $k$ such that $\left( I + r\right)^k =
       I\left( I + r\right)^{k-1}$.
            \item[c)] $I$ is a reduction of $I+<r>$. 
           \item[d)] There exists a finitely generated $R-$module $M$ such that $rM
            \subset IM$ and if there exists $a\in R$ such that $aM=0$, then there exists an integer $\ell$ such that $ar^\ell=0$. .
       \end{itemize}
   \end{proposition}      
      
      A very important corollary of this Proposition is that 
      $\overline{I} \subset R$ is an integrally closed ideal of $R$ and you
      can find a complete proof of this fact in \cite[Cor. 1.3.1]{SwaHu}. \\
      
      We have that $I \subset \overline{I} \subset \sqrt{I}$, but in fact the integral
      closure is much ``closer" to $I$ than to the radical and a very good family 
      of examples in which it is easy to calculate and compare is that of monomial ideals
      in $\C\{z_1,\ldots,z_d\}$, which are the ideals generated by monomials. We begin with an example: \\
           \begin{example}
        For the ideal $I=\langle x^4,xy^2,y^3\rangle\C\{x,y\}$ we have that
        \[\overline{I}=\langle x^4,x^3y, xy^2,y^3\rangle \] and
        \[\sqrt{I}=\langle x,y\rangle.\]
        %Poner el dibujo y tratar de calcular la ecuacion de dependencia integral%
        
     \noindent The exponent set of $I$ consists of all integer lattice points in the yellow region below:
\begin{figure}[h!] 
\begin{center}
\begin{tikzpicture}[x=.8cm,y=.8cm] 
\tikzstyle{every node}=[font=\small]
\fill[fill=yellow!40!white] (0,3) -- (1,3)--(1,2)--(4,2) -- (4,0) -- (7,0) --(7,5) -- (0,5) --cycle;
\foreach \x in {0,1,...,7}{
\foreach \y in {0,1,...,5}{
       \node[draw,circle,inner sep=0.7pt,fill, color=gray!40] at (1*\x,1*\y) {}; }
       }
\draw[->] (0,0) -- (7,0) node[right,below] {${\rm exp.}x$};
\draw[->] (0,0) -- (0,5) node[above,left] {${\rm exp.}y$};
\draw[thick] (4,0) node[below] {(4,0)};
\draw[thick] (0,3) node[left] {(0,3)};
\draw[thick] (1,2) node[left] {(1,2)};
\draw[thick] (3,1) node[right] {(3,1)};
\draw [ultra thick, color=cyan](4,0) -- (1,2);
\draw [ultra thick, color=cyan](1,2) -- (0,3);
\node[draw,circle,inner sep=1.5pt,fill=orange!50,color=orange!50] at (4,0) {};
\node[draw,circle,inner sep=1.5pt,fill=orange!50,color=orange!50] at (1,2) {};
\node[draw,circle,inner sep=1.5pt,fill=orange!50, color=orange!50] at (0,3) {};
\node[draw,circle,inner sep=1.5pt,fill=black!50, color=black!50] at (3,1) {};
\node [above] at (4.2,0) {$x^{4}$};
\node [right] at (0,3.2) {$y^{3}$};
\node [above] at (1.2,2.1) {$xy^{2}$};
\node [above] at (3.1,1.1) {$x^3y$};
 \end{tikzpicture}
\end{center}
\caption{The point $(3,1)$ representing the monomial $x^3y$ is in the convex hull of the yellow region, whose integral points represent monomials in $I$. The integral dependence relation is $(x^3y)^2-x^5.xy^2=0$.}  
   \label{fig:NPfirstexample}
    \end{figure}
 \end{example} 
 Similarly, in $\C\{z_1,\ldots,z_d\}$ we have $\overline{\langle z_1^n,\ldots, z_d^n\rangle}=\langle z_1,\ldots ,z_d\rangle^n$.\par\medskip\noindent      
The \index{exponent vector}\textit{exponent vector} of a monomial $m=z_1^{m_1}\cdots z_d^{m_d}$ is $(m_1,\ldots,m_d)\in \N^d$. For any monomial ideal $I$, the set of all exponent vectors of all the monomials in $I$ is called the \textbf{exponent set of $I$}. Since a monomial $m$ is in $I$ if and only if it is a multiple  in $\C\{z_1,\ldots,z_d\}$ of one of the monomial generators of $I$, the exponent set of $I$ consists of all those points of $\N^d$ which are componentwise greater or equal than the exponent vector of one of the monomial generators of $I$.  Moreover one can prove that $\overline{I}$ is monomial 
      and its exponent set is equal to all the integer lattice points in the convex hull of the set of exponents of elements of $I$. (See \cite[\S 3, \S 4]{Tei04}, \cite[Chap.1, \S 2]{Te1},  \cite[Props 1.4.2 $\&$ 1.4.6]{SwaHu}).

      
    
  % \begin{definition}
  %   Let $O$ be a commutative ring with unity, and let $I,J$ be two ideals of $O$. Set
  %   \[\nu_I(J)= \; \mathrm{sup} \; \{n \; | \; n \in \N, J \subset I^n\}\]
  %   and
  %   \[\overline{\nu}_I(J)= \lim_{k \to \infty} \frac{\nu_I(J^k)}{k}\] 
  %   See \cite[pg 787]{Lej-Te} for a proof that $\overline{\nu}_I$ is well defined.
  % \end{definition}
  
    To understand how this theory can be used in the setting of complex analytic geometry the following result is fundamental. 
   
  \begin{theorem}{\rm (\cite[Thm 2.1, p. 799]{Lej-Te})}\label{Avatarsintegraldependence}
     Let $X$ be a reduced complex analytic space. Let $Y \subset X$ be a closed, nowhere dense,
   analytic subspace of $X$, and $x$ a point in $Y$. Let $\mathcal{I} \subset \Oo_X$ be the coherent ideal 
   defining $Y$, and let $\mathcal{J} \subset \Oo_X$ be another coherent ideal. Let $I$ (resp. $J$) be the
   stalk of $\mathcal{I}$ (resp. $\mathcal{J}$) at $x$. Then the following statements are equivalent:
   \begin{enumerate}
		 			\item $J \subset \overline{I},$
		 			%\item $\overline{\nu}_I(J) \geq 1$
		 			\item For every germ of morphism $\phi:(\C,0) \to (X,x)$
		 			        \[\phi^*J \cdot \Oo_{\C,0} \subset\phi^*I\cdot \Oo_{\C,0}, \]
		 			      
		 			\item For every morphism $\pi: X' \to X$ such that $X'$ is a normal analytic space,
		 			 $\pi$ is proper and surjective, and $\mathcal{I}\cdot \Oo_{X'}$ is locally invertible, there 
		 			 exists an open subset $U \subset X$ containing $x$, such that:
		 			 \[ \mathcal{J} \cdot \Oo_{X'}\vert{\pi^{-1}(U)} \subseteq \mathcal{I} \cdot \Oo_{X'}\vert{\pi^{-1}(U)}, \]
		 			 \item [$3^*$.] If $\Pi: \overline{Bl_{\mathcal{I}} X} \to X$ denotes the normalized blowup
		 			 of $X$ along $\mathcal{I}$, then there exists an open subset $U \subset X$ containing $x$, 
		 			 such that:
		 			 \[ \mathcal{J} \cdot \Oo_{\overline{Bl_{\mathcal{I}} X}}\vert{\Pi^{-1}(U)} \subseteq \mathcal{I} 
		 			 \cdot \Oo_{\overline{Bl_{\mathcal{I}} X}}\vert{\Pi^{-1}(U)}, \] 
		 			 \item Let $V \subset X$ be a neighborhood of $x$, where both $\mathcal{J}$ and $\mathcal{I}$ are
		 			 generated by their global sections. Then for every system of generators $g_1, \ldots,
		 			 g_m$ of $\Gamma (V, \mathcal{I})$ and every $f \in \Gamma (V, \mathcal{J})$, there exist a neighborhood 
		 			 $V'$ of $x$ in $V$ and a constant $C$ such that:
		 			 \[ \left| f(y) \right| \leq C \sup_{i=1,\ldots, m} \left| g_i(y) \right|\]
		 			 for every $y \in V'$.
   \end{enumerate}
  \end{theorem}
  
   Let us take a closer look at statement 2: For any arc $\varphi: (\C,0) \to (X,0)\subset (\C^N,0)$ we have a corresponding morphism of analytic algebras
   \begin{align*}
  \varphi^*: \Oo_{X,0}= \C\{z_1,\ldots,z_N\}/ \mathfrak{a} & \longrightarrow \C\{t\}\\
                          z_i + \mathfrak{a} & \mapsto \varphi_i(t)=t^{m_i}u_i(t)
   \end{align*}
   where $m_i\geq 1$ and  $u_i(t)$ is a unit in $\C\{t\}$. So if $I \subset \Oo_{X,0}$
   is an ideal then $\varphi^*(I)\Oo_{\C,0}=\langle t^k\rangle\C\{t\}$ for some integer $k$ and an element
   $g \in \Oo_{X,0}$ is in $\overline{I}$ if and only if for any such arc $\varphi(t)$
   the order of the series $g\left(\varphi_1(t),\ldots, \varphi_N(t)\right)$ is greater or equal
   than this $k$.\\
  The fact that the normalized blowing-up map is proper implies that the condition of statement 2 needs to be verified only for finitely many arcs. Since the general statement is somewhat cumbersome, let us illustrate how this works in the case where the ideal $I$ a complete intersection defining the origin in $(X,0)$. Let $I=\langle h_1,\ldots ,h_d\rangle \subset \Oo_{X,0}$. The blowing up $Bl_IX$ of $I$ in $X$ is the subspace of $X\times \Pp^{d-1}$ defined by the $d-1$ equations $\frac{h_1}{T_1}=\frac{h_2}{T_2}=\cdots =\frac{h_d}{T_d}$, again a complete intersection. The fiber of the natural projection $Bl_IX\to \Pp^{d-1}$ over a point $t\in \Pp^{d-1}$ with coordinates $(t_1:t_2:\cdots :t_d)$ is a curve in $Bl_IX$ which is isomorphic to its image in $X$ defined by the equations $h_it_j-h_jt_i=0$. So we can view $Bl_IX$ as a family of curves $C_t$ on $X$ parametrized by $\Pp^{d-1}$, which is the exceptional divisor of the map $Bl_IX\to X$. When we pass to the normalization  $n\colon \overline{Bl_I X}\to Bl_IX$, by general Theorems on normalization (see Proposition \ref{constant} and use the fact that by generic flatness there is a dense open $U\subset \Pp^{d-1}$ where $\delta$ is constant), there exists a Zariski dense open subset $U\subset \Pp^{d-1}$ such that $n^{-1}(U)$ is a non singular divisor in a non singular space $n^{-1}((X\times U)\cap Bl_IX)$, and for each point $t\in U$ the map $n$ induces a normalisation of the curve $C_t$. This normalization is then a union of disks, one for each irreducible component of $C_t$, and each disk transversal to $n^{-1}(\Pp^{d-1})$ in $n^{-1}((X\times U)\cap Bl_IX)$. Because a meromorphic function on a normal space is holomorphic if it has no poles in codimension one, to verify that an element $g \in \Oo_{X,0}$ is in $\overline{I}$, it suffices to verify that for some $t\in U$, the order of vanishing of $g$ along each arc parametrizing a branch of $C_t$ is larger than the order of vanishing of the ideal $I$. Because of what we have just seen, the order of vanishing along these arcs will, after lifting to $ \overline{Bl_I X}$, translate as the order of vanishing along some irreducible component of the exceptional divisor in $ \overline{Bl_I X}$. Since the ideal $I$ is locally principal on $ \overline{Bl_I X}$, to prove that $g\in \overline I$ it suffices to prove that after lifting to $ \overline{Bl_I X}$ the function $g$ becomes a multiple of the local equations of the exceptional divisor. But the polar set of the quotient of $g$ by that equation is contained in that exceptional divisor and the inequalities of orders imply that there are no poles at a general point of each irreducible component. Because  $ \overline{Bl_I X}$ is normal, there are no poles anywhere and on $ \overline{Bl_I X}$ the pull back of the function $g$ is indeed in the pull back of the ideal $I$ so that $g$ is in $\overline I$. \par\medskip\noindent
  We shall use this below to describe the saturation.
   \par\medskip
   
   With this at hand we can now characterize $C_3$-general projections in terms of integral closure of ideals. Let $(X,0) \subset (\C^N,0)$ be a reduced germ of analytic space of pure dimension $d$. Let us choose coordinates $z_1,\ldots ,z_N$ on $\C^N$, denote by $L$ the linear subspace of $\C^N$ defined by $z_1=\cdots =z_d=0$ and let $\mathfrak{a}$ be the ideal of $\Oo_{X,0}$ generated by the images of $z_1,\ldots ,z_d$. 
   
   \begin{proposition}\label{C3genintdep}
      The restriction to $(X,0)$
        \[ \pi\vert (X,0): (X,0) \to (\C^d,0) \hspace{0.2in} (z_1,\ldots,z_N) \mapsto (z_1,\ldots,z_d) \]
     of the linear projection $\pi$ with kernel $L$ is $C_3$-general if and only if $\overline{\mathfrak{a}}=\mathfrak{m}$ where
     $\mathfrak{m}=\langle z_1,\ldots,z_N\rangle\Oo_{X,0}$ is the maximal ideal 
     of the analytic algebra $\Oo_{X,0}$.
   \end{proposition}
   \begin{proof}
      Recall that $\pi$ is $C_3$-general if and only if $C_3(X,0) \cap L=\{0\}$.
    Let $\ell=[a_1: \cdots:a_N] \in \mathbf{P}^{N-1}$ be a line in the (projectivized) tangent cone $C_3(X,0)$, then $\ell \not\subset L$ if and only if $a_i \neq 0$ for some $i \in \{1,\ldots,d\}$. Note that any arc $\varphi: (\C,0) \to (X,0)$ determines a line in $C_3(X,0)$, the limit as $t\to 0$ of 
    \[t \longrightarrow \left[\varphi_1(t): \cdots :\varphi_N(t)\right]\in \Pp^{N-1},\]
    and conversely any line in the tangent cone can be obtained through an arc since it corresponds to a point in the fiber over $0$ of the blowing-up $Bl_0X\to X$. On the other hand, for every arc $\varphi: (\C,0) \to (X,0)$ we have that
    \[\varphi^*(\mathfrak{a})\Oo_{\C,0}=\langle \varphi_1(t), \ldots ,\varphi_d(t) \rangle\C\{t\}= \langle t^k\rangle\C\{t\},\] 
    where $k=\mathrm{min}\{\mathrm{ord}_0\varphi_i(t)\,|\, i=1,\ldots,d\}$.
    Finally $a_i \neq 0$ for some $i \in \{1,\ldots,d\}$ if and only if for all 
    $j \in \{d+1, \ldots, N\}$  
    \[\mathrm{ord}_0\varphi_j(t) \geq k=\mathrm{min}\{\mathrm{ord}_0\varphi_i(t)\,|\, i=1,\ldots,d\}\]
    if and only if $\varphi^*(z_j) \in \varphi^*(\mathfrak{a})\Oo_{\C,0}$ if and only if 
    $z_j \in \overline{\mathfrak{a}}$ for all $j \in \{d+1, \ldots, N\}$, that is, $\overline{\mathfrak{a}}=\mathfrak{m}$.
   \end{proof}
   
   By a linear change of coordinates in $\C^N$ we can always place ourselves in the 
   setting of the previous result. But the theory of integral closure also gives us
   an algebraic way to prove that for a given germ $(X,0)$ of pure dimension $d$ 
   almost all linear projections $\pi:(\C^N,0) \to (\C^d,0)$ are $C_3$-general as stated 
   in the following result (For a proof see \cite[Thm 14.14]{Mats1})
   
   \begin{theorem} (Rees-Samuel)\label{Rees-Samuel}
      Let $\Oo_{X,0}$ be a $d$-dimensional analytic algebra with maximal ideal
      $\mathfrak{m}=\langle z_1,\ldots,z_N\rangle$. Then if $y_i=\sum_{j=1}^N \lambda_{ij}z_j$
      for $1\leq i \leq d$ are $d$ ``sufficiently general" $\C$-linear combinations 
      of $z_1,\ldots,z_n$ the ideal $\mathfrak{a}= \langle y_1,\ldots,y_d\rangle$ satisfies
      $\overline{\mathfrak{a}}=\mathfrak{m}$.
   \end{theorem}
   
   We can take this one step further by considering another important aspect of this theory,
   namely its relation with multiplicity. For a local Noetherian ring $(R,\mathfrak{m})$  and an $\mathfrak{m}-$primary ideal $\mathfrak{a}\subset R$ we can define a Hilbert Samuel function
   \[ k \in \N \mapsto {\rm dim}_{R/\mathfrak{m}} R/\mathfrak{a}^k.\]
   The result is that for large enough $k$ the Hilbert-Samuel function behaves like 
   a polynomial of degree equal to the dimension of $R$ and its leading coefficient
   is of the form $e(\mathfrak{a})k^{d}/d!$, where $e(\mathfrak{a})$ is a positive
   integer called the \index{multiplicity of an ideal}multiplicity of the ideal $\mathfrak{a}$. In the case
   $R$ is the analytic algebra $\Oo_{X,0}$ of a germ $(X,0)$ and $\mathfrak{a}=
   \mathfrak{m}$ it IS the multiplicity of the germ. (See \cite[Section 4.2]{DeJo-Pfi})
   
   \begin{theorem}(Rees)\label{reduc}{\rm (see \cite[Thm 3.2]{Ree1},\cite[Thm 11.3.1]{SwaHu})}\\
       Let $(\Oo_{X,0},\mathfrak{m})$ be a reduced and equidimensional analytic
       algebra and $\mathfrak{a}\subset \mathfrak{b}$ two $\mathfrak{m}-$primary
       ideals. Then $\overline{\mathfrak{a}}=\overline{\mathfrak{b}}$ if and only if $e(\mathfrak{a})=e(\mathfrak{b})$.
   \end{theorem}
   
   A geometric interpretation of this result is described by Lipman in \cite{Lip1}. Let $(X,0)$ be a germ of reduced and equidimensional singularity of dimension $d$
   with associated analytic algebra $\left(\Oo_{X,0}, \mathfrak{m}\right)$. Every  $\mathfrak{m}-$primary ideal is generated by at least $d$ elements, and every
   $d-$tuple $(f_1,\ldots,f_d)$ of elements of $\mathfrak{m}$ defines a map-germ
   $F:(X,0) \to (\C^d,0)$.\\  
   
   Now, the ideal $\mathfrak{a}=\langle f_1,\ldots,f_d\rangle$ is $\mathfrak{m}-$primary if and only if $F$ is finite. As we have mentioned before you can prove that such an $F:(X,0) \to (\C^d,0)$ is then a ramified analytic cover of degree equal to $e(\mathfrak{a})$ and by Rees' Theorem this degree will be the multiplicity of $(X,0)$ ($=e(\mathfrak{m}))$
   if and only if $\overline{\mathfrak{a}}=\mathfrak{m}$.\\ 
   
   Moreover using Nakayama's Lemma one checks that $\mathfrak{a}$ is a reduction of $\mathfrak{m}$
   (equivalently $\overline{\mathfrak{a}}=\mathfrak{m}$) if and only if in the graded $\C-$algebra
   \[{\rm gr}_\mathfrak{m}{\mathcal O}=\bigoplus_{k\geq0} \mathfrak{m}^k/\mathfrak{m}^{k+1},\ \ {\rm with}\ \mathfrak{m}^0={\mathcal O}\]
   (which is the homogeneous coordinate ring of the projectivized tangent cone $\mathbf{P}C_3(X,0)$ see \cite[Section 2.4]{GiTei}) the images $\overline{f_i}$ of the $f_i$ 
   in $\mathfrak{m}/\mathfrak{m}^2$ generate an irrelevant ideal (that is, an ideal
   containing all elements of ${\rm gr}_\mathfrak{m}{\mathcal O}$ of sufficiently large degree so that its zero locus
   in projective space is empty).\\
   
   What this last condition means is that first of all the $\overline{f_i}$ are
   linearly independent over $\C$, so that there is an embedding of the germ
   $(X,0)$ into $(\C^N,0)$ for some $N$ and a linear projection $\pi:(\C^N,0) \to (\C^d,0)$ such that its restriction to $(X,0)$ is germwise the $F$ associated above
   to $(f_1,\ldots,f_d)$ and secondly, since $\overline{\mathfrak{a}}=\mathfrak{m}$ by Proposition \ref{C3genintdep} the projection $\pi$ is $C_3$-general. \\
   
   We end this section by establishing a result analogous to Proposition \ref{C3genintdep} but with respect to generic projections of curves. 
   
   \begin{definition}\label{DefTenProd}
      Let $\varphi_1: R \to A_1$ and $\varphi_2:R \to A_2$ be morphisms of $\C$-analytic algebras. There is a unique $\C$-analytic algebra, denoted $A_1{{\widearc{\otimes}}}_RA_2$,
      together with morphisms $\theta_i:A_i \to A_1{\widearc{\otimes}}_RA_2,\ i=1,2 ,$ such that
      $\theta_1 \circ \varphi_1=\theta_2\circ \varphi_2$ and for every pair of morphisms
      of $\C$-analytic algebras $\psi_1:A_1 \to B$, $\psi_2:A_2 \to B$ satisfying
      $\psi_1\circ \varphi_1= \psi_2 \circ \varphi_2$ there is a unique morphism 
      of $\C$-analytic algebras $\psi: A_1{\widearc{\otimes}}_RA_2 \to B$ making 
      the whole diagram commute. The algebra $A_1{\widearc{\otimes}}_RA_2$ is called the 
      analytic tensor product of $A_1$ and $A_2$ over $R$.
      \[\xymatrix{
           & A_1\ar[dr]^{\theta_1}\ar@/^2pc/[drr]^{\psi_1} & & \\
         R\ar[ur]^{\varphi_1}\ar[dr]_{\varphi_2} & & A_1{\widearc{\otimes}}_RA_2 \ar @{.>}[r]^\psi & B\\  
         & A_2\ar[ur]_{\theta_2}\ar@/_2pc/[urr]_{\psi_2} & &\\}\]
   \end{definition}
      
      Geometrically this analytic tensor product is the operation on the analytic algebras
      that corresponds to the fibre product of analytic spaces. Given holomorphic maps
      $\phi_1:(X_1,p_1) \to (Y,q)$ and $\phi_2: (X_2,p_2) \to (Y,q)$ we have the 
      fibre product:
       \[\xymatrix { 
       X_1 \times_Y X_2 \ar[d]^{\Pi_2} \ar[r]^{\Pi_1}  & X_1 \ar[d]^{\phi_1} \\
        X_2 \ar[r]_{\phi_2}&  Y }\]
      which induces the corresponding diagram of analytic algebras
      \[\xymatrix { 
       \Oo_{Y,q}\ar[d]^{\varphi_2} \ar[r]^{\varphi_1}  & \Oo_{X_1,p_1} \ar[d] \\
        \Oo_{X_2,p_2} \ar[r] & \Oo_{X_1\times_Y X_2, (p_1,p_2)} }\]
      that is, the analytic algebra $\Oo_{X_1\times_Y X_2, (p_1,p_2)}$ is isomorphic to $\Oo_{X_1,p_1}{\widearc{\otimes}}_{\Oo_{Y,q}}\Oo_{X_2,p_2}$. 
      
      \begin{remark} \label{TensorProduct}See \cite[Def 1.28, Example 1.46.1 $\&$ Lemma 1.89]{Gre-Pfis} and \cite{Ada1}.
        \begin{enumerate}
            \item  When $R=\C$ in the definition, the analytic tensor product $\Oo_{X_1,p_1} {\widearc{\otimes}}_\C \Oo_{X_2,p_2}$ is the analytic algebra corresponding
   to the product germ $\left(X_1\times X_2,(p_1,p_2)\right)$. Moreover if  $\Oo_{X_1,p_1}=\C\{\mathbf{z}\}/I$ and $\Oo_{X_2,p_2}=\C\{\mathbf{w}\}/J$ with $\mathbf{z}=(z_1,\ldots,z_N)$
     and $\mathbf{w}=(w_1,\ldots,w_M)$ then 
     \[\Oo_{X_1,p_1} {\widearc{\otimes}}_\C \Oo_{X_2,p_2}= \frac{\C\{\mathbf{z,w}\}}{\langle I \C\{\mathbf{z,w}\} + J \C\{\mathbf{z,w}\}\rangle}.\]
     \item In general if $(X_1,p_1) \subset  (\C^N,0)$, $(X_2,p_2) \subset (\C^M,0)$ and
     $(Y,q) \subset (\C^k,0) $ with  $\Oo_{Y,q}=\C\{y_1,\ldots ,y_k\}/K$, denoting by $y_k(\mathbf{z})$ (resp. $y_k(\mathbf{w})$ a representative in $\C\{\mathbf{z}\}$ (resp. $\C\{\mathbf{w}\}$) of the image of $y_k$ in $\Oo_{X_1,p_1}$ (resp. $\Oo_{X_2,p_2}$ by the structure maps $\Oo_{Y,q}\rightarrow \Oo_{X_i,p_i}\ \ i=1,2$, then 
     \begin{align*}
      & \Oo_{X_1,p_1}{\widearc{\otimes}}_{\Oo_{Y,q}}\Oo_{X_2,p_2}= 
     \frac{\C\{\mathbf{z}\}}{I} {\widearc{\otimes}}_{\Oo_{Y,q}}\frac{\C\{\mathbf{w}\}}{J}\\
     & \cong 
     \frac{\C\{\mathbf{z,w}\}}{ I \C\{\mathbf{z,w}\} + J \C\{\mathbf{z,w}\}
    + \langle  y_1(\mathbf{z})-y_1(\mathbf{w}),\ldots,
    y_k(\mathbf{z})-y_k(\mathbf{w})\rangle\C\{\mathbf{z,w}\}}. \end{align*}
        \end{enumerate}
      \end{remark}
      Let $(X,0) \subset (\C^N,0)$ be
   a germ of reduced singular curve. By Proposition \ref{C5curve} the $C_5$ cone is a 
   finite union of $2$-planes of $\C^n$
   \[C_5(X,0)= H_1 \cup \ldots \cup H_r.\]
   If we let $\pi:(\C^N,0) \to (\C^2,0)$ denote the linear projection to the first two coordinates $(z_1,\ldots,z_N) \mapsto (z_1,z_2)$ then $\pi$ is generic if and only 
   if $\pi(H_i)=\C^2$ for $i=1,\ldots,r$. Recall that the construction of $C_5(X,0)$ 
   goes through the blowup of the diagonal of $X\times X$, so let $I_{\Delta} \subset \Oo_{X \times X, (0,0)}$ be the ideal defining this diagonal
   \[I_{\Delta}=\langle z_1-w_1,\ldots, z_N-w_N\rangle\Oo_{X\times X, (0,0)}.\]
   
   \begin{proposition}\label{C5genintdep}
     Let $I_{\Delta_2}\subset \Oo_{X\times X,(0,0)}$ be the ideal coming from the projection $\pi$, that is,
     $I_{\Delta_2}=\langle z_1-w_1,z_2-w_2\rangle\Oo_{X\times X,(0,0)}$. Then $\pi$
     is generic if and only if $\overline{I_{\Delta_2}}=\overline{I_\Delta}$.
   \end{proposition}
   \begin{proof}
    The proof is now very similar to the $C_3$-general case, and since $I_{\Delta_2}\subset I_{\Delta}$ all 
    we have to prove is that genericity is equivalent to the inclusion $I_{\Delta}\subset \overline{I_{\Delta_2}}$.
    
    Let $L=V(z_1,z_2)$ be the kernel of $\pi$. Then  $\pi$ is generic if and only if $C_5(X,0) \cap L=\{0\}$.
    Let $\ell=[a_1: \cdots:a_N] \in \mathbf{P}^{N-1}$ be a line in the (projectivized) cone $C_5(X,0)$, then $\ell \not\subset L$ if and only if $a_i \neq 0$ for some $i \in \{1,2\}$. This time the lines in $C_5(X,0)$ are determined by taking the limit as $t\to 0$ of the secants associated to pairs of arcs
    $(\varphi,\psi): (\C,0) \to (X\times X, (0,0))$
    \[t \longrightarrow \left[\varphi_1(t)-\psi_1(t): \cdots :\varphi_N(t)-\psi_N(t) \right]\in \Pp^{N-1}\]
     Again for every such pair of arcs $(\varphi,\psi): (\C,0) \to  (X\times X, (0,0))$ we have that
    \[(\varphi,\psi)^*(I_{\Delta_2})\Oo_{\C,0}=\langle \varphi_1(t)-\psi_1(t), \varphi_2(t)- \psi_2(t)\rangle\C\{t\}= \langle t^k\rangle\C\{t\}\] 
    where $k=\mathrm{min}\{\mathrm{ord}_0(\varphi_1(t)-\psi_1(t)),\mathrm{ord}_0(\varphi_2(t)-\psi_2(t)) \}$.
    Finally $a_i \neq 0$ for some $i \in \{1,2\}$ if and only if for all 
    $j \in \{3, \ldots, N\}$  
    \[\mathrm{ord}_0(\varphi_j(t) -\psi_j(t))\geq k=\mathrm{min}\{\mathrm{ord}_0(\varphi_1(t)-\psi_1(t)),\mathrm{ord}_0(\varphi_2(t)-\psi_2(t)) \}\]
    if and only if $(\varphi,\psi)^*(z_j-w_j) \in (\varphi,\psi)^*(I_{\Delta_2})\Oo_{\C,0}$ if and only if 
    $z_j-w_j \in \overline{I_{\Delta_2}}$ for all $j \in \{3, \ldots, N\}$, that is, $I_\Delta\subset \overline{I_{\Delta_2}}$.
   \end{proof}
   
   
   \subsection{Lipschitz Saturation}
       Let $n^*\colon \Oo_{X,0} \hookrightarrow \overline{\Oo_{X,0}}$ be the integral closure of a reduced complex analytic algebra which is a quotient of $\C\{z_1,\ldots ,z_N\}$.  Recall that $\overline{\Oo_{X,0}}$ is a direct sum of normal analytic algebras (in particular integral domains), one for each irreducible component of the germ $(X,0)$. 
      By Definition \ref{DefTenProd}\  the following commutative diagram determines a unique morphism $\Psi$ of direct sums of analytic algebras: 
       \[\xymatrix{
         &  &  \overline{\Oo_{X,0}}\ar[dr]^{\theta_1}\ar@/^3pc/[drr]^{\psi_1} & & \\
         \C \ar@/_2pc/[drr] \ar@/^2pc/[urr] \ar[r] & \Oo_{X,0 }\ar[ur]\ar[dr] & & \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\C}\overline{\Oo_{X,0}} \ar @{.>}[r]^\Psi & 
         \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\Oo_{X,0}}\overline{\Oo_{X,0}}\\  
        &  &  \overline{\Oo_{X,0}} \ar[ur]_{\theta_2} \ar@/_3pc/[urr]_{\psi_2} & & \\ } \]
 \noindent       
  where $\theta_1(f)=f {\widearc{\otimes}}_\C 1$ and $\theta_2(f)=1{\widearc{\otimes}}_\C f$. Note that the map $\Psi:\overline{\Oo_{X,0}}{\widearc{\otimes}}_{\C}\overline{\Oo_{X,0}} \to  
   \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\Oo_{X,0}}\overline{\Oo_{X,0}}$ is the morphism of sums of analytic algebras
   corresponding to the inclusion $\left(\overline{X} \times_X \overline{X}, (0,0) \right) \hookrightarrow \left(\overline{X} \times \overline{X}, (0,0) \right)$. 
   By remark \ref{TensorProduct} if we denote by $n:(\overline{X},0) \to (X,0)$ the normalization map  and $\overline{\Oo_{X,0}}=\bigoplus_{i=1}^r\C\{{\underline t}_i\}/J_i({\underline t}_i)$ with ${\underline t}_i=({\underline t}_{i,1},\ldots,{\underline t}_{i,m_i})$, then 
   \[\overline{\Oo_{X,0}}{\widearc{\otimes}}_{\C}\overline{\Oo_{X,0}} = \bigoplus_{i,j}\frac{\C\{{\underline t}_i,{\underline u}_j\}}{\langle  J_i({\underline t}_i), J_j({\underline u}_j)\rangle} \]
 and $\Psi$ is a surjection with kernel $$I_\Delta=\langle z_1{\widearc{\otimes}}_\C1-1{\widearc{\otimes}}_\C z_1,\dots ,z_N{\widearc{\otimes}}_\C 1-1{\widearc{\otimes}}_\C z_N \rangle\overline{\Oo_{X,0}}{\widearc{\otimes}}_{\C}\overline{\Oo_{X,0}}.$$
       
 \begin{definition}\label{Saturationdef}
   Let $I_\Delta$ be the kernel of the morphism $\Psi$ above. We define the \index{Lipschitz saturation}Lipschitz saturation $\Oo_{X,0}^s$ of $\Oo_{X,0}$ as the algebra
   \[\Oo_{X,0}^s:= \left\{ f \in \overline{\Oo_{X,0}} \, | \, \theta_1(f)-\theta_2(f) \in \overline{I_\Delta}\right\}= \left\{ f \in \overline{\Oo_{X,0}} \, | \, f {\widearc{\otimes}}_\C 1-1{\widearc{\otimes}}_\C f \in \overline{I_\Delta}\right\} .\]
 \end{definition}
 
 \begin{example}
   Let $(X,0) \subset (\C^3,0)$ be the irreducible curve with normalization map:
   \begin{align*}
         \eta:(\C,0) &\longrightarrow (X,0) \\
               t & \mapsto (t^4,t^6,t^7)
   \end{align*}
    In this setting the map $\Psi$ above is
   \begin{align*}
       \Psi: \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\C}\overline{\Oo_{X,0}} &  \to  
   \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\Oo_{X,0}}\overline{\Oo_{X,0}}\\ 
    \Psi: \C\{t,u\} &\longrightarrow \frac{\C\{t,u\}}{\langle t^4-u^4,t^6-u^6,t^7-u^7\rangle}   . 
   \end{align*}
   The maps $\theta_i$ are just inclusions, $\C\{t\} \hookrightarrow \C\{t,u\}$,
  $\C\{u\} \hookrightarrow \C\{t,u\}$ and the ideal $I_\Delta=\langle t^4-u^4,t^6-u^6,t^7-u^7\rangle$.  
  By definition $\Oo_{X,0}^s:= \left\{ f \in \C\{t\} \, | \, f(t)- f(u) \in \overline{I_\Delta}\right\}$ and  note that $\Oo_{X,0} \subset \Oo_{X,0}^s$.
   For example $t^5 \in \C\{t\}$ is in $\Oo_{X,0}^s$ if and only if $t^5-u^5$ is in $\overline{I_\Delta}$. By taking the arc $\phi(\tau)=(\tau,-\tau)$ we have that $\phi^*I_\Delta\Oo_{\C,0}
   =\langle \tau^7\rangle$ and $\phi^*(t^5-u^5)=2\tau^5 \notin \phi^*I_\Delta\Oo_{\C,0}$, so by Theorem
   \ref{Avatarsintegraldependence}-2 the element $t^5\in \C\{t\}=\overline{\Oo_{X,0}}$ is not in the Lipschitz saturation $\Oo_{X,0}^s$.
   For this particular arc we have $\phi^*(t^9-u^9)=2\tau^9 \in \phi^*I_\Delta\Oo_{\C,0}$, and one can 
   actually prove that $t^9 \in \Oo_{X,0}^s$. As we shall see later on, in fact $\Oo_{X,0}^s=\C\{t^4,t^6,t^7,t^9\}$.
 \end{example}
 
 We are going to show that the Lipschitz saturation $\Oo_{X,0}^s$ is 
 always an analytic algebra, even if the germ $(X,0)$ is not irreducible. To begin to understand
 why this is true, let's look at the irreducible case. Define the map 
  \begin{align*}
       \alpha: \overline{\Oo_{X,0}} & \to \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\C}\overline{\Oo_{X,0}}\\
                       f &\mapsto \theta_1(f)-\theta_2(f)=f(z)-f(w)
  \end{align*}
  It is not a ring map, however if  $n^*:\Oo_{X,0} \hookrightarrow \overline{\Oo_{X,0}}$ denotes the inclusion
  coming from the normalization map $n:\overline{X} \to X$ then $\alpha(n^*(\Oo_{X,0}))= \alpha(n^*(\mathfrak{m}_{X,0}))$ and $I=\mathrm{Ker}\,\Psi=\langle \alpha(n^*(\mathfrak{m}_{X,0}))\rangle$.\\
  
  By Definition \ref{DefTenProd} $\overline{\Oo_{X,0}}{\widearc{\otimes}}_{\Oo_{X,0}}\overline{\Oo_{X,0}}$ is an 
  $\Oo_{X,0}$-algebra, in particular an $\Oo_{X,0}$-module. However, an interesting point is that since $n:\overline{X} \to X$ is a finite map, by \cite[Lemma 1.89]{Gre-Pfis} this algebra is isomorphic 
  to the algebraic tensor product $\overline{\Oo_{X,0}}\otimes_{\Oo_{X,0}}\overline{\Oo_{X,0}}$, so for instance
  \[\frac{\C\{t,u\}}{\langle t^2-u^2,t^3-u^3\rangle} \cong \C\{t\} \otimes_{\C\{t^2,t^3\}} \C\{u\} \]
  
  \begin{lemma}\label{Sat0}
   The map
  $$ \Psi\circ \alpha: \overline{\Oo_{X,0}} \longrightarrow \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\Oo_{X,0}}\overline{\Oo_{X,0}} $$
 is a morphism of $\Oo_{X,0}$-modules. 
  \end{lemma}
  \begin{proof}
  Indeed for $r \in \Oo_{X,0}$ and $f \in \overline{\Oo_{X,0}}$:
 \[rf \stackrel{\alpha}{\mapsto} r(z)f(z)-r(w)f(w) \stackrel{\Psi}{\mapsto}r(z)f(z) -r(w)f(w) + I\]
 but $r(z)=r(w)$ mod$(I)$ so $r(z)f(z) -r(w)f(w)=(r(z) + I)(f(z)-f(w)+I)=r(\Psi \circ \alpha)(f)$.\\
  \end{proof}
  
 By definition $\overline{I_\Delta}$ is an ideal of $ \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\C}\overline{\Oo_{X,0}}$
 and since $\Psi$ is a surjective ring homomorphism we have that $\Psi(\overline{I_\Delta}) \subset \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\Oo_{X,0}}\overline{\Oo_{X,0}}$ is an ideal, in particular it is an $\Oo_{X,0}$-module. But this implies that 
 \[\left( \Psi \circ \alpha\right)^{-1}(\Psi(\overline{I_\Delta}))=\alpha^{-1}(\overline{I_\Delta})= \Oo_{X,0}^s 
  \subset \overline{\Oo_{X,0}}\]
 is an $\Oo_{X,0}$-module.\hfill$\square$

 \begin{lemma}\label{Sat1}
     The Lipschitz saturation $\Oo_{X,0}^s$ is an $\Oo_{X,0}$-algebra and a direct sum of analytic algebras. 
 \end{lemma}
 \begin{proof}
    Since $\Oo_{X,0}^s$ is a submodule of the  Noetherian module $\overline{\Oo_{X,0}}$, it is a finitely generated $\Oo_{X,0}$-module. Even more, you can easily check 
 that $\Oo_{X,0}^s$ is closed under multiplication, so it is an $\Oo_{X,0}$-algebra and by \cite[Cor. 3.3.25 $\&$ 3.3.26]{DeJo-Pfi} this implies that $\Oo_{X,0}^s$ is a direct sum of analytic algebras. 
 
  Indeed, take $f_1, f_2 \in \Oo_{X,0}^s$ then $(\Psi \circ \alpha)(f_1)=f_1(z)-f_1(w) + I_\Delta \in \Psi(\overline{I_\Delta})$, 
  but it is an ideal so $(f_2(z) + I_\Delta)(f_1(z)-f_1(w) + I_\Delta) \in \Psi(\overline{I_\Delta})$. Analogously 
  $(f_1(w) + I_\Delta)(f_2(z)-f_2(w) + I_\Delta) \in \Psi(\overline{I_\Delta})$ by taking their sum we get 
  that $(\Psi \circ \alpha)(f_1f_2)=f_1(z)f_2(z)-f_1(w)f_2(w) + I_\Delta \in \Psi(\overline{I_\Delta})$ which 
  implies that $f_1f_2 \in \Oo_{X,0}^s$ as claimed. 
 \end{proof}
 
 Before proving that $\Oo_{X,0}^s$ is actually an analytic algebra we would like to give an idea of how things work in the non-irreducible case so suppose there are two irreducible components
 $(X,0)=(X_1,0) \cup (X_2,0)$. As we said before $\overline{X}$ is then a multigerm
 $(\overline{X_1},p) \sqcup (\overline{X_2},q)$ and $\overline{\Oo_{X,0}}= \overline{\Oo_{X_1,0}} \bigoplus \overline{\Oo_{X_2,0}}= \Oo_{\overline{X_1},p} \bigoplus \Oo_{\overline{X_2},q}$. Since the analytic 
 tensor product should be the algebraic counterpart of the fibre product then we should consider/define
$$\overline{\Oo_{X,0}}{\widearc{\otimes}}_{\Oo_{X,0}}\overline{\Oo_{X,0}} =$$
 $$\Oo_{\overline{X}_1,p}{\widearc{\otimes}}_{\Oo_{X,0}}\Oo_{\overline{X}_1,p} \oplus
 \Oo_{\overline{X}_1,p}{\widearc{\otimes}}_{\Oo_{X,0}}\Oo_{\overline{X}_2,q} \oplus
  \Oo_{\overline{X}_2,q}{\widearc{\otimes}}_{\Oo_{X,0}}\Oo_{\overline{X}_1,p} \oplus
  \Oo_{\overline{X}_2,q} {\widearc{\otimes}}_{\Oo_{X,0}}\Oo_{\overline{X}_2,q} $$
  and analogously for $\overline{\Oo_{X,0}}{\widearc{\otimes}}_{\C} \overline{\Oo_{X,0}}$. By componentwise 
  taking the ring maps $\Psi_{ij}$ coming from the universal property of the irreducible case, for example:
  \[\Psi_{12}:\Oo_{\overline{X}_1,p}{\widearc{\otimes}}_{\C} \Oo_{\overline{X}_2,q} \to
    \Oo_{\overline{X}_1,p}{\widearc{\otimes}}_{\Oo_{X,0}}\Oo_{\overline{X}_2,q}\]   
  we get the ring map $\Psi$ as before with kernel $I_\Delta=I_{11} \oplus I_{12} \oplus I_{21} \oplus I_{22}$.
  The map $\alpha$ should now be defined as 
  \begin{align*}
      \alpha:\overline{\Oo_{X,0}} & \longrightarrow \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\C} \overline{\Oo_{X,0}} \\
            (f_1,f_2) &\mapsto \left(f_1(z)-f_1(w), f_1(z)-f_2(w), f_2(z)-f_1(w), f_2(z)-f_2(w) \right)
  \end{align*}
  and we get the same definition for the Lipschitz saturation
  \[\Oo_{X,0}^s:= \left\{ f=(f_1,f_2) \in \overline{\Oo_{X,0}} \, | \, \alpha(f) \in \overline{I_\Delta}\right\}. \]
  More importantly both Lemmas remain valid.  Note that in this context of two irreducible components we have $\alpha(f) \in \overline{I_\Delta}$  if and only if $f_1(z)-f_1(w) \in \overline{I_{11}}$, $f_1(z)-f_2(w) \in \overline{I_{12}}$,  $f_2(z)-f_1(w) \in \overline{I_{21}}$ and $f_2(z)-f_2(w) \in \overline{I_{22}}$. 
  
  
  \begin{proposition}\label{Sat2} {\rm (See \cite[Theorem 1.2]{P-T69}, \cite[Prop. 6.1.1]{Te1})}
     The algebra $\Oo_{X,0}^s$ is the ring of germs of meromorphic functions on $(X,0)$ which are locally 
     Lipschitz with respect to the ambient metric. 
  \end{proposition}
  \begin{proof}
   Recall that $\overline{\Oo_{X,0}}$ is the ring of meromorphic functions on $(X,0)$ that are locally bounded and a Lipschitz meromorphic function is locally bounded. Now if $h \in \Oo_{X,0}^s$ we need
   to prove that there exists a real positive constant $C>0$ such that for every couple $(x_1,x_2) \in X\setminus\mathrm{Sing}\,X \times X \setminus \mathrm{Sing}\,X$ (in a small enough representative)  we  have
   \[ | h(x_1)-h(x_2)| \leq C || x_1-x_2 || .\]
   Let $n:\overline{X} \to X \subset \C^n $ be the normalization map. In the irreducible case where $\Oo_{X,0}=\C\{z_1,\ldots ,z_N\}/\langle f_1,\ldots ,f_s\rangle$ and $\overline{\Oo_{X,0}}=\C\{t_1,\ldots ,t_m\}/J({\underline t})$, the map $n$ induces  a morphism of analytic algebras which may be described by
   \begin{align*}
          n^*: \Oo_{X,0} &\longrightarrow \overline{\Oo_{X,0}} \\
                        z_i & \mapsto z_i(t_1,\ldots,t_m)=z_i({\underline t})
   \end{align*}
   and refering to the maps 
   $\alpha$ and $\Psi$ as above we have that 
   \[I_\Delta = \mathrm{Ker}\, \Psi = \langle z_1(\underline t) - z_1(\underline u) , \ldots, z_N(\underline t) - z_N(\underline u) \rangle .\]
   By definition, $h \in \Oo_{X,0}^s$ if $\alpha(h)=h({\underline t}) -h({\underline u}) \in \overline{I_\Delta}$ and by Theorem 
   \ref{Avatarsintegraldependence} there exists a constant $C$ such that
     \[ |h({\underline t}) -h({\underline u}) | \leq C \, \mathrm{sup} \, |z_i({\underline t})-z_i({\underline u})| 
       = C || z({\underline t}) - z({\underline u}) ||,\]
    with $\left(z({\underline t}) ,z({\underline u}) \right) \in X \times X$ and so $h$ is Lipschitz. Reading the proof in the opposite sense gives that a meromorphic, locally Lipschitz function $h$ is necessarily in $\Oo_{X,0}^s$.
     
     If $(X,0)$ has $r$ irreducible components  then $\overline{X}$ is
   a multigerm and  then we have $r$ maps $n_k:(\overline{X_k},x_k) \to (X_k,0) \subset (X,0)$ with 
   coordinate functions $z_1({\underline t}_k), \ldots, z_N({\underline t}_k)$. Then for $h=(h_1,\ldots,h_r) \in \overline{\Oo_{X,0}}$
   we have that $\alpha(h)=\left(h_i({\underline t}_i){\widearc{\otimes}} 1-1{\widearc{\otimes}}h_j({\underline u}_j)\right)_{i,j}\in\bigoplus_{i,j=1}^{i,j=r}\overline{\Oo_{X_i,0}}{\widearc{\otimes}}_\C\overline{\Oo_{X_j,0}}$, and {\small$$I_\Delta=\bigoplus_{i,j=1}^{i,j=r} I_{ij}\  \ {\rm with}\  
   I_{ij}=\langle  z_1({\underline t}_i){\widearc{\otimes}} 1-1{\widearc{\otimes}}z_1({\underline u}_j), \ldots,  z_N({\underline t}_i){\widearc{\otimes}} 1-1{\widearc{\otimes}}z_N({\underline u}_j)\rangle\overline{\Oo_{X_i,0}}{\widearc{\otimes}}_\C\overline{\Oo_{X_j,0}}$$} and $\alpha(h) \in 
   \overline{I_\Delta}$ if and only if $h_i({\underline t}_i){\widearc{\otimes}} 1 - 1{\widearc{\otimes}}h_j({\underline u}_j) \in \overline{I_{ij}}$ for all $(i,j)$.\par\noindent So in the spirit of Example 
   \ref{WeaklyHolEx} the ``coordinate'' $h_i$ of $h$ indicates you how to evaluate $h$ in points of the 
   corresponding irreducible component $(X_i,0)$ of $(X,0)$ and for $i\neq j$ the condition $h_i({\underline t}_i){\widearc{\otimes}} 1 - 1{\widearc{\otimes}}h_j({\underline u}_j) \in \overline{I_{ij}}$ tells you that the Lipschitz condition must also be satisfied when you take 
   points in different irreducible components. 
  \end{proof}
  
  \begin{corollary}{\rm (See \cite[Corollary 1.3]{P-T69})} 
     Let $(X,0) \subset (\C^N,0)$ be a reduced germ of complex analytic singularity. The ring $\Oo_{X,0}^s$ is 
     an analytic algebra.
  \end{corollary}
   \begin{proof}
      We already proved in Lemma \ref{Sat1} that $\Oo_{X,0}^s$ is a direct sum of analytic algebras, 
      but if there were more than one, the function $(1,0,\ldots,0)\in\Oo^s_{X,0}$ would not be Lipschitz, contradicting
      Proposition \ref{Sat2}. 
   \end{proof}
   \noindent
            From Lemma \ref{Sat1} we have injective ring morphisms 
   \[\Oo_{X,0} \hookrightarrow \Oo_{X,0}^s \hookrightarrow \overline{\Oo_{X,0}}.\]
   Since $\overline{\Oo_{X,0}}$ is contained in the total ring of fractions $Q(\Oo_{X,0})$,
   the total ring of fractions of the Lipschitz saturation $O^s_{X,0}$ coincides  with $Q(\Oo_{X,0})$
   and by transitivity of integral dependence the normalizations also coincides i.e., $\overline{\Oo_{X,0}^s}=
   \overline{\Oo_{X,0}}$. In terms of holomorphic maps we have:
   \[\overline{X}\stackrel{n_s}{\longrightarrow} X^s \stackrel{\zeta}{\longrightarrow} X,\]
   where $X^s$ is the germ of complex analytic singularity corresponding to the analytic algebra
   $\Oo_{X,0}^s$, the map $n_s:\overline{X} \to X^s$ is the normalization map of $X^s$,
   $\zeta:(X^s,0) \to (X,0)$ is finite and induces an isomorphism outside the non-normal locus of $X$,
   and $n=\zeta \circ n_s : \overline{X} \to X$ is the normalization map of $X$.
   
   \begin{definition}
       The germ $(X^s,0)$ together with the finite map $\zeta:(X^s,0) \to (X,0)$ is called the Lipschitz saturation of $(X,0)$.
   \end{definition}
   
   \begin{lemma}\label{Sat3}
      Let $(X,0) \subset (\C^n,0)$ be a reduced germ of complex analytic singularity, then 
      $\left(\Oo_{X,0}^s\right)^s = \Oo_{X,0}^s$.
   \end{lemma}
   \begin{proof}
    Following the notation of Lemma \ref{Sat0} we have the maps:
    \[\Oo_{X,0} \hookrightarrow \Oo_{X,o}^s \hookrightarrow \overline{\Oo_{X,0}} \stackrel{\alpha}{\longrightarrow}
       \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\C}\overline{\Oo_{X,0}}, \]
    and this induces a map 
    \[ \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\Oo_{X,0}}\overline{\Oo_{X,0}} \cong \frac{ \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\C}\overline{\Oo_{X,0}}}{\langle \alpha(\Oo_{X,0})\rangle} \longrightarrow
    \frac{ \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\C}\overline{\Oo_{X,0}}}{\langle \alpha(\Oo_{X,0}^s)\rangle} \cong
     \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\Oo_{X,0}^s}\overline{\Oo_{X,0}}\]
     that makes the following diagram commute
     \[ \xymatrix{  \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\C}\overline{\Oo_{X,0}} \ar[dr]^{\Psi_s} \ar[d]_{\Psi}&  \\
       \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\Oo_{X,0}}\overline{\Oo_{X,0}} \ar[r] & 
        \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\Oo_{X,0}^s}\overline{\Oo_{X,0}}}\]
    If we denote $I_\Delta=\mathrm{Ker}\, \Psi $ and $I_{\Delta^s}=\mathrm{Ker}\, \Psi_s$ we have $I_\Delta \subset I_{\Delta^s}$. Now by definition we have $\Oo_{X,0}^s=\{ h \in \overline{\Oo_{X,0}} \, | \, \alpha(h) \in \overline{I_\Delta}\}$ 
    so $\alpha(\Oo_{X,0}^s) \subset \overline{I_\Delta}$ which implies $\overline{I_\Delta}=\overline{I_{\Delta^s}}$
    and so $\left(\Oo_{X,0}^s\right)^s=\{ h \in \overline{\Oo_{X,0}} \, | \, \alpha(h) \in \overline{I_{\Delta^s}}=\overline{I_\Delta}\}=\Oo_{X,0}^s$.
   \end{proof}
   
   \subsection{The case of dimension 1}\label{dim1}
    
       Let $(X,0) \subset (\C^2,0)$ be a germ of reduced plane curve, and $\zeta: (X^s,0) \to (X,0)\subset (\C^2,0)$ 
       the finite map given by the Lipschitz saturation of $(X,0)$. What we want to emphasize is that
       this map can always be realized as a linear projection on suitable 
       representatives. Indeed, any representative of $(X^s,0) \subset (\C^m,0)$ can be re-embedded
       as the graph of $\zeta$ in $\C^{m+2}$, namely by the map $X^s\to \C^2\times\C^m:\ p\mapsto (\zeta_1(p),\zeta_2(p),p)$. The map $\zeta$ is now the projection of $(X^s,0)$ to $(X,0)$ by the first two coordinates: $(z_1,\ldots,z_{m+2}) \mapsto (z_1,z_2)$.
  
  \begin{proposition}{\rm (See \cite[Proposition 6.2.1]{Te1})}\label{SatisGeneric}
     For a germ of reduced plane curve $(X,0) \subset (\C^2,0)$  the Lipschitz saturation map $\zeta: (X^s,0) \to (X,0)$ is a generic projection. 
  \end{proposition}
   \begin{proof}
      Suppose first that $(X,0)$ is irreducible, in this case we have the holomorphic maps
      \[  (\C,0) \stackrel{\eta_s}{\longrightarrow} (X^s,0) \subset (\C^{m+2},0) \stackrel{\zeta}{\longrightarrow} (X,0) \subset (\C^2,0) \]
            \[t \mapsto (z_1(t),z_2(t), z_3(t),\ldots, z_{m+2}(t)) \mapsto (z_1(t),z_2(t))\]
      By Proposition \ref{C5genintdep} we have to prove that the ideals 
      $I_{\Delta^s}=\langle z_1-w_1, \ldots, z_{m+2}-w_{m+2} \rangle$ and $I_{\Delta_2^s}=\langle  z_1-w_1, z_2-w_2\rangle$ have the same integral closure in $\Oo_{X^s\times X^s,(0,0)}$.
      In this coordinate system we have the normalization map $\eta_s^*: \Oo_{X^s,0} 
      \hookrightarrow \overline{\Oo_{X,0}}$ given by
    \[
    \frac{\C\{z_1,\ldots,z_{m+2}\}}{I} \lhook\joinrel\longrightarrow \C\{t\} \]
    \[  z_i  \mapsto z_i(t) \hspace{0.2in} i=1,2\, ; \hspace{0.5in}
     z_{j+2} \mapsto z_{j+2}(t) \hspace{0.2in} j=1, \ldots m, \]
    which induces the morphism
    \[\theta:\Oo_{X^s \times X^s, (0,0)} = \frac{\C\{z_1,\ldots,z_{m+2},
    w_1, \ldots, w_{m+2}\}}{I(z) + I(w) } 
    \lhook\joinrel\longrightarrow \C\{t,u\} = \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\C}\overline{\Oo_{X,0}}\]
    \[  z_i \mapsto z_i(t) \hspace{0.2in} i=1,2\, ; \hspace{0.5in}
     z_{j+2}  \mapsto z_{j+2}(t) \hspace{0.2in} j=1, \ldots m \]
    \[  w_i  \mapsto z_i(u) \hspace{0.2in} i=1,2\, ; \hspace{0.5in}
     w_{j+2} \mapsto z_{j+2}(u) \hspace{0.2in} j=1, \ldots m .\]   
    But from the proof of Lemma \ref{Sat3} we have that the ideals $I_{\Delta^s_2}=
    \langle z_1(t)- z_1(u),z_2(t)-z_2(u) \rangle$ and \newline $I_{\Delta^s}=\langle z_1(t)- z_1(u),z_2(t)-z_2(u),z_3(t) -z_3(u), \ldots ,z_{m+2}(t)-z_{m+2}(u)\rangle $
    have the same integral closure in $\C\{t,u\}$ and so by remark \ref{RemIntDepId}-5 the ideals 
    $\theta^{-1}(I_{\Delta^s})$ and $\theta^{-1}(I_{\Delta^s_2})$ have the same integral closure in $\Oo_{X^s \times X^s, (0,0)}$, which is what we wanted.\par\medskip
    
    
    In the reducible case the proof works exactly the same way, it is just a lot messier to write down. The only thing 
    you have to keep track off is the following. Suppose $(X,0)$ has two irreducible components $(X_1,0) \cup (X_2,0)$ then
    $(X^s,0)$ also has two irreducible components and $\overline{\Oo_{X,0}} \cong \C\{t_1\} \oplus \C\{t_2\}$. This implies that
    the normalization map $\eta_s^*: \Oo_{X^s,0} 
      \hookrightarrow \overline{\Oo_{X,0}}$ is given by
    \[
    \frac{\C\{z_1,\ldots,z_{m+2}\}}{I} \lhook\joinrel\longrightarrow \C\{t_1\} \oplus \C\{t_2\}\]
    \[  z_i  \mapsto (z_i(t_1),z_i(t_2)) \hspace{0.2in} i=1,2\, ; \hspace{0.5in}
     z_{j+2} \mapsto (z_{j+2}(t_1),z_{j+2}(t_2)) \hspace{0.15in} j=1, \ldots m \]
    In this case $\overline{\Oo_{X,0}}{\widearc{\otimes}}_{\C}\overline{\Oo_{X,0}}\cong \C\{t_1,u_1\} \oplus \C\{t_1,u_2\} 
    \oplus \C\{t_2,u_1\} \oplus \C\{t_2,u_2\}$ and the induced morphism $\theta$ looks like:
     \[\theta:\Oo_{X^s \times X^s, (0,0)} = \frac{\C\{z_1,\ldots,z_{m+2},
    w_1, \ldots, w_{m+2}\}}{I(z) +I(w) } 
    \lhook\joinrel\longrightarrow  \overline{\Oo_{X,0}}{\widearc{\otimes}}_{\C}\overline{\Oo_{X,0}}\]
     \[  z_i  \mapsto (z_i(t_1),z_i(t_1), z_i(t_2),z_i(t_2)) \hspace{0.2in} i=1,2 \] 
     \[ z_{j+2}\mapsto (z_{j+2}(t_1), z_{j+2}(t_1), z_{j+2}(t_2), z_{j+2}(t_2)) \hspace{0.2in} j=1, \ldots m \]
    \[  w_i \mapsto (z_i(u_1), z_i(u_2), z_i(u_1), z_i(u_2)) \hspace{0.2in} i=1,2\]
    \[ w_{j+2} \mapsto z_{j+2}(u_1), z_{j+2}(u_2),z_{j+2}(u_1), z_{j+2}(u_2)) \hspace{0.2in} j=1, \ldots m, \]  
    then you have the map $\alpha$ as in the proof of Proposition \ref{Sat2} and the rest follows through.
   \end{proof}
  
   \begin{remark}
   \begin{enumerate}
    \item  Since the Lipschitz saturation map $\zeta: (X^s,0) \to (X,0)$ is a generic projection the multiplicity
      of $(X^s,0)$ is equal to the multiplicity of $(X,0)$.
    \item Except if the plane branch $(X,0)$ is non singular, the map $(\overline X,0)\to (X,0)$ is never obtained as a generic projection since the multiplicity changes. However, among all germs $(X',0)$ which dominate $(X,0)$ and are dominated by $(\overline X,0)$, and in addition are such that the map $(X',0)\to (X,0)$ can be represented by a \textit{generic} linear projection, there is a unique one, up to isomorphism, which dominates all the others: it is the saturation.
 \end{enumerate}
   \end{remark}
   
   \begin{corollary}\label{Bilipschitz}
       Let $(X,0) \subset (\C^2,0)$ be a reduced plane curve. The Lipschitz saturation map $\zeta: (X^s,0) \to (X,0)$
       is a biLipschitz homeomorphism.
   \end{corollary} 
   \begin{proof}
     We already know that a generic projection induces a homeomorphism with its image (Prop. \ref{C5smooth}), so by Proposition \ref{SatisGeneric}  the map $\zeta$ is a homeorphism and since it is the restriction to $X^s$ of the linear projection $(z_1,\ldots,z_{m+2}) \mapsto (z_1,z_2)$ it is Lipschitz. The inverse of $\zeta$ can be described on each irreducible component $X_k$ by
     \[(z_1(t_k),z_2(t_k)) \mapsto (z_1(t_k),z_2(t_k),z_3(t_k), \ldots, z_{m+2}(t_k)),\]  and since for all $j \in \{1,\ldots,m\}$, $z_{j+2}(\underline t) \in \Oo_{X,0}^s$ Proposition \ref{Sat2} tells us that it is 
     also Lipschitz. 
   \end{proof}
  
     Our main result now follows from the following result.
     
     \begin{theorem}{\rm (See \cite[\S 4]{P-T69},  \cite[Prop. VI.3.2]{BGG})}
        Let $\Oo_{X,0}$ be the analytic algebra of a germ of reduced plane curve $(X,0) \subset (\C^2,0)$. The Lipschitz saturation $\Oo_{X,0}^s$ determines and is determined by the characteristic exponents of its branches (irreducible components) and the intersection multiplicities $m_{ij}=(X_i,X_j)$ of each 
        pair of branches. In particular the saturated curve $(X^s,0)$ is an invariant (up to isomorphism) of the equisingularity class of $(X,0)$.
     \end{theorem}
     
     This implies that every member of the equisingularity class  of a germ of reduced plane curve $(X,0) \subset (\C^2,0)$ can be obtained by a generic projection of the Lipschitz saturation $(X^s,0)$ of any one of them. The proof of the Proposition
     involves a lot of calculations and can be found in the references. For this reason we would rather describe how to calculate the saturated curve $(X^s,0)$. Let us start with the irreducible case:
     
     \begin{definition}
         Let $h \in  \C\{t\}$ be a power series with coprime exponents. If 
          \[h = \sum_{j=0}^\infty a_jt^j,\]
          we define the set of exponents of $f$ as $Ex(f)=\{ j \in \N\, | \, a_j \neq 0 \}$.
        And for a germ of analytically irreducible plane curve $(X,0) \subset (\C^2,0)$ we define the set
        of exponents of $\Oo_{X,0}$ as
        \[E(\Oo_{X,0})= \bigcup_{h \in \mathfrak{m}} Ex(h),\]
        where $ \mathfrak{m}$ is the maximal ideal of $\Oo_{X,0}$.
        Note that the semigroup $\Gamma(X)$ of the plane branch $(X,0)$ is contained in 
        $E(\Oo_{X,0})$.(See \cite[Section 8]{TeiBia}).
     \end{definition}
     
     If $(X,0) \subset (\C^2,0)$ is irreducible then:
     \begin{enumerate}
         \item For every $j \in E(\Oo_{X,0})$ we have that $t^j \in \Oo_{X,0}^s$.
         \item The analytic algebra $\Oo_{X,0}^s$ is monomial, in particular:
            \[\Oo_{X,0}^s=\C\{t^j \, | \, j \in E(\Oo_{X,0}^s)\}.\]
     \end{enumerate}
     
  For a numerical semigroup (i.e., a subsemigroup of $(\N,+)$ with finite complement) there is the concept of saturated semigroup (see \cite[Chapter 3, Section 2]{RosGa}) which is defined as follows:
   
   For $A \subset \N$ and $a \in  A\setminus \{0\}$ define
   \[ d_A(a)= \mathrm{gcd}\{ x \in A \,| \, x \leq a\}.\]
   Then a non-empty subset $A \subset \N$ such that $0 \in A$ and $\mathrm{gcd}(A)=1$ is a saturated numerical semigroup if and only if 
   $a +kd_A(a) \in A$ for all $a \in A\setminus\{0\}$ and $k \in \N$.\par\smallskip\noindent
   The reader can verify that the condition indeed implies that $A$ is a semigroup and that the intersection of two saturated semigroups is again saturated, so that any $A \subset \N$ such that $0 \in A$ and $\mathrm{gcd}(A)=1$ is contained in a smallest saturated semigroup.
  
  \begin{example}\label{op}
     Let $(X,0) \subset (\C^2,0)$ be the cusp singularity defined by $y^2-x^3=0$. Its normalization map 
     is $t \mapsto (t^2,t^3)$ and so its semigroup is generated by $\langle 2,3\rangle$. Since $\Gamma(X)=
     \N \setminus \{1\}$ then $E(\Oo_{X,0})=\Gamma(X)$ is a saturated numerical semigroup.
  \end{example}
  
   This definition also tells us how to obtain the smallest saturated semigroup containing any $A \subset \N$ with $0\in A$ and $\mathrm{gcd}(A)=1$, for example the set of exponents $E(\Oo_{X,0})$. \\
   Let $e_0=\beta_0=\mathrm{min}\, \{ x \in A\}$ and define 
    \[ \widetilde{A_0}:= A \cup \beta_0 \cdot \N .\] 
    In the case of $E(\Oo_{X,0})$ we have that $e_0=\beta_0$ is the multiplicity of $(X,0)$.\par\noindent
  Let $\beta_1:=\mathrm{min} \{x \in A \,|\, e_0\, \mathrm{does} \, \mathrm{not} \, \mathrm{divide} \, x\} $ and $e_1=\mathrm{gcd}\{\beta_0,\beta_1\}$; note that $e_1=d_A(\beta_1)$. Again define 
    \[ \widetilde{A_1}:= \widetilde{A_0} \cup \{ \beta_1 + k e_1\,|\, k \in\N\} .\]
 Continuing this way we obtain two sequences of natural numbers $e_0 > e_1 > \cdots > e_g=1=\mathrm{gcd}(A)$ and $\beta_0 < \beta_1< \cdots < \beta_g$  and an associated sequence
 of subsets of $\widetilde{A_0} \subset \widetilde{A_1} \subset \cdots \subset \widetilde{A_g} \subset \N$
 where $\beta_{i+1}:= \mathrm{min} \{x \in A \,|\, e_i\, \mathrm{does} \, \mathrm{not} \, \mathrm{divide} \, x\}$, $e_i:=\mathrm{gcd}\{\beta_0,\ldots,\beta_i\}=d_A(\beta_i)$ and
 \[\widetilde{A_{i+1}}:= \widetilde{A_i} \cup \{ \beta_{i+1} + k e_{i+1}\,|\, k \in\N\}.\]
 Note that $\widetilde{A}:= \widetilde{A_g}$ is a saturated semigroup which is completely determined by its \emph{characteristic sequence} $\{\beta_0, \ldots, \beta_g\}$. Moreover if 
 $t \mapsto \left(t^n, \sum_{i \geq n} a_it^i\right)$ is a Puiseux parametrization of the plane branch $(X,0) \subset
 (\C^2,0)$, the characteristic sequence of $E(\Oo_{X,0})$ is the set of characteristic exponents of 
 $(X,0)$ and so it determines its equisingularity class. 
 
 \begin{proposition}\label{monom}{\rm (Pham-Teissier), see \cite[\S 4]{P-T69}, \cite[Thm VI.1.6]{BGG}}
     For a germ of irreducible plane curve singularity $(X,0) \subset (\C^2,0)$ the Lipschitz
     saturation $\Oo_{X,0}^s$ is given by
     \[\Oo_{X,0}^s= \C \{t^p \, | \, p \in \widetilde{E(\Oo_{X,0})}\}.\]
     In particular $E\left(\Oo_{X,0}^s\right)=\widetilde{E(\Oo_{X,0})}$.
 \end{proposition}
 Let us give a sketch of the proof: we start from a structured parametrization $(t^n,y(t))$ of our branch $X$ as in subsection \ref{stru} and we have to study integral dependence over the ideal $I_\Delta=(t-u){\mathcal N}:=\langle t^n-u^n, y(t)-y(u)\rangle\subset \C\{t,u\}$. Here ${\mathcal N}$ is the primary ideal $\langle \frac{t^n-u^n}{t-u},\frac{y(t)-y(u)}{t-u}\rangle\C\{t,u\}$. According to what we saw after Theorem \ref{Avatarsintegraldependence}, to verify that $g(t)-g(u)$ is integral over $I$, which is the same as $\frac{g(t)-g(u)}{t-u}$ being integral over ${\mathcal N}$, it suffices to verify that its order along any of the branches of a plane curve $C_T\subset \C^2$ defined by $T_1 \frac{t^n-u^n}{t-u}-T_2\frac{y(t)-y(u)}{t-u}=0$ is larger than that of the ideal $I$ for $T=(T_1:T_2)$ in the open set $U\subset \Pp^1$. Now we claim that the open set $U$ is $T_1\neq 0$. Indeed, since the order of $y(t)$ is $>n$ all the plane curves $C_T$ with $T_1\neq 0$ have a tangent cone consisting of $n-1$ lines in general position. It is not difficult then to show (see \cite[Chap. II, Lemma 2.6, Proposition 2.7]{Te2}) that they are equisingular with their tangent cone, and therefore are all equisingular, with simultaneous normalization. So the curve $\frac{t^n-u^n}{t-u}=0$ is in $U$, and its branches are the lines $u=\omega t,\ \omega\in\mu_n\setminus\{1\}$, which means that a function $g(t)\in \C\{t\}$ is in the saturation if and only if we have
 $${\rm ord}_t(g(t)-g(\omega t))\geq {\rm ord}_t(y(t)-y(\omega t)) \ {\rm for\ all}\ \omega\in\mu_n\setminus\{1\}.$$
 The result now follows easily from what we saw at the end of subsection \ref{stru} about the orders of the $y(t)-y(\omega t)$.\par\noindent It may be interesting to remark here that this construction gives an intrinsic (coordinate free) definition of the Puiseux characteristic as the set of valuations (orders of vanishing) of the ideal ${\mathcal N}$ along the irreducible components of the exceptional divisor of the normalized blowing up of ${\mathcal N}$ in $\overline X\times\overline X$. For more details, see \cite[\S 4]{P-T69} and  \cite[Thm VI.1.6]{BGG}.
 \begin{remark} It is shown in \cite[5.2]{Tei80} that the multiplicity, in the sense we saw after Theorem \ref{Rees-Samuel}, of the primary ideal ${\mathcal N}$ is equal to twice the invariant $\delta$ which appears in Propositions \ref{constant} and \ref{gengen}. It is also shown there that $\delta$ is the maximum number of different singular points (then necessarily ordinary double points) which can appear when deforming the parametrization of the plane branch. Both results extends to reducible curves. One can define an analogous ideal ${\mathcal N}$ for a non-plane branch but then, in view of Theorem \ref{reduc} and Proposition \ref{C5genintdep}, its multiplicity is twice the $\delta$ invariant of a generic plane projection and no longer the classical ${\rm dim}_\C\frac{\overline{\Oo_{X,0}}}{\Oo_{X,0}}$ in this case.
 \end{remark}
 \begin{example}
  Let $(X,0) \subset (\C^2,0)$ be the plane branch with normalization map:
    \begin{align*}
         \eta:(\C,0) &\longrightarrow (X,0) \\
               t & \mapsto (t^4,t^6+t^7)
   \end{align*}
   Then the exponent set $E(\Oo_{X,0})$ contains the semigroup $\Gamma(X)=\langle 4,6,13\rangle\N$ but 
   by definition it also contains $7$. Now $\beta_1=6$ and $e_1=2$ so
   \[\widetilde{E_1}= E(\Oo_{X,0}) \cup \{6 + 2k \, | \, k \in \N\}.\]
   In the next step $\beta_2=7$ and $e_2=1$ so $g=2$ and we get the saturated semigroup
    \[\widetilde{E_2}= \widetilde{E_1} \cup \{7 + k \, | \, k \in \N\}.\]
   Note that $\widetilde{E(\Oo_{X,0})}=\langle 4,6,7,9\rangle\N$ and so we have the normalization map 
   for the Lipschitz saturation $(X^s,0) \subset (\C^4,0)$ given by:
   \begin{align*}
         \eta^s:(\C,0) &\longrightarrow (X^s,0) \\
               t & \mapsto (t^4,t^6,t^7,t^9)
   \end{align*}
   By making the change of coordinates in $(\C^4,0)$, $(x,y,z,w) \mapsto (x,y+z,z,w)$ we can view the Lipschitz saturation map 
   \[ \zeta: (X^s,0) \to (X,0)\]
   as the projection on the first two coordinates as before.
 \end{example}
 \begin{remark}(see \cite[Chap. I, Theorem 6.3.1]{Te1}, \cite[Appendice]{BGG}) A more concrete way of seeing that all plane branches with the same Puiseux characteristic are generic plane projections of a single space curve is to go back to the notations of subsection \ref{stru} to write down explicitly the saturation of a plane branch $(X,0)$ with given characteristic $(n,\beta_1,\ldots ,\beta_g)$: it is isomorphic to the monomial curve with analytic algebra {\small $$\C\{t^n,t^{2n},\ldots ,t^{\beta_1},t^{\beta_1+e_1},\ldots,t^{\beta_2},t^{\beta_2+e_2},\ldots ,t^{\beta_3},\ldots ,t^{\beta_g},t^{\beta_g+1},\ldots\},$$}
 where $n=e_0=\beta_0$ as above. The semigroup generated by these exponents, which are those of  $\widetilde{E(\Oo_{X,0})}$, is finitely generated by Dickson's Lemma and because the Puiseux exponents are coprime its complement in $\N$ is finite. For more details on the saturation of semigroups we refer to \cite[Chapter 3, Section 2]{RosGa}.\par\noindent
As we saw in subsection \ref{stru}, up to isomorphism, the image of this monomial curve by a generic linear projection can be parametrized by $x=t^n,y=\sum_{p\in \widetilde{E(\Oo_{X,0})}\setminus \{n\}}a_pt^p$. Now we see that the generic projections are precisely those which are such that the coefficient of $t^n$ is $\neq 0$ and for $p=\beta_1,\ldots ,\beta_g$ we have $a_p\neq 0$, which means that the projection has characteristic $(n,\beta_1,\ldots ,\beta_g)$.\par
Remark that, except if $n=2$, the semigroup of integers generated by the exponents of the monomials belonging to the saturation $\Oo^s_{X,0}$ is different from the semigroup $\Gamma$ we saw in subsection \ref{stru}. This has the interesting consequence that the specialization of a plane branch to the monomial curve with the same semigroup, which is Whitney equisingular (see \cite[Remark 4.1]{GiTei}; the argument there can be generalized to any plane branch), is not in general biLipschitz trivial.
 \end{remark}
   When $(X,0)$ is not irreducible it is a bit more complicated, nevertheless the Lipschitz saturation
   $\Oo_{X,0}^s$ can be described in the following way:
   
   \begin{theorem} {\rm (see \cite[\S 4]{P-T69} and \cite[Thm VI.2.2]{BGG})} \label{ReducibleSaturation}
     Let $\Oo_{X,0}$ be the analytic algebra of a reduced plane curve $(X,0) = (X_1,0) \cup \ldots 
     \cup (X_r,0)$ with normalization $\overline{\Oo_{X,0}}= \C\{t_1\} \oplus \cdots \oplus \C\{t_r\}$.
     We may assume that the image of $x$ in $\overline{\Oo_{X,0}}$ is $\left(t_1^{n_1}, \ldots,t_r^{n_r}\right)$ where 
     $n_i$ is the multiplicity of the branch $(X_i,0)$. Let $\mu$ be the least common multiple of 
     $\{n_1, \ldots, n_r\}$. Then the element $\underline h=(h_1,\ldots,h_r) \in \overline{\Oo_{X,0}} $ is in
     the Lipschitz saturation $\Oo_{X,0}^s$ if and only if the following two conditions are satisfied:
    \begin{enumerate}
        \item For every $j \in \{1, \ldots, r\}$ we have that $h_j \in \Oo_{X_j,0}^s$.
        \item For every $\mu$-th root of unity $\epsilon$ and every couple $i\neq j$ we have
        the inequality
        \[m_{i,j,\epsilon}(h) \geq m_{i,j,\epsilon}:= \inf_{\underline g \in \overline{\Oo_{X,0}}} 
         \left\{\nu_{\tau}\left(g_i(\tau^{\mu/n_i})-g_j([\epsilon\tau]^{\mu/n_j})\right)\right\},\]
    \end{enumerate}
    where $m_{i,j,\epsilon}(\underline h)=\nu_{\tau}\left(h_i(\tau^{\mu/n_i})-h_j([\epsilon\tau]^{\mu/n_j})\right)$
    and $\nu_{\tau}$ is the valuation of $\C\{\tau\}$ given by the order of the series. The number
    $m_{i,j,\epsilon}$ depends only on the characteristic exponents and the intersection
    multiplicity of the branches $X_i$ and $X_j$.
   \end{theorem}
   
   \begin{example}
      Let $(X,0)= (X_1,0) \cup (X_2,0)$ be the plane curve with normalization map:
    \begin{align*}
         \eta:(\C,0)\sqcup (\C,0) &\longrightarrow (X,0) \\
                t_1 & \mapsto (t_1^4,t_1^6+t_1^7) \\
                t_2 & \mapsto (t_2^3,t_2^5)
   \end{align*}
   In the previous example we already calculated the Lipschitz saturation $\Oo_{X_1,0}^s=\C\{t_1^4,t_1^6,t_1^7,t_1^9\}$
   and following the algorithm we get the Lipschitz saturation $\Oo_{X_2,0}^s= \C\{t_2^3,t_2^5,t_2^7\}$. Since the branches are tangent, their intersection multiplicity is greater than the product of their multiplicities and is 
   equal to the order of the series in $t_1$ obtained by substituting the parametrization of $(X_1,0)$ in the equation $y^3-x^5=0$ defining $(X_2,0)$. In this case it is equal to 18.\\
   
   By definition $\mu=\mathrm{lcm}\{3,4\}=12$ and it is not hard to prove that for any $12-$th root of unity $\epsilon$ 
   \begin{align*}
     m_{1,2,\epsilon} & = \nu_{\tau}\left( y_1(\tau^3) - y_2([\epsilon\tau]^4)\right) \\
                       & = \nu_{\tau}\left( \tau^{18}+\tau^{21} - \epsilon^8\tau^{20}\right) = 18.
   \end{align*}
    So from the Theorem \ref{ReducibleSaturation} we have that $h=(h_1(t_1),h_2(t_2))$ is in $\Oo_{X,0}^s$ if and only if
    $h_1(t_1) \in \Oo_{X_1,0}^s$, $h_2(t_2) \in \Oo_{X_2,0}^s$ and
    $m_{1,2,\epsilon}(h) \geq 18$. For example if $h=(t_1^4,t_2^5)$ then 
    \begin{align*}m_{1,2,\epsilon}(h) & =\nu_{\tau}\left( (\tau^3)^4 - ([\epsilon\tau]^4)^5\right)\\
                        &=\nu_{\tau}(\tau^{12}-\epsilon^8\tau^{20})=12 \Rightarrow h \notin \Oo_{X,0}^s.
    \end{align*}
    On the other hand if $h=(t_1^6+t_1^7,t_2^5)$ then 
    \begin{align*}m_{1,2,\epsilon}(h) & =\nu_{\tau}\left( (\tau^3)^6+(\tau^3)^7 - ([\epsilon\tau]^4)^5\right) \\
                        &=\nu_{\tau}(\tau^{18} + \tau^{21} -\epsilon^8\tau^{20})=18 \Rightarrow h \in \Oo_{X,0}^s .
    \end{align*}                        
   \end{example}

    We will end this section with the following consequence of the Theorem:
    \begin{corollary}{\rm (see \cite[VI.3.7]{BGG})}
       Let $(X,0) = (X_1,0) \cup \ldots \cup (X_r,0)$  be a reduced plane curve with normalization $\overline{\Oo_{X,0}}= \C\{t_1\} \oplus \cdots \oplus \C\{t_r\}$. If $\Pi_j:\overline{\Oo_{X,0}} \to
      \C\{t_j\}$ denotes the canonical projection to the $j-$th factor
      then \[\Pi_j(\Oo_{X,0}^s)= \Oo_{X_j,0}^s.\]
    \end{corollary}
   \subsection{Application to local polar curves}
   Let $(X,0)\subset (\C^N,0)$ be a reduced equidimensional germ of complex analytic space. Consider linear projections $\pi\colon \C^N\to \C^2$ and the critical locus of $\pi$ restricted to the smooth part $X^0$ of $X$. It is proved in \cite{L-T81}, where the theory of (absolute)\footnote{This precision refers to a distinction between absolute and relative polar varieties, which is not relevant here but should be mentioned to avoid confusions. See \cite[Chap. IV, p. 417]{Te1}}  local polar varieties was initiated, that for a Zariski dense open set $U$ in the space $G(N,N-2)$ of linear projection, this critical locus is either empty or a reduced curve. The \textit{closure} of this curve in $X$ is an (absolute) \index{Polar curve}\textit{polar curve}\index{Polar curve} of $X$ and is denoted by $P_{d-1}(X,\pi)$ where $d$ is the dimension of $X$. It is also denoted by $P_{d-1}(X,D)$, where $D={\rm ker}\pi$. These curves play an important role in the local study of singularities, and especially in the study of the Lipschitz geometry of surfaces. See  \cite{L-T81}, \cite[Chap. IV]{Te1} and \cite{N-P1} for more details. \par
Of course, if it is not empty, $P_{d-1}(X,\pi)$ varies with the projection $\pi\in U$ and \textit{a priori} it could be that $\pi$ remains constantly a non generic projection for $P_{d-1}(X,\pi)$. That seems unlikely but still we need a proof for the following:
\begin{theorem}{\rm (See \cite[Chap. V, Lemme 1.2.2]{Te1})}
Given $(X,0)\subset (\C^N,0)$ as above and assuming that $P_{d-1}(X,\pi)$ is a reduced curve for $\pi\in U\subset G(N,N-2)$, there exists a non empty Zariski open set $V\subset U$ such that for $\pi\in V$, the projection $\pi\colon \C^N\to\C^2$ is a generic projection for the curve $P_{d-1}(X,\pi)\subset \C^N$.
\end{theorem}   
The proof, which we only sketch, gives an example of the notion of Lipschitz equisaturation \index{Lipschitz Equisaturation}, which is found in \cite[\S 6]{P-T69}. Fixing coordinates $z_1,\ldots ,z_N$ on $\C^N$ and $x,y$ on $\C^2$, we can parametrize by $\C^{2(N-2)}$ a dense open set of the space of linear projections $\C^N\to\C^2$ as follows:
$$x=z_1+\sum_3^Na_iz_i,\ \ y=z_2+\sum_3^Nb_iz_i\ ,\ (\underline a,\underline b)\in \C^{2(N-2)}.$$
To simplify notations while keeping the ideas, we assume that $X$ is a hypersurface defined by $f(z_1,\ldots ,z_N)=0$. One can also consult \cite[Lemme 3.7]{B-H} which gives the proof for isolated singularities of surfaces in $\C^3$. \par\noindent  For any series $h(z_1,\ldots ,z_N)\in \C\{z_1,\ldots ,z_N\}$ let us denote by $h_{\underline a,\underline b}$ the series $$h_{\underline a,\underline b}(\underline z,\underline a,\underline b)=h(x-\sum_3^Na_iz_i, y-\sum_3^Nb_iz_i, z_3,\ldots ,z_N).$$\par\noindent
The equation $f_{\underline a,\underline b}=0$ defines a germ of hypersurface $\mathcal Z$ in $\C^N\times \C^{2(N-2)}$ and if we consider thefamily of projections $\pi_{\underline a,\underline b}\colon \C^N\times \C^{2(N-2)}\to \C^2\times \C^{2(N-2)}$ defined by 
$$x\mapsto z_1+\sum_3^Na_iz_i,\ \ y\mapsto z_2+\sum_3^Nb_iz_i, \ \underline a\mapsto \underline a, \ \underline b\mapsto \underline b,$$
and the closure of its critical locus on the non singular part of $\mathcal Z$, we obtain a subspace which, over a Zariski open subset of $\C^{2(N-2)}$, contains the family of polar curves associated to the family of projections $\pi_{\underline a,\underline b}$ . Over a possibly smaller Zariski open subset $V$ of $\C^{2(N-2)}$ this family of curves is equisingular in the sense that it has a simultaneous parametrization. The number $r$ of irreducible components of $\mathcal Z$ at points of $\{0\}\times V\subset \C^N\times V$ is constant and after choosing as origin of $\C^{2(N-2)}$ a point of $V$ we can parametrize each irreducible component in a neighborhood of $\{0\}\times\{0\}$ by:
\[ z_1=t_\ell^{n_\ell}, z_2=\upsilon (t_\ell, \underline a,\underline b), z_i=\zeta_i(t_\ell, \underline a,\underline b),\ \]
with $\upsilon (t_\ell, \underline a,\underline b),\zeta_i(t_\ell, \underline a,\underline b)\in \C\{t_\ell,\underline a,\underline b\}$ for $i=3,\ldots ,N$. The normalization of $\Oo_{{\mathcal Z},0}$ being $\overline{\Oo_{{\mathcal Z},0}}=\prod_{i=1}^r\C\{t_\ell,\underline a,\underline b\}$.\par\noindent
By definition of $\mathcal Z$ we have for each $\ell=1,\ldots ,r$ the identity in $\C\{t_\ell,\underline a,\underline b\}$
{\small
 \[f(t^{n_\ell}-\sum_3^Na_i\zeta_i(t_\ell, \underline a,\underline b),\upsilon (t_\ell, \underline a,\underline b)-\sum_3^Na_i\zeta_i(t_\ell, \underline a,\underline b), \zeta_3(t_\ell, \underline a,\underline b),\ldots ,\zeta_N(t_\ell, \underline a,\underline b))\equiv0.\]}
 Differentiating $f_{\underline a,\underline b}=0$ with respect to $z_i$ gives the following equations on $\mathcal Z$:

\[-a_i\frac{\partial f_{\underline a,\underline b}}{\partial z_1} -b_i\frac{\partial f_{\underline a,\underline b}}{\partial z_2}+\frac{\partial f_{\underline a,\underline b}}{\partial z_i}\equiv0,\]
for $i=3,\ldots ,N$.
which by definition are satisfied on the polar curve. \par Differentiating the first identity with respect to $b_k$ and taking into account the second set of identities, we obtain that the equation $$\left(\zeta_k(t_\ell, \underline a,\underline b)-\frac{\partial \upsilon (t_\ell, \underline a,\underline b)}{\partial b_k}\right)\frac{\partial  f_{\underline a,\underline b}}{\partial z_2}=0$$ must be satisfied in each $ \C\{t_\ell,\underline a,\underline b\}$. By general transversality results found in \cite[Cor. 4.1.6]{L-T81} and  \cite[Chap. IV, 5.1]{Te1}, $\frac{\partial  f_{\underline a,\underline b}}{\partial z_2}$ does not vanish because it would entail a lack of $C_3$ transversality (see Definition \ref{C3trans}) of the polar curve with the kernel of the projection which defines it. So we must have on $\overline{\mathcal Z}$ the identity $z_k=\frac{\partial \upsilon}{\partial b_k}$.\par\noindent
By \cite[Proposition 6.4.2]{Te1}, after perhaps shrinking $V$ to a smaller Zariski open dense subset $V_1$ of $\C^{2(N-2)}$ we have that over $V_1$ the family ${\mathcal Z}_1$ of plane curves given parametrically by the parametrizations $x=t^{n_\ell}, y=\upsilon (t_\ell, \underline a,\underline b)$, which consists of the plane projections of our polar curves, is equisaturated. This implies that the derivations $\frac{\partial}{\partial b_k}$ of $\C\{\underline a, \underline b\}$ extend to derivations $D_k$ of $\overline {\Oo_{{\mathcal Z}_1,(0,v)}}=\overline {\Oo_{{\mathcal Z},(0,v)}}=\prod_{i=1}^r\C\{t_\ell, \underline a, \underline b\}$ into itself which preserve the relative saturated ring (see \cite{P-T69}). Since of course the functions $\upsilon (t_\ell, \underline a,\underline b)$ belong to the relative saturation of $\overline {\Oo_{{\mathcal Z}_1,(0,v)}}$, so do the $\zeta_k(t_\ell, \underline a,\underline b))$ which are their images by $D_k$. But $\zeta_k$  belonging to this relative saturation means precisely that for $v\in V_1$, the saturations of the rings $\overline {\Oo_{{\mathcal Z}_1(v)}}$ and 
$\overline {\Oo_{{\mathcal Z}(v)}}$ of the fibers over $v$ of $\mathcal Z$ and $\mathcal Z_1$ are equal for $v\in V_1$, which is the condition for $C_5$ genericity according to Proposition \ref{C5genintdep}. \par\medskip
The fact that the plane projection of a generic polar curve by the map which defines it is generic plays an important role in the following three domains: the comparison of Zariski equisingularity and Whitney equisingularity for surfaces (see \cite{B-H}, \cite{N-P1}), the comparison of Zariski equisingularity and Lipschitz equisingularity for surfaces (see \cite{N-P1}, \cite{PaP}), the numerical characterization of Whitney equisingularity (see \cite[Chap. V]{Te1}) and the valuative study of the metric geometry of surface singularities in view of their biLipschitz classification (see \cite{B-F-P}).\par\medskip\noindent

\noindent\textbf{Aknowledgements}: Arturo Giles Flores was supported by CONACyT Grant 221635.\par\smallskip\noindent
Otoniel Nogueira da Silva would like to thank CONACyT (grant 282937) for the financial support by Fordecyt 265667, and also 
UNAM/DGAPA for support by PAPIIT IN 113817.\par\smallskip\noindent
\textbf{The authors are grateful to the referee for his or her very careful reading and numerous useful observations.}
\bibliographystyle{alpha}
\begin{thebibliography}{Whi65}   

\bibitem[Ada12]{Ada1}
Janusz Adamus,
\newblock {\em Topics in Complex Analytic Geometry Part II}, lecture notes. 
\newblock Available at \url{https://www.uwo.ca/math/faculty/adamus/adamus\_publications/AGII.pdf}

\bibitem[B-H80]{B-H} Jo\"el Brian\c con \& Jean Pierre G. Henry,
\newblock Equisingularit\'e g\'en\'erique des familles de surfaces \`a singularit\'es isol\'ees,
\newblock Bull. S.M.F., 108 (2), 259-281, 1980.

\bibitem[B-G-G80]{BGG}
Jo\"el Brian\c con, Andr\'e Galligo \& Michel Granger,
\newblock D\'{e}formations \'{e}quisinguli\`eres des germes de courbes gauches
              r\'{e}duites
\newblock {\em M\'em. Soc. Math. France, 2\`eme serie (1)}, 69, 1980.

\bibitem[B-F-P]{B-F-P} Andr\'e Belotto da Silva, Lorenzo Fantini, Anne Pichon,
\newblock Inner geometry of complex surfaces: a valuative approach.
\newblock Submitted; see arXiv:1905.01677v1 [math.AG] May 2019.

\bibitem[Br-K86]{Brie}
Egbert Brieskorn \& Horst Kn\"{o}rrer,
\newblock {\em Plane algebraic curves}, 
\newblock Birkh\"{a}user/Springer Basel AG, Basel, 1986.

\bibitem[Bu-G80]{buch}
Ragnar Buchweitz \& Gert Martin Greuel,
\newblock The Milnor number and deformations of complex curve singularities
\newblock {\em Invent. Math.}, 58: 241--281, 1980.

\bibitem[Che78]{Chen}
Alain Chenciner,
\newblock {\em Courbes alg\'{e}briques planes}, 
\newblock Publications Math\'{e}matiques de l'Universit\'{e} Paris VII, 1978.

\bibitem[Chi89]{Chi}
Evgenii M. Chirka,
\newblock {\em Complex Analytic Sets}, 
\newblock Kluwer Academic Publishers, 1989.

\bibitem[D-G-P-S]{ref21}
Wolfram Decker, Gert Martin Greuel, Gerhard Pfister \& Hans Sch{\"o}nemann, 
\newblock {\sc Singular} {4-1-2} --- {A} computer algebra system for polynomial computations.
\newblock {http://www.singular.uni-kl.de} (2019).

\bibitem[De-P00]{DeJo-Pfi}
Theo de Jong \& Gerhard Pfister,
\newblock {\em Local Analytic Geometry: Basic theory and applications}, 
\newblock Advanced Lectures in Mathematics, Springer 2000.

\bibitem[Eis95]{Eis95} David Eisenbud,
\newblock Commutative algebra with a view toward Algebraic Geometry,
\newblock Graduate Texts in Mathematics, n$^\circ$ 150, Springer 1995.

\bibitem[Gi-T18]{GiTei}
Arturo Giles Flores  \& Bernard Teissier,
\newblock Local polar varieties in the geometric study of singularities,
\newblock {\em Ann. Fac. Sci. Toulouse Math. (6)}, 4: 679--775, 2018.

\bibitem[Gr-F02]{Fr-Gr}
Hans Grauert \& Klaus Fritzsche
\newblock {\em From holomorphic functions to complex manifolds }, 
\newblock Graduate Texts in Matematics, Springer-Verlag,  2002.

\bibitem[G-L-S07]{GLS07}
Gert Martin Greuel, Christoph Lossen \& Euguenii Shustin,
\newblock {\em Introduction to singularities and deformations},
\newblock  Springer Monographs in Mathematics. Berlin: Springer 2007.

\bibitem[Gr-P07]{Gre-Pfis}
Gert Martin Greuel \& Gerhard Pfister,
\newblock {\em A Singular introduction to commutative algebra}, 
\newblock  Springer-Verlag,  2007.

\bibitem[H-M-P18]{HMP}
Marcelo E. Hernandes, Aldicio J. Miranda \& Guillermo Pe\~nafort-Sanchis,
\newblock A presentation matrix algorithm for $f_{\ast}\mathcal{O}_{X,x}$,
\newblock {Topology and its Applications}, 234: 440--451, 2018.

\bibitem[Hu-S06]{SwaHu}
Craig Huneke  \& Irena Swanson,
\newblock {\em Integral closure of ideals, rings and modules}, 
\newblock London Mathematical Society Lecture Note Series, n$^\circ$ 336, Cambridge University Press,  2006.

\bibitem[Kau83]{Ka}
Ludger Kaup \& Burchard Kaup,
\newblock {\em Holomorphic functions of several variables. An introduction to the fundamental theory }, 
\newblock De Gruyter Studies in Mathematics, 1983.

\bibitem[L15]{L15}
L\^e C\^ong-Tr\`\i nh,
\newblock Equinormalizability and topological triviality of deformations of isolated curve singularities over smooth base spaces,
\newblock {\em Kodai Math. Journal}, (38), 3, 642--657, 2015.

\bibitem[L-R76]{L-R76}
L\^e D\~ung Tr\'ang \& C.P. Ramanujam,
\newblock The invariance of Milnor's number implies the invariance of topological type,
\newblock {\em Amer. Journ. of Math.}, (91), 1, 67--78, 1976.

\bibitem[L-T81]{L-T81}
L\^e D\~ung Tr\'ang \& Bernard Teissier,
\newblock Vari\'et\'es polaires locales et classes de Chern des vari\'et\'es singuli\`eres,
\newblock{\em Annals of Math.}, (2), 114 n$^\circ$3, 457--491, 1981.

\bibitem[Lej73]{Lej73} Monique Lejeune-Jalabert,
\newblock Sur l'\'equivalence des singularit\'es des courbes alg\'ebriques planes. Coefficients de Newton, \i {\em Introduction \`a la th\'eorie des singularit\'es}, Thesis, Paris 7, 1973. Published in {\em Travaux en Cours}, n$^\circ$ 36, 49--124, Hermann, 1988.

\bibitem[Lej-T08]{Lej-Te}
Monique Lejeune-Jalabert  \& Bernard Teissier,
\newblock  Cl{\^o}ture int{\'e}grale des id{\'e}aux et {\'e}quisingularit{\'e},
\newblock {\em Ann. Fac. Sci. Toulouse Math. (6)}, 4: 781--859, 2008.

\bibitem[Lip82]{Lip1}
Joseph Lipman,
\newblock Equimultiplicity, reduction and blowing up, 
\newblock {\em Commutative algebra: analytical methods, Conf. Fairfax/Va. 1979, Lect. Notes Pure Appl. Math. 68}, 111-147, 1982.

\bibitem[Loj91]{Loj}
Stanis\l aw \L ojasiewicz
\newblock {\em Introduction to complex analytic geometry }, 
\newblock Birkh\"{a}user Verlag, 1991.

\bibitem[Mat89]{Mats1}
Hideyuki Matsumura, 
\newblock {\em Commutative ring theory }, 
\newblock Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1989.

\bibitem[Mir19]{ref10}
Aldicio J. Miranda,  
\newblock {https://sites.google.com/site/aldicio/publicacoes} .

\bibitem[Mo-P89]{Mo-Pe}
David Mond \& Ruud Pellikaan,
\newblock Fitting ideals and multiple points of analytic mappings, 
\newblock {\em Lecture Notes in Math, Springer},  1414: 107--161, 1989.

\bibitem[N-P1]{N-P1} Walter D. Neumann \& Anne Pichon,
\newblock Lipschitz geometry of complex surfaces, analytic invariants and equisingularity,
\newblock In preparation; see arXiv:1211.4897v3 [math.AG] 17 Feb. 2016.

\bibitem[P-T69]{P-T69} Fr\'ed\'eric Pham and Bernard Teissier,
\newblock Lipschitz fractions of a complex analytic algebra and Zariski saturation,
\newblock This volume.

\bibitem[PaP]{PaP} Adam Parusi\'nski and Lauren\c tiu P\u aunescu,
\newblock Lipschitz stratification of complex hypersurfaces in codimension 2,
\newblock In preparation; see arXiv:1909.00296v3 15 January 2020

\bibitem[Ree61]{Ree1}
David Rees,
\newblock  $\mathfrak{a}$-transforms of local rings and a theorem on multiplicities of ideals,
\newblock {\em Proc. Cambridge Philos. Soc.}, 57:  8 -- 17, 1961.

\bibitem[Rim72]{Rim}
D.S. Rim,
\newblock Formal Deformation Theory,
\newblock {\em SGA 7 I, Groupe de Monodromie en G\'eom\'etrie Alg\'ebrique}, Springer Lecture Notes in Math., n$^\circ$ 288, 32-132, 1972.

\bibitem[Ro-G09]{RosGa}
Jos\'e C. Rosales \& Pedro A. Garc\'ia-S\'anchez, 
\newblock {\em Numerical Semigroups }, 
\newblock Developments in Mathematics, Springer, 2009.

\bibitem[Sn]{Sn}
Jawad Snoussi,
\newblock A quick trip into local singularities of complex curves and surfaces,
\newblock This volume.

\bibitem[Stu72a]{Stu1}
John Stutz,
\newblock Analytic sets as branched coverings,
\newblock {\em Trans. Amer. Math. Soc.}, 166: 241--259, 1972.

\bibitem[Stu72b]{Stu2}
John Stutz,
\newblock Equisingularity and equisaturation in codimension 1,
\newblock {\em American Journal of Mathematics}, 94: 1245--1268, 1972.

\bibitem[Stu77]{Stu3}
John Stutz,
\newblock Equisingularity and local analytic geometry,
\newblock {\em Several complex variables ({P}roc. {S}ympos. {P}ure {M}ath.)},  XXX Part 1: 77--84, 1977.

\bibitem[Tei73]{Te2}
Bernard Teissier, 
\newblock Cycles \'evanescents, sections planes et conditions de Whitney,
\newblock {\em Singularit{\'e}s {\`a} Carg{\`e}se, Ast{\'e}risque 7-8}, 285-362, 1973.

\bibitem[Tei77]{hunting}
Bernard Teissier, 
\newblock The Hunting of Invariants in the Geometry of Discriminants,
\newblock {\em Real and Complex singularities (Proc. Ninth Nordic Summer School/ NAVF Sympos. Math. Oslo, 1976)}, 565 -- 678, 1977.

\bibitem[Tei80]{Tei80}
Bernard Teissier, 
\newblock  R\'esolution simultan\'ee, I, II,
\newblock in: {\em S\'eminaire sur les Singularit\'es des Surfaces},  Lecture Notes in Mathematics n$^\circ$ 777. Springer, Berlin, 72 -- 146, 1980.

\bibitem[Tei82]{Te1}
Bernard Teissier, 
\newblock Vari{\'e}t{\'e}s polaires. II: Multiplicit{\'e}s polaires, sections planes et conditions de Whitney,
\newblock {\em Algebraic geometry, Proc. int. Conf., La R{\'a}bida/Spain 1981}, Springer Lect. Notes Math. 961, 314--491, 1982.

\bibitem [Tei04]{Tei04} Bernard Teissier
\newblock  Monomial ideals, binomial ideals, polynomial ideals,
\newblock  Trends in commutative algebra, 211--246, Math. Sci. Res. Inst. Publ., 51, Cambridge Univ. Press, Cambridge, 2004

\bibitem[Tei07]{TeiBia}
Bernard Teissier, 
\newblock Complex curve singularities: a biased introduction,
\newblock {\em Singularities in Geometry and Topology, World Sci. Publ., Hackensack, NJ }, 825 -- 887, 2007.

\bibitem[To72]{To72}
\newblock Jean Claude Tougeron
\newblock {\em Id\'eaux de fonctions diff\'erentiables},
\newblock Ergebnisse der Math., Band 71, Springer 1972.

\bibitem[Wal78]{Wal}
Robert J. Walker, 
\newblock {\em Algebraic Curves }, 
\newblock Springer-Verlag, 1978.

\bibitem[Whi65]{Whi}
Hassler Whitney,
\newblock Local properties of analytic varieties, 
\newblock {\em Differ. and Combinat. Topology, Sympos. Marston Morse, Princeton}, 205--244, 1965.

\bibitem[Za]{Za}
Oscar Zariski,
\newblock Some results in the arithmetic theory of algebraic varieties,
\newblock {\em Amer. Journ. of Math.}, 61, 249--294, 1939.

\bibitem[Za1]{Za1}
Oscar Zariski,
\newblock General Theory of saturation and of saturated local rings, II: Saturated local rings of dimension1,
\newblock {\em Amer. Journ. of Math.}, 936 (4), 872--964, 1971.

\bibitem[Za2]{Za2}
Oscar Zariski,
\newblock {\em The moduli Problem for Plane Branches},
\newblock American Math. Soc., University Lecture Series, Volume 39.

\bibitem[Za3]{Za3}
Oscar Zariski,
\newblock Studies in Equisingularity, III. Saturation of local rings and equisingularity, 
\newblock {\em Amer.J. Math.}, 90, 961-1023, 1968.




\end{thebibliography}
   
%\bibliographystyle{plain}
%\bibliography{bibliothese.bib}
\noindent
\scriptsize{\noindent
Arturo Giles Flores,
Departamento de Matem\'aticas y F\'isica,
Universidad Aut\'onoma de Aguascalientes.\par\noindent
Avenida Universidad 940, Ciudad Universitaria
C.P. 20131,\par\noindent
Aguascalientes, Aguascalientes, M\'exico.\par\noindent
arturo.giles@cimat.mx
\par\medskip\noindent
Otoniel Nogueira da Silva, Instituto de Matem\'aticas UNAM, Unidad Cuernavaca, \par\noindent 
Av. Universidad s/n. Col. Lomas de Chamilpa
C.P. 62210,  \par\noindent Cuernavaca, Morelos, M\'exico.\par\noindent
otoniel@im.unam.mx
\par\medskip\noindent
Bernard Teissier, Institut math\'ematique de Jussieu-Paris Rive Gauche, \par\noindent 
UP7D - Campus des Grands Moulins,
Boite Courrier 7012. \par\noindent B\^at. Sophie Germain, 8 Place Aur\'elie Nemours, 75205 PARIS Cedex 13, France.\par\noindent
bernard.teissier@imj-prg.fr
}


\end{document}


