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\begin{document}
\title{Local polar varieties in the geometric study of singularities}
%\author{Bernard Teissier}
\maketitle
\pagestyle{myheadings}
\markboth{\rm Bernard Teissier}{\rm Local polar varieties}
\vskip.3truecm

These are notes that came out of a course imparted by Dr. Bernard Teissier in the
``2 Congreso Latinoamericano de Matematicos'' celebrated in Cancun, Mexico on
July 20 -26, 2004.\par
The theme of the course was the local study of analytic subsets of $\C^N$. 
That is, we'll consider subsets defined in a neighbourhood of a point $0\in \C^N$ 
by equations:
\[f_1(z_1,\ldots,z_n)=\cdots=f_k(z_1,\ldots,z_n)=0\]
\[f_i \in \C\{z_1,\ldots,z_n\}, \; f_i(0)=0, \, i=1,\ldots,k.\]
Meaning that the subset $X\subset \C^N$ is thus defined in a neighbourhood $U$ of $0$,
where all the series $f_i$ converge.\par
The purpose of the course was to \textit{stratify} $X$. In other words, partition $X$ into
finitely many \textbf{non-singular} analytic manifolds $\{X_{\alpha}\}_{\alpha \in A}$, which will 
be called strata, such that:
\begin{itemize}
\item[i)] The closure $\overline{X_{\alpha}}$ is a closed complex subspace of $X$, for all $\alpha \in A$.
\item[ii)] $\overline{X_{\alpha}} \backslash X_{\alpha}$ is a union of strata $X_{\beta}$.
\item[iii)] Given any $x \in X_{\beta}$, the ``geometry'' of all the closures $\overline{X_{\alpha}}$
            containing $X_{\beta}$ is locally constant along $X_{\beta}$ in a neighbourhood of x. 
\end{itemize}
\par There is a \textbf{minimal} such partition, and it plays an important role in generalizing to singular
spaces, geometric concepts such as Chern classes, integrals of curvature, Pl$\ddot{u}$cker formulas. This partition
can be described by means of \textbf{Polar Varieties},as we will see, however it is also characterized by 
topological properties. We will begin by trying to put into historical context the appearance of the so called 
polar varieties, as a means to give the reader a little insight and intuition into what we will be doing.


\section{Introduction}\label{basic}

\indent It is possible that the first example of a polar variety is in the treatise on 
conics of Apollonius. The cone drawn
from a point $0$ in affine three-space outside of a fixed sphere 
around that sphere meets the sphere along a circle $C$. If we
consider a plane not containing the point, and the projection $\pi$ 
from $0$ of the affine three-space onto that plane, the circle $C$ is
the set of critical points of the restriction of $\pi$ to the sphere. 
By moving the point $0$, we can obtain any circle drawn on the
sphere, the great circles beeing obtained by sending the point to 
infinity in the direction perpendicular to the plane of the 
circle.\\
\par  
Somewhat later, around 1780, Wallis
asked how many tangents can be drawn to a curve of degree
$d$ in the plane frow a point in that plane and conjectured that this 
number should always be $\leq d^2$. In modern terms, he was proposing 
to compare the visual
complexity of a curve (or surface) as measured by the number of 
morphological changes as the line of sight varies with its algebraic 
complexity as measured by the degree;
given an algebraic surface
$S$ of degree $d$ in affine three-space and a point $0$ outside it, 
the lines emanating from
$0$ and tangent to $S$ touch $S$ along an algebraic curve $P$. Taking 
a general hyperplane $H$ through $0$, we see that the number of 
tangents drawn
from $0$ to the curve $S\cap H$ which is of degree $d$ is the number 
of points in $P\cap H$ and therefore is bounded by the degree
of the algebraic curve $P$. This algebraic curve is an example of a 
\textit{polar curve}. Wallis' question was answered by Poncelet, who 
saw (without writing a single
equation) that the natural setting for the problem which had been 
stated in the affine real domain by Wallis was the complex projective 
plane and that the number of
tangents drawn from a points with projective coordinates
$(\xi:\eta:\zeta)$ to the curve $C$ with homogeneous equation 
$F(x,y,z)=0$ is equal to the number of intersection points of $C$ 
with the curve with equation
$$\xi\frac{\partial F}{\partial x}+\eta\frac{\partial F}{\partial 
y}+\zeta\frac{\partial F}{\partial z}=0.\leqno{(1)}$$
This equation is obtained by \textit{polarizing} the polynomial 
$F(x,y,z)$ with respect to $(\xi:\eta:\zeta)$, a terminology which 
comes from the study of conics; it is the
method for obtaining a bilinear form from a quadratic form (in 
characteristic $\neq 2$). By B\'ezout's theorem, except in the case 
where the curve $C$ is not reduced,
i.e., has multiple components, the number of intersection points 
counted with multiplicities is exactly $d(d-1)$ where $d$ is the 
degree of the polynomial $F(x,y,z)$, that is,
the degree of the curve; this is because the \textit{polar curve} of 
$C$ with respect to the point $(\xi:\eta:\zeta)$, defined by equation 
$(1)$, is of degree $d-1$. So we
conclude with Poncelet that the number (counted with multiplicities) of points 
of the non singular curve $C$ where the tangent goes through the 
point with coordinates
$(\xi:\eta:\zeta)$ is equal to $d(d-1)$. As the point varies in the 
projective plane, the equations $(1)$ describe a \textit{linear 
system} of curves of degree $d-1$ in the
plane, which cuts out a linear system of points on the curve
$C$, of degree $d(d-1)$.
       
\section{Limits of tangent spaces, conormal space and the tangent cone.}

    To set the working grounds, let's fix a reduced, and pure-dimensional germ
of analytic subspace $(X,0)\subset (\C ^n,0)$. That is, we are assuming that the ideal 
of $\C \{z_1,\ldots,z_n\}$ generated by $(f_1,\ldots,f_k)$ contains all analytic functions 
vanishing on $X$, and also that all its irreducible components have the same dimension $d$.\\

By definition, a singular point of a  complex analytic space 
is a point where the tangent space cannot be defined as usual. However, as a kind of
substitute, we can look at all limit positions of tangent spaces at non-singular points tending to 
a given singular point. 

\begin{definition}
A d-plane $T$ of the Grassmannian $Gr(d,n)$ is a limit in $x \in X$ of tangent spaces to the analytic space
$X$ if there exists a sequence $\{x_i\}$ of non-singular points of $X$ and a sequence of d-planes $\{T_i\}$ of
$Gr(d,n)$ such that for all i, the d-plane $T_i$ is the direction of the tangent space to $X$ at $x_i$, the sequence 
$\{x_i\}$ converges to $x$ and the sequence $T_i$ converges to $T$.
\end{definition}

Recall that if $X$ is an analytic space then $Sing \: X$, the set of
singular points of X, is also an analytic space and the non singular part $X^\circ=X\backslash Sing \: X$ is
dense in X and has the structure of a complex manifold. How can we determine these limit positions?\\
\par
 
Let $X$ be a representative of $(X,0)$, consider the application
\begin{align*} \gamma_{X^\circ}: X^\circ & \longrightarrow Gr(d,n)\\
                              x  & \longrightarrow T_xX^\circ
\end{align*} 
where $T_xX^\circ$ denotes
the tangent space to the manifold $X^\circ$ at the point $x$. Let $NX$ be the closure
of the graph of $\gamma_{X^\circ}$ in $X\times Gr(d,n)$. It can be proved that $NX$ is an analytic 
subspace of dimension d ([Whi] thm. 16.4). 

\begin{definition}
The morphism $\nu_{X}: NX \longrightarrow X$ induced by the first projection of $X \times Gr(d,n)$, 
is called the \textit{Nash Modification} of $X$.
\[ \xymatrix { & NX\subset X \times Gr(d,n)\ar[dl]_{\nu_{X}}\ar[dr] &  \\
                 X  & & Gr(d,n)} \]
\end{definition} 

 The fiber 
$\nu_X^{-1}(0)$ is a closed algebraic
subvariety of $Gr(d,n)$; set-theoretically, it is the 
set of limit positions of tangent spaces at points of $X^\circ$ 
tending to $0$.\\\par

The notation $NX$ is justified by the fact that the Nash 
transformation is independent, up to a unique $X$-isomorphism, of the 
embedding of $X$ in a nonsingular space. See [Te1],
\S2, where the abstract construction of the Nash modification is 
explained in terms of the Grothendieck Grassmanian associated to the 
module of differentials of $X$.\\\par

The Nash modification, however is a little difficult to handle 
because of the fact that the rich geometry of the Grassmanian entails 
somewhat cumbersone computations.
There is a less intrinsic but more amenable way of encoding the 
limits of tangent spaces. The idea is to replace a tangent space to $X^\circ$ by the 
collection of all the hyperplanes of $\C^n$ which contain it. Tangent hyperplanes live in a 
projective space, namely the dual projective space $\check{\P}^{n-1}$,
which is easier to deal with than the Grassmannian.

\subsection{Some Symplectic Geometry}

In order to describe this set of tangent hyperplanes, we are going to use the 
language of symplectic geometry and lagrangian submanifolds. So let's start
by a couple of definitions.\\

Let $M$ be any n-dimensional manifold, and let $\omega$ be a de Rham 2-form on M,
that is, for each $p \in M$, the map
\[\omega_p:T_pM \times T_pM \rightarrow \R \]
is skew-symmetric bilinear on the tangent space to M at p, and $\omega_p$ varies 
smoothly in p. We say that $\omega$ is symplectic if it is closed and $\omega_p$ 
is non-degenerate for all $p \in M$. Naturally, a symplectic manifold is a pair
$(M,\omega)$, where $M$ is a manifold and $\omega$ is a symplectic form.\\ \par

Now, for any manifold $M$, it's cotangent bundle $T^*M$ has a canonical symplectic
structure as follows. Let
\begin{align*}
      \pi:T^*M & \longrightarrow M \\
       p=(x,\xi) & \longmapsto   x    
\end{align*} 
%\[\xymatrix {T^*M\ar[d]^\pi & & p=(x,\xi)\ar[d] & & \xi \in T_x^*M \\ 
%                M   & &             x } \]
where $\xi \in T_x^*M$, be the natural projection. The \textbf{Liouville 1-form} $\alpha$ on $T^*M$ may be defined pointwise
by:
\[\alpha_{p}(v)=\xi\left( (d\pi_{p})v\right), \; \; for \; v \in T_{p}(T^*M)\]
Note that $d\pi_{p}:T_{p}(T^*M)\rightarrow T_xM$, so that $\alpha$ is well defined. 
Then, the \textbf{canonical symplectic 2-form} $\omega$ on $T^*M$ is defined as 
\[\omega = - d \alpha\]
And it's not hard to see, that if $(U,x_1,\ldots,x_n)$ is a coordinate chart for $M$
with associated cotangent coordinates $(T^*U,x_1,\ldots,x_n,\xi_1,\ldots,\xi_n)$, then
locally:
\[\omega=\sum_{i=1}^n dx_i\wedge d\xi_i \]

\begin{definition}
Let $(M,\omega)$ be a 2n-dimensional symplectic manifold. A submanifold $Y$ of $M$ is a
\textbf{lagrangian submanifold} if, at each $p \in Y$, $T_pY$ is a lagrangian subspace
of $T_pM$, i.e., $\omega_p |_{T_pY}\equiv 0$ and $dim\: T_pY= 1/2 dim\: T_pM$. Equivalently,
if $i:Y\hookrightarrow M$ is the inclusion map, then $Y$ is \textbf{lagrangian} if and only 
if $i^*\omega=0$ and $dim \:Y=1/2 dim \: M$.
\end{definition}

\begin{ex}
The \textbf{zero section} of $T^*M$
\[X:=\{ (x,\xi) \in T^*M | \xi=0 \; in \; T_x^*M\}\]
is an n-dimensional lagrangian submanifold of $T^*M$.
\end{ex}

\subsection{Conormal space.}

Let now $X\subset M$ be a possibly singular complex subspace of pure dimension d, and let as before
$X^\circ =X \backslash Sing \, X$, be the non-singular part of X, so it is a submanifold of M.

\begin{definition}
%The \textbf{conormal space} at $x \in X^\circ$ is
Set
\[N_x^*X^\circ = \{ \xi \in T_x^*M | \xi (v)=0, \; \forall v \in T_xX^\circ\}\]
this means that the hyperplane $\{\xi=0\} $ contains the tangent space to $X^\circ$ at x.\\
The \textbf{conormal bundle} of $X^\circ$ is
\[T^*_{X^\circ}M =\{ (x,\xi) \in T^*M | x \in X^\circ, \; \xi \in N_x^*X^\circ\} \]
\end{definition}

\begin{proposition}\label{si}
Let $i:T^*_{X^\circ}M\hookrightarrow T^*M$ be the inclusion, and let $\alpha$ be the Liouville 1-form in $T^*M$
as before. Then $i^*\alpha =0$. In particular the conormal bundle $T^*_{X^\circ}M$ is a lagrangian submanifold of $T^*M$,
and of course has dimension n.
\end{proposition}
\begin{proof}$\;$\linebreak
See [CdS] proposition 3.6, pg 16.
\end{proof}

In the same context, we can define the \textbf{conormal space of X in M}, denoted as $T_X^*M$, as 
the closure of $T^*_{X^\circ}M$ in $T^*M$, with the \textbf{conormal map} $\kappa_X:T^*_XM \rightarrow X$,
induced by the natural projection $\pi:T^*M\to M$. The conormal space may be singular, but it is
of dimension n, and by proposition \ref{si} $\alpha$ vanishes on every tangent vector at a nonsingular point,
so it is by construction a lagrangian subspace of $T^*M$. \newline

In words, the fiber of the conormal map $\kappa_X\colon 
T^*_XM \to X$ above a point
$x\in X$ consists, if $x\in X^\circ$, of all the equations of 
hyperplanes tangent to $X$ at $x$, in the sense that they contain the 
tangent space $T_xX^\circ$. If $x$ is a singular
point, the fiber consists of all equations of limits of hyperplane 
directions tangent at non singular points of $X$ tending to 
$x$. In addition, if $x\in X^\circ$, the fiber $\kappa_X^{-1}(x)$ is isomorphic to 
$\C^{n-d}$.\newline

Now the fibers of $\kappa_X$ are invariant under multiplication by an 
element of $\C^*$, and we can divide by the equivalence relation this 
defines. The idea is to remember only the directions of tangent 
hyperplanes, and not a specific linear form defining it. That is, the conormal 
space is conical:= stable by vertical homotheities, so we can ``projectivize'' it.
Moreover, we can characterize those subvarieties of the cotangent space which are
the conormal spaces of their images in M.

\begin{proposition}\label{conormalequiv} Let $M$ be a non singular 
analytic variety of dimension n and let $L$ be a closed conical 
irreducible analytic subvariety of $T^*M$. The following conditions 
are equivalent:\par\noindent
1) The variety $L$ is the conormal space 
of its image in $M$ \par\noindent
2) The Liouville 1-form $\alpha$ 
vanishes on all tangent vectors to $L$ at every non singular point of 
$L$.\par\noindent
3) The symplectic 2-form $\omega=-\hbox{\rm 
d}\alpha$ vanishes on every pair of tangent vectors to $L$ at every 
non singular point of $L$.
\end{proposition}

Since conormal varieties are conical, we may as well projectivize with respect
to vertical homotheities of $T^*M$ and work in $\P T^*M$, where it still makes sense
to be lagrangian since $\alpha$ is homogeneous by definition.\\

Now, going back to our original problem we have $X\subset M=\C^n$, so $T^*M=\C^n\times \check{\C}^n$
and $\P T^*M=\C^n \times \check{\P}^{n-1}$. So we have the \textbf{(projective) conormal space}
$\kappa_X:C(X) \to X$ with $C(X)\subset X \times \check{\P}^{n-1}$, where $C(X)$ denotes the projectivization of the
conormal space $T^*_XM$. Note that we have not 
changed the name of the map $\kappa_X$ after projectivizing since there is no 
ambiguity, and that the dimension of $C(X)$ is $n-1$, which shows immediately that it depends on 
the embedding of $X$ in an affine space. We have the following result:

\begin{proposition}
The (projective) conormal space $C(X)$ is a closed, reduced, analytic subspace of
$X \times \check{\P}^{n-1}$ of dimension $n-1$.
\end{proposition}
\begin{proof}$\;$\newline
See [Te2], proposition 4.1, pg 379.
\end{proof}

In fact if $X$ is a hypersurface, the conormal map coincides with the Nash 
modification. In general, while it is true that the geometric 
structure of the inclusion
$\kappa^{-1}(x)\subset \P^{n-1}$ determines the set of limit 
positions of tangent spaces, i.e., the fiber $\nu^{-1}(x)$ of the 
Nash modification, the correspondance is
not so simple: the points of $\nu^{-1}(x)$ correspond to projective 
subspaces $\P^{N-1-d}$ of $\P^{n-1}$ contained in 
$\kappa_X^{-1}(x)$.\Par 

\subsection{Conormal spaces and projective duality}

Let's assume for a moment that $X\subset \P^n(\C)$ is a projective algebraic variety. 
In the spirit of last section, let's take $M=\P^n(\C)$ with coordinates $[x_0:\cdots:x_n]$, and 
consider the dual projective space $\check{\P}^n(\C)$ with coordinates $[\xi_0:\cdots:\xi_n]$, i.e its 
points are the hyperplanes of $\P^n$. 

\begin{definition}\label{incidence}
  Define the \textbf{incidence variety} $I\subset \P^n \times \check{\P}^n$ as the set of 
points satisfying:
\[ \sum_{i=0}^n x_i\xi_i=0\]
where $[x_1:\cdots:x_n,\xi_1:\cdots:\xi_n] \in \P^n \times \check{\P}^n$
\end{definition}

\begin{lemma}\label{inclema}
    The projectivized cotangent bundle of $\P^n$ is naturally isomorphic to $I$.   
\end{lemma}
\begin{proof}$\;$\newline
\large{\textbf{P  E N D I E N T E}}
\end{proof}

Now, by lemma \ref{inclema}, $I$ inherits the Liouville 1-form $\alpha(:=\sum \xi_i dx_i$ locally) 
from it's isomorphism with $\P T^*\P^n$. But, exchanging $\P^n$ and $\check{\P}^n$, $I$ is also
isomorphic to $\P T^*\check{\P}^n$ so it also inherits the 1-form $\check{\alpha}(:=\sum x_i d\xi_i$ 
locally).  

\begin{lemma}
Let I be the incidence variety as above, then
$\alpha + \check{\alpha}= 0$ on $I$.
\end{lemma}
\begin{proof}$\;$\newline
\large{\textbf{P  E N D I E N T E}}
\end{proof}
 
In particular, this lemma is telling us that if at some point $p \in I$, $\alpha=0$,
then $\check{\alpha}=0$ too. This means, that if we have a  a closed conical 
irreducible analytic subvariety of $T^*\P^n$ as in proposition \ref{conormalequiv},
then it is the conormal space of its image in $\P^n$ if and only if it is the
conormal space of its image in $\check{\P}^n$.
So we have $\P T_X^*\P^n \subset I \subset \P^n \times \check{\P}^n$ and the restriction
of the two canonical projections:
\[ \xymatrix{ & \P T_X^*\P^n \subset I \ar[dl]\ar[dr] & \\
	        X\subset \P^n & &\check{\P}^n \supset \check{X}}  \]

\begin{definition}
    The \textbf{dual variety $\check{X}$} is the image of $\P T_X^* \P^n \subset I$ in $\check{\P}^n$.
So by construction $\check{X}$ is the closure in $\check{\P}^n$ of the set of hyperplanes 
tangent to $X^\circ$. 
\end{definition}

    Now, by symmetry, we immediately get that $\check{\check{X}}=X$. But more,
duality is finding a lagrangian subvariety in $I$, its images in $\P^n$ and $\check{\P}^n$
are necessarily dual. \\

    Let's assume now that $(X,0)\subset (\C^n,0)$ is a cone over a projective algebraic 
variety $V\subset \P^{n-1}$.

\begin{lemma}\label{cone}
      Let $x \in X$ be a nonsingular point of $X$. Then the tangent space $T_xX^\circ$,
contains the line $\mathcal{L}$ joining $x$ to the origin. Moreover, the quotient 
$T_xX^\circ / \mathcal{L}$ is precisely the tangent space to the projective variety $V$ 
at the corresponding point.  
\end{lemma}
\begin{proof}$\;$\newline
  This is due to Euler's identity for a homogeneous polynomial of degree $m$:
\[m.f=\sum_{i=1}^n x_i\frac{\partial f}{\partial x_i} \]
and the fact that if $\{f_1,\ldots, f_r\}$ is the set of homogeneous polynomials defining $X$, 
then $T_xX^\circ$ is the kernel of the matrix:
\[\left( \begin{array}{c}\nabla f_1 \\ \cdot \\ \cdot \\ \nabla f_r \end{array} \right)\] 
\end{proof}

     It's also important to note that at all non-singular points of $X$ in the same
generating line the tangent space to $X^\circ$ is constant and, by the lemma \ref{cone},
the quotient by the generating line is the tangent space to $V$. \\

%%%%INSERTAR FIGURA PARA EJEMPLIFICAR EL LEMMA%%

     So, $\P T_X^*\C^n$ has an image in $\check{\P}^{n-1}$ which is the projective dual
of V.\newline
\[\xymatrix {& \P T_V^*\P^{n-1}\ar[dl]\ar[dr] & &
                 \P T_X^*\C^n\subset \check{\P}^{n-1}\times \C^n \ar[dl]\ar[dr] & \\
              V\subset \P^{n-1} & & \check{\P}^{n-1}\supset \check{V}& &
	      X\subset \C^n  }\]

The fiber over 0 of $\P T_X^*\C^n \to X$ as subvariety of $\check{\P}^{n-1}$, is equal
to $\check{V}\colon$ it is the set of limit positions at 0 of hyperplanes tangent to $X^\circ$.

\subsection{Tangent cone}

   At the very beginning we mentioned how the limit of tangent spaces can be thought of as 
a substitute for the tangent space at singular points. However there is another common substitute
for the missing tangent space, the \textbf{tangent cone}.

   Let's start by the geometric definition. Let $X\subset \C^n$ be a representative of
$(X,0)$. The canonical projection $\C^n\setminus \{0\} \to \P^{n-1}$ 
induces the secant map
\begin{align*}
s_X\colon X\setminus\{0\}&\to \P^{N-1}\\ 
x & \mapsto [0x]
\end{align*}  
   Denote by $E_0X$ the closure in $X\times \P^{n-1}$ of the graph of $s_X$. $E_0X$ is an 
analytic subspace of dimension $d$, and the natural projection $e_0X\colon 
E_0X\to X$ induced by the first projection is called the \textbf{blowing up} of $0$ in $X$. The fiber 
$e_0^{-1}(0)$ is a projective subvariety of $\P^{n-1}$ of dimension d-1, not necessarily reduced.  
(See [Whi1]) 

\begin{definition}\label{tanconedef1}
The cone with vertex $0$ in $\C^n$ corresponding to 
this subset of projective space $|e_0^{-1}(0)|$
is the set-theoretic \textbf{tangent cone}.
\end{definition}

Again, set-theoretically, it is the set of limit directions of secant lines $0x$ for points $x\in 
X\setminus\{0\}$ tending to $0$. This means more precisely that for 
each sequence $(x_i )_{i\in \N}$ of points of $X\setminus\{0\}$, tending to $0$ as $i\to\infty$ we can, 
since $\P^{n-1}$ is compact, extract a subsequence such that the directions $[0x_i]$ of the secants
$0x_i$ converge. The set of such limits is the underlying set of $e_0^{-1}(0)$. (See [Whi2]thm 5.8) \\

Now, on to the algebraic definition. Let $O=O_{X,0}= \C\{z_1,\ldots,z_n\}/\left<f_1,\ldots, f_k
\right>$ be the local algebra of $X$ at $0$ and let $\m=\m_{X,0}$ 
be its maximal ideal. There is a natural filtration of $\O_{X,0}$ by 
the powers of $\m$:
\[\O_{X,0}\supset \m \supset\cdots\supset \m^i\supset \m^{i+1}\supset\cdots,\]
which is \textit{separated} in the sense that $\bigcap_{i=o}^\infty 
\m^i=(0)$ because the ring $\O_{X,0}$ is n\oe therian.  Consider the \textit{Rees Algebra} of
$O$ with respect to $\m$. (See [Eis], section 6.5)
\[\mathcal{R} = \bigoplus_{i \in \Z} \m^i t^{-i} \subset O[t,t^{-1}]\]
where $\m^i=O$ for $i\leq 0$. Note that $\mathcal{R}\supset O[t] \supset \C[t]$, in fact it 
is a finitely generated $O-$algebra and consequently a finitely generated $\C$-algebra 
(See [Mat], pg 120-122).\\

\begin{definition}
We define the \textbf{associated graded ring of $O$ with respect to $\m$}, written $gr_{\m}O$ to
be the graded ring
\[gr_{\m}O\colon = \bigoplus_{i\geq0}\m^i/ \m^{i+1}\]
where $\m^0=O$.
\end{definition}

   Note that $gr_{\m}O$ is generated as $\C-$algebra by $\m/\m^2$, which is a finite dimensional
vector space. So, $gr_{\m}O$ is a finitely generated $\C-$algebra, to which we can associate 
a complex analytic space $Specan \; gr_{\m}O$. Moreover, since $gr_{\m}O$ is graded and 
finitely generated in degree one, the associated affine variety $Specan \; gr_{\m}O$ is a cone.
(For more on Specan, see [Ka] pg 172) 

% Given a finitely generated 
% $\C$-algebra, we can associate to it an affine scheme over $\C$, but we can also associate to it an 
% analytic space, with the same underlying set, but a different topology (complex topology 
% versus Zariski topology) and a different sheaf of functions (analytic functions versus algebraic ones). \
\begin{definition}\label{tanconedef2}
   We define the \textbf{tangent cone} $C_{X,0}$ as the complex analytic space $Specan \; (gr_{\m}O)$.
\end{definition}
   
    We have yet to establish the relation between the geometric and algebraic definitions of
the tangent cone. For this purpose we will need the next result.

\begin{proposition}
Let $\mathcal{R}$ be the Rees algebra defined above. Then:
\begin{itemize}
\item[i)] The $\C[t]-$algebra $\mathcal{R}$ is torsion free.
\item[ii)] $\mathcal{R}$ is flat over $\C[t]$
\item[iii)] The map $\phi:\mathcal{R}\to gr_{\m}O$ sending $xt^{-i}$ to the image of $x$ in 
           $\m^i/\m^{i+1}$ is well defined and induces an isomorphism $\mathcal{R}/(t\cdot\mathcal{R}) 
           \to gr_{\m}O$.
\item[iv)] For any $t_0 \in \C\setminus \{0\}$ the map  $\mathcal{R} \to  O$, sending
           $xt^{-i} \mapsto xt_0^{-i}$ induces an isomorphism $\frac{\mathcal{R}}{(t-t_0)\cdot \mathcal{R}}
            \to O$
\end{itemize}
\end{proposition}
\begin{proof}$\;$\newline
See [Eis] pg 171.
\end{proof}

The proposition may be a little technical, but what we are trying to say is that the Rees algebra 
is a way of producing flat families, since in general, the inclusion morphism 
$\C[t] \hookrightarrow \mathcal{R}$ is flat.   Now, given any finitely generated algebra over $\C$, there 
is a ``smallest'' analytic algebra which contains it (it means that any map from our algebra
to an analytic algebra factors uniquely through this ``analytization''. The proof: our algebra is a 
quotient of a polynomial ring by an ideal I; take the quotient of the corresponding convergent 
power series ring by the ideal generated by I. \\

Taking the analytic algebra $\mathcal{R}^h$ associated to $\mathcal{R}$, and the analytic germ 
$Z$ associated to $\mathcal{R}^h$, we have a germ of map induced by the inclusion $\C\{t\} 
\hookrightarrow \mathcal{R}^h$:
\[ p: (Z,0) \longrightarrow (\mathbb{D},0) \] 
such that:
\begin{itemize}
\item [$\cdot$] $p$ is faithfully flat.
\item [$\cdot$] $p^{-1}(0)$ is the analytic space associated to $gr_{\m}O$, that is
                the tangent cone $C_{X,0}$
\item[$\cdot$]  $p^{-1}(t)$ is an analytic space isomorphic to $X$, for all $t\neq 0$.
\end{itemize}
that is, we have produced a 1-parameter flat family specializing $X$ to $C_{X,0}$.

\begin{exercise}$\;$\newline
1) Suppose $(X,0)$ is a germ of hypersurface, then $O=\C\{z_1,\ldots,z_n\}/<f(z_1,\ldots,z_n)>$.
Show that $\mathcal{R}^h=\C\{t,z_1,\ldots,z_n\}/<t^{-m}f(tz_1,\ldots,tz_n)>$. \par
Note that this makes sense since $f=f_m(z_1,\ldots,z_n)+f_{m+1}(z_1,\ldots,z_n)+ \ldots$,
where $f_m$ is an homogeneous polynomial of degree m, then:
\[t^{-m}f(tz_1,\ldots,tz_n)= f_m(z_1,\ldots,z_n)+tf_{m+1}(z_1,\ldots,z_n)+ \ldots\]
2)More generally, take $I\subset \C\{z_1,\ldots,z_n\}$ and choose generators $f_i$ such
that their initial forms $(f_i,m_i)$ generate the ideal of all initial forms of elements
of $I$. Then:
\[\mathcal{R}^h=\C\{t,z_1,\ldots,z_n\}/<t^{-m_i}f_i(tz_1,\ldots,tz_n)>\]
\end{exercise}

\begin{proposition}
Let $|C_{X,0}|$ be the underlying set of the analytic space $C_{X,0}$. Then, generating lines
in $|C_{X,0}|$ are the limit positions of secant lines $0x_i$ as $x_i \in X\setminus \{0\}$ tends
to 0.
\end{proposition}
\begin{proof}$\;$\newline
The important fact is that since $p:(Z,0)\to (\mathbb{D},0)$ is faithfully flat, for every point 
$\overline{x} \in p^{-1}(0)=C_{X,0}$ there are sequences of points $(x_i,t_i) \in X \times \C^*$ 
tending to $\overline{x}$. Then $\overline{x}$ is in the limit of secants $0x_i$.\newline
\end{proof}

   So, we finally know that our two concepts of tangent cone coincide, at least set-theoretically.
However, the tangent cone contains very little information on $(X,0)$.

\begin{ex}$\;$\newline
For all curves $y^2-x^m$, $m\geq 3$, the tangent cone is $y^2=0$ and it's non-reduced. 
\end{ex}

\subsection{Multiplicity}

Nevertheless the analytic structure of $C_{X,0}=Specan \; gr_{\m}O$ does
carry some important piece of information on $(X,0)$, its  multiplicity. \\

For a hypersurface, $f=f_m(z_1,\ldots,z_n)+f_{m+1}(z_1,\ldots,z_n)+ \ldots$, the 
multiplicity at 0 is just $m$=the degree of the initial polynomial. And, from the example
above, it's tangent cone is also a hypersurface with the same multiplicity at 0 in this sense.
In general, it is more complicated. \\ 

Let $O$ be the analytic algebra of $X$ with $\m$ it's maximal ideal as before, then we have
the following results:\\

a) For each $i\geq 0$, the quotient $O/\m^{i+1}$ is a finite dimensional vector space over $\C$,
   and the generating function:
\[\sum_{i\geq0}(dim_{\C}\; O/\m^{i+1}T^i)= \frac{Q(T)}{(1-T)^d} \]
is a rational function with numerator $Q(T)\in \mathbb{Z}[T]$, and $Q(1) \in \mathbb{N}$.
(See [A-M] pg 117-118)\\

b) For large enough $i$: 
   \[dim_{\C} \; O/m^{i+1}=e_{\m}(O)\frac{(i+1)^d}{d!} + lower \; order \; terms\]
   and $e_{\m}(O)=Q(1)$ is called \textbf{the multiplicity of X at 0}.(See [Mat] \S14)\\

c) A linear space $L_{\epsilon}$ of dimension $n-d$ at a sufficiently small distance $\epsilon>0$
   from the origin and of general direction has the property that $L_{\epsilon}\cap X$ consists
   of $e_{\m}(O)$ points which tend to 0 with $\epsilon$. (See [He-Or] pg 510-555)\\

d) The multiplicity of $X$ at 0 coincides with the multiplicity of $C_{X,0}$ at 0.
   (See [He-Or] Thm 5.2.1 \& Cor) \\

\section{Normal Cone and Polar Varieties}
%  Tal vez hay que agregar lo del ideal de formas iniciales al cono tangente y agregarlo 
%aqui tambien para que sea mas bonito y entendible. 
The normal cone is defined with respect to an analytic subspace, say $Y$. If $X$ were non-singular
it would only be the normal bundle of $Y$ in $X$. It will be a very similar construction to what we did 
with the tangent cone. In fact, when $Y=\{0\}$ it is precisely the tangent cone. We will only consider
the case where $Y$ is a non-singular subspace of dimension $t$.\\

Let $I$ be the ideal of $O_X$ defining $Y$: it consists of all elements of $O_X$ vanishing on $Y$,
and it is clear that $O_Y=O_X/I$. Just as in the case of the tangent cone, let's consider $gr_IO_X$, the 
associated graded ring of $O_X$ with respect to $I$:
\[gr_IO_X= \bigoplus_{j\geq0} I^j/I^{j+1}= O_X/I \oplus I/I^2 \oplus \cdots \]

\begin{definition}\label{defnc}
We define the \textbf{normal cone of $X$ along $Y$} $C_{X,Y}$, as the complex analytic space
$Specan \; (gr_IO_X)$.
\end{definition}

   Note that we have a canonical inclusion $O_Y \hookrightarrow gr_IO_X$, which induces the structure of
a $O_Y$-algebra and consequently a canonical analytic projection $C_{X,Y} \to Y$, in which the fibers are
cones. It's also important
to have in mind that since $Y$ is non-singular we can assume it is linear, by choosing a sufficiently 
small representative of $(X,0)$ and adequate local coordinates on $\C^n$.\\
%% PENDIENTE PGS 22-23 %%
   Now if we consider the filtration of $O_X$ by the powers of the ideal $I$ defining $Y$:
\[O_X \supset I \supset  \cdots I^{m} \supset I^{m+1}\cdots \]
and use the Rees Algebra technique of $O_X$ with respect to $I$. We can just as in the case where
$Y=\{0\}$ produce a flat 1-parameter family specializing $X$ to $C_{X,Y}$. That is, we'll obtain a 
germ of map $q:(Z,0)\to (\mathbb{D},0)$ such that:
\begin{itemize}
\item [$\cdot$] $q$ is faithfully flat.
\item [$\cdot$] $q^{-1}(0)$ is the normal cone of X along Y.
\item[$\cdot$]  $q^{-1}(t)$ is an analytic space isomorphic to $X$, for all $t\neq 0$.
\end{itemize}

    Using this it can be shown that after choosing a local retraction $r:(X,0)\to (Y,0)$,
the underlying set of $C_{X,Y}$ can be identified with the set of limit positions of secant
lines $x_ir(x_i)$ for $x_i \in X\setminus Y$ as $x_i$ tends to $y \in Y$.(See [Hi])
\newpage

\begin{proposition}\label{propnc}
Given a t-dimensional nonsingular subspace $(Y,0) \subset (X,0) \subset \C^n$, the following
conditions are equivalent:
\begin{itemize}
\item[i)] The multiplicity $m_y(X)$ of $X$ at the points $y \in Y$ is locally constant on $Y$
near 0.
\item[ii)] The dimension of the fibers of the maps $C_{X,Y}\to Y$ are locally constant on $Y$
near 0.
\item[iii)] There exists a dense open set of $(n-d+t)$-dimensional linear spaces containing
$T_0Y$(the tangent space to $Y$ at 0), such that if $W$ is a non-singular $(n-d+t)$-dimensional
space containing $Y$ and whose tangent space is in that open set, then locally:
\[|W\cap X|=Y\]
\end{itemize}
\end{proposition}
\begin{proof}$\;$\\
See [He-Or] Appendix III, theorem 2.2.2, or additionally [Te2] Chapter I, 5.5 and \cite{Hi}.
\end{proof}

\begin{ex}
Take $(X,0) \subset (\C^3,0)$ defined by $x^2-y^2z=0$, otherwise known as Whitney's Umbrella,
and $Y=y$-axis. \\

Taking $W$ as the non-singular 2-dimensional space defined by $z=ax$, gives
\[x(x-ay^2)=0\]
so that whenever $a\neq 0$, the intersection $W\bigcap X$ has two irreducible components:
the $y$-axis, and the curve defined by the equations $x=ay^2$, $z=ax$.\\

%%%%%%%%INSERTAR IMAGENES%%%%%%
We can do the same for $Y=z$-axis, and $W$ defined by $y=ax$ which gives
\[x^2(1-az^2)\]
that locally defines the $z$-axis.
\end{ex}

     In order to understand better the normal cone and proposition \ref{propnc}, we are going
to introduce the blowing up of $X$ along $Y$. As in the case of the tangent cone, we'll start
by a geometric description. \\

     Let $(Y,0)\subset (X,0)$ be germ of non-singular subspace of dimension $t$ as before. 
We can choose a local analytic retraction $r:\C^n \to Y$, use it to define a map:
\begin{align*}
\phi: X\setminus Y & \longrightarrow \P^{n-1-t}\\
         x   & \longmapsto         direction \; of \; \overline{xr(x)} 
\end{align*}
and then consider it's graph in $(X\setminus Y) \times \P^{n-1-t}$. Note that, since we can assume
$Y$ is a linear subspace, under a suitable set of coordinates the map $r$ is just the canonical 
projection $r:\C^n \to \C^t$. Moreover, the map $\phi$ maps $x=(x_1,\ldots,x_n)\in X\setminus Y 
\mapsto [x_{t+1}:\cdots:x_n]\in \P^{n-1-t}$.\newline
Just as in the case of the blow-up of a point, the closure of the graph is a complex analytic space 
$E_{Y}X \subset X \times \P^{n-1-t}$, and the natural projection map $e_y:= p \circ i$:
\[\xymatrix{ E_YX \ar@{^{(}->}[r]^i\ar[dr]_{e_y}&  X\times \P^{n-1-t}\ar[d]^p\\
                              & X   }\]
is proper. Moreover the map $e_y$ induces an isomorphism $E_YX\setminus e_y^{-1}(Y) \to 
X\setminus Y$.\\

Now to the algebraic description, let $I\subset O_X$ be the ideal defining $Y\subset X$ as before.
In our special case $I=\left< x_{t+1},\ldots,x_n\right>$. 

\begin{definition}
The \textbf{blowup algebra} of $I$ in $O_X$ is the $O_X$-algebra:
\[P(I)=\bigoplus_{j\geq0} I^j=O_X \oplus I \oplus I^2 \oplus \cdots\] 
\end{definition}

Note that $P(I)/IP(I)\cong gr_IO_X $, the associated graded ring of $O_X$ with respect to $I$. Moreover, 
since $I$ is finitely generated, $P(I)$ is a finitely-generated, graded $O_X$-algebra,
generated in degree 1, and as such, it has a presentation:
\[\frac{O_X[z_1,\ldots,z_{n-t}]}{(g_1,\ldots,g_m)}\cong P(I)\]
where the $g_i$ are homogenous polynomilas in $z_1,\ldots,z_{n-t}$ with coefficients in $O_X$.\\

Defining $\tilde{E}_YX$ as the \textbf{projective analytic spectrum} of $P(I)$, $\tilde{E}_YX= 
Projan \; P(I)$ (see [He-Or] Appendix III, 1.2.8), we can view this as defining a family of projective 
varieties parametrized by X, as a result of the inclusion $O_X\hookrightarrow P(I)$.
\[\xymatrix{\tilde{E}_YX \subset X \times \P^{n-1-t}\ar[d]\\ X }\]

\begin{proposition}
Given an ideal $I\subset O_X$, choosing a system of generators $I=\left< h_1,\ldots,h_s \right>$
gives an embedding $E_YX \subset X \times \P^{s-1}$ and an embedding \newline $\tilde{E}_YX \subset 
X \times \P^{s-1}$. Their images are equal.
\end{proposition}
\begin{proof}$\;$\newline
Let $Y\subset X$ be the subspace defined by $I$. Now consider the map:
\begin{align*}
\lambda : X\setminus Y &\longrightarrow \P^{s-1}\\
                 x     &\longmapsto [h_1(x):\cdots:h_s(x)]
\end{align*}
and as before, let $E_YX \subset X\times \P^{s-1}$ be the closure of the graph of $\lambda$. \\
On the other hand, consider the presentation $O_X[z_1,\ldots,z_s]/(g_1,\ldots,g_m)\cong P(I)$
where the isomorphism is defined by $z_i\mapsto h_i$. Note that the $g_i's\in O_X[z_1,\ldots,z_s], 
i=1,\ldots,m$, generate the ideal of all homogeneous relations $g(h_1,\ldots,h_s)=0, \; g\in 
O_X[z_1,\ldots,z_s]$ which are exactly the equations for the closure of the graph. To see why this
last statement is true, recall that:
\[graph(\lambda)=\{(x,z_1:\cdots:z_s)\subset X \times \P^{s-1}| \, [z_1:\cdots:z_s]=[h_1(x):\cdots:
h_s(x)]\}\]
and remember that the coefficients of $g \in \left<g_1,\ldots,g_m\right>$ live in $O_X$, so we can make 
sense of the ``evaluation of'' $g$ in a point $(x,z_1:\cdots:z_s)\in X\times \P^{s-1}$. Moreover, 
if  $(x,z_1:\cdots:z_s)\in graph(\lambda)$, then $[z]=[h(x)]$ thus $g(z)=0$, that is they are the 
equations defining the graph, and consequently its closure.
\end{proof}

Finally, to relate all this to the normal cone, note that in the map:
\[e_y:E_YX \longrightarrow X\]
the inverse image of $Y$ is the projective family associated to the family of cones
\[C_{X,Y}\longrightarrow Y\]
this is clear, set-theoretically, in the geometric description. In the algebraic description, it 
follows from the identity:
\[\bigoplus_{j\geq0}I^j\bigotimes_{O_X}O_X/I \cong \bigoplus_{j\geq0}I^j/I^{j+1}=gr_IO_X\]
and the fact that fiber product corresponds to tensor product.\\

\[\xymatrix{O_X\ar@{^{(}->}[r]\ar[d] & \bigoplus_{j\geq0}I^j \\
            O_X/I &} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
  \xymatrix{ & Y\ar@{^{(}->}[d]\\
            E_YX\ar[r]_{e_y}& X}\] 

\subsection{Polar Varieties}

    We'll get here a first glimpse, into a special case of what will be called the 
normal-conormal diagram. Let's denote by $C(X)$ and $E_0X$ respectively the conormal space of $X$
and the blowup of 0 in $X$ as before, then we have the diagram:
\[\xymatrix{E_0C(X)\ar[r]^{\hat{e}_0}\ar[dd]^{\kappa'}\ar[ddr]^\xi & C(X)\ar@{^{(}->}[r]\ar[dd]^\kappa
            \ar[dr]^{\lambda}& X \times \check{\P}^{n-1}\ar[d]^{pr_2} \\
             & &  \check{\P}^{n-1} \\
             E_0X\ar[r]_{e_0}  & X &}\]
where $E_0C(X)$ is the blowup of $\kappa^{-1}(0)$ in $C(X)$, and as we have mentioned before can
be identified set-theoretically, with the fiber product $$E_0C(X)=C(X)\times_X E_0X$$

Let $D_{d-k+1}\subset \C^n$ be a linear subspace of codimension $d-k+1$, for $0\leq k \leq d-1$,
and let $L^{d-k}\subset \check{\P}^{n-1}$ the dual space of $D_{d-k+1}$, that is the linear space
of hyperplanes of $\C^n$ that contain $D_{d-k+1}$.

\begin{proposition}\label{polarv}
For a sufficiently general $D_{d-k+1}$, the image $\kappa(\lambda^{-1}(L^{d-k}))$ is the closure in 
$X$ of the set of points of $X^0$ which are critical for the projection $\pi|_{X^\circ}:X^\circ \to \C^{d-k+1}$, induced
by the projection $\C^n \to \C^{d-k+1}$ with kernel $D_{d-k+1}=\check{L}^{d-k}$.
\end{proposition}
\begin{proof}$\;$ \newline
We will only give an idea of the proof. Note that $x \in X^0$ is critical for $\pi$, iff
the tangent map  $d_x\pi:T_xX^0\to \C^{d-k+1}$ is not onto, that is iff dim $ker \; d_x \pi \geq k$ 
since 
dim $T_xX^0=d$, and $ker \; d_x{\pi}=D_{d-k+1}\cap T_xX^0$.\\
Now, by definition:
\[\lambda^{-1}(L^{d-k})=\{(x,H) \in C(X) | H \in L^{d-k}, \; T_xX^0 \subset H \}\]
equivalently:
\[\lambda^{-1}(L^{d-k})=\{(x,H) \in C(X)| H \in \check{D},\; H \in \check{T_x}X^0\} \]
thus $H \in \check{D}\cap \check{T_x}X^0$, and from the equation $\check{D}\cap \check{T_x}X^0=
(D+T)^{\check{}}$ we deduce that the intersection is not empty iff $D+T\neq \C^n$, which
implies that dim $D\cap T\geq k$, and consequently $\kappa(H)=x$ is a critical point.\\
(See [Te2] Chap. 4, 4.1.1. for a complete proof of a more general statement.) 
\end{proof}

\begin{definition}Under the notation and hypothesis of proposition \ref{polarv}, define
the \textbf{local polar variety}.
\[P_k(X;L^{d-k})= \kappa(\lambda^{-1}(L^{d-k})) \]
\end{definition}

Note that $\polar \subset X$ is either an analytic subspace of ¿pure? dimension $d-k$, or empty.

\begin{ex} 
%EL DE LA PAG 42. INCLUIR CON TODO Y FIGURA%%%%
\end{ex}

\textbf{INCLUIR COSAS DE LA MULTIPLICIDAD DE LAS VARIEDADES POLARES Y SU INVARIANZA}

\subsection{Limits of tangent spaces}
      
      We have talked before of the limits of tangent spaces, but with the help of the 
normal/conormal diagram and the polar varieties we'll be able to describe the limits of tangent spaces
to $X$ at 0,  as long as $(X,0)$ is reduced and purely d-dimensional. This method, is based 
on the following two results and Whitney's lemma:

\begin{lemma}\textbf{Whitney's lemma.-}\label{wl}
 Let $(X,0)$ be a pure dimensional germ of analytic subspace, choose a representative $X$ and let 
 $\{x_n\}\subset X^\circ$ be a sequence of points tending to $0$, such that 
 \[\lim_{n\to \infty}[0x_n]=l \;\; and \;\;
 \lim_{n\to \infty}T_{x_n}X=T\] Then $l \subset T$.
\end{lemma} 
      
     This lemma originally appeared in [Whi1] theorem 22.1, and you can also find a proof due to
Hironaka in [L\^e].\\

\begin{theorem}$\;$\\

\textbf{I)} In the normal/conormal diagram
\[\xymatrix{E_0C(X)\ar[r]^{\hat{e}_0}\ar[dd]^{\kappa'}\ar[ddr]^\xi & C(X)\ar[dd]^\kappa
             \\
             & &  \\
             E_0X\ar[r]_{e_0}  & X }\]
consider the irreducible components $\{D_{\alpha}\}$ of $D=|\xi^{-1}(0)|$. Then:
\begin{itemize}
\item[a)] Each $D_{\alpha}\subset \P^{n-1} \times \check{\P}^{n-1}$ is in fact contained
          in the incidence variety $I\subset \P^{n-1} \times \check{\P}^{n-1}$.\\
        \textnormal{  Note that, using the interpretation of $E_0C(X)$ as the fibered product $C(X)\times_X E_0X$,
	  what we are stating here is immediate from Whitney's lemma \ref{wl} since a point $(0,l,H) \in \xi^{-1}(0)$
	  means that both $l$ and $H$ are limits obtained from the same sequence $\{x_n\}\subset X^\circ$, and 
	  as such this implies $l\subset H$.}

\item[b)] Each $D_\alpha$ is lagrangian in $I$ and therefore establishes a projective duality 
          of its images:
\[\xymatrix{ D_\alpha \ar[rr]\ar[dd]& &  W_\alpha \subset \check{\P}^{n-1}\\
                                    & & \\
             V_\alpha \subset \P^{n-1} & & }\]
\end{itemize}

     \textnormal{Note that, from commutativity of the diagram we obtain  $\kappa^{-1}(0)=\bigcup W_{\alpha}$, and 
                 $e_0^{-1}(0)=\bigcup V_{\alpha}$. It's important to notice that this expressions are not the 
                 irreducible decompositions of $\kappa^{-1}(0)$ and $e_0^{-1}(0)$ respectively, since we can't assure
		 that all the $V_\alpha \;(W_\alpha)$ are of the correct dimension. However, it is true that they contain
                 the respective irreducible decompositions.} \\

		 \textnormal{In particular, note that if dim$V_{\alpha_0}=d-1$, then the cone $O(V_{\alpha_0})\subset \C^n$ is 
                 an irreducible component of the tangent cone $C_{X,0}$ and its projective dual $W_{\alpha_0}=
		 \check{V_{\alpha_0}}\subset \kappa^{-1}(0)$. That is, any tangent hyperplane to the tangent cone is 
                 a limit of tangent hyperplanes to $X$ at $0$.} \\

\textbf{II)} For any integer $k$, $0\leq k \leq d-1$ and sufficiently general $L^{d-k}\subset \check{\P}^{n-1}$
             the tangent cone $C_{P_k(X,L),0}$ of the polar variety $P_k(X,L)$ at the origin consists of:
\begin{itemize}
\item[$\bullet$]The union of the cones $O(V_\alpha)$ which are of dimension d-k (= dim $P_k(X,L)$).
\item[$\bullet$]The polar varieties of dimension d-k, for the projection $p$ associated to $L$, of the
              cones $O(V_\beta)$, for dim $O(V_\beta)> d-k$, that is $P_r(O(V_\beta),L)$.
\end{itemize}

\textnormal{ Note that $P_k(X,L)$ is not unique, since it varies with $L$, but we are saying that their tangent cones have things 
            in common. The $V_\alpha$'s are fixed, so the first part is the fixed part of $C_{P_k(X,L),0}$ because it is independent 
            of $L$, the second part is the mobile part, since we are again talking of polar varieties of certain spaces which of
            course depend on $L$.}\\
\end{theorem}
\begin{proof}$\;$\\
You can find a proof of this results in [L-T1], [Te2] and [Te3].
\end{proof}
       So for any complex germ $(X,0)$ reduced and purely $d-$dimensional, we have a method to ``compute'' or rather
       describe, the set of limiting positions of tangent hyperplanes:
\begin{itemize}
\item[1)] For all integers $k$, $0\leq k \leq d-1$, compute the ``general'' polar varieties 
          $P_k(X,L)$, leaving in the computation the coefficients of the equations of L as indeterminates.
          (Partial derivatives)  
\item[2)] Compute the tangent cones  $C_{P_k(X,L),0}$. (Grobner basis)
\item[3)] Sort out those components of the tangent cone of each $P_k(X,L)$ which are independent of L.
          (Irreducible components)
\item[4)] Take the projective duals of the corresponding projective varieties. (Elimination)  
\end{itemize}

       We have noticed, that among the $V_\alpha$'s, there are those which are irreducible components of $Proj C_{X,0}$
       and those that don't reach the dimension.

\begin{definition}
       The cones $O(V_{\alpha})$'s such that 
       \[ dim\; V_\alpha < dim\; Proj C_{X,0}\]
       are called exceptional cones. 
\end{definition}

\textbf{INCLUIR DEFINICION DE WHITNEY EQUISINGULAR}
       
       So, now one may wonder if having no exceptional tangents makes $X$ look like a cone. For this question, 
       we can give a partial answer in the case of surfaces.

\begin{proposition}
       Let $(X,0)$ be a germ of a reduced, 2-dimensional analytic subspace of $\C^n$. If $X$ has either an isolated
       singularity at $0$ or the tangent cone $C_{X,0}$ is reduced. Then, having no exceptional tangents makes $X$
       Whitney-equisingular with its tangent cone.
\end{proposition}
\textbf{FALTA REFERENCIA} 

\section{Whitney Stratifications}

       Whitney had observed, as we can see from the statement of lemma \ref{wl}, that ``asymptotically'' a germ $(X,0)$ 
       behaves like a cone with vertex $0$, near $0$. Suppose now, that we replace $0$ by a non-singular subspace
       $Y\subset X$, and we want to force $X$ to ``look like a cone with vertex $Y$''.

\begin{definition}
       A cone with vertex $Y$ is a space $C$ equipped with a map \[C \longrightarrow Y\] and homotheities in the fibers.
\end{definition}

      Let's take a look at the basic example we have thus far constructed.

\begin{example}$\;$\\
 The reduced normal cone $|C_{X,Y}|\longrightarrow Y$, with the canonical analytic projection mentioned after definition
   \ref{defnc}.
\end{example}

      Now, making an analogy with the opening statement of this section we can ask ourselves: What does it mean that
      ``asymptotically'' $X$ is cone-like over $Y$? Well, here is Whitney's answer:

\subsection{Whitney's conditions}

     Let $X$ be a reduced, pure dimensional analytic space of dimension d, let $Y\subset X$  be a nonsingular 
     analytic subspace containing $0$ of dimension t. Choose a local embedding $(X,0)\subset (\C^n,0)$ around $0$, and a local 
     retraction $\rho:(\C^n,0)\longrightarrow (Y,0)$. Note that, since $Y$ is non-singular we can assume it is an open
     subset of $\C^t$, $(X,0)$ is embedded in an open subset of $\C^t \times \C^{n-t}$ and the retraction $\rho$ coincides with the
     first projection.
 
     We say that $X^\circ$ satisfies Whitney's conditions along $Y$ at $0$ if for any sequence of pairs of points
     $\{(x_i,y_i)\}_{i\in \N} \subset  X^\circ \times Y$ tending to $(0,0)$ we have:
     \[\lim_{i\to \infty} [x_iy_i] \subset \lim_{i\to \infty} T_{x_i}X\]
     where $[x_iy_i]$ denotes the line passing throught these two points. If we compare this to Whitney's lemma,
     it is just spreading out along $Y$ the fact observed when $Y=\{0\}$.
     
     In fact, Whitney stated 2 conditions, which together are equivalent to the above. Letting $\rho$ be the aforementioned 
     retraction, here we state the 2 conditions:

\begin{itemize}
     \item[a)] For any sequence $\{x_i\}_{i \in \N}\subset X^\circ$, tending to $0$ we have:
               \[ T_0Y \subset \lim_{i\to \infty} T_{x_i}X \]
     \item[b)] For any sequence $\{x_i\}_{i \in \N}\subset X^\circ$, tending to $0$ we have:
               \[ \lim_{i\to \infty} [x_i\rho(x_i)] \subset \lim_{i\to \infty} T_{x_i}X\]        
\end{itemize}

\subsection{Stratifications}

     Recall, that our objective is to ``stratify'' $X$. What exactly do we mean by stratify, and how do Whitney
     conditions relate to this? What follows in this section consists mostly of material from [Lip] 
     and [Te2].

\begin{definition}
     A \textbf{stratification} of $X$ is a decomposition into a locally finite disjoint union $X=\bigcup X_\alpha$, of non-empty, 
     connected, locally closed subvarietes called \emph{strata}, satisfying:
\begin{itemize}
   \item [(i)] For any stratum $X_\alpha$, with closure $\overline{X_\alpha}$, the boundary $\partial X_\alpha:= 
           \overline{X_\alpha} \setminus X_\alpha$ is a union of strata (and therefore is a closed subvariety of $X$).
   \item [(ii)] Every stratum $X_\alpha$ is smooth (and therefore an analytic manifold).	     
\end{itemize}
\end{definition}

    Stratifications can be determined by \emph{local stratyfing conditions} as follows. 
    We consider conditions $C=C(W_1,W_2,x)$ defined for all $x \in X$ and all pairs $(W_1,x)\supset (W_2,x)$ of
    subgerms of $(X,x)$ with $(W_1,x)$ equidimensional and $(W_2,x)$ smooth. For example, $C(W_1,W_2,x)$ could signiffy
    that the Whitney conditions hold at x.

    For such a $C$ and for any subvarieties $W_1,W_2$ of $X$ with $W_1$ closed and locally equidimensional, and $W_2$ 
    locally closed, set
    \begin{align*}
         C(W_1,W_2) & :=\{x \in W_2 | \, W_2 \; \mathit{smooth \;at \; x, \; and \; if} \; x \in W_1 \; \mathit{then} 
         \; (W_1,x)\supset (W_2,x) \; \mathit{and}\\ & \; C(W_1,W_2,x) \} \\
         \widetilde{C}(W_1,W_2) & := W_2 \setminus C(W_1,W_2)
    \end{align*}  

    The condition C is called \emph{stratifying} if for any such $W_1$ and $W_2$, $\widetilde{C}(W_1,W_2)$ is contained in a 
    nowhere dense closed subvariety of $W_2$. In fact, it suffices that this be so whenever $W_2$ is smooth, connected,
    and contained in $W_1$.

    A stratification is a \emph{$C$-stratification} if for any strata $X_\alpha$, $X_\beta$ with 
    $X_\beta \subset \overline{X_\alpha}$ it holds that $\widetilde{C}(\overline{X_\alpha},\overline{X_\beta})^{Zar}$
    is a union of strata.\\

    Going back to our case, it is true that Whitney's conditions are stratifying. The key point is to prove
    Given $Y\subset X$ as in section 4.1, the set of points of $Y$ where the couple $(X^\circ,Y)$ satisfies Whitney's 
    conditions contains the complement of a strict closed analytic subspace of $Y$. For a proof of this see [Te2] and
    [Whi1].\\

    We will now state two fundamental theorems concerning Whitney's conditions, the first of which was proved by
    Whitney himself and the second one by R. Thom and J. Mather. The proofs can be found in [Whi1], and [Ma] 
    respectively.

\begin{theorem}
    Let $M$ be a reduced complex analytic space and let $X\subset M$ be a locally closed analytic subspace of $M$.
    Then, there exists a Whitney stratification of $M=\bigcup M_\alpha$ such that:
    \begin{itemize}
     \item[i)] $X$ is a union of strata. 
     \item[ii)] If $M_\beta \cap \overline{M_\alpha} \neq \emptyset$ then $M_\beta \subset \overline{M_\alpha}$.
     \item[iii)] If $M_\beta \subset \overline{M_\alpha}$ the pair $(\overline{M_\alpha}^\circ,M_\beta)$ satisfies
                 Whitney's conditions at every point of $M_\beta$.
    \end{itemize}  
\end{theorem}
       
\begin{theorem}\label{T-M}\textnormal{(Thom-Mather)}$\;$\\
    Let $X=\bigcup M_\alpha$ be a Whitney stratification, and let $x \in X$ such that $X_\alpha$ is the stratum
    that contains $x$. Then, for any local embedding $(X,x)\subset (\C^n,0)$ there exists a local retraction
    $r:(\C^n,0)\to (X_\alpha,x)$ and a real number $\epsilon_0>0$ such that for all $0<\epsilon < \epsilon_0$ 
    there exists $\eta_\epsilon$ such that for any $0<\eta < \eta_\epsilon$ there is a homeomorphism h
    \[\xymatrix{B_\epsilon \cap r^{-1}(B_\eta \cap X_\alpha) \ar[rr]^h \ar[ddr]^r  & &
                (r^{-1}(x)\cap B_\epsilon) \times (X_\alpha \cap B_\eta)\ar[ddl]^{pr_2} \\ \\
              & X_\alpha \cap B_\eta & }\]
    compatible with the retraction $r$, and inducing for each stratum $X_\beta$ such that $\overline{X_\beta}
    \supset X_\alpha$ a homeomorphism 
    \[\overline{X_\beta}\cap B_\epsilon \cap r^{-1}(B_\eta \cap X_\alpha) \longrightarrow
               (\overline{X_\beta}\cap r^{-1}(x)\cap B_\epsilon)\times (X\alpha \cap B_\eta)\]
    where $B_\epsilon$ denotes the ball in $\C^n$ with center in the origin and radius $\epsilon$.
\end{theorem}

    In short, each $\overline{X_\beta}$, or if you prefere, the stratified set $X$, is locally topologically trivial
    along $X_\alpha$ in $x$. A natural question arises then, is the converse to the Thom-Mather theorem true? That
    is, does local topological triviality implies Whitney? 

    The answer is \emph{NO}, in [BS] Briancon and Speder showed that the family of surface germs
    \[z^5+ty^6z+y^7x+x^{15}=0\]
    (each member, for small t, having an isolated singularity at the origin) is locally topologically trivial, 
    but not Whitney.\\

    However there is a converse, proved by L\^e and Teissier (see [L-T3], and [T2]). Let's refere to the 
    conclusion of the Thom-Mather theorem, as the condition $(TT)$ (local topological triviality), so we can restate theorem 
    \ref{T-M} as Whitney implies $(TT)$. Let's recall the notations of the theorem and let $d_\alpha=dim \; X_\alpha$. We say 
    that a stratification satisfies the condition $(TT)^*$ (local topological triviality for the general sections) if in 
    addition to the condition $(TT)$, for every $x \in X_\alpha$, there exists for every $k> dim \; X_\alpha$ a Zariski open
    set $\Omega$ in $G(n-d_\alpha,k-d_\alpha)$ such that for any non-singular space $E$ containing $X_\alpha$ and such
    that $T_xE \in \Omega$, the intersection $\overline{X_\beta}\cap E$ satisfies $(TT)$ for all $X_\beta$ such that
    $\overline{X_\beta}\supset X_\alpha$. 

\begin{theorem}\textnormal{(L\^e-Teissier)}$\;$\\
    For a stratification  $X=\bigcup X_\alpha$ of a complex analytic space $X$, the following conditions are equivalent:
    \begin{itemize}
          \item [1)] $X=\bigcup X_\alpha$ is a Whitney stratification.
	  \item [2)] $X=\bigcup X_\alpha$ satifies condition $(TT)^*$.
    \end{itemize}	    
\end{theorem}

\subsection{Whitney's stratifications and polar varieties}    

     We now have all the ingredients, so it's time to put them all together. Let's fix a nonsingular subspace 
     $Y \subset X$ through $0$ of dimension $t$ as before, recall that we are assuming $X$ is a reduced, pure dimensional analytic 
     space of dimension $d$. Let's recall the notations of section 3.1  and take a look at the normal/conormal diagram:

     \[\xymatrix{E_YC(X)\ar[r]^{\hat{e}_Y}\ar[dd]^{\kappa'}\ar[ddr]^\xi & C(X)\ar[dd]^\kappa
             \\
             & &  \\
             E_YX\ar[r]_{e_Y}  & Y\subset X }\]

     Remember that $C(X)\subset X \times \check{\P}^{n-1}$, $E_YC(X)$ is the blowup of $\kappa^{-1}(Y)$ in $C(X)$
     and by identifying it with the fibered product we can consider $E_YC(X)\subset X\times \P^{n-1-t}\times \check
     {\P}^{n-1}$ and $E_YX$ is the blowup of $Y$ in $X$. 

     Looking at the definitions, one can prove that if we consider the divisor:
     \[D=|\xi^{-1}(Y)| \subset E_YC(X), \;\; D \subset Y \times \P^{n-1-t}\times \check {\P}^{n-1}\]
     we have that, denoting by $\check{\P}^{n-1-t}$ the space of hyperplanes containing $T_0Y$:

     \begin{itemize}
     \item[$\bullet$)] The couple $(X^\circ,Y)$ satisfies Whitney's condition a) at $0$ if and only if 
                       $\xi^{-1} (0) \subset \P^{n-1-t}\times \check {\P}^{n-1-t}$
     \item[$\bullet$)] The couple $(X^\circ,Y)$ satisfies Whitney's condition b) at $0$ if and only if
                       they satisfy condition a) and $\xi^{-1}(0)$ is contained in the incidence variety
		       $I\subset \P^{n-1-t}\times \check {\P}^{n-1-t}$.
     \end{itemize}

     So, in order to verify the Whitney conditions, it is important to control the geometry of the divisor
     $D \subset E_YC(X)$.\\

     \textbf{FALTA AQUI EXPLICAR COSAS ANTES DE ENUNCIAR EL TEOREMA}

     \begin{theorem}\label{prin}
            Given $0 \in Y \subset X$ as before, the following conditions are equivalent:
     \begin{itemize}
        \item[1)] The couple $(X^\circ,Y)$ satisfies Whitney's condition at $0$. 
	\item[2)] The local polar varieties $P_k(X,L)$ $0\leq k \leq d-1$ are equimultiple
	          along $Y$ (at $0$), for general $L$.
        \item[3)] $dim \; \xi^{-1}(0)=n-2-t$
     \end{itemize}	  
     \end{theorem}  

      Note that since $dim \; D=n-2$ condition 3) is open and the theorem implies that
      $(X^\circ,Y)$ satisfies Whitney's conditions at $0$ if and only if it satisfies Whitney's 
      conditions in a neighborhood of $0$.\\

      Note also that by semicontinuity of fiber dimension, condition 3) is satisfied outside of a closed
      analytic subspace of $Y$, which shows that Whitney's is a stratifying condition.\\

      Moreover, since a blowup does not lower dimension, the condition $dim \;\xi^{-1}(0)=n-2-t$ implies 
      $dim \; \kappa^{-1}(0)\leq n-2-t$. So that, in particular $\kappa^{-1}(0) \not\supset \check{P}^{n-1-t}$,
      where $\check{P}^{n-1-t}$ denotes as before the space of hyperplanes containing $T_0Y$. This tells us 
      that a general hyperplane containing $T_0Y$ is not a limit of tangent hyperplanes to $X$.\\

      Finally, a consequence of the theorem is that given a complex analytic space $X$, there is a unique
      coarsest Whitney stratification; any other Whitney stratification of $X$ is obtained by adding strata
      inside the strata of the coarsest one.\\

\subsection{Relative Duality}

      \textbf{FALTAN REFERENCIAS PARA TODA ESTA SECCION} 
 
      There still is another result which can be expressed in terms of the relative conormal space
      and as consequently of course, in terms of relative duality. What does we mean by this?\\

      Let $f:X\to X$ be a morfism of reduced analytic spaces such that there exists a closed
      subanalytic space $F\subset X$ where the restriction :
      \[f|_{X \setminus F} : X\setminus F \longrightarrow S\]
      has all of its fibers smooth and of dimension equal to dim $X-$ dim $S$. 
      For $x \in X\setminus F$, denote by $M_x$ the submanifold $f^{-1}(f(x))\subset X\setminus F$ and 
      now using this submanifolds we will build the \textbf{conormal space of X relative to f}, denoted
      by $C_f(X)$, by setting
      \[N_x^*M_x = \{ \xi \in T_x^*\C^n | \xi (v)=0, \; \forall v \in T_xM_x\}\] and
      \[T^*_{X\setminus F}\C^n =\{ (x,\xi) \in T^*\C^n | x \in X\setminus F, \; \xi \in N_x^*M_x\} \] 
      finally taking the closure of $T^*_{X\setminus F}\C^n$ in $T^*\C^n$ and projectivizing.\\

      Going back to our normal-conormal diagram:

      \[\xymatrix{E_YC(X)\ar[r]^{\hat{e}_Y}\ar[dd]^{\kappa'}\ar[ddr]^\xi & C(X)\ar[dd]^\kappa
             \\
             & &  \\
             E_YX\ar[r]_{e_Y}  & Y\subset X }\]

      Consider the irreducible components $D_\alpha\subset Y\times \P^{n-1-t} \times \check{\P}^{n-1}$ of \newline
      $D=|\xi^{-1}(Y)|$, that is $D=\bigcup D_\alpha$, and its images:
      \begin{align*}
      V_\alpha & = \kappa'(D_\alpha)\subset Y \times \P^{n-1-t}\\
      W_\alpha  & = \hat{e}_Y (D_\alpha)\subset Y \times \check{\P}^{n-1}
      \end{align*}	
      Then it can be proved that the equivalent statements of theorem \ref{prin} are also equivalent to:\\

      For each $\alpha$, $D_\alpha$ is the relative conormal space of its image $V_\alpha$ with respect
      to the canonical analytic projection $C_{X,Y}\to Y$ restricted to $V_\alpha$, and all the fibers
      of the restriction $\xi:D_\alpha \to Y$ have the same dimension near $0$.\\

      In particular, Whitney's conditions are equivalent to the equidimensionality of the fibers and a relative
      duality:
      \[\xymatrix{ D_\alpha \ar[r]\ar[d] &   W_\alpha = Y-dual \:of\:  V_\alpha\\
                   V_\alpha   & }\]

      The proof is similar to the case $Y={0}$ and the condition $dim \; \xi^{-1}(0)=n-2-t$ gives exactly
      what is needed for $D_\alpha$ to be $Y$-lagrangian. As a final result, we will state a corollary of
      this fact.

      \begin{corollary} Using the notations above we have:
      \begin{itemize}
            \item[1)] The couple $(X^\circ,Y)$ satisfies Whitney's conditions at $0$ is and only if for each
	              $\alpha$ the dimension of the fibers of $W_\alpha \to Y$ is locally constant near $0$.	      
 	    \item[2)] The couple $(X^\circ,Y)$ satisfies Whitney's conditions at $0$ is and only if for each
	              $\alpha$ the dimension of the fibers of $V_\alpha \to Y$ is locally constant near $0$. 
      \end{itemize}	      
      \end{corollary} 	
      
      
 %    It must be noted that the multiplicities of the polar varieties depend only on the analytic type of 
 %    $(X,x)$ and not on the embedding. 

%-----------------------------------------------
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\end{document}





