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\markboth{\rm Plane branches and toric maps}
{\rm Rebecca Goldin and Bernard Teissier}
\title{Resolving singularities of plane analytic branches with one toric morphism}
\author{ Rebecca Goldin and Bernard Teissier}
\date{}
\begin{document}
\maketitle
%-----------------------------------------------
\hfill { To Heisuke Hironaka}\par\noindent
\bigskip 
%\footnote{A.M.S 1991 Classification: 32S45, 32S30}{}}
\begin{abstract}
\noindent Let $(C,0)$ be an irreducible germ of complex plane curve. Let $\Gamma\subset \N$ be
the semigroup associated to it and $\cgam\subset \C^{g+1}$ the corresponding
monomial curve, where $g$ is the number of Puiseux exponents of $(C,0)$. We
show, using the specialization of $(C,0)$ to $(\cgam ,0)$, that the same toric morphisms
$Z(\Sigma)\to \C^{g+1}$ which induce an embedded resolution of singularities of $(\cgam ,0)$
also resolve the singularities of
$(C,0)\subset (\C^{g+1},0)$, the embedding being defined by elements of the analytic algebra ${\cal O}_{C,0}$
whose valuations generate the semigroup $\Gamma$.\par\vskip.2truecm
\end{abstract}

%-----------------------------------------------

\section{Introduction}
\textcolor{red}{Some corrections to the original text are made in red.}\par

In the last few years Mark Spivakovsky has proposed a program to prove the
resolution
of singularities of excellent schemes. A part of this program is a new
look at Zariski's local uniformisation theorem for arbitrary valuations. A
fundamental
object of study in this approach is the graded ring associated to the
filtration of a local
domain $R$ naturally provided by a valuation of $R$.

In the special case where $R$ is the local analytic algebra $\O$ of a plane
branch $C$,
with its unique valuation, this graded ring was studied in detail by Monique Lejeune-Jalabert and the second author (see [L-T], [T1]), as a special case of the
$\overline{\hbox{\rm gr}}_IR$ appearing in their study of the ''$\overline\nu_I$-filtration'' of a ring (roughly, by the integral closures of
the fractional powers of an ideal $I$ of $R$). \par\noindent
Recall that one calls {\it branch} a germ of analytically irreducible excellent curve. We will in
this paper deal only with complex analytic branches.

The following facts which appeared at that time in that special case may be
relevant to the understanding of the role of the graded ring in the general case:

-- The graded ring is the ring of the monomial curve $\cgam$ with the same
semigroup $\Gamma$ as the given plane branch (see definitions below).

-- The generators of the semigroup are the intersection numbers with the
given plane
branch $C$ of a transversal non-singular germ $x=0$ and of plane branches
$f_j(x,y)=0$ with
a smaller number $j\ ,\ 0\leq j\leq g-1$, of Puiseux exponents and having
with $C$ maximal
contact in the sense defined by M. Lejeune-Jalabert (see [Z], pp.16-17,
[L-J]). In particular
the initial forms in the graded ring of the images in the algebra $\O$ of
the equations of
these branches generate it as a $\C$-algebra.

-- There exists a one parameter deformation of $\cgam$ having all its
fibers except the
special one isomorphic to $C$, and this deformation is equisingular in the
sense that it
has a simulateous resolution of singularities by normalization; the
normalization of the
total space of the family is non-singular and induces normalization for
each fiber (see [T1]).

We may therefore hope that a process of resolution for $\cgam$ will
induce resolution for $C$. In this paper we show that the curve $\cgam$ can be
resolved by a single toric modification of its ambient space $\C^{g+1}$ (where
$g$ is the number of characteristic Puiseux exponents of the plane branch $C$) 
and that \textcolor{red}{some of the toric modifications which resolve $\cgam$ also resolve} the curve $C$ if we view it as 
embedded in $\C^{g+1}$ by $g+1$ elements of its maximal ideal whose valuations 
generate the semigroup $\Gamma$. This is shown by a generalization of the usual non
degeneracy argument, which can be found also in [O2], [O3]. We need more details than
in these papers because we want to show that the toric modification resolves not only
$\cgam$ but also $C$.

From this follows the rather interesting fact that \it any plane branch with
$g$ Puiseux exponents can be embedded in  $\C^{g+1}$ in such a way as to be
resolved by one single toric modification, i.e., it is
in some sense non-degenerate with respect to its Newton polyhedron\rm . 
 Moreover, if we denote by $\pi\colon Z\to
\C^{g+1}$ a toric map resolving $C$, and by $\pi\vert S'\colon S'\to S$ the
strict transform by this
toric map of a non-singular surface $S$ containing our plane branch $C$, we
can identify
$\pi\vert S'$ to the composition of $g$ toric modifications of non-singular
surfaces, thus
recovering the known fact that a plane branch can be resolved by a
composition of $g$ toric morphisms. The first non trivial example, the simplest
branch with two characteristic exponents, is computed in an Appendix.\par The
description of this composition of toric maps of surfaces obtained in this way
has the advantage over the ``static'' one of [O1] of showing explicitly its analytic
dependence on the coefficients of the equation or the
parametrization of the curve, in view of the results of [T1], and also of
giving a geometric vision of the relationship with the resolution process of the ``singular curves
with maximal contact'' of M. Lejeune-Jalabert, and probably also of the approximate roots \`a la
Abhyankar. Indeed, the embedding of the plane branch $C$ in $\C^{g+1}$ is obtained
by adding to the coordinates $x,y$ the images  in ${\cal O}$ of the equations of
plane branches with $<g$ characteristic exponents having maximal contact with $C$.  The
disadvantage is that the construction given here is for the time being restricted to irreducible
germs.

It is tempting to ask whether in general, \it given any germ of an
algebraic or analytic space and a valuation of its local algebra, the germ
can be embedded in an affine space in such a way as to be non-degenerate with
respect to its Newton polyhedron and to the given valuation, in the sense that
its strict transform is non-singular at the point specified by the valuation in a
toric modification of the ambient space subordinate to the
Newton polyhedron.\rm \par This would be an effective local uniformization
theorem and the results of this paper indicate that perhaps the specialization to
the graded algebra is the key to such a result. 

In the general case the graded algebra associated with a valuation is not
even N\oe therian, and as the reader will see, the proofs in this paper rely on
the extensive
knowledge we have of the structure of the semigroup of a plane branch, a
knowledge
essentially lacking in dimension $\geq 3$ (the case of dimension 2 is
currently under study). In the paper ``Valuations, Deformations, and toric Geometry '' following
this one the second author begins to prepare the way for a proof of the local uniformization
theorem along these lines. It is worth noting that one of the results of [L-T] is that the graded
algebra associated with the
$\overline{\nu}_I$ filtration, i.e.,
$\overline{\hbox{\rm gr}}_IR$, is an $R$-algebra of finite type. Some very interesting
finiteness results on the graded algebra associated with a valuation and its relation with
Abhyankar's inequality have recently been obtained by O. Piltant ([P]).

This text is based on a lecture given at the singularities Seminar by the
second author in January 1994, where examples largely stood in for proofs. The
ideas are presented in the framework of complex geometry for simplicity, but the interested
reader will see how to transform this into a characteristic-blind resolution for
one-dimensional excellent henselian equicharacteristic local integral domains with an algebraically
closed residue field.\par We are grateful to Tadao Oda and Michel Vaqui\'e for pointing out some
errors and  imprecisions in preliminary versions of this text.

%-----------------------------------------------
\section{Puiseux expansion and the semigroup of a curve}
%-----------------------------------------------

\begin{definition}
Let $(C,0)\subset (\C^2,0)$ be a germ of an irreducible analytic curve (a
plane branch) defined by the
 equation $f(X,Y) = 0,\ \ f\in \C\{X,Y\}$.
We will call  $\O = \C\{X,Y\}/(f)$ the {\em local algebra of the curve}
$C$, and denote by ${\bf m}_{\cal O}$ its maximal ideal.
Note that $\O = \C\{x,y\}$ where $x$ and $y$ are the residue classes in
$\C\{X,Y\}/(f)$ of $X$ and
$Y$ respectively.

According to Newton and Puiseux, there is a parametric representation of
$C$, called the
{\em Newton-Puiseux expansion}, i.e an injection of the algebra $\O$ into 
$\C\{ t\}$
described in suitable coordinates $X,Y,t$ by:
$$                                        
\begin{array}{ccl}                        
 X &= & x(t) =  t^n  \mbox{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \
(*)}\\                    
 Y &= & y(t) = \sum_{i\geq m} a_i t^i \ \  \mbox{with}\ \mbox{m}>\mbox{n.}
\end{array}
$$
The Puiseux expansion of $x$ and $y$; since the two series converge for small
enough $\| t\|$, say $\| t\|<\epsilon$. Denoting by $\D (0,\epsilon)$ the disk with center $0$
and radius $\epsilon$ in $\C$,  the image of the map
$\D (0,\epsilon)\to
\C^2$ defined by $x(t),y(t)$ is a representative of the germ $(C,0)\subset (\C^2,0)$. From
the integer
$n$ and the exponents appearing in the expansion
of $y(t)$ one can extract the Puiseux characteristic exponents, which characterize
the topological type of a small representative of our branch. (see [Z]). In
particular, we may choose the coordinates $X,Y$ in such a way that the
exponent $m$ appearing in the expansion of $y(t)$ is not divisible by the
multiplicity $n$. It is then generally denoted by $\beta_1$. We shall do so
from
now on.
 More generally, any analytically irreducible germ of a curve has a
normalization isomorphic to
$\C\{t\}$, and its algebra is an analytic subalgebra of
$\C\{t\}$.
\end{definition}


Given this parametrization, certain properties of an analytic branch $C$,
plane or not, can
be described using the valuation on the algebra $\O$ induced via the
injection
$\O\hookrightarrow
\C\{t\}$ by the $t$-adic valuation of $\C\{t\}$. 
\begin{definition}
Given a branch $(C,0)$ and its algebra $\O\hookrightarrow \C\{t\}$, let $\nu$
be the $t$-adic valuation on $\C\{ t \}$.  We define the {\em semigroup}
$\Gamma$ of $\O$ to be
$$\Gamma = \{ \nu (\xi ) :\  \xi \in \O \setminus \{ 0\}\}.$$
We follow here the common usage, which is to denote by the same letter $\Gamma$ the
semigroup deprived of its zero element, i.e., $\{ \nu (\xi ) :\  \xi \in   {\bf
m}_{\cal O} \}$ We say that the branch $(C,0)$ {\em has the semigroup}
$\Gamma$.
\end{definition}

The semigroup $\Gamma$ is finitely generated and has finite complement in
$\N$ (see [Zariski,
II.1] for the case of plane branches), and if $(C,0)$ is a plane branch,
its minimal set of generators
$\{\b_0,\dots ,\b_g\}$ can be uniquely determined by:
\begin{enumerate}
        \item[(a)] $\b_0 = n$
        \item[(b)] $\b_i = \mbox{min}\{ z \in \Gamma \mid \ z\ \notin <\b_0, \dots
,\b_{i-1}>\}$
\end{enumerate}
where $<\b_0, \dots ,\b_{i-1}> \subset \N$ is the semigroup generated by
$\{\b_0,
\dots ,\b_{i-1}\}$ (in particular, $\b_1=\beta_1$).   We then have that
\begin{enumerate}
        \item $\{\b_0, \dots ,\b_g\}$ generates $\Gamma$,
        \item $\b_0 < \dots < \b_g$, and
        \item $\gcd(\b_0, \dots ,\b_g) = 1$
\end{enumerate}
We call the vector $(\b_0, \dots ,\b_g) \subset \Z^{g+1}$ the {\em weight
vector}.
Of course the datum of its semigroup $\Gamma$ does {\em not} determine a
branch up
to analytic isomorphism. In fact, the datum of the semigroup $\Gamma$ of a
plane branch is equivalent to the datum of its Puiseux characteristic
exponents
(see [Z]). Among all the branches, plane or not, having a given semigroup,
one may single
out the affine curve
$\cgam
\subset
\C^{g+1}$ defined by the parametrization:
\begin{equation}
\cgam :\  u_i = t^{\b_i} \ \ \ \ 0\leq i \leq g.   \label{parametrization}
\end{equation}
Consider $(\cgam , 0)$ as the germ of a curve.  Its algebra
$$
\O_{\cgam ,0} = \C\{ t^{\b_0}, \dots ,t^{\b_g}\}\subset \C\{t\}$$
clearly has semigroup $\Gamma$. It is shown in [T1] that all branches
having a given
semigroup $\Gamma$ are complex analytic deformations of $(\cgam , 0)$ and these
deformations are in some sense {\em equisingular}.

\begin{definition}
Given a branch $(C,0)$ and its semigroup $\Gamma$, we call the curve
$\cgam$ described above by the parametrization (\ref{parametrization}) the
{\em monomial curve} associated to $C$. If $(C,0)$ is a plane germ, the
monomial curve $\cgam$ is a complete intersection in $\C^{g+1}$, where $g$ is
the number of Puiseux characteristic exponents of $(C,0)$.
\end{definition}

%-----------------------------------------------
\section{Deforming Curves} \label{mini}
%-----------------------------------------------

The curve $\cgam$ mentioned above, as we shall see, is the center of a
finite-dimensional flat
family  of branches which contains (up to complex analytic isomorphism of germs) every
germ of analytically irreducible curve (or  branch) with semigroup $\Gamma$. \par
We begin by showing (see [T1]) that given any branch $(C,0)$, one can
construct explicitly a one-parameter analytic
deformation of
$(\cgam,0)$, whose general fibres are isomorphic to the curve $(C,0)$. Let
us take for
simplicity the case of a plane branch. Consider the parametric
representation of the curve
$C$ as in (*).  After a change of coordinates we may assume that it has the
form :
$$
\begin{array}{ccl}
x(t) &=& t^{\b_0} \\
y(t) &=& t^{\b_1} + \sum_{j>\b_1}c_j^{(1)}t^j.
\end{array}$$
Note that  $\b_0$ is the multiplicity $n$ of $C$.
By the definition of the $\b_i's$, there exist elements $\xi_i(t) \in \O =
\C\{ x,y \}\ ,\ 2\leq i\leq
g$ such that
$$
\xi_i(t) = t^{\b_i} + \sum_{j>\b_i}c_j^{(i)}t^j.
$$
where $\xi_0 = x(t)$.
Now consider the family, parametrized by $v$, of curves parame\-trized by
$t$, in $\C^{g+1}\times
\C$:
$$
\X (v)\left\{
\begin{array}{rcl}
x&=& t^{\b_0} \\
y &=& t^{\b_1} + \sum_{j>\b_1}c_j^{(1)}v^{j-\b_1}t^j\\
u_2 &=& t^{\b_2} + \sum_{j>\b_2}c_j^{(2)}v^{j-\b_2}t^j\\
\cdot&=&\cdot \\
\cdot&=&\cdot \\
\cdot&=&\cdot \\
u_g &=& t^{\b_g} + \sum_{j>\b_g}c_j^{(g)}v^{j-\b_g}t^j\ .\\
\end{array}
\right.
$$

%-----------------------------------------------

\begin{proposition} \label{specific_fibre}
The branch $(\X(0),0)$ is isomorphic to $(C^{\Gamma},0)$, and for any
$v_0\neq 0$, the
branch $(\X(v_0),0)$ is isomorphic to $(C,0)$ by the isomorphism
$u_i(t)\mapsto
v_0^{-\b_i}u_i(v_0t)$ (where we set $x=u_0,\ y=u_1$).
\end{proposition}

%-----------------------------------------------

\begin{proof}
For $v=0$ it is obvious, and for $v\neq 0$, since $\O$ is generated as an
analytic subalgebra of
$\C\{t\}$ by $x(t),y(t)$, it follows from the fact that the
$\xi_i(t)$ are in $\O$ (see [T1]). \end{proof}

%-----------------------------------------------

\noindent
The semigroups of plane branches are characterized by the following
arithmetical
properties (see [T1]); let us set $e_i=\hbox{\rm gcd}(\b_0,\b_1,\ldots
,\b_i),
e_{i-1}=n_ie_i$. Then
$$\begin{array}{lclclccr}
n_i\b_i\in &<\b_0,\ldots ,\b_{i-1}>\ \ \ \ \ \ \\
n_i\b_i<&\b_{i+1}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{array}$$
The first relation implies the existence of integers $\ell^{(i)}_j\ , 1\leq
i\leq g\ ,
0\leq j\leq i-1$, such that
$$n_i\b_i=\ell^{(i)}_0\b_0+\cdots +\ell^{(i)}_{i-1}\b_{i-1}$$
So that on the curve $\cgam$ as parametrized, we have equations:
$$
\begin{array}{lclclccr}
f_1 &=& u_1^{n_1} &-& u_0^{\ell_0^{(1)}}&=&0&\\
f_2 &=& u_2^{n_2} &-& u_0^{\ell_0^{(2)}}u_1^{\ell_1^{(2)}}&=&0&\\
\cdot&&&\cdot&&&\cdot&\mbox{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (**)}\\         
            
\cdot&&&\cdot&&&\cdot&\\
\cdot&&&\cdot&&&\cdot&\\
f_g &=& u_g^{n_g} &-& u_0^{\ell_0^{(g)}}\cdots
u_{g-1}^{\ell_{g-1}^{(g)}} &=&0.&       
\end{array}
$$
It is shown in [T1] that these equations define the curve $\cgam$, which is
therefore a complete
intersection in $\C^{g+1}$. Now using the theory of miniversal deformations (see [T3]),
we have:
\begin{theorem}\label{theorem1}{\em \bf [Teissier]}
There exists a germ of a flat morphism $$p\colon \ (\X_u,0) \rightarrow
(\C^{\tau_-}, 0)$$ 
endowed with a section $\sigma$ such that $(p^{-1}(0), 0)$ is analytically
isomorphic to
$(\cgam, 0)$, and for any representative of the germ of the
morphism $p$ and any branch $(C,0)$ with semigroup $\Gamma$, there exists
 $v_C\in \C^{\tau_-}$ such that
$(p^{-1}(v_C),\sigma(v_C))$ is analytically isomorphic to $(C,0)$.
Moreover,
$(\X_u,0)$ is embedded in $(\C^{g+1}\times\C^{\tau_-},0)$ in such a way
that $p$ is induced by the second projection.
\end{theorem}

%-----------------------------------------------

\begin{proof}  See [T1]; this is the miniversal constant semigroup
deformation of $\cgam$, and it
corresponds to the part of the base of the miniversal (equivariant)
deformation of $\cgam$ which
is spanned by the coordinates with negative weight. It is shown in [T1],
4.4 that $\tau_-\leq \#
(\N\setminus \Gamma)$. \par More precisely the miniversal constant
semigroup
deformation of $\cgam $ is a map
$p\colon \X_u\to
\C^{\tau_-}$, where $\X_u$ is embedded in $\C^{g+1}\times \C^{\tau_-}$
in such a way that $p$ is induced by the second projection, and defined by
the equations
$$
\begin{array}{lclcr}
F_1 &=& f_1 + \sum_{r=1}^{\tau_-} v_r\phi_{r,1}(u_0,\dots,u_g) &=& 0\\
\cdot&&\cdot &=& \cdot\\
\cdot&&\cdot &.& \cdot\\
\cdot&&\cdot &.& \cdot\\
F_g &=& f_g + \sum_{r=1}^{\tau_-} v_r\phi_{r,g}(u_0,\dots,u_g) &=& 0
\end{array}
$$
where the $\phi_{r,j}$ are polynomials and each monomial in
$(u_0,\dots,u_g)$ appearing in
$\phi_{r,j}$ is of weight $>n_j\b_j$ when each $u_k$ is given the weight
$\b_k$; the equation
$f_j$ is then homogeneous of weight $n_j\b_j$. In fact one may choose the
vectors
$\phi_{r}$ to have only one nonzero coordinate which is a monomial
$\phi_{r,j}$. Moreover, the images vectors $\phi_1, \ldots ,\phi_{\tau_-}$ in
$\C[C^\Gamma ]^{g+1}/N$, where $N$ is a certain jacobian submodule of $\C[C^\Gamma ]^{g+1}$,
are linearly independant over $\C$.
\par The basic property of the miniversal deformation is that for any deformation  $d\colon
({\cal Y},y)\rightarrow (S,0)$ of the germ
$(\cgam ,0)$ such that all the fibers
have a singular point with semigroup $\Gamma$ there exists an analytic map
germ
$h\colon (S,0)\rightarrow  (\C^{\tau_-},0)$ such that $d$ is isomorphic to
the deformation
obtained from $p$ by pull-back by $h$. Moreover, since $\cgam$ is quasi-homogeneous, it is
also the case for $p$ in the sense that there are weights on the $v_i$ such
that the equations
$F_j$ are homogeneous of degree $n_j\b_j$. These weights are negative,
which is the reason for
the notation $\C^{\tau_-}$; in fact the weight $w_r$ of $v_r$ is $n_j\b_j$
minus the weight of
$\phi_{r,j}$, this difference being independent of $j$. \par 
The recipe to find vectors $\phi_r$ making the deformation miniversal is
due to J. Mather; it is
the following (see [T1]): Consider the ring of algebraic functions $\ \C[\cgam]$ on the
monomial
curve, and the submodule
$N$ of  $\C[\cgam]^g$ generated by the vectors $\partial_{u_i}f$ where
$f=(f_1,\ldots ,f_g)$. The quotient $ \C[\cgam]^g/N$ is a finite
dimensional vector space over $\C$,
and so we may choose vectors $\psi_i\in \C[u_0,\ldots ,u_g]^g$, each having
a single nonzero
coordinate which is a monomial, such that the natural images of the
$\psi_i$ in $ \C[\cgam]^g/N$
form a basis of this vector space. The $\phi_r$ are those among the $\psi_i
=(0,\ldots ,0,
u^{q_{i,j}},0,\ldots ,0) $ such that if $u^{q_{i,j}}$ is at the $j$-th
line, the weight of $u^{q_{i,j}}$ is
$>n_j\b_j$, so that in the corresponding deformation the only equation
modified,
$f_j+v_iu^{q_{i,j}}$ is modified by a term of weight greater than that of
$f_j$.\par  In
particular, the one-parameter specialization of a branch $C$ to $\cgam$
which we have
seen above can be obtained in this way, up to isomorphism, by a map
$h_1\colon (\C ,0)\to (\C^{\tau_-},0)$. The map $p$ is equivariant with
respect to the action of the
group
$\C^*$ on
$\X_u$ (resp
$\C^{\tau_-}$) which is described by $u_k\mapsto\lambda ^{\b_j}u_k,\
v_r\to\lambda^{w_r}v_r$
(resp. only the action on the $v_r$). If $v_C\in \C^{\tau_-}$ is a point
corresponding to a branch
isomorphic to $(C,0)$, the image of $h_1$ is contained in the orbit of
$v_C$ under this action of
$\C^*$. The action of
$\C^*$ on $\C^{\tau_-}$ ensures that any branch with semigroup $\Gamma$
appears up to
isomorphism as a fiber in any representative of the germ $p\colon
(\X_u,0)\to
(\C^{\tau_-},0)$.\end{proof}

%-----------------------------------------------
\section{Resolution Using Toric Morphisms}
%-----------------------------------------------

We produce  {\em toric morphisms} $\pi(\Sigma)\colon Z(\Sigma) \rightarrow
\C^{g+1}$ to
resolve the singularities of $\cgam \subset \C^{g+1}$, and then show that
the same morphisms
which resolves
$\cgam$ resolve any fibre of the miniversal constant semigroup deformation
$p\colon
\X_u\rightarrow\C^{\tau_-}$. In particular, since all curves with semigroup
$\Gamma$ are represented as fibers of this deformation, Theorem
\ref{theorem1} implies that
$\pi(\Sigma)$ resolves $(C,0)$. The toric resolution of
the curve $\cgam$ will be a consequence of a generalization of the work of
Varchenko [V]
(expounded also in [M]) for ``convenient" or ``commode'' functions, whose
Newton polyhedron
intersects each axis and which are non-degenerate.

%-----------------------------------------------
\subsection{The Toric Morphism}
%-----------------------------------------------

For the notions and basic results of toric geometry used in this section, we refer to
[C].  A toric morphism is locally described by monomial maps
$\pi(a)\colon\C^{g+1}\rightarrow\C^{g+1} $ where $a=(a^0,\dots,a^g)$
with $a^j\in\N_0^{g+1}$. for all
$j$ and $\hbox{\rm span}\{a^0, \ldots ,a^g\} = \R^{g+1}$:
$$
\begin{array}{llrcl}
u_0 &=& {y_0}^{a_{0}^0} \cdot\cdot\cdot\cdot {y_g}^{a_{0}^g} \\
 u_1 &=& {y_0}^{a_{1}^0} \cdot\cdot\cdot\cdot {y_g}^{a_{1}^g} \\
\cdot &&\cdot && \\
\cdot &&\cdot && \\
\cdot &&\cdot && \\
 u_g &=& {y_0}^{a_{g}^0} \cdot\cdot\cdot\cdot {y_g}^{a_{g}^g}
\end{array}
$$
More precisely, a fan (see [Oda], Chap. 1) $\Sigma$ with support $
\R_+^{g+1}$ is a decomposition
of the positive quadrant $ \R_+^{g+1}$ into rational simplicial cones
$\sigma_\alpha$ with the
properties that any face of such a cone is also a part of the fan and that
the intersection of two of
them is a face of each. A fan is \it non-singular \rm if the primitive
integral vectors of
the 1-skeleton of each cone $\sigma$ of dimension $g+1$ form a basis of the
integral lattice
$\N_0^{g+1}$. We have set $\N_0=\N\cup\{0\}$.\par To a fan $\Sigma$ are associated a toric variety
$Z(\Sigma) $ and a toric
(equivariant) map
$\pi (\Sigma)\colon Z(\Sigma) \to \C^{g+1}$. The variety $Z(\Sigma)$ is 
obtained by glueing up affine varieties $Z(\sigma)$ corresponding to each cone of
maximum dimension, and if the fan is non-singular, so
is $Z(\Sigma)$; in this case, for each cone $\sigma$ of maximal dimension
$Z(\Sigma)$ is
isomorphic to $\C^{g+1}$ and the map
$\pi (\sigma)$ induced by $\pi (\Sigma)$ is equal to $\pi(a^0,\dots,a^g)$ where the
$a^j$ are the primitive integral vectors of the 1-skeleton of the cone $\sigma$.
An upper convex map (see [C], [Oda] A.3 p. 182-185) $m\colon
\R_+^{g+1} \to \R_+$ taking integral values on the integral points and
linear in each cone
$\sigma\in \Sigma$ determines a divisor on $Z(\Sigma)$, which in the chart
$\pi (a^0,\dots,a^g) \colon Z(\Sigma)\to \C^{g+1}$ has the equation
$$y_0^{m(a^0)}\ldots y_g^{m(a^g)}=0.$$
We will call it the exceptional divisor of the toric map $\pi (\Sigma)$
corresponding to the upper convex function $m$. Just like in the absolute case, the
datum of such a map is equivalent, in the case where the function $m$ is \it strictly convex,
\rm  to the datum of an embedding $Z(\Sigma)\subset \C^{g+1}\times \P^N$
such that the exceptional divisor is the pull back of a hyperplane section of 
$\P^N$. Such an upper
convex map is called a \it support function \rm .\par The \it Newton polyhedron
\rm of a function
$f\colon  \C^{g+1}\to \C$ can be used to define fans and a support
function.\par \begin{definition}
For a function $f=\sum_{p\in \N^{g+1}}f_pu^p$, let ${\rm supp}\ f = \{p \in
\N^{g+1}: f_p
\neq 0\}$ and
${\cal N}_+(f) = \mbox{\em boundary of the convex hull of }(\{{\rm supp}\ f\} +
\R_+^{g+1})\mbox{ in}\ \R_+^{g+1}$.
 We call ${\cal N}_+(f)$
the {\em Newton polyhedron} of the function $f$. Note that ${\cal N}_+(f)$
has finitely many
compact faces and that its non-compact faces of dimension $\leq g$ are
parallel to coordinate
hyperplanes.
\end{definition}
\noindent
Define the function $m$ by:
$$
m(q) = \mathop{\inf}_{p \in {\cal N}_+(f)} <q, p>
$$
and define an equivalence relation: two vectors are equivalent if and
only if the same elements of ${\cal N}_+(f)$ minimize the inner product
with each vector.
In other words,
$$
\begin{array}{rcl}
q &\sim& q' \\   
&\Updownarrow & \\
\{p \in {\cal N}_+(f) \mid <q,p> = m(q)\} &=&
\{p \in {\cal N}_+(f) \mid <q',p> = m(q')\}\ \ \ \mbox{         (***)}
\end{array}
$$
The function $m$ is homogenous and piecewise linear, thus the equivalence
classes (***)
form a fan; to each class (and corresponding set of $p \in
{\cal N}_+(f)$), we associate a cone $\sigma$ which
consists of those vectors whose inner product is minimized by (and only by)
these $p$.  We obtain
a convex rational fan $\Sigma_0$ in $\R_+^{g+1}$ with vertex $0$. By
construction, the function $m$ is linear on each cone $\sigma \in \Sigma$, and it is
easy to verify that it is strictly upper convex. The non-compact faces of the boundary ${\cal
N}(f)$ of ${\cal N}_+(f)$ correspond to cones $\sigma$ in the fan which are contained in
a coordinate hyperplane of $\R^{g+1}$.  Moreover, given a subset $J\subset \{0,\ldots
,g\}$, we can consider the projection of the Newton polyhedron to the subspace
$\R^J=\{p\in
\R^{g+1}\mid p_k=0\ \hbox{\rm for}\ k\notin J\}$ of $\R^{g+1}$. If that
projection contains the origin, then for all $a\in \check\R^{g+1}\mid a_k=0\ 
\hbox{\rm for}\ k\notin J$, we have $m(a)=0$, and therefore we may
assume that the cone $\R_+^J$ is in the fan, so that the basis vectors of
$\R_+^J$ are in the 1-skeleton of the fan. \par By a theorem of Kempf,
Mumford, et al. (see [TE], pp. 32-35, [Oda], p.23), any fan can be refined
into a non-singular fan, still containing as faces the spaces $\R_+^J$ such
that $m(a)=0$ for $a\in \R_+^J$. The function $m$ is of course linear in
each cone of the finer fan and upper convex (but not strictly so in general), and we are in the
situation described above.\par\medskip\noindent If we now consider several functions $f_j;
1\leq j\leq k$, to each of them corresponds a Newton polyhedron, and
therefore a fan $\Sigma_0^{(j)}$ and a support function $m_j$. 
\begin{definition} Let $\Sigma_0$ be the fan consisting of the intersections of the
cones of the fans $\Sigma_0^{(j)}$. The fan $\Sigma_0$ is the least fine common
refinement of all the $\Sigma^{(j)}_0$.\end{definition} It is also the fan associated in the
manner we have just seen to a single Newton polyhedron, which is the
{\em Minkowski sum} of the Newton polyhedra of the $f_j$ and is
also the Newton polyhedron of the product $f_1\ldots f_k$.\par Recall
that the Minkowski sum ${\cal N}_1+{\cal N}_2$ of two Newton polyhedra
${\cal N}_1,\ {\cal N}_2$ is the convex domain spanned by vector sums
$\{p_1+p_2\ \mid p_1 \in {\cal N}_1,\ p_2
\in{\cal N}_2\}$. It is a commutative and associative operation. If ${\cal
N}_k$ is the Newton polyhedron of $f_k$, ${\cal N}_1+{\cal N}_2$ is the Newton
polyhedron of $f_1f_2$.
\par All the functions $m_j$ are linear in each cone of $\Sigma_0$. Taking a
non-singular refinement $\Sigma$ of $\Sigma_0$ as described above, we finally have a
non-singular fan with support functions $m_j$ and thus, since this fan is also a
refinement of the ``trivial'' fan of $\R_+^{g+1}$, we get a proper toric map
$\pi (\Sigma)\colon Z(\Sigma)\to \C^{g+1}$ where $Z(\Sigma)$ is a
non singular toric variety, and each function $f_j\circ\pi (\Sigma)$ defines a
map from $Z(\Sigma)$ to $\C$.\par

To study the effect of the modification $\pi(\Sigma)$ on the functions
$f_j$, we restrict
ourselves to a chart $Z(\sigma)$ of $Z(\Sigma)$ corresponding to a cone
$\sigma$ whose primitive
integral vectors are denoted by
$a=(a^0,\dots,a^g)$. We assume that they form a basis of the integral
lattice, and we shall write $\sigma =<a^0,\ldots
,a^g>$ for the convex cone spanned by the vectors $(a^0,\ldots ,a^g)$. If we write the
function  $$
f_j = \sum_{p \in \N^{g+1}} f^{(j)}_p u^p\ ,
$$
where
$ u^p = u_0^{p_0}u_1^{p_1}\cdots u_g^{p_g},$
then the composition
\begin{equation}
f_j \circ \pi(a^0,\dots,a^g) = \sum_{p \in \N^{g+1}} f_p^{(j)}
y_0^{<a^0,p>}
\cdots y_g^{<a^g,p>}, \label{comp} 
\end{equation}
where $<,>$ is the standard inner product, can be written
$$\begin{array}{ccl} f_j \circ \pi(a)&=&y_0^{m_j(a^0)}\ldots
y_g^{m_j(a^g)}\sum_{p \in
\N^{g+1}} f_p^{(j)} y_0^{<a^0,p>-m_j(a^0)}
\cdots y_g^{<a^g,p>-m_j(a^g)}\\&=&y_0^{m_j(a^0)}\ldots y_g^{m_j(a^g)}\tilde
{f_j}(y_0,\ldots
,y_g)\ .\\
\end{array}
$$ The function $\tilde{f_j}$ is called the \it strict transform \rm of
$f_j$; it satisfies
$\tilde{f_j}(0)\neq 0$.\par
We use the following fact, which is obvious from the
definitions:\par\noindent
 For each $j,\ 1\leq j\leq k$, there is a unique $p^{(j)}\in {\cal
N}_+(f_j)$ such that
$<a^i,p^{(j)}>=m_j(a^i),\ 0\leq i\leq g$, and the strict transforms take
the form
$$\tilde f_j=f^{(j)}_{p^{(j)}}+\sum_{p \in {\cal
N}_+(f_j)\setminus\{p^{(j)}\}} f_p^{(j)}
y_0^{<a^0,p>-m_j(a^0)}
\cdots y_g^{<a^g,p>-m_j(a^g)}.$$
Moreover we can compute the critical locus of $\hbox{\rm crit} \pi (\Sigma)$ of $\pi
(\Sigma)$, which locally is the critical locus of $\pi(\sigma)$ for each
$\sigma=<a^0,\dots,a^g>\in \Sigma$.\par\noindent By direct inspection we find the
following relation for the jacobian matrix $\hbox{\rm jac}\pi
(\sigma)$ of $\pi (\sigma)$:
$$y_0\cdots y_g\  \hbox{\rm jac}\pi (\sigma)=u_0\ldots u_g\ \hbox{\rm
det}(a^0,\ldots , a^g),$$ 
setting $\alpha_j=\big(\sum_{i=0}^ga^j_i\big)-1$ this means since $\hbox{\rm
det}(a^0,\ldots , a^g)=\pm 1$,
$$\hbox{\rm jac}\pi (\sigma)=\pm y_0^{\alpha_0}\ldots y_g^{\alpha_g}.$$
So the divisor $y_j=0$ is contained in the critical locus of $\pi (\sigma)$
if and only if $a^j$ is not
a coordinate vector.\par\medskip\noindent
Note that the divisor $y_j=0$ is contained in $\pi (\sigma)^{-1}(0)$ if
and only if the vector
$a^j$ has all its coordinates different from zero.

%-----------------------------------------------
\section{Existence of a toric resolution}
%-----------------------------------------------

In this section, we study a germ at $0$ of a complete
intersection $X\subset \C^{g+1}$  defined by a set of equations $\{f_1 = f_2 = \dots = f_k = 0\}$
with $f_j\in \C\{u_0,\ldots ,u_g\},\ j=1,\ldots k$. We assume that the series $f_j$ have no
constant term. We use the notations introduced above and consider a toric map of
non-singular spaces
$\pi (\Sigma)\colon Z(\Sigma )\to \C^{g+1}$. We make our computations in a
chart corresponding to a regular simplicial cone $\sigma=<a^0,\ldots
,a^g>$ of $\Sigma$.

%-----------------------------------------------

\begin{definition} \rm Given a family of functions $\{f_j\}_{1\leq j \leq k}$
as above, and a toric morphism associated to a regular fan $\Sigma$ such that all the
support functions $(m_j)_{1\leq j \leq k}$ are linear in each cone $\sigma$ of
$\Sigma$, for each map $\pi (\sigma)\colon \C^{g+1}(\sigma)\to\C^{g+1}$, we
will call the divisor $y_0^{m(a^0)}\ldots y_g^{m(a^g)}$, where
$m(a)=\sum_{j=1}^km_j(a)$, the {\em toric exceptional divisor of $\pi (\sigma)$
associated to the support functions $m_j$.} This definition globalizes to $\pi
(\Sigma)$.\rm \end{definition} 

%-----------------------------------------------

From what we have seen one deduces for each $\pi (\sigma)$ with $\sigma =<a^0, 
\ldots ,a^g>$ the following statements:

\begin{proposition}\label{fiber} The fiber $\pi (\sigma)^{-1}(0)\subset \C^{g+1}$ is the
union of the intersections $y_{i_1}=\cdots =y_{i_t}=0$ over all minimal
families $J=(i_1,\ldots ,i_t)$ of indices such that for each $\ell \ ,0\leq
\ell \leq g$, there is
an $i\in J$ such that the $\ell$-th coordinate of $a^i$ is $\neq 0$. In
particular the divisorial
part of $\pi (\sigma)^{-1}(0)$ is the union of the divisors
$y_i=0$ for those  $i$ such that the vector $a^i$ has no zero
component. 

\noindent The critical locus $\hbox{\rm crit}(\pi (\sigma))$ 
is the union of the divisors $y_s=0$ for
those $s$ such that $a^s$ is not a coordinate vector of
$\Z^{g+1}$. \end{proposition}
%
%-----------------------------------------------
%
\noindent For any set of functions
$\{f_j\}$, we label the compact faces of ${\cal N}_+(f_j)$ by $\gamma_j$.  Any
compact face of the Newton polyhedron ${\cal N}_+=\sum_{j=1}^k {\cal
N}_+(f_j)$ which is the Minkowski sum of the Newton polyhedra of the
$\{f_j\}$
 will be of the form
$\gamma = \gamma_1 +\cdots+\gamma_k$. Each face $\gamma_j$ in turn is of
the form
$$\gamma_j=\gamma_j(I)=\{p\in {\cal N}_+(f_j)\mid <a^h,p>=m_j(a^h)\ \hbox{\rm for}\ h\in
I\}\ ,$$ where $I$ is a subset of $\{0,\ldots ,g\}$. If the face $\gamma_j(I)$
is not compact, it contains a line parallel to a coordinate axis, say the $\ell^{th}$, so
that there is an $\ell$, $0\leq \ell \leq g$, such that for all $h\in I$ we have
$a^h_\ell=0$. 

%-----------------------------------------------

\begin{definition} \label{non-deg}  We call the set of functions
$\{f_j\}_{1\leq j\leq k}$ {\em non-degenerate} if for all compact faces 
$\gamma = \gamma_1 + \cdots + \gamma_k$ of ${\cal N}_+=
\sum_{j=1}^k {\cal N}_+(f_j)$, denoting by $f_j|_{\gamma_j}$, the sum
$$f_j|_{\gamma_j}=\sum_{p\in \gamma_j}f^{(j)}_pu^p,$$ the $k \times (g+1)$
matrix $$
\left(
        \begin{array}{ccccc}
                {\partial_{u_0}f_1|_{\gamma_1}} & \cdot&\cdot&\cdot
                         & {\partial_{u_g} f_1|_{\gamma_1}} \\
                {\partial_{u_0} f_2|_{\gamma_2}} & \cdot&\cdot&\cdot
                         & {\partial_{u_g} f_2|_{\gamma_2}} \\
                \cdot&&&&\cdot\\
                \cdot&&&&\cdot\\
                \cdot&&&&\cdot\\
               {\partial_{u_0} f_k|_{\gamma_k}} & \cdot&\cdot&\cdot
                         & {\partial_{u_g} f_k|_{\gamma_k}}        
        \end{array}
\right)
$$
has maximal rank $k$ on $(\C^*)^{g+1}$. This means  that the equations
$$f_1|_{\gamma_1}=\cdots = f_k|_{\gamma_k}=0$$
define a non singular complete intersection in $(\C^*)^{g+1}$. For
example  a single function $f$ is non-degenerate if
for each compact face $\gamma$ of ${\cal N}_+(f)$, the $(g+1)$-vector
$\left({\partial_{u_i}
f|_{\gamma}}\right)_i$ does not vanish in $(\C^*)^{g+1}$; this is the
original definition of Varchenko [V]. \end{definition}

%-----------------------------------------------

\begin{definition}
The morphism $\pi: \tilde{X} \longrightarrow X$ is a {\em resolution} if
the following conditions
hold:
\begin{itemize}
\item $\pi$ is a proper morphism,
\item $\tilde{X}$ is non-singular, and
\item $\tilde{X}\setminus\pi^{-1}({\rm Sing} X) \longrightarrow X \setminus{\rm
Sing} X$ is an isomorphism.
\end{itemize}
\end{definition}
%-----------------------------------------------
%
\noindent If the morphism $\pi$ is a toric morphism $\pi(\Sigma)$, the last condition
is a consequence of the inclusion
$$
\tilde{X} \cap {\rm crit}(\pi(\Sigma)) \subset \pi(\Sigma)^{-1}({\rm Sing}
X).$$
 However, this condition is difficult to check if $X$ is not itself a toric
subvariety, which motivates the following definition of a toric pseudo-resolution,
after recalling that the strict transform $\tilde X$ of $X\subset \C^{g+1}$ by $\pi
(\Sigma)$ is the closure in $Z(\Sigma )$
of $\pi(\Sigma)^{-1}(X)\setminus {\rm crit}(\pi(\Sigma))$.

%-----------------------------------------------

\begin{definition}
A toric morphism $\pi(\Sigma)\colon Z(\Sigma)\to \C^{g+1}$ is a {\em toric
(embedded) pseudo-resolution} of a subvariety $X\subset \C^{g+1}$ if the strict
transform
$\tilde X$ of $X$ by $\pi(\Sigma)$ is smooth and
transversal to a stratification of the critical locus of $\pi(\Sigma)$.
\end{definition}
Note that a toric pseudo-resolution is not necessarily a resolution of singularities in
the usual sense since it may not induce an isomorphism
$$\tilde X\setminus \pi(\Sigma)^{-1}(\hbox{\rm Sing} X)\to X\setminus
\hbox{\rm Sing}X.$$
A toric pseudo-resolution only induces an isomorphism $$\tilde X\setminus
{\rm crit}(\pi(\Sigma))\to X\setminus {\rm disc}(\pi(\Sigma)),$$ where
${\rm disc}(\pi(\Sigma))$ is the image by $\pi(\Sigma)$ of ${\rm crit}(\pi(\Sigma))$,
and since $\tilde X$ is non singular, the inclusion $\hbox{\rm
Sing}X\subset {\rm disc}(\pi(\Sigma))$ holds. In the case of a single function $f$,
Varchenko introduced in [V] the condition of being \it commode \rm which is
equivalent to asking that if a primitive vector $a^i$ of a cone $\sigma$ (of a fan
compatible with the Newton polyhedron of $f$) is contained in a hyperplane (which
means that if in the chart $\C^{g+1}(\sigma)$ the divisor $y_i=0$ is not contained in
$\pi (\sigma)^{-1}(0)$), then we have $m(a^i)=0$; it follows then from the conditions
satisfied by our fans that $a^i$ must in fact be a basis vector, so that $y_i=0$ is not in
the critical locus. For commode functions, in fact, the toric exceptional divisor and
the critical locus of $\pi (\Sigma)$ both coincide set theoretically with
$\pi(\Sigma)^{-1}(0)$ so that a toric pseudo-resolution of $f(u_0,\ldots ,u_g)=0$ is
also a resolution in the usual sense. \par For complete intersections, Oka introduced in
[O3] a notion of
\it convenient
\rm generalizing Varchenko's definition. It is much too strong for our purposes. Our
monomial curve is only 1-convenient in the sense of [O3], and the product
of its equations is far from being commode in the sense of [V]. \par However, in the
case of a curve, a toric embedded pseudo-resolution is an embedded
resolution in the usual sense unless the curve is contained in a coordinate hyperplane.
\vfill\eject
%----------------------------------------------- 
\subsection{The Inverse Image of $X$ by the Morphism $\pi(\Sigma)$.} \par 
%-----------------------------------------------

\begin{theorem} \label{adherence} If the set of functions $\{f_j\}_{1\leq j
\leq k}$ defining the complete intersection
$X\subset \C^{g+1}$ is non-degenerate at the origin, there exists a neighborhood $U$ of $0$ in $\C^{g+1}$
 such that in $\pi(\Sigma)^{-1}(U)$ the strict transform of $X$ by
$\pi(\Sigma)$ is non-singular and transversal in $Z(\Sigma )$ to the strata
of a stratification of the
divisor $\pi(\Sigma)^{-1}(0)$ .
\end{theorem}

%-----------------------------------------------

\begin{proof} We consider for each $\pi (\sigma)$ a natural stratification
of $\C^{g+1}(\sigma)$ such that $\pi
(\sigma)^{-1}(0)$ is a union of strata, as follows: For each $I\subset \{0,\ldots ,g\}$,
define $S_I$ to be the constructible subset of
$\C^{g+1}(\sigma)$ defined by $y_i=0\ \hbox{\rm for}\ i\in I,\ y_i\neq 0\ 
\hbox{\rm for}\ \ i\notin I$. The sets $S_I$ form a partition of $\C^{g+1}(\sigma)$ by
non-singular varieties.\par
If the subset $I$ is such that for each $\ell \ ,0\leq \ell \leq g$, there is
a $j\in I$ such that the $\ell$-th coordinate of $a^j$ is $\neq 0$ (see
\ref{fiber}), the stratum $S_I$ is contained in $\pi(\sigma )^{-1}(0)$, and conversely,
so that $\pi(\sigma )^{-1}(0)$ is a union of strata. This stratification of each
$\C^{g+1}(\sigma)$ is compatible with the chart decomposition of $Z(\Sigma)$ and so
gives a stratification of $Z(\Sigma)$. Moreover, this stratification satisfies the
Whitney conditions, so that to check the transversality of a non-singular subspace to
every stratum in a neighborhood it is sufficient to check transversality to the strata
contained in $\pi(\sigma )^{-1}(0)$; as we have seen, these strata correspond to
compact faces of the Newton polyhedra. Now we can compute in each chart
$\C^{g+1}(\sigma)$ the restriction of each $\tilde{f_j}$ to $S_I$: $$\tilde{f_j}\vert
_{S_I}=\sum_{p\mid <a^i,p>=m_j(a^i)\ \hbox{\rm for}\ i\in
I}f^{(j)}_py_{j_1}^{<a^{j_1},p>-m_j(a^{j_1})}\ldots
y_{j_\ell}^{<a^{j_\ell},p>-m_j(a^{j_\ell})},$$ where $\{j_1,\ldots ,j_\ell\}=\{0,\ldots
,g\}\setminus I$.\par\noindent The set $\{p\mid <a^i,p>=m_j(a^i)\ \hbox{\rm for}\ i\in
I\}$ is by definition a face $\gamma_{j,I}$ of the Newton polyhedron ${\cal
N}_+(f_j)$, and this face is compact by the assumption on $I$. \par
 The function $\tilde{f_j}\vert _{S_I}$ is the restriction to $S_I$ of a
function on $\C^{g+1}(\sigma)$ which is independent of the $y_i\ ;\ i\in I$.
Moreover, this last function differs from the composition with $\pi (\sigma)$ of the
function $$f_{j,\gamma_{j,I}}=\sum_{p\in \gamma_{j,I}}f^{(j)}_pu^p$$ by
multiplication by a monomial in $(y_0,\ldots ,y_g)$. Since on $S_I$ the coordinates
$y_k;\ k\notin I$ are $\neq 0$, the behavior of the functions $\tilde{f_j}\vert _{S_I}$
is determined by the behavior of the $f_{j,\gamma_{j,I}}\circ \pi(\sigma)$ at points
where all the $y_i$ are $\neq 0$.  It follows that if the jacobian determinant of the
functions $f_{1,\gamma_{1,I}},\ldots f_{k,\gamma_{k,I}}$ has a $k\times k$ minor
which does not vanish on $(\C^*)^{g+1}$ then the functions $\tilde{f_1}\vert
_{S_I},\ldots ,\tilde{f_k}\vert _{S_I}$ define a non-singular subspace of codimension
$k$ in $S_I$ (in particular it is empty if $k>\hbox{\rm dim}S_I$).  This implies that
the strict transform of $X$ by $\pi (\Sigma)$ is non-singular and transversal to all
the strata $S_I$ in a neighborhood of $\pi(\sigma )^{-1}(0)$; in particular it is
transversal to the critical locus of $\pi (\sigma)$.\end{proof}

Applying this to each chart of $\pi (\Sigma)$ gives: 

%-----------------------------------------------

\begin{corollary} If the set of functions $\{f_j\}_{1\leq j \leq
k}$ defining the complete intersection $X\subset \C^{g+1}$ is non-degenerate at $0$, there 
exists a neighborhood $U$ of $0$ in $\C^{g+1}$ such that the strict transform
$\tilde X\to X$ of $X$ by $\pi(\Sigma)$  induces in $\pi(\Sigma)^{-1}(U)$ an embedded toric 
pseudo-resolution of singularities of $X\cap U$.
\end{corollary}

%-----------------------------------------------
\subsection{The Toric Resolution of the Monomial Curve}
%-----------------------------------------------

Let $\cgam$ be the monomial curve derived from $C$. 
In this section, we study the embedded resolution of the curve $\cgam$ by a toric morphism from two
different viewpoints, which correspond to its parametric and equational presentations
respectively.

 First, the monomial curve is the closure in $\C^{g+1}$ of an orbit of $\C^*$
described by $$t\mapsto (t^{\overline \beta_0},t^{\overline \beta_1},\ldots ,t^{\overline \beta_g}).$$ By the
theory of toric varieties (see [O], Chap 1) this orbit is described combinatorially by the linear map $\Z\to
\Z^{g+1}$ such that the image of $1$ is the \it weight vector \rm 
$$w=(\overline \beta_0,\overline \beta_1,\ldots ,\overline \beta_g)\in \Z^{g+1}.$$
This means that the corresponding morphism of semigroup algebras 
$$ \C[u_0,\ldots ,u_g] \to \C[t]$$ obtained from the dual map $\check\Z^{g+1}\to \check \Z$ by restriction to
$\N^{g+1}$ is the morphism of algebras which corresponds to our parametrization of 
the monomial curve.

 We use the notations of \S 4.
Now if we take a regular fan $\Sigma$ in $\R_+^{g+1}$ which is compatible with the weight vector $w$,
in the sense that $w$ is contained in an edge of any simplicial cone it meets outside $0$, we obtain a toric
map
$$\pi (\Sigma )\colon Z(\Sigma )\to \C^{g+1}.$$ To compute the strict transform of $\cgam$ by $\pi (\Sigma
)$, we need to find out first which charts $Z(\sigma )$ of $Z(\Sigma )$, corresponding to $(g+1)$-dimensional simplicial
cones $\sigma$ of $\Sigma$, have an image which contains $\cgam$. Seeking solutions of the form
$y_i(t)=c_it^{\alpha_i}+\cdots
$ to the equations $y_0^{a_i^0}.\ldots .y_g^{a_i^g}=t^{\overline \beta_i}$, we see that the vector
$(\alpha_0,\ldots ,\alpha_g)$ must satisfy $M.\alpha=w$ where $M=(a^j_i)$ is the matrix of the
column vectors $(a^0,\ldots ,a^g)$ which are the generators of the simplicial cone $\sigma$. Since
we want the $\alpha$'s to be positive, this implies that $\sigma$ contains $w$, so that by
our assumption that the fan is compatible with $w$, the vector $w$ must be equal to one of the
$a^j$. We see that we need to consider only those simplicial cones $\sigma=~<~a^0,\ldots ,a^g>$,
where up to reordering we have $w=a^0$. Then,
$w$ is the image of the coordinate vector $(1,0,\ldots ,0)$ by the linear map $p(\sigma )\colon \Z^{g+1}\to
\Z^{g+1}$ described by the matrix $M$, i.e., sending the $i$-th basis vector to $a^i$. The strict transform of
$\cgam$ by the map $\pi (\sigma )$ is again a monomial curve, but its weight vector is the pull back of $w$
by $p(\sigma )$, which is the basis vector $(1,0,\ldots ,0)$; in fact, the simplicial cone $\sigma$ is the image
by $p(\sigma )$ of the cone generated by the basis vectors. So the strict transform of $\cgam$ has the
parametric representation
$$y_0=t,y_1=\cdots =y_g=1,$$
and it is indeed non singular and transversal to the exceptional divisor.

%-----------------------------------------------

\begin{remark} 1) The regular simplicial fan compatible with $w$ is
by no means unique; however, there are, at least in the case of plane branches,
algorithms which produce such fans. Their description is beyond the intent of this
text, but the reader can get some idea of the problem by looking at the example which
concludes it.\par\noindent  2) This resolution process works for any monomial curve:
the embedded toric resolution reduces entirely to the combinatorial problem of
finding a regular simplicial fan $\Sigma $ in
$\R_+^{g+1}$ compatible with the weight vector. We can summarize this in:\end{remark}

%-----------------------------------------------

\begin{theorem} Let $\cgam\subset \C^{g+1}$ be a monomial curve with weight vector $w\in \N^{g+1}$. For
any regular simplicial fan $\Sigma$ in $\R_+^{g+1}$ compatible with $w$, the toric map $\pi (\Sigma)\colon Z(\Sigma )\to
\C^{g+1}$ has the property that the strict transform of $\cgam$ is non singular and transverse to the
exceptional divisor. In fact, it appears only in the charts corresponding to simplicial cones
containing $w$ and there it is defined parametrically, up to reordering of the coordinates, by
$y_0=t, y_1=\cdots =y_g=1$. \hfill\qedsymbol\end{theorem} 

%-----------------------------------------------

 This construction of the resolution is simpler than the following one, but not as easily adaptable
to the study of the effect of the toric map $\pi (\Sigma )$ on deformations of $\cgam$ such as our
curve $C\subset\C^{g+1}$. The other viewpoint is to explore the implications of the
previous section on
$\cgam$. We reprove the result above from this viewpoint.  As specified in
Equations (**), $\cgam$ is defined by equations $\{f_j=0\}_{1\leq j\leq g}$, where each
$f_j = u_j^\alpha - M_j$ and $M_j$ is a monomial
in $u_0,\dots,u_{j-1}$.  Note that ${\rm supp}f_j$ has exactly two
elements, corresponding
to the exponents of the two monomials in $f_j$, and each Newton polyhedron ${\cal N}_+(f_j)$ has only one
compact face, which
is a segment.
  For each function $f_j$, let $\Sigma_0^{(j)}$ be the fan representing the
equivalence classes derived from the function $m_j$.  Let $\Sigma$ be a
non-singular fan obtained
by refining $\Sigma_0 = \mathop{\bigcap}_j\Sigma_0^{(j)}$, and ${\cal N}_+$
be the
corresponding Newton polyhedron.  We refer to the compact face of the
Newton
polyhedron ${\cal N}_+(f_j)$ as $\gamma_{j}$.

%-----------------------------------------------

\begin{proposition}\label{away}
The curve defined by $\{\tilde{f_1} = \tilde{f_2} = \dots = \tilde{f_g} =
0\}$
intersects the divisor $\pi(\sigma)^{-1}(0)$ only in the charts 
$\C^{g+1}(\sigma)$, $\sigma
=<a^0,\dots ,a^g>$, in which for some $b$ the vector $a^b$ is the weight
vector $(\b_0,\dots
,\b_g)$. If $a^b$ is the weight vector, then $(\Pi_{i=0}^gy_i=0 )
\bigcap \{\tilde{f_1} = \tilde{f_2} = \dots = \tilde{f_k} = 0\}$ is
contained in
$\{y_b=0\}$ but not in any other hyperplane $y_i=0$ for $i\neq b$.
\end{proposition}

%-----------------------------------------------

\begin{proof} Each strict transform $\tilde{f_j}$ has only two
terms 
$$\tilde{f_j}=f^{(j)}_{p^{(j)}_0}+f^{(j)}_{p^{(j)}_1}y_0^{<a^0,p^{(j)}_1>-m_
j(a^0)}\ldots
y_g^{<a^g,p^{(j)}_1>-m_j(a^g)}$$ since each equation $f_j$ has only two
terms. The only possibility
for the $g$ equations $\tilde{f_j}=0$ to have a common root with $y_i=0$
for some $i$ is
that
$<a^i,p^{(j)}_1>=m_j(a^i)=<a^i,p^{(j)}_0>$. But this implies that $a^i$ is
constant on the
Minkowski sum of the
$g$ segments constituting the Newton polyhedra of the $f_j$. By the
structure of the equations,
this sum is of dimension $g$; $a^i$ is uniquely determined as the primitive
normal vector of
this face, and it has to be the weight vector $a^b$.\par\noindent
 Indeed, the monomial map $\pi (\sigma)\colon\ Z(\sigma)\to \C^{g+1}$ maps
the strict
transform
$\tilde\cgam$ to $\cgam$; if the coordinate $y_b$ corresponds to the weight
vector, we see
that
$y_b$ must be a coordinate on the strict transform of $\cgam$, and all
other coordinates
equal to $1$. In other words, the equations in $Z(\sigma)$ of the
desingularization of
$\cgam$ are $(y_i=1,\ \hbox{\rm for}\ i\neq b)$; they are the strict
transforms of the
equations of $\cgam$, which have only $\pm 1$'s as coefficients.\end{proof}

%-----------------------------------------------
 
\begin{theorem}
Let $\cgam$ be a monomial curve defined by $\{f_j\}_{1\leq j\leq g}$ as
above. The strict transform of $\{f_1 = \cdots = f_g =0\}\cap (\C^*)^{g+1}$ by the
morphism $\pi(\Sigma)$ is non-singular and transversal in $Z(\Sigma )$ to the strata
of $\pi(\Sigma)^{-1}(0)$.
\end{theorem}

%-----------------------------------------------

\begin{proof} This follows directly from Theorem \ref{adherence} if we can
prove the fact that the
family of functions
$\{f_j\}_{1\leq j\leq g}$ is non-degenerate. \par
But there is only one compact face for the Newton polyhedron
$\sum_{j=1}^g{\cal N}_+(f_j)$; it is
the (Minkowski) sum $\sum_{j=1}^g\gamma_{j}$. By Proposition \ref{away}, this
compact face
corresponds to the subset $I=\{j\}\subset \{0,\ldots ,g\}$ such that $a^j$ is the weight vector.
Moreover, for each
$j$, the function $f_j$ is equal to $f_{j,\gamma_j}$, so that we only have to check that
the equations of the monomial curve define a non-singular complete intersection in
$(\C^*)^{g+1}$ and we have
$$df_1\wedge \cdots \wedge df_g=n_1\ldots n_g u_1^{n_1-1}\ldots
u_g^{n_g-1}du_1\wedge \cdots \wedge du_g+\cdots$$ so the differential form does
not vanish outside of the coordinate hyperplanes: the set of equations $f_1,\ldots
,f_g$ is non-degenerate.\par\noindent
Note that unlike the first proof, this one encounters a (minor) difficulty if we work over a field of
characteristic dividing one of the $n_i$. 
\end{proof}

%-----------------------------------------------
\section{Simultaneous Resolution}
%-----------------------------------------------

We will now show that \textcolor{red}{some toric morphisms} not only resolves $\cgam$,
but simultaneously resolves all curves in the miniversal deformation with
constant semigroup $\X_u$ of $\cgam$ which we saw in section \ref{mini}.

\begin{definition}
Let $f: (X,0) \rightarrow (Y,0)$ be a flat map with reduced fibres and $Y$
reduced. 
We say (see [T2] ) that $f$ admits a {\em very weak simultaneous
resolution} if, for all
sufficiently small representatives, there exists a proper morphism $\pi:~
\tilde{X} \rightarrow X$
such that:
%-----------------------------------------------
\begin{enumerate}
\item The composition $q=f\circ \pi : \tilde{X} \rightarrow Y$ is an
analytic
submersion, i.e., $q$ is flat, and for all $y \in Y$, the fiber
$\tilde{X}(y) =
q^{-1}(y)$ is non-singular.
\item For all $y \in Y$, the induced morphism $ \tilde{X}(y) \rightarrow
X(y)$ is a
resolution of the singularities of $X(y)$.
\noindent
Let $Y_1$ be the image of a section $\sigma: Y \rightarrow X$.\end{enumerate}

%-----------------------------------------------

We say that $f$ admits a {\em weak simultaneous resolution along} $Y_1$
if it also satisfies the condition:

\begin{enumerate}
\item The morphism $q_{Y_1}: (\pi^{-1}(Y_1))_{\rm red}
\rightarrow Y$ induced by $q$ is locally topologically a fibration, in the sense that 
every point
$\tilde{x} \in
\pi^{-1}(Y_1)$, has an open neighborhood $U \in (\pi^{-1}(Y_1))_{\rm red}$
such
that $U _{Y_1}\simeq V \times \tilde{Y}_0$, where $V$ is an open set in
$Y$, $\tilde{Y}_0$ is an
open neighbourhood of $\tilde{x}$ in the fibre of $q_{Y_1}$ passing through
$\tilde{x}$, and $\simeq$ is a $Y$-homeomorphism.\end{enumerate}

%-----------------------------------------------

We say that $f$ admits a {\em strong simultaneous resolution along} $Y_1$
if it also satisfies the condition:
\item The morphism $q_{Y_1}: (\pi^{-1}(Y_1))
\rightarrow Y$ induced by $q$ is locally analytically trivial (before reduction), in the sense that 
every point
$\tilde{x} \in
\pi^{-1}(Y_1)$, has an open neighborhood $U \in (\pi^{-1}(Y_1))$
such
that $U _{Y_1}\simeq V \times \tilde{Y}_0$, where $V$ is an open set in
$Y$, $\tilde{Y}_0$ is an
open neighbourhood of $\tilde{x}$ in the fibre of $q_{Y_1}$ passing through
$\tilde{x}$, and $\simeq$ is a $Y$-isomorphism.

%-----------------------------------------------

An {\em embedded resolution} for $\X\subset \C^N$ is a bimeromorphic map $\pi
\colon Z\rightarrow
\C^N$ where $Z$ is smooth, the exceptional locus $E$ of $\pi$ is a divisor with normal crossings in
$Z$ and the strict transform of $\X$ is smooth and transversal to the canonical stratification of $E$.
As usual, $\C^N$ stands for an open subset of $\C^N$.
\end{definition}
%
%-----------------------------------------------
%
 Recall from Section \ref{mini} that the miniversal deformation of the
curve $\cgam$ yields a
family of curves, $p\colon \X_u\to
\C^{\tau_-}$, embedded in $\C^{g+1}\times \C^{\tau_-}$ in such a way that
$p$ is induced by the
second projection, and where $\X_u$ is defined by the equations
$$
\begin{array}{lclcr}
F_1 &=& f_1 + \sum_{r=1}^{\tau_-} v_r\phi_{r,1}(u_0,\dots,u_g) &=& 0\\
\cdot&&\cdot && \cdot\\
\cdot&&\cdot && \cdot\\
\cdot&&\cdot && \cdot\\
F_g &=& f_g + \sum_{r=1}^{\tau_-} v_r\phi_{r,g}(u_0,\dots,u_g) &=& 0
\end{array}
$$
where the $\phi_{r,j}$ are polynomials and each monomial in
$(u_0,\dots,u_g)$ appearing in
$\phi_{r,j}$ is of weight $>n_j\b_j$ when each $u_k$ is given the weight
$\b_k$; the equation
$f_j$ is then homogeneous of weight $n_j\b_j$. We have $\X_u(0) \simeq
\cgam$.   Let $\pi (\Sigma )\colon Z(\Sigma )\rightarrow
\C^{g+1}$ be
the toric modification associated to a regular fan in $\R_+^{g+1}$ compatible with
the sum of the
Newton polyhedra of the equations $f_j$ of $\cgam$. By construction,
$\cgam$ is resolved
by the morphism
$\pi(\Sigma)$. 

%-----------------------------------------------

\begin{theorem}\label{toricres}
If $\Sigma$ is a regular fan compatible with the Newton polyhedra of the
equations $(**)$ of
$\cgam$, the morphism
$$\Pi(\Sigma)=\pi (\Sigma)\times \hbox{\rm Id}_{\C^{\tau_-}}\colon
Z(\Sigma)\times
\C^{\tau_-}\rightarrow
\C^{g+1}\times
\C^{\tau_-}$$ induces by restriction to the strict transform $\tilde\X_u$ a
resolution of
singularities of
$\X_u$, which is a strong simultaneous resolution for $p\colon
\X_u\rightarrow \C^{\tau_-}$ with
respect to the subspace
$Y_1=\{0\}\times
\C^{\tau_-}$.  In addition, it is an embedded resolution for $\X\subset \C^{g+1}\times
\C^{\tau_-}$.
\end{theorem}

%-----------------------------------------------

\begin{proof}  Consider in a chart $\C^{g+1}(\sigma)\times \C^{\tau_-}$ the
composition
$$F_j \circ \Pi (\sigma) = f_j \circ \Pi(\sigma) +  \sum_{r=1}^{\tau_-}
v_r(\phi_{r,j}\circ \pi(\sigma)).$$
Since for each support function $m(a)$ and fixed $q$ the function
$<a,q>/m(a)$ is a continuous
function of $a$, and since the $\phi_{j,r}$ are polynomials, after possibly
refining the fan
$\Sigma$ \textcolor{red}{(this is where we have to impose conditions on the toric morphisms which resolve $\cgam$)}, we may assume that for each cone $\sigma =<a^b,a^1,\ldots ,a^g>$ containing the
weight vector $a^b$, for each $j,\ 1\leq j\leq g$, and for each monomial
appearing in one of the
polynomials $\phi_{j,r}$, the inequality $ <a^b,q>\  > m_j(a^b)$, which is
equivalent to the weight
inequalities mentioned above, implies the inequalities $ <a^s,q>\  \geq
m_j(a^s)\ \mbox{for}\ 1\leq s\leq g$. We can therefore
write the composition $F_j\circ \Pi (\sigma )$ as follows:
$$F_j\circ \Pi (\sigma )
= y_0^{m_j(a^0)}\ldots
y_g^{m_j(a^g)}(\tilde{f_j}+\sum_{r=1}^{\tau_-} v_r\tilde\phi_{r,j}),$$
where
$\tilde\phi_{r,j}=\sum_q\tilde\phi_{r,j,q}y_0^{<a^b,q>-m_j(a^b)}\ldots
y_g^{<a^g,q>-m_j(a^g)}$. From this follows, since it is true for the
$\tilde{f_j}$, that at least for
small $\|v\|$ the
$\tilde{F_j}=\tilde{f_j}+\sum_{r=1}^{\tau_-} v_r\tilde\phi_{r,j}$ define a
non-singular complete
intersection $\tilde \X_u$ in $Z(\sigma )\times \C^{\tau_-}$: the strict
transform of $\X_u$;
moreover, each fiber over a point
$v\in
\C^{\tau_-}$ for sufficiently small $\|v\|$ is the resolution of the
corresponding fiber of $\X_u(v)$.
Let us now consider $\tilde Y_1=(\pi (\sigma )
\times \hbox{\rm Id}_{\C^{\tau_-}}\vert _{\tilde{\X_u}})^{-1}(Y_1) $. Its
equations are
$y_b=0, \tilde F_j=0$; by the Implicit Function theorem, it is non-singular
and admits at least
for small $\|v\|$ the coordinates $v_1,\ldots ,v_{\tau_-}$; the morphism
$\tilde Y_1\to Y_1$ is
not only a homeomorphism, but a local analytic isomorphism. Since there is
an equivariant
action of $\C^*$, all this is in fact true for all $v$. \par\noindent
Let us now consider a cone $\sigma$ not adjacent to the weight vector. Let
$u^q$ be a
monomial appearing as a $\phi_{r,j}$, set $G_j=f_j+v_ru^q$,
and let $K_j\subset
\{0,\ldots g\}$ be the set of those indices $s$ such that $<a^s,q>\ <m_j(a^s)$.
Set $n_j(a^s)=<a^s,q>$ for $s\in K_j$. Then 
$G_j\circ \Pi (\sigma )$ equals 
$$
\Pi_{s\notin K_j}y_s^{m_j(a^s)}\Pi_{s\in
K_j}y_s^{n_j(a^s)}\big(\tilde f_j\Pi_{s\in
K_j}y_s^{m_j(a^s)-n_j(a^s)}+v_r\Pi_{s\notin K_j}y_s^{<a^s,q>-m_j(a^s)}\big).
$$
By our choice of $\sigma$ we know that the $\tilde f_j$ do not vanish
together on the exceptional
divisor. The common zeroes of the strict transforms $$\tilde G_j=\tilde
f_j\Pi_{s\in
K_j}y_s^{m_j(a^s)-n_j(a^s)}+v_r\Pi_{s\notin K_j}y_s^{<a^s,q>-m_j(a^s)}$$
constitute in the chart
$\C^{g+1}(\sigma)\times \C^{\tau_-}$ the strict transform $\tilde\X_u$ of
the space $\X_u$,
therefore this strict transform meets the exceptional divisor only for
$v_r=0$ and then it coincides with the union of some components in
$\C^{g+1}(\sigma)\times \{0\}$ of the exceptional divisor
of $\pi (\sigma)$. However, these components do not meet the strict transform (see
\ref{away}) of the curve
$\cgam$, and therefore the inverse image of $\{0\}$ in $\tilde\X_u$ is not
connected. Since $\X_u$ is the total space of an equisingular deformation of $\cgam$
it is analytically irreducible at the origin and by Zariski's main theorem the inverse
image of $\{0\}$ must be connected. We therefore obtain a contradiction, which shows
that  for each cone $\sigma=<a^0,\ldots ,a^g> \in
\Sigma$ all the monomials $u^q$ appearing in the miniversal equisingular
deformation of
$\cgam$ satisfy the inequalities $<a^s,q>\ \geq m_j(a^s)\ \mbox{for}\ 0\leq
s\leq g$. We are in the situation described at the beginning of the proof, and
this shows that the map
$\tilde\X_u\rightarrow \X_u$ induced by
the toric modification $Z(\Sigma)\times \C^{\tau_-}\rightarrow \C^{g+1}\times
\C^{\tau_-}$ is a weak embedded
simultaneous resolution along $Y_1$. \par
Finally we remark that a refinement $\Sigma '$ of our original regular fan
corresponds to a
birational toric map $Z(\Sigma ')\rightarrow Z(\Sigma)$ which is an
isomorphism outside the
exceptional divisor. Moreover, in a chart corresponding to a cone $\sigma$
adjacent to the weight vector, it is an isomorphism outside $\Pi_{s=1}^gy_s=0$; this
open set contains the strict transform of $\cgam$ since we saw that it meets only
$y_0=0$ (see \ref{away}). It follows from
this that if the strict transform of $\X_u$ in $Z(\Sigma
')\times\C^{\tau_-}$ is a
simultaneous resolution of $\X_u$, such was already the case for the strict
transform in
$Z(\Sigma)\times \C^{\tau_-}$. This concludes the proof.\end{proof}

%-----------------------------------------------

\begin{corollary}\label{newemb} For a plane branch $(C,0)$ with $g$
characteristic
exponents, with semi group
$\Gamma=<\b_0,\ldots ,\b_g>$ and given parametrically by $x=x(t),\ y=y(t)$, with $\nu (x)=\b_0$, $\nu (y)=\b_1$,
let
$\xi_i (t)\in {\cal O}_{C,0}=\C\{x(t),y(t)\}\subset \C\{ t\}$ for $2\leq
i\leq g$ be
series such that $$(\nu (x),\nu (y), \nu (\xi_2),\ldots ,\nu
(\xi_g))=(\b_0,\ldots
,\b_g),$$ where $\nu$ is the $t$-adic valuation. Then the embedding
$(C,0)\to (\C^{g+1},0)$ given by $$u_0=x(t),u_1=y(t),u_2=\xi_2(t),\ldots
,u_g =\xi_g(t)$$ has the
property that its image is resolved by one toric
modification.\end{corollary}

%-----------------------------------------------
\section{The transforms of plane curves}
%-----------------------------------------------

Let us return to the construction of the miniversal constant semigroup
deformation in \S
\ref{mini}. It is easy to check that the vectors
$\phi_j\in \C[\cgam]^g
\ ,1\leq j\leq g-1,$ containing
$u_{j+1}$ at the
$j$-th line and zero elsewhere, are independent modulo $N$ with the
notations of \S 3. In the miniversal constant semigroup deformation,
the plane curves appear as
those for which the coefficients of all these vectors are $\neq 0$ (see
[T1]). For each point
$v_0$ in the corresponding open set of $\C^{\tau_-}$, the equation of the
corresponding
plane branch is obtained as follows, in a small neighborhood of $0$ :

One uses the first
equation $F_1$ to express
$u_2$ as a series in
$u_0,u_1$, using the implicit function theorem. Then one substitutes
this value in
$F_2$ and uses this equation to express $u_3$ as a function of $u_0,u_1$,
and so on. Finally
the equation
$F_{g-1}=0$ allows us to express $u_g$ as a series in $u_0,u_1$, and at
this point we have
the equations of a non-singular surface $S_{v_0}$ containing our branch,
thus explicitly
shown to be planar.

The next step is to consider the strict transform $\tilde S_{v_0}$ of
$S_{v_0}$ in
$\C^{g+1}\times \lbrace v_0\rbrace$ and the induced map $\tilde S_{v_0}\to
S_{v_0}$
induced by $\Pi(\Sigma)$. One shows that, in a neighborhood of the point of
the
exceptional divisor picked by the strict transform of $C$, it factors
through the toric map
$S^1_{v_0}\to S_{v_0}$ which on $S_{v_0}$ with the coordinates $u_0,u_1$
resolves the
singularities of the branch $u_1^{n_1}-u_0^{\ell^{(0)}_1}=0$. One then
takes the strict
transform of this equation as a new coordinate on $S^1_{v_0}$ and then the
strict
transform of $C$ appears as a branch on $S^1_{v_0}$ with one less
characteristic exponent;
its equations are the strict transforms of $F_2,\dots ,F_g$ on $S^1_{v_0}$,
in the new
coordinates. Again one must look at the toric map (in the new coordinates)
which resolves
the first characteristic pair and show that it is dominated by
$\tilde S_{v_0}$, and so on; finally we have factored the map $\tilde
S_{v_0}\to S_{v_0}$
into the composition of $g$ toric maps. This is similar to the process
described by
Spivakovsky in [S] of approximating a given valuation by a sequence of
monomial
valuations. However we see that if we allow a change in ambient space, the
uniformization
of a single monomial valuation, namely the
$t$-adic valuation on the monomial curve, gives the uniformization of the
$t$-adic
valuation of $C$.

On the surface $S$, each of the coordinates $u_i,\ 2\leq i\leq g$ is
expressed as a series in
$u_0,u_1$. For each $i,\ 2\leq i\leq g$, the equation $u_i=0$ on $S$
defines a curve with $i$ characteristic exponents having at the origin maximal contact
with our original plane branch; this is a consequence of what we have seen in Corollary
\ref{newemb}, and gives a geometric construction of curves (singular or not)
having maximal contact in the sense of [L-J] with our plane branch; they are the
curves on $S_{v_0}$ defined by the equations $u_i=0, \ i\geq 1$. Remark that for
$i\geq 2$ they can also be written $f_{i-1}(u_0,\dots ,u_g)+\hbox{\rm higher degree}$,
and with a little more work we find the structure of the approximate roots. 

The construction just outlined needs to be detailed but we will not do it here since
it is not part of the main purpose of this text. It is performed in the first non
trivial special case (two characteristic exponents $(3/2, 7/4) $, i.e., semigroup
$<4,6,13>$)  at the end of the example to which we now turn.

%-----------------------------------------------
\section{Appendix : An example}
%-----------------------------------------------

\begin{example}
Suppose we have a plane branch $C$ with semigroup $<4,6,13>$. 
The corresponding monomial curve $\cgam$ is described (see [T]) by the
polynomials
$$
\begin{array}{llrcl}
f_1 &=& {u_1}^2 - {u_0}^3 &=& 0\\
f_2 &=& {u_2}^2 - {u_0}^5u_1 &=& 0.
\end{array}
$$
As specified above, we construct ${\cal N}_+(f_1)$ and ${\cal N}_+(f_2)$,
the Newton
polygons, by the convex hull of the union of positive quadrants beginning
at points corresponding
to the exponents of the
$\{u_i\}$'s .  Thus,
$$
\begin{array}{rcl}
{\cal N}_+(f_1) &=& \mbox{convex hull of }\ \{\{(3,0,0),
(0,2,0)\}+\R_+^3\}\\
{\cal N}_+(f_2) &=& \mbox{convex hull of }\ \{\{(5,1,0), (0,0,2)\}+\R_+^3\}
\end{array}
$$
and the corresponding fans are defined, respectively, by
$$
\begin{array}{rcl}
\Sigma_0^{(1)} &=& \mbox{the first quadrant of}\ \R^3\mbox{
cut by the plane}\ 3x=2y,\mbox{and }\\
\Sigma_0^{(2)} &=& \mbox{the first quadrant of}\ \R^3 \mbox{ cut by
the plane}\ 5x + y =
2z.
\end{array}
$$
Then the equivalence class fan of $f_1$ and $f_2$ together is the
intersection
 $$\Sigma_0 = \Sigma^{(1)}_0 \cap \Sigma^{(2)}_0$$
which has four maximal dimension cones:
\begin{enumerate}
\item  $\langle (2,3,0),\ (0,1,0),\ (4,6,13),\ (0,2,1)\rangle$
\item  $\langle (0,2,1),\ (0,0,1)\ (4,6,13) \rangle$
\item  $\langle (2,0,5),\ (1,0,0),\ (2,3,0),\ (4,6,13)\rangle$
\item  $\langle (0,0,1),\ (2,0,5),\ (4,6,13)\rangle$

\end{enumerate} 
We build here by an inductive method a refinement which resolves
the fan $\Sigma_0 = \Sigma^{(1)}_0 \cap \Sigma^{(2)}_0.$  Notice that each
maximal dimension 
cone spans $\Z^3$ (since the matrix of
the spanning skeleton is unimodular).   For each cone, we have listed the
corresponding
morphism $\pi(\sigma)$ in terms of the resulting $\{u_i\}$, the composition
$f_i\circ\pi(\sigma)(y_0,y_1,y_2)$ for $i = 1,2$, and the exceptional
divisor
$\pi(\sigma)^{-1}(0)$. We show that the two surfaces are transverse to each
other
and to the exceptional divisor.
We start from a refinement for the fan in two dimensions associated to the
first equation
$u_1^2-u_0^3=0$. The weight vector is $(2,3)$ and from the geometric
interpretation of the continued fraction expansion we find that a regular
fan subdividing it is
composed of the following four 2-dimensional cones and their faces:\par
\begin{enumerate}
\item  $\sigma^{(2)}_1$ =$\langle (1,0),\ (1,1)\rangle$
\item  $\sigma^{(2)}_2$ =$\langle (1,1),\ (2,3)\rangle$
\item  $\sigma^{(2)}_3$ =$\langle (2,3),\ (1,2)\rangle$
\item  $\sigma^{(2)}_4$ = $\langle (1,2),\ (0,1)\rangle$
\end{enumerate}
Now we have to lift the vectors of the 1-skeleton to $\R^3$ in such a way
that they form the
1-skeleton of a regular fan subdividing $\Sigma_0$. It suffices to show
that in each cone of this
fan both functions $m_1$ and $m_2$ are linear, i.e., that all linear forms
take their minimum at a
vertex of the sum ${\cal N}$ of the Newton polyhedra of $f_1$ and $f_2$.
This Newton polyhedron
has four vertices: $(3,0,2),(8,1,0), (0,2,2), (5,3,0)$, so we are
especially interested in finding
four cones having the weight vector as one of their faces.\par Let us begin
by lifting the weight
vector
$(2,3)$ :
\par\noindent We seek integral vectors of the form $(2k,3k,z)$ with $k>0,\
z>0$; we know that
they take their minimum value on the compact segment of the Newton
polyhedron of $f_1$. On
${\cal N}_+(f_2)$ they take their minimum at $((5,1,0)$ if $13k\leq 2z$, at
$(0,0,2)$ if $13k\geq
2z$, at both if
$13k=2z$. Remembering that we seek primitive vectors, we find three natural
possibilities: $k=1,z=6$, giving the vector $a^1=(2,3,6)$, which takes its
minimum at
$(0,0,2)$, $k=1, z=7$ giving
$a^2= (2,3,7)$, which takes its minimum at $(5,1,0)$, and of course the
weight vector
$a^0=(4,6,13)$ itself. We remark that $a^0=a^1+a^2$.\par\noindent
Let us now try to lift $(1,1)$; we seek vectors of the form $(k,k,z)$; on
${\cal N}_+(f_1)$ they
take their minimum value at $(0,2,0)$, and on ${\cal N}_+(f_2)$ at
$(5,1,0)$ if $3k\leq z$, at
$(0,0,2)$ if $3k\geq z$. In this case we can take $k=1,z=3$, which gives
the vector $a^3=(1,1,3)$
which takes its minimum on the compact segment of ${\cal N}_+(f_2)$.
\par\noindent
Finally take $(1,2)$; we seek vectors $(k,2k,z)$ and there is one taking
its minimum on the
compact face of ${\cal N}_+(f_2)$: it corresponds to $k=1, z=3$, giving the
vector
${a'}^4=(1,2,3)$. We remark that $a^0=a^2+a^3+{a'}^4$, and it is therefore
tempting to take ${a'}^4$
as our fourth vector. However, it is not on the hyperplane $5x+y=2z$ so
that the support
function $m_2$ will \it not \rm be linear on a cone spanned by
$\{a^0,a^2,{a'}^4\}$. A better choice
is to take $a^4=a^0-a^3=(3,5,10)$.\par Now we check that the four cones
(which we have
decorated with the points of the Newton polyhedra of $f_1$ and $f_2$ where
the support
functions $m_1(a),\ m_2(a)$ take their minimum for $a\in \sigma_i$):
\begin{enumerate}
\item  $\sigma^{(3)}_1$ = $\langle a^0,a^1,a^3\rangle$; \ \ 
$p_0^{(1)}=(0,2,0);\ p_0^{(2)}=(0,0,2)$
\item  $\sigma^{(3)}_2$ =$\langle a^0,a^2,a^3\rangle$; \ \ 
$p_0^{(1)}=(0,2,0);\ p_0^{(2)}=(5,1,0)$
\item  $\sigma^{(3)}_3$ =$\langle a^0,a^2,a^4\rangle$; \ \ 
$p_0^{(1)}=(3,0,0);\ p_0^{(2)}=(5,1,0)$
\item  $\sigma^{(3)}_4$ =$\langle a^0,a^1,a^4\rangle$; \ \ 
$p_0^{(1)}=(3,0,0);\ p_0^{(2)}=(0,0,2)$
\end{enumerate}
are unimodular and such that their union is a neighborhood of $\R_+ a^0$ in
$\R^3$; the second
fact is obvious since the weight vector is in the interior of the convex
hull of the $\sigma_i\ ,
1\leq i\leq 4$, and to check the first it suffices to check that one of
them is unimodular.
 Now we can complete this subfan into a regular fan of $\R^3_+$, either by
the
same method or by invoking a general theorem, but we do not much care about
it, since we have
seen that the only charts where something interesting happens are those
which correspond to a
cone containing the weight vector in its 1-skeleton, and we have those
above. Moreover, 
the projections in $\R^2$ of the $\sigma_i$ form part of a regular subfan
of the
$\sigma^{(2)}_i$.\par
Let us now study the behavior of $f_1,f_2$ under the monomial maps
corresponding to the
$\sigma^{(3)}_i\ , 1\leq i\leq 4$. For economy we write $\sigma_i$ for
$\sigma^{(3)}_i$.

\begin{enumerate}
\item The cone $\sigma_1$ spanned by $\{(4,6,13),\ (2,3,6),\ (1,1,3) \}$
$$
\begin{array}{lcllcl}
\pi(\sigma_1):  &u_0 &=& {y_0}^4{y_1}^2{y_2}\\
      &u_1 &=& {y_0}^6{y_1}^3{y_2}\\
           &u_2 &=& {y_0}^{13}{y_1}^{6}{y_2}^3
\end{array}
$$
Then,
$$
\begin{array}{lcl}
f_1\circ\pi(\sigma_1)(y_0,y_1,y_2) &=&{y_0}^{12}{y_1}^6{y_2}^2(1-y_2) \\
f_2\circ\pi(\sigma_1)(y_0,y_1,y_2) &=&{y_0}^{26}{y_1}^{12}{y_2}^{6}(1-y_1)
\end{array}
$$
and
$$
\pi(\sigma_1)^{-1}(0) = \{y_0=0\}\cup\{y_1=0\}\cup\{y_2=0\}.
$$
Here the strict transforms are
$$
\begin{array}{lcl}
\tilde{f}_1 &=& 1-y_2\\
\tilde{f}_2 &=& 1-y_1.
\end{array}
$$
It is clear that the equations $\tilde{f}_1= \tilde{f}_2=0$ define a
non-singular complete
intersection meeting the exceptional divisor transversally at the point
$y_0=0,
y_1=y_2=1$.\par\noindent To make the same computation for the other charts
$\pi (\sigma_i)$ is
in fact superfluous, since by construction of the toric modification, we
will only observe the same
phenomenon in a different chart.\par
For verification's sake let us compute for $\pi (\sigma_3)$:\par
\item The cone $\sigma_3$  spanned by $\{(4,6,13),\ (2,3,7),\ (3,5,10) \}$
$$
\begin{array}{lcllcl}
\pi(\sigma_3):  &u_0 &=& {y_0}^4{y_1}^2{y_2}^3\\
&u_1 &=& {y_0}^6{y_1}^3{y_2}^5\\
&u_2 &=& {y_0}^{13}{y_1}^{7}{y_2}^{10}
\end{array}
$$
Then,
$$
\begin{array}{lcl}
f_1\circ\pi(\sigma_3)(y_0,y_1,y_2) &=&{y_0}^{12}{y_1}^6{y_2}^9(y_2-1) \\
f_2\circ\pi(\sigma_3)(y_0,y_1,y_2) &=&{y_0}^{26}{y_1}^{13}{y_2}^{20}(y_1-1)
\end{array}
$$
\end{enumerate}
And indeed it is the same situation viewed in another chart.
\vskip.5truecm
\noindent
The miniversal deformation with constant semigroup of $\cgam$ is computed
in [T1]: here
$\tau_-=2$ and $\X_u$ is defined in $\C^3\times \C^2$ by the equations
$$
\begin{array}{lcl}
F_1 &=&u_1^2-u_0^3+v_1u_2+v_2u_0u_2=0\\
F_2 &=&u_2^2-u_0^5u_1=0
\end{array}
$$
If we compose $F_1$ and $F_2$ with
$\Pi(\sigma_1)=\pi(\sigma_1)\times\hbox{\rm
Id}_{\C^2}\colon\ 
\C^{g+1}(\sigma_1)\times \C^2\to \C^{g+1}\times \C^2$ we get
$$
\begin{array}{lcl}
F_1\circ\Pi (\sigma_1)(y_0,y_1,y_2) &=&{y_0}^{12}{y_1}^6{y_2}^2(1-y_2
+v_1y_0y_2+v_2y_0^5y_1^2y_2^2)\\ F_2\circ\Pi (\sigma_1)(y_0,y_1,y_2)
&=&{y_0}^{26}{y_1}^{13}{y_2}^{20}(1-y_1)
\end{array}
$$
The strict transform of $\X_u$ is defined in this chart by 
$$
\begin{array}{lcl}
\tilde F_1(y_0,y_1,y_2) &=&1-y_2 +v_1y_0y_2+v_2y_0^5y_1^2y_2^2=0\\ 
\tilde F_2(y_0,y_1,y_2)
&=&1-y_1=0
\end{array}
$$
It is indeed a simultaneous resolution for $\X_u$.
There remains to check that the strict transform of $\X_u$ does not meet
the charts
corresponding to cones that are not adjacent to the weight vector. This
depends on our
construction of a regular fan and will not be done here.
\vskip.5truecm 
Let us now consider the plane branch with equations in $\C^3$ (for $v=v_0\neq
0$):
$$
\begin{array}{lcl}
 F_1(v_0;u_0,u_1,u_2) &=&u_1^2-u_0^3-v_0u_2=0\\ 
 F_2(u_0,u_1,u_2)&=&u_2^2-u_0^5u_1=0
\end{array}
$$
It lies on the non-singular surface $S_{v_0}$ with equation $F_1(v_0;u_0,u_1,u_2)
=0$. Let $\tilde
S_{v_0}$ be the strict transform of $S_{v_0}$ by the toric map $Z(\Sigma)\to
\C^{g+1}$. As before
we need to examine the situation only in a chart $Z(\sigma)$ where $\sigma$
is adjacent to
the weight vector. We take $a^0=(4,6,13),\ a^1=(2,3,7),\ a^2=(3,5,10)$. So
$\pi (\sigma) $ is
described as:
$$
\begin{array}{lcllcl}
\pi(\sigma):  &u_0 &=& {y_0}^4{y_1}^2{y_2}^3\\
      &u_1 &=& {y_0}^6{y_1}^3{y_2}^5\\
           &u_2 &=& {y_0}^{13}{y_1}^{7}{y_2}^{10}
\end{array}
$$
and we have
$$(u_1^2-u_0^3+v_0u_2)\circ \pi
(\sigma)=y_0^{12}y_1^6y_2^9(1-y_2+v_0y_0y_1y_2)$$
so that the equation of $\tilde S_{v_0}$ in this chart is
$$y_2(1-v_0y_0y_1)=1,$$
and the mapping $\tilde S_{v_0}\to S$ is described by
$$
\begin{array}{lcll}
  &u_0 &=& {y_0}^4{y_1}^2(\frac{1} {1-v_0y_0y_1})^3\\
      &u_1 &=& {y_0}^6{y_1}^3(\frac{1}{1-v_0y_0y_1})^5\\
           \end{array}
$$
All computations are made in a neighborhood of the exceptional divisor, i.e.,
for $\vert
y_0\vert$ small.\par
 Now the toric map $S^1_{v_0}\to S_{v_0}$ which resolves the plane branch
with one characteristic pair $u_1^2-u_0^3=0$ has a chart:
$$
\begin{array}{lcll}
  &u_0 &=& x_0x_1^2\\
      &u_1 &=& x_0x_1^3\\
           \end{array}
$$
and we can check that there is a factorization $\tilde  S_{v_0}\to S^1_{v_0}$ given
as follows in the
coordinates $y_0,y_1$ on $\tilde S_{v_0}$:
$$
\begin{array}{lcll}
  &x_0 &=& 1-v_0y_0y_1\\
      &x_1 &=& y_0^2y_1(\frac{1}{1-v_0y_0y_1})^2\\
           \end{array}
$$
and we can view this factorization itself as composed of the maps 
$$
\begin{array}{lcll}
  &x_0 &=& w_0\\
      &x_1 &=& w_0^2w_1\\
           \end{array}
$$
and 
$$
\begin{array}{lcll}
  &w_0 &=& 1-v_0y_0y_1\\
      &w_1 &=& y_0^2y_1\\
           \end{array}
$$
Now this last map is monomial after a change of the coordinates $w_0,w_1$
and moreover
the substitution of the $w$'s in the $x$'s gives a monomial map $S^2_{v_0}\to
S_{v_0}$ described
as follows
$$
\begin{array}{lcll}
  &u_0 &=& w_0^5w_1^2\\
      &u_1 &=& w_0^7w_1^3\\
           \end{array}
$$ 
which is again a monomial map, so that we have factorized our map $\tilde
S_{v_0}\to S_{v_0}$ as
a composition of two monomial maps, up to a change of variables
(essentially a translation
$w_0\mapsto w_0-1$). Note the fact that it was convenient to refine part of
the fan of the
plane branch $u_1^2-u_0^3=0$ from $(1,1),(2,3)$ to $(5,7) ,(2,3)$ before
writing the
translation, but that we could also have made a change in the variables
$y_0,y_1$ (for
$\vert y_0\vert$ small) to bring directly the map $\tilde S_{v_0}\to S^1_{v_0}$
into monomial form.
\par Finally, note that the strict transform on $\tilde S_v$ of our plane
curve $C_{v_0}\subset
S_{v_0}$ has equation $y_1-1=0$ In the general case, since we do not deform the last equation
$f_g$ of the monomial curve, the last equation of the strict transform of $C_{v_0}$, in the proper
chart will still be of the form $y_k-1=0$.
\end{example}

%-----------------------------------------------
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\end{thebibliography}

%-----------------------------------------------

\medskip
\noindent
Dept. of Mathematics\newline 
M.I.T. \newline 
Cambridge, MA 02139, USA\newline 
goldin@math.mit.edu

\bigskip\noindent 
Laboratoire de Math\'ematiques de  l'E.N.S.,\newline 
URA 762 du C.N.R.S.,\newline
Ecole Normale Sup\'erieure de Paris\newline 
45 Rue d'Ulm\newline
75005 Paris, France\newline
teissier@dmi.ens.fr


\end{document}

                  

                                                                            
                                                                            
                                                                            
                                                                            
                             




