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\title{Partial embedded resolution of \\
non necessarily normal toric varieties}
\author{Gonz\'alez P\'erez, P.,
         Teissier, B.}



\date{April 2001}

\begin{document}
\maketitle 

\centerline{\textbf {R\'esum\'e}}\par\medskip
Nous pr\'esentons une m\'ethode de construction d'une {\it r\'esolution plong\'ee partielle} d'une vari\'et\'e torique affine non
n\'ecessairement normale $Z^\Gamma$ plong\'ee de mani\`ere \'equivariante dans une vari\'et\'e torique affine normale $Z_\rho$. Cette
r\'esolution partielle est une {\it normalisation plong\'ee} de $Z^\Gamma$ dans un un espace ambiant torique normal et une r\'esolution des
singularit\'es de l'espace ambiant fournit une une r\'esolution plong\'ee des singularit\'es. L'avantage est que cette r\'esolution partielle
est enti\`erement d\'etermin\'ee par l'un quelconque des ensembles d'\'equations d\'efinissant le plongement $Z^\Gamma\subset Z_\rho$. Une
cons\'equence est la construction de la normalisation sans calcul de la saturation du semigroupe $\Gamma$ de la vari\'et\'e torique (voir
\cite{Fulton}). Ce r\'esulat est valide sur un corps
$\C$ alg\'ebriquement clos de caract\'eristique quelconque. \par\medskip
\centerline{\textbf {Abstract}} \footnotetext[1]{The first author is financed by La Caixa and the
French Embassy in Spain.}
We give a method to construct a {\it partial embedded resolution} of  a non necessarily normal 
affine toric variety $Z^\Gamma$ equivariantly embedded in a normal 
affine toric variety $Z_\rho$.
This partial resolution is an {\it embedded normalization} 
inside a normal toric ambient space and 
a resolution of singularities of the ambient space provides 
an embedded resolution.
The advantage is  that this partial resolution is 
completely determined by  
any set of equations defining the embedding $Z^\Gamma\subset Z_\rho$.
As a by-product, the construction of the normalization
is made without an explicit computation of 
the saturation of the semigroup $\Gamma$ of the toric variety (see \cite{Fulton}). 
This result is valid for a base field $\C$ algebraically closed
of arbitrary characteristic.


\section*{Introduction}
The existence  of a resolution of singularities of any variety 
 was proved by Hironaka on characteristic zero and is still open 
in positive characteristic.
For the class of toric varieties the situation is much simpler.
We can find resolutions of singularities of normal toric varieties
which are toric morphisms and these toric resolutions  
admit a completely combinatorial description
independent of the characteristic of the algebraically closed base 
field (see \cite{TE}, \cite{Cox}).

In this note we will consider an affine (not necessarily normal) toric variety $Z^\Gamma$
{\it equivariantly embedded} in an affine  normal toric variety  $Z_\r$, i.e., the embedding is 
compatible with the actions of their tori.
%\foootnote{See the definition in  section \ref{Par}}.
For instance we have {\it monomial curves}, the affine curves defined by monomial parametrizations, 
(see \cite{Rebeca} for their resolutions).
This note deals with {\it embedded toric resolutions} of $Z^\Gamma \subset Z_\r$, 
roughly speaking, toric resolutions of $Z_\r$ that provide  
resolutions of $Z^\Gamma$. If the dimension   $Z_\r$ is  $>2$ there are many different 
toric resolutions and  no generalization of the minimal toric resolution
existing  in the two dimensional case.
For this reason, 
we introduce the notion of {\it partial resolution} (inspired by some ideas of Lejeune and Reguera
in a different context (see \cite{LR})).  
Partial resolutions of $Z^\Gamma $ will be completely determined by the equations of the embedding by a simple 
combinatorial method and will provide an {\it embedded normalization} of $Z^\Gamma$ inside a normal toric variety $Z_{\Sigma_0}$ 
in such a way that any toric resolution of $Z_{\Sigma_0}$ will provide an embedded 
resolution of $Z^\Gamma$.
We give an example for a {\it monomial surface}.


The results of this note  are included in a work of 
the second author which deals with local uniformization of a valuation of a noetherian local integral domain
by using a geometric specialization of this ring to the graded ring determined by the valuation  
and the tools of toric geometry (see  \cite{Teissier}, \S 6, Proposition 6.4.).
They are also included in a work of the first author, based on \cite{GP}, 
extending them to quasi-ordinary singularities of hypersurfaces.





\section{A reminder of toric geometry} \label{toricg}


We give some definitions and notations (see \cite{Fulton}, \cite{Ewald} and \cite{Oda} for proofs).
If $N \cong \Z^{d+1}$ is a lattice we denote by $M$ the dual lattice, by 
$N_\R$ the real vector space spanned by $N$ and 
by $N_\ell$ the sub-lattice generated by  $\ell \cap N$  for 
$\ell \subset N_\R$.
A {\it rational convex polyhedral cone} $\s$  in  $N_\R$ is the set non
negative linear combinations of vectors $a^1, \dots, a^s  \in N$.
In what follows a {\it cone} will mean a  rational convex polyhedral cone.
The cone $\s$ is {\it strictly convex} if 
$\s$ contains no linear subspace of dimension $>0$;
the
cone $\s$ is {\it regular\index{regular cone}} if 
the primitive integral vectors defining the $1$-dimensional faces 
belong to a basis of the lattice  $N$. 
We denote by $\stackrel{\circ}{\s}$ the { \it relative interior} of a cone $\s$.
The {\it dual} cone  $\s^\vee$ (resp. {\it orthogonal} cone $\s^\bot$) of $\s$ is the set
$ \{ w  \in M_\R/ \langle w, u \rangle \geq 0,$  (resp. $ \langle w, u \rangle = 0$)  $ \; \forall u \in \s \}$. 
%i.e., cone $\s$ minus the union of its proper faces. 
A {\it fan\index{fan}} $\Sigma$ is a family of {\it strictly convex 
 cones\index{convex rational polyhedral cone}} in $N_\R$ 
such that any face of such a cone is in the family and 
the intersection of any two of them is a face of each.
% For instance the set of faces of a strictly convex cone is a fan. 
The {\it support\index{support of a fan}} of the fan $\Sigma$ is the set 
$ \bigcup_{\s \in \Sigma} \s \subset  N_\R$.
 The fan $\Sigma$ is {\it regular} if all its cones are regular; 
it is {\it compatible} 
with  $\ell \subset N_\R$ 
if $\s \cap \ell \in \Sigma, \; \forall \s \in \Sigma$.
% We  denote $\Sigma^{(i)}$ the $i$-skeleton of the fan $\Sigma$, 

 
%f the cone $\s$ belongs to a fan $\Sigma$, then the cone $\s^\vee$ is of dimension $d+1$.
% The semigroup defined by $\s$ is  $\s^\vee  \cap M$, where $M$
% is the dual lattice of $N$. 
% If $\Phi$ is a monoid of finite type, we 
% denote by 
% $\C[ \Phi ]$ the associated $\C$-algebra and by $Z^\Phi$ the variety $\makebox{ Spec } \C
% [\Phi]$.

 Any non necessarily normal affine toric
 variety is of the form
$Z^{\Gamma} = \makebox{Spec} \, \C [ \Gamma ]$ where $\Gamma $ is a
sub-semigroup of finite type of a lattice $\Gamma - \Gamma$
which it generates as a group.
% The closed points of $Z$ correspond to homomorphisms of semigroups  $\Gamma \rightarrow \C$ where $\C$ is considered as a semigroup with respect to multiplication. The torus embedded in $Z$ is the group of homomorphisms of semigroups $\Gamma \rightarrow \C- \{ 0 \}$ and acts naturally on the closed points of  $Z$.
The {\it normalization} of $Z^{\Gamma}$  is obtained from  the
semigroup inclusion $\Gamma  \rightarrow \t \cap (\Gamma - \Gamma) $ where $\t$ is the 
cone spanned by the elements of  $\Gamma$ (see \cite{TE}). 
% i.e.,  $\C[ \Phi ] = \{ {\sum_{{\small finite}}} c_\a X^{\a} \, / \, c_\a \in \C,\a \in \Phi \}.$
In particular, if  $\s$ is a cone in the fan $\Sigma$, the semigroup $\s^\vee  \cap M$  is of finite type, 
it spans the lattice $M$ 
and the variety $Z^{\s^\vee \cap M}$, which 
we denote also by  $Z_{\s, N}$ or by $Z_\s$ when the lattice is clear from the context, 
is normal.

If $\s \subset \s'$ are cones in the fan $\Sigma$ then
we have an open immersion $Z_\s \subset Z_\s' $;  
the affine varieties $Z_\s$  corresponding to cones in a fan
$\Sigma$ 
glue up to define the {\it toric variety\index{toric variety}} 
$ Z_\Sigma$. The torus,   $(\C- \{ 0 \})^{d+1} = Z_{ (0) } $,   
is an open dense subset of  each chart
$ Z_\sigma$ acting  on $ Z_\sigma$; these actions
paste into an  action on $ Z_\Sigma$ that extends
the  product on the torus.
General toric varieties are defined by this property (see \cite{Sturmfels}).
The toric varieties which can be defined using fans are precisely the 
normal ones (see \cite{TE}).
The toric variety $Z_\Sigma$ is non singular if and only
if the fan $\Sigma$ is regular.
We define for each $\s \in \Sigma$, the   closed subset $\O_{\s, N }$ ($\O_{\s}$ for short)  of $Z_\s$ defined 
by the ideal $(X^w/ w  \in (\s^\vee - \s^\bot) \cap M
)$ of $\C[\s^\vee  \cap M ]$.
% where $\s^\bot = \{ w  \in M_\R /\langle w, u \rangle =  0, \; \forall u \in \s \}$.
The coordinate ring of $\O_\s$ is $\C [ \s^\bot \cap M]$.
The map that applies a  cone $\s$ in $\Sigma$  to 
$\O_\s \subset Z_\Sigma$ is a bijection between the cones of the fan and 
the orbits of the action, each $\O_\s $ being the orbit of the {\it special point} $\o_\s $
defined by $X^u (\o_\s) = 1 $ for all $u \in \s^\bot \cap  M $. 
% If $\dim \s = \mbox{rk} N$ the set $\O_{\s}$ has only one point $o_\s$,
% otherwise we have an isomorphism defined by: 
% \begin{equation} \label{product} 
% Z_{\s, N} \cong \O_{\s, N } \times Z_{\s, N_\s} 
% \end{equation}
% where $N_\s = -\s \cap N + \s \cap N$ (see \cite{Fulton}). 

We say that a fan $\Sigma'$ is a {\it subdivision\index{fan subdivision}} of the fan
$\Sigma$ if both fans have the same support and if any cone of
$\Sigma'$ is contained in a cone of $\Sigma$;
this subdivision defines the   
 {\it toric
  modification\index{modification}} 
$ \p_{\Sigma'} : Z_{\Sigma '} \rightarrow   Z_\Sigma$  
which is equivariant and  induces an isomorphism between the tori.


% A primitive vector $a$ in   $\Z^{d+1}$ is {\it
% essential\index{essential vectors}} for the fan $\Sigma$ if it
% defines a cone which belongs to any regular subdivision of $\Sigma$
% (see \cite{Aguzzoli} and \cite{Bouvier}).  In particular, if a
% $2$-dimensional cone $\s \in \Sigma$  is not regular, it admits a
% unique minimal  regular subdivision (see \cite{Oda} Prop. 1.19).  Any primitive vector defining an edge in this subdivision is essential since any regular subdivision of $\Sigma$ provides a regular subdivision of the cone $\s$.
    

%\begin{Exem}
% If the fan $\Sigma$ is regular and supported on $\R^{d+1}_{\geq 0}$, then we have a subdivision of the fan defined by the rational cone $\D_{d+1} := \R^{d+1}_{\geq 0}$ for $N:= \Z^{d+1}$. This subdivision defines a modification  $ \p_\Sigma : Z_\Sigma \rightarrow Z_{\D_{d+1}} = \C^{d+1} $ which  we describe in detail:
%\end{Exem}

% \noindent
% The variety $Z_\Sigma$ is non singular, 
% for each cone $\s$ of maximal dimension the variety $Z_\s $ is
% isomorphic to $\C ^{d+1}$ and the restriction  $\p_\s : Z_\sigma \rightarrow \C^{d+1} $ of the morphism   $\p_\Sigma $ is induced by the semigroup inclusion $ \D_{d+1} \cap M  \rightarrow \s^\vee  \cap M$. The primitive vectors in the $1$-skeleton of each regular  $(d+1)$-dimensional $\s$ are a basis of $N$, and its dual basis  is a minimal set of generators of the semigroup $\s^\vee  \cap M$. These generators give us coordinates to describe the map  $\p_\s : Z_\sigma \rightarrow \C^{d+1} $ in the form:
% \begin{equation}  \label{chart}
% \begin{array}{ccc}
% X_1 &=& U_1^{a^{1}_1} U_2^{a^{2}_1} \cdots  U_{d+1}^{a^{d+1}_1} \\
% X_2 &=& U_1^{a^{1}_2} U_2^{a^{2}_2} \cdots  U_{d+1}^{a^{d+1}_2}\\
% & \dots &\\
% X_{d+1} &=&  U_1^{a^{1}_{d+1}} U_2^{a^{2}_{d+1}} \cdots  U_{d+1}^{a^{d+1}_{d+1}}   \\
% \end{array}
% \end{equation}
% where $(a^{i}_1, a^{i}_2, \dots, a_{d+1}^{i})$ are the 
% coordinates of the primitive vectors $a^{i}$ in the  $1$-skeleton of $\s$,
% for $i =1, \dots, d+1$. Since the fan $\Sigma$ is regular, it is easy
% to see directly from formula (\ref{chart}) that 
% the map $ \p_\Sigma$ is an isomorphism over the torus $X_1 \cdots
% X_{d+1} \ne 0$ of $\C^{d+1}$.




%  We describe the orbits of the action of the torus on the variety $Z_\Sigma$.
% The closed orbit $\O_{\s, N}$ (which we denote also  $\O_{\s}$) in $Z_\s$  is defined by the ideal 
% $(X^w/ w  \in (\s^\vee - \s^\bot) \cap M)$ of $\C[\s^\vee  \cap M ]$ where 
% $\s^\bot = \{ w  \in \R^{d+1}/ \langle w, u \rangle =  0, \; \forall u \in \s \}$.
% This orbit is a torus for $0 \leq \dim \s < d+1$,
%  since the associated coordinate ring is the $\C$-algebra of the sub-lattice
% $M(\s) :=  M \cap \s^\bot$ of $M$ of codimension  equal to $\dim \s$. 
% On the closed orbit $\O_\s$ we consider the {\it special point} $\o_\s $
% defined by $X^u (\o_\s) = 1 $ for all $u \in M(\s) $. 
% If $\dim \s = \makebox{rk} \, N$ the orbit $\O_\s$ is reduced to the special point.
% If $\dim \s < \, \makebox{rk} \, N$ we have an exact sequence of lattices:
% $$ 0 \rightarrow M(\s) \rightarrow   M  \stackrel{j}{\rightarrow}
% M_\s \rightarrow 0.$$
% If we choose a splitting $M = M(\s) \oplus M'$ we have the lattice $M'$ is isomorphic to $ M_\s$.
% If $ 0 \rightarrow N_\s  \stackrel{j^*}{\rightarrow} N  \rightarrow N(\s) \rightarrow 0$ is the dual exact sequence the lattice  $N_\s = \s \cap N $ is of dimension equal to $\dim \s$  and the semigroup associated to $\s$ with respect to $N_\s$ is  $ \s^\vee_{N_\s}   \cap M_\s = j(\s^\vee  \cap M)$.   We obtain a semigroup isomorphism $\s^\vee  \cap M  \cong M(\s) \oplus (\s^\vee_{N_\s}   \cap M_\s) $ inducing an isomorphism of $\C$-algebras $\C[\s^\vee  \cap M] \cong \C[M(\s)] \otimes_\C \C  [ \s^\vee_{N_\s} \cap M_\s ]$ which induces  (non canonically) the product structure
% \begin{equation} \label{product} 
% Z_{\s, N} \cong \O_{\s, N} \times Z_{\s, N_\s}. 
% \end{equation}
% The map that applies a  cone $\s$ in $\Sigma$  to $\O_\s \subset Z_\Sigma$ is a bijection. If $\s $ is a face of $\t$ then  $Z_\s$ is an open subset of $ Z_\t$ and the orbit $\O_\t$ is contained in the closure of $\O_\s$ in $ Z_\t$ since $\t^\bot \subset \s^\bot$, thus the closure of the orbit of $\s$ in $Z_\Sigma$ is $\overline{ \O _\s } = \bigcup \O_\t$ where $ \t$ runs through the cones of $\Sigma$ which have $\s$ as a face. The orbit closures are normal toric varieties by themselves with respect to the lattice $N(\s)$. The cones of the fan associated to $\overline{ \O_\s }$ are of the form  $\t + (N_\s)_\R \subset N_\R / (N_\s)_\R $ for $\t $ containing  $\s$ as a face.

% The singular locus of $Z_\Sigma$ is the union of those orbits  $\O_\s$ for $\s \ne \{ 0 \} $ or $\s$ a non regular cone. This follows from formula (\ref{product}) by noticing that the orbit $\O_\s$ is contained in the singular locus of $Z_\s$ if and only if $\o_{\s, N_\s}$ is a singular point of $Z_{\s, N_\s}$ if and only the cone $\s$ is not a regular cone.
%Description of the singular locus of a toric variety as the union of those orbits
% for $\s$ such that the cone $ (\s, N_\s)$ is not regular.



% We describe now the exceptional locus associated to a subdivision
% $\Sigma'$ of a fan $\Sigma$.
% Taking away  $\s$ from the fan of the cone $\s$ means geometrically to
% take away the orbit $\O_\s$ from the variety $Z_\s$. We deduce that:
% \begin{equation} \label{exceptional}
% \p^{-1}( \O_\s ) = \bigcup_{\t \in \Sigma', \stackrel{\circ}{\t} \subset 
%   \stackrel{\circ}{\s} } \O_\t
% \end{equation}
% In particular, formula (\ref{exceptional}) describes the {\it exceptional locus},  i.e., the union of subvarieties of dimension $>1$  that are mapped to points, which is equal to  $$ \bigcup_{\dim \s = \mbox{{\small rk}} N , \s \notin \Sigma' } \p^{-1} ( \O_\s)$$ and the {\it critical locus}, i.e., the subvarieties that are mapped on a variety of smaller dimension,  which is $$ \bigcup_{\s \notin \Sigma' } \p^{-1} ( \O_\s) =  \bigcup_{\t \mbox{ {\small minimal }} \in \Sigma' - \Sigma } \overline{\O_\t}.$$ The {\it discriminant locus}, i.e., the image of the critical locus,  is  the  subvariety of codimension two equal to   $$ \bigcup_{\s \mbox{ {\small minimal }} \in \Sigma - \Sigma' } \overline{\O_\s}. $$ 

\section{Partial embedded resolution of toric varieties} \label{par}

 


 
A {\it resolution of singularities\index{resolution of singularities}} of a variety $Z$ is a smooth variety $Z'$ 
with
a modification  $ Z' \rightarrow Z$ which is an isomorphism outside the singular locus of $Z$.
Toric resolutions of singularities of normal toric varieties
have a completely combinatorial description:
\begin{The} (see \cite{Cox}, Theorem 5.1) \label{regfan}
Given any fan $\Sigma$, there is a regular fan $\Sigma'$ subdividing
$\Sigma$, which contains the regular cones of $\Sigma$.
The associated toric morphism $ Z_{\Sigma '} \rightarrow   Z_\Sigma$
is a resolution of singularities of the variety  $Z_\Sigma$.
\qed
\end{The} 

We say that the fan  $\Sigma'$  in the theorem above is a {\it regular
subdivision\index{regular subdivision}} of the fan $\Sigma$.





% and the analytic
% algebra $\C \{ \Gamma \}$ of $Z$ at this point
% can be viewed as
% a subring of the ring $\C [[ \Gamma ]]$ of formal complex power series with exponents 
% in the semigroup $\Gamma$ (see \cite{GP} lemme 1.1).
% If we fix a base $\e_1, \dots, \e_{d+1}$ of the lattice $M$ we can embed the semigroup $\Gamma$ as a sub-semigroup of $\Z^{d+1}$. This embedding defines an homomorphism $\C[ \Gamma ] \rightarrow  \C[ X^{\pm}_1, \dots, X_{d+1}^{\pm}]$ and geometrically an embedding of the torus $(\C - \{ 0 \} )^{d+1} \rightarrow  Z$. 

Let $\Sigma$ be any fan supported on the cone $\r$, defining the modification
 $\p_\Sigma: Z_\Sigma \rightarrow Z_\r$. 
The {\it strict transform} ${\cal V}_\Sigma$  
of the closed subscheme ${\cal V}$ of  $Z_\r$ defined by an 
ideal $I$,   is defined on each  chart $Z_\s$ by the ideal generated by 
$\{ X^u \f  / \f  \in I,  u \in M \} \cap \C[\s^\vee \cap M ]$, for $\s \in \Sigma$.
% The restriction ${\cal V}_\Sigma \rightarrow {\cal V}  $ is a
% modification if the intersection of the torus $ Z_{(0)}$ with each  irreducible
% component ${\cal  V}_i $ of $ {\cal  V}$ is dense in ${\cal V}_i$.
If the fan  $\Sigma$ is  regular, we say that 
the toric map $\p_\Sigma: Z_\Sigma \rightarrow Z_\r$
is an {\it embedded pseudo-resolution\index{toric embedded
    pseudo-resolution}} of ${\cal V}$ if the
restriction
 ${\cal V}_\Sigma \rightarrow {\cal V}$
is a modification such that the strict transform ${\cal V}_\Sigma$ is
non singular and transversal to the orbit stratification of the 
exceptional locus of 
$Z_\Sigma$ (see \cite{Rebeca}).
If, in addition the restriction of the map $\p_\Sigma $ to the strict
transform $  {\cal V}_\Sigma  \rightarrow  {\cal V}$ is an isomorphism outside the
singular locus of ${\cal V}$, then the modification $\p_\Sigma $ is  an {\it
embedded resolution\index{embedded resolution}} of ${\cal
V} $ (see  \cite{Rebeca}).  
%  If $\p_\Sigma $ is only a pseudo-resolution we can only guarantee that
% the map $  {\cal V}_\Sigma  \rightarrow  {\cal V}$ is an isomorphism outside the intersection
% of ${\cal V}$ with the discriminant
% locus of $\p_\Sigma $. In this case, this set  contains the singular locus of ${\cal V} $
% but it is not necessarily equal to it. 
 For a general fan  $\Sigma$, we say that $\p_\Sigma: Z_\Sigma \rightarrow Z_\r$
is a {\it partial embedded pseudo-resolution \index{partial embedded pseudo-resolution}} (resp. 
{\it partial embedded resolution}) of ${\cal V}$  
if for any regular subdivision $\Sigma'$ of  $\Sigma$ the map $\p_{\Sigma'} \circ \p_\Sigma $
is an embedded pseudo-resolution (resp. embedded resolution) of ${\cal V}$.


% The action of the torus has a $0$-dimensional orbit if and only if the
% cone $\t$  is strictly convex. In this case this orbit is reduced
% to the closed point defined by
%  $0$-dimensional orbit

Let $\Gamma$ be a semigroup as above, we will suppose that the cone $\t$  is strictly convex
 thus there is a unique  $0$-dimensional orbit in $Z^\Gamma$.
% defined by the ideal $(X^u / u \in \Gamma - \{ 0 \})$ of $\C [ \Gamma ]$.
The  embedding  $Z^\Gamma \subset Z_\r$
is equivariant if $Z_{ \{ 0 \}} \cap  Z^\Gamma$ is the torus of $ Z^\Gamma$ and the actions 
of the two tori coincide on $Z^\Gamma$.  An equivariant embedding is defined  
by a surjective  homomorphism of  semigroups:
\begin{equation}
\label{eqemb}
\psi : \r^\vee \cap M \rightarrow \Gamma
\end{equation}
which extends to a lattice homomorphism
% (resp. vector space homomorphism)  
$\psi:  M \rightarrow \Gamma - \Gamma$.
%(resp.   $ \psi_\R: M_\R \rightarrow (-\Gamma + \Gamma)_\R $).
The ideal of $Z^\Gamma$ in $Z_\r$ is generated by a finite set of binomials
$\{ X^{u_i} - X^{v_i} \}_{i \in I} $ such that $ u,v \in  \r^\vee \cap M  $ and 
$\psi(u) = \psi(v) $ (see  \cite{Sturmfels} Chapter 4).
% \noindent {\it Proof.} See  \cite{Sturmfels} Chapter 4. {\qed}
% It is clear that the ideal $(J)$ is contained in the ideal $J'$ of $Z^\Gamma$ in $Z_\r$. By induction on the number of terms of a polynomial $\f= \sum c_\a X^a$ in the kernel of $\psi$. It is clear that if $\f \in J'$ is a binomial then $\f \in (J)$. Suppose the result for polynomials with $k-1$ terms.  If $\f \in J'$ has $k$ terms and  $c_{\a_0} \ne 0$  then we have $\sum_{ \psi(\a_k) = \psi(\a_0)} c_{\a_k} X^{\psi( \a_k )} =0$ thus the summand  $\sum_{\psi(\a_k) = \psi(\a_0)} c_{\a_k} X^{\a_k} =0$ of $\f$ is in $(J)$ and we finish by induction. {\qed}
The vectors $u_i -v_i$ span $\makebox{ Ker } \psi$.
We denote by $\ell$  the intersection of the linear subspaces 
$(N_\R) \supset \ell_i: \langle u_i - v_i, w \rangle = 0 $, for $i \in I$.
%  equal to 
%  the orthogonal of $\makebox{ Ker } \psi_\R$.
Our main result is:





\begin{Pro} \label{binomial}

Let $\Sigma_0$ be  the least fine  subdivision of $\r$  compatible with the linear subspaces 
$\ell_i$. The cone $\r \cap \ell$ belongs to $\Sigma_0$.
Let $\Sigma$ be any subdivision of $\Sigma_0$, and
$ {\p_{\Sigma}} : Z_{\Sigma}  {\rightarrow} Z_\r$ the corresponding toric modification.
% We denote by $\tilde{Z}^\Gamma$ the strict transform of $Z^\Gamma $ by $\p_{\Sigma}$.
\begin{enumerate}
\item If   $\s \in \Sigma $ with $\s \subset \ell$ and $\dim \s = \dim
  \ell$,  the intersection  of the strict transform ${Z}_\Sigma^\Gamma$
of $Z^\Gamma$  and the orbit $\O_\s $  as schemes is the simple point $ o_\s $.

\item
If the fan  $\Sigma $  contains the cone $\r \cap \ell$  then 
${Z}_\Sigma^\Gamma $
is isomorphic to $ Z_{ {\r \cap \ell }, N_ {\r \cap \ell}  }$ and 
the restriction $ {Z}_\Sigma^\Gamma \rightarrow
Z^\Gamma$ of $ \p_{\Sigma}$ to $ {Z}_\Sigma^\Gamma$  is the normalization map.
\item
The modification $\p_{\Sigma_0}$ is a partial embedded resolution of singularities of $Z^\Gamma \subset Z_\r$.
\end{enumerate}{\qed}
\end{Pro}


% The {\it Newton polyhedron\index{Newton polyhedron}}, ${\cal N} (\f)$,
% of a formal power series 
% $0 \ne f = \sum c_a X^{a} \in \C\{ X \} $ with $X=(X_1, \dots, X_{d+1})$
% is the convex hull of the set 
% $\bigcup_{c_a \ne 0} \  a  + \Z_{\geq 0}^{d+1}$.
% More generally we define the  Newton polyhedron of  any non-zero
% germ $\f$ of
% holomorphic function at the especial point $\o_\r$ of a normal
% $(d+1)$-dimensional affine toric
% variety $Z_\r = \makebox{Spec} \C [\r^\vee \cap M  ]$ (thus for a $(d+1)$-dimensional
% strictly convex cone  $\r$). The germ $\f$ can be viewed as a power series $\f = \sum  c_a
% X^a$ with exponents in the semigroup $ \r^\vee \cap M  $. 
% The {\it Newton polyhedron} is the convex hull of
% the set $\bigcup_{c_a \ne 0} \  a  + \r^\vee$. We denote it by  ${\cal
%   N}_\r (\f)$  or by  ${\cal N} (\f)$ if the cone $\r$ is clear.
% Many of the properties associated with classical Newton polyhedra hold
% in this case for instance: if $0 \ne \f = \f_1 \cdots \f_s$  we have that ${\cal N}_\r (\f) $ is the Minkowski sum  ${\cal N}_\r (\f_1) +  \dots + {\cal N}_\r (\f_1) $ since the series $\f_i $ have coefficients on a domain. 

% \noindent
% The {\it face ${\cal F}_u$ of the polyhedron ${\cal N}_\r (\f)$  defined by a vector\index{face (of a polyhedron}} in  $ u \in \r$ is the set of vectors $  v \in {\cal N}_\r (\f)$ such that   $\langle u, v \rangle   = \inf_{v' \in {\cal N}_\r (\f)} \langle u, v' \rangle$$. It is easy to see that the face of ${\cal N}_\r (\f)$  defined by $ u$ is compact if and only if  we have $u \in \stackrel{\circ}{\r} $. All faces of the polyhedron ${\cal N}_\r (\f)$ can be recovered in this way.

% 
% \noindent
% The {\it cone $\s( {\cal F} ) \subset \r $  associated to the face $\cal F$}
% of the polyhedron ${\cal N}_\r (\f)$ is 
% $$\s( {\cal F} ) := \{ u \in \r  \; /
% \; \forall v \in {\cal F}, \makebox{ we have } 
% \langle u, v \rangle   = \inf_{v' \in {\cal N}_\r (\f)} \langle u, v'
% \rangle \}.$$
% We have that   $\dim {\cal F} + \dim \s( {\cal F}) = d+1$.


 
% \noindent
% The cones $\s( {\cal F})$, for $\cal F$ running through the set
% of faces of the polyhedron ${\cal N}_\r (\f)$, define a  subdivision $\Sigma ({\cal N}_\r (\f))$ of
% the fan of the  cone $\r$.
% The relative
% interiors of the cones in the fan $\Sigma ({\cal N}_\r (\f))$ are
% equal to the equivalence classes of vectors in $\r$ by the relation: 
% $$ u \sim u'  \Leftrightarrow {\cal F}_u ={\cal F}_{u'}.$$


% \noindent
% We say that a fan $\Sigma$  supported on the cone $\r$ is 
% {\it compatible\index{fan compatible with a polyhedron}} with a set of series
% $\f_1, \dots, \f_s \in \C \{ \r^\vee \cap M \} $
% if it  subdivides the fan $\Sigma ({\cal N}_\r (\f))$ with $\f = \f_1 \cdots \f_s$.
% This implies that a cone in the fan $\Sigma ({\cal N}_\r (\f_1 \cdots \f_s))$ is  intersection
% of cones of the fans $\Sigma ({\cal N}_\r (\f_i))$, in particular 
% $\Sigma$ is compatible with all the polyhedra ${\cal N}_\r (\f_i)$.
% If $\Sigma $ is compatible with  ${\cal N}_\r (\f)$ all vectors  in $\stackrel{\circ}{\s}$  
% define the same face ${\cal F}_\s$ of  ${\cal N}_\r (\f)$ for $\s \in \Sigma$. 
% We have the inequality $\dim {\cal F}_\s + \dim  \s \leq d+1 $,  
% which implies that a face associated to a cone of maximal dimension is a vertex of ${\cal N}_\r (\f)$.

%  \noindent
% The {\it symbolic restriction\index{symbolic restriction}} of $\f$ to
% the compact face $\cal F$ of its  Newton  
% polyhedron is the polynomial $\f_{| \cal F} = \sum_{a \in {\cal F}} c_a X^{a} \in \C[ \r^\vee \cap M] $.


% Let $\Sigma$ a  fan supported on $\r$.
% Suppose that ${\cal V}$ is a reduce hypersurface 
% such that its singular locus does not meet the torus.
% The toric map $\p_\Sigma: Z_\Sigma \rightarrow Z_\r$
% is a {\it partial embedded resolution of singularities of ${\cal V}$} if the pair
% $(Z_\Sigma,  {\cal U})$ defines a 
% toroidal embedding without self intersection
% where ${\cal U}$ is the complement in $Z_\Sigma$
% of the hypersurface
% ${\cal V}_\Sigma \cup \bigcup_{ \t \in \Sigma^{(1)}} \overline{ \O_\t } $.
% This implies that the strict transform  ${\cal V}_\Sigma$ is normal.
% A toric embedded pseudo-resolution of ${\cal V}$ is obtained by 
% composing $\p_\Sigma$ with any modification defined by a regular subdivision of 
% the associated conic polyhedral complex.
% Let $ {\cal V} \subset Z_\r$ a subvariety defined by $\f_1, \dots, \f_{s}$.
% Let $\Sigma$  be a subdivision $\r$ compatible with $\f_1, \dots, \f_{s} $. We give a necessary condition on $\f_1, \dots, \f_{s}$ for $\O_\s \cap {\cal V}_\Sigma \ne \emptyset$ (for $\s \in \Sigma$ defining an exceptional  orbit).



% \begin{Lem} \label{nondeg}
% Let  $\s$ a cone in $ \Sigma$ such that
% $\stackrel{\circ}{\s} \subset \stackrel{\circ}{\r}$.
% If $\O_\s \cap {\cal V}_\Sigma \ne
% \emptyset$ then  the face ${\cal F}_i$ of the Newton polyhedron
% ${\cal N}_i$ of $\f_i$ defined by $\s$ is  of dimension $\geq 1$ for  $1\leq i\leq s$.
% \end{Lem}
% \noindent {\it Proof.} We have that $ \f_{ i} -  \f_{ i {| \cal F}_i}  $ belongs to the ideal generated by
% $\{ X^u /  u \in ({\cal N}_\r (\f_i) - {\cal F}_i ) \cap M \}$. 
% Since $\Sigma$ is compatible with the $\f_i$  the cone $\s^\vee$ contains
% the cone spanned by elements in the polyhedron  $- u_0 + {\cal N}_i$ for any $u_0 \in {\cal F}_i$. 
% Take $u_i $ a vertex of ${\cal F}_i$  then 
% we can factor in the ring  $\C [ \s^\vee \cap M ]$: 
%  $$ \f_{ i} \circ \p_\s = X^{u_i} \psi_i \makebox{ and } 
%  \f_{ i {| \cal F}_i} \circ \p_\s  = X^{u_i} \psi_{ i {| \cal F}_i}
% \makebox{ with } \psi_{ i {| \cal F}_i} \in \C[\s^\bot \cap M]$$
% in such a way that the exponent of a term appearing in 
% $ X^{-u_i} ( \f_{ i} \circ \p_\s 
% - \f_{ i {| \cal F}_i} \circ \p_\s )$ belongs to  $ (\s^\vee -
% \s^\bot) \cap M$ and thus this term  vanishes on the orbit  $\O_\s$.
% By definition the elements 
% $X^{-u_i} \f  \circ \p_\s $ for $1\leq i\leq s$
% belong to the ideal defining the strict transform of ${\cal V}$.
% If the face ${\cal F}_i$ is a vertex
% for some $i$ the ideal of $\O_\s \cap {\cal V}_\Sigma$ in $Z_\s$ is
% equal to $(1)$ thus ${\cal V}_\Sigma \cap \O_\s$ is empty.
% {\qed}

% For example, if we have $f= X_{d+1}^m - X_1^{k_1} \dots X_d^{k_d}$
% we see that  a fan $\Sigma$ supported on $(\R^{d+1})^*_{\geq 0}$ is
% compatible with $f$ if and only if  the intersection of any cone $\s
% \in \Sigma$ with the linear  subspace  $\ell$ of $(\R^{d+1})^*$ with
% equation  $m u_{d+1} = k_1 u_1 +\dots+ k_d u_d $ is a cone of
% $\Sigma$. We will say that the fan $\Sigma$ is compatible with the linear subspace $\ell$.


\section{An example}

 

We give an example of this construction for a ``monomial surface'' in ${\mathbf A}^4(\C)$.
The monomial parametrization
$
U_1 =  T_1^{2}, \, 
U_2 =     T_2^{2}, \, 
U_3 =    T_1^3, \, 
U_4 =    T_1^7  T_2 $
defines an equivariant embedding of a toric surface in ${\mathbf A}^4(\C)$, 
where the affine space with the coordinates $U_i$ is considered as a toric variety in a natural way.
The semigroup homomorphism  (\ref{eqemb}) corresponding to the embedding is   $\Z^4_{\geq_0} \rightarrow \Gamma$ where:
$$e_1 \mapsto \g_1 = (2,0), e_2 \mapsto \g_2 = (0,2), e_3 \mapsto \g_3 = (3,0),  e_4 \mapsto \g_4=(7,2),$$
 for $e_i$ the elements of the canonical basis and $\Gamma$ the semigroup generated by $\g_1, \dots, \g_4$.
The relations 
$
2 \g_3 =    3 \g_1, \, 
2 \g_4 =    4 \g_1 + \g_2 + 2 \g_3 
$
provide the following 
elements in the ideal of the embedding of the monomial surface:
$ h_1 := U_3^2 - U_1^3, \, h_2 := U_4^2 - U_1^{4} U_2 U_3^{2}$.
We can prove that
$h_1, h_2$ generate the ideal of the embedding
by using 
Euclidean division of polynomials  
and that the order of $\g_3 $ (resp. $\g_4$) modulo $\g_1 \Z + \g_2 \Z$ 
(resp. modulo $\g_1 \Z + \g_2 \Z + \g_3 \Z$) is equal to two.

The fan $\Sigma_0$ defining a partial embedded resolution of 
the monomial surface is the subdivision of $\R^4_{\geq 0}$
defined by the linear subspaces:
$ \ell_1 : -2 u_3 + 3 u_1 = 0, \, \ell_2  : - 2 u_4 + 4 u_1+  u_2+
2 u_3=0.$
The cone  $\s:=\ell_1 \cap \ell_2 \cap \R^4_{\geq 0}$ belongs to $\Sigma_0$,
and by proposition \ref{binomial} 
the chart defined by this cone contains the strict transform of 
the monomial surface (which is equal to its normalization).

We describe now a regular subdivision $\Sigma$ of $\Sigma_0$
and we show that the map $\p_\Sigma \circ \p_{\Sigma_0} $ is an embedded 
resolution of 
the monomial surface.
We do not define all the cones of the fan $\Sigma$
but just the regular cones of dimension four that contains $\s$.
The reason is that we can define a suitable fan 
$\Sigma$ in many different ways 
but in any case the orbits corresponding to cones in $\Sigma$ 
which are not contained in $\s$ do not intersect the strict transform
of the monomial surface by proposition \ref{binomial}.

\begin{Rem}
In this example the cone $\s$ above is generated by the the vectors 
$a^1 = (0, 2, 0, 1)$ and $a^2 = (2, 0, 3, 7)$; it is a regular cone thus it must belong to $\Sigma$.
Geometrically this means that the normalization of $Z^\Gamma $ is smooth.
In general we have to find a regular subdivision of $\s$; in this $2$-dimensional 
situation 
there exist a unique minimal regular subdivision of $\s$ which corresponds 
geometrically to the minimal resolution of the 
normalization of $Z^\Gamma$, and this subdivision can be
determined  algorithmically from the pair $(\s, (-\s+\s) \cap N)$
(see \cite{Oda}).  
\end{Rem}

The intersection of $\ell_1$ with $\R^4_{\geq 0}$ is the regular cone 
generated by the vectors: 
$ (2, 0, 3, 0)$, $ (0, 1, 0, 0 )$, $  (0, 0, 0,1)$.
With respect to this basis the equation of the plane $\ell_1 \cap
\ell_2$ is $14 s_1 + s_2 -2 s_3 = 0$.
This equation allows us to choose one vector in each half-space  $(\ell_1 \cap
\ell_2) ^\pm$ of $\ell_1$, such that with $a^1, a^2$ define a regular
basis of $\ell_1 \cap \Z^3$. 
We can take for instance 
$c^1 := (0, 3, 0, 1)$ and $c^2 = (0, 1, 0, 1)$.
The intersection of $\ell_2$ with $\R^4_{\geq 0}$ is the regular cone 
generated by the vectors: 
$ (0, 0, 1, 1),  \,  (1, 0, 0, 2),  \, (0, 2, 0, 1) $.
With respect to this basis the equation of the plane $\ell_1 \cap
\ell_2$ is $2 t_1 -3 t_2 = 0$.
This equation allows us to choose one vector in each half-space  $(\ell_1 \cap
\ell_2 )^\pm$ of $\ell_2$, such that with $a^1, a^2$ define a regular
base of $\ell_2 \cap \Z^3$. 
We can take for instance 
$d^1 := (1, 0, 2, 4)$ and $d^2 = (1, 0, 1, 3)$.
Then we verify easily that 
the cones $\s_{ij} := \langle a^1, a^2, c^i, d^j \rangle $
are regular for $i, j = 1, 2$ and that they are part of 
a regular fan $\Sigma$ compatible with the equations of monomial surface.
For instance, the morphism  $\p(\Sigma)$ is defined on the chart 
$Z(\s_{11})$ by: 
\[
%\begin{array}{ccl}
U_1 = W_2 ^2  W_4, \;\;
U_2 = W_1^2 W_3^3,  \;\;
U_3 = W_2^3 W_4^2,  \;\;
U_4 = W^1 W_2^7 W_3 W_4^4
%\end{array}
\]
By proposition \ref{binomial} the strict transform of the monomial surface 
is defined by the strict transforms of the polynomials $h_1$ and $h_2$.
The transform of  $h_1$ is $ W_2^6 \, W_4^3 ( W_4 -1 )$ and the transform of 
$h_2$ is $ W_1^2 \,  W_2^{14} \, W_3 ^2 \, W_4 ^8  ( 1 - W_3 )$.
The strict transform of the monomial surface is defined in this chart by $W_4 = 1$ and $W_3 = 1$.
We remark immediately that it is smooth and transversal to the exceptional divisors in this chart
which are defined by $W_i = 0$, for $i= 1, \dots, 4$. 






%{\small 

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\bibitem[G-T]{Rebeca}{\sc  Goldin, R., Teissier, B.,} 
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% , A research textbook in tribute to Oscar Zariski}.
% Edited by H. Hauser, J. Lipman, F.Oort and A. Quiros. 
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315-340.
\bibitem[GP]{GP}{\sc  Gonz\'alez P\'erez, P.D.}
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Tesis doctoral,  Universidad de La Laguna, 2000.
 
\bibitem[KKMS]
{TE}{\sc Kempf, G., Knudsen, F., Mumford D. , St Donat, B.,} {\em Toroidal    Embeddings},
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\bibitem[L-R]{LR}{\sc  Lejeune-Jalabert,  M.,  Reguera L\'opez, A.,} 
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\bibitem[T]{Teissier}{\sc  Teissier, B.,} Valuations, Deformations and Toric Geometry,
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\end{thebibliography}

\noindent
{\tt gonzalez@math.jussieu.fr, \quad teissier@math.jussieu.fr}

\medskip

\noindent
Universit\'e de Paris 7,
Institut de Math\'ematiques,
Equipe G\'eom\'etrie et Dynamique\\
Case 7012;
2, Place Jussieu,
75005 Paris, France. 




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