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\Volume{332}
\Year{2001}
\FirstPage{1}
\LastPage{??}
\AuteurCourant{Gonz\'alez P\'erez, P.D., Teissier, B.}
\TitreCourant{Embedded resolution of affine toric varieties} %less than 60 characters
\Journal
\Rubrique{G\'eom\'etrie alg\'ebrique}{Algebraic Geometry}
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\PresentePar{Michel}{Raynaud}
\Recu{18/12/2001}{apr\`es r\'evision}{14/01/2002}
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\begin{document}
\selectlanguage{english}
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\title{Embedded resolution of non necessarily normal affine toric varieties
}
%
\author{%
Gonz\'alez P\'erez, P.D.$^{\text{a}}$,\ \
 Teissier, B. $^{\text{b}}$,\ \
}
%
\address{
Universit\'e de Paris 7,
Institut de Math\'ematiques,
Equipe G\'eom\'etrie et Dynamique, 
UMR CNRS 7586,\\
Case 7012,
2, Place Jussieu,
75005 Paris, France. 
\begin{itemize}\labelsep=2mm\leftskip=-5mm
\item[$^{\text{a}}$]
E-mail: gonzalez@math.jussieu.fr
\item[$^{\text{b}}$]
E-mail: teissier@math.jussieu.fr
\end{itemize}
}
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%%%  Abstract  %%%
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\begin{Abstract}{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, which always exists,  
provides
an embedded resolution.
The advantage is  that this partial resolution is
completely determined by
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 $k$ algebraically closed
of arbitrary characteristic.
}\end{Abstract}
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%%%  French title  %%%
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\selectlanguage{french}
\begin{Ftitle}{
R\'esolution plong\'ee d'une vari\'et\'e torique affine non n\'ecessairement normale
}\end{Ftitle}
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%%%  R\'esum\'e  %%%
%%%%%%%%%%%%%%%%%%%%
\begin{Resume}{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, qui existe toujours,  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 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\'esultat est valide sur un corps
$k$ alg\'ebriquement clos de caract\'eristique quelconque.
}\end{Resume}

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%%%  Version fran\c{c}aise abr\'eg\'ee  %%%
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%\AFv
% The abridged French version.

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\section*{Introduction} 
The existence  of a resolution of singularities of any variety
  was proved by Hironaka in 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} and \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 
$Z_\rho$, i.e., the embedding is
compatible with the actions of their tori.
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 $Z^\Gamma 
\subset Z_\rho$,
roughly speaking, toric resolutions of $Z_\rho$ that provide
resolutions of $Z^\Gamma$. If the dimension of   $Z_\rho$ 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})).
A partial resolution of $Z^\Gamma $ is completely characterized by 
a simple
combinatorial property of the embedding, and it defines an  {\it embedded normalization} 
of the variety $Z^\Gamma$ as a {\it non degenerate section} 
of a normal toric variety $Z_{\Sigma}$, i.e., 
any toric resolution of $Z_{\Sigma}$ provides also an embedded
resolution of $Z^\Gamma$.
We give an example for a monomial surface.


The results of this note  are included in a work of
the first author
on quasi-ordinary singularities of hypersurfaces and their embedded resolutions
(see \cite{GP}, Proposition 5)
and 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}, Proposition 6.4).






\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 $N \cong \bZ^{d+1}$ is a lattice we denote by $M$ the dual lattice, by
$N_\bR$ the real vector space spanned by $N$ and
by $N_\ell$ the sub-lattice $\ell \cap N$
of $N$ associated to a  vector subspace $\ell $ of $N_\bR$.
A {\it rational convex polyhedral cone} $\sigma$  in  $N_\bR$ 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 $\sigma$ is {\it strictly convex} if
$\sigma$ contains no linear subspace of dimension $>0$;
the
cone $\sigma$ 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 $\stackrel{\circ}{\sigma}$ the { \it relative interior} of a 
cone $\sigma$.
The {\it dual} cone  $\sigma^\vee$ (resp. {\it orthogonal} cone 
$\sigma^\bot$) of $\sigma$ is the set
$ \{ w  \in M_\bR / \langle w, u \rangle \geq 0,$  (resp. $ \langle w, 
u \rangle = 0$)  $ \; \forall u \in \sigma \}$.
A {\it fan\index{fan}} $\Sigma$ is a family of strictly convex
  cones  in $N_\bR$
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.
The {\it support\index{support of a fan}} of the fan $\Sigma$ is the set
$ \bigcup_{\sigma \in \Sigma} \sigma \subset  N_\bR$.
  The fan $\Sigma$ is {\it regular} if all its cones are regular;
it is {\it compatible}
with the subspace $\ell \subset N_\bR$
if $\sigma \cap \ell \in \Sigma, \; \forall \sigma \in \Sigma$.

  A non necessarily normal affine toric
  variety is of the form
$Z^{\Gamma} = \makebox{Spec} \, k [ \Gamma ]$ where $\Gamma $ is a
sub-semigroup of finite type of a lattice $(\Gamma - \Gamma)$
which it generates as a group.
The {\it normalization} of the variety $Z^{\Gamma}$  is obtained from  the
semigroup inclusion $\Gamma  \rightarrow \tau \cap (\Gamma - \Gamma) $ 
where $\tau$ is the
cone spanned by the elements of  $\Gamma$ (see \cite{TE}).
In particular, if  $\sigma$ is a cone in the fan $\Sigma$, the semigroup 
$\sigma^\vee  \cap M$  is of finite type,
it spans the lattice $M$
and the variety $Z^{\sigma^\vee \cap M}$, which
we denote also by  $Z_{\sigma, N}$ or by $Z_\sigma$ when the lattice is clear from the context, is normal.

If $\sigma \subset \sigma'$ are cones in the fan $\Sigma$ then we have an open immersion $Z_\sigma \subset Z_\sigma' $;
the affine varieties $Z_\sigma$  corresponding to cones in a fan $\Sigma$  
glue up to define the {\it toric variety\index{toric variety}}
$ Z_\Sigma$. The torus   $(k^*)^{d+1} $ is an open dense subset $Z_{ \{ 0 \} }$ of each chart 
$ Z_\sigma$ acting on $ Z_\sigma$; these actions
paste into an  action on $ Z_\Sigma$ which extends
the action of the torus on itself by multiplication.
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 $\sigma \in \Sigma$, the   closed subset 
$\bO_{\sigma}$  of $Z_\sigma$ defined
by the ideal $(X^w/ w  \in (\sigma^\vee - \sigma^\bot) \cap M)$ 
of $k[\sigma^\vee  \cap M ]$.
The coordinate ring of $\bO_\sigma$ is $k [ \sigma^\bot \cap M]$.
The map that applies a cone $\sigma$ in the fan  $\Sigma$  to the set
$\bO_\sigma \subset Z_\Sigma$ is a bijection between the cones of the fan and
the orbits of the torus action, each $\bO_\sigma $ being the orbit of the {\it 
special point} $o_\sigma $
defined by $X^u (o_\sigma) = 1 $ for all $u \in \sigma^\bot \cap  M $.

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 every cone of
$\Sigma'$ is contained in a cone of $\Sigma$;
this subdivision defines the {\it toric
modification\index{modification}}
$ \pi_{\Sigma'} : Z_{\Sigma '} \rightarrow   Z_\Sigma$
which is equivariant and  induces an isomorphism between the tori.



\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:
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 then  a resolution of singularities of the variety  $Z_\Sigma$
(see \cite{Cox}, Theorem 5.1).
We call such a fan  $\Sigma'$  a {\it
resolution} of the fan $\Sigma$.

Let $\Sigma$ be any fan supported on $\rho$ defining the
modification $\pi_\Sigma: Z_\Sigma \rightarrow Z_\rho$. 
Let ${\cal V}$ be a subvariety of $Z_\rho$ such that the intersection with 
the torus is a non singular dense open subset of ${\cal V}$. 
The {\it strict transform} ${\cal V}_\Sigma \subset Z_\Sigma $  
is the subvariety of $\pi_\Sigma^{-1} ({\cal V})$ 
such that the restriction $ {\cal V}_\Sigma  \rightarrow {\cal V}$ is a modification. 
If the fan  $\Sigma$ is  regular, we say that
the toric map $\pi_\Sigma: Z_\Sigma \rightarrow Z_\rho$
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}).
The modification $\pi_\Sigma $ is  an {\it
embedded resolution\index{embedded resolution}} of ${\cal
V} $ if in addition the restriction of the map $\pi_\Sigma $ to the strict
transform $  {\cal V}_\Sigma  \rightarrow  {\cal V}$ is an 
isomorphism outside the
singular locus of ${\cal V}$ (see  \cite{Rebeca}).
For a general fan  $\Sigma$, we say that $\pi_\Sigma: Z_\Sigma 
\rightarrow Z_\rho$
is a {\it partial embedded pseudo-resolution \index{partial embedded 
pseudo-resolution}} (resp.
{\it partial embedded resolution}) of ${\cal V}$
if for any resolution $\Sigma'$ of the fan $\Sigma$ the map 
$\pi_{\Sigma'} \circ \pi_\Sigma $
is an embedded pseudo-resolution (resp. embedded resolution) of the variety ${\cal V}$.
Any fan $\Sigma$ admits resolutions (see \cite{TE} and \cite{Cox}), so that any partial embedded 
resolution in our sense extends to an embedded resolution. 


Let $\Gamma$ be a semigroup as above, we will suppose that the cone 
$\tau$  is strictly convex
so that  there is a unique  $0$-dimensional orbit in $Z^\Gamma$.
An embedding  $Z^\Gamma \subset Z_\rho$
is equivariant if $Z_{ \{ 0 \}} \cap  Z^\Gamma$ is the torus of $ 
Z^\Gamma$ and the actions
of the two tori coincide on $Z^\Gamma$. In this case 
the embedding 
 $Z^\Gamma \subset Z_\rho$ 
is defined by a surjective  homomorphism of  semigroups:
\begin{equation}
\label{eqemb}
 \rho^\vee \cap M \rightarrow \Gamma
\end{equation}
which extends to a lattice homomorphism
$\psi:  M \rightarrow (\Gamma - \Gamma)$.
The ideal of $Z^\Gamma$ in $Z_\rho$ is generated by a finite set of binomials
$\{ X^{u_i} - X^{v_i} \}_{i \in I} $ such that $ u,v \in  \rho^\vee \cap M  $ and
$\psi(u) = \psi(v) $ (see  \cite{Sturmfels} Chapter 4).
The linear subspace $\ell$ orthogonal to $\makebox{Ker} ( \psi) $ is completely determined by the embedding
and coincides with the 
intersection of the linear subspaces
$(N_\bR) \supset \ell_i: \langle u_i - v_i, w \rangle = 0 $ for $i \in I$,
since the set of vectors $\{ u_i -v_i \}_{i \in I}$ generates the kernel of $ \psi$.
Our main result is the following:




\begin{theorem} \label{binomial}
Let $\Sigma$ be a subdivision of $\rho$ containing the cone $\sigma_0 :=\rho \cap \ell$.
\begin{enumerate}
\item
The strict transform $Z_{\Sigma}^\Gamma$ of $Z^\Gamma$ by the morphism $\pi_{\Sigma}$ is contained
in $Z_{\sigma_0}$, it is  
isomorphic to $Z_{\sigma_0, N_{\sigma_0}}$ and the restriction  
$\pi_{\Sigma} | Z_\Sigma^\Gamma :  Z_\Sigma^\Gamma  \rightarrow Z^\Gamma$
is the normalization map.
\item
The morphism 
$\pi_{\Sigma}$ is a partial embedded resolution of $Z^\Gamma \subset Z_\rho$.
\end{enumerate}
\end{theorem}


\section{An example}



We give an example of this construction for a   {\it monomial surface}   in 
${\mathbf A}^4 _k$.
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 ${\mathbf A}^4 _k$,
where the affine space 
is considered as 
toric variety.
The semigroup homomorphism   $\bZ^4_{\geq_0} \rightarrow \Gamma$  corresponding to the 
embedding (see (\ref{eqemb})) is defined by:
$e_1 \mapsto \gamma_1 = (2,0)$, $e_2 \mapsto \gamma_2 = (0,2)$, $ e_3 \mapsto 
\gamma_3 = (3,0)$,  $e_4 \mapsto \gamma_4 =(7,2)$,
  for $\{ e_i \}_{1}^4$ the canonical basis and $ \{ \gamma_i \}_{1}^4$ 
generating the semigroup
$\Gamma$.
The ideal of the embedding of the monomial surface is generated by
$ h_1 := U_3^2 - U_1^3, \, h_2 := U_4^2 - U_1^{4} U_2 U_3^{2}$.
The subdivision $\Sigma$  of $\bR^4_{\geq 0}$ defined by the linear subspaces:
$ \ell_1 : 3 u_1 -2 u_3  = 0, \, \ell_2  :  4 u_1+  u_2+
2 u_3 -2 u_4 =0$ contains the cone $\sigma_0 := \bR^4_{\geq 0} \cap \ell_1 \cap \ell_2$.
By theorem \ref{binomial}, the morphism $\pi_\Sigma$
defines a partial embedded resolution of the surface $Z^\Gamma$ and 
the open set $Z_{\sigma_0}$ 
contains the strict transform $Z^\Gamma_\Sigma$ of $Z^\Gamma$.

We describe now a resolution ${\Sigma}'$ of the fan $\Sigma$
and we show that the map $\pi_{\Sigma'} \circ \pi_{\Sigma} $ 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 containing  $\sigma_0$.
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 $\sigma_0$ do not intersect the strict transform
of the monomial surface.

\begin{remark}
In this example the cone $\sigma_0$ 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 
and therefore 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 $\sigma_0$; in this 
two dimensional
situation
there exists a unique minimal regular subdivision of $\sigma_0$ which corresponds
geometrically to the minimal resolution of the
normalization of $Z^\Gamma$, and this subdivision can be
determined  algorithmically from the pair $(\sigma, (-\sigma_0 +\sigma_0) \cap N)$
(see \cite{Oda}).
\end{remark}

The intersection of $\ell_1$ with $\bR^4_{\geq 0}$ 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 each of them  
 defines, with $a^1$ and $ a^2$, a basis of the lattice $\ell_1 \cap \bZ^4$.
 We can take for instance
 $c^1 := (0, 3, 0, 1)$ and $c^2 = (0, 1, 0, 1)$.
 The intersection of $\ell_2$ with $\bR^4_{\geq 0}$ 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 each of them defines, with $a^1$ and 
$ a^2$,  a 
 basis of the lattice $\ell_2 \cap \bZ^4$.
 We can take for instance
$d^1 := (1, 0, 2, 4)$ and $d^2 = (1, 0, 1, 3)$.
We can verify that
the cones $\theta_{ij} := \langle a^1, a^2, c^i, d^j \rangle $
are regular for $i, j = 1, 2$ and they are part of 
a resolution $\Sigma' $ of the fan $\Sigma$.
The toric morphism  $\pi_{\Sigma'} \circ \pi_\Sigma$ is defined on the chart
$Z_{ \theta_{11} }$ by:
\[
U_1 = W_2 ^2  W_4, \;\;
U_2 = W_1^2 W_3^3,  \;\;
U_3 = W_2^3 W_4^2,  \;\;
U_4 = W^1 W_2^7 W_3 W_4^4
\]
The total transform of  $h_1= 0$ (resp.\ of  $h_2 =0$) on this chart is defined by
$ W_2^6 \, W_4^3 ( W_4 -1 ) =0$ (resp.\  by  $ W_1^2 \,  W_2^{14} \, W_3 ^2 \, W_4 ^8  ( 1 - W_3 )=0$).
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 canonical 
stratification of the exceptional divisor (defined by $W_i = 0$, for $i= 1, \dots, 4$  in this chart).







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%  Acknowledgements  %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%
\Acknowledgements{The first author is supported by a Marie Curie Fellowship of the European Community
programme ``Improving Human Research Potential''
% and the Socio-economic Knowlegde Base'' 
under contract number HPMF-CT-2000-00877.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{thebibliography}
%
\end{document}

