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\title{TORIC GEOMETRY AND THE SEMPLE-NASH MODIFICATION}
\author{Pedro D. Gonz\'alez P\'erez}

\address{Instituto de Ciencias Matem\'aticas-CSIC-UAM-UC3M-UCM. Dpto. de Algebra. Facultad de Ciencias Matem\'aticas.
Universidad Complutense de Madrid.
Plaza de las Ciencias 3. 28040. Madrid. Spain.}

\email{pgonzalez@mat.ucm.es}
\thanks{Gonz\'alez P\'erez is supported by {\em Programa Ram\'on y Cajal} and
MTM2007-6798-C02-02 grants of MEC Spain. Teissier is partially supported by grant MTM2007-64704 of MEC Spain}

\author{Bernard Teissier}
\address{
Institut Math\'{e}matique de Jussieu, UMR 7586 du CNRS, 175 Rue du Chevaleret, 75013 Paris, France}

\email{teissier@math.jussieu.fr}

\keywords{Toric geometry, Semple-Nash modification, logarithmic
jacobian ideal}


\subjclass[2000]{14M25, 14E15, 14B05}





\begin{document}


\begin{abstract}
This paper proposes some material towards a theory of general
toric varieties without the assumption of normality. Their
combinatorial description involves a fan to which is attached a
set of semigroups subjected to gluing-up conditions. In particular
it contains a combinatorial construction of the blowing up of a
sheaf of monomial ideals on a toric variety. In the second part
this is used to show that iterating the Semple-Nash modification
or its characteristic-free avatar eventually resolves the
singularities of any toric variety. This solves for toric
varieties an old problem of singularity theory.
\end{abstract}
\maketitle

\section* {Introduction}
In the first part of this paper we study  abstract  toric
varieties without the assumption of normality. Since Sumihiro's
Theorem on the existence of a covering of a toric variety by
invariant affine varieties fails without the assumption of
normality, we have to set the existence of such a covering as part
of the definition of a toric variety. Then an abstract toric
variety has a combinatorial description: it corresponds to certain
semigroups in the convex duals of the cones of a fan, which
satisfy a natural gluing-up condition. This generalizes the
definition of \cite{GKZ} which concerns toric varieties
equivariantly embedded in projective space. In spirit it is also a
continuation of our previous work \cite{GP-T} on embedded
normalization and embedded toric resolution of singularities of
affine toric varieties. We can then define blowing-ups of sheaves
of monomial ideals as toric varieties, and describe the
corresponding operations on semigroups. We also provide the
combinatorial description of torus-invariant Cartier divisors on a
toric variety and the general versions of the classical criteria
for ampleness and very-ampleness.\par In the second part of the
paper we use the description of blowing-ups given in the first
part to show that one can, over an algebraically closed field,
resolve the singularities of a toric variety by iterating the
blowing-up of the logarithmic jacobian ideal introduced in
\cite{GS-ENS}. If the field is of characteristic zero, this
blowing-up is isomorphic to the Semple-Nash modification, so that
a consequence of our result is that in characteristic zero one can
resolve the singularities of a toric variety by iterating the
Semple-Nash modification. Recall that this is a canonical modification of a reduced equidimensional space which replaces each point by the set of limit positions of tangent spaces at nearby non singular points. See the second part for details.
\par  This answers in the special case of
toric varieties an interesting question apparently first asked by
Semple in \cite{S}. One can present the relation of this proposed
resolution process with "classical" resolution of singularities as
follows: two types of proper birational correspondances naturally
associate a singular variety to a non singular one: proper
birational projections of an embedded non-singular algebraic
variety to a smaller dimensional ambient space, and the taking of
the \textit{envelope} of a family of linear subspaces (of an
affine or projective space) whose parameter space is a non
singular algebraic variety. Hironaka's resolution shows that in
characteristic zero all singularities may be created by the first
process, and Semple-Nash resolution in general would show that at least all
singularities of quasi-projective varieties in characteristic zero may be created by iterating
the second process, if we allow singular spaces as parameter
spaces. Moreover it would produce a canonical process for (non embedded) resolution in characteristic zero, as our results here do for toric varieties in a characteristic-free manner. \newpage\centerline{\textbf{Part I: Toric
varieties}}\par\medskip

\Par\noindent
The purpose of this part is to develop the combinatorial theory of toric varieties
without the assumption of normality. We refer to \cite{Da},  \cite{Fulton}, \cite{TE}, \cite{Oda-M}, and \cite{Oda} for background on normal toric varieties, and to the books of Oda-Miyake (\cite{Oda-M}), Gel{\cprime}fand, Kapranov, and  Zelevinsky (\cite{GKZ}) and Sturmfels (\cite{Sturmfels}) for certain classes of non necessarily normal toric varieties. We also point to previous work by H.M. Thompson towards the development of a general theory of toric varieties, see \cite{Thompson1}, \cite{Thompson2} and, from the perspective of Log Schemes, \cite{Thompson3}. We recommend \cite{Ewald} as a particularly accessible introduction to (normal) toric varieties, and mention the forthcoming book of D. Cox, J. Little and H. Schenk on the subject. \large\par\medskip

\begin{section}{Semigroups and semigroup algebras}\label{semi}
The theory of affine toric varieties over a field $k$ is the geometric version of the
theory of semigroup algebras over $k$. For part of the theory, one can omit the
assumption that the semigroup is finitely generated, and replace the field $k$
by a commutative ring.
\begin{definition} A (commutative) semigroup $\Gamma$ is a set equipped with
an operation  $+\colon\Gamma\times \Gamma\to \Gamma$ such that
$\epsilon_1+\epsilon_2=\epsilon_2+\epsilon_1$, which satisfies the associativity property and is cancellative ($\epsilon_1+\epsilon_2=\epsilon_1+\epsilon_3$ implies $\epsilon_2=\epsilon_3$). We shall assume
that $\Gamma$ contains a zero element $0$ such that
$\epsilon+0=\epsilon$ and that no multiple of an element of $\Gamma$ is zero. A system of generators of a semigroup is a
subset $(\gamma_i)$ of $\Gamma$ such that each element of $\Gamma$
is a (finite) linear combination of the $\gamma_i$ with non
negative integral coefficients. We denote by $\Z \Gamma$ the group
generated by $\Gamma$ (defined in a similar way as the field of
fractions of an integral domain). The elements of $\Z \Gamma$ are
finite linear combinations of the $\g_i$ with integral
coefficients. If the semigroup $\Gamma$ is finitely generated, the group $\Z\Gamma$ is a lattice.
\end{definition}

\noindent \textbf{Examples of semigroups}: \Par  \noindent
$\bullet$ Given finitely many coprime integers the set of all
combinations of these integers with non negative integral
coefficients is a subsemigroup $\Gamma$ of the semigroup $\mathbf
N$ of integers, and ${\mathbf N}\setminus \Gamma$ is finite. In
fact any semigroup of integers is finitely generated.\Par
\noindent $\bullet$ Let $(s_i)_{i\geq 1}$ be a sequence of
integers such that $s_i\geq 2$ for $i\geq 2$. Define a sequence of
rational numbers $\gamma_i$ inductively by:
$$\gamma_1=\frac{1}{s_1},\ \ \ \gamma_{i+1}=s_i\gamma_i+\frac{1}{s_1\ldots s_{i+1}}.$$
The set of integral linear combinations of the $\gamma_i$ is a
subsemigroup of ${\mathbf Q}_{\geq 0}$, which is not finitely
generated. In fact the $\gamma_i$ form a minimal set of
generators.\Par \noindent $\bullet$ Let $d$ be an integer and let
$\check\sigma$
 (the reason for the dual notation will appear below) be a convex cone of dimension $d$ in $\check\R^d$.
Denote by $M$ the integral lattice of $\check\R^d$. Then the
intersection $\check\sigma\cap M$ is a subsemigroup of the group
$M$, which generates $M$ as a group.\Par  \noindent By a Theorem
of Gordan, if the convex cone $\check\sigma$ is rational in the
sense that it is the intersection of finitely many half spaces
determined by hyperplanes with integral coefficients, then the
semigroup $\check\sigma\cap M$ is finitely generated.


\begin{definition} If $\Delta$ is a
subsemigroup of $\Lambda$ the saturation of $\Delta$ in $\Lambda$
is the semigroup  $\Theta$ consisting of those elements of
$\Delta$ which have a multiple in $\Lambda$. The semigroup
$\Delta$ is saturated in $\Lambda$ if $\Delta = \Theta$.
\end{definition}
\begin{lemma}\label{reduc}
Let $\check\t$ be a rational convex cone in $\check\R^d$ for
the lattice $M$. The semigroup $\check\t\cap M$ is saturated
in $M$ and the saturation of a subsemigroup $\Gamma$ of $M$ is
$\check\s\cap M$ where $\check{\s} = \R_{\geq 0} \Gamma$ is
the closed convex cone generated by $\Gamma$.
\end{lemma}
\textit{Proof.}: The first statement is clear. If $\R_{\geq 0}
\Gamma= \check\sigma$, any element of
 $\check\sigma\cap M$ is a combination with rational coefficients of elements of $\Gamma$.
 Chasing denominators shows that an integral multiple of this element is in $\Gamma$. The converse is clear.
$\Box$

\begin{definition}
Let $\Gamma$ be a finitely generated commutative semigroup and
$A$ a commutative ring. The semigroup algebra $A[t^\Gamma]$ of $\Gamma$ with coefficients in $A$
is the ring consisting of finite sums $\sum_\gamma a_\gamma t^\gamma$ with $a_\gamma\in A$,
endowed with the multiplication law
$$(\sum_\gamma a_\gamma t^\gamma)(\sum_\delta b_\delta t^\delta)\
=\  \sum_\zeta(\sum_{\gamma+\delta=\zeta} a_\gamma b_\delta)t^\zeta .$$
\end{definition}

\begin{proposition} \label{normal} If $\Gamma$ is a finitely generated subsemigroup of the lattice
$M\subset \R^d$ such that $\Z \Gamma  = M$  and $\check\s=\R_{\geq
0}\Gamma$ is the rational convex cone generated by $\Gamma$,  the
integral closure of $k[t^\Gamma]$ in its field of fractions is
$k[t^{\check\s\cap M}]$.\end{proposition} This follows directly
from Lemma \ref{reduc}.



\begin{remark}
Quite generally, if $k$ is a field the Krull dimension of $k
[t^\Gamma]$ is equal to the \textit{rational rank} of the
semigroup $\Gamma$, which is the integer $\hbox{\rm dim}_{\Q}
\Gamma \otimes_\Z\Q$ (see \cite{T-valuations}, Proposition 3.1).
\end{remark}



\begin{remark}       \label{special-point}
If $\Gamma$ is a semigroup the ideal of $A[t^\Gamma]$
generated by the $(t^\gamma)_{\gamma\in \Gamma\setminus\{0\}}$ is
non trivial if and only if the cone  $ \R_{\geq 0} \Gamma$ is
strictly convex. If $k$ is a field, it is then a maximal ideal. We
shall mostly be interested in the local study of the spectrum of
semigroup algebras in the vicinity of the origin of coordinates,
which corresponds precisely to that ideal.
\end{remark}

The semigroup algebra has the following universal property:
any semigroup map from $\Gamma$ to the multiplicative semigroup of an $A$-algebra $B$ extends uniquely to an
homomorphism $A[t^\Gamma]\to B$ of $A$-algebras.

 An additive map of semigroups $\phi\colon
\Gamma\to \Gamma'$ induces a graded map of $A$-algebras
$A[\phi]\colon A[t^\Gamma]\to A[t^{\Gamma'}]$ which is injective
(resp. surjective) if $\phi$ is. If the semigroup $\Gamma$ is
torsion-free, the semigroup algebra $A[t^\Gamma]$ injects into
$A[t^{\Z^d}]=A[t_1^{\pm 1},\ldots ,t_d^{\pm 1}]$ and therefore is
an integral domain if $A$ is.
\begin{proposition} \label{product}
Let $\Gamma, \Gamma'$ be two semigroups.
The map of $A$-algebras
$$A[t^{\Gamma\times\Gamma'}]\rightarrow A[u^\Gamma]\otimes_\A A[v^{\Gamma'}];\
\ t^{(\gamma,\gamma')}\mapsto u^\gamma\otimes_A v^{\gamma'}$$ is
an isomorphism.
\end{proposition}
\textit{Proof.} This follows immediately from the universal
property. \hfill $\ {\Box}$




\section{Algebraic tori} \label{tori}

Let $k$ be a field. The  multiplicative group  $k^*$ of non zero
elements of $k$ is equipped with the structure of algebraic group
over $k$, usually denoted by $\mathbf{G}_m := \mbox{Spec} k
[t^{\pm 1}]$. A $d$-dimensional \textit{algebraic torus}  over $k$
is an algebraic group isomorphic to a $(k^*)^d$.



If $M$ is a rank $d$ lattice then $T^M := \mathrm{Spec } k [t^M]$
is an algebraic torus over $k$. If we fix a basis $m_1, \dots,
m_d$ of the lattice $M$ we get a group isomorphism
\[ \Z^d \rightarrow
M, \quad a= (a_1, \dots, a_d) \mapsto \sum_{i=1, \dots d}  a_i m_i \]
 and
isomorphism of $k$-algebras $k [t_1^{\pm 1}, \dots, t_d^{\pm 1} ]
\rightarrow k [t^M]$ which induces an isomorphism $T^M (k)
\rightarrow (k^*)^d$.

\begin{remark}
More generally one can consider the scheme $\mathrm{Spec }  A
[t^M]$, which is an algebraic torus over $\mathrm{Spec } A$ for
any commutative ring $A$.
\end{remark}


A \textit{character} of the torus     $T(k) $ is a group
homomorphism $T(k) \rightarrow k^*$. The set of characters
$\mbox{\rm Hom }_{\mathrm{alg. groups}}({T}^M, k^*)$ of $T^M(k)$
is a multiplicative group isomorphic to the lattice $M$ by the
homomorphism given by $m \mapsto t^m$ for $m \in M$.  We identify
the monomials $t^m$ of the semigroup algebra $k[t^M]$ with the
characters of the torus.


By the universal property of the semigroup algebras applied to $k
[t^M]$ we have a representation of $k$-rational points of $T^M$ as
group homomorphisms:
\[
T^M (k) = \mathrm{Hom}_{\mathrm{groups}} (M , k^*) = N \otimes_\Z
k^*,
\]
where $N := \mbox{\rm Hom }(M, \Z)$  is the dual lattice of $M$.
We denote by  $\langle \, , \, \rangle \colon N \times M
\rightarrow \Z$ the duality pairing between the lattices $N$ and
$M$.


A  one parameter subgroup of  $T^M (k)$ is  group  homomorphism
$k^* \rightarrow  T^M (k) $. Any vector $\nu \in N$ gives rise to
a one parameter subgroup  $\l_{\nu} $ which maps $z \in k^*$ to
the closed point of    $T^M (k)$ given by the homomorphism of
semigroups $M \rightarrow k^*$, $m \mapsto z^{\langle \nu, m
\rangle}$.  The set of one parameter subgroups $\mbox{\rm Hom
}_{\mathrm{alg. groups}}(k^* , {T}^M)$  forms a multiplicative
group, which is isomorphic to $N$ by the homomorphism given by
$\nu \mapsto \l_\nu$.








 \section{Affine toric varieties}\label{affine}






In this section  we consider a finitely generated subsemigroup
$\Gamma$ of a free abelian group $M$ of rank $d$. We assume in
addition that the group $\Z \Gamma$ generated by $\Gamma$ is equal
to $M$. We denote by $N$ the dual lattice of $M$. We introduce some useful
 notations.

\begin{notation}
We denote by $M_\R$ the $d$-dimensional real vector space $M
\otimes_\Z \R$. The semigroup $\Gamma$, viewed in $M_\R$, spans
the cone $\R_{\geq 0} \Gamma \subset M_\R$  which we denote also
by $\check{\sigma}$. The dual cone
of $\check{\s}$ is the cone $\s:= \{ \nu \in N_\R \mid \langle
\nu, \g \rangle \geq 0,\, \forall \g \in \check{\s} \}$.  We use
the notation $\t \leq \s$ to indicate that $\t$ is a \textit{face}
of $\s$.
 Any face of $\check{\s}$ is of the form
$\check{\s} \cap \t^\bot$ for a unique face $\t$ of $\s$, where
$\t^\bot$ is the linear subspace $ \{ \g \in M_\R \mid \langle
\nu, \g \rangle = 0, \, \forall \nu \in \t \}$.
\end{notation}

Let $\gamma_1,\ldots ,\gamma_{r}$ be generators of $\Gamma$.  Then the semigroup $\Gamma$ is the image of
$\N^{r}\subset \Z^r$ by the surjective linear map
$b\colon \Z^{r}\to M$ determined by $b(e_i)=\gamma_i$ where the $e_i,\ 1\leq i\leq {r}$
form the canonical basis of $\N^{r}$. The    kernel $\L$ of $b$ is isomorphic to $\Z^{{r}-d}$.

Let
us consider the map of semigroup algebras associated to the map
$b\vert \N^{r}\colon\N^{r}\to \Z^d$, whose image is $\Gamma$. It is a
map of $A$-algebras $A[U_1,\ldots ,U_{r}]\to A[t_1^{\pm 1},\ldots
,t_d^{\pm 1}]$. Its image is the subalgebra $A[t^\Gamma]$ of
$A[t_1^{\pm 1},\ldots ,t_d^{\pm 1}]$.

  An element $m\in \Z^{r}$
can be written uniquely $m=m_+-m_-$ where $m_+$ and $m_-$ have non
negative entries and disjoint support.

By construction, the
kernel of the surjection $A[U_1,\ldots ,U_{r}]\to A[t^\Gamma]$ is
the ideal generated by the binomials $(U^{m_+}-U^{m_-})$ where
$b(m_+)=b(m_-)$. It is the \textit{toric ideal} associated to the
map $ b$. Note that it is not in general generated by the
binomials associated to a basis of $\L$.
Since the algebra
$A[t^\Gamma]$ is an integral domain if $A$ is, the toric ideal is
a prime ideal in that case.

Conversely, assuming now that $A$ is an algebraically closed field
$k$, an ideal generated by binomials in $k[U_1,\ldots ,U_{r}]$ is
called a \textit{binomial ideal}. Those ideals are studied in
\cite{E-S}, where it is shown that a prime binomial ideal
$I\subset k[U_1,\ldots ,U_{r}] $ gives rise to a semigroup algebra
$k[U_1,\ldots ,U_{r}] /I\simeq k[t^\Gamma]$, where
$\Gamma=\N^{r}/_\sim$, and $\sim$ is an equivalence relation
associated to the binomial relations.   The \textit{affine toric
variety} ${T}^\Gamma : =\hbox{\rm Spec } k[t^\Gamma]$ is the
subvariety of the affine space $\A^{r} (k)$ defined by the
binomial equations generating the toric ideal.  By the universal
property of the semigroup algebra, there is a bijection
$$\{\hbox{\rm Closed points of } \hbox{\rm Spec }k[t^\Gamma]\}\leftrightarrow
\{ \hbox{\rm semigroup homomorphisms} \ \Gamma\to k\},$$  where
$k$ is considered as semigroup with respect to multiplication.

In particular, the torus  ${T}^M (k) = \hbox{\rm
Hom}_{\mathrm{groups}} (M,k^*)$ is embedded in ${T}^{\Gamma}$, as
the principal open set where $t^{\g_1 } \cdots t^{\g_r } \ne 0$.

From the description of closed points of ${T}^\Gamma$ in terms of
homomorphisms of semigroups we have an action of the torus ${T}^M
(k) $ on ${T}^\Gamma (k)$.
Another way to describe this action, which shows that it is
algebraic, is to say that thanks to the universal property of
semigroup algebras it corresponds to the composed map of
$k$-algebras
$$k[t^\Gamma]\rightarrow k[t^\Gamma]\otimes_k k[t^\Gamma]\rightarrow k[t^M]\otimes_k k[t^\Gamma]$$
where the first map is determined by $t^\gamma\mapsto
t^\gamma\otimes_k t^\gamma$ and the second by the inclusion
$\Gamma\subset M$. The corresponding map ${T}^M \times {T}^\Gamma\to
{T}^\Gamma$ is the action.



 Let us now seek the invariant
subsets of $T^{\Gamma}$ under the torus action.







\begin{definition}
Given a semigroup $\Gamma$, a subsemigroup $F\subset \Gamma$ is a
\textit{face} of $\Gamma$ if whenever $x,y\in \Gamma$ satisfy
$x+y\in F$, then $x$ and $y$ are in $F$.
\end{definition}
Let us remark that this condition is equivalent to the fact that
the vector space of finite sums $\sum_{\delta\in \Gamma\setminus
F}a_\delta t^\delta$ is in fact a prime ideal $I_F$ of
$k[t^\Gamma]$. It also implies that $\Gamma\setminus F$ is a
subsemigroup of $\Gamma$ (which in general is not finitely generated) and that the Minkowski sum $\Gamma +
(\Gamma \setminus F) $ is contained in $\Gamma \setminus F$.


\begin{lemma} \label{face}
The faces of the semigroup $\Gamma$ are of the form $\Gamma \cap
\t^\bot$, for $\t \leq \s$.
\end{lemma}
\textit{Proof.} Let $F$ be a face of the semigroup $\Gamma$. Then
there is a face $\check{\s} \cap \t^\bot$ of $\check{\s}$  which
contains $F$ and is of minimal dimension. Then $F$ is also a face
of the semigroup $\Gamma \cap \t^\bot$ and there is an element
$\g_0 \in F$ which belongs to the relative interior of the cone
$\check{\s} \cap \t^\bot$. Under these conditions is enough to
prove that if $\t= 0$ then $F = \Gamma$.

Notice that if $\g \in \Gamma $ and if $(\g + \Gamma) \cap
\Z_{\geq 0} \g_0 \ne \emptyset$ then $\g \in F$ since $F$ is a
face and $\g_0 \in F$. By Theorem 1.9 \cite{K-K} there is $\d_0
\in \Gamma \cap \int (\check{\s} ) $ such that $\d_0 + \check{\s}
\cap M \subset \Gamma$.  We deduce that the intersection $(\g +
\d_0 + \check{\s} \cap M ) \cap \Z_{\geq 0} \g_0$ is non-empty,
for any $\g \in \Gamma$,
 since $\g_0 \in \int (\check{\s}) \cap \Gamma$. \hfill ${\Box}$


\begin{notation}
If $\t \leq \s$ the set $\Gamma \cap \t^\bot$ is a subsemigroup of
finite type of $\Gamma$. If $\t \leq \s$ the lattice $M (\t,
\Gamma)$  spanned by
 $\Gamma \cap \t^\bot$
is a sublattice of finite index of $M (\t) := M \cap
 \t^\bot$.
\end{notation}

\begin{remark} \label{orb-clos}
The torus of the affine toric variety ${T}^{\Gamma \cap \t^\bot}$
is $ T^{M(\t, \Gamma)}$. If $A$ is a commutative ring, the
homomorphism of $A$-algebras $ A[\Gamma] \rightarrow A[\Gamma \cap
\t^\bot] \cong A [\Gamma] / I_{\Gamma \cap \t^\bot}$,   is
surjective and defines a closed embedding \[ i_\t: {T}^{\Gamma
\cap \t^\bot} \hookrightarrow {T}^{\Gamma} \]  over $\mathrm{Spec
} A$.  If $k =A$ the image by the embedding $i_\t$ of a closed
point $u \in T^{\Gamma \cap \t^\bot} (k)$ (or $u \in  T^{M( \t,
\Gamma)} (k) $) is the semigroup homomorphism $i_\t (u) : \Gamma
\rightarrow k$ given by
\[
\g \mapsto \left\{
\begin{array}{cr}
u(\g) & \mathrm{if} \g \in \t^\bot,
\\
 0  & \mathrm{otherwise}.
\end{array}
\right.
\]
\end{remark}

\begin{proposition} \label{orbit-aff}
The map
\[
\t \mapsto \orb (\t, \Gamma):= i_\t( {T}^{M(\t, \Gamma)})  \quad
(\mbox{ resp. } \t \mapsto i_\tau ({T}^{\Gamma \cap \t^\bot})\,) \] defines
a bijection (resp. inclusion-reversing bijection) between
 the faces of $\s$ and the
orbits (resp. the closures of the orbits) of the torus action on
${T}^\Gamma$.
\end{proposition}
\textit{Proof.}  Let $u: \Gamma \rightarrow k$ be a semigroup
homomorphism. Then $u^{-1} (k^*)$ is a face of $\Gamma$, hence of
the form $\Gamma \cap \t^\bot$ for some face $\t $ of $\s$. Any
such $u$ extends in a unique manner to a group homomorphism $M(\t,
\Gamma) \rightarrow k^*$ defining an element of the torus
${T}^{M(\t, \Gamma)}$ of the affine toric variety ${T}^{\Gamma
\cap \t^\bot}$. Conversely, given   a group homomorphism  $u
\colon M(\t, \Gamma) \rightarrow k^*$ we define a semigroup
homomorphism $ i_\t (u) : \Gamma \rightarrow k$  as indicated
above.

It follows that the orbit of the point defined by $u$ by the
action of $T^{M}$ coincides with the image by $i_\t$ of the orbit
${T}^{M(\t, \Gamma)}$ of the point $u_{| \Gamma \cap \t^\bot}
\colon \Gamma \cap \t^\bot \rightarrow k^*$ on the toric variety
${T}^{\Gamma \cap \t^\bot}$.  The rest of the assertion follows
from Remark \ref{orb-clos}. \hfill $\ {\Box}$



The partition induced by the orbits of the torus action on
${T}^{\Gamma}$  is of the form:
\begin{equation} \label{orbit}
{T}^{\G} = \bigsqcup_{\t \leq \s} \orb (\t, \G).
\end{equation}

\begin{proposition} \label{affine-inv}
If $X$ is an affine toric variety with torus ${T}^M$ then $X$ is
${T}^M$-equivariantly isomorphic to ${T}^{\Gamma}$, where $\Gamma
\subset M$ a semigroup  of finite type such that $\Z \Gamma =M$.
\end{proposition}
{\textit Proof.} This is well-known (see  Proposition
2.4, Chapter 5 of \cite{GKZ}). \hfill $\ {\Box}$



We characterize the affine $T^M$-invariant open subsets of
$T^{\Gamma}$.
\begin{definition}
For any face $\t$ of $\s$ the set
\begin{equation} \label{Gamma_tau}
 \Gamma_\t := \Gamma + M ( \t,\Gamma)
\end{equation} is a semigroup of finite type
generating the lattice $M$.
\end{definition}
Notice that the cone $\Gamma_\t \R_{\geq 0} $ is equal to
$\check{\t}$ and  if
 $\t \leq \s$ the set $\int ( \check{\s} \cap
\t^\bot ) \cap \Gamma$ is non empty ($\int$ denotes relative
interior).


\begin{lemma} $\,$ \label{conditions-aff}
\begin{enumerate}
\item[{\rm i.}] The minimal face of the semigroup $\Gamma$ is a
sublattice of $M$  equal to $\Gamma \cap \s^\bot$.

\item[{\rm ii.}] For any $m \in \Gamma$ in the relative interior
of $( \check{\s} \cap \t^\bot )$  we have that
\[
\Gamma_\t = \Gamma + \Z_{\geq 0} (-m).
\]
\item[{\rm iii.}] If $\t \leq \theta \leq \s$ we have that $M (\t,
\Gamma_\theta) = M(\t , \Gamma_\t)$ and $\Gamma_\t = \Gamma_\theta
+  M(\t, \Gamma_\theta)$.
\end{enumerate}
\end{lemma}
\textit{Proof.} i.~By Lemma \ref{face} the correspondence $\t
\mapsto \Gamma \cap \t^\bot$ is a bijection between the  faces of
the cone $\s$  and the faces of the semigroup $\Gamma$. By duality
the minimal face of $\Gamma$ is equal to $\Gamma \cap \s^\bot$. It
is enough to prove that if $\Gamma$ is a semigroup such that
$\Z\Gamma = M$ and $\Gamma \R_{\geq 0} = M_\R$ then
$\Gamma = M$. Since $M$ is the saturation of $\Gamma$ the
assertion reduces to the case of rank one semigroups, for which it
is elementary by Bezout identity.

ii.~If $m  \in \int ( \check{\s} \cap \t^\bot ) \cap \Gamma$ then
the semigroup $\Gamma + \Z_{\geq 0} (-m) \subset M$ spans the cone
$\check{\t} = \check{\s} + \t^\bot \subset M_\R$. By i.~the
minimal face of this semigroup is the lattice $(\Gamma + \Z_{\geq
0} (-m)) \cap \t^\bot$ which coincides by definition with the
lattice $ M(\t, \Gamma)$.

iii.~The lattices $M(\t, \Gamma_\theta)$ and $M(\t, \Gamma)$ are
both generated by $\Gamma \cap \t^\bot$ hence are equal. We have
that $\Gamma_\t = \Gamma_\theta +  M(\t, \Gamma_\theta)$ since
$\theta^\bot \subset \t^\bot$. \hfill $\ {\Box}$

\begin{lemma} \label{open-embedding}
If  $\t \leq \s$ the inclusion of semigroups $\Gamma \subset
\Gamma_\t$ determines a ${T}^M$-equivariant  embedding
${T}^{\Gamma_{\t}} \subset {T}^{\Gamma}$ as an affine open set.
Conversely, if $X \subset T^{\Gamma}$ is  a ${T}^M$-equivariant
embedding of an affine open set then there is a unique  $\t \leq
\s$ such that $X$ is ${T}^M$-equivariantly isomorphic to
$T^{\Gamma_{\t}}$.
\end{lemma}
\textit{Proof.} By Lemma \ref{conditions-aff} we have that
$\Gamma_\t = \Gamma + \Z_{\geq 0} (-m)$. More generally  if $\g
\in \Gamma$ and  $f = t^\g$,  the localization ${T}^{\Gamma}_f =
\mbox{\rm Spec \, } k [ \Gamma ]_f $ is equal to ${T}^{\Gamma +
(-\g)\Z_{\geq 0}}$ and it is embedded in ${T}^{\Gamma}$ as a
principal open set.

Conversely, an affine ${T}^M$-invariant open subset  of
${T}^{\Gamma}$ is an affine toric variety for the torus ${T}^M$
hence it is of the form ${T}^{\Lambda}$, for $\Lambda \subset M$ a
subsemigroup of finite type, such that $\Z \Lambda  = M$
(see Proposition \ref{affine-inv}). We denote the cone $ \R_{\geq
0} \Lambda$ by $\check{\theta}$. Since the embedding
${T}^{\Lambda} \subset {T}^{\Gamma}$ is ${T}^M$ equivariant it is
defined by the inclusion of algebras $k [t^{\Gamma}] \rightarrow k
[ t^{\Lambda} ]$ corresponding to the inclusion of semigroups
$\Gamma \subset \Lambda$. We deduce that $\check{\s} \subset
\check{\theta}$ and hence that $\theta \subset \s$ by duality. We
prove that if $\t$ is the smallest face of $\s$ which contains
$\theta$ then $\Lambda = \Gamma_{\t}$. It is enough to prove that
if $\int{( \theta )} \cap \int (\s)  \ne \emptyset$ then $\Lambda =
\Gamma$.

Notice that the lattice $F= \s^\bot \cap M$ is the minimal face of
$\Gamma$ and the prime ideal $I_{F}$ of $k[ t^{\Gamma} ]$ defines
the orbit $\orb (\s, \Gamma)$, which is embedded as a closed
subset of ${T}^{\Gamma}$. Let us consider a vector $\nu$ such that
$\nu \in \int(\theta) \cap \int (\s)$. Then we get that  $ \s^\bot
\cap M = \check{\s}  \cap \nu^\bot  \cap M $ is contained in  $
\check{\theta}\cap \nu^\bot \cap M   = \theta^{\bot} \cap M $
hence $\Gamma \setminus (\s^\bot \cap M)$ is contained in $
\Lambda \setminus (\theta^\bot \cap M)$ and therefore $1 \notin
I_F k [t^{\Lambda}]$. Since ${T}^{\Lambda} \subset {T}^{\Gamma}$
is an open immersion  $\orb (\s, \Gamma)$ is contained in
${T}^{\Lambda}$. By  (\ref{orbit}) and Proposition \ref{orbit-aff}
the closure of any orbit contained  $  {T}^{\Gamma} $ contains
$\orb (\s, \Gamma)$ thus  ${T}^{\Gamma} \subset {T}^{\Lambda}$.
\hfill $\ {\Box}$

\begin{remark} \label{open-nor}
The immersion of ${T}^M$-invariant affine open subsets is
compatible with normalization. By Lemma \ref{open-embedding} any
$T^{M}$-invariant affine open set of ${T}^{\Gamma}$ is of the form
$X^{\Gamma}_f$ for $f = t^\g$, $\g \in \Gamma$. Then the following
diagram commutes:
\[
\begin{array}{ccc}
{T}^{\check{\sigma} \cap M } &  \hookrightarrow &  {T}^{\Gamma}
\\
\uparrow &  & \uparrow
\\
{T}^{\check{\sigma} \cap M }_f  & \hookrightarrow &
{T}^{\Gamma}_f,
\end{array}
\]
since $\Gamma + (-\g )\Z_{\geq 0}$ is saturated in $\check{\sigma}
\cap M + (-\g )\Z_{\geq 0}$. The vertical arrows are embeddings as
principal open sets while the horizontal arrows are normalization
maps (see Proposition \ref{normal}).
\end{remark}




\end{section}


\begin{section}{Toric varieties}


Recall that a \textit{fan} is a finite set $\Sigma$  of strictly
convex polyhedral cones rational for the a lattice $N$, such that
if $\s \in \Sigma$ any face $\t$ of $\s$ belongs to $\Sigma$ and
if $\s, \s' \in \Sigma$ the cone $\t= \s \cap \s'$ is in $\Sigma$.
If $j \geq 0$ is an integer the subset of $\Sigma (j)$ of
$j$-dimensional cones of $\Sigma$ is called the
\textit{$j$-skeleton} of the fan. The \textit{support} of the fan
$\Sigma$ is the set $|\Sigma | = \cup_{\s \in \Sigma} \s \subset
N_\R$.


We give first a \textit{combinatorial definition} of toric
varieties.

\begin{definition} \label{def-general-toric}
A toric variety is given by the datum of a triple $(N, \Sigma,
\Gamma)$ consisting of lattice $N$,  a fan $\Sigma$ in $N_{\mathbf
R}$ and a family of finitely generated subsemigroups $\Gamma = \{
\Gamma_\s \subset \mid \s \in \Sigma \}$ of a lattice $M =
\mbox{Hom} (N, \Z)$ such that:
\begin{enumerate}
\item[{\rm i.}]  $\Z \Gamma_\s = M$, for $\s  \in \Sigma$.
\item[{\rm ii.}] $\Gamma_\t = \Gamma_\s + M (\t, \Gamma_\s)$, for
a each $\s \in \Sigma$ and  any face $\t$ of $\s$.
\end{enumerate}
 The  corresponding toric variety ${T}_{\Sigma}^{\Gamma}$
 is the union of the affine varieties ${T}^{\Gamma_\s}$
for $\s \in \Sigma$ where for any pair $\s, \s'$ in $\Sigma$ we
glue up ${T}^{\Gamma_\s}$ and ${T}^{\Gamma_{\s'}}$ along their common
open affine variety ${T}^{\Gamma_{\s \cap \s'}}$.
\end{definition}

\begin{remark}
The lattice $N$ in the triple $(N, \Sigma, \Gamma)$ is determined
by $\Gamma$. We recall it by convenience. We omit the reference to
the lattice $N$ in the notation $T^\Gamma_\Sigma$.
\end{remark}

\begin{remark}
This definition is consistent with the case of affine toric
varieties. Let  ${T}^\Gamma$ be an affine toric variety in the
sense of Section \ref{affine}.  If $\s' := \{ \t \mid \t \leq \s
\} $ and  $\Gamma' := \{ \Gamma_\t \mid \t \leq \s \}$, where
$\Gamma_\t$ is the semigroup defined by (\ref{Gamma_tau}) for $\t
\leq \s$ then the conditions i.~and ii.~are satisfied by Lemma
\ref{conditions-aff}. Then ${T}^\Gamma $ is $T^M$-invariantly
isomorphic to ${T}^{\Gamma'}_{\s'}$.
\end{remark}


\begin{remark}
 A triple $(N, \Sigma, \Gamma)$
determines similarly a toric scheme over $\mathrm{Spec } A$, for
any commutative ring $A$.
\end{remark}


\begin{lemma}\label{sum}
Let $(\Sigma, \Gamma)$ as in Definition \ref{def-general-toric}
define a toric variety ${T}_{\Sigma}^{\Gamma}$. Then we have:
\begin{enumerate}
\item[{\rm i.}]
 If
$\s, \theta \in \Sigma$ and if $\t = \s \cap \theta$ then
$\Gamma_\t = \Gamma_\s + \Gamma_\theta$. \item[{\rm ii.}] The
variety  ${T}_{\Sigma}^{\Gamma}$ is separated.
\end{enumerate}
\end{lemma}
{\textit Proof.} The intersection $\t=\s \cap \theta$ is a face of
both $\s$ and $\theta$. By Lemma \ref{conditions-aff} we have that
$M(\t, \Gamma_\t) = M(\t, \Gamma_\s) = M (\t, \Gamma_\theta)$. By
axiom i.~in the Definition \ref{def-general-toric} we get
$\Gamma_\theta, \Gamma_\s \subset \Gamma_\t$ and $\Gamma_\theta +
\Gamma_\s \subset \Gamma_\t$. Conversely, by the separation lemma
for polyhedral cones, for any  $u \in \int (\check{\s} \cap
(-\check{\theta}))$ we have that $\t = \s \cap u^\bot = \theta
\cap u^\bot$.  Notice that we can assume that $ u \in \Gamma_\s
\cap (-\Gamma_\theta) \cap \int (\check{\s} \cap
(-\check{\theta})) \ne \emptyset$. Then by Lemma
\ref{conditions-aff} we obtain $\Gamma_\t = \Gamma_\s + \Z_{\geq
0}  (-u)$. Hence $\Gamma_\t$ is contained in $\Gamma_\s +
\Gamma_\theta$ since $-u \in \Gamma_\theta$.



 The homomorphism $k
[t^{\Gamma_\theta}] \otimes_k k [t^{\Gamma_\s}] \rightarrow k
[t^{\Gamma_\t}]$ which sends $t^{\gamma} \otimes t^{\gamma'}
\mapsto t^{\gamma+ \gamma'}$ is surjective since $\Gamma_\s +
\Gamma_\theta = \Gamma_\t$. In geometric terms this implies that
the diagonal map ${T}^{\Gamma_\t} \rightarrow {T}^{\Gamma_\theta}
\times {T}^{\Gamma_\s}$ is a closed embedding for any $\theta, \s
\in \Sigma$ with $\t = \theta \cap \t$, hence the variety
${T}^{\Gamma}_\Sigma$ is separated (see Chapter 2 of \cite{Ha}).
\hfill $\ {\Box}$







\begin{remark}\label{norma} The morphisms
corresponding to the inclusions $k[t^{\Gamma_\sigma}]\to
k[t^{\Gamma_\sigma+\Gamma_{\sigma'}}]$ are open embeddings
compatible with the normalization maps. The normalization of the
toric variety ${T}_{\Sigma}^{\Gamma}$ is the toric variety
${T}_{\Sigma}$ corresponding to the fan $\Sigma$ and the
normalization map is obtained by gluing-up normalizations
${T}_{\s}:= T^{\check{\s} \cap M} $ of the charts
${T}^{\Gamma_\s}$, for $\Gamma_\s \in \Gamma$ and $\s \in \Sigma$.
\end{remark}

\begin{lemma} Let $\l_v$ be a one-parameter subgroup of the
torus $T^M$ for some $v \in N$. Then $\lim_{z \rightarrow 0} \l_v
(z)$ exists in the toric variety ${T}^{\Gamma}_\Sigma$ if and only
if $v$ belongs to $\vert\Sigma\vert \cap N $.
\end{lemma}
{\em Proof.} The statement is well-known in the normal case (see
Proposition 1.6 \cite{Oda}).  The normalization map $n \colon
{T}_\Sigma \rightarrow {T}^\Gamma_\Sigma$ is an isomorphism over
the torus $T^{M}$. If $\l_v \colon k^* \rightarrow T^{M} \subset
{T}^\Gamma_\Sigma$ is a one-parameter subgroup defined by $v \in
N$ it lifts to the normalization, i.e., there is a morphism
$\bar{\l}_v \colon k^* \rightarrow T^{M} \subset {T}_\Sigma$ in
such a way that $n
 \circ \bar{\l}_v = \l_v$.
Since the normalization is a proper morphism we get by the
valuative criterion of properness that $\lim_{z \rightarrow 0}
\l_v (z)$ exists in the toric variety ${T}^{\Gamma}_\Sigma$ if and
only if $\lim_{z \rightarrow 0} \bar{\l}_v (z)$ exists in
${T}_\Sigma$. \hfill $\ {\Box}$




\begin{lemma} \label{orbit-gen}
Let $X^\Gamma_\Sigma$ be a toric variety. Then the map
\[
\t \mapsto \orb (\t, \Gamma_{\t}):= i_\t( {T}^{M(\t, \Gamma_\t)})
\] defines a bijection
between the faces of $\Sigma$ and the orbits of the torus action
on ${T}^\Gamma_\Sigma$.
\end{lemma}
\textit{Proof.} This is consequence of the definitions and Lemma
\ref{orbit-aff}. \hfill $\ {\Box}$

In order to illustrate the combinatorial definition of a toric
variety we describe the orbit closures as toric varieties.



\begin{notation}   \label{ind}
If $\t \in \Sigma$ we denote by $N_\t$ the sublattice of $N$
spanned by $\t \cap N$ and by $N(\t)$ the quotient $N /
N_\t$. The lattice $N (\t)$ is the dual lattice of $M(\t) = M \cap
\t^\bot$. Since $M(\t, \Gamma_\t)$ is a sublattice of finite index
$i (\t, \Gamma_\t)$ of $M(\t)$ then the dual lattice $N(\t,
\Gamma_\t)$ of $M(\t, \Gamma_\t)$ contains $N(\t)$ as a sublattice
of  finite index equal to  $i (\t, \Gamma_\t)$.
\end{notation}

If $\s \in \Sigma$ and $\t \leq \s$ the image $\s(\t)$ of $\s$ in
$N(\t)_\R = N_\R / (N_\t)_\R$ is a polyhedral cone, rational for
the lattice $N(\t, \Gamma_\t)$. The set $\Sigma(\t) := \{ \s(\t)
\mid \s \in \Sigma, \t \leq \s \}$ is a fan in $N(\t)_\R$. If
$\s(\t) \in \Sigma (\t)$ we set $\Gamma_{\s(\t)} := \Gamma_\s \cap
\t^\bot$.  The set  $\s(\t) \subset N(\t)_\R$ is
 the dual cone   of the cone spanned by $\Gamma_\s
\cap \t^\bot$ in $M(\t)_\R$. Let us denote by $\Gamma(\t)$ the set
$\{ \Gamma_{\s(\t)} \mid \s(\t) \in \Sigma(\t) \}$.





\begin{lemma} \label{orbit-closure}
Let $X^\Gamma_\Sigma$ be a toric variety. If $\t \in \Sigma$ the
triple  $( N(\t, \Gamma_\t), \Sigma(\t), \Gamma(\t) )$ defines a
toric variety $T_{\Sigma (\t)}^{\Gamma(\t)}$. We have a closed
embedding $i_\t : T_{\Sigma (\t)}^{\Gamma(\t)} \to
T^{\Gamma}_{\Sigma}$. The map
\[
\t \mapsto i_{\t}({T}_{\Sigma(\t)}^{\Gamma(\t)})
\] defines a bijection
between the faces of $\Sigma$ and orbit closures of the action of
${T}^M$ on ${T}^\Gamma_\Sigma$.
\end{lemma}
\textit{Proof.} If $\t$ is not a face of $\s$, for $\s \in \Sigma$
then $\orb (\t, \Gamma_{\t})$ is does not intersect the affine
invariant open set ${T}^{\Gamma_\s}$; if $\t \leq \s$, for $\s \in
\Sigma$  the closure of the orbit $\orb (\t, \Gamma_{\t})$ in the
affine open set ${T}^{\Gamma_\s}$ is equal to ${T}^{\Gamma_\s \cap
\t^\bot}$ (see  Lemma \ref{orbit-aff}).



If $\t \leq \theta \leq \s$ then $\theta(\t) \leq \s(\t)$ and
$\theta^\bot \subset \t^\bot$ hence $M(\theta, \Gamma_\s) = M (
\theta(\t), \Gamma_\s \cap \t^\bot)$ is the sublattice spanned by
$\Gamma_\s \cap \theta^\bot$.

If $\t \leq \s, \s' $ and if $\theta = \s \cap \s'$ then we deduce
from condition ii.~in Definition \ref{def-general-toric} that:
\[
\Gamma_\theta \cap \t^\bot = \Gamma_\s \cap \t^\bot + M(\theta
(\t) , \Gamma_{\s (\t)} ) = \Gamma_{\s'} \cap \t^\bot +
M(\theta(\t), \Gamma_{\s'(\t)}).
\]
We obtain that  the triple $( N(\t, \Gamma_\t), \Sigma(\t),
\Gamma(\t) )$ satisfies the axioms in Definition
\ref{def-general-toric} with respect to the torus $T^{M(\t,
\Gamma_\t)}$.

We have also described an embedding ${T}_{\Sigma(\t)}^{\Gamma(\t)}
\hookrightarrow {T}^\Gamma_\Sigma$ in such a way that the
intersection of this variety with any affine chart containing
$\orb(\t, \Gamma)$ is the closure of the orbit   $\orb(\t,
\Gamma)$ in the chart. The conclusion follows from Lemma
\ref{orbit-gen}.
 \hfill $\ {\Box}$
\begin{remark} The non singular locus of the toric variety
${T}^\Gamma_\Sigma$ is the union of the orbits $\orb(\t, \Gamma)$
corresponding to \textit{regular} cones $\t \in \Sigma$ such their
index $i (\t, \Gamma_\t)$ is equal to $1$.

\end{remark}



\end{section}




\begin{section} {Blowing ups}\label{blow}

The theory of normal toric varieties deals with normalized
equivariant blowing ups, i.e., blowing ups of equivariant ideals
followed by normalization. In this section we build blowing ups of
equivariant ideals in toric varieties.

Let $\sigma$ be a strictly convex rational cone in $N_{\mathbf R}$
and $\Gamma$ a subsemigroup of finite type of the lattice $M$ such
that $\Z \Gamma = M$ and the saturation of $\Gamma$ in $M$ is
equal to $\check\sigma\cap M$. For simplicity we assume that the
cone $\s$ is of dimension $d$ hence $\check \s$ is strictly
convex.

Let us consider a graded ideal ${\mathcal I}$  in $A[t^\Gamma]$,
which is necessarily generated by monomials $t^{m_1},\ldots
,t^{m_k}$. We build the corresponding \textit{Newton polyhedron}
$\Newton_\s ({\mathcal I})$, by definition the convex hull in
$M_{\mathbf R}$ of the $m_i+\check\sigma$, which is also the
convex hull of the set  $|{\mathcal I}|$ of exponents of monomials
belonging to the ideal ${\mathcal I}$ of $A[t^{\Gamma}]$. It is
quite convenient to denote with the same letter $\mathcal{I}$ the
set $\{ m_1, \dots m_k \}$.



The set  $\mathcal{I}$ determines the \textit{order function}:
\begin{equation} \label{ord}
\ord_{\mathcal{I}} \colon \s \rightarrow \R, \quad  \nu \mapsto
\min_{m \in \mathcal{I}} \langle \nu, m \rangle.
\end{equation}
The order function $\ord_{\mathcal{I}}$ coincides with the support
function of the polyhedron $\Newton_\s (\mathcal{I})$. It is a
gauge ($\ord_{\mathcal{I}}(\lambda u)=\lambda
\ord_{\mathcal{I}}(u)\ \ \hbox{\rm for }\lambda>0$) which is
piecewise linear. The maximal cones of linearity of the function
$\ord_{\mathcal{I}}$ form the $d$-skeleton  of the fan
$\Sigma(\mathcal{I})$ subdividing $\s$. Each such cone $\s_i $ in
the $d$-skeleton of  $\Sigma(\mathcal{I}) $ is the convex dual of
the convex rational cone generated by the vectors $(m -m_i)_{m \in
\Newton_\s (\mathcal{I})}$, where $m_i$ is a vertex of $\Newton_\s
(\mathcal{I})$. The correspondence $m_i \mapsto \s_i$
 is a bijection  between the set of vertices $\{m_1, \dots, m_s
\} \subset \mathcal{I}$ of the polyhedron $\Newton_\s
(\mathcal{I})$ and the $d$-skeleton of $\Sigma(\mathcal{I})$, such
that
\[
 m_i \mapsto \s_i  \mbox{ if and only if  } \ord_{\mathcal{I}} (\nu) =
\langle \nu, m_i \rangle \mbox { for all } \nu \in \s_i.
\]



 Note that $\Gamma\subset \check\sigma \cap M\subset
\check \sigma_{i} \cap M$. In each of the cones $\check\sigma_{i}$
we consider  the semigroup
\begin{equation} \label{Gamma}
\Gamma_i= \Gamma+\langle m_1-m_i,\ldots,
m_{i-1}-m_i,m_{i+1}-m_i,\ldots, m_k-m_i\rangle \subset
\check\sigma_i \cap M. \end{equation} By Lemma \ref{reduc}, the
saturation in $M$ of this semigroup is equal to $\check\sigma_i
\cap M$. We denote by $\Gamma(\mathcal{I})$ the set consisting of
the semigroups $\Gamma_i$,  together with $\Gamma_{i, \tau}$
(defined by equation (\ref{Gamma_tau})) for $\tau \leq \s_i$,
$i=1, \dots, s$.


\begin{proposition} \label{blowing}
The triple $(N, \Sigma  (\mathcal{I}), \Gamma(\mathcal{I}))$
defines a toric scheme $B$ over $\hbox{\rm Spec }A$. The
inclusions $\Gamma \subset \Gamma_i$, $i=1, \dots,s $, determine a
map of schemes
\[ \p \colon B  \rightarrow \hbox{\rm Spec }A[t^{\Gamma}]
\]
over $ \hbox{\rm Spec }A$, which is the blowing up of the ideal
$\mathcal{I}$.
\end{proposition}
\textit{Proof.} We prove  first that  the triple $(N, \Sigma
(\mathcal{I}), \Gamma(\mathcal{I}))$ verifies the compatibility
conditions stated in Definition \ref{def-general-toric}. By  Lemma
\ref{conditions-aff} it is enough to check them for the affine
open sets corresponding to two vertices, say $m_1$ and $m_2$, of
$\Newton_\s( \mathcal{I})$. Then, if $\t = \s_1 \cap \s_2$ the
condition we have to prove is that $\Gamma_{1, \t} = \Gamma_{2,
\t}$.


Notice that the vector $m := m_2 - m_1 \in \Gamma_1$ belongs to
the interior of $\check\s_1 \cap \t^\perp$. By Lemma
\ref{conditions-aff}  and the definitions we get $\Gamma_{1, \t} =
\Gamma_1 + \Z_{\geq 0} (- m)$ and similarly $\Gamma_{2, \t} =
\Gamma_2 + \Z_{\geq 0} m $.  Then the assertion follows since
$\Gamma_{1, \t}$, which  is equal to
\[
  \Gamma +   \Z (m_2 - m_1) + \sum_{j=2, \dots, k}
\Z_{\geq 0} (m_j - m_1) = \Gamma +   \Z (m_1 - m_2) + \sum_{j=2,
\dots, k}  \Z_{\geq 0} (m_j - m_2),
\]
is the same semigroup as $\Gamma_{2, \t}$.


It follows that the scheme $B$ is covered by the affine sets
$\hbox{\rm Spec }A[t^{\Gamma_i}]$ for $i =1, \dots, s$.
 Since each
$\Gamma_i$ contains $\Gamma$, there is a natural map
$\pi \colon   \hbox{\rm Spec }A[t^{\Gamma_i}]    \to \hbox{\rm Spec }A[t^\Gamma]$. The sheaf of
ideals on ${B}$ determined by the compositions with $\pi$ of the
generators of ${\mathcal I}$ is generated by $t^{m_i}\circ \pi$ in
the chart $\hbox{\rm Spec}A[t^{\Gamma_i}]$.


It is not difficult to prove that any  semigroup  $\Gamma_{i}$ defined
by (\ref{Gamma}), for $i
> s$, that is when $m_i$ is not a vertex of $\Newton_\s
(\mathcal{I})$, is of the form $\Gamma_{j, \t}$ for some $1 \leq j
\leq s$ and $\t \leq \s_j$. This means that the corresponding
affine chart $\hbox{\rm Spec }A[t^{\Gamma_i}]$ of the blowing up
of $\mathcal{I}$ is in fact an affine open subset of $\hbox{\rm
Spec }A[t^{\Gamma_j}]$, where $m_j$ is a vertex of $\Newton_\s
(\mathcal{I})$. \hfill $\ {\Box}$
\begin{corollary} The blowing-up of an equivariant sheaf of ideals on a toric variety
$T_\Sigma^\Gamma$ is a toric variety. Its description above each
equivariant open affine chart of $T_\Sigma^\Gamma$ is given by
Proposition \ref{blowing}.
\end{corollary}

\end{section}



\section{Toric morphisms}

Recall that a morphism $\phi \colon T^{M'} \to T^{M}$ of algebraic
tori gives rise to  two group homomorphisms
\[ \phi^*: M \to M' \mbox{ and } \phi_*:
N' \to N
\]
between the corresponding lattices of characters and  between the
corresponding lattices of one-parameter subgroups. The
homomorphisms $\phi^*$ and $\phi_*$ are mutually dual and
determine the morphism $\phi \colon T^{M'} \to T^{M}$ of algebraic
tori. Note that $\phi$ is defined algebraically by
\[
k[t^M] \to k [t^{M'}], \quad t^m \mapsto t^{\phi^* (m)}, \quad  m
\in M.
\]


Now suppose that we have two toric varieties $T^{\Gamma}_{\Sigma}$
and $T^{\Gamma'}_{\Sigma'}$ with respective tori $T^{M}$ and
$T^{M'}$ defined by the  combinatorial data given by the triples
 $(N, \Sigma,
\Gamma)$ and $(N', \Sigma', \Gamma')$ (see Definition
\ref{def-general-toric}).

\begin{definition} \label{map-fan}
The homomorphism $\phi_*$ is a \textit{map of fans with attached semigroups} $(N, \Sigma, \Gamma) \to (N', \Sigma', \Gamma ')$ if for
any $\s' \in \Sigma'$ there exists $\s \in \Sigma$ such that
$\phi^* ( \Gamma_\s) \subset \Gamma_{\s'} '$.
\end{definition}

Note then that  $\phi_*$  is a \textit{map of fans}, that is, for
any $\s' \in \Sigma'$ there is a cone $\s \in \Sigma$ such that
image by $\s'$ by the $\R$-linear extension of $\phi_*$ is
contained in $\s$. See Section 1.5 \cite{Oda}.

\begin{proposition} \label{morphism}
Let $\phi \colon T^{M'} \to T^M$ be a morphism of algebraic tori.
If $\phi_*$ defines a map of fans with attached semigroups $(N,
\Sigma, \Gamma) \to (N', \Sigma', \Gamma')$ then it gives rise to
a morphism: $ \bar{\phi} \colon T_{\Sigma'}^{\Gamma'} \to
T_{\Sigma}^{\Gamma} $ which extends $\phi \colon T^{M'} \to T^M$
and is equivariant with respect to $\phi$. Conversely, if $f
\colon T_{\Sigma'}^{\Gamma'} \to T_{\Sigma}^{\Gamma}$ is an
equivariant morphism with respect to $\phi$ then $\phi_*$ defines
a map of fans with  attached semigroups  $(N', \Sigma', \Gamma')
\to (N, \Sigma, \Gamma)$ and $f= \bar{\phi}$. In addition we have
a commutative diagram
\[
\begin{array}{ccc}
T_{\Sigma'} & \longrightarrow  & T_{\Sigma}
\\
\downarrow && \downarrow
\\
T_{\Sigma'}^{\Gamma'} & \longrightarrow & T_{\Sigma}^{\Gamma}
\end{array}
\]
 where the vertical arrows are
normalizations and the horizontal ones are the toric morphisms which
extend $\phi: T^{M'} \rightarrow T^M$.
\end{proposition}
\textit{Proof.} For any $\s' \in \Sigma'$ there exists a cone $\s
\in \Sigma$ such that the restriction of  $\phi^*$ determines a
semigroup homomorphism $\Gamma_\s \to \Gamma_{\s'}'$. The
corresponding homomorphism of $k$-algebras $k [\Gamma_\s] \to k
[\Gamma_{\s'}']$ defines a morphism:
\[
\bar{\phi}_{\s',\s} \colon T^{\Gamma_{\s'}'} \to T^{\Gamma_\s}
\mbox{ given on closed points by } \bar{\phi}_{\s',\s} (x) = x
\circ \phi^*_{| \Gamma_{\s}},
\]
where $x \in T^{\Gamma_{\s'}'}; x\colon \Gamma_{\s'}' \to k$  is a
homomorphism of semigroups. The morphism $\bar{\phi}_{\s',\s}$ is
equivariant through $\phi$ since for any  $y \in T^{M'}$, $y
\colon M' \rightarrow k^*$ group homomorphism  and  any $x \in
T^{\Gamma_{\s'}'}$ we get:
\[
\bar{\phi}_{\s',\s}  (y \cdot x) = (y\cdot x) \circ  \phi^*_{|
\Gamma_{\s}} = (y \circ \phi^*) \cdot (x \circ \phi^*  _{|
\Gamma_{\s}}) = \phi (y) \cdot \bar{\phi}_{\s',\s} (x).
\]
By gluing-up the affine pieces together we get a morphism
$\bar{\phi} \colon  T_{\Sigma'}^{\Gamma'} \to T_{\Sigma}^{\Gamma}$
which is equivariant with respect to $\phi$.


For the converse, since $f$ is assumed to be equivariant through
$\phi$ the image by $f$ of each orbit of the action of $T^{M'}$ on
$T^{\Sigma'}_{\Gamma'}$ is contained in one orbit of the action of
$T^M$ on $T_{\Sigma}^{\Gamma}$. If $\t' \leq \s'$ and $\s' \in
\Sigma'$ then the orbit $\orb (\s', \Gamma_{\s'}')$ is contained
in the closure of $\orb (\t', \Gamma_{\t'}')$ by Proposition
\ref{orbit-aff}. Then there exist $\s, \t \in \Sigma$ such that
\[ f (\orb (\s', \Gamma_{\s'}')) \subset \orb (\s, \Gamma_{\s}) \mbox{ and }
f (\orb (\t', \Gamma_{\t'}')) \subset \orb (\t, \Gamma_{\t}).\]
Since $f$ is continuous $\orb (\s, \Gamma_{\s})$ must be contained
in the closure of $\orb (\t, \Gamma_{\t})$, hence $\t$ is a face
of $\s$  by Proposition \ref{orbit-aff} and Lemma \ref{orb-clos}.
By (\ref{orbit}) it follows that $f(T^{\Gamma_{\s'}'}) \subset
T^{\Gamma_\s}$. The restriction $f_{|T^{\Gamma_{\s'}'}} \colon
T^{\Gamma_{\s'}'} \to T^{\Gamma_\s}$ is  equivariant with respect
to $\phi: T^{M'} \rightarrow T^{M}$. Hence $f_{|T^{\Gamma_{\s'}'}}
\colon T^{\Gamma_{\s'}'} \to T^{\Gamma_\s}$ is defined
algebraically by the homomorphism of $k$-algebras $k
[t^{\Gamma_\s}] \rightarrow k [t^{\Gamma_{\s'}'}]$, which is
obtained by restriction from the homomorphism of $k$-algebras $k
[t^{M}] \rightarrow k [t^{M'}]$ which maps $t^m \mapsto t^{\phi^*
(m)}$ for $m \in M$. This implies that $\phi^* (\Gamma_\s) \subset
\Gamma_{\s'}'$ and also that $f= \bar{\phi}$.

Since $\phi_*$ is a map of fans it defines a toric
morphism between the normalizations of $T^{\Gamma'}_{\Sigma'}$ and
 $T^{\Gamma}_{\Sigma}$. Finally, it is easy to check  that the diagram above
 is commutative.
 \hfill $\ {\Box}$

It is sometimes useful to consider morphisms of toric varieties which send
the torus of the source into a non dense orbit of the target:
Let $(N, \Sigma, \Gamma)$ and $( N', \Sigma', \Gamma')$ be two
triples defining toric varieties $T^{\Gamma}_{\Sigma}$ and
$T^{\Gamma'}_{\Sigma'}$. Let $\t $ be a cone of  $\Sigma$. Suppose
that we have  a morphism of algebraic tori
$ \phi:
T^{M'} \to T^{M(\t, \Gamma_\t)}$
such that $\phi_* : N' \to N(\t, \Gamma_\t)$ defines a map of fans with attached semigroups $(N', \Sigma', \Gamma') \to ( N   (\t,
\Gamma_\t) , \Sigma (\t) , \Gamma(\t))$. Then by Proposition
\ref{morphism} and Lemma \ref{orb-clos} we have a toric morphism
\[
\bar{\phi}: T_{\Sigma'}^{\Gamma'} \to T_{\Sigma(\t)}^{\Gamma(\t)}.
\]
Let us denote by $n \colon  T_{\Sigma} \rightarrow
T^{\Gamma}_{\Sigma}$ the normalization map and by  $\bar{i}_\t
\colon  T_{\Sigma(\t)} \to T_\Sigma $ the closed embedding of the
closure of $\orb (\t)$ in $T_\Sigma$. The following Proposition is
consequence of Proposition \ref{morphism} and Lemma
\ref{orb-clos}.


\begin{proposition} \label{morphism-emb}
The composite of $\bar{\phi}$ with the closed embedding $i_\t:
T_{\Sigma(\t)}^{\Gamma(\t)} \hookrightarrow T_\Sigma^\Gamma$ lifts
to the normalization of  $T_\Sigma^\Gamma$, i.e., there exists a
toric morphism $\psi: T_{\Sigma'}^{\Gamma'} \to  T_{\Sigma (\t)} $
such that
$
 i_\t \circ \bar{\phi} = n \circ \bar{i}_\t \circ \psi$
 if and only if
there is a lattice homomorphism $ \varphi^*: M(\t) \rightarrow M'
$ such that  $\varphi^*_{|  M(\t, \Gamma_\t)} = \phi^*$ and then
$\psi= \bar{\varphi}$.
\end{proposition}

\begin{example}
By Proposition \ref{morphism-emb} the map $ u \mapsto (u,0,0) $,
which parametrizes the singular locus of the Whitney umbrella $\{
x_1^2 x_2 - x_3^2 =0 \}$  does not lift to the normalization while
$u \mapsto (u^2,0,0) $ does.
\end{example}








\section{Abstract toric varieties}

We recall the \textit{usual} definition of toric variety.
\begin{definition} \label{usual}
A toric variety $X$ is an irreducible (separated) algebraic
variety equi\-pped with an action of an algebraic torus $T$
embedded in $X$ as a Zariski open set such that the action of $T$
on $X$ is morphism which extends the action of $T$ over itself by
multiplication.
\end{definition}

As stated in Proposition \ref{affine-inv} any affine toric variety
is  the spectrum of certain semigroup algebra. Gel{\cprime}fand,
Kapranov, and Zelevinsky have defined  and studied those
projective toric varieties which are equivariantly embedded in the
projective space, which is viewed as a toric variety, see
\cite{GKZ}, Chapter 5.




The following Theorem, which is consequence of a more general
result of Sumihiro, provides the key to establish a combinatorial
description of normal toric varieties.
\begin{theorem} (see \cite{Sumihiro-I}) \label{Sh}
Any normal toric variety $X$ has a finite  covering by $T$-invariant
affine normal toric varieties.
\end{theorem}
The statement of Theorem \ref{Sh} does not hold  if the normality
assumption is dropped.
\begin{example}
Let $C \subset \P^2_\C$ be the projective nodal cubic with
equation $y^2 z - x^2( x+ z)$. It is a rational curve with a node
singularity at $P = (0:0:1)$ and only one point $Q = (0:1:0)$ at
the line of infinity  $z=0$. The curve $C$ is rational and has a
parametrization $\p: \P^1_\C \rightarrow C$ such that $\p(0) =
\p(\infty) = P$ and $\p (1) = Q$. Then we have that $\p_{| \C^* }
\colon \C^* \rightarrow C \setminus \{ P \}$ is an isomorphism.
The multiplicative action of $\C^*$ on $\P^1_\C$ corresponds by
$\p$ to the group law action on the cubic hence it is algebraic.
It follows that $C$ is a toric variety with respect to Definition
\ref{usual}. Notice that $C$ is the only open set containing $P$
which is invariant by the action of $\C^*$. This example is also a
projective toric curve which does not admit any equivariant
embedding in the projective space (see \cite{Oda-M} page 4 and
\cite{GKZ} Chapter 5, Remark 1.6).
\end{example}


We modify the abstract definition of toric varieties as follows:
\begin{definition}
\label{new} A toric variety $X$ is an irreducible separated
algebraic variety equi\-pped with an action of an algebraic torus
$T$ embedded in $X$ as a Zariski open set such that the action of
$T$ on $X$ is morphism which extends the action of $T$ over itself
by multiplication and $X$ has a finite covering by affine
$T$-invariant Zariski open sets.
\end{definition}


\begin{theorem} \label{cate}
If $X$ is a toric variety in the sense of Definition \ref{new}
with torus $T$, then there exists a triple $(N, \Sigma, \Gamma)$
as in Definition \ref{def-general-toric} and an isomorphism
$\varphi \colon T \to T^M$ such that the pair $(T,X)$ is equivariantly isomorphic
to $(T^M, T^{\Gamma}_\Sigma)$ with respect to $\varphi$.
\end{theorem}
\textit{Proof.} We denote by $M$ the lattice of characters of the
torus $T$ hence $T= T^M$ and $N$ is the dual lattice of $M$.

By Proposition \ref{affine-inv} an affine $T^M$-invariant open
subset is of the form
 $T^{\Gamma_\s}$ where ${\Gamma_\s}$ is a subsemigroup of finite type of $M$ such that
$\Z {\Gamma_\s} = M$, and $\s  \subset N_\R$ is the dual
 cone of $\check\s =  \R_ {\geq 0}  {\Gamma_\s} \subset M_\R$.
 By Lemma  \ref{open-embedding} the open affine $T^M$-invariant subsets of  $T^{ \Gamma_\s} $ are
$T^{ \Gamma_\t} $, for $\t \leq \s$, where
 $\Gamma_\t = \Gamma_\s + M (\t, \Gamma_\s )$.


 By definition $X$ is covered
by a finite number of $T^M$-invariant affine open subsets of the form
$\{ T^{\Gamma_\s} \}_{\s \in \Sigma }$. We can assume that if $\s \in \Sigma$ and if
$\t \leq \s$ then $\t \in \Sigma$. We are going to show that $\Sigma$ is a fan in $N_\R$,  hence
$T^ {\Gamma_\s} \ne T^{\Gamma_{\s'}}$ if  $\s \ne \s'$.

We have that for any $\s, \s' \in \Sigma$ the intersection $T^
{\Gamma_\s} \cap T^{\Gamma_{\s'}}$ is an affine open subset of the
separated variety $X$ (see Chapter 2 of \cite{Ha}). It is also a
$T^M$-invariant affine subset of both $T^ {\Gamma_\s} $ and $
T^{\Gamma_{\s'}}$, hence it is of the form $T^ {\Gamma_\t}$. By
Lemma \ref{open-embedding}  we obtain two inclusion of semigroups
$\Gamma_\s \to \Gamma_\t$ and $\Gamma_{\s'} \to \Gamma_{\t}$.
  Since $X$ is separated the diagonal map
$T^ {\Gamma_\t} \to  T^ {\Gamma_\s} \times T^{\Gamma_{\s'}}$  is  a closed embedding (see Chapter 2 of\cite{Ha}).
Algebraically, this implies the surjectivity of the homomorphism
\[ k [ t^{\Gamma_\s} ] \otimes_k k [t^{\Gamma_{\s'}} ] \to k [ t^{\Gamma_\t} ]
\mbox{ ,  determined by } t^\gamma \otimes t^{\gamma'} \mapsto t^ {  \gamma + \gamma'}.
\]
It follows that the homomorphism of semigroups $
 {\Gamma_\s}  \times {\Gamma_{\s'}} \to {\Gamma_\t} $,
 $(\gamma, {\gamma'}) \mapsto {  \gamma + \gamma'}$
is surjective.
This proves that $\Gamma_\t = {\Gamma_\s}  + {\Gamma_{\s'}}$ thus
\[
\R_{\geq 0} \Gamma_\t = \check{\t} =  \R_{\geq 0} ({\Gamma_\s}  + {\Gamma_{\s'}}) = \check{\s} + \check{\s'}
\]
By duality we deduce that $\t = \s \cap \s'$.  By Proposition
\ref{open-embedding}  we obtain
\[
 \Gamma_\t = \Gamma_\s + M (\t, \Gamma_\s) =  \Gamma_{\s'} + M (\t, \Gamma_{\s'}).
\]

In conclusion, $\Sigma$ is a fan in $N_\R$ and if $\Gamma := \{
\Gamma_\s \mid \s \in \Sigma  \}$ the triple $(N, \Sigma, \Gamma)$
verifies the compatibility properties of Definition
\ref{def-general-toric} and the variety $X^\Gamma_\Sigma$ is
$T^M$-equivariantly isomorphic to $X$.
 \hfill $\ {\Box}$

The following corollary is consequence of  Proposition
\ref{morphism} and Theorem \ref{cate}.

\begin{corollary}  The category with objects the triples $(N,
\Sigma, \Gamma)$ of Definition \ref{def-general-toric} and
morphisms those maps of fans with attached semigroups of
Definition \ref{map-fan} is equivalent to the category with objets
the toric varieties of Definition \ref{new} and morphisms those
equivariant morphism which extend morphisms of the corresponding
algebraic tori; see Proposition \ref{morphism}.
\end{corollary}

\section{Invertible sheaves on toric varieties}

In this section we describe how some of the classical results in the
study of invariant invertible sheaves on a normal toric variety
extend to the general case.



Let $T_\Sigma^\Gamma$ denote a toric variety defined by the triple
$(N, \Sigma, \Gamma)$. Recall that if $\s \in \Sigma$ we denote by
$T_{\s} = T^{\check{\s} \cap M}$ the normalization of the chart
$T^{\Gamma_\s}$ and by $T_\Sigma$ the normalization of
$T^{\Gamma}_\Sigma$.

A \textit{support function}  $ h : |\Sigma| \to
\R$ is a continuous function such that for each $\s \in \Sigma$
the restriction $h_{| \s}  \colon \s \to \R$ is linear.  We say
that $h$ is \textit{integral} with respect to $N$ if $h ( |\Sigma|
\cap N) \subset \Z$. We denote by $\mbox{SF} (N, \Sigma)$  the set
of support functions integral with respect to $N$. If $h$ is a
support function integral with respect to $N$ then for any $\s \in
\Sigma$ there exists $m_\s \in M$ such that
\[
h (\nu) = \langle \nu, m_\s \rangle, \quad \mbox{ for all } \nu
\in \s.
\]
Notice that by continuity we have that
\begin{equation}  \label{car-con-nor}
 m_\t = m_\s \mod M(\t)  = M \cap \t^\perp, \mbox{ for } \t \leq \s, \, \s \in \Sigma.
\end{equation}
The set $\{ m_\s \mid \s \in \Sigma \}$ determines $h$ but may not
be uniquely determined.

\begin{definition} \label{sup}
 A support function for the   triple $(N, \Sigma, \Gamma)$
is a support function $ h : |\Sigma| \to \R$ integral with respect
to $N$ which in addition verifies the compatibility property
\begin{equation} \label{car-con}
 m_\t = m_\s \mod M(\t,
\Gamma_\t) , \mbox{ for } \t \leq \s, \, \s \in \Sigma.
\end{equation}

\end{definition}
We denote by  $\mbox{SF} (N, \Sigma, \Gamma)$ the additive group
of support functions for the triple $(N, \Sigma, \Gamma)$. It is a
subgroup of $\mbox{SF} (N, \Sigma)$. A vector $m \in M$ defines an
element of $\mbox{SF} (N, \Sigma, \Gamma) $ hence we have a
homomorphism $M \to \mbox{SF} (N, \Sigma, \Gamma) $, which is
injective if the support of $\Sigma$ spans $N_\R$ as a real vector
space.

Any $h \in \mbox{SF} (N, \Sigma)$ determines
 $T^M$-invariant \textit{ Cartier divisor} $D_h$
on $T_\Sigma$ by \begin{equation} \label{cartier}
D_{h | T_\s } =
\mbox{{\rm div}} (t^{-m_\s}) \mbox{ for }\s \in \Sigma,
\end{equation}
 where $\mbox{{\rm div}} (g)$ denotes
the \textit{principal Cartier divisor} of the rational function
$g$ on an irreducible variety. Notice that $D_h$ is independent of
the possible choices of different \textit{Cartier data} $\{ m_\s
\mid \s \in \Sigma \}$ defining $h$. If $\s, \s' \in \Sigma$, $\t
= \s \cap \s'$ then $T_\t = T_\s \cap T_{\s'}$ and
(\ref{car-con-nor}) guarantees that $ t^{-m_\s + m_{\s'}}$ and $
t^{m_\s - m_{\s'}}$ are both regular functions on $T_\t$. Any
$T^M$-invariant Cartier divisor on $T_\Sigma$ is of the form $D_h$
for $h \in \mbox{SF} (N, \Sigma)$, i.e., it is defined by Cartier
data.

\begin{lemma} If $h \in  \mbox{SF} (N, \Sigma)$ is defined by the Cartier data $\{ m_\s \mid \s \in \Sigma \}$
then it defines a $T^M$-invariant Cartier divisor on
$T^{\Gamma}_\Sigma$ if and only if (\ref{car-con}) holds, that is,
if and only if $h \in  \mbox{SF} (N, \Sigma, \Gamma)$.
\end{lemma}
\textit{Proof.} The condition to determine a Cartier divisor is
that for any $\s, \s' \in \Sigma$, $\t = \s \cap \s'$ the
transition function $ t^{-m_\s + m_{\s'}}$ is an invertible
regular function on $T^{\Gamma_\t} = T^{\Gamma_\s} \cap
T^{\Gamma_{\s'}}$. By Lemma \ref{conditions-aff} this is
equivalent to (\ref{car-con}).
 \hfill $\ {\Box}$


We have shown that the group  $\mbox{\rm CDiv}_{T^M}
(T^\Gamma_\Sigma)$ of $T^M$-invariant Cartier divisors on
$T^\Gamma_\Sigma$ can be seen as a subset of  $\mbox{\rm
CDiv}_{T^M} (T_\Sigma)$. The set $\{ \mathrm{div} (t^m) \}_{m \in
M}$ is a subgroup of $\mbox{\rm CDiv}_{T^M} (T^\Gamma_\Sigma)$
consisting of principal Cartier divisors.

 The map
\[
\mbox{SF} (N, \Sigma, \Gamma) \longrightarrow  \mbox{\rm
CDiv}_{T^M} (T^\Gamma_\Sigma), \quad h \mapsto D_h.
\]
is a group isomorphism. The  inverse map sends a Cartier divisor
$D$ on $T_\Sigma^\Gamma$,  given by the Cartier data $\{ m_\s \mid
\s \in \Sigma \}$, to the function
\[
h_D := |\Sigma| \to \R, \quad h_D (\nu) = \langle \nu, m_\s
\rangle \mbox { if } \nu \in \s.
\]


A Cartier divisor on $T_\Sigma$  determines an \textit{invertible
sheaf} $\mathcal{O}_{T_\Sigma} (D)$. If $U$ is an affine open set
in which $D =  \mbox{{\rm div}} (g_U)$ for some rational function
$g_U$  then the set of sections $ H^0 ( U, \mathcal{O}_{T_\Sigma}
(D))$  consists of those rational functions $f$ which verify that
$fg_U$ is a regular function on $U$.

We denote by $\mathcal{O}_{T^\Gamma_\Sigma}$ the structure sheaf
on the toric variety ${T^\Gamma_\Sigma}$. The \textit{invertible
sheaf} of a $T^M$-invariant Cartier divisor  $D$ on
$T_\Sigma^\Gamma$ is the sheaf  of
$\mathcal{O}_{T^\Gamma_\Sigma}$-modules
$\mathcal{O}_{T_\Sigma^\Gamma} (D)$. By (\ref{cartier}) the set of
sections of this sheaf  on  $T^{\Gamma_\s}$ is
\begin{equation} \label{section}
H^0 ( {T^{\Gamma_\s}}, \mathcal{O}_{T_\Sigma^\Gamma} (D) )  =
t^{m_\s} k [t^{\Gamma_\s} ].
\end{equation}




We denote by $P_\Sigma^\Gamma$ the following subset of $M$:
\begin{equation} \label{PDG}
P_D^\Gamma : = \bigcap_{\s \in \Sigma} {m_\s} + \Gamma_\s.
\end{equation}


The set of global sections of the sheaf
$\mathcal{O}_{T^\Gamma_\Sigma}$ is equal to
\begin{equation} \label{global-sec}
H^0 (T^\Gamma_\Sigma, \mathcal{O}_{T^\Gamma_\Sigma} (D) ) =
\bigcap_{\s \in \Sigma} t^{m_\s} k [t^{\Gamma_\s}] =  \bigoplus_{m
\in P_D^\Gamma} k t^m.
\end{equation}




\begin{remark} As in the normal case, a $T^M$-invariant Cartier divisor $D$ defines an equivariant line bundle $\mathcal{L}_{D}$ whose sections coincide with those of the invertible sheaf $\mathcal{O}_{T^\Gamma_\Sigma} (D)$. See \cite{Oda}, Chapter 2.\end{remark}
The \textit{Picard group} $\mathrm{Pic} (X)$ of a variety $X$
consists of the isomorphism classes of invertible sheaves in $X$.


\begin{lemma} Suppose that $|\Sigma | = N_\R$.
 For any Cartier divisor $D$ on the toric variety
$T^{\Gamma}_\Sigma$ we have an
$\mathcal{O}_{T^{\Gamma}_\Sigma}$-module isomorphism $
\mathcal{O}_{T^{\Gamma}_\Sigma} (D) \cong
\mathcal{O}_{T^{\Gamma}_\Sigma} (D_h)$ for some $h \in \mathrm{SF}
(N, \Sigma, \Gamma)$. The following are equivalent for $h \in
\mathrm{SF} (N, \Sigma, \Gamma)$.
 \begin{enumerate}
\item[{\rm i.}] $h \in M$

\item[{\rm ii.}] $D_h$ is a principal Cartier divisor.

\item[{\rm iii.}] $\mathcal{L}_{D_h}$ is a trivial line bundle.

\item[{\rm iv.}] The sheaf $\mathcal{O}_{T^{\Gamma}_\Sigma} (D_h)
$ is isomorphic to $\mathcal{O}_{T^{\Gamma}_\Sigma}$ as
$\mathcal{O}_{T^{\Gamma}_\Sigma}$-module.
 \end{enumerate}
\end{lemma}
\textit{Proof.} See Proposition 2.4 of \cite{Oda}.  \hfill
${\Box}$


\begin{proposition}
Suppose that $|\Sigma | = N_\R$. Then we have canonical
isomorphisms
\[
 \mathrm{SF}
(N, \Sigma, \Gamma) / M  \to \mathrm{Pic} (T^\Gamma_\Sigma)  \to
\mbox{\rm CDiv}_{T^M} (T^\Gamma_\Sigma) / \{ \mathrm{div} (t^m)
\}_{m \in M},
\]
from which we deduce a canonical injection $\mathrm{Pic} (T^\Gamma_\Sigma)
\rightarrow  \mathrm{Pic} (T_\Sigma)$.
\end{proposition}
\textit{Proof.} This follows by using the same arguments as in
Corollary 2.5 \cite{Oda}.  \hfill ${\Box}$


If $\r $ belongs to the $1$-skeleton
$\Sigma(1)$ of the fan $\Sigma$ we denote by $\nu_\r$ the
primitive integral vector for the lattice $N$ in the ray $\r$,
that is the generator of the semigroup $\r \cap N$. We associate
to  $h \in \mbox{SF} (N, \Sigma) $ the polyhedron
\begin{equation} \label{PD}
P_h: = \{ m \in M_\R \mid \langle \nu_\r , m \rangle \geq h
(\nu_\r),  \, \r \in \Sigma (1) \}.
\end{equation}
Recall that
\begin{equation}
P_{l h } = l P_h \mbox{ and } P_h + P_h = P_{h+h'} \label{Plh}
\end{equation}
for any integer $l \geq 1$ and $h, h'  \in \mbox{SF} (N, \Sigma)
$.

\begin{proposition} \label{base-free}
Suppose that
 $|\Sigma |= N_\R$. The following are equivalent for $h \in \mbox{SF} (N, \Sigma,
 \Gamma)$ defining a Cartier divisor $D = D_h$.
 \begin{enumerate}
\item[{\rm i.}] The $\mathcal{O}_{T^\Gamma_\Sigma}$-module
$\mathcal{O}_{T^\Gamma_\Sigma} (D)$ is generated by its global
sections.


\item[{\rm ii.}] $h$ is upper convex, i.e., $h (\nu) + h(\nu')
\leq h (\nu + \nu')$ for all $\nu, \nu' \in N_\R$.

\item[{\rm iii.}]  The polytope $P_h$ has vertices  $\{ m_\s \mid
\s \in \Sigma \}$.
 \end{enumerate}
 If these conditions hold  the convex hull of the set $P_D^\Gamma$ is the
polytope $P_h$ and $h$ is the support function of the polytope
$P_h$.
\end{proposition}
\textit{Proof.}  The proof follows as in the normal case  (see Theorem 2.7 \cite{Oda}). \hfill
${\Box}$

If $|\Sigma| = N_\R$ the support function $h \in \mbox{SF} (N,
\Sigma,
 \Gamma)$, defined by the Cartier data $\{ m_\s \mid \s \in \Sigma
 \}$,
  is \textit{strictly upper convex}
if it is upper convex and in addition
\[
h( \nu) = \langle \nu, m_\s \rangle \mbox{ if and only if } \nu
\in \s \mbox{ , for } \s \in \Sigma.
\]


 Suppose that $h \in \mbox{SF}
(N, \Sigma,
 \Gamma)$ verifies the equivalent conditions of
Proposition \ref{base-free}. Set $D = D_h$. If $P_{D}^\Gamma = \{
u_1, \dots, u_s \}$ we have a morphism
\begin{equation}\label{embedding}
\Phi_D \colon T^\Gamma_\Sigma \longrightarrow \P^{s-1}_k, \quad
\Phi_D = (t^{u_1} \colon \cdots \colon t^{u_s} ).
\end{equation}
(defined in homogeneous coordinates of $\P^{s-1}_k$). The morphism
$\Phi_D$ is equivariant with respect the  map of tori $\Phi_{|T^M}
\colon T^M \to T^{M'}$, where $T^{M'}$ denotes the torus of
$\P^{s-1}_k$ with respect to the fixed coordinates.

\begin{proposition} \label{very-ample}
Suppose that
 $|\Sigma| = N_\R$. The following are equivalent for $h \in \mbox{SF} (N, \Sigma,
 \Gamma)$ defining a Cartier divisor $D = D_h$.
  \begin{enumerate}
\item[{\rm i.}] $D$ is very ample.


\item[{\rm ii.}]  $h$ is strictly upper convex and for all $\s \in
\Sigma (d)$ the set $\{ m - m_\s \mid m \in P_D^\Gamma \}$
generates the semigroup $\Gamma_\s$.
 \end{enumerate}
\end{proposition}
\textit{Proof.} Suppose that $h$ is not strictly upper convex.
Then there exists $d$-dimensional cones $\s, \s' \in \Sigma$ such
that $\t = \s \cap \s'$ is of dimension $d-1$ and $ m_{\s} =
m_{\s'}$. This implies that the section defined by $t^{m_\s}$ in
the open set $ U= T^{\Gamma_\s} \cup T^{\Gamma_{\s'}}$ is no where
vanishing.

By definition there exists $1 \leq i \leq s$ such that $m_\s =
u_i$.

 The restriction of $\Phi_D$ to $U$
 factors through the
affine open set $\C^{s-1}$, where the $i$-th homogeneous
coordinate does not vanish. It is of the form:
\[
\Phi_{|U} : U \to \C^{s-1} \mbox{, with }  \Phi_{|U} = (t^{u_1 -
m_\s} , \dots, t^{u_{i-1} - m_{\s}},  t^{u_{i+1} - m_{\s}}, \dots,
t^{u_s - m_{\s} }).
\]
 By Lemma \ref{orb-clos} the closure
of the orbit $\orb (\t, \Gamma)$ is a complete one-dimensional toric variety
contained in $U$. The restriction $\Phi_{| \overline{\orb (\t,
\Gamma)} }$ must be constant hence $\Phi$ is not an embedding.
This implies that if $D_h$ is very ample $h$ is strictly upper
convex.

Suppose that $h$ is  strictly upper convex.  If $\s \in \Sigma$ is
a $d$-dimensional cone then $m_\s$ belongs to  $\{u_i\}_{i=1}^s$,
say $m_\s = u_s$. The restriction of $\Phi$ to $T^{\Gamma_\s}$
factors though the affine open set of $\P^{s-1}_k$  where the last
homogeneous coordinate does not vanish. It is described
algebraically by the homomorphism of $k$-algebras:
\[
k [ y_1, \dots, y_{s-1} ] \to k [t^{\Gamma} ], \quad y_i \mapsto
t^{u_i - m_\s}, \, i =1, \dots, s-1.
\]
This maps defines a closed immersion if and only if it is
surjective. This happens if and only if the set of vectors $\{ u_i
- m_\s \}_{1 \leq i \leq s-1 }$ generate the semigroup
$\Gamma_\s$.
 \hfill ${\Box}$






\begin{proposition}
Suppose that
 $|\Sigma | = N_\R$. The following are equivalent for $h \in \mbox{SF} (N, \Sigma,
 \Gamma)$.
 \begin{enumerate}
\item[{\rm i.}]  $D_h$ is ample

\item[{\rm ii.}] $h$ is strictly upper convex.
\end{enumerate}
\end{proposition}
\textit{Proof.}  If $D$ is ample then $l D$ is very ample for $l
\gg 0$. Since $l D = D_{l h}$ it follows that $h$ is strictly
upper convex if $l h$ is and  the assertion holds by Proposition
\ref{very-ample}.

Conversely suppose that $h$ is strictly upper convex. We prove
that $l D_h$ is very ample for $l \gg  0$. By Proposition
\ref{very-ample} it is sufficient to prove that there exists an
integer $l \gg 0$ such that for each $d$-dimensional cone $\s \in
\Sigma$ the semigroup $\Gamma_\s$ is generated by $\{ m - l m_\s
\mid m \in P^\Gamma_{D_{lh}} \}$.

If $\s' \in \Sigma$, $\dim \s' = d$,  $\t = \s' \cap \s$   we have
that $\Gamma_\t = \Gamma_\s + \Z_{\geq 0} (-u)$
for any $u \in \Gamma_\s$ in the relative interior of the cone
$\t^\perp \cap \check{\s}$ (see Lemma \ref{conditions-aff}). For
instance we take $u = m _{\s'} - m_{\s}$. We obtain similarly that
$
\Gamma_\t = \Gamma_{\s'} + \Z_{\geq 0} (u).
$

If $\g \in \Gamma_\s$ then $\g$ belongs to $\Gamma_\t$ and there
exists $\g' \in \Gamma_{\s'}$ and an integer $p \geq 0$  such that
$ \g = \g'   + p u$.  If $l \geq p $ we obtain:
\begin{equation} \label{g}
l m_{\s'} + \g' + (l - p) (m_\s - m_{\s'}) =  l m_\s + \g.
\end{equation}
If $l$ is big enough, a formula of the form (\ref{g}) holds for
any $\g$ in a finite set $G_{\s}$ of generators of $\Gamma_\s$
(where $p$ and $\g'$  vary with $\g$) and for any cone $\s' \in
\Sigma(d)$. Since $\g'$ and $m_\s - m_{\s'}$ belong to
$\Gamma_{\s'}$ this implies $t^{ l m_\s + \g }$ defines a section
in $H^0 ( {T^{\Gamma_{\s'}}}, \mathcal{O}_{T_\Sigma^\Gamma}
(D_{lh}) ) $ (see (\ref{section})) for  any cone $\s' \in \Sigma
(d)$. We deduce that for any $\g \in G_\s$ the vector $l m_\s + \g
$ belongs to the set $P^\Gamma_{D_{l h}}$ and $t^{l m_\s + \g}$
defines a global section of $\mathcal{O}_{T_\Sigma^\Gamma}
(D_{lh}) $. \hfill ${\Box}$


\begin{remark}
 Let $\mathcal{A} = \{ u_1, \dots, u_s \} $ be a subset of a
lattice $M$ such that $\Z \mathcal{A} = M$, i.e., $\mathcal{A}$
spans $M$ as a lattice. Gel{\cprime}fand, Kapranov, and Zelevinsky
\cite{GKZ} define a projective
 toric variety $X_\mathcal{A}$ as the closure of the image of the
 map
 \[
\varphi_{\mathcal{A}} =(t^{u_1} \colon \dots \colon t^{u_s})
\colon T^M \to \P^{s-1}_k.
\]
\end{remark}
Let us explain how their definition fits with our notion of
projective toric variety.  Let $P$ be the convex hull of
$\mathcal{A}$ in $M_\R$ and $\Sigma$ de dual fan of $P$. Each $\s
\in \Sigma$ of maximal dimension determines a vertex $m_\s$ of
$P$, which is necessarily an element of $\mathcal{A}$. We
associate to $\s$ the semigroup $\Gamma_\s : = \sum \Z_{\geq 0}
(u_i - m_\s)$. If $\t \leq \s$ we define $\Gamma_\t$ by
(\ref{Gamma_tau}). The set $\Gamma := \{ \Gamma_\theta \mid \theta
\in \Sigma \}$ is well-defined and the triple $(N, \Sigma,
\Gamma)$ defines a toric variety $T^{\Gamma}_\Sigma$ (the argument
is the same as the one used in the proof of Proposition
\ref{blowing}). The support function $h$ of $P$ belongs to
$\mbox{SF} (N, \Sigma, \Gamma)$ and is strictly upper convex. If
$D = D_h$ we deduce from the definitions that $P_D^\Gamma =
\mathcal{A}$. By Proposition \ref{very-ample} the Cartier divisor
$D$ is very ample, and the morphism (\ref{embedding}) is an
equivariant embedding of $T^\Gamma_\Sigma$ in the projective space
$\P^{s-1}_k$ such that $(\Phi_{D })_{| T^M } =
\varphi_{\mathcal{A}}$. It follows that $ X_{\mathcal{A}} =
\Phi_{D} (T^\Gamma_\Sigma)$.

\begin{remark}
If $F = \sum_{i=1}^s c_i t^{u_i} \in k [t^M]$ is a polynomial with
$c_1 \dots c_s \ne 0$, then $F$ defines a global section of
$\mathcal{O}_{T^\Gamma_\Sigma} (D)$ such that the closure of $\{ F
=0 \} \cap T^M$ in $T^\Gamma_\Sigma$ does not meet any
zero-dimensional orbit of $T^\Gamma_\Sigma$.
\end{remark}



\centerline{\textbf{Part II: Resolution of toric varieties by Semple-Nash
modifications}}
\par\medskip
In this Part we prove that any toric variety has a canonical
resolution of singularities by iterated blowing ups of
logarithmic jacobian ideals. Recall that, as shown below, if $k$ is
an algebraically closed field of characteristic zero the blowing
up of the logarithmic jacobian ideal of an affine toric variety
$T^\Gamma$  coincides with the Semple-Nash
modification. This fact, originally due to Gonzalez Sprinberg in the normal case (\cite{GS-ENS}), is our starting point.\par\medskip

The sequence of logarithmic jacobian blowing-ups of a toric
variety $T_\Sigma^\Gamma$ is a sequence
of toric varieties $T_{\Sigma^{(i)}}^{\Gamma^{(i)}}$ defined by a sequence $\Sigma^{(i)}$ of refinements (or subdivisions) of $\Sigma$ with attached families $\Gamma^{(i)}$ of semigroups and we have
to show that it eventually stabilizes. In most proofs of
resolution the strategy is to attach to points an invariant which
takes its minimal value only for regular points  and then show
that it can be made to decrease by successive blowing-ups. Our
strategy is different: we show that the very nature of the
blowing-up of the logarithmic jacobian ideals forces cones in the
successive refinements $\Sigma^{(i)}$ of $\Sigma$ to stabilize for
$i$ large enough, meaning that they are not subdivided in the
$\Sigma^{(j)}$ for $j\geq i$. If one can stabilize the cones of
maximal dimension the logarithmic jacobian blowing-ups are finite
morphisms from then on, and it is easy to show that they resolve
in finitely many steps (see Proposition \ref{stable}). This stabilization is not measured by the
constancy of some local invariant. The basic idea is to show
stabilization by extending it from lower-dimensional cones to
higher-dimensional ones, so that if one really insists on having
an invariant, it should be the maximal codimension of stable cones
of $\Sigma^{(j)}$; it is at most $d-1$ since edges are stable, and
if it is zero, we are essentially done. \par
Here is a quick description of the structure of the proof: first we study a "local" problem with respect to the monomial valuation associated to a vector $\nu\in \s\cap N$, where $\s\in\Sigma(d)$. Such a vector determines a unique cone $\theta^{(j)}\in\Sigma^{(j)}$ for all $j$ containing $\nu$ in its relative interior, and the first observation is that this sequence $(\theta^{(j)})_{j\geq 0}$ stabilizes for $j\geq j_1$ say; the limit $\theta^{(\infty)}=\theta^{(j_1)}$ is by definition a stable cone of $\Sigma^{(j_1)}$. This implies that the chart $T^{\Gamma^{(j)}}_{\theta^{(\infty)}}$ is non singular for $j\gg 0$ (see Propositions \ref{stable} and \ref{prev}).\par\noindent One of the difficulties is
that the logarithmic jacobian blowing-up of the variety does not
induce the logarithmic jacobian blowing-up of its
lower-dimensional orbit closures. Nevertheless it is possible to show that the sequence of the fans of the corresponding orbit closures in the successive blowing-ups also stabilizes (Proposition \ref{eta}). A key point in the proof of this result is that given a stable cone $\eta$, a nested sequence of cones $\zeta^{(j)}\in\Sigma^{(j)}$ containing $\eta$ as a codimension one face necessarily stabilizes (Proposition \ref{uno}). This uses the fact that for every one dimensional orbit closure associated to a stable cone of codimension one, the effect of the ambient blowing-up is very similar that of its logarithmic jacobian blowing-up (see Claim \ref{oneorbit}). \par\noindent
This controls the proliferation of the cones containing a given a stable cone $\eta$ and corresponding to "new" orbits appearing in the successive blowing-ups.\par\noindent
By induction on the codimension of $\eta$, this enables us to prove that given a stable cone $\eta$, the set of cones of $\Sigma^{(j)}$ which contain it stabilizes. The final step is to globalize this argument to prove the stabilization of the sequence of fans $\Sigma^{(j)}$ thanks to the compactness of the projectivization of $\vert\Sigma\vert$ (Proposition \ref{equ}).



 \section{The Semple-Nash modification: preliminaries}

 In \cite{S}, Semple introduced the Semple-Nash modification
 of an algebraic variety and asked whether a finite number of iterations
 would resolve the singularities of the variety.
 The same question was apparently rediscovered by Chevalley and Nash in the 1960's,
 and studied notably by Nobile (see  \cite{No}), Gonzalez Sprinberg (see  \cite{GS} and \cite{GS-ENS}),
  Hironaka (see  \cite{H}),  and Spivakovsky (see  \cite{Spivakovsky}).
  The best consequence so far of all this work is the Theorem, due to Spivakovsky,
  stating that by iterating the operation consisting of the Semple-Nash modification followed
  by normalization one eventually resolves singularities of surfaces over an algebraically
  closed field of characteristic zero.\par
Let $X$ be a reduced algebraic variety or analytic space,  which
we may assume of pure dimension $d$ for simplicity. Whenever we
speak of the Semple-Nash modification, we assume that we are
working over an algebraically closed field $k$ of characteristic
zero. Consider the Grassmanian $g\colon \hbox{\rm
Grass}_d\Omega^1_X\to X$; it is a proper algebraic map, which has
the property that its fiber over a point of $x$ is the Grassmanian
of $d$-dimensional subspaces of the Zariski tangent space
$E_{X,x}$. The map $g$ is characterized by the fact that
$g^*\Omega^1_X$ has a locally free quotient of rank $d$ and $g$
factorizes in a unique manner every map to $X$ with this property.
Let $X^o$ denote the non singular part of $X$, which is
$d$-dimensional and dense in $X$ by our assumptions. Since the
restriction $\Omega^1_X\vert X^o$ is locally free the map $g$ has
an algebraic section over $X^o$ and the Semple-Nash modification
is defined as the closure $NX$ of the image of this section,
endowed with the natural projection $n_X\colon NX\to X$ induced by
$g$. The map $n_X$ is proper and is an isomorphism over $X^o$; it
is a modification. Like the Grassmanian of $\Omega^1_X$, it is
defined up to a unique $X$-isomorphism.\par A local description
can be given for a chart $X\vert U$ of $X$ embedded in affine
space ${\mathbf A}^N(k)$ by taking the closure in $(X\vert U)\times {\mathbf G}(N,d)$ of the graph of the
Gauss map $\gamma\colon (X\vert U)^o\to {\mathbf G}(N,d)$ sending each non singular point to the class of its tangent space in the Grassmanian of $d$-dimensional vector subspaces in ${\mathbf A}^N(k)$. For any
point $x\in X$ the fiber $n_X^{-1}(x)$ is the subset of ${\mathbf
G}(N,d)$ consisting of limit positions at $x\in X$ of tangent
spaces to $X$ along sequences of non singular points tending to
$x$. In this guise, the Semple-Nash modification appears in a complex-analytic
framework in the paper \cite{Wh} of Hassler Whitney in connection with equisingularity problems.


 \begin{proposition}\label{iso}{\rm (Nobile), see \cite{No} and \cite{T-hunting}}.
 Let $X$ be a reduced equidimensional space; if the map $$n_X\colon NX\to X$$ is an isomorphism,
 the space $X$ is non singular.
\end{proposition}
For the convenience of the reader, we sketch the proof found in
\cite{T-hunting}:\par\noindent If the map $n_X$ is an isomorphism,
the sheaf $\Omega^1_X$ has a locally free quotient of rank $d$.
The problem is local, so it is enough to prove that the existence
of a surjective map $\phi\colon \Omega^1_{X,x}\to {\mathcal
O}^d_{X,x}$ implies, in characteristic zero,
 that ${\mathcal O}_{X,x}$
 is regular.
Passing to the completion and tensoring $\Omega^1_{X,x}$ by
$\hat{\mathcal O}_{X,x}$ we may assume that ${\mathcal O}_{X,x}$
is complete. We consider the linear map $e\colon{\mathcal
O}^d_{X,x}\to{\mathcal O}_{X,x} $ sending the first basis vector
to $1$ and the others to $0$. The composition of $e$ with the map
$\phi$ gives a surjective map, so that there has to be an element
$h\in {\mathcal O}_{X,x}$ such that the image of $dh$ in
${\mathcal O}_{X,x}$ by $e\circ\phi$ is equal to $1$, and then the
$k$-derivation $D\colon{\mathcal O}_{X,x}\to {\mathcal O}_{X,x}$
corresponding to $e\circ\phi$ is such that $Dh=1$. In
characteristic zero one can formally integrate this non vanishing
vector field using the formal expansion of $\mathrm{exp}(-hD)$ to
get an isomorphism ${\mathcal O}_{X,x}\simeq {\mathcal O}_1[[h]]$
where $ {\mathcal O}_1\simeq {\mathcal O}_{X,x}/(h)$. By
construction $ {\mathcal O}_1$ satisfies the same assumptions as
${\mathcal O}_{X,x}$ in one less dimension. By induction we are
reduced to dimension zero, but a reduced zero dimensional complete
equicharacteristic local ring is $k$ in our case. We refer to
\cite{T-hunting} for details, and to \cite{No} for the original
proof.
\begin{remark} We will see below in Section \ref{iteration}, Proposition \ref{jacoiso}  the characteristic-free version of this statement, which is that if the blowing-up of the logarithmic jacobian ideal is an isomorphism, the toric variety is smooth. Note that the Semple-Nash modification is defined in any characteristic but its being an isomorphism does not imply regularity in positive characteristic; it is the case for $y^p-x^q=0$ with $(p,q)=1$ in characteristic $p$. See \cite{No}.\end{remark}
  \section{The Semple-Nash modification in the toric case}
 The following is an extension to the case of not necessarily normal toric varieties  of a result of
Gonzalez Sprinberg (\cite{GS-ENS}; a summary of this work appeared in  \cite{GS-CRAS})
which was revisited by Lejeune-Jalabert and Reguera in the appendix to \cite{LJ-R}.
 \par\noindent
 Let $X$ be an affine toric variety over an algebraically closed field $k$. Using the notations of Section \ref{affine} we write its ring $$R=k[U_1,\ldots ,U_{r}]/P,$$ where $P$ is a prime binomial ideal $(U^{m^\ell}-U^{n^\ell})_{\ell\in\mathbf L}$ of the polynomial ring $k[U_1,\ldots ,U_{r}]$. Let $d$ be the dimension of $X$ and  denote by  ${\mathcal L}\subset \Z^r$ the lattice generated by the differences $(m^\ell -n^\ell)_{\ell\in \mathbf L}$; by [E-S], it is a direct factor of $ \Z^r$ since $X$ is irreducible and $k$ is algebraically closed. Setting $c=r-d$, we may identify $\mathbf L$ with $\{1,\ldots ,L\} $ with $L=\vert \mathbf L\vert$ in such a way that the lattice generated by $(m^1-n^1,\ldots ,m^c-n^c)$ has rank $c$.
 The quotient $\Z^r/\mathcal L$ is isomorphic to $\Z^d$ and we have an exact sequence
\begin{equation}\label{star} 0\to {\mathcal L}\stackrel{\psi}\rightarrow \Z^r\to \Z^d\to 0.\end{equation}
Our affine toric variety $X$ is $\hbox{\rm Spec} k[t^\Gamma]$,
where $\Gamma$ is the semigroup generated in $\Z^d$ by the images
$\gamma_1,\ldots ,\gamma_r$ of the basis vectors of $\Z^r$. \par


 \begin{proposition}\label{Nashbl}{\rm (Generalizing \cite{GS-ENS} \cite{GS-CRAS} and  \cite{LJ-R})} Keeping the notations just introduced, let $X$ be an affine toric variety over an algebraically closed field of characteristic zero. The Semple-Nash modification of $X$ is isomorphic to the blowing-up of the ideal of $R$ generated by the images of the products $U_{i_1}\dots U_{i_d}$ such that $\hbox{\rm Det}(\gamma_{i_1},\ldots ,\gamma_{i_d})\neq 0$.\end{proposition}
\begin{proof} A straightforward computation using logarithmic differentials shows that the jacobian determinant
$J_{K,\mathbf L'}$ of rank
$c=r-d$ of the generators $(U^{m^\ell}-U^{n^\ell})_{\ell\in\{1,\ldots , L\}}$ of our prime binomial ideal $ P\subset k[U_1,\ldots
,U_r]$, associated to a sequence $K=(k_1,\ldots , k_c)$ of distinct elements of $\{1,\ldots, r\}$ and a subset $\mathbf L'\subseteq \{1,\ldots , L\}$ of
cardinality $c$, satisfies the congruence
$$U_{k_1}\ldots U_{k_c}.J_{K,\mathbf L'}\equiv
\big(\prod_{\ell\in \mathbf L'}U^{m^\ell}\big)\hbox{\rm Det}_{K,\mathbf L'}\big( (\langle m-n\rangle )\big) \ \
\hbox{\rm mod.} P,$$ where $\big(\langle m-n\rangle\big)$ is the matrix of the
vectors $(m^\ell-n^\ell)_{\ell\in \{1,\ldots , L\}}$,  and $\hbox{\rm Det}_{K,\mathbf L'}$ indicates the minor
in question. By Lemma 6.3 of \cite{T-valuations}, the rank of the image in $k^{r\times L}$ of the  matrix $\big(\langle m-n\rangle\big)$ is equal to $c$.
 Now we know that we may assume that the lattice generated by $(m^1-n^1,\ldots ,m^c-n^c)$ has rank $c$, so that the binomial variety corresponding to the ideal $(U^{m^\ell}-U^{n^\ell})_{1\leq \ell \leq c}$ is a complete intersection $X_1$ of dimension $r-c$ containing our binomial variety $X$.
 By \cite{No}, the Nash modification of $X$ is isomorphic to the blowing up of the restriction to $X$ of the jacobian ideal of $X_1$. Therefore the Semple-Nash modification of our binomial variety coincides with the blowing-up in $X$ of the ideal generated as $K=(k_1,\ldots , k_c)$ runs through the sets of $c$ distinct elements of $(1,\ldots ,r)$ by the elements $J_{K,\mathbf L_0}$ satisfying the congruences
$$U_{k_1}\ldots U_{k_c}.J_{K,\mathbf L_0}\equiv
\big(\prod_{\ell\in\mathbf L_0}U^{m^\ell}\big)\hbox{\rm Det}_{K,\mathbf L_0}\big( (\langle m-n\rangle )\big) \ \
\hbox{\rm mod.} P,$$ where $\mathbf L_0=(1,\ldots ,c)$ and with the necessity that $J_{K,\mathbf L_0}=0$ whenever the determinant  on the right side is zero.\par
Now for each $K$ let us multiply both sides by  $U_{i_1}\ldots U_{i_d}$, where $I=(i_1,\ldots ,i_d)=\{1,\ldots ,r\}\setminus K$. We obtain for each $K$ the equality:\par
\begin{equation}\label{(**)}
U_1\ldots U_{r}.J_{K,\mathbf L_0}\equiv U_{i_1}\dots
U_{i_d}
\big(\prod_{\ell\in \mathbf L_0}U^{m^\ell}\big)\hbox{\rm Det}_{K,\mathbf L_0}\big( (\langle m-n\rangle )\big) \ \
\hbox{\rm mod.} P.
\end{equation}

Taking exterior powers for the map $\psi$ in the sequence (\ref{star}) gives an injection
$$0\to\stackrel{r-d}\Lambda {\mathcal L}\ \stackrel{\stackrel{r-d}\Lambda \psi}\longrightarrow \ \ \stackrel{r-d}\Lambda \Z^r$$
whose image is a primitive vector in $\stackrel{r-d}\Lambda \Z^r$ since it is a direct factor.\par\noindent Let $\L_0\subset \L$ be the lattice generated by the differences $(m^1-n^1,\ldots ,m^c-n^c)$, that is, corresponding to the first $c$ binomial equations. The image of its $(r-d)$-th exterior power is a non zero multiple of the primitive vector $\stackrel{r-d}\Lambda {\mathcal L}$; all the $c\times c$  minors of the matrix $\big(\langle m-n\rangle\big)$ involving vectors $m^\ell-n^\ell$ with $\ell>c$ are rationally dependent upon those which do not.  Consider now the $d$-th exterior power of the map dual to the surjection $ \Z^r\to \Z^d\to 0$ of (\ref{star}):
$$0\to \stackrel{d}\Lambda \check{\Z}^d\to  \stackrel{d}\Lambda\check{\Z}^r.$$
The image of $\stackrel{d}\Lambda \check{\Z^d}$ is a primitive vector in
$ \stackrel{d}\Lambda\check{\Z^r}$.
 \par\noindent
 By the natural duality isomorphism between $ \stackrel{d}\Lambda\check{\Z^r}$ and $\stackrel{r-d}\Lambda \Z^r$ (see  \cite {B}  \S 11, No. 11, Prop. 12) deduced from the pairings
 $$\begin{array}{lr}\stackrel{d}\Lambda\check{\Z}^r\otimes \stackrel{d}\Lambda{\Z}^r\to \Z,\ \ \stackrel{d}\Lambda{\Z}^r\otimes\stackrel{r-d}\Lambda \Z^r\to \Z,\cr\end{array}$$ this vector correspond to the image of $\stackrel{r-d}\Lambda {\mathcal L}$ in such a way that the coordinate which corresponds to the determinant of the vectors $\gamma_{i_1},\ldots ,\gamma_{i_d}$ in $\Z^d$ is a rational multiple of the determinant $\hbox{\rm Det}_{K,\mathbf L_0}\big( (\langle m-n\rangle )\big)$, which is non zero since our base field is of characteristic zero. \par
Equation \ref{(**)} now shows that the ideal of $R$ generated by the $J_{K,\mathbf L_0}$ differs from the ideal generated by the images of the products $U_{i_1}\dots U_{i_d}$ such that $\hbox{\rm Det}(\gamma_{i_1},\ldots ,\gamma_{i_d})\neq 0$  only by the product by invertible ideals, so that these two ideals determine isomorphic blowing ups, which proves the Proposition.\end{proof}
\begin{remark} The proof found in [LJ-R] is valid in the non-normal case; the proof given here makes explicit the connection of the logarithmic jacobian ideal with the usual one.\end{remark}
\begin{remark}\label{idproduct}
The isomorphism of Proposition \ref{product} carries the
logarithmic jacobian ideal of $k[t^{\Gamma\times\Gamma'}]$ onto
the tensor product of the logarithmic jacobian ideals of the
factors.
\end{remark}
\begin{remark}\label{dimone}
In the one-dimensional case the logarithmic jacobian ideal is the maximal ideal corresponding to the closed orbit. It is a classical fact that iterating the blowing-up of the singular point resolves the singularities of any branch.
\end{remark}



\section{The sheaf of logarithmic jacobian ideals on a toric variety}       \label{sheaf}

Let the pair $(\Sigma, \Gamma)$ define a toric variety
$T_{\Sigma}^\Gamma$ as in Definition \ref{def-general-toric}.



On the affine open set $T^{\Gamma_\s}$, $\s \in \Sigma$  we consider the
 ideal $\J_\s$ of $k [t^{\Gamma_\s}]$  generated by
monomials of the form  $t^{\a}$, where $\a$ belongs to the set
\[
| \J_\s | = \{  \a_1 + \cdots  + \a_d \, \mid \,  \a_1, \dots,
\a_d \in \Gamma_\s \mbox{ and } \a_1 \y  \cdots \y \a_d \ne 0 \}.
\]
The ideal $\J_\s$ is called the \textit{logarithmic jacobian ideal of
$T^{\Gamma_\s}$}.

\begin{remark} \label{gen}
If $\g_1, \dots, \g_r$ are generators of $\Gamma_\s$ the semigroup
$\Gamma_\s$ then the monomials $t^\a$, for $\a$ in
\begin{equation} \label{set-j}
 \{  \g_{i_1} + \cdots +  \g_{i_d} \, \mid \, \g_{i_1}
 \y  \cdots \y \g_{i_d} \ne 0, \, 1 \leq  i_1, \dots, i_d \leq r
 \},
\end{equation}
generate the ideal  $\J_\s$. Abusing notation we denote the
set (\ref{set-j})  with the same letter $\J_\s$, whenever the set
of generators of $\Gamma_\s$ is clear from the context.
\end{remark}


\begin{proposition}
The family $\{ \J_\s \mid \s \in \Sigma \}$ defines a
$T^M$-invariant sheaf of ideals  $\J$ on $T_\Sigma^\Gamma$, which
is called the sheaf of logarithmic jacobian ideals of
$T_\Sigma^\Gamma$.
\end{proposition}

{\textit{Proof.}} It is sufficient to check that if $\t \leq \s$,
$\s \in \Sigma$, then the ideal $\J_\t$ coincides with the
extension $\J_\s k [t^{\Gamma_\t}]$, induced by the inclusion
$k[t^{\Gamma_\s}] \hookrightarrow k [t^{\Gamma_\t}]$ defined by
$\Gamma_\s \subset \Gamma_\t$.

By Lemma \ref{conditions-aff} if $m \in \Gamma_\s$ belongs
to the relative interior of the cone $\check{\s} \cap \t^\perp$ then we
have that $\Gamma_\t = \Gamma_\s + \Z_{\geq 0} (-m)$ (such a
vector $m$ always exists).

If  $\g_1, \dots, \g_r$ are generators of $\Gamma_\s$ then $\g_1,
\dots, \g_r, -m$ are generators of $\Gamma_\t$. This implies the
inclusion $\J_\s \subset \J_\t$. By Remark \ref{gen}  an exponent
${\a}$ in $\J_\t$ which does not belong to the set $\J_\s$ is of
the form: $ \a= \g_{i_1} + \cdots + \g_{i_{d-1}} - m, \mbox{ with
} \g_{i_1} \y  \cdots \y \g_{i_{d-1}}  \y (-m) \ne 0$. Then, the
element $\beta:= \g_{i_1} + \cdots + \g_{i_{d-1}} +  m$ belongs to
$\J_\s$ and we obtain that: $ t^{\a} = t^{-2m} t^{\b} \in \J_\s k
[t^{\Gamma_\t}], \mbox{ and } \J_\s k [t^{\Gamma_\t}] = \J_\t$.
\hfill $\ {\Box}$


\begin{lemma} \label{pw}
There is a continuous  piecewise linear function $\ord_{\J} \colon
|\Sigma | \rightarrow \R$ such that for each $\t \in \Sigma$ the
function $\ord_{\J_{\t}}$ is the restriction of $\ord_{\J}$ to
$\t$.
\end{lemma}
{\textit{Proof.}} This follows from the definition of
$\ord_{\J_\s}$ (see (\ref{ord})), by using that $\J$ is a sheaf of
monomial ideals.

\begin{remark}
Note that Lemma \ref{pw} holds more generally if we replace $\J$
by any sheaf of monomial ideals $\mathcal{I}$ on
$T_\Sigma^\Gamma$.
\end{remark}


The will need the following lemma  in Section \ref{iteration}.
\begin{lemma}\label{induced}
Let $\theta^\perp\cap\Gamma$ be a face of the finitely generated
semigroup  $\Gamma\subset M$. The logarithmic jacobian ideal
$\tilde\J$ of the image $\tilde\Gamma$ of $\Gamma$ in the lattice
$M/M(\theta)$ is equal to the image of the logarithmic jacobian
ideal $\J$ of $\Gamma$.
\end{lemma}
\textit{Proof} Let us denote by $\tilde\gamma_i$ the images in
$M/M(\theta)$ of the generators $\gamma_i$ of $\Gamma$ and by $p$
 the rank of the lattice $M/M(\theta)$. If $\tilde\gamma_{i_1}, \ldots
,\tilde\gamma_{i_{p}}$ are linearly independent in $M/M(\theta)$,
then $\gamma_{i_1}, \ldots ,\gamma_{i_{p}}$ must be linearly
independent from $\theta^\perp$.  Remark that
  since $\theta^\perp\cap\Gamma$ is a face
it must contain $d-p$ generators of $\Gamma$ which are linearly
independent, since  $\theta^\perp\cap\Gamma$ spans the rank $d-p$
lattice $M(\theta, \Gamma)$.  Choosing linearly independent
generators $\gamma_{i_{p+1}} , \dots,
\gamma_{i_d}\in\theta^\perp\cap\Gamma$ of $\Gamma$ gives us a
generator $\gamma_{i_1}+ \cdots +\gamma_{i_{p}}+ \gamma_{i_{p+1}}
+ \cdots + \gamma_{i_d}$ of $\J$ whose image is
$\tilde\gamma_{i_1}+ \cdots +\tilde\gamma_{i_{p}}$, showing that
$\tilde\J$ is contained in the image of $\J$. If we now take $d$
independent generators $\gamma_{i_1}, \ldots
,\gamma_{i_{p}},\gamma_{i_{p+1}} , \dots, \gamma_{i_d}$ of
$\Gamma$, since they generate $M$, there must exist $d-p$
independent elements in their images, say $\tilde\gamma_{i_1},
\ldots ,\tilde\gamma_{i_{p}}$. Then the image $\tilde\gamma_{i_1}+
\cdots +\tilde\gamma_{i_{p}}+\tilde\gamma_{i_{p+1}} + \cdots +
\tilde\gamma_{i_d}$ belongs to the logarithmic jacobian ideal
$\tilde\J$, which shows that the image of $\J$ is equal to
$\tilde\J$.\hfill $\ {\Box}$



\section{Iterating the blowing-up of the logarithmic jacobian ideal}\label{iteration}

Let $\Gamma \subset M$ a finitely generated subsemigroup of a rank
$d$ lattice $M$ such that $\Z \Gamma =M$. We assume in addition
that the convex rational cone $\check{\sigma}: = \R_{\geq 0} \Gamma$, which is $d$-dimensional since $\Z \Gamma =M$, is strictly convex, which is equivalent to saying that the dual cone $\s \subset N_\R$ is strictly convex
of dimension $d$. The semigroup $\Gamma$ determines the affine
toric variety $T^{\Gamma} = \mathrm{Spec } k [ t^{\Gamma} ]$.
We fix a finite set of generators $\g_1, \dots, \g_r$ of $\Gamma$.
We consider the set
\[
\J := \{ \g_{i_1} + \cdots + \g_{i_d} \mid \g_{i_1} \y \dots \y
\g_{i_d} \ne 0, \, 1 \leq i_1, \dots, i_r \leq r  \}
\] defining the logarithmic jacobian ideal of $T^\Gamma$.



The Newton polyhedron $\Newton_\s (\J)$ of the monomial ideal
$\J$ (see Section \ref{blow}),  is contained in the interior of $\check{\s}$, since
the elements of $\J$ are sums of $d$-linearly independent elements
in the $d$-dimensional cone $\check{\s}$. The set $\J$ determines
the \textit{order function} defined by (\ref{ord}).
The maximal cones  $\t \subset \s$ of linearity of the function
$\ord_{\J}$ form the $d$-skeleton of a fan $\Sigma$ supported on $\s$.
The map
\begin{equation} \label{m-tau}
\t \mapsto m  \mbox{ if } \ord_\J (\nu) = \langle \nu, m \rangle
\mbox { for all } \nu \in \t.
\end{equation} is a bijection between the set $\Sigma(d)$ of $d$-dimensional cones of $\Sigma$ and the set of vertices of the polyhedron $\Newton_\s (\J)$.



We now consider the blowing up of the monomial ideal $\J$. A cone
$\t^{(1)} \in \Sigma (d)$ determines a vertex $m^{(1)}$ of
$\Newton_\s (\J)$ by (\ref{m-tau}) and also the finitely generated semigroup
\[
\Gamma_{\t^{(1)} }^{(2)} : = \Gamma + \sum_{m \in \J} \Z_{\geq 0 }
(m-m^{(1)})\subset \check\t^{(1)}\cap M.
\]
In view of the description recalled above of $\Sigma(d)$ in terms of $\Newton_\s (\J)$, the cone $\R_{\geq 0} \Gamma_{ \t^{(1)} }^{(2)} $ is
$\check{\t}^{(1)}$. The affine toric variety $T^{ \Gamma_{\t^{(1)}}^{(2)}}$ is a chart of the blowing up of $\J$ and this
toric variety is covered by charts of this form (see Section \ref{blow}).




The semigroup $\Gamma_{\t^{(1)}}^{(2)}$ is generated by $\{ \g_1,
\dots, \g_r\} \cup \{  m -m^{(1)} \}_{m \in \J}$. We denote also  by
$\J_{\t^{(1)}}^{(2)}$ the finite subset of
$\Gamma_{\t^{(1)}}^{(2)}$ corresponding to the monomials
generating the logarithmic jacobian ideal of
$T^{\Gamma_{\t^{(1)}}^{(2)}}$, by the same symbol this last ideal of $k[t^{\Gamma_{\t^{(1)}}^{(2)}}]$ , and by
$ \ord_{\J_{\t^{(1)}}^{(2)}} \colon \t^{(1)} \rightarrow \R $ the
corresponding order function.
\begin{remark} \label{principal}
  On the chart $T^{  \Gamma_{ \t^{(1)} }^{(2)}  }$
the pull back of the ideal $\J$ by the blowing up of $\J$ is the
principal ideal $t^{m^{(1)}} k [ t^{\Gamma_{ \t^{(1)} }^{(2)}}] =
t^\J k [ t^{\Gamma_{ \t^{(1)} }^{(2)}}]$. The Newton polyhedron
\[
\Newton_{\t^{(1)}} ( \J) := \J +  \check{\t}^{(1)} = m^{(1)} +
\check{\t}^{(1)}
\]
 of $t^\J k [ t^{\Gamma_{\t^{(1)} }^{(2)}}]$ is \textit{principal}, i.e.,
 it has only one vertex $m^{(1)}$.
\end{remark}


\begin{lemma} \label{global}
There is a continuous  piecewise linear function $\ord_{\J^{(2)}} \colon
\s \rightarrow \R$ such that for each $\t^{(1)}\in \Sigma(d)$
the function
$\ord_{\J_{\t^{(1)}}^{(2)}}$ is the restriction of $\ord_{\J^{(2)}}$ to
$\t^{(1)}$.
\end{lemma}
\textit{Proof.} This follows from Lemma \ref{pw}.
 \hfill $\
{\Box}$


As above, the maximal cones  $\t \subset \s$ of linearity of the function
$\ord_{\J^{(2)}}$ form the $d$-skeleton of a fan $\Sigma^{(2)}$
supported on $\s$ and subdividing the fan $\Sigma$.
 In particular, if
 $\t^{(2)}  \in \Sigma^{(2)} (d)$ is contained in $ \t^{(1)}   \in \Sigma (d)$
 then we denote by $m^{(2)}$ the vertex of the
Newton polyhedron  $\Newton_{\t^{(1)}} ( \J_{\t^{(1)}}^{(2)} ) $
of $\J_{\t^{(1)}}^{(2)}$ such that
\[
\ord_{\J^{(2)}}  (\nu) =  \langle \nu, m^{(2)} \rangle \mbox{ for
all } \nu \in \t^{(2)}.
\]




By iterating this construction we obtain a sequence of piecewise
linear functions $\ord_{\J^{(j)}}$ on $\s$, together with the
corresponding  fans $\Sigma^{(j)}$, with $\J = \J^{(1)}$ and
$\Sigma^{(1)} = \Sigma$, and such that $\Sigma^{(j)}$ is a
subdivision of $\Sigma^{(j-1)} $ for all $j \geq 2$.

By definition a cone $\t^{(j)} \in \Sigma^{(j)}(d)$ is contained
in a unique cone $\t^{(l)} \in \Sigma^{(l)} (d)$, for $0 \leq l
\leq j-1$, where we set $\t^{(0)} := \s$. Then we have unique
vectors $m^{(l)} \in M$ such that
\[
\ord_{\J^{(l)}}  (\nu) =  \langle \nu, m^{(l)} \rangle \mbox{ for
all } \nu \in \t^{(j)} \mbox{ and } 1 \leq l \leq j.
\]

The cone $\t^{(j)}$ corresponds to a chart of the  blowing up of
the logarithmic jacobian ideal $\J^{(j)}_{\t^{(j-1)}}$ of $k [ t
^{ \Gamma_{\t^{(j-1)}}^{(j)}}]$. This chart is the affine toric
variety defined by the semigroup
\[
\Gamma_{\t^{(j)}}^{(j+1)} = \Gamma_{\t^{(j-1)}}^{(j)} + \sum_{m
\in \J^{(j)}_{\t^{(j-1)}} } \Z_{\geq 0} (m - m^{(j)} ).
\]
By induction this procedure also provides a system of generators
of each semigroup $\Gamma_{\t^{(j)}}^{j+1}$. We  use also the
notation $\J_{\t^{(j)}}^{(j+1)}$ to refer to the finite set of
generators of the logarithmic jacobian ideal of $k [ t^{
\Gamma_{\t^{(j)}}^{(j+1)} }]$ (see Remark \ref{gen}). The
following inclusions, for $j\geq 2$, are consequence of the
definitions:
\begin{equation} \label{essential}
\Gamma_ {\t^{(j-1)}} ^{(j)} \subset \Gamma_{\t^{(j)}}^{(j+1)},
\quad  k[t^{\Gamma_ {\t^{(j-1)}} ^{(j)}}]\subset k[t^{
\Gamma_{\t^{(j)}}^{(j+1)}}], \quad \J_{\t^{(j-1)}}^{(j)} k[t^{
\Gamma_{\t^{(j)}}^{(j+1)}}] \subset \J_{\t^{(j)}}^{(j+1)} k[t^{
\Gamma_{\t^{(j)}}^{(j+1)}}].
\end{equation}
By
(\ref{essential}) we have that
\begin{equation} \label{essential2}
\ord_{\J^{(j+1)}} (\nu) \leq  \ord_{\J^{(j)}} (\nu)  \mbox{ for
all } \nu \in \s.
\end{equation}


\begin{remark} \label{principal-k}
For $1 \leq l \leq j$ we deduce from Remark \ref{principal} that $
\J^{(l)}_{\t^{(l-1)}}  k [ t^{ \Gamma^{(l+1)}_{\t^{(l)} }} ] =
t^{m^{(l)}}  k [ t^{ \Gamma^{(l+1)}_{\t^{(l)} } } ]$, hence the
Newton polyhedron
 $\Newton_{\t^{(l)}} ( \J^{(l)}_{\t^{(l-1)}} ) =
\J^{(l)}_{\t^{(l-1)}} + \check{\t}^{(l)}  =   m^{(l)} +
\check{\t}^{(l)}   $ has only one vertex $m^{(l)}$. \end{remark}

\begin{notation} We denote the Newton polyhedron $\Newton_{\t^{(j-1)}} ( \J^{(j)}_{\t^{(j-1)}} )$ simply by $\Newton( \J^{(j)}_{\t^{(j-1)}} )$ since there is no risk of confusion.
\end{notation}

\begin{proposition} \label{jacoiso} The affine toric variety $T^\Gamma$ is non singular
 if and only if the blowing up of the logarithmic jacobian ideal is an isomorphism.
\end{proposition}
\textit{Proof.} We only have to prove that if the blowing up is an
isomorphism the variety is smooth.  Remembering that $\Gamma$
generates the group $M$, let us choose a minimal system of
generators $\gamma_1, \ldots , \gamma_d,\gamma_{d+1},\ldots $ such
that $m^{(1)}=\gamma_1+\cdots +\gamma_d$ corresponds to the
generator of the logarithmic jacobian ideal which, by our
assumption, is principal. If there are more than $d$ generators,
then $\gamma_{d+1}$ is linearly dependent on the previous ones
which gives us another element $m=\gamma_1+\cdots
+\gamma_{i-1}+\gamma_{d+1}+\gamma_{i+1}+\cdots +\gamma_d$ of our
ideal. Our assumption ensures that
$m-m^{(1)}=\gamma_{d+1}-\gamma_i\in \Gamma$ which contradicts the
assumption of minimality. Therefore $\Gamma$ has $d$ independent
generators which generate $M$ and $k[t^\Gamma]$ is a polynomial
ring.\hfill $\ {\Box}$


\begin{proposition} \label{finite}
The following assertions are equivalent:
\begin{enumerate}
\item[{\rm i.}] $\t^{(j)} = \t^{(j-1)}$

\item[{\rm ii.}] The blowing up of the ideal $\J _ {\t^{(j-1)}} ^
{(j)} $ of $T^{ \Gamma_ {\t^{(j-1)}} ^ {(j)} }$ is a finite
morphism.
\end{enumerate}


\end{proposition}
\textit{Proof.} The hypothesis {\rm i.} is equivalent to the
following fact: the semigroups $\Gamma_ {\t^{(j-1)}} ^ {(j)}$ and
$\Gamma_ {\t^{(j)}} ^ {(j+1)}$ have the same saturation in the
lattice $M$; it is equal to $\check{\t}^{(j-1)} \cap M   =
\check{\t}^{(j)} \cap M $. This is equivalent to the following
geometric statement: the composite of the normalization of $T^{
\Gamma_ {\t^{(j)}} ^ {(j+1)} }$ with the blowing up of the
logarithmic jacobian ideal of $T^{ \Gamma_ {\t^{(j-1)}} ^ {(j)} }$
is the normalization map of $T^{ \Gamma_ {\t^{(j-1)}} ^ {(j)} }$
and therefore this blowing up is finite. Conversely, if {\rm ii.}
holds, the blowing up morphism $T^{ \Gamma_ {\t^{(j)}} ^ {(j+1)}
}\to T^{ \Gamma_ {\t^{(j-1)}} ^ {(j)} }$ induces an isomorphism of
the normalizations, from which {\rm i.} follows in view of Remark
\ref{norma}. \hfill $\ {\Box}$
\begin{remark}\label{vertex}
The conditions of the Lemma are also equivalent to the fact that the Newton polyhedron of the ideal $\J _ {\t^{(j-1)}} ^
{(j)} $ has only one vertex $m^{(j)}$.
\end{remark}
\begin{definition} \label{function}
For any integer $j \geq 1$ we introduce  a function
\[
f^{(j)} : \{ \t \subset \s \mid  0 \ne \t \mbox { convex rational
polyhedral cone } \} \rightarrow \Z_{\geq 1}.
\]
If $\nu_1, \dots, \nu_s$ are the primitive integral vectors for
the lattice $N$ which span the edges of $\t$, then the value of
$f^{(j)}(\t)$ is defined by
\[
f^{(j)} (\t) := \sum_{i=1}^s \ord_{\J^{(j)}} (\nu_i).
\]
\end{definition}

\begin{remark} \label{key}
Notice that if $0 \ne \t$ is any rational  polyhedral cone
contained in $\t^{(j)} \in \Sigma^{(j)} (d)$ then $f ^{(j)} (\t) =
\sum_{i=1}^s \langle \nu_i , m^{(j)} \rangle$ and if $\t$ is of
dimension $d$ then $f ^{(j)} (\t) \geq d$. Moreover, by
(\ref{essential2}) we obtain that
\begin{equation} \label{ine}
f^{(j)} (\t) \leq  f^{(j-1)} ( \t).
\end{equation}
\end{remark}



\begin{lemma} \label{smooth}
The following conditions are equivalent for $j\geq 1$:
\begin{enumerate}
\item[{\rm i.}]  The equality $f^{(j)} ( \t^{ (j-1) }) = d$ holds.

\item[{\rm ii.}] The cone $\t^{ (j-1) }$ is regular for the
lattice $N$ and $\Gamma ^{(j)}_{\t^{ (j-1) } }  = \check{\t}^{
(j-1) } \cap M$.

\item[{\rm iii.}] The toric variety $T^{\Gamma^{(j)}_{\t^{ (j-1)
}}} $ is smooth.
\end{enumerate}
\end{lemma}
Note that if the conditions of the Lemma are satisfied, the polyhedron $\Newton_{\t^{(j-1)}}
(\J^{(j)}_{\t^{ (j-1) } } )$ has only one vertex $m^{(j)}$.\par\noindent
\textit{Proof.} It is clear that {\rm ii.} and {\rm iii.} are
equivalent. It is enough to prove the result for $j=1$. Suppose
first that {\rm i.} holds. By hypothesis the fan $\Sigma^{(1)}$ is
the fan consisting of the faces of $\s$. If $\nu_1, \dots, \nu_s$
are the primitive vectors for the lattice $N$ which span the cone
$\s $ then $\langle \nu_i, m \rangle
>0$, $i=1, \dots, s$ since $m=m^{(1)} $ belongs to the interior of
$\check{\s}$. Since $f(\s) = d = \sum_{i=1}^s \langle \nu_i, m
\rangle$ we get that $s= d$ and $\langle \nu_i, m \rangle = 1$.

By definition of $\J$ the vector $m$ is sum of $d$ generators of
$\Gamma$ which are linearly independent, say $m = \g_1 + \dots +
\g_d$. Since $\sum_{j=1}^d \langle \nu_i, \g_j \rangle = 1$ for
$i=1, \dots, d$ we obtain that, up to relabelling the  $\nu_i$,
the vectors  $\nu_1, \dots, \nu_d$ in $N_\R$ form the dual basis
of $\g_1, \dots, \g_d$ in $M_\R$. Finally, notice that the
parallelogram generated by the  primitive vectors $\g_1, \dots,
\g_d$  in $M_\R$ contains no integral points different from the
vertices. It follows that $\g_1, \dots, \g_d$ form a basis of $M$.

 Conversely, if {\rm ii.} holds then we check from the
 definitions that
{\rm i.} holds.
 \hfill $\ {\Box}$



\begin{proposition} \label{eq} Suppose that $\t^{(j)} \in \Sigma^{(j)} (d)$ is
contained in $\t^{(j-1)} \in \Sigma^{(j-1)} (d)$. The following
equalities are equivalent:
\begin{enumerate}
\item[{\rm i.}] $f^{(j)} ( \t^{(j)} ) = f^{(j-1)} ( \t^{(j)} )$,

\item[{\rm ii.}] $m^{(j)} = m^{(j-1)}$.
\end{enumerate}
\end{proposition}
\textit{Proof.} Notice that if $m^{(j)} = m^{(j-1)}$ then {\rm i.}
follows by Remark \ref{key}.

Suppose that the equality {\rm i.} holds. By Remark
\ref{principal-k} we have that
\[
\Newton_{ \t^{(j-1)} } ( \J^{(j-1)}_{\t^{(j-2)}} ) = m ^{(j-1)} +
\check{\t}^{(j-1)} \mbox{ and } \Newton_{ \t^{(j)} } (
\J^{(j)}_{\t^{(j-1)}} ) = m ^{(j)} + \check{\t}^{(j)}.
\]
Since $\t^{(j)}$ is contained in $\t^{(j-1)}$ we get that
$\check{\t}^{(j-1)} \subset  \check{\t}^{(j)} $ and then
$
\Newton_{ \t^{(j)} } ( \J^{(j-1)}_{\t^{(j-2)}} ) = m ^{(j-1)} +
\check{\t}^{(j)}
$.
By (\ref{essential}) we get
\begin{equation} \label{inc}
 m ^{(j-1)} +
\check{\t}^{(j)} \subset  m ^{(j)} + \check{\t}^{(j)}.
\end{equation}
Let $\nu_1, \dots, \nu_s$ be the primitive integral vectors for
the lattice $N$ which span the cone $\t^{(j)}$. The vector $\nu: =
\sum_{i=1}^s \nu_i$ belongs to the interior of the cones
$\t^{(j)}$ and $\t^{(j-1)}$. By Remark \ref{key} and the
hypothesis we deduce
\[
f^{(j-1)} ( \t^{(j)} ) = \langle \nu, m^{(j-1)} \rangle = f^{(j)}
( \t^{(j)} ) = \langle \nu, m^{(j)} \rangle.
\]
This equality and the inclusion (\ref{inc}) imply that
$m^{(j-1) } = m^{(j)}$.
 \hfill $\ {\Box}$

\begin{proposition} \label{bound}
There exists an integer $l \geq 1$ such that for any cone $\t \in
\Sigma^{(l)} (d) $ if  $f^{(1)} (\t) > d$ then $f^{(1)} (\t)
> f^{(l)} (\t)$.
\end{proposition}
\textit{Proof.} Let us assume that the assertion of the
Proposition does not hold. This implies that there exists a
infinite sequence of convex polyhedral cones
\begin{equation} \label{sequence}
\s = \t^{(0)} \supset \t^{(1)} \supseteq  \t^{(2)} \supseteq
\cdots \supseteq \t^{(j)} \supseteq \cdots,
\end{equation}
such that $\t^{(j)} \in \Sigma^{(j)} (d)$ and

\begin{equation} \label{assumption}
 f^{(j)} (\t^{(j)}) = f^{(1)} (\t^{(j)} )
> d  \mbox{,  for all }j \geq 2.
\end{equation}

By Remark \ref{key} we have that $f^{(j)} (\t^{(j)}) = f^{(j-1)}
(\t^{(j)})$ for all $j \geq 2$. Proposition \ref{eq} implies  then
that $m^{(j)} = m^{ (j-1)}$ for all $j \geq 2$.

\begin{claim} \label{strict}
There exists a strictly increasing sequence $(i_j)_{j \geq 1}$ of
integers $\geq 0$ such that $\t^{(i_{j})} \ne \t^{(i_{j}+1)}$,
that is, the inclusion $\t^{(i_{j})} \supset \t^{(i_{j}+1)}$ is
strict, for $j \geq 1$.
\end{claim}
\textit{Proof of the claim.} Assume that the claim does not hold.
This implies that $\t^{(j)} = \t^{(j-1)}$ for all $j \geq 1$. By
Proposition \ref{finite} the  blowing up of the ideal $\J _
{\t^{(j-1)}} ^ {(j)} $ of $T^{ \Gamma_ {\t^{(j-1)}} ^ {(j)} }$ is
a finite morphism, dominated by the normalization of $T^\Gamma$,
 for all $j \geq 1$. It follows that for $j \gg
0$ the variety $T^{ \Gamma_
{\t^{(j-1)}} ^ {(j)} }$ is normal. By Proposition \ref{jacoiso}, this variety is also smooth.
 By Lemma
\ref{smooth} it follows that $f^{(j)} (\t^{(j)}) = d$ for $j \gg
0$. This is a contradiction with (\ref{assumption}). \hfill $\
{\Box}$


\medskip



 Let us fix a
representation for $m = m^{(1)}$ in terms of the generators of $\Gamma$:
\[
m  = \g_1 + \cdots  + \g_d \mbox{ with } \g_1 \y \cdots \y \g_d
\ne 0,
\]
(up to an eventual relabelling  of the generators $\{\g_i
\}_{i=1}^r$ of the semigroup  $\Gamma$).



By Claim \ref{strict} we can suppose without loss of generality
that $i_1 =0$, that is, the Newton polyhedron $\Newton_\s (\J)$
has at least two different vertices $m$ and $n$.



\begin{lemma}\label{end}
Given one of the $\gamma_j$ which appear in the decomposition of
$m$, say $\gamma_d$, for any $j \geq 0$ the vector $n_j := n -
j \g_d$  verifies that
\[
n_j  \in \J_{\t^{(j)}}^{(j+1)} \mbox{ and } (n_j -m) \y \g_1
\y  \cdots \y \g_{d-1} \ne 0.
\]
\end{lemma}
\textit{Proof} We prove the assertion by induction on $j$. Notice
that for $j=0$ the vector $n_0 = n$ belongs to $\J$ by
hypothesis. We suppose by induction that $n_l \in
\J_{\t^{(l)}}^{(l+1)}$, $1 \leq l \leq j$.

We  prove first that:
\begin{equation} \label{independent}
(n_l -m) \y \g_1  \y \cdots \y \g_{d-1} \ne 0 \mbox{,  for } 0
\leq l \leq j.
\end{equation}

Assume on the contrary that (\ref{independent}) does not hold for
some $0 \leq l \leq j$.  After relabelling the vectors $\g_1,
\dots, \g_{d-1}$ if necessary, we have an expansion of the form:
\begin{equation} \label{decomposition}
n_l -m = a_1 \g_1 + \cdots + a_h \g_h \mbox{ with } h \leq d-1,
\end{equation}
and in addition the coefficients of (\ref{decomposition}) are
non-zero rational numbers which are not of the same sign, that is,
\[
\left\{
\begin{array}{ccccccl} a_i & > & 0 & \mbox{for} & i & = & 1, \dots, s
\\
a_i & < &  0 & \mbox{for} & i &  = & s+1, \dots, h.
\end{array}
\right.
\]
Indeed, if all coefficients $a_i$ in (\ref{decomposition}) are
$\geq 0$  we obtain that
\[
n = m + a_1 \g_1 + \cdots + a_h \g_h + l \g_d \subset m + \check\s,
\]
contradicting that $n \ne m$ is a vertex of $\Newton_\s (\J)$. In
particular, we have that $n_l \ne m$. Similarly, if all the
coefficients $a_i$ are smaller than zero we get that
\[
m = n_l -  a_1 \g_1 - \cdots - a_h \g_h \subset  n_l +
\check\s \subset n_l +  \check{\t}^{(l)}.
 \]
This implies that $m$ is not a vertex of the Newton polyhedron
$\Newton_{\t^{(l)}}  (\J_{\t^{(l)}}^{(l+1)} )$,  since $n_l \in
\J_{\t^{(l)}}^{(l+1)}$ and $n_l \ne m$.



If $\a \in \R$ we denote by  $\lceil \a \rceil$ the smallest
integer $p$ such that $\a \leq p$.

\begin{claim} \label{Q}
If ${q} := \sum_{i=1}^s \lceil a_i \rceil$, $0 \leq {p} \leq q$
and $b_i$ are integers such that $0 \leq b_i \leq \lceil a_i
\rceil$, $i=1, \dots, s$ and $\sum_{i=1}^s b_i = {p}$ then the
vector $\beta_{l,{p}} := n_l - \sum_{i=1}^s b_i \g_i$ belongs
to $\J_{\t^{(l+{p} )}}^{(l+{p}+1)}$.
\end{claim}
\textit{Proof of the claim.}  We prove the assertion by induction
on ${p}$. For ${p} =0$ we have $\beta_{l, 0} = n_{l}$ hence
the assertion holds by assumption. Suppose that $\beta_{l,{p}}
\in \J_{\t^{(l+{p} )}}^{(l+{p}+1)}$ for $0 \leq {p} < {q}$. The
vector
\[
\beta_{l,{p}}  - m = n_l - m - \sum_{i=1}^s b_i \g_i =
\sum_{i=1}^s (a_i - b_i) \g_i + \sum_{i =s+1}^h a_i \g_i
\]
belongs to $\Gamma_{\t^{(l+{p}+1 )}}^{(l+{p}+2)}$. Since ${p} <
{q}$ there is a strictly positive coefficient in this expansion of
$\beta_{l,{p}}  - m $, say $a_1 - b_1$, for instance. We get $
(\beta_{l,{p}}-m) \y \g_2 \y \cdots \y \g_d \ne 0$, hence the vector
\[ \beta_{l,{p}+1} := \beta_{l,{p}} - m + \g_2 + \cdots + \g_{d} =
\beta_{l,{p}} - \g_1
\] belongs to $\J_{\t^{(l+{p}+1 )}}^{(l+{p}+2)}$.  \hfill $\ {\Box}$

\medskip


By Claim \ref{Q}  the expansion
\[
\beta_{l, {q}} - m = \sum_{i=1}^s  (a_i - \lceil a_i \rceil) \g_i
+ \sum_{i = s+1}^h a_i \g_i
\]
has only coefficients $\leq 0$ and  $\beta_{l, {q}} \in
\J_{\t^{(l+{q}+1 )}}^{(l+{q}+2)}$. We get also that $m \ne
\beta_{l, {q}}$ since the coefficients $a_{s+1}, \dots, a_h$ are
non-zero and the vectors $\g_1, \dots , \g_h$ are linearly
independent. We deduce from this that
\[
m = \beta_{l, {q}} - \sum_{i=1}^s  (a_i - \lceil a_i \rceil) \g_i
-  \sum_{i = s+1}^h a_i \g_i \, \in \,  \beta_{l, {q}} + \check\s\, \subset \, \beta_{l, {q}} + {\check{\t}^{(j+{q}+1 )}}.
\]
This contradicts the assumption, $m$ being a vertex of the Newton
polyhedron of $\J_{\t^{(l+{q}+1 )}}^{(l+{q}+2)}$.

Finally, we have proven that (\ref{independent}) holds hence
\[ n_{j+1} = (n_j -m) + \g_1 +  \cdots +  \g_{d-1}  \in  \J_{\t^{(j+1)}}^{(j+2)}.\]
 This concludes the induction in the proof of Lemma \ref{end}. \hfill $\ {\Box}$

\medskip



The cone
\[
\t^{(\infty)} = \bigcap_{l \geq 1} \t^{(l)} = \bigcap_{j \geq 1}
\t^{(i_j)}
\]
is a closed convex subset of $\s$ different from $ 0 $. A vector
$0 \ne w \in \t^{(\infty)}$ defines a \textit{monomial valuation}
$\w$ of the field of fractions of $k [t^\Gamma]$, which verifies
that if $ 0 \ne \sum a_\g t^\g \in k [t^\Gamma]$ then
$
\w (  \sum a_\g t^\g  ) = \min_{a_\g \ne 0}  \langle w, \g
\rangle
$.
By definition this valuation is non-negative in the subrings $k [
t^{\Gamma_ {\t^{(j-1)}} ^ {(j)}}]$ for all $j \geq 1$. Notice that
the vector $w \in N_\R$ is not necessarily an element of $N_\Q$
and it may lie in a face of $\s$ (different from $0$).
We remark that for all $j\geq 1$ we have that
$ \hbox{\rm min}  \{ \langle w,\gamma\rangle   \mid   \gamma\in
\J^{(j)}_{\t^{(j-1)}}
   \} = \langle w,m\rangle  $
 since $w$ takes non negative values on $\J^{(j+1)}_{\t^{(j)}}$
 which contains the  set $\{\gamma-m\mid \gamma\in\J^{(j)}_{\t^{(j-1)}} \}$.

Since $\g_1 \dots, \g_d$ span $M_\R$ at least one of the vectors
$\g_i$ verifies that  $\langle w, \g_i \rangle \ne 0$. Suppose for
instance that $\langle w, \g_d \rangle > 0$.

By Lemma \ref{end}, for any integer $j\geq 0$  the vector $ n_j
= n - j \g_{d}$ belongs to $\J_{\t^{(j)}}^{(j+1)}  \subset
\Gamma_{\t^{(j)}}^{(j+1)} $. This implies that
\[
\w( t^{n_j}) = \langle w, n_j \rangle = \langle w, n \rangle
- j \langle w, \g_d \rangle
\]
becomes strictly negative for $j$ large enough. This is a
contradiction since $t^{n_j} \in k [
t^{\Gamma_{\t^{(j)}}^{(j+1)}}]$ and the valuation $\omega$ is non
negative on  the ring $k [ t^{\Gamma_{\t^{(j)}}^{(j+1)}}]$. \hfill
$\ {\Box}$

\begin{corollary} \label{bound2}
With the previous notations, given any sequence of the form
(\ref{sequence}) if $T^{\Gamma}$ is not smooth is not possible
that $m^{(1)} = m^{(j)}$ for all $j \geq 2$.
\end{corollary}
This is now a consequence of Proposition \ref{eq}.


%% ADDITION: Stability problem
\begin{definition} \label{def-sta}\par\noindent
\begin{enumerate}
\item[i.] If $0 \ne \eta$ is a cone we denote by  $\nu_{\eta}$ the
sum of the primitive  vectors, for the lattice $N$
 in the edges of the cone $\eta$.
\item[ii.]A cone $\eta \subset \s$ is \textit{stable} if there is
an integer $I \geq 1$ such that $\eta \in \Sigma^{(j)}$ for all $j
\geq I$. The \textit{stability problem} for the toric variety
$T^\Gamma$ consists of determining if the sequence of fans
$(\Sigma^{(j)})_{j\geq 0}$ stabilizes, that is $\Sigma^{(j)} =
\Sigma^{(j+1)}$ for $j \gg 0$. \item[iii.] If $\theta \in
\Sigma^{(l)}$ the \textit{stability problem} for the cone $\theta$
consists of determining if the sequence of fans,
 $\{ \theta^{(j)} \in \Sigma^{(j)} \mid \theta^{(j)} \subset   \theta \}$ $j \geq l$, stabilizes.
\end{enumerate}
\end{definition}
For instance,  if $\r \in \Sigma^{(j)}$ is of dimension one then  $\r$ is stable.

%%%%% ADDITION
The solution of the stability problem is the essential part of our proof that the iteration of Semple Nash modifications
eventually resolves singularities of toric varieties.


\begin{lemma} \label{qzero}
If  $\theta    \in \Sigma^{(j-1)}$ is a cone of codimension $>0$ and
 $M ( \theta, \Gamma^{(j)}_\theta) = M(\theta)$ (see Notation \ref{ind}), then the stability problem
for the cone $\theta \subset N_\R$ is equivalent to a  stability problem for the cone $\theta$, viewed in $(N_\theta)_\R$,
with respect to the sequence
of iterated Semple-Nash modifications of another toric variety of dimension equal to $\dim \theta$.
\end{lemma}
\textit{Proof}.
The sublattice $M (\theta) = M \cap \theta^\perp$ of $M$ is obviously saturated, so that
it is a direct summand of $M$. Let us consider a sublattice $M'$
of $M$ such that $M = M (\theta) \oplus M'$. Such a sublattice
$M'$ is spanned by vectors $v_{q_0 +1 }, \dots, v_d$, completing a
basis $v_1, \dots, v_{q_0}$ of $M (\theta)$ to a basis of $M$. Any
$\g \in M$ can be written in a unique way as $\g = \a_1(\g)  +
\a_2(\g) $ with $\a_1 (\g)  \in M (\theta)$ and $\a_2 (\g) \in
M'$. The restriction $\beta \colon M' \mapsto M/ M (\theta)$ of
the canonical map $M \to M/M (\theta)$ to the sublattice $M'
\subset M$ is an isomorphism. The image
$\tilde{\Gamma}_{\theta}^{(j)}$ of ${\Gamma}_{\theta}^{(j)}$ by the canonical map $M \mapsto M/ M (\theta)$ is
a semigroup of finite type, generates the lattice $M/ M (\theta)$
and spans a strictly convex cone $\R_{\geq 0}
\tilde{\Gamma}_{\theta}^{(j)}$.

The semigroup  $\Gamma^{(j)}_\theta  \cap  \theta^{\perp}$, which  is the minimal face of
$\Gamma^{(j)}_\theta$, is a rank $q_0$ lattice by Lemma \ref{conditions-aff}.
Using the equality    $\Gamma^{(j)}_\theta  \cap  \theta^{\perp} = M(\theta)$
one checks directly that the map
\begin{equation}       \label{spliting}
{\Gamma}_{\theta}^{(j)} \longrightarrow  M (\theta) \times
\tilde{\Gamma}_{\theta}^{(j)},  \quad  \g \mapsto (\a_1
(\g), \beta \circ \a_2 (\g) )
\end{equation}
is a semigroup isomorphism.
 The map (\ref{spliting}) determines an
isomorphism $ T^ {{\Gamma}_{\theta}^{(j)}}
\stackrel{\simeq}{\longrightarrow } \orb (\theta,
{\Gamma}_{\theta}^{(j)} ) \times
\tilde{\Gamma}_{\theta}^{(j)} $.
 In particular, the
variety $T^{{\Gamma}_{\theta}^{(j)}}$ is smooth if and
 only if $T^{\tilde{\Gamma}_{\theta}^{(j)}}$ is smooth.

 According to Remark
\ref{idproduct} and Lemma \ref{induced} the blowing up of
logarithmic jacobian ideals commutes with the splitting defined by
(\ref{spliting}).      The assertion follows from this since the cone $\theta$,  viewed in the $\R$-linear subspace
$(N_\theta)_\R$ it spans in $N_\R$, is the dual cone of $\R_{\geq 0}
\tilde{\Gamma}_{\theta}^{(j)}$.
\hfill $\ {\Box}$



 \begin{remark}
Geometrically, we see that the semigroup   $\tilde\Gamma_{\theta}^{(j)}$  in Lemma \ref{qzero}
corresponds to
 the toric variety of dimension $\dim \theta$ which is
 a transverse linear section of $T^{\Gamma^{(j)}_\theta}$
 at the point $(1,\ldots, 1)$ of the orbit corresponding to $\theta$.
 \end{remark}


\begin{proposition}  \label{stable}
If   $\theta$ is a stable cone then there is an integer $I
\geq 1$ such that the variety $T^{\Gamma^{(j)}_\theta}$
 is smooth
for all $j \geq I$. If $T^{\Gamma^{(j)}_\theta}$
 is smooth the cone $\theta$ is stable.

\end{proposition}
\textit{Proof}. By Lemma \ref{qzero} we can assume that $\codim \theta = 0$.
The  blowing up of the ideal $\J _ {\theta} ^ {(j)} $ of $T^{
\Gamma_ {\theta} ^ {(j)} }$ is a finite morphism, dominated for all $j \gg 0$
by the
normalization of $T^{ \Gamma_ {\theta} ^ {(j)} }$, which is equal to that of $T^{ \Gamma_ {\theta} ^ {(1)} }$
 (see Proposition \ref{finite}). It follows that for $j \gg
0$ the map $T^{ \Gamma_ {\theta} ^ {(j+1)} }\to T^{ \Gamma_ {\theta} ^ {(j)} }$ is an isomorphism.
 By
Proposition \ref{jacoiso}, this variety is smooth. The converse follows directly from the definitions.
 \hfill $\ {\Box}$


\begin{proposition} \label{prev}
Given $0 \ne \nu \in \s \cap N$,   for any $j \geq 1$ there exists a unique cone $\theta^{(j)}$ such that
$\nu \in \int(\theta^{(j)} )$.  Then for $j \gg 0$ we have:
\begin{enumerate}
\item[{\rm i.}] If $q_0 = \codim \theta^{(j-1)}$ and if $m^{(j)} \in \J^{(j)}_{\theta^{(j-1)}}$ is such that
$\ord_{\J^{(j)}} (\nu) = \langle \nu , m^{(j)} \rangle$, then for any representation
\begin{equation}      \label{repre}
m^{(j)} = \g_1^{(j)} + \cdots + \g_d^{(j)}
\end{equation}
as a sum of linearly independent elements in the
semigroup $\Gamma_{\theta^{(j-1)}}^{(j)}$, there are exactly
$q_0$ of them in $ M(\theta^{(j-1)})$.
\item[{\rm ii.}]  $M(\theta^{(j-1)} , \Gamma_{\theta^{(j-1)}}^{(j)}) =  M(\theta^{(j-1)}) = M( \theta^{(j)} ) $ (cf. Notation \ref{ind}).
\end{enumerate}
Moreover, the sequence of cones   $(\theta^{(j)})$ stabilizes.
\end{proposition}
\textit{Proof}.   Notice that the vector
$\nu$ is in a subdivision of $\theta^{(j)}$ induced by
$\Sigma^{(j+1)}$, so that we have $\theta^{(j+1)}\subseteq
\theta^{(j)}$.
   The sequence $(\codim_{N_\R}  \ \theta^{(j)} )_j$ is increasing, thus there exists an integer $0 \leq q_0 \leq d-1$
such that  $ \codim  \ \theta^{(j)} = q_0 $, for $j \gg 0$.

Since $\nu \in \int (\theta^{(j)}) \cap N$, we have the equalities
\begin{equation} \label{perp}
 \Gamma_{\theta^{(j-1)}}^{(j)} \cap ({\theta^{(j-1)}})^\perp  = \Gamma_{\theta^{(j-1)}}^{(j)} \cap  \nu^\perp =
   M( \theta^{(j-1)} , \Gamma_{\theta}^{(j)} ).
\end{equation}
The lattice $M( \theta^{(j-1)} , \Gamma_{\theta^{(j-1)}}^{(j)} ) $
is a sublattice of finite index $i ( \theta^{(j-1)} ,
\Gamma_{\theta^{(j-1)}}^{(j)} )$ of $M( \theta^{(j-1)})$ (see
Notation \ref{ind} and Lemma \ref{conditions-aff}). By definition
$M( \theta^{(j-1)} )$ is  a saturated subsemigroup of $M$, and it
is also a rank $q_0$ lattice. By (\ref{essential})  we deduce that
$M( \theta^{(j-1)} ) = M (\theta^{(j)} )$ and $ M( \theta^{(j-1)}
, \Gamma_{\theta}^{(j)} ) \subset M( \theta^{(j)} ,
\Gamma_{\theta}^{(j+1)} )$. Then, the sequence of indices  $( i (
\theta^{(j-1)} , \Gamma_{\theta^{(j-1)} }^{(j)} ))_j$ stabilizes,
hence          $ M( \theta^{(j-1)} , \Gamma_{\theta}^{(j)} ) = M(
\theta^{(j)} , \Gamma_{\theta}^{(j+1)} )$ for $j \gg 0$.
 The
lattice  $ M( \theta^{(j-1)} , \Gamma_{\theta}^{(j)} )$  is
\textit{a priori} a sublattice of finite index
 of $M(\theta^{(j-1)})$.

We deal first with the proof of (i).
Up to relabelling the vectors we can assume that those $\g_i^{(j)}$ appearing in (\ref{repre}),
which belong to $M(\theta^{(j-1)})$ are
 $\g_1^{(j)} \dots, \g_{s}^{(j)}$ for $0 \leq s \leq q_0$. By (\ref{perp})
 these vectors belong to $M(\theta^{(j-1)}, \Gamma_{\theta^{(j-1)}}^{(j)} )$.
Suppose that $s\ne q_0$.
 Since, the images of the $\g_i^{(j)}$,
$i=s+1 , \dots,d$,
  generate the rank $d- q_0$ lattice
 $M/M(\theta^{(j-1)})$, we get that $d- q_0$ of them, say for $i=q_0 +1 ,
 \dots,d$, are linearly independent modulo $M(\theta^{(j-1)})$.
%%%%
We can find vectors $\tilde{\g}^{(j)}_{s+1}, \dots, \tilde{\g}^{(j)}_{q_0}
\in \Gamma_{\theta^{(j-1)}} ^{(j)} \cap \nu^\perp $, such that
$\g_1^{(j)}\y \dots \y \g_s^{(j)} \y \tilde{\g}^{(j)}_{s+1}\y  \dots \y
\tilde{\g}^{(j)}_{q_0} \ne 0$.
 Then the
vector
\[ m' := \g_1^{(j)}+ \dots +  \g_{s}^{(j)}+  \tilde{\g}^{(j)}_{s+1} + \dots +
\tilde{\g}^{(j)}_{q_0}  + \g_{q_0+1}^{(j)} + \dots +  \g_{d}^{(j)}
\]
verifies that  $  m'  \in \J_{\theta^{(j-1)}}^{(j-1)}$. Since $0 =
\langle \nu,  \tilde{\g}^{(j)}_j \rangle <  \langle \nu, \tilde{\g}^{(j)}_j
\rangle $, for $ s+1\leq j\leq q_0$
 we would have $\langle \nu, m' \rangle  < \langle \nu,m^{(j)} \rangle$, a contradiction.


Suppose that (ii) does not hold. Then, since $
\Gamma_{\theta^{(j-1)}}^{(j)}$ generates the lattice $M$, there
exist $\g, \g' \in  \Gamma_{\theta^{(j-1)}}^{(j)} $ such that $\g
-\g ' \in M(\theta^{(j-1)}) \setminus
   M(\theta^{(j-1)}, \Gamma_{\theta^{(j-1)}}^{(j)} )
$. In view of (\ref{perp}) we get $0 \ne \langle \nu, \g  \rangle
= \langle \nu, \g' \rangle$. Since $\g_{q_0+ 1}^{(j)}, \dots,
\g_d^{(j)}$ define linearly independent vectors in the lattice $M
/ M(\theta^{(j-1)})$ which is of rank $d - q_0$, there exists an
integer $q_0+ 1 \leq i_0 \leq d$ such that $ \g$ (resp. $\g'$)
together with  $\g_{q_0+ 1}^{(j)} ,\dots,  \g_{i_0-1}^{(j)} ,
\g_{i_0+ 1}^{(j)} , \dots, \g_d^{(j)} $ are linearly independent
modulo $M(\theta^{(j-1)})$. Suppose without loss of generality
that  $i_0 = d$. Then the vectors $ \tilde{\g} :=  \g + \sum_{i =
1}^{d-1} \g_i^{(j)}  $ and $ \tilde{\g}' :=  \g' + \sum_{i =
1}^{d-1} \g_i^{(j)} $ belong to $\J_{\theta^{(j-1)}}^{(j)} $,
hence $\tilde{\g} -m^{(j)} = \g - \g_d^{(j)}$ and $\tilde{\g}
'-m^{(j)} = \g' - \g_d^{(j)}$  are both elements of
$\Gamma_{\theta^{(j)}}^{(j+1)} $. Since $\langle \nu,
\g_d^{(j)} \rangle  > 0$, 
 it follows that $
\tilde{\g} - \tilde{\g}' = \g - \g' $ and $ \langle \nu,
\tilde{\g} \rangle  = \langle \nu, \tilde{\g}' \rangle  < \langle
\nu, {\g} \rangle  = \langle \nu, {\g}' \rangle$. By repeating
this construction, since $\nu\in \s\cap N$, in a finite number of
steps we reduce to the case when   $\langle \nu, \g \rangle =
\langle \nu, \g' \rangle     = 0 $.
 As we remarked before this implies that $\g - \g' \in  M(\theta^{(j-1)}, \Gamma_{\theta^{(j-1)}}^{(j)} )$, a contradiction.



By Lemma \ref{qzero} and ii. we can assume that $q_0 = 0$. By
(\ref{essential2})  the sequence  of positive integers  $( \langle
\nu, m^{(j)} \rangle)_{j \geq 1} $ is decreasing, hence it
stabilizes.
 Suppose that the sequence $(\theta^{(j)})$ does not
stabilize. By Proposition \ref{stable} this implies that the toric
variety $T^{\Gamma_{\theta^{(j-1)}}^{(j)}}$ is not smooth for any
$j\geq 1$.

 Let us fix an integer $j_1 \gg 0$. By  Proposition
\ref{bound} there exists a
 smallest
 integer $j_2 > j_1$ such that $m^{(j_2)} \ne m^{(j_1)}$.
We consider a representation of $m^{(j_2)} \in \J_{\theta
^{(j_2-1)}}^{(j_2) }$ of the form (\ref{repre}). The vector
$m^{(j_1)} - m^{(j_2)}$  belongs to the semigroup $\Gamma_{\theta
^{(j_2)}}^{(j_2 + 1) }$  since   $m^{(j_1)} \in \J_{\theta
^{(j_2-1)}}^{(j_2) }$ by (\ref{essential}). We obtain that
$\langle \nu, m^{(j_1)} - m^{(j_2)} \rangle =0$, hence $ m^{(j_1)}
- m^{(j_2)} \in \Gamma_{\theta ^{(j_2)}}^{(j_2 + 1) } \cap
\nu^\perp$. Since $\nu \in \int \theta^{(j_2)}$ and $\dim
\theta^{(j_2)} = d$ we deduce also that $\Gamma_{\theta
^{(j_2)}}^{(j_2 + 1) } \cap \nu^\perp = \{ 0 \}$, a contradiction,
which ends the proof of Proposition \ref{prev}. \hfill $\ {\Box}$
\par\medskip

If $\eta \subset N_\R$ is a rational convex polyhedral cone the
duality between the lattices $N$ and $M$ induces a duality between
the lattices $N_\eta$ and $M_\eta = M/ M(\eta)$ and also a duality
between $N(\eta) = N/ N_\eta$ and $M(\eta) = M \cap \eta^\perp$
(cf. Notations \ref{ind}).



If $0 \ne \eta$ is a stable cone one can consider for $j \gg 0$
the orbit closure $T^{\Gamma^{(j)}}_{\Sigma^{(j-1)}} (\eta) $
associated to $\eta \in \Sigma^{(j-1)}$. By convenience we recall the notations to
describe this toric variety in this case. See Lemma
\ref{orbit-closure} and Notations \ref{ind}.
\begin{notation} \label{clo-st} If $\eta\in \Sigma^{(j-1)}$ is a stable cone the variety
$T^{\Gamma^{(j)}}_{\Sigma^{(j-1)}} (\eta)$ is covered by charts
defined by the semigroups $\Gamma^{(j)}_{\theta^{(j-1)}} \cap
\eta^\perp$, for $\eta \leq \theta^{(j-1)}$ and $\theta^{(j-1)}
\in \Sigma^{(j-1)}$.  Notice that the semigroup
$\Gamma^{(j)}_{\theta^{(j-1)}} \cap \eta^\perp$ is a face of
$\Gamma^{(j)}_{\theta^{(j-1)}}$, and it spans the lattice $M(\eta,
\Gamma_\eta^{(j)}) = \Gamma^{(j)}_\eta \cap \eta^\perp$ by Lemma
\ref{conditions-aff}. By Proposition \ref{prev} ii. this lattice
is equal to $M(\eta)$ if $j \gg 0$. The fan $\Sigma^{(j-1)}
(\eta)$ consists of the images $\theta^{(j-1)} ( \eta) $ of cones
$\theta^{(j-1)} \in \Sigma^{(j-1)}$ in $N(\eta)_\R$.  The cone
$\theta^{(j-1)} (\eta)$ is the dual cone of $\R_{\geq 0} (
\Gamma^{(j)}_{\theta^{(j-1)}} \cap \eta^\perp)$. We denote by
$\J^{(j)}_{\theta^{(j-1)}} ( \eta ) $ the logarithmic jacobian
ideal of $k [ t^{\Gamma^{(j)}_{\theta^{(j-1)}} \cap \eta^\perp
}]$.
\end{notation}



The following technical lemma will be useful.
\begin{lemma}      \label{basis}
Let $0 \ne \eta$ be a stable cone of codimension $d_0 < d$. We
denote by
\[ \p: M_\R \to M_\R / M(\eta)_\R, \quad \a \mapsto \tilde{\a} \] the canonical
projection. If $ (\theta^{(j)})_{j}$ is a sequence  such that
\begin{equation}           \label{cond}
          \theta^{(j)} \in
\Sigma^{(j)}, \ \theta^{(j)} \supset \theta^{(j+1)}  \  \mbox{ and
}  \ \eta \leq  \theta^{(j)},
\end{equation}
then  for $j \gg 0$ we have:
\begin{enumerate}
\item[i.]
$\p ( \Gamma_{\theta^{(j-1)}}^{(j)} ) = \p (
\Gamma_{\theta^{(j)}}^{(j+1)}    )$
 and
the semigroup $\tilde{\Gamma}_\eta := \p (
\Gamma_{\theta^{(j-1)}}^{(j)}    )  $ is generated by a basis
$\tilde{e}_{d_0 +1} , \dots, \tilde{e}_{d}$ of $M/M(\eta)$.

\item[ii.] If $m^{(j)}$ is the vertex of the polyhedron
$\mathcal{N} (\J^{(j)}_{\theta^{(j-1)}})$ such that
\begin{equation} \label{numj}
\ord_{{\J}^{(j)}} (\nu) = \langle \nu, m^{(j)} \rangle, \quad
\forall  \nu \in \theta^{(j)}, \end{equation} then for any
representation of the form (\ref{repre}) of $m^{(j)}$ as a sum of
linearly independent vectors in $\Gamma^{(j)}_{\theta^{(j-1)}}$
then exactly $d_0$ of the $\g_i^{(j)}$ belong to
$\Gamma^{(j)}_{\theta^{(j-1)}}\cap\eta^\perp$, say for $i =1, \dots, d_0$, while
$\langle \nu_\eta, \g_i^{(j)} \rangle = 1$ for $i =d_0 +1, \dots,
d$.

\item[iii.] The vector $m^{(j)}$ belongs to the face
$\mathcal{F}_{\theta^{(j-1)}}^{(j)}$ of $\mathcal{N}
(\J^{(j)}_{\theta^{(j-1)}})$ determined by $\nu_\eta$ (cf.
Definition \ref{def-sta}). The face
$\mathcal{F}_{\theta^{(j-1)}}^{(j)}$ is the Minkowski sum
\[ \mathcal{N} (\J^{(j)} _{\theta^{(j-1)}} (\eta) ) +
\mathcal{P}^{(j)}_{\theta^{(j-1)}},\]
 where
$\mathcal{P}^{(j)}_{\theta^{(j-1)}}$ is the convex hull of the set
$ \bigcup \, \delta_{i_{d_0 +1}}^{(j)}  + \cdots + \delta_{i_{d}}^{(j)}
\, + \, (\check\theta^{(j-1)}
  \cap \eta^{\perp})$,
for    $\delta_{i_{d_0 +1}}^{(j)} , \dots, \delta_{i_{d}}^{(j)} \in
\Gamma_{\theta^{(j-1)}}^{(j)}$ such that
$
       \delta_{i_{d_0 +1}}^{(j)} \y \cdots \y \delta_{i_{d}}^{(j)} \ne 0$,   and   $\langle \nu_\eta, \delta_{i_{l}}^{(j)} \rangle =
       1 $ for $l=d_0+1,\ldots, d $.
\end{enumerate}
\end{lemma}
\textit{Proof}. Since
$\eta$ is a stable cone we get that $M(\eta) = M(\eta,\Gamma_\eta^{(j)})$,
by applying Proposition \ref{prev} to the constant sequence of cones $\eta_j :=\eta$.
By Lemma \ref{qzero} the cone $\eta \subset (N_\eta)_\R$ is a stable cone for the semigroup
$     \p ( \Gamma_{\eta}^{(j)}    ) $ for $j \geq j_0 \gg 0$.  By Proposition \ref{stable}
the sequence of semigroups $(\p ( \Gamma_{\eta}^{(j)}    ))_{j\geq j_0}$ stabilizes and
$\p ( \Gamma_{\eta}^{(j)}    )$  is generated by a basis of $M/M(\eta)$,
for $j \gg 0$.
By Lemma \ref{conditions-aff}
we have that
$ \Gamma_\eta^{(j)} = \Gamma_{\theta^{(j-1)}}^{(j)}  + \Z_{\geq 0} (-u_j) $, for any
 $u_j \in \Gamma_{\theta^{(j-1)}}^{(j)}$  which belongs to $\int (\check{\theta}^{(j-1)} \cap \eta^\perp)$.
We deduce that   $\p ( \Gamma_{\theta^{(j-1)}}^{(j)}    )  = \p (
\Gamma_{\eta}^{(j)}) $ for $j \gg 0$.


Since $\eta \leq \theta^{(j)}$ we get from (\ref{numj}) that
$\ord_{\J^{(j)}} (\nu_\eta) = \langle \nu_\eta, m^{(j)} \rangle$.
This implies that $m^{(j)}$ belongs to
$\mathcal{F}_{\theta^{(j-1)}}^{(j)}$. Notice also that $m^{(j)}$
belongs to $\J_{\eta}^{(j)}$. By Proposition \ref{prev} i., for
any representation of $m^{(j)}$ of the form (\ref{repre}), $d_0$
of the $\g_i^{(j)}$ belong to $\eta^{\perp}$, say for $i =1,
\dots, d_0$. By Lemma \ref{induced} the vectors $\p (\g^{(j)}_i) =
\tilde{\g}^{(j)}$, $i =d_0 +1, \dots, d$ are linearly independent
elements of $\Gamma_\eta$, such that $\tilde{m}^{(j)} = \sum_i
\tilde\g^{(j)}_i $ belongs to the logarithmic jacobian ideal $
\tilde{\J}_\eta$ of $k[t^{\tilde{\Gamma}_\eta}]$. We know that
$\langle \nu, \g \rangle = \langle \nu, \tilde{m} \rangle$ for any
$m \in M$ and $\nu \in \eta$,  thus the vector $\tilde{m}^{(j)}$
is in the face of $\mathcal{N} ( \tilde{\J}_\eta)$ determined by
$\nu_\eta$. By i. the semigroup $\tilde{\Gamma}_\eta$ is regular,
hence we deduce that $\tilde{m}^{(j)} = \sum_{i=d_0+1}^d
\tilde{e}_i$ and, up to relabelling,  $\tilde{e}_i =
\tilde{\g}^{(j)}_i$, $i = d_0 +1, \dots,d$. This ends the proof of
ii.~ and also shows that $m^{(j)} \in \mathcal{N} (\J^{(j)}
_{\theta^{(j-1)}} (\eta) ) + \mathcal{P}^{(j)}_{\theta^{(j-1)}}$.

We have also shown the equalities
\[ \ord_{\J^{(j)}} ( \nu_\eta) = \langle
\nu_\eta, m^{(j)} \rangle = \langle \nu_\eta, \tilde{m}^{(j)}
\rangle = d-d_0.  \] Finally, we remark that the argument given
above applies more generally for any vector $m \in
\J^{(j)}_{\theta^{(j-1)}}$ in the face
$\mathcal{F}^{(j)}_{\theta^{(j-1)}}$. This implies that
iii.~holds. \hfill $\ {\Box}$


\begin{proposition} \label{uno}
If $0 \ne \eta$ is a stable cone and if  $ (\theta^{(j)})_{j}$ is a sequence of cones of the form (\ref{cond})
such that $ \theta^{(j)} $ contains $\eta$ as a face of codimension one,  then
\begin{equation} \label{ret}
M(\theta^{(j-1)}, \Gamma_{\theta^{(j-1)}} ^{(j)} ) = M (\theta^{(j-1)}) \mbox{ for } j \gg 0,
\end{equation}
and the sequence of cones $(\theta^{(j)})_{j
\geq I}$ stabilizes.
\end{proposition}
\textit{Proof}.  We denote by $q_0$ the integer $\codim_{N_\R}  \ \theta^{(j)}$ for $j \gg 0$ and
by $d_0$ the codimension of $\eta$.
Notice that $0 \leq q_0 < d-1$ since $0 \ne \eta \leq \theta^{(j)}$.

We deal first with the proof of (\ref{ret}). By the argument given
in the proof of Proposition  \ref{prev} we get $M(\theta^{(j-1)} )
= M(\theta^{(j)})$ and $M(\theta^{(j-1)} ,
\Gamma^{(j)}_{\theta^{(j-1)}}) = M(\theta^{(j)} ,
\Gamma^{(j+1)}_{\theta^{(j)}})$ for $j \gg 0$.
 We recall that for $j \gg 0$
the lattice spanned by the face $\Gamma^{(j)}_{\theta^{(j-1)}}
\cap \eta^\perp$ is equal to $M(\eta)$ (see Notations
\ref{clo-st}).

If (\ref{ret}) were not true then there exists $\g, \g' \in   \Gamma^{(j)}_{\theta^{(j-1)}}$ such that
$\g - \g' \in   M (\theta^{(j-1)}) \setminus  M(\theta^{(j-1)}, \Gamma_{\theta^{(j-1)}} ^{(j)} )$. Notice then that
$\langle \nu_\eta, \g \rangle = \langle \nu_\eta, \g' \rangle $ since $M (\theta^{(j-1)}) \subset M(\eta)$ by duality.


By Lemma \ref{basis}, if  $m^{(j)} \in \J_{ \theta^{(j-1)}}^{(j)}$
is such that $\ord_{\J^{(j)}} (\nu) = \langle \nu, m^{(j)}
\rangle$ for any $\nu \in \theta^{(j)}$ then for any
representation of $m^{(j)}$ of the form (\ref{repre}),  $d_0$ of
the $\g_i^{(j)}$ belong to $M(\eta)$.

By applying the argument of Proposition \ref{prev} we can assume,
replacing $j$ by a bigger number, that $\langle \nu_\eta, \g
\rangle  =0$, that is $\g$ and $\g'$ belong to the face $
\Gamma^{(j)}_{\theta^{(j-1)}} \cap \eta^\perp$.

By hypothesis $\eta \leq \theta^{(j-1)}$ is a face of codimension
one. The image of
 $\check{\theta}^{(j-1)} \cap \eta^\perp$    in $M(\eta)_\R / M(\theta^{(j-1)})_\R$ is the dual cone of
the image   $\bar{\theta}^{(j-1)}$  of $\theta^{(j-1)}$  in $ (N_{\theta^{(j-1)})_\R} / (N_\eta)_\R$.
Notice that the lattice  $N_{\theta^{(j-1)}} / N_{\eta}$  and the cone $\bar{\theta}^{(j-1)}$
  are independent of $j$ for
$j\gg 0$.
We denote by  $\bar{\nu}$ the generator of the semigroup $\bar{\theta}^{(j-1)} \cap (N_{\theta^{(j-1)}} / N_{\eta})$
and by $\bar{\a}$ the class of $\a \in M(\eta)$ modulo $M(\theta^{(j-1)})$.

Among those $\g_i^{(j)}$ which belong to $\eta^{\perp}$ there exists at least one, say $\g_d^{(j)}$, which does not belong to
$M(\theta^{(j-1)})$, hence $ \langle \bar{\nu}, \bar{\g}_d^{(j)} \rangle \ne 0$. Notice also that
$0 \ne \langle \bar{\nu}, \bar{\g} \rangle =  \langle \bar{\nu}, \bar{\g}' \rangle$. Then,
we apply the same argument as in
the proof of Proposition \ref{prev} i. to get that  $\g - \g_d^{(j)}$ and $\g' - \g_d^{(j)}$ belong to
$\Gamma_{\theta^{(j)}}^{(j+1)} \cap \eta^\perp$ and $\langle \bar{\nu},  \bar{\g} - \bar{\g}_d^{(j)} \rangle < \langle
\bar{\nu}, \bar{\g} \rangle$. By iterating this procedure,  replacing $\g$ and $\g'$ by $\g - \g_d^{(j)}$ and
$\g - \g_d^{(j)}$, respectively, we reduce to the case $0 =
\langle \bar{\nu}, \bar{\g} \rangle  =   \langle \bar{\nu}, \bar{\g}' \rangle$. But this implies that
$\g, \g'$ belong to $M( \theta^{(j-1)}, \Gamma^{(j)}_{\theta^{(j-1)}} )$, contradicting  the hypothesis.
This ends the proof of the equality
(\ref{ret}).

\medskip

We prove now that the sequence $(\theta^{(j)})$ stabilizes. Since
(\ref{ret}) holds, by Lemma \ref{qzero} we can assume that $q_0 =
0$. Then, by hypothesis the lattice $M(\eta)$ is of rank one. The
cone $\check\theta^{(j-1)}\cap \eta^{ \perp} \subset M(\eta)_\R$
is a one dimensional face  of $\check\theta^{(j-1)}$ for all $j
\gg 0$. Since $\check{\theta}^{(j-1)} \subset
\check{\theta}^{(j)}$ we get that the cone $\check\vartheta:=
\check\theta^{(j-1)}\cap \eta^{ \perp}$ is independent of $j$, for
$j \gg 0$.


With notations of Lemma \ref{basis}, the Minkowski sum $\p^{-1}
(\tilde{e}_i) \cap \Gamma^{(j)}_{\theta^{(j-1)}} +
\check\vartheta$ is an affine one dimensional cone with only one
vertex $\g_i^{(j)}$, which belongs to
$\Gamma^{(j)}_{\theta^{(j-1)}}$, for $i=2, \dots, d$. We denote
by $\g_1^{(j)}$ the smallest generator of the rank one semigroup
$\Gamma^{(j)}_{\theta^{(j-1)}}\cap \check\vartheta$. Notice that
$\g_1^{(j)}$ is independent of $j$ for $j \gg 0$.

By Lemma \ref{basis} the vector $m^{(j)} = \g_1^{(j)} + \cdots +
\g_{d}^{(j)} \in \J^{(j)}_{\theta^{(j-1)}}$  is the unique vertex
of the face $\mathcal{F}^{(j)}_{\theta^{(j-1)}} = m^{(j)}+
\check\vartheta$. Since $\nu_\eta \in \theta^{(j)}$ the cone
$\theta^{(j)}$ is dual to the cone spanned by $\{ \g - m^{(j)}
\mid \g \in \J^{(j)}_{\theta^{(j-1)}} \}$.





\begin{claim}\label{oneorbit} The semigroup
$\Gamma^{(j)}_{\theta^{(j-1)}} \cap \eta^\perp $  is equal to
$M(\eta)$, for $j \gg 0$.
 \end{claim}
\textit{Proof of the claim.}  By Proposition \ref{prev} the
semigroup $\Gamma^{(j)}_{\theta^{(j-1)}}\cap\eta^{ \perp} =
\Gamma^{(j)}_{\theta^{(j-1)}}\cap \check\vartheta$
 generates the  group
$M(\eta)$,  for $j \gg 0$. If it has only one generator the result
follows directly. Assume that it has at least two generators. Let
us denote by $\d > \g_1^{(j)}$ the second element by order of
size. The element $n^{(j)}= \gamma^{(j)}_2+\cdots
+\gamma^{(j)}_1+ \delta$ belongs to
$\J^{(j)}_{\theta^{(j-1)}}$ so that $n^{(j)}-m^{(j)}=\delta
-\gamma_1^{(j)}$ is in $ \Gamma^{(j+1)}_{\theta^{(j)}}$. We see
that after finitely many steps the smallest generator of our
semigroup has decreased, so in the end we reach the generator of
the regular semigroup $\check\vartheta \cap M$, which proves the
result. This is similar to the resolution process for one
dimensional affine toric varieties by Semple-Nash
modifications.\hfill $\ {\Box}$


By Lemma \ref{basis} and Claim \ref{oneorbit} the elements
$\g_1^{(j)}, \dots, \g_d^{(j)}$ form a basis of $M$.



The expansion of $\g \in \Gamma^{(j)}_{\theta^{(j-1)}}$  in terms of
the basis  $\g_1^{(j)}, \dots, \g_{d}^{(j)}$ is of the form
\begin{equation} \label{exp-gamma}
\g = a_1\g_1^{(j)} + \cdots + a_d \g_d^{(j)},
\end{equation}
where $a_i \in \Z_{\geq 0}$ and $a_d \in \Z$. If $\g$ is of the
form (\ref{exp-gamma}) then we have $\langle \nu_\eta, \g \rangle
= \sum_{i=2}^{d} a_i$. In particular, we get that
$\ord_{\J^{(j)}} (\nu_\eta) = \langle \nu_\eta, m^{(j)} \rangle =
d-1$.


We denote by $G^{(j)}$ the minimal generating system of the
semigroup $\Gamma^{(j)}_{\theta^{(j-1)}}$, and by   $g^{(j)}$ the
maximum of $\nu_\eta$ on the set $G^{(j)}$.

Notice that the elements $\g_1^{(j)}, \dots, \g_d^{(j)}$ belong to
$G^{(j)}$ by our assumptions.  If a generator $\g \in \G^{(j)}$ is different
from $\g_1^{(j)}, \dots, \g_d^{(j)}$ then the coefficient $a_1$
 in (\ref{exp-gamma}) is $< 0$, and $\langle \nu_\eta, \g \rangle  \geq 2$. Otherwise $\g$ would
be in the semigroup generated by the $\g_i^{(j)}$, contradicting
the minimality of the generating system $G^{(j)}$. We deduce that
the equality $g^{(j)} = 1$  implies that $G^{(j)} = \{ \g_i^{(j)}
\}_{i=1}^d$, hence in this case the semigroup
$\Gamma^{(j)}_{\theta^{(j-1)}}$ is regular.

\begin{claim}
If $g^{(j)} > 1$ there exists an integer $t_0 \geq 1$ such that
$g^{(j+t_0)} < g^{(j)}$.
\end{claim}
\textit{Proof of the claim}. If $\g \in G^{(j)}$ with $\langle
\nu_\eta, \g \rangle  \geq 2$ is of the form (\ref{exp-gamma}) and
if $a_l \ne 0$ then $\g + \sum_{i=1, \dots, d}^{i\ne l}
\g_i^{(j)}$ belongs to $\J^{(j)}_{\theta^{(j-1)}}$ hence $\g -
\g_l^{(j)} = \g + \sum_{i=1, \dots, d}^{i\ne l} \g_i^{(j)} -
m^{(j)}$ belongs to the semigroup $\Gamma^{(j+1)}_{\theta^{(j)}}$
by definition. We call these elements \textit{followers} of $\g$.
Notice that $\g - \g_1^{(j)}$ is always a follower of $\g$ with
$\langle \nu_\eta, \g \rangle  = \langle \nu_\eta,  \g -
\g_1^{(j)} \rangle$, while $\langle \nu_\eta, \g - \g_l^{(j)}
\rangle  < \langle \nu_\eta, \g \rangle$ if $l \ne 1$.

We show first that the semigroup $\Gamma_{\theta^{(j)}}^{(j+1)}$
is generated by $\g_1^{(j)}, \dots, \g_d^{(j)}$ and the followers
of $\g$, for $\g$ in $G^{(j)}$ with $\langle \nu_\eta, \g \rangle
\geq 2$. Let $\g_1', \dots, \g_d'$ be linearly independent
elements  in $G^{(j)}$. Up to relabelling we can assume that
$\g_i' = \g_i^{(j)}$ for $i= s+1, \dots, d$,  and $\g_i' \notin \{
\g_1^{(j)}, \dots, \g_s^{(j)} \}$, in particular $\langle
\nu_\eta,  \g_i' \rangle \geq 2$, for $i=1, \dots,s$. Then, there
is a permutation $(l_1, \dots, l_s)$ of $(1, \dots,s)$ such that
the $i$-th coeeffient of the expansion of $\g_{l_i}$ in terms of
the basis $\g_1^{(j)}, \dots, \g_d^{(j)}$ is non zero, for $i =1 ,
\dots, s$ (otherwise we would get  $\g_1' \y \dots \y \g_s' = 0$,
which is contrary to the assumption). We deduce that $\g_{l_i}
-\g_i$ are followers of $\g_{l_i}'$ for $i=1, \dots, s$, and  the
element $\sum_{i=1}^d \g_i' - m^{(j)} = \sum_{i=1}^s (\g_{l_i}' -
\g_i^{(j)})$  of $\Gamma_{\theta^{(j)}}^{(j+1)}$  is in the
semigroup generated by $\g_1^{(j),} \dots, \g_d^{(j)}$ and the
followers of $\g$, for $\g$ in $G^{(j)}$ with $\langle \nu_\eta,
\g \rangle  \geq 2$ as stated.

We deduce  also that $m^{(j)} \ne m^{(j+1)}$ if and only if there
is an element $\g \in G^{(j)}$ with $\langle \nu_\eta, \g \rangle
= 2$. Indeed, if $\langle \nu_\eta, \g \rangle  = 2$ then there is
$2 \leq i \leq d$ such that $\g - \g_i^{(j-1)}$ is a follower of
$\g$ with $\langle \nu_\eta,  \g - \g_i^{(j-1)} \rangle =1$.  This
implies that $\g_i^{(j)}$ is in the semigroup generated by $\g -
 \g_i^{(j)}$ and $\g_d^{(j)}$, thus $\g_i^{(j)}$ does not belong
 to $G^{(j+1)}$ hence $\g_i^{(j)} \ne \g_i^{(j+1)}$ and $m^{(j)}
 \ne m^{(j+1)}$. The converse is deduced similarly.





Assume that $m^{(j)} = m^{(j+1)} = \cdots  = m^{(j+t_0 -1)}$ for
some $t_0 \geq 1$. Let $\g \in G^{(j)}$  be such that $\langle
\nu_\eta, \g \rangle  = g^{(j)}$. If $\g$ is of the form
(\ref{exp-gamma}) the followers of $\g$ after at most  $t_0$
iterations are $\g_\a = \sum_{i=1}^d (a_i - \a_i) \g_i^{(j)}$,
where $\a= (\a_1, \dots, \a_d) \in \Z_{\geq 0}^d$ verify that
$\sum_{i=1}^{d} \a_i \leq t_0$, $\a_i \leq a_i$ for $i=2, \dots,
d$ and $\langle \nu_\eta, \g_\a \rangle = \sum_{i=2}^{d} (a_i
-\a_i) \geq 1$. Those $\g_\a$ with $\langle \nu_\eta, \g_\a
\rangle = \langle \nu_\eta, \g \rangle$ are precisely $\g - l
\g_1^{(j)}$ for $0 \leq l \leq t_0$. The elements  $\g - l
\g_1^{(j)}$, $l =0, \dots, t_0 -1$ do not belong to $G^{(j+t_0)}$
since they are in the semigroup generated by $\g_1^{(j)}$ and $\g
- t_0 \g_1^{(j)}$.


Assume that $t_0
> 1$ is  the smallest integer such that $m^{(j)} \ne m^{(j+t_0)}$.
Notice that  $t_0 \leq  g^{(j)} -1$, otherwise we would get a
follower $\g_\a$ of $\g$ with $\langle \nu_\eta, \g_\a \rangle  =
1$, which is necessarily different from the  $\g_i^{(j)}$, $i= 2,
\dots, d$, and then $m^{(j)} \ne m^{(j+ g^{(j)} - 1 )}$, a
contradiction.

There exists $0 \ne (p_2, \dots, p_d) \in \Z^{d-1}_{\geq 0}$
such that
\[
\g_i^{(j + t_0 ) } = \g_i^{(j)} - p_i \g_1^{(j)}, \ i =2, \dots,
d, \  \mbox{ and } \ \g^{(j+ t_0)}_1  = \g^{(j)}_1.
\]
We deduce the expansion
\begin{equation} \label{g-alpha}
\g_\a = \sum_{i=2}^{d} (a_i - \a_i) \g_i^{(j + t_0 ) } + (a_1 -
\a_1  + \sum_{i=2}^{d} p_i (a_i - \a_i) ) \g_1^{(j + t_0 ) }.
\end{equation}

If $a_i p_i = 0$ for $i=2, \dots, d$ we iterate this procedure
replacing $\g$ by $\g - t_0 \g_1^{(j)}$. In at most $\langle
\nu_\eta, \g \rangle -1$ steps we get to the situation where at
least one of the $a_i p_i$ is non zero, say $a_d p_d \ne 0$ for
simplicity. We prove that $\a_0 = (t_0 -1, 0, \dots, 0, 1)$
defines a follower of $\g$ such that $\g - t_0 \g_1^{(j)}$ belongs
to the semigroup generated by $\g_{\a_0} $ and $\g_i^{(j + t_0 )
}$, $i=1, \dots, d$, in particular $\g - t_0 \g^{(j)}_1$ does not
belong to $G^{(j+t_0)}$. We check this assertion by verifying that
the coefficient of the term $\g_i^{(j+t_0)}$ in the expansion
(\ref{g-alpha}) of $\g_{\a_0}$ is less than or equal to the
corresponding coefficient in the expansion (\ref{g-alpha}) of $\g
- t_0 \g_1^{(j)}$ for $i=1, \dots, d$. If $i =d$ we get a strict
inequality. The inequality is trivial for $i=2, \dots, d-1$. If
$i= 1$ we have to show the inequality:
\[
a_1 - (t_0 -1) - p_d + \sum_{i=2}^{d} p_i a_i \leq a_1 - t_0 +
\sum_{i=2}^{d} p_i a_i.
\]
This inequality is equivalent to  $p_d \geq 1$, which holds since
$a_d p_d \ne 0$.

Since there is a finite number of $\g \in G^{(j)}$ with $\langle
\nu_\eta, \g \rangle = g^{(j)}$ there exist an integer $t_2 \geq
0$ such that $g^{(j+t_2) } < g^{(j)}$ as claimed.
 \hfill $\ {\Box}$

Using this claim and induction there exists an integer $t_1 \geq
1$  such that $g^{(j+t_1)} = 1$ hence the semigroup
$\Gamma^{(j)}_{\theta^{(j-1)}}$ is regular. This ends the proof of Proposition \ref{uno}. \hfill $\ {\Box}$




We say that the nested sequence of cones $(\theta^{(j)})_j$ is
\textit{distinguished} if there exist a stable cone $0 \ne \eta$
and an integer $I $ such that for any $j \geq I$ there
exists a sequence of faces of $\theta^{(j)}$
\begin{equation} \label{faces}
\eta = \z_0^{(j)}  \leq \z_1^{(j)} \leq \cdots \leq \z_{l_0}^{(j)}
= \theta^{(j)}
\end{equation}
such that $\dim \z_i^{(j)} = \dim \eta + i$ and $\z_i^{(j)}
\supset \z_i^{(j+1)}$ for $i =0, \dots, l_0$ for $l_0 \leq \codim
\eta$.

 \begin{proposition}\label{fin}
If the sequence $ (\theta^{(j)})_j$ is distinguished then it
stabilizes.
 \end{proposition}
\textit{Proof}. Let $(\theta^{(j)})_j$ be a distinguished sequence
as above. We consider the sequence of faces $(\z_1^{(j)})_j$.
Since $\eta $ is a face of codimension one of $\z_1^{(j)}$ we get
that the sequence $(\z_1^{(j)})_j$ stabilizes by Proposition
\ref{uno}. We proceed replacing $\eta = \z_0^{(j)}$ and
$\z_1^{(j)}$ by $\z_1^{(j)}$ and $\z_2^{(j)}$ respectively, in the
previous argument, and then the result follows by induction on the
length $l_0$ of the sequence (\ref{faces}).
 \hfill $\ {\Box}$

\medskip




\begin{proposition} \label{equ}
The following  assertions are equivalent:
\begin{enumerate}
\item[(i)] The sequence of fans $(\Sigma^{(j)})_{j\geq 1}$
stabilizes.

\item[(ii)] If $0 \ne \nu \in \s \cap N$ then there exists  an
integer $l=l(\nu) \geq 1$ such that if $\nu \in \t^{(l)}\in
\Sigma^{(l)} (d)$ then ${\tau^{(l)}}$ is a stable cone.

\item[(iii)] If $0 \ne \eta \in \Sigma^{(j)}$, $j \geq I$ is a stable
cone then the sequence of fans $(\Sigma^{(j)}( \eta))_{j \geq I}$
stabilizes.
\end{enumerate}
\end{proposition}
\textit{Proof}. $(ii) \Rightarrow (i)$. Let us fix a vector $m \in
M$ in the interior of the cone $\check{\s}$ and the convex
polytope $\Theta:= \{ \nu \in \s \mid \langle \nu, m \rangle = 1
\}$ which is rational for the lattice $N \cap m^\perp$. If $0 \ne
\nu \in \s$ the ray $\nu \R_{\geq 0}$ intersects the polytope
$\Theta$ exactly at one point $\xi(\nu)$.

For  $0 \ne \nu \in \s \cap N$ we denote by $l(\nu)$ the integer
verifying the hypothesis. The set
\[ U_\nu := \int ( \bigcup_{\nu \in \t, \t \in \Sigma^{ (l(\nu) ) }}  \t )
\cap \Theta \] is an open subset of $\Theta$ which contains $\xi
(\nu)$. It follows that $\{ U_\nu \mid 0 \ne \nu \in \s \cap N \}$
is an open covering of the compact set $\Theta$; there exists a
finite subcovering $\{ U_{\nu_1}, \dots, U_{\nu_s} \}$.

If $l = \max_{i=1, \dots, s} \{ l(\nu_i) \}$ then any cone $\t
\in \Sigma^{(l)} (d)$ is stable hence  the sequence $(\Sigma^{(j)})$ stabilizes.


 $(i) \Rightarrow (iii)$. It is trivial from the definitions.

 $(iii) \Rightarrow (ii)$. If $0 \ne \nu \in \s \cap N$ and if
 $\theta_j \in \Sigma^{(j)}$ is such that $\nu \in \int
 \theta^{(j)}$ then the sequence $(\theta^{(j)})$
 stabilizes by Proposition \ref{prev}. We
 denote by $\vartheta $
  the cone $ \theta^{(j)},\  j\gg 1$. If the codimension $q_0$ of $\vartheta $
  is
 zero then we have finished.


Suppose that $q_0 > 0$.
Let $\t^{(j)}  \in \Sigma^{(j)} (d) $ be a cone
containg $\vartheta $
 as a face.  If    $j \gg 0 $ the cone
$\t^{(j)} (\vartheta)$ is stable, since the sequence
$\Sigma^{(j)} (\vartheta)$ stabilizes by hypothesis.
Then there is a unique cone
$\t^{(j+1)} \in  \Sigma^{(j+1)}$ with
\[
 \vartheta  \leq \t^{(j+1)}, \    \t^{(j)} \supset \t^{(j+1)} \mbox{ and }    \t^{(j)} (\vartheta ) =
\t^{(j+1)} ( \vartheta )  .
\]
For  $j \gg 0 $ we fix  a  chain of faces of  $ \t^{(j)} (
\vartheta )$
\[
0 = \z_0 (\vartheta)  \leq \z_1 (\vartheta)  \leq \dots \leq \z_{q_0} (\vartheta)  = \t^{(j)} (
\vartheta )
\]
with $\dim \z_i (\vartheta) = i$ we get  a chain of faces
\[
\eta = \z_0 ^{(j)} \leq \z_1^{(j)}  \leq \dots \leq \z_{q_0}^{(j)}
=  \t^{(j)}
\]
such that  $\z_i^{(j)} \supset
\z_i^{(j+1)}$,  $\z_i^{(j)} (\vartheta ) = \z_i (\vartheta)$, and  the cone $\z_i^{(j)}$ is  a face of codimension
one of $\z_{i+1}^{(j)}$. That is, the sequence $(\t^{(j)})$
 is distinguished.
By Proposition \ref{fin} the sequence $\t^{(j)}$ stabilizes. Since
there are finitely many $q_0$ dimensional faces in the fan
$\Sigma^{(j)} (\vartheta )$
  it follows that for $j \gg 0$
the cone $\vartheta $,
and hence the vector $\nu$, is contained in the
interior of the union of stable cones  $\bigcup_{\tau ^{(j)} (\vartheta )
\
\, \in \, \Sigma^{(j)} (\vartheta ) }  \t^{(j)}$.
 \hfill $\ {\Box}$

\begin{remark} It is interesting to verify that propositions \ref{equ}, \ref{uno} and \ref{prev} imply resolution by logarithmic jacobian blowing-ups in the two-dimensional case.
\end{remark}


\begin{proposition} \label{equ-eta}
Let $\eta \in \Sigma^{(j)}$, $j \geq I$, be a stable cone of
codimension $q_0 >0$. The following assertions are equivalent.
\begin{enumerate}
\item[(i)] The sequence of fans $(\Sigma^{(j)} (\eta) )_{j\geq I}$
stabilizes.

\item[(ii)] If $0 \ne w \in | \Sigma^{(j)} (\eta) | \cap N (\eta)$
then there exists an integer $l=l(w) \geq I$ such that if $w \in
\t^{(l)} (\eta) $, $ \t^{(l)} (\eta) \in \Sigma^{(l)} (\eta)$ then
$\t^{(l)}$ is stable.
\end{enumerate}
\end{proposition}
\textit{Proof}.  The proof  of   $(ii) \Rightarrow (i)$is
analogous to the implication $(ii) \Rightarrow (i)$ in Proposition
\ref{eq}. The implication $(i) \Rightarrow (ii)$ is trivial.
\hfill $\ {\Box}$

\medskip



\begin{remark} \label{inclu}  If $0 \ne \eta$ is a stable cone, for $j \gg 0$,
the blowing up of the logarithmic
jacobian ideal induces a proper toric modification $\xi_\eta^{(j)}
: T^{\Gamma^{(j+1)}}_{\Sigma^{(j+1)}} (\eta) \to
T^{\Gamma^{(j)}}_{\Sigma^{(j)}} (\eta)$.
 As we explain below, the map $\xi_\eta^{(j)}$  is not
necessarily equal to the
 blowing up of the logarithmic jacobian ideal of
 $    T^{\Gamma^{(j)}}_{\Sigma^{(j)}}  (\eta)$.
\end{remark}

\begin{proposition}     \label{orbitq}
Given a stable cone $\eta$ of codimension $d_0$, a vector $0 \ne w
\in N(\eta)$ and a sequence of cones $\theta^{(j)} (\eta) \in
\Sigma^{(j)} (\eta)$ such that $\theta^{(j)} (\eta) \supset
\theta^{(j+1)} (\eta)$ for $j \gg 0$.  Then, if $w \in \int (
\theta^{(j)} (\eta) )$ for $j \gg 0$, we have that
\begin{equation} \label{ret2}
M(\theta^{(j-1)} (\eta) , \Gamma_{\theta^{(j-1)}} ^{(j)}  \cap
\eta^\perp ) = M (\theta^{(j-1)} (\eta) ) \mbox{ for } j \gg 0,
\end{equation}
and the sequence $\theta^{(j)} (\eta)$ stabilizes.
\end{proposition}
\textit{Proof}. We use notations \ref{clo-st}. We denote by $q_0$
the codimension of $\theta^{(j)} (\eta)$ for $j\gg 0$. If
$\theta^{(j-1)} \in \Sigma^{(j-1)}$ is a cone such that  $\eta
\leq  \theta^{(j-1)} $ and its image in $N(\eta)$ is equal to
$\theta^{(j-1)} (\eta)$ then by definition we get
$M(\theta^{(j-1)} (\eta) , \Gamma_{\theta^{(j-1)}} ^{(j)} \cap
\eta^\perp ) = M(\theta^{(j-1)} , \Gamma_{\theta^{(j-1)}} ^{(j)}
\cap \theta^{{(j-1)}\perp} ) \subset M( \eta, \Gamma_{\eta} ^{(j)})
$ and $ M (\theta^{(j-1)} (\eta) ) = M (\theta^{(j-1)}) \subset
 M (\eta)$.

We obtain that $M(\theta^{(j-1)} (\eta) , \Gamma_{\theta^{(j-1)}}
^{(j)} \cap \eta^\perp ) = M(\theta^{(j)} (\eta) ,
\Gamma_{\theta^{(j)}} ^{(j+1)}  \cap \eta^\perp )$ and  $ M
(\theta^{(j-1)} (\eta) )  = M (\theta^{(j)} (\eta) )$ for $j \gg
0$. Then, the proof of formula (\ref{ret2}) follows similarly as
that of (\ref{ret}) and of Proposition \ref{prev} ii.

We deduce from (\ref{ret2}) and Lemma \ref{basis} that it is
enough to prove the result in the case $\codim \theta^{(j)} (\eta)
= 0$. In this case, the cone $\theta^{(j-1)} (\eta)$ is the image
of a cone $\theta^{(j-1)} \in \Sigma^{(j)} (d)$ by the projection
$\p \colon N_\R \to N(\eta)_\R$.


The charts of the blowing up of the logarithmic jacobian ideal of
$T^{\Gamma^{(j)}}_{\theta^{(j-1)}}$ which are defined by those $d$
dimensional cones which contain $\eta$, are in bijection with the
vertices  of the
 face
$\mathcal{F}^{(j)}_{\theta^{(j-1)}}$  of the Newton polyhedron of
the logarithmic jacobian ideal $\J^{(j)}_{\theta^{(j-1)}}$
determined by the vector $\nu_\eta$ (see Lemma \ref{basis}).

One of these charts  $T^{\Gamma^{(j+1)}}_{\theta^{(j)}}$ ,
corresponding to a vertex  $m^{(j)}$ of
$\mathcal{F}^{(j)}_{\theta^{(j-1)}}$ is such that the image of the
cone $\theta^{(j)}$ in $N(\eta)$ is equal to the cone $\theta^{(j)}
(\eta)$ in the sequence. By hypothesis $w \in N(\eta)$ is in the
interior of $\theta^{(j)} (\eta)$, hence the face of
$\mathcal{F}^{(j)}_{\theta^{(j-1)}}$ supported by the vector $w$
is equal to the vertex $m^{(j)}$.

 By Lemma \ref{basis} we
get that $ m^{(j)} = \bar{m}^{(j)} +   m'^{(j)}$ where $m^{(j)}$
is a vertex of the Newton polyhedron of the  logarithmic jacobian
ideal $\J^{(j)}_{\theta^{(j-1)}} (\eta)$ of $k
[t^{\Gamma^{(j)}_{\theta^{(j-1)}}}]$ and $m'^{(j)} $ is a vertex
of $ \mathcal{P}^{(j)}_{\theta^{(j-1)}}$. We choose
representations
\begin{equation}  \label{repre2}
\bar{m}^{(j)} = \g_1^{(j)} + \cdots + \g_{d_0}^{(j)} \mbox{ and }
m'^{(j)} =  \g^{(j)}_{d_0 +1} + \cdots + \g^{(j)}_{d},
\end{equation}
as sum of linearly independent elements of
$\Gamma^{(j)}_{\theta^{(j-1)}}$.



Remark that the chart associated to $\bar{m}^{(j)}$ of the blowing
up of the logarithmic jacobian ideal $\J^{(j)}_{\theta^{(j-1)}}
(\eta)$ of $k [t^{\Gamma^{(j)}_{\theta^{(j-1)}}}]$ is defined by
the semigroup $S^{(j+1)}_{j, w}$ generated by
$\Gamma^{(j)}_{\theta^{(j-1)}} \cap \eta^\perp$ and vectors of the
form $\g - \bar{m}^{(j)}$ for $\g$ in $\J_{\theta^{(j-1)}}^{(j)}
(\eta)$. The inclusion
\begin{equation} \label{inclusion2}
  S^{(j+1)}_{j, w} \subset \Gamma^{(j+1)}_{\theta^{(j)}} \cap
\eta^\perp,
\end{equation}
may be strict (cf. Remark \ref{inclu}). The reason is the
following. If $\g \in \Gamma^{(j)}_{\theta^{(j-1)}}$ and if $\g \y
\g_1^{(j)} \y \cdots \y \g_{i-1} ^{(j)} \y \g_{i+1}^{(j)} \y
\cdots \y \g_{d}^{(j)} \ne 0$ then $\d := \g + m^{(j)} -
\g_i^{(j)}$ belongs to $\J_{\theta^{(j-1)}}^{(j)}$ and  $\d -
m^{(j)} = \g - \g_i^{(j)} \in \Gamma^{(j+1)}_{\theta^{(j)}}$. If
in addition, $\langle \nu_\eta, \g \rangle = \langle \nu_\eta,
\g_i^{(j)} \rangle$ we obtain that $\d -  m^{(j)} $ belongs to $
\Gamma^{(j+1)}_{\theta^{(j)}} \cap \eta^\perp$, even if $\g \notin
\eta^\perp$.








Notice that the sequence $(\langle w, \bar{m}^{(j)} \rangle )$
becomes stationary for $j \gg 0$, since $w$ belongs to the
interior of $\theta^{(j-1)} (\eta)$.

 If the sequence
$(\theta^{(j)} (\eta))$ does not stabilize then by Proposition \ref{stable} the toric variety
$T^{\Gamma_{\theta^{(j-1)}}^{(j)}}$ is not smooth for any integer
$j\geq 1$. Let us fix some $j_1 \gg 0$. By Proposition
\ref{bound}, if the toric variety
$T^{\Gamma_{\theta^{(j_1-1)}}^{(j_1)}}$ is not smooth then there
is a smallest $j_2 > j_1$ such that $m^{(j_1)} \ne m^{(j_2)}$. By
(\ref{essential}) we get that ${m}  ^{(j_1)} - {m}^ {(j_2)} \in
\Gamma^{(j_2+1)}_{\theta^{(j_2)}} $, which can be written
\[
   {m}  ^{(j_1)} - {m}^ {(j_2)}  = ( \bar{m}  ^{(j_1)} - \bar{m}^ {(j_2)} )  + ( {m'}  ^{(j_1)} - {m'}^ {(j_2)}),
\]
belongs to the face  $ \Gamma^{(j_2+1)}_{\theta^{(j_2)}} \cap \eta^\perp$ of  $\Gamma^{(j_2+1)}_{\theta^{(j_2)}} $.
It follows that both terms     $\bar{m}  ^{(j_1)} - \bar{m}^ {(j_2)}$ and      ${m'}  ^{(j_1)} - {m'}^ {(j_2)}$
belong to         the face  $ \Gamma^{(j_2+1)}_{\theta^{(j_2)}} \cap \eta^\perp$, and at least one of them is non zero.

If   $\bar{m}  ^{(j_1)} - \bar{m}^ {(j_2)} \ne 0 $ then we get
that $\langle w , \bar{m}  ^{(j_1)} - \bar{m}^ {(j_2)} \rangle  =
0$. This implies that $\bar{m}  ^{(j_1)} - \bar{m}^ {(j_2)} \in
(\Gamma^{(j_2+1)}_{\theta^{(j_2)}} \cap \eta^\perp) \cap w^\perp$,
but since $w  \in \int (\theta^{(j_2)} (\eta))$ the face
$\Gamma^{(j_2+1)}_{\theta^{(j_2)}} \cap \eta^\perp \cap w^\perp$
is reduced  to zero, a contradiction.

We deduce that $\bar{m}^{(j)} = \bar{m}^{(j_1)}$ for all $j \geq
j_1$. We distinguish two cases:

If $\Gamma^{(j)}_{\theta^{(j- 1)}} \cap \eta^\perp$ is a regular
semigroup for all  $j \geq j_1$ then it follows that $\g_i^{(j)}$,
$i =1, \dots, d_0$ is a basis of $\Gamma^{(j)}_{\theta^{(j- 1)}}
\cap \eta^\perp$. Since $\bar{m}^{(j)} = \bar{m}^{(j+1)}$ and
$\Gamma^{(j)}_{\theta^{(j- 1)}} \cap \eta^\perp \subset
\Gamma^{(j+1)}_{\theta^{(j)}} \cap \eta^\perp$ the equality
$\Gamma^{(j)}_{\theta^{(j- 1)}} \cap \eta^\perp =
\Gamma^{(j+1)}_{\theta^{(j)}} \cap \eta^\perp$ holds, for all $j
\geq j_1$. That means that  the sequence $\theta^{(j)} (\eta)$
stabilizes, contradicting the assumption.



If $\Gamma^{(j)}_{\theta^{(j- 1)}} \cap \eta^\perp$ is not regular
for some $j_0 \geq j_1$ then we define  inductively a sequence of
semigroups $S_{j_0, w}^{(j)}$, $j > j_0$ such that
$
 S_{j_0, w}^{(j)} \subset \Gamma^{(j)}_{\theta^{(j- 1)}} \cap
 \eta^\perp
$
and $\bar{m}^{(j_1)}$ is the vertex supported by $w$ of the Newton
polyhedron of the logarithmic jacobian ideal of $k[t^{S_{j_0, w}^{(j)}}]$. For $j =j_0 +1 $ the assertion holds for $ S_{j_0, w}^{(j_0+1)}$
 Assume then by induction the assertion for $j-1$
and define $S_{j_0, w}^{(j)}$ the semigroup generated by $S_{j_0,
w}^{(j-1)}$ and $\g -\bar{m}^{(j_1)}$, with $\g$ running through
the set defining the logarithmic jacobian ideal of $k[ t^{S_{j_0,
w}^{(j-1)}}]$. We deduce similarly as (\ref{inclusion2}) that the
inclusion   $
 S_{j_0, w}^{(j)} \subset \Gamma^{(j)}_{\theta^{(j- 1)}} \cap
 \eta^\perp
$  holds.  Since the vector $w$ supports the vertex
$\bar{m}^{(j_1)}$ of the polyhedron $\mathcal{N}
(\J^{(j)}_{\theta^{(j-1)}} (\eta))$ it follows that the face
supported by $w$ of the Newton polyhedron of the logarithmic
jacobian ideal of $k[ t^{S_{j_0, w}^{(j)}}]$ is a vertex equal to
$\bar{m}^{(j_1)}$ and this holds by induction  for all $j > j_1$.
This contradiction with Corollary \ref{bound2} ends the
proof of Proposition \ref{orbitq}.
  \hfill $\ {\Box}$















\begin{proposition} \label{eta}
Let $\eta \in \Sigma^{(j)}$, $j \geq I$, be a stable cone. Then
there exists an integer $l (\eta) \geq I$ such that the sequence
of fans $(\Sigma^{(j)} (\eta))_{j \geq I}$ is constant for $j \geq
l(\eta)$.
\end{proposition}
\textit{Proof}. We prove the it by induction on the codimension
$q_0$ of $\eta$. There is nothing to prove if $q_0 =0$. If $q_0 =
1$ the fan $\Sigma^{(j)} (\eta)$ has at most two cones which are
one dimensional and then the assertion follows by Claim
\ref{oneorbit}.

We suppose the assertion true for stable cones of codimension less
than  $q_0 \leq   d-1 $ and we prove it for an stable cone $\eta$
of codimension $q_0$.

We show  that if $0 \ne w \in N(\eta)$ and if $\t^{(j)} (\eta) \in
\Sigma^{(j)} (\eta)$ is a cone of dimension $q_0$ such that $ w
\in \t^{(j)} (\eta)$  and $\t^{(j)}(\eta) \supset \t^{(j+1)}(\eta)$ for all $j
\gg I$, then the sequence of cones $(\t^{(j)} (\eta) )$
stabilizes. Then we prove that $\Sigma^{(j)} (\eta)$ stabilizes.

Let $\theta^{(j)} (\eta) $ be the face of $\t^{(j)} (\eta)$ such
that $w \in \int ( \theta ^{(j)} (\eta)) $. By Proposition
\ref{orbitq} the sequence $\theta^{(j)} (\eta)$ stabilizes for $j
\gg 0$. We denote by $\vartheta (\eta)$ the cone $\theta^{(j)}(\eta)$ for $j\gg 0$. We distinguish
two cases.

If $\dim \vartheta (\eta) = q_0$ then we have finished.

If $ 1 \leq \dim \vartheta (\eta) < q_0$ then we consider a sequence $\t^{(j)}
\in \Sigma^{(j)}$ such that $\eta \leq \t^{(j)} \in \Sigma^{(j)}$,
$\t^{(j)} \subset \t^{(j+1)}$ and the image of $\t^{(j)}$ in $N(\eta)$ is $\t^{(j)}(\eta)$. We denote also by $\theta^{(j)}$
the unique face of $\t^{(j)}$ such that $\theta^{(j)} (\eta) =
\vartheta (\eta)$ for $j \gg 0$.  By the same construction as in the
proof of implication $(iii) \Rightarrow (ii)$ in Proposition \ref{equ} we
get that the sequence $\theta^{(j)}$ is distinguished for $j \gg
0$ hence it stabilizes by Proposition \ref{fin}. We denote by
$\vartheta$ the the cone $\theta^{(j)}$ for $j\gg 0$. By the induction hypothesis we get that the
sequence $(\Sigma^{(j)} (\vartheta))_j$ stabilizes for $j \gg 0$,
since $\vartheta$ is of codimension less than $q_0$.\par Notice that
for $j \gg 0$ we have that $\Sigma^{(j)} (\vartheta) =
\Sigma^{(j)} (\eta) (\vartheta (\eta))$; this implies that the set $\{ \t^{(j)}(\eta) \in \Sigma^{(j)} (\eta)
\mid \vartheta (\eta) \leq \t^{(j)} (\eta) \}$ is stabilized for
$j \gg 0$.

This shows that assertion $(ii)$ in Proposition \ref{equ-eta}
holds hence also assertion $(i)$ as required.
 \hfill $\ {\Box}$

\begin{theorem} \label{main}
Any toric variety has a canonical resolution of singularities by
iteration of blowing ups of the sheaf of logarithmic jacobian
ideals.
\end{theorem}
\textit{Proof}. It is enough to prove it in the affine case. We
can assume in addition (see Remark \ref{idproduct}) that the
affine toric variety has no torus factors, i.e., it is defined by
a semigroup ${\Gamma}$ such that the cone $\check{\s} = \R_{\geq
0} \Gamma$ is \textit{strictly convex} and of dimension  $d = \dim
M_\R$.


By Propositions \ref{equ} and \ref{eta} we get that there exists
an integer $l \geq 1$ such that  the sequence of fans
$(\Sigma^{(j)})$ stabilizes for $j \geq I$. Then applying
Proposition \ref{stable} to the finitely many cones of
$\Sigma^{(I)}$ we deduce that there exists an integer
 $l' \geq I$
such that the iterated blow up of logarithmic jacobian ideals
$T_{\Sigma^{(l')}}^{ \Gamma^{(l')}}$ is smooth.
 \hfill $\ {\Box}$

\begin{remark} The recent paper \cite{ALPPT} suggests that it would be interesting to develop
 an approach from a computational viewpoint to the iteration of Semple-Nash modification.
\end{remark}\par\noindent
 {\bf Acknowledgements.} We are grateful to Monique Lejeune-Jalabert for
introducing us to the reference \cite{S}, to Ezra Miller for bringing to our attention the work of H.M. Thompson,
%CORREC
to Michael Thaddeus for his perspicacious and useful remarks on a previous version of this work,
and to the  \textit{Institut
Math\'ematique de Jussieu} and  the \textit{Dpto.~\'Algebra,
Universidad Complutense de Madrid} for their hospitality.


\bibliographystyle{amsplain}


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\end{document}
