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\begin{document}

% Title

\title[Fans]{Combinatorial Fans,
lattice-ordered groups, and their neighbours: a short excursion}

% Authors

\author[Vincenzo Marra]{Vincenzo Marra}
\address[V. Marra]{Department of Computer Science\\ University of Milan\\
Italy}
\email{marra@dsi.unimi.it}
%\thanks{Support First Author}

\author[Daniele Mundici]{Daniele Mundici}
\address[D. Mundici]{Department of Computer Science\\ University of
Milan\\Italy}
\email{mundici@mailserver.unimi.it}
%\thanks{Support Second Author}

% Abstract, Keywords, AMS Classification

\keywords{Fans, polyhedral cones, convex geometry,
lattice-ordered groups, toric varieties.}
\date{}
\begin{abstract}
Just like the celebrated dictionary between integral convex
geometry and toric varieties bridged until then remote
mathematical worlds, the less well-known duality between (certain)
lattice-ordered groups and rational polyhedral sets brought
combinatorial techniques to bear on the investigation of these
abstract algebraic structures. We offer a brief excursion into
these relatively new territories, with special emphasis on the
novel and admittedly unclear r\^{o}le played by order-theoretical
considerations.
\end{abstract}
\maketitle
%%%%%%%%%%%%%% INTRODUCTION %%%%%%%%%%%%%%

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\font\rms=cmr8 
\font\its=cmti8 
\font\bfs=cmbx8
\markright{\its S\'eminaire Lotharingien de
Combinatoire \bfs 47 \rms (2002), Article~B47f\hfill}
\def\thepage{}
%

\section{Introduction}
\label{S: Intro}

In an inaugural address (\cite{rota}) delivered in Florence,
Italy, in 1993, G.-C. Rota  maintained that

\begin{quotation}
[while] one of the leading trends that is visible in present-day
mathematics [is] `the return to concreteness', [...] the
relationship between combinatorics and algebra today [...] is not
a forgetting of the past. [...] some outstanding work in
combinatorics that is going on today is greatly benefiting the
algebra of yesterday.
\end{quotation}

\noindent
As a prominent example, Rota mentioned

\begin{quotation}
the discovery of toric varieties, [...] a dictionary whereby
results of algebraic geometry can be translated into results of
convex integral geometry in the style of Minkowski and Hadwiger.
[...] The theorems of algebraic geometry now find new and
unexpected life in the newly added class of toric varieties,
coming from an altogether different source, a combinatorial
source to be sure.
\end{quotation}

The main theme of this paper is a similar interaction between
another realm of abstract algebra, the theory of lattice-ordered
groups, and a version of ``convex integral geometry in the style
of Minkowski and Hadwiger'' (\cite{rota}), the eminently
combinatorial theory of rational polyhedral sets. {\it Prima
facie}, ordered groups are no less far removed from polyhedra
than  algebraic geometry
--- and yet, much as in the case of toric varieties, the
connection is so intimate as to be a genuine duality.

We begin with a survey of basic results on complexes of polyhedral
cones and their subdivisions. We then introduce lattice-ordered
abelian groups, and sketch enough of their theory to establish the
relationship with polyhedral complexes. Along the way, as a sort
of counterpoint to our  {\it leitmotiv}, we indicate the
algebro-geometric significance of some of the material
introduced. It is only in the last section that we bring this
second theme into the foreground  for a necessarily short
excursion into the world of toric varieties.

%%%%%%%%%%%%% FINIS INTRODUCTION %%%%%%%%%%


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\markboth{\SMALL VINCENZO MARRA AND DANIELE MUNDICI}{\SMALL 
COMBINATORIAL FANS AND LATTICE-ORDERED GROUPS}
%
%

%%%%%%%%%%%%%%%%% FANS %%%%%%%%%%%%%%%

\section{Combinatorial Convexity: Fans}
\label{S: Fans}

We introduce a linear version of polyhedral complexes known as
fans. The terminology is due to M. Demazure, who called these
objects \textit{eventails}. It is quite a descriptive name,
provided it is interpreted as `possibly torn fans' (\cite{oda}). A
comprehensive treatment may be found in \cite{ewa}.

Throughout this paper, we fix a free abelian group $N$ of rank
$n$. Upon tensoring $N$ with $\mathbb{R}$ we obtain a real
$n$-dimensional vector space $N_{\mathbb{R}} = \mathbb{R} \otimes
N$. We let $M = \Hom_\mathbb{Z}(N,\mathbb{Z})$ denote the
$\mathbb{Z}$-module dual to $N$, and $M_{\mathbb{R}} = \mathbb{R}
\otimes M$ the vector space dual to $N_{\mathbb{R}}$.

In down-to-earth terms, $N \cong \mathbb{Z}^n$, $N_{\mathbb{R}}
\cong \mathbb{R}^n$ and $N$ is the group of integral $n$-tuples
in  $\mathbb{R}^n$. Similarly, $M$ can be identified with
$\mathbb{Z}^n$ and  $M_{\mathbb{R}}$ with $\mathbb{R}^n$. These
identifications, however, only make sense if we choose a basis of
$N$ (and therefore of $M$,  $N_{\mathbb{R}}$, and
$M_{\mathbb{R}}$).\footnote{It turns out that this is equivalent
to a choice of coordinates on an algebraic variety --- see Section
\ref{S: TV}.} Whenever convenient, we shall tacitly assume we have
fixed one such basis.

Let $\{\vec{v}_1, \ldots, \vec{v}_m \} \subseteq N$ be a finite
set of integral vectors. The \emph{rational polyhedral cone
$\sigma$}, or simply the \emph{cone $\sigma$} generated by
$\{\vec{v}_1, \ldots, \vec{v}_m\}$ is their positive linear hull
over $\mathbb{R}$, i.e.,
$$
\sigma = \langle \vec{v}_1, \ldots, \vec{v}_m \rangle =
\{p_1\vec{v}_1 +\cdots + p_m\vec{v}_m \mid 0 \leq p_1, \ldots,
p_m \in \mathbb{R} \}.
$$

A cone $\sigma$ is \emph{$k$-dimensional} if{f} the
$\mathbb{R}$-linear space it spans is $k$-dimensional. A
\emph{face} of $\sigma$ is any nonempty convex subset $\tau
\subseteq \sigma$ such that every line segment in $\sigma$ which
has an interior point in $\tau$ lies entirely in $\tau$. Lest we
burden this paper with an inordinate amount of detail, we shall
assume a number of intuitive and easily proved statements about
cones --- e.g., the faces of a cone are finite in number, the
set of faces is a (finite) lattice under inclusion (whence every
face is contained in some maximal proper face, a \emph{facet}),
and so on.

Upon intersecting a finite number of integral half-spaces (i.e.,
sets of the form $\{(x_1, \ldots , x_n) \in \mathbb{R}^n \mid
z_1x_1 + \cdots + z_nx_n \geq 0\}$, $z_i \in \mathbb{Z}$) one
clearly obtains a cone in $N_\mathbb{R}$. That this is no
accident is a basic result in convex geometry.

\medskip
\noindent{\bf The Fundamental Theorem of Polyhedra.} {\it A subset
$\sigma \subseteq N_\mathbb{R}$ is a cone if{f} it is the
intersection of finitely many integral half spaces of
$N_\mathbb{R}$.}
\medskip

A  cone $\sigma = \langle \vec{v}_1, \ldots, \vec{v}_m \rangle$
is \emph{simplicial} if{f} $\{\vec{v}_1, \ldots, \vec{v}_m \}$ is
linearly independent over $\mathbb{R}$. A nonzero vector $\vec{v}
\in N$ is \emph{primitive} if{f} its coordinates (with respect to
the canonical basis) are relatively prime. The \emph{vertices} of
a simplicial cone $\sigma$ are the (uniquely determined)
primitive vectors $\vec{v}_i \in N$ such that $\sigma = \langle
\vec{v}_1, \ldots, \vec{v}_m \rangle$.

A \emph{rational polyhedral fan} (in $N_\mathbb{R}$), in short, a
\emph{fan}, is a finite set $\Sigma$ of polyhedral cones such that

\begin{enumerate}
\item Every face of every cone of $\Sigma$ belongs to $\Sigma$.
\item Any two cones of $\Sigma$ intersect in a common face.
\end{enumerate}
The \emph{support} of $\Sigma$, denoted $|\Sigma|$, is the union
of all its cones. A fan $\Sigma$ is \emph{simplicial} if{f} all
its cones are simplicial.

A polyhedral cone $\sigma$ carries a natural fan structure
$\Sigma_\sigma$ which is given by the collection of all its faces
(including $\emptyset$ and $\sigma$ itself). With this notation,
$|\Sigma_\sigma| = \sigma$. We shall not be pedantic about the
distinction between $\Sigma_\sigma$ and $\sigma$.

\medskip
\noindent{\sc Subdivisions. \,\,} Subdivisions of rational
polyhedral cones, besides being of interest in themselves, encode
the birational geometry of toric varieties. Thus, we discuss
various notions of refinement for fans (see Figure~1). Given two
fans $\Sigma$ and $\Delta$ in $N_\mathbb{R}$, $\Delta$ is a
\emph{subdivision} of $\Sigma$, or it \emph{refines} $\Sigma$,
if{f} $|\Sigma| = |\Delta|$ and every cone of $\Delta$ is
contained in some cone of $\Sigma$. We write $\Delta \leq \Sigma$
to denote the fact that $\Delta$ refines $\Sigma$. In this case,
every cone of $\Sigma$ is a union of cones of $\Delta$. More
generally, following Danilov (\cite{dansur}) we say that $\Sigma$
is \emph{inscribed} in $\Delta$ if{f}  every cone of $\Sigma$ is
a union of cones of $\Delta$, whence we have the (possibly
strict) inclusion $|\Sigma| \subseteq |\Delta|$.

Just like a polyhedron can be triangulated without adding new
vertices (say, by generalising the elementary plane construction
of `adding diagonals'), a fan can always be subdivided into a
simplicial fan without adding $1$-dimensional cones. The
\emph{$m$-skeleton} $\Sigma^{(m)}$ of a fan $\Sigma$ is the set
of $m$-dimensional cones of $\Sigma$. (It is not itself a fan,
for it does not contain, together with a cone, all of its faces).
Then there exists a simplicial fan $\Delta_1$ with the same
$1$-skeleton of $\Sigma$ such that $\Delta_1 \leq \Sigma$.

In many contexts, simplicial subdivisions are not fine enough ---
it turns out that a certain proper subclass of simplicial cones
plays a key r\^{o}le in the theory of toric varieties. A cone
$\sigma = \langle \vec{v}_1, \ldots, \vec{v}_m \rangle \subseteq
N_\mathbb{R}$ is \emph{unimodular} (alternative terminology,
\emph{regular}) if{f} $\{\vec{v}_1, \ldots, \vec{v}_m \}$ can be
completed to a basis of $N$. Hence, in particular, $\sigma$ is
simplicial and the set of its vertices is $\{\vec{v}_1, \ldots,
\vec{v}_m \}$. A fan $\Sigma$ is unimodular if{f} all its cones
are unimodular.

Clearly, not every simplicial cone is unimodular. However, it is
always possible to subdivide a given  cone into unimodular
subcones. Further, this can always be done through \emph{stellar
subdivisions}. Let $\Sigma$ be a fan in $N_\mathbb{R}$, and let
$\sigma \in \Sigma$ be an $m$-dimensional cone. The \emph{star}
in $\Sigma$ of a face $F \subseteq \sigma$, denoted $\st F$, is
the set of cones in $\Sigma$ having $F$ as a face; the
\emph{closed star} of $F$ is the fan $\cstar F = \{\tau \mid \tau
\mbox{ is a face of a cone } \theta \in \st F \}$. The
\emph{relative interior} of $\sigma$, denoted $\rel \sigma$, is
the topological interior of $\sigma \cap V$, where $V$ is the
linear subspace of $N_\mathbb{R}$ spanned by $\sigma$. Let
$\vec{w} \in \rel \sigma \cap N$. Then there is a unique minimal
face $F$ of $\sigma$ such that $\vec{w} \in \rel F$, namely, the
intersection of all faces containing $\vec{w}$. The \emph{stellar
subdivision}\footnote{Stellar subdivisions were introduced by
James W. Alexander in \cite{ale30} for abstract simplicial
complexes, and in \cite{ale26} for geometric complexes. Their
importance in algebraic topology is well-known
--- see e.g. \cite{alexandrov}.}
of $\Sigma$ along $\sigma$ through $\vec{w}$ is the fan obtained
from $\Sigma$ removing $\st F$ and joining $\vec{w}$ to the
boundary $(\cstar F - \st F)$ of the star of $F$. If $\Delta$ is
obtained from $\Sigma$ via a finite number (may be zero) of
stellar subdivisions, we write $\Sigma \leq^* \Delta$. Clearly,
$\Sigma \leq^* \Delta$ implies $\Sigma \leq \Delta$, but not
conversely.

\medskip
\noindent{\bf Unimodular Refinements.} {\it Let $\Sigma$ be a fan.
There exists a simplicial fan $\Delta_1$ with the same
$1$-skeleton of $\Sigma$ such that $\Delta_1 \leq \Sigma$.
Further, there exists a unimodular fan $\Delta_2$ such that
$\Delta_2 \leq^* \Delta_1 \leq \Sigma$, that is, $\Delta_2$ is
obtained from $\Delta_1$ by a finite number of stellar
subdivision.}

\begin{remark} Unimodularity is the combinatorial counterpart of
the central algebro-geometric notion of \emph{smoothness}
(\cite{oda}, \cite{ful}, \cite{ewa}).
\end{remark}

For self-contained proofs of the existence of simplicial and
unimodular refinements, see e.g. \cite{ewa}.

A special sort of stellar subdivision is particularly important
for the contents of this paper. Let $\sigma = \langle
\vec{v}_1,\ldots,\vec{v}_m \rangle$ be an $m$-dimensional cone of
a fan $\Sigma$ in $N_\mathbb{R}$. Suppose $\tau = \langle
\vec{v}_{i_1}, \ldots\vec{v}_{i_d} \rangle$ is a $d$-dimensional
face of $\sigma$. We call $\vec{w} = \sum \vec{v}_{i_j}$ the
\emph{barycenter} of $\tau$. The fan obtained from $\Sigma$ by
stellar subdivision along $\tau$ through $\vec{w}$ is a
\emph{barycentric stellar subdivision of $\Sigma$}. We write
$\Delta \preceq \Sigma$ to denote the fact that $\Delta$ is
obtained from $\Sigma$ by a finite number (maybe zero) of stellar
barycentric subdivisions; if all barycenters belong to
$2$-dimensional cones, we write $\Delta \preceq_2 \Sigma$ and
speak of \emph{binary starring}. It is easy to show but important
to realize  that if $\Delta \preceq \Sigma$ and $\Sigma$ is
unimodular, then $\Delta$ is unimodular
--- \emph{barycentric subdivisions preserve unimodularity}. This
fails for general stellar subdivisions.\footnote{The barycenter
of a rational simplex  is known to geometers of numbers as the
\emph{(multidimensional) Farey mediant} of its vertices, a
classical notion intimately connected with (multidimensional)
continued fraction expansions. See \cite{CDM}, \cite{M11-ADV} and
references therein for further information.}

\unitlength 1mm \linethickness{0.4pt}
\begin{figure}[tb]\label{F: Sub}
\begin{picture}(50,50)(50,60)

%%%%%%%%%%%%% A cone C %%%%%%%%%%%%%%%%%
\put(18.67,65.00){(a)}
\put(18.67,72.00){\vector(1,2){9.50}}
\put(18.67,72.00){\vector(2,1){13.67}}
\put(19.00,72.33){\vector(1,1){22.33}}
\put(19.00,72.33){\vector(3,1){29.00}}
\put(28.00,90.67){\line(1,-3){4.11}}
\put(32.11,78.33){\line(4,1){15.89}}
\put(48.00,82.31){\line(-1,2){6.18}}
\put(41.82,94.67){\line(-4,-1){14.15}}

%%%%%%%%%%% A subdivision (triangulation) of C %%%%%%%%%%%%
\put(48.67,64.00){(b)} \put(48.67,72.00){\vector(1,2){9.50}}
\put(48.67,72.00){\vector(2,1){13.67}}
\put(49.00,72.33){\vector(1,1){22.33}}
\put(49.00,72.33){\vector(3,1){29.00}}
\put(58.00,90.67){\line(1,-3){4.11}}
\put(62.11,78.33){\line(4,1){15.89}}
\put(78.00,82.31){\line(-1,2){6.18}}
\put(71.82,94.67){\line(-4,-1){14.15}}
\put(62.33,78.67){\line(3,5){9.80}}

%%%%%%%%%%%% A stellar subdivision of C %%%%%%%
\put(78.67,64.5){(c)}
\put(78.67,72.00){\vector(1,2){9.50}}
\put(78.67,72.00){\vector(2,1){13.67}}
\put(79.00,72.33){\vector(1,1){22.33}}
\put(79.00,72.33){\vector(3,1){29.00}}
\put(88.00,90.67){\line(1,-3){4.11}}
\put(92.11,78.33){\line(4,1){15.89}}
\put(108.00,82.31){\line(-1,2){6.18}}
\put(101.82,94.67){\line(-4,-1){14.15}}
\put(79.00,72.00){\vector(4,3){20.00}}
\put(92.33,79.00){\line(5,6){6.67}}
\put(88.33,91.00){\line(5,-2){11.00}}
\put(101.33,94.67){\line(-1,-4){1.92}}
\put(108.00,82.33){\line(-2,1){9.00}}

%%%%%%%% A barycentric stellar subdivision of C %%%%%%
\put(108.67,64.5){(d)}
\put(108.67,72.00){\vector(1,2){9.50}}
\put(108.67,72.00){\vector(2,1){13.67}}
\put(109.00,72.33){\vector(1,1){22.33}}
\put(109.00,72.33){\vector(3,1){29.00}}
\put(118.00,90.67){\line(1,-3){4.11}}
\put(122.11,78.33){\line(4,1){15.89}}
\put(138.00,82.31){\line(-1,2){6.18}}
\put(131.82,94.67){\line(-4,-1){14.15}}
\put(109.00,72.33){\vector(3,4){23.00}}
\put(119.67,86.33){\line(4,-1){18.33}}
\put(119.67,86.33){\line(3,2){12.00}}
\end{picture}
\caption{Subdivisions of a cone. (a) A cone $C$.\, (b) A
subdivision (triangulation) of $C$.\, (c) A stellar subdivision of $C$.\, (d) A
barycentric stellar subdivision of $C$.}
\end{figure}

If $\Sigma$ is simplicial but not unimodular, a fan $\Delta$ such
$\Delta \preceq \Sigma$  cannot be unimodular. Thus barycentric
subdivisions cannot be used to refine
 a given fan into a unimodular one. On the other hand, if $\Sigma$ happens
to be unimodular, binary starrings produce most general
subdivisions, in the following sense.

\medskip
\noindent{\bf The De Concini-Procesi Lemma.} {\it Let $\Sigma$,
$\Delta$ be two unimodular fans in $N_\mathbb{R}$, and assume
$|\Sigma| = |\Delta|$. There exists a unimodular fan $\Sigma^*$ in
$\mathbb{R}^n$ such that
\begin{enumerate}
\item $\Sigma^* \prec_2 \Sigma$  ;
\item $\Sigma^* \leq \Delta$ \, .
\end{enumerate}}
\medskip

The  De Concini-Procesi Lemma was first established in \cite{DP},
and may be interpreted as a statement on the birational geometry
of toric varieties. The interested reader can find a
self-contained elementary proof in Panti's paper \cite{panti}.

\medskip
\noindent{\sc Morphisms of fans. \,\,} Let $\Sigma$ be a fan in
$N_\mathbb{R}$. Let $N'$ be a free abelian group of rank $n'$,
$N'_{\mathbb{R}} = \mathbb{R} \otimes N'$, $\Sigma'$ a fan in
$N'_\mathbb{R}$. A \emph{map of fans}  is an element of
$\Hom_{\mathbb{Z}}(N, N')$ --- a group homomorphism $\phi : N
\rightarrow N'$ --- such that its (unique) scalar extension
$\phi_\mathbb{R} : N_\mathbb{R} \rightarrow N'_\mathbb{R}$ maps
cones into cones; explicitly, for each $\sigma \in \Sigma$ there
is $\sigma' \in \Sigma'$ such that $\phi_\mathbb{R}(\sigma)
\subseteq \sigma'$. In the fan-toric dictionary included in
Section~\ref{S: TV}, maps of fans correspond to morphisms of
toric varieties. The latter provided the original motivation for
the definition of a map of fans.

Let us restrict attention to an endomorphism $\phi$ of fans with
the same support
--- that is, assume $N' = N$ and $|\Sigma'| = |\Sigma|$. A
moment's reflection shows that $\phi$ induces a subdivision of
$\Sigma$ into $\Sigma'$. Conversely, any fan $\Sigma'$ in
$N_\mathbb{R}$ such that $\Sigma' \leq \Sigma$ canonically
induces an endomorphism $\phi$ of fans which fixes the support
$|\Sigma|$. Thus, all forms of subdivisions we have introduced
above may be regarded as $\emph{morphisms}$ (of fans or of toric
varieties). The combinatorially fundamental notion of stellar
subdivision gives rise to morphisms of great interest in
algebraic geometry, namely $\emph{blow-ups}$ --- the basic tool
used to resolve singularities of algebraic varieties. As we have
seen above, every fan admits a unimodular refinement. This
translates to the  statement that \emph{every toric variety
admits a resolution of singularities.} While we refer to
\cite{oda}, \cite{ful} and \cite{ewa} for further information on
the algebro-geometric aspects of desingularizations, we include a
fan-theoretic treatment of resolutions of two-dimensional toric
singularities, both by way of illustration and with the intent of
showing interesting connections with the geometry of numbers.

A cone $\sigma \subseteq N_{\mathbb{R}}$ is \emph{strongly convex}
(alternative terminology, has \emph{apex} at the origin) if{f} it
does not contain a linear subspace of $N_\mathbb{R}$, except
$\{0\}$.\footnote{This is an interesting notion for many reasons,
amomgst which is the following characterization: for $m \geq 2$,
an $m$-dimensional cone is strongly convex if{f} it is the
positive linear hull of a polytope of dimension $m-1$. A polytope
(= bounded polyhedron) is the convex hull of a finite set of
points in $\mathbb{R}^n$.} For the remaining part of this
section, set $n = 2$, and let $\Sigma$ be a fan in $N_\mathbb{R}
\cong \mathbb{R}^2$. To avoid tedious degenerate cases, we assume
all cones of $\Sigma$ are strongly convex. Let
$\mathscr{S}_\Sigma$ be the set of all unimodular fans with the
same support as $\Sigma$. Clearly, refinement is a partial order
on $\mathscr{S}_\Sigma$. We now claim that the poset $\langle
\mathscr{S}_\Sigma, \leq \rangle$ is in fact a
lattice.\footnote{With maximum, but with no minimum, if $\Sigma$
has at least one $2$-dimensional cone.} Furthermore, we claim we
can \emph{compute}  finite lattice joins and meets in $\langle
\mathscr{S}_\Sigma, \leq \rangle$. It suffices to show this much:
Given unimodular fans $\Sigma_1$,$\Sigma_2 \in
\mathscr{S}_\Sigma$, one can compute $\Delta \in
\mathscr{S}_\Sigma$ with the property that if $\Sigma' \leq
\Sigma_1$, $\Sigma_2$ for some $\Sigma' \in \mathscr{S}_\Sigma$,
then $\Delta \leq \Sigma'$.  The argument for the existence and
computability of joins is similar.

Let $\Sigma_1$,$\Sigma_2 \in \mathscr{S}_\Sigma$, and consider
the set $\Theta = \{\sigma_1 \cap \sigma_2 \mid \sigma_i \in
\Sigma_i, i = 1,2\}$. It is promptly seen that $\Theta$ is a fan
such that $|\Theta| = |\Sigma|$. Further, $\Theta$ jointly refines
$\Sigma_1$ and $\Sigma_2$, and it is clear that it is the
coarsest such refinement. Unfortunately, $\Theta$ need not be
unimodular, whence in general $\Theta \not \in
\mathscr{S}_\Sigma$. Our claims are settled if we can show that
\emph{it is possible to compute a coarsest (hence, canonical)
unimodular refinement of  $\Theta$}.

\begin{figure}[ht]
\center
\includegraphics[scale = .8]{Figura.eps}
\caption{Canonical unimodular refinement of a $2$-dimensional
cone.}
\end{figure}

Clearly, it suffices to proceed locally and restrict attention to
a single strongly convex $2$-dimensional cone $\sigma$ in
$N_\mathbb{R}$, whence, in particular, $\sigma$ is simplicial.
Consider the convex hull $P$ in $N_\mathbb{R}$ of all integral
points $(\sigma \cap N) - \{0\}$. (See Figure~2.) This is an
unbounded polygon with a finite number of compact edges. Let us
display the (necessarily integral) endpoints of such edges as the
finite set of vectors
$$\{\vec{v}_0, \ldots, \vec{v}_m\}$$ such that $\vec{v}_i$ and
$\vec{v}_{i+1}$ are adjacent\footnote{That is, their convex hull
in $N_\mathbb{R}$ does not contain any other $\vec{v}_k$, $k \neq
i$, $k \neq i+1$.}, for $0 \leq i < m$. To fix ideas, suppose
that the order in which we have listed the $\vec{v}_i$'s is
counterclockwise. For each  $i$, $\sigma_{i} = \langle \vec{v}_i,
\vec{v}_{i+1} \rangle$ is a $2$-dimensional simplicial cone whose
vertices are precisely the vectors $ \vec{v}_i$, $\vec{v}_{i+1}$
--- if some $\vec{v}_i$ were not primitive, the primitive generator of
$\langle \vec{v}_i \rangle$ would not belong to $P$, a
contradiction. Furthermore, each $\sigma_i$ is unimodular, as is
easy to check. For $0 \leq i < m$, the collection of all cones
$\sigma_{i}$, together with their $1$-dimensional faces
$\vec{v}_i$ and the origin of $N_\mathbb{R}$, is a unimodular fan
$\Sigma^\sigma$ refining $\sigma$. A moment's reflection shows
that \emph{any unimodular subdivision of $\sigma$ must refine
$\Sigma^\sigma$}. We have thus shown the existence of the coarsest
desingularization of $\sigma$. Finally, we exhibit an explicit
algorithm to compute $\Sigma^\sigma$, given a presentation
$\sigma = \langle \vec{u}_1, \ldots, \vec{u}_t\rangle$ of the
cone $\sigma$. Notice that from any such  presentation  one can
compute the vertices of $\sigma$
--- first compute rational vectors $\vec{r}_1$, $\vec{r}_2$ such
that $\sigma = \langle \vec{r}_1$, $\vec{r}_2 \rangle$, and then
use the Euclidean algorithm to compute the vertices. Without loss
of generality, then, we may assume that $\sigma$ is presented as
$\sigma = \langle \vec{w}_1, \vec{w}_2\rangle$, where
$\{\vec{w}_1, \vec{w}_2\}$ is the set of its vertices, again
listed in counterclockwise order. Notice that $\vec{w}_1 =
\vec{v}_0$ and $\vec{w}_2 = \vec{v}_m$. Since these vertices are
primitive linearly independent vectors, another application of
the Euclidean algorithm yields a primitive vector $\vec{n}_1$ and
integers $p$, $q$ such that

\begin{itemize}
\item $\{\vec{w_1}, \vec{n}_1\}$ is a basis of the $\mathbb{Z}$-module
$N$ --- in other words, $\langle \vec{w_1}, \vec{n}_1 \rangle$ is
unimodular;
\item $0 \leq p < q$ with $p$ and $q$ relatively prime;
\item $\vec{w_2} = p\vec{w_1} + q\vec{n}_1$.
\end{itemize}

\noindent In case $q=1$, we are done, for $p=0$ and $\sigma$ is
unimodular. Otherwise, we consider the \emph{Hirzebruch-Jung
continued fraction expansion}\footnote{This is different from the
usual continued fraction expansion, sometimes called
\emph{regular}, where minus signs are replaced by plus signs. See
\cite{ful}.}

\medskip

$$\frac{q}{q-p} = b_1 -  \frac{1}{b_2 - \frac{1}{b_3 - \cdots}}$$

\medskip

\noindent with $b_j \geq 2$, which must terminate since
$\frac{q}{q-p}$ is rational. Then it can be proved that, upon
inductively defining

$$\vec{n}_{i+1} = (b_i-2)\vec{v}_{i-i} + (b_i-1)\vec{n}_i \mbox{ ,}$$

\medskip

\noindent the following equalities hold:

$$\vec{v}_i = \vec{v}_{i-1}+\vec{n}_i \mbox{ ,}$$

\medskip
\noindent whence the vectors $\vec{v}_i$ can be computed, as
claimed. Thus, the Hirzebruch-Jung continued fraction expansion is
precisely what is needed for the effective desingularization of
$2$-dimensional toric varieties.
\begin{remark}In higher dimensions, things are far more
complicated and not yet well-understood. Already in dimension
$3$, in general there is no coarsest unimodular refinement of a
given strongly convex cone $\sigma \subseteq \mathbb{R}^3$.
However, the set of maximally coarse refinements is finite, and
its classification was  carried out by M. Reid and V. I. Danilov
(see \cite{oda} for details and references). However, the
Danilov-Reid treatment is non-effective. Algorithmic
desingularization of $3$-dimensional toric variety was attained
in the paper \cite{agumun}, to which we refer the interested
reader for further information.
\end{remark}

%%%%%%%%%%% FINIS FANS %%%%%%%%%%%%%

\section{Abelian lattice-ordered groups as functions on fans}
\label{S: Groups}

This section is devoted to show that bringing together ordered
algebraic structures and fans is in the nature of things ---
indeed, there is an intimate connection between (finitely
presented) lattice-ordered groups and integral piecewise linear
geometry. We begin with an admittedly cursory introduction to
ordered groups. Some standard references are \cite{glass} and
\cite{darnel}.

A \emph{partially ordered abelian group} is an abelian group $G$
together with a submonoid $G^+$ of $G$ such that the induced
binary relation
\[ a \geq b \mbox{\quad if{f} \quad} a-b \in G^+
\]
is a partial  order which is translation invariant with respect to
addition, i.e.,
\[ a \geq b \mbox{\quad implies \quad} a+t \geq b+t \mbox{\quad for all
\quad} a,b,t \in G
\]

The submonoid $G^+$ is the \emph{positive cone} of $G$. Notice
that $0 \in G^+$. An element $p \in G$ is \emph{positive} if{f} $p
\in G^+$, and it is \emph{strictly positive} if{f} $p \in G^+ -
\{0\} = G_+$. Accordingly, $\mathbb{Z}^+ = \mathbb{N} \cup \{0\}$ and
$\mathbb{Z}_+ = \mathbb{N}$.

A partially ordered abelian group $G$ is \emph{directed} if{f }any
two elements of $G$ have a lower and an upper bound, and it is
\emph{unperforated} if{f} $ng \in G^+$, $n \in \mathbb{N}$,
implies $g \in G^+$. Further, $G$ has the \emph{Riesz Interpolation
Property} if{f} for any $x_1,x_2,y_1,y_2 \in G$, $x_i \leq y_j$
implies that there exists $z \in G$ such that $x_i \leq z \leq y_j
\quad (i,j=1,2)$. Homomorphisms of partially ordered abelian
groups are order-preserving group homomorphisms. A subgroup $J
\subseteq G$ is an \emph{$o$-ideal} if{f} it is \emph{convex} ($a
\leq b\leq c$ and $a, c \in J$ imply $b \in J$) and directed.
Kernels of homomorphisms, then, are precisely $o$-ideals.

A \emph{lattice-ordered abelian group} ($\ell$-group, for short)
is a partially ordered abelian group which is a lattice with
respect to the order relation. Equivalently, an $\ell$-group is a
structure $(G,+,0,\wedge,\vee)$ such that $(G,+,0)$ is an abelian
group, $(G,\wedge,\vee)$ is a lattice, and $+$ distributes over
$\vee$ and $\wedge$. By an \emph{$\ell$-subgroup} of an $\ell$-group
we mean a subgroup which is also a sublattice. Given a subset $S
\subseteq G$ of an $\ell$-group, we denote by $\langle S \rangle$,
$[S]$ and $[[S]]$ the $\ell$-subgroup, the subgroup and the
submonoid generated by $S$.

Homomorphisms of $\ell$-groups (\emph{$\ell$-homomorphisms}) are
maps preserving both the group and the lattice structure. We call
their kernels \emph{$\ell$-ideals}. It is easy to check that
$\ell$-ideals are the same thing as convex $\ell$-subgroups. One
easily deduces appropriate versions of the usual isomorphisms
theorems.  Let $G$ be an $\ell$-group, $J$ an $\ell$-ideal of
$G$, $\phi_J$ the canonical $\ell$-homomorphism associated to
$J$.  We denote by $G/J$ the $\ell$-group quotient of $G$ by $J$,
or, equivalently, the image of $G$ under $\phi_J$. We use the
symbol `$\cong$' to denote isomorphism of $\ell$-groups.

\medskip
\noindent {\sc Free $\ell$-groups.\,\,} Since $\ell$-groups are
equationally definable, they are a variety and thus admit free
(more generally, finitely presented) objects. We denote by
$\mathscr{F}_\kappa$ the free $\ell$-group over $\kappa$
generators, for any cardinal $\kappa$. A very strong
representation theorem for $\mathscr{F}_\kappa$ holds. Its
interesting but slightly involved history is more thoroughly
traced in \cite{MM}, where precise references to the original
results can be found
--- here we summarize the mathematical facts. Let $n \in
\mathbb{N}$. By an \emph{$\ell$-function} (of $n$ variables) we
mean a continuous piecewise linear homogeneous map $f :
\mathbb{R}^n \rightarrow \mathbb{R}$ with integral
coefficients.\footnote{In detail, a function $f : \mathbb{R}^n
\rightarrow \mathbb{R}$ is an $\ell$-function if{f} it is
continuous and there exist finitely many homogeneous linear
functions $l_i : \mathbb{R}^n \rightarrow \mathbb{R}$ $\,(i \in \{
1, \ldots, m\}$) with integer coefficients, the \emph{linear
constituents} of $f$, such that for every $\vec{x} \in
\mathbb{R}^n$ there is some $i \in \{1, \ldots, m\}$ such that
$f(\vec{x}) = l_i(\vec{x})$.} The set $\mathscr{C}_n$ of all
continuous functions from $\mathbb{R}^n$ to $\mathbb{R}$ is an
$\ell$-group under pointwise addition and order. Let
$\mathscr{L}_n \subseteq \mathscr{C}_n$ denote the
$\ell$-subgroup of all $\ell$-functions of $n$ variables.
Further, let $\mathscr{D}_n \subseteq \mathscr{C}_n$ denote the
$\ell$-subgroup generated by the projection functions $\pi_i :
\mathbb{R}^n \rightarrow \mathbb{R}$, $i = 1, \ldots, n$.
Clearly, $\mathscr{D}_n \subseteq \mathscr{L}_n \subset
\mathscr{C}_n$. Besides, a trivial universal-algebraic argument
shows that $\mathscr{D}_n$ is a homomorphic image of
$\mathscr{F}_n$. Non-trivial geometric and algebraic arguments
yield the following strengthening of the previous
statements.\footnote{Generalization to an arbitrary number of
generators is straightforward.}

\medskip
\noindent {\bf Weinberg-Beynon/Chang-McNaughton Representation.}
{\it $\mathscr{F}_n \cong \mathscr{D}_n = \mathscr{L}_n$.}
\medskip

\begin{remark} One half of the theorem ($\mathscr{F}_n \cong \mathscr{D}_n$,
or $\mathbb{Z}$ generates the variety of abelian $\ell$-groups),
is due to Weinberg (\cite{wein}). The other half ($\mathscr{D}_n =
\mathscr{L}_n$, or the $\ell$-group of $\ell$-functions is
generated by the projection functions) is due to Beynon
(\cite{bey1}). As proved by the second author (\cite{M11-JFA};
also see \cite{CDM} for self-contained proofs), abelian
$\ell$-groups with strong order unit\footnote{An element $u \in
G$  is  a \emph{strong order unit} of the $\ell$-group $G$ if{f}
for every $p \in G_+$ there exists $n \in \mathbb{N}$ such that
$nu \geq p$.} are categorically equivalent to
\emph{M(any)V(alued)-algebras}, a (natural) generalization of
boolean algebras introduced by C. C. Chang in his algebraic
analysis of \L ukasiewcz infinite-valued propositional logic.
MV-algebraic versions of Weinberg's and Beynon's aforementioned
results were proved by Chang (\cite{chang}) and McNaughton
(\cite{McN}), respectively. For the interplay between ordered
groups and MV-algebras see \cite{MM} and \cite{CDM}.
\end{remark}

On the basis of the above representation theorem, we shall
tacitly identify $\mathscr{F}_n$ and  $\mathscr{L}_n$
(equivalently, $\mathscr{D}_n$) whenever convenient.

\medskip
\noindent {\sc Finitely presented $\ell$-groups.\,\,} Let $G$ be
an $\ell$-group. An $\ell$-ideal $J$ of $G$ is \emph{generated}
by a set $S \subseteq G$, written $J = \langle\langle S
\rangle\rangle$, if{f} the intersection of all $\ell$-ideals of
$G$ containing $S$ is precisely $J$. We say that $J =
\langle\langle S \rangle\rangle$ is \emph{finitely} generated
if{f} $S$ can be chosen finite. Further, $J$ is
$\emph{principal}$ if{f} it is generated by a singleton $\{g\}
\subseteq G$. In this case, we write $J = \langle\langle g
\rangle\rangle$. It is an exercise to check that every finitely
generated $\ell$-ideal is principal. We call an $\ell$-group $G$
\emph{finitely presented} if{f} $G \cong
\mathscr{F}_n/\langle\langle g \rangle\rangle$ for some $n \in
\mathbb{N}$. A useful description of the principal ideal
$\langle\langle g \rangle\rangle$ is the following: $f \in
\langle\langle g \rangle\rangle$ if{f} there is some $n \in
\mathbb{N}$ such that $ng \geq f$ (in $\mathscr{F}_n$). Given $f
\in \mathscr{F}_n$, the \emph{zeroset} of $f$ is the set
$\mathbb{V}(f) = \{\vec{x} \in \mathbb{R}^n \mid f(\vec{x})=0\}$.
More generally, for any subset $S \subseteq \mathscr{F}_n$, let us
define $\mathbb{V}(S) = \bigcap_{f \in S }\mathbb{V}(f)$. From the
description of $\langle\langle g \rangle\rangle$ given a moment
ago, it follows that $\mathbb{V}(\langle\langle g \rangle\rangle)
= \mathbb{V}(g)$. We now claim that the zeroset of an
$\ell$-function $g$ is the support of a fan. Let $g \in
\mathscr{F}_n$ be an $\ell$-function, and let $\{l_i\}$  ($i \in
\{1, \ldots, m\}$) be the set of its linear constituents (cfr.
footnote~8). Let $\mathscr{S}_m$ denote the group of permutations
of $m$ letters $\{1, \ldots, m\}$ (the symmetric group). For each
$\sigma \in \mathscr{S}_m$, we define the \emph{basic cone}
$$C_\sigma = \{\vec{x} \in \mathbb{R}^n \mid l_{\sigma(1)}(\vec{x}) \geq
  \ldots \geq l_{\sigma(m)}(\vec{x})\} \, .$$ It is
clear that $C_\sigma$ is a closed set; also notice that it may
well be the singleton $\{0\}$. In fact, $C_\sigma$ is the
intersection of finitely many integral half-spaces, as is read off
directly from its definition. Hence, by the Fundamental Theorem
of Polyhedra, it is  a (rational polyhedral) cone. Furthermore,
the set of all such $C_\sigma$ and their finite intersections
$C_{\sigma_1} \cap \cdots \cap C_{\sigma_s}$, as $\sigma_1,
\ldots, \sigma_s$ vary in $\mathscr{S}_m$, is a fan $\Sigma_g$ in
$N_\mathbb{R} \cong \mathbb{R}^n$. Since $g$ is linear over each
$C_\sigma$, its zeroset $\mathbb{V}(g)$ can be described as the
support of a fan $\Theta_g$ inscribed in $\Sigma_g$, as we
claimed. In analogy with the time-honored tradition of classical
algebraic geometry, it makes sense to investigate the relationship
between the $\ell$-group of all $\ell$-functions in
$\mathscr{F}_n$ restricted to $|\Theta_g|$
--- a geometric entity --- and the $\ell$-group
$\mathscr{F}_n/\langle\langle g \rangle\rangle$ --- an algebraic
object. In order to do this at the natural level of generality,
we introduce some further notions from the theory of ordered
groups.

Let $G$ be an $\ell$-group and $J \subseteq G$ an $\ell$-ideal. We
say that $J$ is \emph{maximal} if{f} for every $\ell$-ideal $I
\subseteq G$, $J \subseteq I$ implies $J=I$. Further, $J$ is
\emph{proper} if{f} $J \neq G$. The intersection of any
collection of $\ell$-ideals is an $\ell$-ideal. The
\emph{radical} of $G$ is the intersection of all its maximal
$\ell$-ideals. Finally, $G$ is \emph{archimedean} if{f} for every
$p, q \in G_+$ such that $p \leq q$, there is $n \in \mathbb{N}$
such that $np \not \leq q$. Assume that $G$ is finitely generated.
Then $G$ is archimedean if{f} its radical is $\{0\}$, as is easy
to see.

Quite generally, if $J \subseteq \mathscr{F}_n$ is an $\ell$-ideal
of the free $\ell$-group over $n$ variables, then $\mathbb{V}(J)$
is a closed conical\footnote{A subset $C \subseteq \mathbb{R}^n$
is \emph{conical} if{f} it contains, together with every point
$\vec{x} \in C$, the ray $\{p\vec{x} \mid p \in \mathbb{R}^+\}$
generated by that point.} subset of $N_\mathbb{R}$. Conversely,
given a closed conical subset $C \subseteq N_\mathbb{R}$, let
$\mathbb{I}(C)$ denote the set of all $\ell$-functions of $n$
variables vanishing over $C$. Then $\mathbb{I}(C)$ is an
$\ell$-ideal. Clearly, $\mathbb{V}(\mathbb{I}(C)) = C$ for any
closed conical set $C$. However, $J \subseteq
\mathbb{I}(\mathbb{V}(J))$ may well be a strict inclusion. The
following result, characterizing those $\ell$-ideals $J$ such
that $J = \mathbb{I}(\mathbb{V}(J))$, is the analogue of
Hilbert's Nullstellensatz.

\medskip
\noindent {\bf Archimedean representation.}\footnote{Fair
attribution of this result would take up an amount of space we
are not ready to devote to the matter here. See \cite{CDM} and
\cite{MM}.}
 Let $G$ be a finitely
generated $\ell$-group, and assume $G \cong \mathscr{F}_n/J$. Let
$\mathscr{F}_n\upharpoonright\mathbb{V}(J)$ denote the
$\ell$-group of $\ell$-functions of $n$ variables restricted to
$\mathbb{V}(J)$. Then $G \cong
\mathscr{F}_n\upharpoonright\mathbb{V}(J)$ if{f} $G$ is
archimedean if{f} $J = \mathbb{I}(\mathbb{V}(J))$.
\medskip

Thus, returning to the finitely presented $\ell$-group $G \cong
\mathcal{F}_n/\langle\langle g \rangle\rangle$, we need to
ascertain whether $G$ can violate the archimedean condition. It
cannot.

\medskip
\noindent {\bf W\'ojcicki-Baker's Theorem.}\footnote{This is due
to Baker ($\ell$-group version, \cite{bak}) and W\'ojcicki
(MV-algebraic version, \cite{woj}).} A finitely presented
$\ell$-group is archimedean.
\medskip

As an immediate consequence of the above results, a finitely
presented $\ell$-group can be identified with the $\ell$-group of
$\ell$-functions of $n$-variables restricted to the support of a
fan. Conversely, given a fan $\Sigma$ in $N_\mathbb{R}$, it is
possible to describe its support as the zeroset of an
$\ell$-function $g$, thus obtaining a finitely presented
$\ell$-group $\mathscr{F}_n/\langle\langle g \rangle\rangle$
associated with $\Sigma$. It turns out that this duality between
fans and $\ell$-groups is the germ of a full-scale categorical
equivalence.



\medskip
\noindent {\sc Morphisms of finitely presented
$\ell$-groups.\,\,} The maps of fans we have hitherto considered
are motivated by algebro-geometric considerations --- they are the
direct combinatorial translation of the notion of toric morphism.
We now introduce another class of morphisms of fans which is of
equal importance for the contents of this paper. Let $N'$ be a
free abelian group of rank $n'$, $N'_{\mathbb{R}} = \mathbb{R}
\otimes N'$, $\Sigma'$ a fan in $N'_\mathbb{R}$. Let $\Sigma$ be
a fan in $N_\mathbb{R}$ and $\Sigma'$ a fan in $N'_\mathbb{R}$. A
function $\phi : |\Sigma| \rightarrow |\Sigma'|$ is an
\emph{$\ell$-map} if{f} $\phi(\vec{x}) = (f_1(\vec{x}), \ldots,
f_{n'}(\vec{x }))$ for every $\vec{x} \in |\Sigma|$, where $f_i$
is an $\ell$-function of $n$ variables. The fan $\Sigma$ is
\emph{$\ell$-embedded} in $\Sigma'$ if{f} $\phi$ is injective;
$\Sigma'$ is an \emph{$\ell$-homomorphic image} of $\Sigma$ if{f}
$\phi$ is surjective; $\Sigma$ and $\Sigma'$ are
\emph{$\ell$-homeomorphic} if{f} there is an $\ell$-map $\psi :
|\Sigma'| \rightarrow |\Sigma|$ such that $\psi \circ \phi$ is
the identity function on $\Sigma$.

\medskip
\noindent {\bf Baker-Beynon Duality.}\footnote{The original
references are \cite{bak},\cite{bey1},\cite{bey2}.}

\noindent{ \begin{enumerate}
\item An abelian $\ell$-group $P$ is finitely presented if{f} it
is isomorphic to  the $\ell$-group of $\ell$-functions of $n \in
\mathbb{N}$ variables restricted to the support of some fan
$\Sigma$ in $N_\mathbb{R}$. We call $\Sigma$ a fan
\emph{associated} with $P$.
\item Two finitely presented $\ell$-groups $P_1$, $P_2$
are isomorphic if{f} for every two fans $\Sigma_1$, $\Sigma_2$
associated with $P_1$, $P_2$, respectively, one has that
$\Sigma_1$ and $\Sigma_2$ are $\ell$-homeomorphic.
\end{enumerate}}
\medskip

We add a few remarks on onto and into morphisms to round off the
above statement.\footnote{Some of the assertions we are about to
make are not trivial to prove.  Some of the material may be
novel, whence the absence of detailed references. What is not new
can be found in \cite{bak},\cite{bey1},\cite{bey2}.} Let $P
\twoheadrightarrow Q$ be a surjective homomorphism of finitely
presented $\ell$-groups $P$ and $Q$,\footnote{Notice that it is
not possible to do without the assumption that $Q$ be finitely
presented  --- $P$ may have non-archimedean (let alone
non-finitely presentable) homomorphic images.} and let $\Sigma$
be a fan associated with $P$. Then there is a fan $\Sigma'$
associated with $Q$ such that $|\Sigma'| \subseteq |\Sigma|$, the
latter inclusion being  an  $\ell$-embedding of fans. (However,
it can also be shown that there exists a surjective $\ell$-map of
fans $\rho : |\Sigma| \twoheadrightarrow |\Sigma'|$ which
\emph{retracts} $\Sigma$ onto $\Sigma'$, i.e., $\rho$ restricted
to $\Sigma'$ is the identity.) Conversely, consider an
$\ell$-embedding of fans $\phi : |\Sigma| \rightarrowtail
|\Sigma'|$, and let $P$ and $Q$ be the finitely presented
$\ell$-groups associated with $\Sigma$ and $\Sigma'$. We identify
$|\Sigma|$ with a subset of $|\Sigma'|$. Restricting
$\ell$-functions on $|\Sigma'|$ (= elements of $Q$) to the subset
$|\Sigma|$, we obtain that $P$ is a homomorphic image of $Q$.
(However, it can indeed be shown that $P$ is also a subalgebra of
$Q$, because there is an $\ell$-map $\rho : |\Sigma'|
\twoheadrightarrow |\Sigma|$ which retracts $|\Sigma'|$ onto
$|\Sigma|$ --- that is, $\rho \circ \phi$ is the identity on
$|\Sigma|$.) Hence we see that $\ell$-embeddings in the category
of fans correspond to surjections  of finitely presented
$\ell$-groups. The analogous results for onto $\ell$-maps of fans
can also be established. Finally, in the light of (2) above,
these statements hold for \emph{any} two fans associated with $P$
and $Q$. The foregoing could be easily recast into a categorical
equivalence between fans and their $\ell$-maps, on the one hand,
and finitely presented $\ell$-groups and their homomorphisms, on
the other.

\medskip
\noindent {\sc $\Delta$-linear support functions.\,\,} Perusal of
any textbook on toric varieties will reveal no reference to
lattice-ordered groups
--- in its long and distinguished history, geometry has seldom
taken advantage of order-theoretical considerations.\footnote{One
notable exception being real semi-algebraic geometry.} We shall
now argue that finitely presented $\ell$-groups do arise in a
natural manner from the theory of toric varieties.

Let $\Delta$ be a fan in $N_\mathbb{R}$. Let $|\Delta|_\mathbb{Z}
= N\cap|\Delta|$ be the set of integral points in $|\Delta|$. A
function $f : |\Delta| \rightarrow \mathbb{R}$ is a
\emph{$\Delta$-linear support function} if{f} it is linear on each
$\sigma \in \Delta$, and $f\upharpoonright{|\Delta|_\mathbb{Z}}$
is integer-valued. We shall denote by ${\rm SF}(\Delta)$ the set
of $\Delta$-linear support functions. It is an abelian group under
addition. (We remark in passing that each element of the dual
lattice $M$ yields an integral linear --- {\it a fortiori}
$\Delta$-linear --- support function, but this correspondence is
injective if{f} $|\Delta|$ spans the whole space $N_\mathbb{R}$.)
>From what we expounded above, up to isomorphism there is a unique
finitely presented $\ell$-group $P_\Delta$ associated with
$\Delta$, namely $P_\Delta \cong
\mathscr{F}_n/\mathbb{I}(|\Delta|)$.  It is easy to check that a
$\Delta$-linear support function is the same thing as an
$\ell$-function on $|\Delta|$ which is linear over each $\sigma
\in \Sigma$. In other words, ${\rm SF}(\Delta) \subseteq
P_\Delta$. We thus see that ${\rm SF}(\Delta)$ inherits a partial
order from $P_\Delta$ turning ${\rm SF}(\Delta)$ into a partially
ordered abelian group. Notice  that ${\rm SF}(\Delta)$ is not an
$\ell$-subgroup of $P_\Delta$ --- it is not closed under lattice
operations. Given $s \in {\rm SF}(\Delta)$, to each ray (=
$1$-dimensional cone) $\rho_i = \langle \vec{r}_i \rangle \in
\Delta^{(1)}$ generated by its (unique) vertex $\vec{r}_i$, we can
associate the integral value $s(\vec{r}_i)$; if the $1$-skeleton
$\Delta^{(1)}$ has cardinality $t$, we thus obtain for each $s$
an element $(s(\vec{r}_1), \ldots,s(\vec{r}_t))$ of the free
abelian group $\mathbb{Z}^t$. Further, the sum of two
$\Delta$-linear support functions corresponds to the
coordinatewise sum of the associated elements of $\mathbb{Z}^t$.
In other words, we have an embedding of abelian groups $\phi :
{\rm SF}(\Delta) \rightarrowtail \mathbb{Z}^t$. An $\ell$-group is
\emph{simplicial} if{f} it is isomorphic (as an $\ell$-group) to
$\langle\mathbb{Z}^k, \leq\rangle$, where $(v_1, \ldots, v_k)
\leq (w_1, \ldots, w_k)$ if{f} $v_i \leq w_i$ for all $i \in \{1,
\ldots, k\}$ (\emph{coordinatewise} or \emph{simplicial order}).
Through the group embedding $\phi$, the simplicial order on
$\mathbb{Z}^t$ induces a partial order on ${\rm SF}(\Delta)$. It
is an exercise to check that the latter coincides with the order
induced on ${\rm SF}(\Delta)$ by the inclusion ${\rm SF}(\Delta)
\subseteq P_\Delta$. Thus $\phi$ is naturally an embedding of
partially ordered abelian groups. It is not hard to show that
\emph{$\phi$ is onto if{f} $\Delta$ is unimodular}. To put it
differently, when $\Delta$ is unimodular, for every assignment
$\alpha : \{\vec{r}_1, \ldots, \vec{r}_k\} \rightarrow
\mathbb{Z}$ of integral values to the vertices $\vec{r}_i$, there
is a unique  $\Delta$-linear support function $s_\alpha \in {\rm
SF}(\Delta)$ such that $s_\alpha(\vec{r}_i) = \alpha(\vec{r}_i)$
for all $i \in \{1, \ldots, k\}$; whereas when $\Delta$ is not
unimodular, there exists an assignment $\alpha$ which cannot be
thus realized by any $s \in {\rm SF}(\Delta)$.

Now assume $\Delta$ is unimodular and \emph{complete}, that is,
$|\Delta| = N_\mathbb{R}$, whence $P_\Delta \cong \mathscr{F}_n$
is the free abelian $\ell$-group over $n$ generators. We remark in
passing that any (unimodular) fan $\Sigma$ can be extended to a
complete (unimodular) fan --- intuitive as this may sound, the
proof is not trivial, see \cite{ewa}.\footnote{Completeness is
the exact counterpart of algebro-geometric compactness, and
completion of a fan corresponds to compactification of the
associated toric variety.} Then $\phi : {\rm SF}(\Delta)
\rightarrowtail \mathbb{Z}^t$ is an isomorphism of partially
ordered groups. In any simplicial group $S$, there exists a unique
set of strictly positive linearly independent generators. We call
such a set the \emph{simplicial basis} of $S$. Let  $\{\vec{h}_1,
\ldots, \vec{h}_t\} \subseteq \mathbb{Z}^t$ be the simplicial
basis of $\mathbb{Z}^t$. Via the injection ${\rm SF}(\Delta)
\subseteq P_\Delta$ and the isomorphism $\phi$, we identify
$\{\vec{h}_i\}$ with a subset of $P_\Delta$, of which we are now
going to provide an explicit combinatorial description. Let $\rho
= \langle \vec{r} \rangle \in \Delta^{(1)}$, where $\vec{r}$  is
the vertex of $\rho$. The \emph{Schauder hat} at $\rho$
(\cite{M11-ADV}, \cite{CDM}) is the unique continuous
piecewise-linear homogeneous function $h_\rho \colon N_\mathbb{R}
\rightarrow \mathbb{R}$ such that
%
\begin{enumerate}
\item $h_\rho(\vec{r}) = 1$,
\item $h_\rho(\vec{v}) = 0$ for every
vertex $\vec{v} \neq \vec{r}$ of any cone in $\Delta^{(1)}$,
\item $h_\rho$ is
 homogeneous linear on each cone of $\Sigma.$
\end{enumerate}
%
As a consequence of the unimodularity of $\Delta$, a Schauder hat
is guaranteed to have linear pieces with integral coefficients.
Thus, $h_\rho$ is an element of $\mathscr{F}_n$. We denote by
$\mathbf{H}_\Delta$ the set of all Schauder hats at the rays of
the complete unimodular fan $\Delta$, and call it a
\emph{Schauder basis} of $\mathscr{F}_n$.\footnote{Generalizing
this construction to finitely presented $\ell$-groups is
straightforward.} The subgroup $[\mathbf{H}_\Delta]$ generated by
$\mathbf{H}_\Delta$ in $\mathscr{F}_n$ is easily seen to be free
(as a group). In fact, ${\rm SF}(\Delta) = [\mathbf{H}_\Delta]
\cong \mathbb{Z}^t$ as groups. But $[\mathbf{H}_\Delta]$ inherits
a partial order from $\mathscr{F}_n$, and ${\rm SF}(\Delta) =
[\mathbf{H}_\Delta] \cong \mathbb{Z}^t$ as simplicial groups. The
positive cone in $[\mathbf{H}_\Delta]$ is precisely the monoid
$[[\mathbf{H}_\Delta]]$. Notice that from $\mathbf{H}_\Delta$ it
is possible to recover $\Delta$ completely, and conversely.
Further, this correspondence is algorithmic --- see \cite{CDM} and
references therein. Thus Schauder bases are the exact
$\ell$-group-theoretical analogue of complete unimodular fans.

\medskip
\noindent {\sc Simplicial approximations. \,\,} We remark that
$\Delta$-linear support functions are the combinatorial
counterpart of linear systems and divisors --- formal integral
linear combinations of subvarieties of codimension $1$ --- on
toric varieties (see \cite{oda} or \cite{ewa}). Thus, we are
considering a partial order on (what essentially is) the group of
divisors ${\rm SF}(\Delta)$ of a toric variety. This is not
entirely foreign to algebraic geometers, who call an element of
${\rm SF}(\Delta)$ \emph{effective} if{f}, in our terminology, it
is positive in the simplicial order of ${\rm SF}(\Delta)$.
However, we are also \emph{order}-embedding\footnote{But not
lattice-embedding.} ${\rm SF}(\Delta)$ into a larger
lattice-ordered group $P_\Delta$ associated with $\Delta$,
something which, to the best of our knowledge, has no established
counterpart in the algebro-geometric language. We shall now show
fan-theoretically that $P_\Delta$ --- a finitely presented
structure amenable to computation --- completely encodes the
birational geometry of the associated toric variety. We continue
to use the notation introduced in the previous subsection, and to
assume that $\Delta$ is complete and unimodular. Let $p \in
\mathscr{F}_n^+$ be a positive $\ell$-function such that $p
 \not \in {\rm SF}(\Delta)$, and suppose
$\Sigma$ is a complete fan in $N_\mathbb{R}$ such that $p \in {\rm
SF}(\Sigma)$. As we have seen, there exists a complete unimodular
refinement $\Sigma' \leq \Sigma$. Clearly, $p \in {\rm
SF}(\Sigma')$. By the De Concini-Procesi Lemma, there exists a
complete unimodular fan $\Delta_1 \preceq_2 \Delta$ such that
$\Delta_1 \leq \Sigma' \leq \Sigma$, whence $p \in {\rm
SF}(\Delta_1)$.\footnote{As shown in \cite{panti}, such a
$\Delta_1$ can be effectively computed.} Equivalently, $p \in
[[\mathbf{H}_{{\Delta}_1}]]$. In any $\ell$-group $G$, every $g
\in G$  decomposes uniquely as $g = g^+ - g^-$ for elements $g^+
, g^- \in G^+$ such that $g^+ \wedge g^- = 0$; thus, the fact
that we have argued for $p \in \mathscr{F}_n^+$ is not
restrictive. Iterating this construction we obtain:


\medskip \noindent Let $\Delta$ be a complete unimodular fan in $N_\mathbb{R}$.
There exists a sequence of complete unimodular fans
$\{\Delta_i\}_{i \in \mathbb{Z}^+}$ such that
\begin{enumerate}
\item $\Delta_0 = \Delta$ ;
\item $\Delta _{i+1} \preceq_2 \Delta_i$ for all $i \in
\mathbb{Z}^+$ ;
\item For all $p \in \mathscr{F}_n^+$, there exists $i \in
\mathbb{Z}^+$ such that $p \in [[\mathbf{H}_{\Delta_i}]]$, hence
for all $f \in \mathscr{F}_n$ there is $i \in \mathbb{Z}^+$ such
that $f \in [\mathbf{H}_{\Delta_i}]$ .
\end{enumerate}
\medskip

\noindent We call a sequence $\{\Delta_i\}_{i \in \mathbb{Z}^+}$
satisfying the properties of the above statement a \emph{De
Concini-Procesi sequence} for $\Delta$.

A moment's reflection shows that $[[\mathbf{H}_{\Delta_0}]]
\subseteq [[\mathbf{H}_{{\Delta}_1}]]$, whence
$[\mathbf{H}_{{\Delta_0}}] \subseteq [\mathbf{H}_{{\Delta}_1}]$.
Let $t_0$ and $t_1$ be the cardinalities of $\Delta_0^{(1)}$ and
$\Delta_1^{(1)}$, respectively. Since $[\mathbf{H}_{{\Delta_0}}]
\cong \mathbb{Z}^{t_0}$ and $[\mathbf{H}_{{\Delta_1}}] \cong
\mathbb{Z}^{t_1}$, we obtain an embedding of free abelian groups
$\phi_1 : \mathbb{Z}^{t_0} \rightarrowtail \mathbb{Z}^{t_1}$;
since $[[\mathbf{H}_{{\Delta_0}}]] \subseteq
[[\mathbf{H}_{{\Delta}_1}]]$, $\phi_1$ is order-preserving (with
respect to the simplicial order), that is, it sends positive
elements to positive elements. In other words, with respect to
simplicial bases in $\mathbb{Z}^{t_0}$ and $\mathbb{Z}^{t_1}$,
$\phi_1$ is represented by a unique full-rank integral $t_1 \times
t_0$ matrix with nonnegative entries. Given a De Concini-Procesi
sequence $\{\Delta_i\}$ as in the above, we thus obtain a
sequence of simplicial groups $\{\mathbb{Z}^{t_i}\}$ and injective
order-preserving group homomorphisms $\{\phi_{i+1} :
\mathbb{Z}^{t_i} \rightarrowtail \mathbb{Z}^{t_{i+1}}\}$. Quite
generally, given a sequence of partially ordered abelian groups
$\{G_i\}$ and order-preserving (not necessarily injective) group
homomorphisms $\{\phi_{i+1} : G_i \rightarrow G_{i+1}\}$, the
couple

$$\langle G_i,\phi_{i+1}\rangle$$

\noindent is called a \emph{direct system} of partially ordered
abelian groups. Every such direct system admits a unique
\emph{limit group} $G = \lim G_i$, which is again a partially
ordered abelian group. When all homomorphisms are injective, the
limit is simply the group $G = \bigcup_{i \in \mathbb{Z}^+}G_i$,
with positive cone $G^+ = \bigcup_{i \in \mathbb{Z}^+}G^+_i$. In
this case, $G$ is called \emph{ultrasimplicial}. Thus:

\medskip \noindent {\bf Simplicial approximations of free $\ell$-groups.}
Let $\Delta$ be a complete unimodular fan in $N_\mathbb{R}$. The
$\ell$-group $P_\Delta \cong \mathscr{F}_n$ is ultrasimplicial.
More precisely, given any De Concini-Procesi sequence
$\{\Delta_i\}$ for $\Delta$, $P_\Delta$ is the limit group of the
direct system

$$\langle {\rm SF}(\Delta_i),\zeta_{i+1}\rangle$$

\noindent of simplicial groups of $\Delta_i$-linear support
functions and injective order preserving homomorphisms $\zeta_i :
{\rm SF}(\Delta_i) \rightarrowtail {\rm SF}(\Delta_{i+1})$ induced
by binary starrings $\Delta_{i+1} \preceq_2 \Delta_i$.


\medskip
\noindent {\it Remarks.}

\noindent {\it 1.} Amongst other things, simplicial
approximations of free $\ell$-groups via  groups of support
functions give rise to a very interesting combinatorics of
positive integral matrices. The key point is that the connecting
morphisms $\zeta_{i+1} : {\rm SF}(\Delta_i) \rightarrowtail {\rm
SF}(\Delta_{i+1})$ are of a very special kind indeed ---
geo\-me\-tri\-cal\-ly, they are binary starrings. This allows for
the definition of a purely matrix-theoretical analogue of binary
starring which is the subject of current research, and still far
from being well-understood.

\smallskip

\noindent {\it 2.} The above result admits far-reaching
generalizations. Techniques already developed in \cite{M11-ADV}
allow to prove that every finitely presented $\ell$-group
$P_\Delta$ is the limit of a direct system of simplicial groups
of support functions, in close analogy with what we have just
shown. The first author proved that, quite generally,
\emph{every} abelian $\ell$-group is ultrasimplicial
(\cite{marra}). While the fan-theoretic interpretation of the
latter result is the subject of ongoing research,  applications
of the ultrasimplicial property to other mathematical realms are
described in \cite{MM}.

\smallskip

\noindent {\it 3.} Direct systems of simplicial groups were
originally investigated  by functional analysts because of their
relevance for the classification of certain $C^*$-algebras. In
general, the limit need not be lattice-ordered --- not even when
all maps are binary starrings. However, the limit of a direct
systems of simplicial groups always is a \emph{dimension group},
namely a directed, unperforated partially ordered abelian group
with the Riesz Interpolation Property. See \cite{MM} for more
details and references.


%%%%%%%%%%%%% TORIC VARIETIES %%%%%%%%%%

\section{Toric varieties}
\label{S: TV}

We briefly introduce the basic correspondence between algebraic
varieties and fans. See \cite{oda}, \cite{ewa} and \cite{ful} for
a comprehensive account. Our terminology and notation for
commutative algebra follows \cite{AM}.

We shall work over a fixed algebraically closed ground field $k =
\overline{k}$ of characteristic zero, even though it is known
that many results remain true in positive characteristic. We
continue to use our previous notation --- recall that $N$ is a
free abelian group of rank $n$, $M$ is its dual
$\mathbb{Z}$-module, $N_\mathbb{R} = \mathbb{R} \otimes N$ and
$M_\mathbb{R}$ is the dual of $N_\mathbb{R}$.

A \emph{lattice} in $\mathbb{R}^n$ is a free abelian group $L
\subseteq \mathbb{R}^n$ of rank $n$. A \emph{lattice point} (with
respect to $L$) in $\mathbb{R}^n$ is simply an element of $L$. In
this terminology --- a legacy of Minkowski's geometry of numbers
--- $N$ is a lattice in $N_\mathbb{R}$. Let $\sigma$ be a cone in
$N_\mathbb{R}$, and consider the set $L_\sigma = \sigma \cap N
\subseteq N_\mathbb{R}$ of lattice points in $\sigma$. Since
$\sigma$ is convex, $S$ is a semigroup under addition. More is
true.

\medskip
\noindent {\bf Gordan's Lemma.} {\it The lattice points $L_\sigma$
within a cone $\sigma$ are a finitely generated semigroup. }
\medskip

While Gordan's Lemma is not difficult to prove (see e.g.
\cite{ewa}), it is an essential result for  the definition of
affine toric varieties, as we now show.

Again, let $\sigma = \langle \vec{v}_1, \ldots, \vec{v}_m \rangle$
be a cone in $N_\mathbb{R}$, and set $L_\sigma = \sigma \cap N$.
By Gordan's Lemma, $L_\sigma$ is in an additive finitely generated
semigroup; let $g_1, \ldots, g_s$ be a set of generators of $S$.
(Each $g_i$ is a vector in $N_\mathbb{R}$, but we drop the vector
notation for simplicity.) We switch to multiplicative notation, so
that an element of $L_\sigma$ is a \emph{Laurent monomial}
$g_1^{m_1} \cdots g_s^{m_s}$ for (not necessarily unique) natural
numbers $m_i \in \mathbb{N}$. (Since $m_i \in \mathbb{N}$, it
might seem sufficient to speak of \emph{monomials}. However, the
$g_i$'s are not free variables, and because of possible relations
amongst them it is actually necessary to deal with Laurent
monomials --- cfr. Example $4$ below.) Consider the semigroup ring
$k[L_\sigma]$. Elements of $k[L_\sigma]$ are \emph{Laurent
polynomials}, that is, finite sums of Laurent monomials in
$L_\sigma$ with coefficients in $k$. Clearly, then, $k[L_\sigma]$
is a $k$-algebra.\footnote{Recall that a ring $R$ is a
\emph{$k$-algebra} if{f} $k$ is a subring of $R$; that $R$ is
\emph{reduced} if{f} it has no nilpotent elements; and that
$\Spec$ is a  categorical equivalence between finitely generated
reduced $k$-algebras and affine algebraic varieties over $k$
(\cite{AM}, \cite{Dan}).} In fact, it is a finitely generated
$k$-algebra, because $L_\sigma$ is finitely generated. It is
almost immediate that $k[L_\sigma]$ is reduced. (Indeed, if the
$n$-th power of a nonzero Laurent monomial is zero, then the
$nm_i$-th power of some variable $g_i$ appearing in that monomial
must be zero; but this implies that $g_i$ is the zero vector, a
contradiction. Extension to Laurent polynomials is trivial.) Thus
we can define the \emph{affine toric variety}
$$X_{\sigma} = \Spec k[L_\sigma] \, .$$

\unitlength 1mm \linethickness{0.4pt}
\begin{figure}[t]\label{F: Duals}
\begin{picture}(30,30)(0,0)
%A cone <(0,1),(1,1)>
\put(-15,10){\vector(0,1){15.00}}
\put(-15,10){\vector(1,1){15.00}} \put(-12,17){$\sigma$}%21.32

%Its dual <(1,0),(-1,1)>
\put(45,10){\vector(1,0){15.00}} \put(45,10){\vector(-1,1){15.00}}
\put(45,17){$\sigma^\vee$}
\end{picture}
\caption{A cone $\sigma = \langle (0,1),(1,1) \rangle$ and its
dual $\sigma^\vee = \langle (1,0),(-1,1) \rangle$.}
\end{figure}

\begin{examples}

\noindent{

\smallskip \noindent {\it 1.} If $\sigma = \langle \vec{e}_1,
\ldots, \vec{e}_n \rangle$ is the cone generated by the canonical
basis of $N_\mathbb{R}$, $X_{\sigma} = \Spec k[L_\sigma] = \Spec
k[x_1, \ldots, x_n]$ is affine $n$-space $k^n$.

\smallskip \noindent {\it 2.} If $\sigma = \langle (a,b)\rangle
\subseteq \mathbb{R}^2$, $(a,b) \neq 0$, $X_{\sigma} = \Spec
k[L_\sigma] = \Spec k[x]$ is the affine line $k$.

\smallskip \noindent {\it 3.} If $\sigma = \langle (1,0), (1,2)
\rangle \subseteq \mathbb{R}^2$, then $\{(1,0), (1,2)\}$ is
\emph{not} a set of generators for $L_\sigma$, because e.g.
$(1,1)$ is not in the submonoid spanned by $\{(1,0), (1,2)\}$.
However, $\{(1,0), (1,1),(1,2)\}$ is a set of generators for
$L_\sigma$. Hence, $X_{\sigma} = \Spec k[L_\sigma] = \Spec k[x_1,
x_1x_2, x_1x_2^2]$. But $k[x_1, x_1x_2, x_1x_2^2] \cong k[X, Y,
Z]/J$, where $J$ is the ideal of $k[X, Y, Z]$ generated by the
polynomial $Y^2-XZ$. Thus, $X_{\sigma}$ is the standard cone with
vertex at the origin. In particular, $X_{\sigma}$ is not
``smooth'' (at the origin) in any reasonable sense of the word.
We shall not define smoothness in this paper; as we already
mentioned, an affine toric variety $X_\sigma$ is smooth if{f}
$\sigma$ is unimodular.}

\smallskip \noindent {\it 4.} If $\sigma = N_\mathbb{R}$, $X_\sigma
= \Spec k[x_1,x_1^{-1}, \ldots, x_n,x_n^{-1}]$. The latter is
known as the \emph{$n$-dimensional algebraic torus}, and it plays
an important r\^{o}le in the theory of toric varieties --- see
\cite{oda}. If $\sigma$ is not strongly convex, it contains a
nontrivial linear subspace $V \subseteq N_\mathbb{R}$ of dimension
$d \geq 1$. In this case, it is not hard to see that the
$d$-dimensional torus $X_V$ splits off as a direct factor of
$X_\sigma$, whence the widespread technical device of restricting
attention to strongly convex cones.
\end{examples}

\smallskip
The next natural step is to construct general toric varieties
gluing together affine toric varieties. The data for the gluing
morphisms come from a fan $\Sigma$. In carrying out this
programme, it turns out that it is convenient to consider dual
cones of the elements of $\Sigma$.

Let $\sigma \subseteq N_{\mathbb{R}}$ be an $m$-dimensional cone.
We set

$$
\sigma^{\vee} = \{f \in M_{\mathbb{R}} \mid f(\vec{u}) \geq 0
\mbox{\, for every \,} \vec{u} \in \sigma\}
$$

It is not hard to check that if $\sigma = \langle \vec{v}_1,
\ldots, \vec{v}_r \rangle$, then $\sigma^\vee$ is the
intersection of the half-spaces $\{f \in M_{\mathbb{R}} \mid
f(\vec{v_i}) \geq 0\} \subseteq M_{\mathbb{R}}$. Thus, by the
Fundamental Theorem of Polyhedra, it is a cone in $M_\mathbb{R}$,
the \emph{dual cone} of $\sigma$. The terminology is appropriate
in that $\sigma^{\vee \vee} = \sigma$ for any cone $\sigma$.
Notice that faces $\tau$ of $\sigma$ coincide with sets of the
form $\tau = \sigma \cap \ker{f}$, for some $f \in \sigma^{\vee}$.

{\small
\begin{table}[t]
\centering
\begin{tabular}{ll}
\hline
{\sc Combinatorial Convexity} & {\sc Algebraic Geometry}\\
\hline
cone &  affine toric variety\\

%\smallskip

face & subvariety\\

%\smallskip

lattice $N \subseteq N_\mathbb{R}$ & torus = $\Spec k[x_1,x_1^{-1}, \ldots,
x_n,x_n^{-1}]$\\

%\smallskip

fan&toric variety\\

%\smallskip

--- complete&--- compact\\

%\smallskip

--- unimodular &--- smooth\\

%\smallskip

--- simplicial & --- quasi-smooth\\

%\smallskip

--- induced by a polytope & --- projective\\

%\smallskip

barycentric stellar subdivision & equivariant blow up\\

%\smallskip

unimodular map  & algebraic isomorphism\\

%\smallskip

map of fans & equivariant toric morphism \\

cardinality of $\Sigma^{(n)}$, $\Sigma$  a   & Euler characteristic
$\chi(X_\Sigma)$  \\
\quad complete unimodular fan & \quad of a  smooth compact variety\\

\hline \vspace{1pt}
\end{tabular}
\caption{Fan-Toric Vocabulary.} \label{table:ft}
\end{table}
} If $\sigma \subseteq N_{\mathbb{R}}$ is an $n$-dimensional
strongly convex cone, its dual $\sigma^\vee$ can be
geo\-me\-tri\-cal\-ly constructed as follows. For each facet (=
maximal proper face) $F_i$ of $\sigma$, let $\vec{p}_i$ be the
inner normal to $F_i$. Then $\sigma^\vee$ is the positive linear
hull of the finite set $\{\vec{p}_i\}$, that is, $\sigma^\vee =
\langle \vec{p}_1, \ldots, \vec{p}_r\rangle$, where $r$ is the
number of facets of $\sigma$. Further, $\sigma^\vee$ is itself
$n$-dimensional and strongly convex (see Figure~3).

Let $\Sigma \subseteq N_\mathbb{R}$ be a fan of strongly convex
cones. For any $\sigma \in \Sigma$, we defined $L_\sigma = \sigma
\cap N$; now, dually, set $L^\sigma = \sigma^\vee \cap M$.  For
$\sigma$, $\tau \in \Sigma$, consider the affine toric varieties
$X_{\sigma^\vee}$ and $X_{\tau^\vee}$
--- again, notice that we are passing to duals. If $\tau$ is a face
of $\sigma$, then $L^\sigma \subseteq L^\tau$, so that
$k[L^\sigma]$ is a subalgebra of $k[L^\tau]$, and
$X_{\sigma^\vee}$ is an open subset of $X_{\tau^\vee}$. Thus, in
general, the common face $\sigma \cap \tau$ corresponds to a
common open subset $X_{(\sigma \cap \tau)^\vee}$ of
$X_{\sigma^\vee}$ and $X_{\tau^\vee}$, along which we can glue
these two affine varieties together. The fan $\Sigma$, then,
encodes a recipe to patch together the affine pieces
$X_{\sigma^\vee}$ corresponding to the \emph{dual} cones of cones
$\sigma \in \Sigma$, thus yielding (the most general instance of)
a \emph{toric variety}, denoted $X_\Sigma$. This construction can
be carried out without passing to duals, and produces an
algebraic variety, say $X^\Sigma$. However, the combinatorics of
the faces of the cones in $\Sigma$ would be contravariant with
respect to the structure of subvarieties of $X^\Sigma$ --- that
is,  cones of smaller dimension in $\Sigma$ would correspond to
\emph{larger} subvarieties of $X^\Sigma$. Dualization gives a
covariant correspondence: cones of smaller dimension in $\Sigma$
correspond to smaller subvarieties of $X_\Sigma$.

Building on this basic construction, it is possible to carry out
an extensive translation of algebro-geometric concepts into
fan-theoretical language, and vice-versa. This has been done in
the last decades. While we refer to \cite{ewa}, \cite{oda} and
\cite{ful} for further information, we collect the very first few
entries of the fan-toric dictionary in Table~1.

%%%%%%%%%%% FINIS TORIC VARIETIES %%%%%%%

\begin{thebibliography}{99}

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\newblock Springer-Verlag, 1994.

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\bibitem{ewa}
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\end{thebibliography}

\end{document}


