with a semigroup morphism $x:S_{\s}\rightarrow\C$ assigned by thinking of $\C$ as a multiplicative semigroup. In particular the *characteristic morphism* $$\label{caratteristico}
\begin{array}{cccc}
x_{\s}&:\s^{\vee}\cap M & \longrightarrow & \C\\
      &      \uu & \longmapsto & \left\{\begin{array}{cc}
                                         1 & \text{if $\uu\in\s^{\bot}$} \\
                                         0 & \text{otherwise}
                                       \end{array}
      \right.
\end{array}$$ which is well defined since $\s^{\bot}<\s^{\vee}$, defines a *characteristic point* $x_{\s}\in U_{\s}$ whose torus orbit $O_{\s}$ turns out to be a $(n-\dim(\s))$–dimensional torus embedded in $U_{\s}$.

Fans and toric varieties {#ssez:fans&tv}
------------------------

A *fan* $\Si$ is a finite set of cones $\s\subset N_{\R}$ such that

1.  for any cone $\s\in\Si$ and for any face $\tau<\s$ then $\tau\in\Si$,

2.  for any $\s,\tau\in\Si$ then $\s\cap\tau<\s$ and $\s\cap\tau<\tau$.

For every $i$ with $0\leq i\leq n$ denote by $\Si(i)\subset \Si$ the subset of $i$–dimensional cones, called the *$i$–skeleton of $\Si$*.

A fan $\Si$ is called *simplicial* if every cone $\s\in\Si$ is rational and simplicial, and is called *non-singular* if every such cone is non-singular. The *support* of a fan $\Si$ is the subset $|\Si|\subset N_{\R}$ obtained as the union of all of its cones i.e. $$|\Si|:= \bigcup_{\s\in\Si} \s \subset N_{\R}\ .$$ If $|\Si|=N_{\R}$ then $\Si$ will be called *complete*.

Since for any face $\tau <\s$ the semigroup $S_{\s}$ turns out 