\operatorname{Cl}(X)$.

A toric variety $X=X(\Si)$ is called *non-degenerate* if the support $|\Si|$ spans $N_{\R}$: in particular this means that it cannot admit torus factors, or, equivalently, that $H^0(X,\cO_X^*)\cong\C^*$. Then, the cardinality of the 1-skeleton is given by $$|\Si(1)|=n+r$$ where $r:=\operatorname{rk}\operatorname{Pic}(X)\geq 1$ is the *Picard number of* $X$, also called *the rank of $X$*, in the following.

\[def:Fmatrice\] An *$F$–matrix* is a $n\times (n+r)$ matrix $V$ with integer entries, satisfying the conditions:

-   $\operatorname{rk}(V)=n$;

-   $V$ is *$F$–complete* i.e. $\langle V\rangle=N_{\R}\cong\R^n$ ;

-   all the columns of $V$ are non zero;

-   if ${\bf  v}$ is a column of $V$, then $V$ does not contain another column of the form $\lambda  {\bf  v}$ where $\lambda>0$ is a real number.

A $F$–matrix $V$ is called *reduced* if every column of $V$ is composed by coprime entries .

For instance, a fan matrix of a complete toric variety $X(\Si)$ is always a reduced $F$–matrix.

### Notation {#sssez:SF}

Given a reduced $F$-matrix $V$, in the following $\SF(V)$ will denote the set of all complete and simplicial fans whose 1-skeleton is given by all the rays generated by the columns of $V$. Moreover, $$\P\SF(V)\subset\SF(V)$$ will denote the subset of those fans whose associated toric variety 