to be a sub-semigroup of $S_{\tau}$, there is an induced immersion $U_{\tau}\hookrightarrow U_{\s}$ between the associated affine toric varieties which embeds $U_{\tau}$ as a principal open subset of $U_{\s}$. Given a fan $\Si$ one can construct *an associated toric variety $X(\Si)$* by patching all the affine toric varieties $\{U_{\s}\ |\ \s\in\Si \}$ along the principal open subsets associated with any common face. Moreover *for every toric variety $X$ there exists a fan $\Si$ such that $X\cong X(\Si)$*. It turns out that:

-   *$X(\Si)$ is non-singular if and only if the fan $\Si$ is non-singular,*

-   *$X(\Si)$ is complete if and only if the fan $\Si$ is complete.*

Let $\v_\rho$ be a primitive generator of the ray $\rho\in \Si(1)$. Up to an identification $N\cong\Z^n$, where $n:=\dim X$, and setting $m:=|\Si(1)|$ $$V=\left(\v_\rho\,|\,\rho\in\Si(1)\right)=(\v_1\,\cdots\,\v_m)$$ gives a $n\times m$ integer matrix called *a fan matrix of $\Si$*. Notice that $\Si$ determines $V$ up to the choice of a basis of $N$ and of a permutation of columns (i.e. generators $\v_\rho$), that is, $V$ and $V'$ are *equivalent* fan matrices if $$\label{M-equivalenza}
  \exists\,A\in\GL(n,\Z)\,,\ \exists\,B\in\mathfrak{S}_m\leq \GL(m,\Z)\quad V'=A\cdot V\cdot B$$

Divisors on Toric varieties
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Let $\mathcal{W}(X)$ denote the group of Weil divisors of a toric variety $X=X(\Si)$. 