Then its subgroup of *torus–invariant Weil divisors* is given by $$\mathcal{W}_T(X)=\left\langle D_{\rho } \mid \rho \in \Sigma (1)\right\rangle_{\Z} =
    \bigoplus_{\rho \in \Sigma (1)}\Z\cdot D_{\rho }$$ where $D_{\rho}=\overline{\T\cdot x_\rho}$, being $\T\cong\operatorname{Hom}(N,\C^*)$ the acting torus and $x_\rho$ the distinguished point of $\rho$, as defined in (\[caratteristico\]). Let $\mathcal{P}(X)\subset\mathcal{W}(X)$ be the subgroup of *principal divisors* and $\v_{\rho}$ be the generator of the monoid $\rho\cap N$. Then the morphism $$\label{div}
\begin{array}{llll}
div : & M & \longrightarrow & \mathcal{P}(X)\cap \mathcal{W}_{T}(X)=:
\mathcal{P}_{T}(X) \\
& \uu & \longmapsto & div(\uu):=\sum_{\rho \in \Sigma (1)}\langle \uu,\v_{\rho }\rangle
D_{\rho }
\end{array}$$ is surjective. Let $V=(\v_1,\ldots,\v_{n+r})$ be a fan matrix of $\Si$, with respect to a chosen identification $N\cong\Z^n$. Then the transposed matrix $V^T$ is a representative matrix of the $\Z$-linear morphism $div$ defined in (\[div\]), with respect to the basis $\{D_1,\ldots,D_{n+r}\}$ of $\mathcal{W}_T(X)$.

Let $\operatorname{Pic}(X)$ be the group of line bundles modulo isomorphism. It is well known that for an *irreducible* variety $X$ the map $D\mapsto\mathcal{O}_X(D)$ induces an isomorphism $\mathcal{C}(X)/\mathcal{P}(X)\cong\operatorname{Pic}(X)$, where $\mathcal{C}(X)\subset\mathcal{W}(X)$ denotes the subgroup of Cartier divisors. The divisor class group is defined as the group of Weil divisors modulo rational (hence linear) equivalence, i.e. $\operatorname{Cl}(X):=\mathcal{W}(X)/\mathcal{P}(X)$. Then the inclusion $\mathcal{C}(X)\subset\mathcal{W}(X)$ passes through the quotient giving an immersion $\operatorname{Pic}(X)\hookrightarrow 