Let $X$ be a complex toric variety and let $X_\mathbb R$ be its real part, that is the $X_\mathbb R$ consists of all the real valued points of $X$.
I would like to learn a little more about the real part and would appreciate some references.
Specifically I was looking at the following theorem (Theorem 4.1.3 of this book)-
Theorem : We have the exact sequence $$M\to \bigoplus_{\rho\in\Delta(1)}\mathbb Z D_\rho\to\text{ Cl }(X)\to 0$$ where the first map is $m\to\text{ div }(\chi^m)$ and the second sends a $T_N$ - invariant divisor to its divisor class in $\text{ Cl }(X)$. Furthermore, we have a short exact sequence $$0\to M\to\bigoplus_{\rho\in\Delta(1)}\mathbb Z D_\rho\to\text{ Cl }(X)\to 0$$ iff $\{ u_\rho\ |\ \rho\in\Delta(1)\}$ spans $N\otimes_\mathbb Z \mathbb R$.
Where $N$ is a lattice, $M$ is its dual, $T_N=\text{ Hom}_\mathbb Z(M,\mathbb C^*)$ is the torus, $\Delta$ is a fan in $N$, $\Delta(1)$ is the set of one dimensional cones (edges) in $\Delta$, $u_\rho$ is the primitive vector along the edge $\rho$, $D_\rho$ is the $T_N$ - invariant prime divisor given by the closure of the orbit corresponding to $\rho$, and $X$ is the toric variety given by the fan $\Delta$.
Now my question is whether I can prove this theorem for $X_\mathbb R$? For this I need a proper understanding of what prime divisors and the divisor class etc are. For example are prime divisors in $X_\mathbb R$ the real points of prime divisors of $X$ or are there other prime divisors of $X_\mathbb R$? I haven't been very successful in finding reading material online for this (real part of a toric variety) and would appreciate some suggestions. Fulton's book has very little.
Thank you.