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For a toric variety $X_{\Sigma}$ over the complex numbers associated to a fan $\Sigma$ there is a simple short exact sequence which computes the divisor class group. To each one dimensional cone $\rho$ in the fan is a torus invariant Weil divisor $D_{\rho}$ (which is the closure of the associated torus orbit in $X_{\Sigma}$). The short exact sequence is $M\rightarrow \bigoplus_{\rho\in \Sigma(1)} \mathbb{Z}\cdot D_{\rho}\rightarrow \text{Cl}(X_{\Sigma})\rightarrow 0$ (where $M$ is the character lattice and the sum in the middle is over all rays associated to the fan).

This description relies on the Orbit Cone correspondence which is proved in Cox Little and Schenck's book (which deals only with toric varieties over $\mathbb{C}$). The proof they present does not directly generalize to characteristic zero fields and even though I have seen it mentioned that many theorems carry over to algebraically closed fields it is not clear to me that this description does generalize. Let $\lambda^n$ denote the cocharacter associated to $n$ in the cocharacter lattice. For instance the correspondence requires that the intersection $U_{\sigma_1}\cap U_{\sigma_2}=U_{\sigma_1\cap \sigma_2}$ and this in turn relies on the Proposition that $n$ lies in a cone $\sigma$ if and only if $\lim_{t\rightarrow 0}\lambda^n(t)$ converges in $U_{\sigma}$. This seems to make use of the fact that $\mathbb{C}$ has a suitable topology.

Is there an example of a smooth fan $\Sigma$ and a finite field $\mathbb{F}$ such that the associated toric variety $X_{\Sigma,\mathbb{F}}$ has a Picard group which is not isomorphic to the Picard group of $X_{\Sigma,\mathbb{C}}$, or is it true that the Picard group in this case just depends on the fan and not on the field of definition. Also, what about over $\bar{\mathbb{F}}$?

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