For a toric variety $ X _ {Sigma} $ about the complex numbers associated with a fan $ Sigma $ There is a simple short and exact sequence that calculates the divisor class group. To each dimensional cone $ rho $ in the fan there is a divisor of Weil invariant of the bull $ D _ { rho} $ (which is the closure of the associated bull orbit in $ X _ {Sigma} $) The exact short sequence is

$ M rightarrow bigoplus _ { rho in Sigma (1)} mathbb {Z} cdot D _ { rho} rightarrow text {Cl} (X _ { Sigma}) rightarrow 0 $ (where $ M $ it is the network of characters and the sum in the middle is over all the rays associated with the fan).

This description is based on the correspondence of Orbit Cone which is tested in the book by Cox Little and Schenck (which deals only with toric varieties on $ mathbb {C} $) The proof they present does not directly generalize to the characteristic zero fields and, although I have seen that many theorems move to algebraically closed fields, it is not clear to me that this description does generalize. Leave $ lambda ^ n $ denote the cocaracter associated with $ n $ in the network of characters. For example, correspondence requires that the intersection $ U _ { sigma_1} cap U _ { sigma_2} = U _ { sigma_1 cap sigma_2} $ and this in turn is based on the Proposition that $ n $ is in a cone $ sigma $ If and only if $ lim_ {t rightarrow 0} lambda ^ n (t) $ converges on $ U _ {sigma} $. This seems to make use of the fact that $ mathbb {C} $ It has an appropriate topology.

Is there any example of a soft fan? $ Sigma $ and a finite field $ mathbb {F} $ such that the associated toric variety $ X _ { Sigma, mathbb {F}} $ has a Picard group that is not isomorphic to the Picard group of $ X _ { Sigma, mathbb {C}} $, or it is true that the Picard group in this case only depends on the fan and not on the definition field. Also, what about more than $ bar { mathbb {F}} $?