from application fulton
The class group Cl(X) of Weil divisors on the toric variety defined by the fan is a finitely generated abelian group of rank N_RAYS-DIM. It usually contains torsion. The rational divisor class group is the tensor product of Cl(X) with Q over Z. This group is torsion free and corresponds to the Picard group if the variety is non-singular.
Properties from algebraic geometry.
LIFTINGLifts a divisor in the class group onto a divisor on the fan.
PROJECTIONMaps a divisor on the variety onto its representation in the class group.