We have defined the toric variety obtained by a fan in N_R, which can be view as a fan in N’_R, where N’ is the sublattice of N generated by the lattice points of the fan. In section 3.3, we first realize the relation between the two toric varieties w.r.t. two lattices. Next we consider the refinement of a fan, and we may find something interesting between the two toric varieties w.r.t. two fans. In the section 3.4, we will concentrate on the complete varieties and the proper morphism, and compare with the similar concept in the classical topology. In the end, we will see a criterion to determine whether a morphism is a proper morphism.

Reference:

Cox, Little, Schenck, Toric Varieties