Jumping deformations of complete toric varieties

We construct one-parameter complex analytic families whose special fibers are complete toric varieties. Under appropriate assumptions, the general fibers of these families also become toric varieties, and the corresponding fans are explicitly described by the data of the fans associated to the special fibers. Using these families, we construct a deformation family for a certain toric weakened Fano 3-fold. Moreover, we obtain certain examples of toric weakened Fano 4-folds.