Dualities between K 3 fibered Calabi – Yau three - folds with two

We propose a way to examine N=1 and N=2 string dualties on Calabi-Yau three-folds and their basis or extensions. Our way is to find out or to construct two types of toric representations of a Calabi-Yau three-fold, which contain phases of topologically equivalent or phases connected by flops. We discuss how to find relations among Calabi-Yau three-fold realized in different toric representations. We examine several examples of Calabi-Yau three-folds which have the Hodge numbers, (h, h)=(5, 185) and the various numbers of K3 fibers. We observe that each phase of our examples contains Del Pezzo 4-cycles, B8 in many ways. [email protected]