We consider the construction of Calabi-Yau varieties recently generalized to
where the defining equations may have negative degrees over some projective
space factors in the embedding space. Within such "generalized complete
intersection" Calabi-Yau ("gCICY") three-folds, we find several sequences of
distinct manifolds. These include both novel elliptic and K3-fibrations and
involve Hirzebruch surfaces and their higher dimensional analogues. En route,
we generalize the standard techniques of cohomology computation to these
generalized complete intersection Calabi-Yau varieties.
