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

On Calabi-Yau generalized complete intersections from Hirzebruch varieties and novel K3-fibrations