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We study the elliptic fibrations of some Calabi-Yau three-folds, includingthe $Z_2\times Z_2$ orbifold with $(h_{1,1},h_{2,1})=(27,3)$, which isequivalent to the common framework of realistic free-fermion models, as well asrelated orbifold models with $(h_{1,1},h_{2,1})=(51,3)$ and (31,7). However,two related puzzles arise when one considers the $(h_{1,1},h_{2,1})=(27,3)$model as an F-theory compactification to six dimensions. The condition for thevanishing of the gravitational anomaly is not satisfied, suggesting that theF-theory compactification does not make sense, and the elliptic fibration iswell defined everywhere except at four singular points in the base. Wespeculate on the possible existence of N=1 tensor and hypermultiplets at thesepoints which would cancel the gravitational anomaly in this case.Comment: 19 pages. Standard Latex. The lack of global section for the (27,3)  model is emphasized, as well as a possible relation with the gravitational  anomal

ON ELEVATING FREE-FERMION Z2×Z2 ORBIFOLDS MODELS TO COMPACTIFICATIONS OF F THEORY

ON ELEVATING FREE-FERMION Z₂×Z₂ ORBIFOLDS MODELS TO COMPACTIFICATIONS OF F THEORY