Elevating the Free-Fermion $Z_{2} \times Z_{2}$ Orbifold Model to a Compactification of $F$-Theory

We study the elliptic fibrations of some Calabi-Yau three-folds, including the $Z_2\times Z_2$ orbifold with $(h_{1,1},h_{2,1})=(27,3)$, which is equivalent to the common framework of realistic free-fermion models, as well as related models with $(h_{1,1},h_{2,1})=(51,3)$ and $(31,7)$. 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. One is that the condition for the vanishing of the gravitational anomaly is not satisfied. This suggests that either a new feature must appear in the F-theory limit of the corresponding four-dimensional type-IIA vacuum, or that the F-theory compactification does not make sense. However, the elliptic fibration is well defined everywhere except at four singular points in the base. We speculate on the possible existence of N=1 tensor and hypermultiplets at these points which would cancel the gravitational anomaly.