  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 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 the
vanishing of the gravitational anomaly is not satisfied, suggesting that the
F-theory compactification does not make sense, and 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 in this case.
