We study the elliptic fibrations of some Calabi–Yau threefolds, including the Z2×Z2 orbifold with (h1,1,h2,1)=(27, 3), which is equivalent to the common framework of realistic free-fermion models, as well as related orbifold models with (h1,1,h2,1)=(51, 3) and (31, 7). However, two related puzzles arise when one considers the (h1,1,h2,1)=(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.
