On the prevalence of elliptic and genus one fibrations among toric hypersurface Calabi-Yau threefolds

We systematically analyze the fibration structure of toric hypersurface Calabi-Yau threefolds with large and small Hodge numbers. We show that there are only four such Calabi-Yau threefolds with h(1,1) 140 or h(2,1) 140 that do not have manifest elliptic or genus one fibers arising from a fibration of the associated 4D polytope. There is a genus one fibration whenever either Hodge number is 150 or greater, and an elliptic fibration when either Hodge number is 228 or greater. We find that for small h(1,1) the fraction of polytopes in the KS database that do not have a genus one or elliptic fibration drops exponentially as h(1,1) increases. We also consider the different toric fiber types that arise in the polytopes of elliptic Calabi-Yau threefolds.