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

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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 $h^{1, 1} \geq 140$ or $h^{2, 1} \geq 140$ 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.