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

Abstract

Abstract 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.

Keywords

• Differential and Algebraic Geometry
• F-Theory