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

TL;DR

This paper investigates the fibration structures of toric hypersurface Calabi-Yau threefolds, revealing that most have elliptic or genus one fibrations, especially at higher Hodge numbers, with only a few exceptions.

Contribution

It provides a systematic analysis of fibration prevalence in toric hypersurface Calabi-Yau threefolds, identifying thresholds for elliptic and genus one fibrations based on Hodge numbers.

Findings

Only four Calabi-Yau threefolds with large Hodge numbers lack manifest fibrations.

Genus one fibrations occur when Hodge numbers reach 150 or more.

Elliptic fibrations are present when Hodge numbers reach 228 or more.

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 $h^{1, 1} \geq 140$ 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. We also consider the different toric fiber types that arise in the polytopes of elliptic Calabi-Yau threefolds.