A Picard rank bound for base surfaces of elliptic Calabi-Yau 3-folds

TL;DR

This paper establishes an explicit upper bound on the Picard rank of surfaces serving as bases for elliptic Calabi-Yau threefolds, using a new birational geometry approach inspired by F-theory physics.

Contribution

It introduces a novel birational geometric strategy to bound the Picard rank of base surfaces in elliptic Calabi-Yau threefolds, connecting mathematical bounds with physical insights.

Findings

Derived an explicit Picard rank bound for base surfaces.

Developed a new birational geometry method for bounding Picard groups.

Clarified the geometric and physical origins of the boundedness.

Abstract

We compute an explicit rank bound on the Picard group of the compact surfaces, which can serve as the base of an elliptic Calabi-Yau variety with canonical singularities. To bound the Picard rank from above, we develop a novel strategy in birational geometry, motivated in part by the physics of six-dimensional vacua of F-theory as discussed in a companion paper [1] to this one. The derivation of the concrete bound illustrates the strategy and clarifies the origin of the boundedness.