Sections of Calabi-Yau threefolds with K3 fibration

TL;DR

This paper investigates sections of K3-fibered Calabi-Yau threefolds, demonstrating the existence of infinitely many isolated sections and non-finitely generated Néron-Severi groups, with implications for K3 surfaces over function fields.

Contribution

It provides new examples of Calabi-Yau threefolds with infinitely many isolated sections and non-finitely generated Néron-Severi groups, advancing understanding of their geometric and arithmetic properties.

Findings

Existence of infinitely many isolated sections on certain Calabi-Yau threefolds

The subgroup generated by these sections is not finitely generated

Examples of K3 surfaces over function fields with Zariski dense rational points

Abstract

We study sections of a Calabi-Yau threefold fibered over a curve by K3 surfaces. We show that there exist infinitely many isolated sections on certain K3 fibered Calabi-Yau threefolds and the subgroup of the N\'eron-Severi group generated by these sections is not finitely generated. This also gives examples of K3 surfaces over the function field F of a complex curve with Zariski dense F-rational points, whose geometric model is Calabi-Yau.