Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

Abstract

Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $X_{\overline {K}}$ has infinitely many rational curves or X has infinitely many unirational specialisations.

Keywords

• 11G18
• 14J28
• 11G10