Picard rank jumps for families of K3 surfaces in positive characteristic

TL;DR

This paper demonstrates that under certain conditions, a family of K3 surfaces over a curve in positive characteristic exhibits infinitely many fibers with increased Picard rank, revealing new geometric phenomena related to Frobenius and monodromy.

Contribution

It establishes conditions under which Picard rank jumps occur infinitely often in families of K3 surfaces over positive characteristic fields.

Findings

Infinitely many fibers have larger Picard rank than the generic fiber.

Picard rank jumps are linked to Frobenius obstructions and monodromy conditions.

The results apply to non-isotrivial families over characteristic p > 2.

Abstract

Let X/C be a non iso-trivial family of K3 surfaces over a curve C defined over characteristic p > 2 field. We show that if X avoids a necessary and structural obstruction coming from Frobenius, and satisfies a big monodromy condition, then there are infinitely may geometric fibers that have larger Picard rank than the geometric generic fiber.