Existence of supersingular reduction for families of K3 surfaces with large Picard number in positive characteristic

TL;DR

This paper proves that certain non-isotrivial families of K3 surfaces in positive characteristic with high Picard number and specific height conditions have potential supersingular reduction, advancing understanding of their geometric properties.

Contribution

It demonstrates potential supersingular reduction for families of K3 surfaces with large Picard number and height in positive characteristic, using moduli space techniques and deformation theory.

Findings

Families with $ ho \,\geq\, 21-2h$ have potential supersingular reduction.

Constructs non-isotrivial K3 families with $ ho=22-2h$ for large $p$ and $h$ between 2 and 10.

Utilizes Maulik's results, Hodge bundle sections, and deformation methods.

Abstract

We study non-isotrivial families of $K3$ surfaces in positive characteristic $p$ whose geometric generic fibers satisfy $\rho\geq21-2h$ and $h\geq3$, where $\rho$ is the Picard number and $h$ is the height of the formal Brauer group. We show that, under a mild assumption on the characteristic of the base field, they have potential supersingular reduction. Our methods rely on Maulik's results on moduli spaces of $K3$ surfaces and the construction of sections of powers of Hodge bundles due to van der Geer and Katsura. For large $p$ and each $2\leq{h}\leq10$, using deformation theory and Taelman's methods, we construct non-isotrivial families of $K3$ surfaces satisfying $\rho=22-2h$.