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

Abstract

We study non-isotrivial families of K3 surfaces in positive characteristic p whose geometric generic fibers satisfy ρ ≥ 21 − 2h and h ≥ 3, where ρ 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 ≤ h ≤ 10, using deformation theory and Taelman's methods, we construct non-isotrivial families of K3 surfaces satisfying ρ = 22 − 2h.