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

Abstract

We focus on non-isotrivial families of \(K3\) surfaces in positive characteristic \(p\) whose geometric generic fibers satisfy \(\rho \geq 21-2h\) and \(h \geq 3\), 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 potentially supersingular reduction. Our methods depend 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 \(\rho=22-2h\).