## Hiroshima Mathematical Journal

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

Kazuhiro Ito

#### Abstract

We study non-isotrivial families of $K3$ surfaces in positive characteristic $p$ whose geometric generic fibers satisfy $\rho \ge 21 - 2h$ and $h \ge 3$, 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 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 $2 \le h \le 10$, using deformation theory and Taelman’s methods, we construct non-isotrivial families of $K3$ surfaces satisfying $\rho = 22 - 2h$.

#### Article information

Source
Hiroshima Math. J., Volume 48, Number 1 (2018), 67-79.

Dates
Revised: 16 June 2017
First available in Project Euclid: 8 March 2018

https://projecteuclid.org/euclid.hmj/1520478024

Digital Object Identifier
doi:10.32917/hmj/1520478024

Mathematical Reviews number (MathSciNet)
MR3772001

Zentralblatt MATH identifier
06901788

#### Citation

Ito, Kazuhiro. Existence of supersingular reduction for families of $K3$ surfaces with large Picard number in positive characteristic. Hiroshima Math. J. 48 (2018), no. 1, 67--79. doi:10.32917/hmj/1520478024. https://projecteuclid.org/euclid.hmj/1520478024