Hiroshima Mathematical Journal
- Hiroshima Math. J.
- Volume 48, Number 1 (2018), 67-79.
Existence of supersingular reduction for families of $K3$ surfaces with large Picard number in positive characteristic
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 $\rho = 22 - 2h$.
Hiroshima Math. J., Volume 48, Number 1 (2018), 67-79.
Received: 21 December 2016
Revised: 16 June 2017
First available in Project Euclid: 8 March 2018
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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