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March 2018 Existence of supersingular reduction for families of $K3$ surfaces with large Picard number in positive characteristic
Kazuhiro Ito
Hiroshima Math. J. 48(1): 67-79 (March 2018). DOI: 10.32917/hmj/1520478024

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 $\rho = 22 - 2h$.

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Kazuhiro Ito. "Existence of supersingular reduction for families of $K3$ surfaces with large Picard number in positive characteristic." Hiroshima Math. J. 48 (1) 67 - 79, March 2018. https://doi.org/10.32917/hmj/1520478024

Information

Received: 21 December 2016; Revised: 16 June 2017; Published: March 2018
First available in Project Euclid: 8 March 2018

zbMATH: 06901788
MathSciNet: MR3772001
Digital Object Identifier: 10.32917/hmj/1520478024

Subjects:
Primary: 14J28
Secondary: 14C22, 14D05

Rights: Copyright © 2018 Hiroshima University, Mathematics Program

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Vol.48 • No. 1 • March 2018
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