K3 surfaces over small number fields and Kummer constructions in families

TL;DR

This paper constructs K3 surfaces over specific small number fields with good reduction everywhere, introduces a new theory of Kummer constructions in families, and develops methods for resolving singularities in these families.

Contribution

It develops a novel theory of Kummer constructions in families and applies it to construct K3 surfaces over certain quadratic and S_3 number fields.

Findings

Existence of K3 surfaces over three quadratic number fields and an infinite family of S_3-number fields.

Development of a theory of Kummer constructions in families based on Romagny's effective models.

Construction of simultaneous resolutions of singularities in families of K3 surfaces.

Abstract

We construct K3 surfaces over number fields that have good reduction everywhere. These do not exists over the rational numbers, by results of Abrashkin and Fontaine. Our surfaces exist for three quadratic number fields, and an infinite family of $S_3$-number fields. To this end we develop a theory of Kummer constructions in families, based on Romagny's notion of the effective models, here applied to sign involutions. This includes quotients of non-normal surfaces by infinitesimal group schemes in characteristic two, as developed by Kondo and myself. By the results of Brieskorn and Artin, the resulting families of normal K3 surfaces admit simultaneous resolutions of singularities, at least after suitable base-changes. These resolutions are constructed in two ways: First, by blowing-up families of one-dimensional centers that acquire embedded components. Second, by computing various l-adic local systems in terms of representation theory, and invoking Shepherd-Barrons results on the resolution functor, which is representable by a highly non-separated algebraic space.