There is no Enriques surface over the integers

We show that there is no family of Enriques surfaces over the ring of integers. This extends non-existence results of Minkowski for families of finite e ' tale schemes, of Tate and Ogg for families of elliptic curves, and of Fontaine and Abrashkin for families of abelian varieties and more general smooth proper schemes with certain restrictions on Hodge numbers. Our main idea is to study the local system of numerical classes of invertible sheaves. Among other things, our result also hinges on counting rational points, Lang's classification of rational elliptic surfaces in characteristic two, the theory of exceptional Enriques surfaces due to Ekedahl and ShepherdBarron, some recent results on the base of their versal deformation, Shioda's theory of Mordell-Weil lattices, and an extensive combinatorial study for the pairwise interaction of genus-one fibrations.

Automated summary

This article proves the non-existence of families of Enriques surfaces over the ring of integers, expanding on existing non-existence results for families of various geometric objects. The author's main approach involves studying the local system of numerical classes of invertible sheaves, and the results also depend on counting rational points, classification of rational elliptic surfaces, theory of exceptional Enriques surfaces, theory of Mordell-Weil lattices, and combinatorial study on genus-one fibrations.