There is no Enriques surface over the integers
Abstract
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 \'etale 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 Shepherd-Barron, 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.
Cite
@article{arxiv.2004.07025,
title = {There is no Enriques surface over the integers},
author = {Stefan Schröer},
journal= {arXiv preprint arXiv:2004.07025},
year = {2022}
}
Comments
52 page; error in Proposition 7.1 and handling of weights in Section 7 corrected; proof for Proposition 9.5 simplified; parts of Section 14 moved to Section 12; a major mistake concerning two-sections corrected, by using Nikulin's theory of discriminant forms; many other corrections and improvements of minor nature; to appear in Ann. of Math