On the Shafarevich conjecture for Enriques surfaces
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Mathematische Zeitschrift
On the Shafarevich conjecture for Enriques surfaces Teppei Takamatsu1 Received: 27 May 2020 / Accepted: 23 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Enriques surfaces are minimal surfaces of Kodaira dimension 0 with b2 = 10. If we work with a field of characteristic away from 2, Enriques surfaces admit double covers which are K3 surfaces. In this paper, we prove the Shafarevich conjecture for Enriques surfaces by reducing the problem to the case of K3 surfaces. In our formulation of the Shafarevich conjecture, we use the notion “admitting a cohomological good K3 cover”, which includes not only good reduction but also flower pot reduction. Keywords Enriques surfaces · Shafarevich conjecture Mathematics Subject Classification 14J28 · 11G35
1 Introduction The Shafarevich conjecture for abelian varieties, which is proved by Faltings [5, VI, §1, Theorem 2] states the finiteness of isomorphism classes of abelian varieties of a fixed dimension over a fixed number field admitting good reduction away from a fixed finite set of finite places. In [11], Javanpeykar and Loughran conjectured that the Shafarevich conjecture hold for more general families of varieties. Moreover, they showed that the Lang–Vojta conjecture on integral points of hyperbolic varieties implies the Shafarevich conjecture for hypersurfaces and complete intersections of general type [11, Theorem 1.5]. The Shafarevich conjecture is proved in many cases (e.g. for del Pezzo surfaces [21]), certain Fano threefolds [12], proper hyperbolic polycurves [10,19], hypersurfaces in abelian varieties [15], but it is still open in general. Enriques surfaces are one of the deformation classes of minimal surfaces of Kodaira dimension 0. We note that the Shafarevich conjecture holds for other classes of minimal surfaces of Kodaira dimension 0 (abelian surfaces, K3 surfaces [22,23], and bielliptic surfaces [24]). The purpose of this paper is to formulate and prove this conjecture for Enriques surfaces. Our main theorem is the following.
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Teppei Takamatsu [email protected] Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
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Theorem 1.1 (Theorem 3.7) Let R be an integrally closed Z-finitely generated integral domain of characteristic zero with fraction field F. Then, the set ⎫ ⎧ ⎬ ⎨ X : Enriques surface over F, /F-isom X for any height 1 prime ideal p ∈ Spec R, ⎭ ⎩ X admits a cohomological good K3 cover at p is finite. Here, we say that X admits a cohomological good K3 cover at p if there exists a K3 double cover of X whose -adic cohomologies are unramified at p (see Definition 3.2). In particular, we prove the original Shafarevich conjecture for Enriques surfaces over finitely generated fields over Q. Corollary 1.2 (Corollary 3.9) Let R be an integrally closed Z-finitely generated integral domain of characteristic zero with fraction field F. Then, the set
X : Enriques surface over F, /F-isom X X
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