Enriques surfaces with finite automorphism group in positive characteristic
We classify Enriques surfaces with smooth K3 cover and finite automorphism group in arbitrary positive characteristic. The classification is the same as over the complex numbers except that some types are missing in small characteristics. Moreover, we give a complete description of the moduli of these surfaces. Finally, we realize all types of Enriques surfaces with finite automorphism group over the prime fields F p and Q whenever they exist. curve with self-intersection (−2), and a
... ), and a codimension 1 subvariety of the boundary of the period domain parametrizing Coble surfaces, that is, smooth rational surfaces X with | − K X | = ∅ and | − 2K X | = ∅. Both of these codimension 1 subvarieties are rational [DK13]. Thus, one expects that a 1-dimensional family of Enriques surfaces degenerates to a Coble surface at some point. We will also see this kind of behaviour for our examples in positive characteristic. In positive and mixed characteristic, a similar picture has been established by C. Liedtke in [Lie15] and T. Ekedahl, J. Hyland and N. Shepherd-Barron in [EHS12]: The moduli space of Cossec-Verra polarized Enriques surfaces is a quasi-separated Artin stack of finite type over Spec Z, which is irreducible, unirational, 10-dimensional, smooth in odd characteristics and consists of two connected components with these properties in characteristic 2. These two connected components parametrize singular and classical Enriques surfaces, respectively. Their 9dimensional intersection parametrizes supersingular Enriques surfaces. The stack of unpolarized Enriques surfaces is very badly behaved [Lie15, Remark 5.3], because the automorphism group of a generic Enriques surface X is infinite and, moreover, discrete unless X is supersingular or an exceptional [ES04] and classical Enriques surface in characteristic 2. The automorphism group of a general complex Enriques surfaces was computed by W. Barth and C. Peters [BP83], independently also by V. V. Nikulin [Nik83], and is equal to the 2congruence subgroup of the group of positive-cone-preserving automorphisms of the E 10 lattice. However, an Enriques surface may acquire additional (−2)-curves under specializations, causing the automorphism group to become smaller. Therefore, it is a natural question whether this group can degenerate to a finite group. Enriques surfaces with finite automorphism group Convention 2.2. From now on, we will drop the "with smooth K3 cover", and we will always assume that the Enriques surfaces we talk about have such a cover. Definition 2.3. An elliptic fibration (with base curve P 1 ) of a smooth surfaceX is a surjective morphismπ :X → P 1 such that almost all fibers are smooth genus 1 curves,π * OX = O P 1 and no fiber contains a (−1)-curve. We do not require thatπ has a section. 597 G. Martin Proposition 2.4 (Bombieri and Mumford [BM76, Theorem 3]). Every Enriques surface admits an elliptic fibration. The reason why we do not assume that elliptic fibrations have a section is that this is never the case for Enriques surfaces. Proposition 2.5 (Cossec and Dolgachev [CD89, Theorems 5.7.2, 5.7.5 and 5.7.6]). Let π be an elliptic fibration of an Enriques surfaces. Then, if char(k) = 2, then π has exactly two tame double fibers, both of which are either of multiplicative type or smooth, and if char(k) = 2, then π has exactly one wild double fiber, which either is of multiplicative type or is a smooth ordinary elliptic curve. Remark 2.6. Since being supersingular is an isogeny-invariant, one can check the type of the double fiber on the K3 cover.