Enriques surfaces in characteristic 2 with a finite group of automorphisms

Abstract

Complex Enriques surfaces with a finite group of automorphisms are classified into seven types. In this paper, we determine which types of such Enriques surfaces exist in characteristic 2. In particular we give a 1-dimensional family of classical and supersingular Enriques surfaces with the automorphism group A u t ( X ) \mathrm {Aut}(X) isomorphic to the symmetric group S 5 \mathfrak {S}_5 of degree 5.

Q_l-cohomology projective planes and singular Enriques surfaces in characteristic two

“…To this end, the following table lists the maximal root types R together with the classical Enriques surfaces S with finite automorphism group, in the notation from [12], which support them. Of course, the Enriques surfaces S need not be unique as we are merely concerned with the existence of R. Along the same lines, one can easily verify that the root type R = 2A 4 + A 1 is supported on the onedimensional family of classical and supersingular Enriques surfaces constructed by Katsura and Kondō in [11]. In total this allows us to realize 30 maximal root types on classical Enriques surfaces.…”

“…Note that no matter what field S and X are defined on, we can always specialize to the situation where they are defined over some finite field F q (which we increase, if necessary, to ensure that NS(X) is defined over F q ). Then X is ordinary by [11,Cor. A.2], and since the Tate conjecture holds for X by [1], we infer ρ(X) = 20.…”

“…Remark 1.2. Thirteen of the types in Theorem 1.1 are supported on singular Enriques surfaces with finite automorphism group in characteristic two, as classified in [11], [20] (compare the approach over C in [8]). This settles the existence part of Theorem 1.1 for these types, but not the stated properties.…”

Classification of Enriques surfaces with finite automorphism group in characteristic 2

“…Very recently, the first and the second author [18] determined the existence or nonexistence of Enriques surfaces in characteristic 2 whose dual graphs of all (−2)-curves are of type I, II, . .…”

“…, or VII as in the following Theorem: Theorem 1.1. (Katsura, Kondo [18]) The existence or non-existence of Enriques surfaces in characteristic 2 whose dual graphs of all non-singular rational curves are of type I, II, . .…”