Surfaces of type K3 over fields of finite characteristic

MODULI CURVES OF SUPERSINGULAR K3 SURFACES IN CHARACTERISTIC 2 WITH ARTIN INVARIANT 2

“…In other words, the automorphism group Aut(X GA [λ] ) of the non-polarized supersingular K3 surface X GA [λ] jumps at λ = η, even though the numerical Néron-Severi lattice of X GA [λ] is constant around λ = η. Note that the automorphism group of a supersingular K3 surface is always embedded into the orthogonal group of its numerical Néron-Severi lattice [8,§ 8,Proposition 3].…”

A $1$-dimensional family of Enriques surfaces in characteristic $2$ covered by the supersingular $K3$ surface with Artin invariant $1$

“…Recall that the moduli space of supersingular K3 surfaces is 9-dimensional and is stratified by Artin invariant σ, 1 ≤ σ ≤ 10. Each stratum has dimension σ − 1 (Artin [1], Rudakov-Shafarevich [20]). …”

On Kummer type construction of supersingular K3 surfaces in characteristic 2

“…Even though the moduli curve of marked supersingular K 3 surfaces with Artin invariant ≤ 2 is constructed [Rudakov and Shafarevich 1981;Ogus 1983], it is not separated. Hence the existence of the complete family of Schröer's Kummer surfaces of dimension 1 does not imply Theorem 1.2 immediately.…”

“…The main ingredient of the proof is the following structure theorem for Néron-Severi lattices of supersingular K 3 surfaces: Theorem 1.3 [Rudakov and Shafarevich 1981]. Let X and X be supersingular K 3 surfaces defined over the same algebraically closed field.…”

“…If p = 2, then NS(X ) is of type I [Rudakov and Shafarevich 1981, Proposition in Section 5]. Hence the classification theorem of even hyperbolic p-elementary lattices [Rudakov and Shafarevich 1981, Theorem in Section 1] implies Theorem 1.3.…”