Enriques Surface - Characteristic 2

Characteristic 2

In characteristic 2 there are some new families of Enriques surfaces, sometimes called quasi Enriques surfaces or non-classical Enriques surfaces or (super)singular Enriques surfaces. In characteristic 2 the definition of Enriques surfaces is modified: they are defined to be minimal surfaces whose canonical class K is numerically equivalent to 0 and whose second Betti number is 10. (In characteristics other than 2 this is equivalent to the usual definition.) There are now 3 families of Enriques surfaces:

• Classical: dim(H1(O)) = 0. This implies 2K=0 but K is nonzero, and Picτ is Z/2Z. The surface is a quotient of a reduced singular Gorenstein surface by the group scheme μ₂.
• Singular: dim(H1(O)) = 1 and is acted on non-trivially by the Frobenius endomorphism. This implies K=0, and Picτ is μ₂. The surface is a quotient of a K3 surface by the group scheme Z/2Z.
• Supersingular: dim(H1(O)) = 1 and is acted on trivially by the Frobenius endomorphism. This implies K=0, and Picτ is α₂. The surface is a quotient of a reduced singular Gorenstein surface by the group scheme α₂.

All Enriques surfaces are elliptic or quasi elliptic.