Kummer surfaces associated with group schemes

Abstract

We introduce Kummer surfaces $$X={\text {Km}}(C\times C)$$ with the group scheme $$G=\mu _2$$ acting on the self-product of the rational cuspidal curve in characteristic two. The resulting quotients are normal surfaces having a configuration of sixteen rational double points of type $$A_1$$ , together with a rational double point of type $$D_4$$ . We show that our Kummer surfaces are precisely the supersingular K3 surfaces with Artin invariant $$\sigma \le 3$$ , and characterize them by the existence of a certain configuration of thirty curves. After contracting suitable curves, they also appear as normal K3-like coverings for simply-connected Enriques surfaces.