Kummer surfaces associated with group schemes

TL;DR

This paper constructs and characterizes special Kummer surfaces in characteristic two, linking them to supersingular K3 surfaces with specific configurations and their role as coverings for Enriques surfaces.

Contribution

It introduces a new class of Kummer surfaces associated with group schemes in characteristic two and characterizes their geometric and algebraic properties.

Findings

Kummer surfaces are supersingular K3 surfaces with Artin invariant ≤ 3.

They have a configuration of thirty curves that characterizes them.

These surfaces serve as normal K3-like coverings for certain Enriques surfaces.

Abstract

We introduce Kummer surfaces X=Km(CxC) 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\leq 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.