Zariski K3 surfaces

TL;DR

This paper constructs Zariski K3 surfaces with low Artin invariants across various characteristics, demonstrating that many supersingular Kummer surfaces are Zariski under certain conditions, using diverse algebraic geometry techniques.

Contribution

It introduces new constructions of Zariski K3 surfaces with specific Artin invariants and characterizes when supersingular Kummer surfaces are Zariski, expanding understanding of their properties.

Findings

Constructed Zariski K3 surfaces of Artin invariant 1, 2, and 3.

Proved that supersingular Kummer surfaces are Zariski if characteristic ≠ 1 mod 12.

Combined methods like quotients by α_p, Kummer surfaces, and automorphisms.

Abstract

We construct Zariski K3 surfaces of Artin invariant 1, 2 and 3 in many characteristics. In particular, we prove that any supersingular Kummer surface is Zariski if the characteristic is not congruent to 1 modulo 12. Our methods combine different approaches such as quotients by the group scheme $\alpha_p$, Kummer surfaces, and automorphisms of elliptic and hyperelliptic curves.