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 $p\not\equiv 1$ mod 12. Our methods combine different approaches such as quotients by the group scheme $\alpha\_p$, Kummer surfaces, and automorphisms of hyperelliptic curves.
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