# What are sufficient and necessary conditions to be a generalized Zariski surface over a finite field?

Let $X$ be an absolutely irreducible reduced surface over a finite field $k$ of characteristic $p$. What are sufficient and necessary conditions for $X$ to be a generalized Zariski surface over $k$ (there is a purely inseparable rational dominant map $\varphi\!: \mathbb{P}^2 \dashrightarrow X$ over $k$)?

More precisely, I am interested in the case when $X$ is the Kummer surface of an Jacobian $J$, which is also a generalized Zariski surface over the algebraic closure $\overline{k}$ (there is a purely inseparable rational dominant map $\psi\!: \mathbb{P}^2 \dashrightarrow X$ only over $\overline{k}$).

K3 surfaces are unirational if and only if they have Picard number $22$ (This was a conjecture of M. Artin, now proved by Liedtke, Inv Math 2015). Unirational means that there is a surjective rational map from $\mathbb{P}^2$ to $X$, which is necessarily inseparable. I am not sure if it has to be purely inseparable, though.
• RE: Artin Conjecture  arxiv.org/abs/1904.04803 – Benjamin Dickman Apr 21 '19 at 16:48