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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 $\overline{k}$).

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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.

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  • $\begingroup$ This is only over an algebraically closed field. $\endgroup$
    – Dimitri Koshelev
    Commented Sep 13, 2016 at 8:43
  • $\begingroup$ @DimaKoshelev Good point. Do you have an example of a surface over a finite field that can only be described as a Zariski surface over an extension field? $\endgroup$
    – Felipe Voloch
    Commented Sep 14, 2016 at 0:51
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    $\begingroup$ I don't have, so I asked this question. $\endgroup$
    – Dimitri Koshelev
    Commented Sep 14, 2016 at 5:56
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    $\begingroup$ RE: Artin Conjecture $$ $$ arxiv.org/abs/1904.04803 $\endgroup$
    – Benjamin Dickman
    Commented Apr 21, 2019 at 16:48
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Is Every K3 Surface With Rho 22 Zariski? Piotr Blass True In Char Two

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