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Suppose I am given a projective plane $P \cong \mathbb{P}^2(k)$ over a (commutative) field $k$.

With "projective plane," I mean the point-line geometry (and not, for instance, the scheme): $k$-rational points and $k$-rational lines.

Is there a combinatorial characterization (only using the points and lines), which tells me that $k$ is algebraically closed ?

If I would have asked that $k$ is a division ring, then Pappus's Theorem tells us when precisely $k$ is commutative through a configurational property.

So I wonder whether similar things can be formulated for the case when $k$ would be algebraically closed.

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    $\begingroup$ In principle, yes, since one can encode algebraic equations using projective line arrangements. $\endgroup$
    – Moishe Kohan
    Commented Jul 19, 2021 at 23:59
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    $\begingroup$ @MoisheKohan Do you know a reference for that? $\endgroup$
    – arsmath
    Commented Jul 20, 2021 at 18:18
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    $\begingroup$ @arsmath It's in Mnev's proof of his universality theorem. I will add a reference later. Mnev works over the reals but the argument is general. $\endgroup$
    – Moishe Kohan
    Commented Jul 20, 2021 at 20:26

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