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I am looking for a reference for the following admittedly imprecise statement:

Any projective invariant of n points in the projective plane may be expressed as a function of well-chosen cross-ratios.

By projective invariant I mean a rational function defined on the set of $n$-tuple of distinct points in the projective plane on an arbitrary field $K$, invariant under the action of the projective group of transformations. This is a folklore result that is often stated without proof (e.g. in Efimov, higher geometry) but I can't find a reference providing a precise statement and a proof.

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    $\begingroup$ Maybe I am missing something, but cross-ratio is usually defined for $4$ points on a line. Here you have $n$ general points in the projective plane, so I expect that non even three of them are collinear. What cross-ratios are you talking about? $\endgroup$
    – Francesco Polizzi
    Commented May 4, 2021 at 8:05
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    $\begingroup$ @Polizzi Well, that's part of the question. For five points $A_1$... $A_5$, an explicit formula is to be found in the book of Efimov, higher geometry. The cross ratios involved depend on $A_1$, $A_5$ and one of the intersection points of the lines $A_2A_3$, $A_2A_4$ and $A_3A_4$ with $A_1A_5$. $\endgroup$
    – coudy
    Commented May 4, 2021 at 8:32

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I believe the precise statement you are looking for is proposition 3.3 on page 26 of Danylo Radchenko's doctoral dissertation, Higher cross-ratios and geometric functional equations for polylogarithms (Rheinischen Friedrich-Wilhelms-Universität Bonn, 2016): the proof of the proposition explains how to view the variables of the invariant field as projected cross-ratios of the points.

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    $\begingroup$ I struggled a bit to find a geometric interpretation of the quantities $\langle \{2\} \mid 1,3,5,4 \rangle$ considered in the reference. Now I understand that this is the cross-ratio of the four lines from $M_1$ to the other points $M_2$, $M_3$, $M_4$ and $M_5$. Fine! $\endgroup$
    – coudy
    Commented May 5, 2021 at 16:40

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