# Projective invariants of the plane and cross ratio

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.

• 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? May 4, 2021 at 8:05
• @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 $A_1A_5$. May 4, 2021 at 8:32

• 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! May 5, 2021 at 16:40