First order joint differential projective invariants

We present a complete algebraic description of the field of first-order joint projective invariants for configurations of $n$ points in the plane, under the natural diagonal action of the projective group $PGL(3,\mathbb{R})$. For $n>1$, we construct an explicit minimal generating set for the field of absolute invariants and prove its algebraic independence. We further determine the structure of the full field of invariants as a simple algebraic extension of field of absolute invariants, generated by a single primitive relative invariant of weight~$-1$, for which we provide a closed-form expression valid for all $n>1$.