2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71214Consider the diagonal action of the projective group $\PGL_3$ on $n$ copies of ${\mathbb P}^2$. In addition, consider the action of the symmetric group $Σ_n$ by permuting the copies. In this paper we find a set of generators for the invariant field of the combined group $Σ_n \times \PGL_3$. As the main application, we obtain a reconstruction principle for point configurations in ${\mathbb P}^2$ from their sub-configurations of five points. Finally, we address the question of how such reconstruction principles pass down to subgroups.Commutative AlgebraAlgebraic Geometry68U;14Ltext