On reconstructing configurations of points in \(\mathbb P^{2}\) from a joint distribution of invariants (Q1778105)

The diagonal action of the projective group \(\text{PGL}_3\) on \(n\) copies of \(\mathbb{P}^2\) and the action of the symmetric group \(\Sigma_n\) by permuting the copies are considered. The authors find a set of generators for the invariant field of the combined group \(\Sigma_n\times \text{PGL}_3\). The result yields a reconstruction principle for point configurations in \(\mathbb{P}^2\) from their subconfigurations of five points. Further, this approach is generalized and reconstruction theorems for arbitrary subgroups of \(\text{PGL}_3\) and of area preserving transformations are obtained.