On the classification of certain geproci sets (Q6645678)

Recall that a set of points \(Z \subset \mathbb{P}^{N}_{\mathbb{C}}\) with \(N\geq 3\) is geproci if its general projection onto a hyperplane is a complete intersection. So far we know geproci sets only in \(\mathbb{P}^{3}_{\mathbb{C}}\), they project onto a plane where their images are the intersection points of two curves of degrees \(a\) and \(b\). One can assume that \(a\leq b\), and we refer to such sets as \((a,b)\)-geproci. We say that a non-trivial \((a,b)\)-geproci set \(Z \subset \mathbb{P}^{3}_{\mathbb{C}}\) is a half-grid if it is not a grid (i.e., is not an intersection set of two sets of suitably chosen lines) and one of the curves defining its general projection as a complete intersection can be taken as a union of lines. The main result of the paper is a classification result on \((4,4)\) half-grids, namely that up to the projective change of coordinates there are two such sets, one called the anharmonic case and the second called the harmonic case.\N\NFor the entire collection see [Zbl 1544.13003].