Generic projections of the \(\mathrm{H}_4\) configuration of points (Q2081916)

In the paper under review, the authors provide the very first example of a finite set of points in \(\mathbb{P}^{3}_{\mathbb{C}}\) that has the so-called geproci property, but it is neither a grid nor a half-grid. Recall that a finite set of points in \(\mathbb{P}^{3}_{\mathbb{C}}\) having the property that its general projection to a plane is a complete intersection is called a geproci set (which comes from a general projection complete intersection set). For example, grids have the geproci property; recall that by a grid in \(\mathbb{P}^{3}_{\mathbb{C}}\) we mean a set \(Z\) consisting of \(a\cdot b\) points such that there exist two sets of lines \(L_{1}, \dots, L_{a}\) and \(M_{1}, \dots, M_{b}\) with the properties that lines in each of the sets are pairwise skew and such that \(Z = \{L_{i}\cap M_{j} : i \in \{1, \dots,a\}, j \in \{1, \dots,b\}\}.\) Moreover, half-grids have also the geproci property; here by a half-grid we mean a set \(Z\) consisting of \(a\cdot b\) points in \(\mathbb{P}^{3}_{\mathbb{C}}\) such that there exists a set of mutually skew lines \(L_{1}, \dots, L_{a}\) covering the set \(Z\) that a general projection of \(Z\) to a hyperplane is a complete intersection of the images of the lines with a possibly reducible curve of degree \(b\). The main result of the paper tells us that the set of \(60\) points in \(\mathbb{P}^{3}_{\mathbb{C}}\) that comes from the root system \(H_{4}\) has the geproci property, but it is not a grid, neither a half grid. This result is proved by direct computations.