Sets of points which project to complete intersections

The motivating problem addressed by this paper is to describe those non-degenerate sets of points $Z$ in ${\mathbb{P}}^3$ whose general projection to a general plane is a complete intersection of curves in that plane. One large class of such $Z$ is what we call $(m,n)$-grids. We relate this problem to the {\em unexpected cone property} $\mathcal{C}(d)$, a special case of the unexpected hypersurfaces which have been the focus of much recent research. After an analysis of $\mathcal{C}(d)$ for small $d$, we show that a non-degenerate set of $9$ points has a general projection that is the complete intersection of two cubics if and only if the points form a $(3,3)$-grid. However, in an appendix we describe a set of $24$ points that are not a grid but nevertheless have the projection property. These points arise from the $F_4$ root system. Furthermore, from this example we find subsets of $20$, $16$ and $12$ points with the same feature.