Sets of points which project to complete intersections, and unexpected cones

The paper is devoted to the description of those non-degenerate sets of points Z in P3 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 (a, b)-grids. We relate this problem to the unexpected cone property C(d), a special case of the unexpected hypersurfaces which have been the focus of much recent research. After an analysis of 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 F4 root system. Furthermore, from this example we find subsets of 20, 16 and 12 points with the same feature.