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The motivating problem addressed by this paper is to describe thosenon-degenerate sets of points $Z$ in $\mathbb P^3$ whose general projection toa general plane is a complete intersection of curves in that plane. One largeclass 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 theunexpected hypersurfaces which have been the focus of much recent research.After an analysis of ${\mathcal C}(d)$ for small $d$, we show that anon-degenerate set of $9$ points has a general projection that is the completeintersection 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 gridbut 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.

Sets of points which project to complete intersections