On Hadamard product of linear varieties

In this paper, we address the Hadamard product of linear varieties not necessarily in general position. In P2, we obtain a complete description of the possible outcomes. In particular, in the case of two disjoint finite sets X and X′ of collinear points, we get conditions for X X′ to be either a collinear finite set of points or a grid of |X||X′| points. In P3, under suitable conditions (which we prove to be generic), we show that X X′ consists of |X||X′| points on the two different rulings of a nondegenerate quadric and we compute its Hilbert function in the case |X| = |X′|.