On Hadamard products of linear varieties

In this paper we address the Hadamard product of linear varieties not necessarily in general position. In the projective plane 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 P^3; 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 non-degenerate quadric and we compute its Hilbert function in the case |X|=|X'| .