We prove that a product of $m>5$ copies of $\mathbb {P}^1$ , embedded in the projective space $\mathbb {P}^r$ by the standard Segre embedding, is $k$ -identifiable (i.e. a general point of the secant variety $S^k(X)$ is contained in only one $(k+1)$ -secant $k$ -space), for all $k$ such that $k+1\leq 2^{m-1}/m$ .

In this paper we address the question if, for points $$P, Q \in \mathbb {P}^{2}$$P , Q ∈ P 2 , $$I(P)^{m} \star I(Q)^{n}=I(P \star Q)^{m+n-1}$$I ( P ) m ⋆ I ( Q ) n = I ( P ⋆ Q ) m + n - 1 and we obtain different results according to the number of zero coordinates in P and Q. Successively, we use our results to define the so called Hadamard fat grids, which are the result of the Hadamard product of two sets of collinear points with given multiplicities. The most important invariants of Hadamard fat grids, as minimal resolution, Waldschmidt constant and resurgence, are then computed.