Hadamard star configurations

Bocci, Carlini and Kileel have shown that the square-free Hadamard product of a finite set of points $Z$ that all lie on a line $\ell $ in $\mathbb {P}^n$ produces a star configuration of codimension $n$. In this paper, we introduce a construction using the Hadamard product to construct star configurations of codimension $c$. In the case that $c = n= 2$, our construction produces the star configurations of Bocci, Carlini and Kileel. We will call any star configuration that can be constructed using our approach a Hadamard star configuration. Our main result is a classification of Hadamard star configurations.