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 $\mathrm{\ell }$ 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.