Weak Hadamard star configurations and apolarity

Abstract

Recently, Carlini, Catalisano, Guardo and Van Tuyl introduced a new construction using the Hadamard product to present star configurations of codimension $c$ of ${\mathbb{P}}^n$ and which they called Hadamard star configurations. We introduce a more general type of Hadamard star configuration; any star configuration constructed by our approach is called a weak Hadamard star configuration. We classify weak Hadamard star configurations, and in the case $c=n$, we investigate the existence of a (weak) Hadamard star configuration which is apolar to the generic homogeneous polynomials of degree $d$.