Hadamard Star Configurations

TL;DR

This paper introduces a Hadamard product-based method to construct and classify star configurations of various codimensions in projective space, generalizing previous results for codimension two.

Contribution

It presents a new construction approach for star configurations using Hadamard products and provides a complete classification of these Hadamard star configurations.

Findings

Construction method for star configurations of any codimension

Equivalence to known star configurations when codimension equals dimension

Complete classification of Hadamard star configurations

Abstract

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.