Star Configurations are Set-Theoretic Complete Intersections

TL;DR

This paper proves that star configurations formed by certain hyperplane arrangements are set-theoretic complete intersections, revealing a new algebraic property of these geometric objects.

Contribution

It establishes that star configurations are set-theoretic complete intersections for all relevant subspace arrangements, extending understanding of their algebraic structure.

Findings

Star configurations are set-theoretic complete intersections.

This property holds for all subspace arrangements generated by products of hyperplane forms.

The result applies to k-generic hyperplane arrangements in projective space.

Abstract

Let $\mathcal A\subset\mathbb P^{k-1}$ be a rank $k$ arrangement of $n$ hyperplanes, with the property that any $k$ of the defining linear forms are linearly independent (i.e., $\mathcal A$ is called $k-$generic). We show that for any $j=0,\ldots,k-2$, the subspace arrangement with defining ideal generated by the $(n-j)-$fold products of the defining linear forms of $\mathcal A$ is a set-theoretic complete intersection, which is equivalent to saying that star configurations have this property.