Star configurations are set-theoretic complete intersections

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.