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 $j=0,\dots ,k-2$, 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.

5 pages. In this version we present an update with an example answering negatively the question asked previously whether or not, for ANY hyperplane arrangement, all the ideals generated by a-fold products of the defining linear forms are set-theoretic complete intersections