A counterexample to Purdy's inequality for hyperplane arrangements in projective three-space

Abstract

We record an explicit counterexample to a refined form of Purdy's inequality for essential hyperplane arrangements in projective three-space. Let $\mathcal{A}$ be an arrangement of $n$ hyperplanes in ${\mathbb{P}}_{\mathbb{C}}^3$. Let $\ell$ be the number of distinct intersection lines of $\mathcal{A}$, and let $p$ be the number of intersection points, where an intersection point means a point at which at least three hyperplanes meet. The expected inequality is \[ p-\ell+n+2\geq 0. \] The classical obstruction is the rank $2+2$ product arrangement, or dually a configuration of points contained in two skew lines. We explain this obstruction first, and then show that it is not the only one. The reflection-arrangement search leads naturally to a subarrangement of the monomial reflection arrangement of type $G(3,3,4)$. Looking dually, this configuration is not contained in two skew lines, and has \[ f_0(S)=12,\qquad f_1(S)=58,\qquad f_2(S)=43. \] Therefore its dual arrangement has \[ n=12,\qquad \ell=58,\qquad p=43, \] and hence \[ p-\ell+n+2=-1. \] Thus the refined statement excluding only the two-skew-lines obstruction is false.