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

Technical Summary: A Counterexample to Purdy's Inequality for Hyperplane Arrangements in Projective Three-Space

Problem Statement
The paper addresses a numerical inequality proposed for essential hyperplane arrangements in complex projective three-space ($\mathbb{P}^3_{\mathbb{C}}$). Let $\mathcal{A}$ be an arrangement of $n$ hyperplanes. The authors define $\ell$ as the number of distinct intersection lines and $p$ as the number of distinct intersection points (where at least three hyperplanes meet). The central question is whether the expression
$$p - \ell + n + 2 \geq 0$$
holds for all essential arrangements. This inequality is the projective arrangement counterpart to Purdy's point-line-plane problem in incidence geometry.

It is well-established that this inequality fails for a specific classical obstruction: the "rank 2 + 2 product" arrangement. Dually, this corresponds to a configuration of points contained in the union of two skew lines. It was previously conjectured that excluding this specific obstruction might render the inequality true. The primary objective of this paper is to demonstrate that this refined statement is also false.

Methodology
The authors employ a duality dictionary to translate the problem from hyperplane arrangements to finite point configurations in $\mathbb{P}^3_{\mathbb{C}}$.

• Dual Formulation: An arrangement of $n$ hyperplanes corresponds to a set of $n$ points $S$. The number of intersection lines $\ell$ corresponds to the number of lines spanned by pairs of points ($f_1(S)$), and the number of intersection points $p$ corresponds to the number of planes spanned by triples of non-collinear points ($f_2(S)$). The inequality becomes $\Delta(S) = f_2(S) - f_1(S) + f_0(S) + 2 \geq 0$.
• Search Strategy: The authors systematically investigate reflection arrangements, a class of highly symmetric configurations derived from complex reflection groups. They examine:
  1. Irreducible finite Coxeter arrangements of rank four ($A_4, D_4, B_4, F_4, H_4$).
  2. Irreducible monomial complex reflection arrangements ($G(m, p, 4)$).
  3. Specific subarrangements of imprimitive reflection arrangements.
• Construction of Counterexample: The search identifies a specific subarrangement within the monomial reflection arrangement of type $G(3, 3, 4)$. This subarrangement is defined by a "gain-graphic" structure supported on a complete bipartite graph $K_{2,2}$ (a four-cycle) with coefficients involving a primitive third root of unity ($\zeta$).

Key Contributions and Results
The paper constructs an explicit counterexample using a specific set of 12 points $S$ in $\mathbb{P}^3_{\mathbb{C}}$, defined as:
$$S = \{ [e_i - \zeta^a e_j] \mid i \in \{0, 1\}, j \in \{2, 3\}, a \in \mathbb{Z}/3\mathbb{Z} \}$$
where $\zeta$ is a primitive third root of unity.

1. Verification of Non-Obstruction: The authors prove that $S$ spans $\mathbb{P}^3_{\mathbb{C}}$ and is not contained in the union of two skew lines. This is established by showing that $S$ contains four distinct lines (the coordinate lines $L_{02}, L_{03}, L_{12}, L_{13}$), whereas a set contained in two skew lines could have at most two such lines containing three points.
2. Exact Enumeration: Through combinatorial counting, the authors determine the exact values for the dual configuration:
   • $f_0(S) = 12$ (the number of points).
   • $f_1(S) = 58$ (the number of spanned lines). This is calculated by taking the total pairs $\binom{12}{2}=66$ and subtracting the overcounted pairs on the four collinear triples.
   • $f_2(S) = 43$ (the number of spanned planes). This count includes four coordinate face planes, 12 planes formed by a line and a point on the opposite line, and 27 "balanced" planes formed by one point from each of the four lines satisfying the congruence $a+d \equiv b+c \pmod 3$.
3. Violation of the Inequality: Substituting these values into the dual expression:
   $$\Delta(S) = 43 - 58 + 12 + 2 = -1$$
   Consequently, for the dual arrangement $\mathcal{A}_{\mu_3}$, the value is $p - \ell + n + 2 = -1$, which is strictly negative.

Significance and Claims
The paper claims to provide the first counterexample to the refined version of Purdy's inequality. While the classical obstruction (points on two skew lines) was known to violate the inequality, the mathematical community had hoped that removing this specific case would restore the inequality's validity.

The authors demonstrate that the refined statement is false. The counterexample is notable for being:

• Small: It involves only 12 hyperplanes.
• Structured: It arises naturally from the theory of reflection arrangements, specifically as a $\mu_3$-balanced subarrangement of type $G(3, 3, 4)$.
• Critical: The authors note that the choice of the third root of unity is critical. A generalization to $m$-th roots of unity yields a defect $\Delta(S_m) = m^3 - 6m^2 + 8m + 2$. This expression is negative only for $m=3$ (yielding $-1$); for $m=2, 4$ and $m \geq 5$, the defect is non-negative. Thus, this specific configuration is the unique critical case in this family.

The paper concludes that the "top-heavy" principle, which suggests constraints on the numbers of flats spanned by configurations, does not hold in this specific low-dimensional numerical refinement without further exclusions beyond the two-skew-lines case.