A semicontinuous relaxation of Saito's criterion and freeness as angular minimization

Abstract

We introduce a nonnegative functional $\mathfrak{S}$ on the space of line arrangements in ${\mathbb{P}}^2$ that vanishes precisely on free arrangements, obtained as a semicontinuous relaxation of Saito's criterion. Given an arrangement $\mathcal{A}$ of $n$ lines with candidate exponents $(d_1,d_2)$, we parameterize the spaces of logarithmic derivations of degrees $d_1$ and $d_2$ via the null spaces of the associated derivation matrices and express the Saito determinant as a bilinear map into the space of degree-$n$ polynomials. The functional admits a natural geometric interpretation: it measures the squared sine of the angle between the image of this bilinear map and the direction of the defining polynomial $Q(\mathcal{A})$ in coefficient space, providing a computable measure of how far an arrangement is from admitting a free basis of logarithmic derivations of the expected degrees. We prove that $\mathfrak{S}$ is upper semicontinuous on natural strata, and use this to give a functional reformulation of Terao's conjecture. Beyond its theoretical interest, $\mathfrak{S}$ provides a viable computational handle on the landscape of free arrangements. We illustrate this through two complementary roles: as a smooth reward signal driving a reinforcement learning search for moderate $n$, and as a fast pre-filter accelerating an algebraic extension procedure for larger $n$. For $n\le 13$, the reinforcement learning system discovers hundreds of verified free arrangements spanning all admissible exponent types. For $n\ge 14$, where the reinforcement learning reward signal becomes insufficient, the hybrid extension procedure -- combined with classical supersolvable constructions -- produces at least one verified free arrangement for every admissible exponent pair $(d_1,d_2)$ with $n\le 20$.