Addition–deletion conjecture for free hyperplane arrangements

Let ${\mathcal{A}}$ be a hyperplane arrangement and let $H\in{\mathcal{A}}$. Set ${\mathcal{A}}':={\mathcal{A}}\setminus\{H\}$. Write $\chi_0({\mathcal{A}};t)$ for the reduced characteristic polynomial, and let “globally divisional along $H$” have the meaning used in the paper.

Addition–deletion conjecture. The following assertions should hold:

1. ${\mathcal{A}}$ is free if ${\mathcal{A}}'$ is free and ${\mathcal{A}}$ is globally divisional along $H$.
2. If, for some integers $d_2,\ldots,d_\ell$,

$$\chi_0({\mathcal{A}};t)=(t-d_2-1)\prod_{i=3}^\ell(t-d_i),$$ $$\chi_0({\mathcal{A}}';t)=(t-d_2)\prod_{i=3}^\ell(t-d_i),$$

then ${\mathcal{A}}$ and ${\mathcal{A}}'$ are both free.

The conjecture proposes a freeness criterion extending the usual addition–deletion framework for arrangements. The source presents it as a conjecture related to the paper’s main theorem; no resolution is supplied here.