Divisionally free arrangements of hyperplanes

We consider the triple $(\mathcal{A},{\mathcal{A}}^{\mathrm{\prime }},{\mathcal{A}}^H)$ of hyperplane arrangements and the division of their characteristic polynomials. We show that the freeness of ${\mathcal{A}}^H$ and the division of $\chi (\mathcal{A};t)$ by $\chi ({\mathcal{A}}^H;t)$ confirm the freeness of $\mathcal{A}$. The key ingredient of this “division theorem” on freeness is the fact that, if $\chi ({\mathcal{A}}^H;t)$ divides $\chi (\mathcal{A};t)$, then the same holds for the localization at the codimension three flat in $H$. This implies the local-freeness of $\mathcal{A}$ in codimension three along $H$. Based on these results, several applications are obtained, which include a definition of “divisionally free arrangements”. It is strictly larger than the set of inductively free arrangements. Also, in the set of divisionally free arrangements, the Terao’s conjecture is true.