Divisionally free arrangements of hyperplanes

Abstract

We consider the triple $(\mathcal{A},{\mathcal{A}}^{\prime },{\mathcal{A}}^H)$ 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.

26 pages (version 01). 32 pages (version 02), 33 pages (version 03), 33 pages (version 04). In version 04, Section 4 is removed. An error in Theorem 6.2 is corrected. In version 03: Title is changed. With minor revisions. In version 02:Orders of results are changed. Previous section 5 is divided into sections 5 and 6. New main results (Theorems 1.4, 6.4 and 7.2) and minor results are added

References in corpus (1)

• Free filtrations of affine Weyl arrangements and the ideal-Shi arrangements

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