Let (Formula presented.) be the triple of hyperplane arrangements. We show that the freeness of (Formula presented.) and the division of (Formula presented.) by (Formula presented.) imply the freeness of (Formula presented.). This “division theorem” improves the famous addition–deletion theorem, and it has several applications, which include a definition of “divisionally free arrangements”. It is a strictly larger class of free arrangements than the classical important class of inductively free arrangements. Also, whether an arrangement is divisionally free or not is determined by the combinatorics. Moreover, we show that a lot of recursively free arrangements, to which almost all known free arrangements are belonging, are divisionally free.
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