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In the theory of hyperplane arrangements, the most important and difficultproblem is the combinatorial dependency of several properties. In this atricle,we prove that Terao's celebrated addition-deletion theorem for freearrangements is combinatorial, i.e., whether you can apply it depends only onthe intersection lattice of arrangements. The proof is based on a classicaltechnique. Since some parts are already completed recently, we prove the restpart, i.e., the combinatoriality of the addition theorem. As a corollary, wecan define a new class of free arrangements called the additionally freearrangement of hyperplanes, which can be constructed from the empty arrangementby using only the addition theorem. Then we can show that Terao's conjecture istrue in this class. As an application, we can show that every ideal-Shiarrangement is additionally free, implying that their freeness iscombinatorial.

Addition-deletion theorem for free hyperplane arrangements and combinatorics