We show that the deletion theorem of a free arrangement is combinatorial, i.e., whether we can delete a hyperplane from a free arrangement keeping freeness depends only on the intersection lattice. In fact, we give a sufficient and necessary condition for the deletion theorem in terms of characteristic polynomials. As a corollary, we prove that whether a free arrangement has a free filtration is also combinatorial. The proof is based on the result about a minimal set of generators of a logarithmic derivation module of a multiarrangement which satisfies the b 2 -equality.
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