  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 an explicit
sufficient and necessary condition for the deletion theorem in terms of
characteristic polynomials. This gives a lot of corollaries including the
existence of free filtrations. The proof is based on the result about the form
of minimal generators of a logarithmic derivation module of a multiarrangement
which satisfies the $b_2$-equality.
