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. In this paper, we classify essential arrangements A ofhyperplanes in R 3 with jAj = 5 and give a full description of eachcase in terms of freeness. 1. IntroductionLet V be an li dimensional vector space over a eld K. A hyperplanein V is a codimension one subspace of V . A central li arrangement A in V is a nite set of hyperplanes in V such that each hyperplane containsthe origin.Let V ⁄ denote the dual space of V and S = S ( V ⁄ ) be the symmetricalgebra of V ⁄ . Choose a basis fe 1 ;¢¢¢ ;e l g in V and let fx 1 ;¢¢¢ ;x l g bethe dual basis in V ⁄ so that x i ( e j ) = – ij . If we identify S ( V ⁄ ) with thepolynomial algebra K[ x 1 ;¢¢¢ ;x l ], each hyperplane H in V containing theorigin is the set of zeros of a polynomial  H = a 1 x 1 + ¢¢¢ + a l x l ; a i 2 K.Givenacentral li arrangement A in V , theproduct Q ( A ) =Q H2A  H is called dening polynomial of A . Suppose µ : S ! S is derivation. It iscalled an Ai derivation if µ ( Q ( A )) 2 Q ( A ) S . The set of Ai derivationsis an

ARRANGEMENTS OF HYPERPLANES IN ℝ3AND THEIR FREENESS

ARRANGEMENTS OF HYPERPLANES IN ℝ³AND THEIR FREENESS