A rank two vector bundle associated to a three arrangement, and its chern polynomial

We prove that the Poincaré polynomial π(A, t) of an essential, central three arrangement A over a field K of characteristic zero is (1+t)·c_t(D₀(A)^∨), where D₀(A) is the sheaf associated to the kernel of the Jacobian ideal of A, and c_t is the Chern polynomial. This shows that a version of Terao's theorem [Invent. Math.63 (1981), 159-179] on free arrangements also holds for all three arrangements. We also prove that for such an arrangement D₀(A)^∨ is a vector bundle on P² and derive an algorithm which computes c_t(D₀(A)^∨) from a free resolution of the Jacobian ideal of the defining polynomial of A.