A Rank Two Vector Bundle Associated to a Three Arrangement, and its Chern Polynomial

Abstract

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)·ct(D0(A)∨), where D0(A) is the sheaf associated to the kernel of the Jacobian ideal of A, and ct 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 ct(D0(A)∨) from a free resolution of the Jacobian ideal of the defining polynomial of A.