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

摘要

We prove that the Poincare polynomial pi(A, t) of an essential, central three arrangement A over a field K of characteristic zero is (1 + t), e(t)(F-0(A)(v)), where F0*A) is the sheaf associated to the kernel of the Jacobian ideal of A, and e(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 L-0(A)(0) is a vector bundle on P-2 and derive an algorithm which computes c(t)(L-0(A)(0)) from a free resolution of the Jacobian ideal of the defining polynomial of A. (C) 2000 Academic Press. [References: 12]