A Geometric Deletion-restriction Formula

In this paper, we recover the characteristic polynomial of an arrangement of hyperplanes by computing the rational equivalence class of the variety defined by the logarithmic ideal of the arrangement. The logarithmic ideal was introduced in [CDFV] in a study of the critical points of the master function. The above result is used to understand the asymptotic behaviour the Hilbert series of the logarithmic ideal. As an application, we note that a well-known formula due to Solomon and Terao may be expressed as an Euler characteristic and, at least in the case of tame arrangements, deduced from our main theorem.