Free arrangements of hyperplanes and supersolvable lattices

Abstract

A finite family of hyperplanes through the origin in C’+’ is called an Iarrangement of hyperplanes, or simply an l-arrangement. An l-arrangement is said to be free if the module of logarithmic vector fields (=holomorphic vector fields tangent to all hyperplanes in the arrangement) is a free module (the accurate definition will be given in (2.1)). Let A be a free I-arrangement. Then a set {do,..., d,} of I + 1 nonnegative integers can be defined (2.3). These (d, ,..., d,) are called the exponents of A. On the other hand, one can associate a geometric lattice L(A) with any arrangement A (3.1). The characteristic polynomial XLca,(t) E Z[t] is defined by using the Mobius function (3.4). The second author [7] proved that