HIGHER MULTIPLICITIES AND ALMOST FREE DIVISORS 3

higher multiplicities. By explicitly computing the higher multiplicities for "almost

free arrangements", we are able to give an algebraic formula for Poincar6

polynomials for such arrangements in terms of the exponents of the associated free

arrangement. These include the free arrangements but also other arrangements such

as the "generic arrangements"; in fact, each free arrangement has its own types of

generic arrangements. In the case of free arrangements this gives a new proof of

the factorization theorem ofTerao [Toll [To3] [OTI.

These methods have the advantage of being applicable to nonlinear

arrangements. There are several possible interpretations of nonlinear arrangements:

arrangements of nonlinear hypersurfaces, arrangements of hyperplanes restricted to

nonsingular spaces or their Milnor fibers, or a mixture of these. For all of these

cases, we can again compute the singular Milnor numbers and higher multiplicities.

Using the computations of the Macaulay-Bezout numbers, we compute these

numbers for: generic nonlinear arrangements of hypersurfaces of fixed degree

based on arbitrary free arrangements A, and generic arrangements of weighted

homogeneous hypersurfaces of varying degrees. However, relating them to the

Poincare polynomial for the complement becomes more difficult and is postponed

forlaterconsideration.

In part in, We obtain formulas for the singular Milnor numbers and higher

multiplicities of nonisolated complete intersection singularities which are "almost

free". Such complete intersections are represented as algebraically transverse

intersections of almost free divisors and include as special cases the ICIS (i.e.

isolated complete intersection singularities). Surprisingly, it is the various unions of

the almost free divisors whose singular Milnor numbers can be directiy computed

as lengths of determinantal modules. The singular Milnor number for the almost

free complete intersection is given as an alternating sum of singular Milnor numbers

for various transverse unions. This becomes very computable because of an

observed "principle of linear combination"; namely, each such term can be

expressed as a linear combination of fixed terms with varying coefficients. Hence,