2

JAMES DAMON

classes and deduce new results for each of them.

Saito [Sa] intoduced the notion of a free divisor. In joint work [DM], David

Mond and this author defined for nonlinear sections fQ: CS, 0 c C^O of free divisors

V,0 cz c',0 a singular Milnor fibration. Using a result of Le, it was proven [DM,§4]

that the Milnor fiber has the homotopy of a bouquet of spheres of real dimension

s-1. Furthermore, the number of such spheres, the "singular Milnor number",

could be computed algebraically as a certain codimension [DM,thm 5]. In this paper

we will show how these results may be applied to the above examples in

unexpected ways.

First, just as Teissier did for isolated singularities [T], it is possible to define

higher multiplicities for nonisolated hypersurface singularities V,0 c Cs,0 by

generic intersection with k-dimensional linear subspaces. In [LeT], Le and Teissier

related the "vanishing Euler characteristics" of generic projections to the properties

of the polar curves. Our goal is instead to directly compute the multiplicities as the

singular Milnor numbers of the sections. If V is almost free then these

multiplicities can be explicitly computed using [DM] as lengths of certain

determinantal modules.

A major part of our calculations involve the weighted homogeneous case.

In [D4], we gave formulas for the lengths of certain determinantal Cohen-Macaulay

modules, the "Macaulay-Bezout numbers", to yield a Bezout-type theorem

involving the elementary symmetric functions of the homogeneous degrees in the

homogeneous case. For the general weighted homogeneous case, we expressed the

weighted Macaulay-Bezout numbers in terms of a universal function x applied to a

"degree matrix" for the module, and deduced algebraic formulas.

In part II, we apply the preceding results to compute the Poincare

polynomials for the complements of arrangements of hyperplanes. We show the

fundamental role higher multiplicities play in the topology by giving a general

formula for the Poincar6 polynomial for hyperplane arrangements in terms of the