4

JAMES DAMON

it is a question of determining by algebraic methods the coefficients of the sum.

In the weighted homogeneous case this can be carried out with Macaulay-

Bezout numbers. Using them, we deduce the formulas obtained by Greuel-Hamm

[GrH] and Giusti [Gi] for weighted homogeneous ICIS's. We also compute singular

Milnor numbers for almost free arrangments on complete intersections, and for the

fibers of singular projections of discriminants.

An alternate way to compute singular Milnor numbers is obtained by an

analogue of the Le-Greuel formula for relative Milnor fibers. We again obtain a

formula for almost free complete intersections in terms of the length of a

determinantal module.

In Part IV, we complete the calculation of "vanishing Euler characteristics"

for the pullbacks of almost free divisors and complete intersections. In part I, the

computation is given in the case "n p". If "n p" then the pullback need no

longer be almost free. Nonetheless, theorem 3 gives a general composition formula

for the "vanishing Euler characteristics" in the case "n p". This is applied to

obtain formulas for the Euler characteristic of nonisolated hypersurface singularities

defined via a composition of a germ defining an ICIS with a plane curve singularity;

these are the "generalized Zariski examples". Nemethi [N] and Massey-Siersma

[MS] obtained formulas involving specific intersection invariants of the curve

singularity. We give an alternative general composition formula which involves

only the 3 main "Milnor numbers" of the various objects, which can be

algebraically computed. Moreover, this composition formula also applies to other

situations such as pullbacks of almost free divisors by finite map germs; when

applied to modified Zariski examples it provides examples of nonREALizable

singular complex cycles, which also occur for bifurcation questions in §11.

In the last section, we consider the bifurcation problems for unfoldings of

isolated hypersurface singularities of finite bifurcation codimension and compute

the singular Milnor numbers for the bifurcation set. We show that there are

universally nonREALizable vanishing cycles for bifurcation problems.