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,
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