Because of the enormous success achieved in understanding the structure of
isolated hypersurface singularities, one goal has been to study larger classes of
singularities which share important natural features with these singularities.
Examples of such classes include isolated complete intersection singularities and
Gorenstein surface singularities. The properties of such classes are understood by
associating smooth objects such as Milnor fibers or resolutions. For nonisolated
singularities, the preceding methods have been applied when the singular behavior
can be kept relatively simple (e.g. Siersma, Pellikan et al). However, in many
important situations which arise, there are nonisolated singularities with quite
complicated singular behavior. It is reasonable to again ask whether there are not
natural classes of highly nonisolated singularities which, in fact, share certain nice
properties even with isolated hypersurface singularities.
In this paper we shall describe such a class, the class of almost free divisors.
In addition to containing isolated hypersurface singularities, this class will also
include: discriminants of finitely determined map germs, generalized Zariski
examples, bifurcation sets of certain unfoldings of hypersurfaces, certain central
arrangements of hyperplanes such as reflection hyperplane arrangements for
Coxeter groups or generic arrangements, and nonlinear arrangements of isolated
hypersurface singularities.
This class of "almost free divisors" is a natural class of varieties which
contains the free divisors and behaves well under pull-backs by algebraically
transverse maps and "transverse unions". Furthermore, the intersection of a finite
number of such divisors which are in "algebraic general position" leads to a class of
almost free complete intersections which includes the isolated complete
intersections. In this paper, we shall investigate topological properties for these
Partially supported by a grant from the National Science Foundation
Received by the editors November 20, 1993
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