Almost free divisors and complete intersections form a general class of
nonisolated hypersurface and complete intersection singularities which
simultaneously extend the free divisors introduced by K. Saito and the isolated
hypersurface and complete intersection singularities. They also include
discriminants of mappings, bifurcation sets, and certain types of arrangements of
hyperplanes such as Coxeter arrangements and generic arrangements.
Associated to the singularities of this class is a "singular Milnor fibration"
which has the same homotopy properties as the Milnor fibration for isolated
singularities. The associated "singular Milnor number" can be computed as the
length of a determinantal module using a Bezout-type theorem. This allows us to
define and compute higher multiplicities along the lines of Teissier's |i* -constants.
These are applied to deduce topological properties of singularities in a
number of situations including: complements of hyperplane arrangements, various
nonisolated complete intersections, nonlinear arrangements of hypersurfaces,
functions on discriminants, singularities defined by compositions of functions, and
bifurcation sets.
1991 Mathematics subject classification : Primary 32S30
Secondary 14B05, 58C10
Key words and phrases : almost free divisors, singular Milnor fiber, singular
Milnor number, algebraic transversality, hyperplane arrangements, Poincare
polynomials, higher multiplicities
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