**Memoirs of the American Mathematical Society**

1996;
113 pp;
Softcover

MSC: Primary 32;
Secondary 14; 58

Print ISBN: 978-0-8218-0481-0

Product Code: MEMO/123/589

List Price: $45.00

AMS Member Price: $27.00

MAA Member Price: $40.50

**Electronic ISBN: 978-1-4704-0174-0
Product Code: MEMO/123/589.E**

List Price: $45.00

AMS Member Price: $27.00

MAA Member Price: $40.50

# Higher Multiplicities and Almost Free Divisors and Complete Intersections

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*James Damon*

In this book, the author considers a general class
of nonisolated hypersurface and complete intersection
singularities called “almost free divisors and complete
intersections”, which simultaneously extend both 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.

Topological properties of these singularities are studied via
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 one to define and compute higher multiplicities along the
lines of Teissier's \(\mu ^*\)-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.

Features:

- Treats nonisolated and isolated singularities together
- Uses the singular Milnor fibration with its simpler homotopy structure as an effective tool
- Explicitly computes the singular Milnor number and higher multiplicities using a Bezout-type theorem for modules

#### Readership

Graduate students and research mathematicians interested in local analytic singularities, algebraic geometry, singularity theory, topology of arrangements and their applications.