Part I Almost Free Divisors

In this first part we introduce the notion of an almost free divisor as one

obtained by the pullback of a free divisor by a germ algebraically transverse to it off

the origin. In §1 we recall the principal examples of free divisors and describe the

corresponding induced classes of almost free divisors. In §2 we recall certain

codimensions which are needed to relate algebraic and geometric transversality. We

add for almost free divisors a codimension which provides sufficient numerical

conditions for the algebraic properties. In the course of this we compute the

"logarithmic tangent space" and deduce fiber square properties for algebraically

transverse maps. In §3 we establish several key properties. These include: the

"product union" of free divisors is free, the "transverse union" of almost free

divisors is almost free, and almost free divisors are preserved under pullbacks by

algebraically transverse finite map germs. In §4 we define higher multiplicites for

almost free divisors in analogy with Teissier's original definition for isolated

hypersurface singularities. Using the results from the preceding sections, we are

able to give a simple extension of the results in [DM, thms 5 and 6] to almost free

divisors and use it to compute the higher multiplicities in terms of the singular

Milnor numbers given by explicit algebraic formulas. We also indicate how these

computations can be used to compute the higher multiplicities up to the "free

codimension" of a non-almost free divisor.

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