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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