10
JAMES DAMON
Remark : There is also a class of geometrically almost free divisors defined as
in (1.3) but using geometric rather than algebraic transversality. There are some
situations where generically these are the only divisors which are guaranteed to
exist. However, at this point there is no known way to compute their vanishing
topology and higher multiplicities algebraically.
Examples 1.4:
a) Discriminants D(fg) of finitely (A-) determined germs fQ:
Cs,0

€l,0
with s t are almost free divisors by [D2].
b) Bifurcation Varieties B(F) of "finite bifurcation-codimension" unfoldings
of isolated hypersurface singularities are almost free divisors (see §11).
c) Isolated Hypersurface Singularities are almost free divisors because {0} c
is a free divisor (Derlog({0}) is freely generated by z j - ) .
d) Generalized Zariski Examples are obtained as hypersurface singularities
defined by compositions h o fQ where h defines an isolated curve singularity in
2 s 2
C ,0, fg: ,0 —• C ,0 defines an isolated complete intersection singularity, and
fQ is transverse to
h-1(0)
off 0. Such examples are almost free divisors. Note that
each point of fQ-1(0) is a singular point with transverse type h_1(0).
e) An Almost Free Arrangement of Hyperplanes A'a C based on a free
arrangement A c C is defined via A'=
(p-1G4)
for a linear 1-1 map cp:
n

Cp which is transverse to all intersections of subsets of hyperplanes of A(except
possibly (0)). Such almost free arrangements are almost free divisors. Then, for
example, "generic arrangements" as defined in [OT,5.1] are almost free
arrangements based on the Boolean arrangement defined by fl Zi = 0.
An arrangement A = u Hj is called essential [OT,2.1] if n H^ = (0). Any
arrangement A - B x T, for an essential arrangement B and T = n Hj, and r(A)
= codim(T) is called the rank of A. More generally given a free arrangement A,
we shall say that an arrangement A 'is A-generic if A' - B xT where B is an
almost free arrangement based on A.
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