§1 Free and Almost Free Divisors
We begin by recalling the notion of Free Divisor due to Saito [Sa], giving a
number of examples, and extending it to the notion of an Almost Free Divisor.
First, we establish some notation.
s t
For a holomorphic germ fQ: ,0 » C ,0, the tangent space to the space of
germs C(s A at fQ consists of germs of vector fields C,: Cs,0 » TCl such that K
o £ = fQ (for n: TC* »
l
the projection), and is denoted by 8(f()). Thus,
e(fo-
Voti'•••'!;} Vol!;}
(we shall denote the R module generated by (p^...,^ by R{91,...,cpk}, or Ricpj}
if k is understood). Also, we let 9S = 9(id s) 3 0
s
|— J and similarly for
9t. We also denote the maximal ideal of 0
s
by ms.
If M c 9n is an 0
n
-module, let {^} denote a set of generators for M.
We let M/X) denote the subspace of
Tx€n
spanned by {^i(xvK This is only well-
defined for x in some sufficiently small neighborhood of 0; however, since all of
our statements will only involve results true on a sufficiently small neighborhood,
this notation makes sense.
If (V,0) c C^O is a germ of a variety, then we consider the module of
vector fields tangent to V. Let I(V) denote the ideal of germs vanishing on V.
Then (following Saito [Sa]) we let
Derlog(V) = e 9t: Cd(V)) s I(V)} .
It extends to a sheaf of vector fields tangent to V, 'Duiotfy) which is easily seen to
be coherent [Sa]. Then, V,0 is called a Free Divisorby Saito if Derlog(V) is a free

t
-module. Its rank is then necessarily t.
Finally it is convenient to define the logarithmic tangent space to V at x e
Vby
7
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