§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* — » €

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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

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