HIGHER MULTIPLICITIES AND ALMOST FREE DIVISORS 9

fQ rBa|q V (the V1 indicates that the transversality is in a punctured neighborhood).

This is related to fQ being geometrically transverse to (V,0) of/0, which

means that fQ is transverse to the strata of the canonical Whitney stratification of V

o

at all x0 in a punctured neighborhood of 0. We denote this by fQ ffSgeomV.

By the observation in [DM, prop.3.11], algebraic transversality implies geometric

transversality (but not conversely, see lemma 2.10).

o

Remark: Then, fQ rfSa( V iff fQ is finitely X -determined.

Here ^-equivalence is defined via the action of a subgroup of the contact group

X

XY = (O E 3C: 0(CS xV)cC S x V}.

This ^Cv-equivalence captures the isomorphism classes of the germs of varieties

fQ-^V) [Dl] [D2].

Although this characterization was stated in [Dl] for finite map germs fQ, the

proof given there works in general. In fact, by the graph trick, every map is

equivalent to an embedding by replacing the map fQ by its graph map fy: C ,0 —

Cs

x

€l,0

and replacing V by

Cs

x V. If V is free then so is

Cs

x V and fQ has

finite X -codimension iff fy has finite X

s

-codimension by [D2,§1].

Throughout this paper we will repeatedly reduce proofs to the case of germs of

embeddings by the graph trick.

Definition 1.3: A hypersurface (V',0) a

Cs,0,

is an almost free divisor (based

on V) if there exists a free divisor (V,0) c €l,0 and a germ fQ: Cs,0 — Cl,0

with f0 r#»alg V such that V =

f0_1(V).

Every class of free divisor gives rise to a corresponding class of almost free

divisors