8 JAMES DAMON
Tlog(v)(x)
= Derlog(V»(x).
This agrees with the usual tangent space at the smooth points of V. As observed in
[DM,prop. 3.11], since the elements of Derlog(V) are tangent to the strata of the
canonical Whitney stratification of V, if Sj is the canonical Whitney stratum
containing x then Tj0g(V)(x) Q TXSj.
Examples 1.1:
s t
For any germ fQ : C ,0 » C ,0 we let D(fQ denote its discriminant.
S+Q S+Q
a) If F: C ,0 » C ,0 denotes the versal unfolding of an isolated
hypersurface singularity, then D(F) is a free divisor by Saito [Sa].
b) More generally, if F denotes the versal unfolding of an isolated complete
intersection singularity, thenD(F) is a free divisor by Looijenga [L].
c) If B(F) denotes the bifurcation variety of the versal unfolding of an isolated
hypersurface singularity, then B(F) is a free divisor by Terao [To2] and Bruce [Br],
2
d) Isolated curve singularities in C are free divisors by Saito [Sa].
e) Central arrangements of hyperplanes in which are free divisors when
viewed as hypersurfaces are called free arrangements by Terao. He has proven, for
example, that the reflection hyperplanes for Coxeter groups are free [Tol] [To3], and
more generally proves by an inductive process that many other non-reflection
arrangements are free.
Next, we enlarge the class of varieties which we consider to the class of
almost free divisors. For this, we recall [D1],[D2], and [DM] that fQ:
Cs,0
C ,0
is algebraically transverse to V,0 c Dl, 0 of/0 if
(1.2)
df0(TX(Cs)
+
Tlog(V)f(xo) = T ^ C *
for all XQ in a punctured neighborhood of 0. We emphasize that (1.2) says
(1.2') df0(T
Cs)
+ C«: ,
V
. . . , C ,f, J = T„ C
l
.
U x
0
l(f(x())) "m(f(x0)) f(x0)
where {Cj}^\ denotes a set of generators for Derlog(V) (by coherence they also
generate the sheaf (DerfogiV) in a neighbourhood of 0). We denote this by writing
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