Content uploaded by James Montaldi
Author content
All content in this area was uploaded by James Montaldi on Jul 16, 2019
Content may be subject to copyright.
ropoloq?' vol. 19. No. 4 pp. 501-510. 1990. w4@9383 90 s0300+.00
Pnnted I" Great Bntam. (: 1990 Pergamon Press plc
ONE-FORMS ON SINGULAR CURVES AND THE TOPOLOGY OF
REAL CURVE SINGULARITIES
JAMES MONTALDI and Duco VAN STRATEN
(Received in revised form 16 April 1989)
INTRODUCTION
LETS: UP, 0 + P-l, 0 be a real analytic map germ, withf-‘(0) a reduced curve germ. In a
recent paper [I], Aoki, Fukuda and Nishimura produced a remarkable algebraic method
for computing the number branches of this curve. Their method is, briefly, to associate tofa
map-germ F : Iw”, 0 + [w”, 0 whose topological degree is equal to the number of branches of
f- l(O), and then to use the Eisenbud-Levine theorem [73 to calculate the degree of F as the
signature of a quadratic form on the local algebra of F. (We describe it in more detail in 92.)
The aim of this paper is to generalize the method of Aoki et al. to apply to the case where
the curve is not a complete intersection. In the case of a complete intersection, local duality
comes into play in the use of the Eisenbud-Levine theorem on F. However, since there is no
such map F in the general case, we were led to use local duality and residues on the curve. As
usual, the more general setting clarifies the special one.
Given any meromorphic form c1 on a curve 59, we use the module of Rosenlicht
difirentials ow of the curve to define two “ramification modules” which measure in some
sense the zeros and poles of the form respectively. These modules are finite dimensional
vector spaces, and we prove in $1 that the difference in dimension is preserved under
deformation of both the form and the curve. In the case that r = dg for some holomorphic
function g, this enables us to find the number of critical points of a small generic
deformation ofg. Further, we give a simple proof of the fact that the jump in Milnor number
(as defined in [3] and [6]) in a flat family of curve singularities is equal to the vanishing
Euler characteristic.
In the case that V and a are real, the l-form defines an orientation on each connected
component of V - {p} ( = half-branch), where p is the base point of %, Some of these half-
branches will be oriented outwards and some inwards. Moreover, the two ramification
modules come with real valued non-degenerate quadratic forms. We show in $2 that the
sum of signatures of these two forms is equal to the difference between the numbers of
branches oriented outwards and those oriented inwards. This is related to the classical
method of Hermite for calculating the number of real roots of a polynomial as the signature
of a quadratic form (see[12]). We remark that it seems surprising that the two important
features of these ramification modules are the difference of the dimensions, but the sum of
the signatures.
81. THE RAMIFICATION MODULES OF A ONE-FORM
Usually, V will be a germ of a reduced analytic curve with base point p and local ring 0~.
in this section defined over @ but in $2 defined over Iw. The normalization of the curve V will
5nl
502 James Montaldi and Duco van Straten
be denoted by n: @ + g. We will be interested in three basic local invariants of curve
singularities. First, r is the number of irreducible components of V, or what is the same, the
number of points in n-‘(p) c G?. Second, 6 = dimc,(C;“g/c& which can be interpreted
heuristically as the number of double points concentrated at p. Third is the Milnor number
~1. which is equal to 26 - r + 1. We extend 6 and p additively to multigerms and to
global curves (with finitely many singular points), which means that the relation becomes
p = 26 - E(r - l), where the summation runs over the singular points of Q?.
Let 0% be the &module of Kihler one-forms on ‘+? and let G(*) = C,(*) 0 06 be the
cd*)-module of meromorphic one-forms on G$ (or, what is the same, on @), where &&*) is
the total fraction ring of 0~. There is a (weakly) non-degenerate bilinear form:
Res: fLg(*) x I!!&(*) + @
w h I-+ Rq,,(h, 4 (1.1)
which we call the residue pairing. The residue can be defined as
Res&a) = (2xi)-’ s a
i’6
where 8% is the boundary of an appropriately small representative of @?. If V? is a multigerm
then the total residue is the sum of the residues at the base points. Indeed, another way to
define the residue is to pull back the form to the normalization and then add the residues
over the points lying above p (see [ 111).
The Cfg-molecule of Rosenlicht differentials 0% is defined by
0%: = 0: = {oERw(*)IRes(o. cry) = 0).
This module is the dualizing module of the curve. If %? is mapped finitely to C”’ ‘, n 2 0, then
this module is naturally isomorphic to &z/~(O~., Q”+l), where R”+l is the Cr = c%.+,-
module of (n + I)-forms. The curve V? is said to be Gorensrein if cuu is generated over Crw by
one element. Such a generating element is called a Gorenstein generator.
The pairing (1.1) descends to a non-degenerate pairing between CUK/O~ and c’g/c’,, so
the former also has dimension 6. Furthermore, by reducing to a finite dimensional situation
(essentially working modulo the conductor ideal) one can show that GU = 0:.
Definition (1.2). (i) Let ac:Ryi*). We say that z is a finite form (or a is finite) if
its restriction to each branch is not identically zero. Such a form induces an isomorphism
h(*) 4 C;,(*); WHO/~. Composing this with the residue pairing (1.1) gives a symmetric
bilinear form:
Y =V,:Ry(*) x Q,(*)-+C
.
(ii) Let z be a finite form. We define the ramification modules:
Ri = R+(a): = oV/O~ n 0%. a
R- = R-(a): = O~.a/~~ n CZ..a
and the integer p(a): = dimcR+(a) - dimcR_(a).
(iii) For a finite form a, Y descends to two bilinear forms:
t,b.f:R*(a) x R*(a)-+@.
(1.3)
ONE-FORMS ON SINGULAR CURVES 503
Remarks (1.4). A finite form a has on each branch an expansion as
a = C r,t’dt
for some NE Z (the order of a on that branch), where t is a local parameter and aN # 0. It is
easy to see that for a finite form the ramification modules are finite dimensional vector
spaces over C. It is worth noting that the number p(a) can also be computed as:
p(a) = dimc(o%/Y) - dimc(fiV. x/Y),
where 2 is any subspace 0~ n 01. a of finite codimension. Finally, since 0% = I& it follows
that the quadratic forms ++ are well-dejned and non-degenerate.
Special cases (1.5). (i) Let %’ be smooth, so 0g = C(t) and of6 = C(t). dt, and let x be a
meromorphic 1 -form of order N. Then Ou. a = tN @{tjdt, so if N 2 0 then dim R’ (2) = N,
dim R+(r) = 0, while if N 5 0 then dim R+ = 0 and dim R- = - N. In particular
p(r) = N.
(ii) It is clear from the definitions that R+(a) = R-(r) = 0 if and only if o,~ = 6%. z, that
is Y? is Gorenstein with generator a.
(iii) Suppose z = dg, with glow Then a is finite precisely when g defines a finite map
g : V + 9, where 3 is a germ of the complex line, and
R +(&) = o-k/&. da
R-(dg) = 0.
In this situation 0% is equal to Y,, . dg, where _Yipgiy = {fo C,(*)(trace(f. 0%) c CC21 is the
classical complementary module, so R+(dg) z _T’~&B~. In [8] Herzog and Waldi relate the
dimension of this space to the cotangent complex of V -+ 8.
(iv) Now suppose that %cJ is Gorenstein with generator w and g E Lo, In this case g has a
Jacobian defined as Jac(g) = dg/w. Then
R+(&) = WJac(d,
so the choice of o gives R+(dg) an algebra structure. In particular, suppose
f: C”, 0 + P-i, 0 defines an isolated complete intersection singularity. Then ~0% is gener-
ated by any o satisfying w A f *CD,, _ 1 = co, (where CD, is a holomorphic volume form on C’),
so
Thus,
&I zxi
Jac(g) = (dg A f *a,_ 1)/o” = det
[ I
. . . . . . . .
&7/h
R+(dg) z C{xi,. . . , x.}/Ui,. . . ,_Ll, Jac(d),
which is precisely (the complexification of) the local algebra of F considered in [l].
LEMMA (1.6). Let n: %? --, V be the normalization map and a anyfinite l-form on %. Then
p(n*a) = p(a) - 26
where 6 is the b-invariant of W.
504 James Montaldi and Duco van Straten
Proof Let A, B, C, D be finite dimensional vector spaces then in the ring of formal
vector spaces:
A - D = (B - C) + (A - B) + (C - D).
Apply this to A = u&/M, B = wg/lM, C = Co,.n*r/M, D = Olg. afM, where M is
w,-n &. 2. Note that dim (A - B) = dim (C - D) = 6, so taking dimensions gives the
result. [XI
We now consider the behaviour of p under deformations of the curve % and the form a.
So we have a Cartesian diagram of germs:
(% P)‘-- (% P)
1 ln
(0) - (SV 0)
Here S is a smooth curve germ, II is a flat map, and X is the total space of the deformation of
%, i.e. 3E is a surface germ. We choose good representatives of all these germs (for good
representatives see [9], 2.B or [6]), and we will be sloppy with the distinction between germs
and global section of sheaves. The fibres of II are curves, which we denote by qs: = II-l(s).
Consider further an analytic family of l-forms a, on the fibres ‘3,, i.e. an element A c&p(*)
such that AIWs = a,. Because the form a = a,, is finite, we may assume after a possible
shrinking of S that A has no vertical zero or pole components, i.e. all a, are finite forms
everywhere on V,. Define the following function on S:
Here P(ar, q) is the p invariant of (1.4) (iii) of the form a, on the curve germ (ws, q).
THEOREM (1.7). The function p : S + Z is constant.
Proof First we choose a function HE 0~ such that A. HE ~~~~ and lI.+(~r.~~~. H) is a
free &s-module of finite rank. This is possible as A restricts to finite forms on the fibres and
the deformation is flat. Here ozis is the so-called relative dualizing module, which can be
considered as a subsheaf of flz,s(*). For a flat family X + S, the sheaf oXis is S-flat and
restricts to Ok, on the fibre ?Zs (see for example [6]). Since now both ox/s and Cx are flat
over S and specialize to oq and 0, on the special fibre V we see that lI,(~r,s/Cr. H.A) is
also a free C”s-module of finite rank. Hence, because L’, z (SE. A etc., one has for all s E S:
rank &(&Jr. A/Ox. H. A) = dimc(c‘,,. aJOy,. H,. a,)
rank Il.+(or&5’, . H . A) = dim&u,,/@&, . H, . a,)
where H, = H,,. But 0,. aL,. H, c my, n O,, . as, so using Remark (1.4) we see that the
value of p(s) is independent of s. El
Remark. It is clear that the dimensions of R+ and R- are not themselves constant unde<
deformation: even on a smooth curve poles and zeros can annihilate.
As might be guessed from the invariance of p under deformation, this number has a clear
topological meaning, which we now explain. Let %? be a curve-either a small representative
of a (multi-)germ or a global curve-with boundary 3%‘. (a%? must be disjoint from the
singularities of %.) Let a be a finite meromorphic l-form on V. At a smooth point p of V, the
ONE-FORMS ON SINGULAR CURVES 505
real part &(a) has a singularity if and only if a has a zero or a pole. A simple calculation
shows that if a has order N at p, then the index i(Re(a); p) of &(a) at p (the usual winding
number of the associated section of the circle bundle) is - N. That is, at a smooth point of W
(cf. 1.5(i)) W(a); p) = - p(a; p). (1.4)
In [2], Arnol’d introduced a local index i, associated to a boundary singularity of a l-form.
(A form has a boundary singularity if its restriction to the boundary has a singularity, but
the form itself does not.) He showed that if M is any compact manifold with boundary, and
fl is any l-form on M with only boundary singularities at the boundary, then
X(M) = W) + 1 +(B), (1.5)
Here I and I+ are the sums of the indices i and i, respectively and x the Euler characteristic.
To apply this to our case, we consider the normalization @ of V, which is smooth. Then
from (1.4) and (1.5) we obtain:
x(%?) = - p(n*a) + Z+(Re(n*a)). (1.6)
Now by Lemma (1.6), &*a) = p(x) - 26, and for topological reasons, x(g) = x(W) +
c(l- 1). Moreover, away from the singular points of V? the normalization is an iso-
morphism. In particular this is the case in a neighbourhood of 8%. So (1.6) becomes:
x(V) + I@- 1) = - p(a) + 26 + Z+(Re(a)),
or, by definition of /.c
XV) - PW’) = - p(a) + I+(Re(a)). (1.7)
In particular:
PROPOSITION (1.8). Let V be a compact curve without boundary, and a any-finite I-form on
V. Then
o(a) = 10) - x(W,
where ,u(%‘) is the sum of the local Milnor numbers at the singular points of ‘3’ and ~(93) is the
Euler characteristic. [XI
Now let g : V + .2 be a finite map of degree d on all irreducible components of 9, and
suppose that g is unramified over &’ = g- l ad. Then for any finite l-form a on S, non-
singular in a neighbourhood of da, Z+(Re(g*a)) = d.Z+(Re(a)). Thus (1.7) gives the
following singular analogue of the Zeuthen-Hurwitz formula of Riemann:
PROPOSITION (1.9). With g : W + 9 as above,
o(g*a) - dWo(a) = PW - XV) - d*(Z@) - x(3).
In particular, if% is a germ, 9 is smooth and a = dt, then one has:
Nd = M9 + d - 1. IXI
Remark. The invariant p(W) + d - 1 has also been considered in [6], Lemma 6.2.8 and
in [S], “On Zariski’s criterion . . . “, prop. 2.2, which say that it is constant under
deformations of V. (In [6] for V a complete intersection and in [S) for general ‘3.) Here the
constancy follows from Theorem (1.7) and the above proposition. The number is equal to
the multiplicity of the discriminant of g, as defined in [S].
506 James Montaldi and Duco van Straten
This differential-topological point of view gives another proof of a theorem of Milnor
[lo] for plane curves, Bassein [3] for smoothable curves and Buchweitz and Greuel [6] in
the general case. (In fact, it is close to Milnor’s original proof.)
THEOREM (1.10). Let lT:x + S be (a good representative of) a flat deformation of the
curve germ V = %,-,. Then for all s E S
,n(Ws) - @0) = x(W,) - x(~g,),
where 1 is the topological Euler characteristic and p is the sum of the local Milnor numbers
over the curve.
Remark. In fact, it is not hard to show that under a flat deformation a reduced curve
germ remains connected (see e.g. [6]), so one can replace x by - dim H’ in the formula
above.
Proof After possibly shrinking 3E and S we can assume that:
(a) III,, is a smooth fibration (8x = UNs);
(b) we have a holomorphic l-form A on 3E whose restrictions a, are finite;
(c) for all s the zeros of a, do not meet &fZs.
The theorem then follows from (1.7) since Z+(Re(a,)) is constant (Arnol’d’s boundary index
i, is constant under homotopy provided no singular points of the l-form cross the
boundary), and p(a,) is constant by Theorem (1.7). IXI
We turn to another consequence of the deformation Theorem (1.7) concerning the
multiplicity of the critical point of a function on a singular curve germ. In [4], Bruce and
Roberts define for certain singular spaces and functions g on them a “stratified Milnor
number” p(g) in terms of the Jacobian ideal generated by vector fields tangent to X acting
on g. They show that if the so-called logarithmic characteristic variety K-(X) is Cohen-
Macaulay, then this Milnor number is continuous under deformation of g ([4], prop. 5.4).
However, even in the simplest examples, p(g) is not constant under deformations of X:
Example. X = {(x, y)~@‘jx, y = 01; g = x + y. Then
p(g) = dim @{x, y}/(xy, x&Ax + Y), yd,(x + Y)) = 1,
whereas the number of critical points of g on xy = E is clearly 2. (And of course, p(dg) = 2.)
COROLLARY (1.11) (of Theorem (1.7) and Proposition (1.9)). Let g E mq define a finite
mapping of degree d. Then for generic L E S+.HQ - SW& and sujiciently small A # 0 the function
g + R.L has d - m(U) critical points away from 0, where m(q) is the multiplicity of %9.
Proof For a finite mapping h:W + C, we have by Proposition (1.9): p(dh) = ,u(Gf?) +
degree(h) - 1. Thus for generic h E mrg - m& p(dh) = p(W) + m(W) - 1. Thus for generic L
this equation is satisfied by both L and i..L + g for sufficiently small 1. By Theorem (1.7),
p(dg + I..dL) summed over critical points is independent of 1. A further genericity condition
on L ensures that all the critical points away from 0 have multiplicity one. Thus the number
of these critical points is: p(dg) - p(dg + A.dL, 0) = (p(%) + d - 1) - (/.I(%?) + m(U) - l).[xI
Remark. The number deg(g) - m(W) is called the number of vertical tangents of g, see
also [6].
ONE-FORMS ON SINGULAR CURVES 507
$2. REAL CURVE SINGULARITIES
In this section we consider real l-forms on real analytic curve germs. We find that the
signatures of the quadratic forms defined in Definition (1.2) are related to the orientation
induced on the curve. Throughout this section V denotes a germ of a real analytic curve.
The local ring 0% now is an R-algebra of the form:
O$f z R{x,, . . . ,x,)/l
for some n and ideal 1. The complexification ‘Xc has local ring UV @ R@. The complexifi-
cation of an R-algebra or module has a natural complex conjugation on it, and one can
identify the original with the subspace of fixed points of this conjugation. Whenever we say
that ‘1p is reduced, irreducible or Gorenstein, we mean that %‘c is. Note that all the Ok-
modules defined in $1 are already defined over IF!. Let a E C&(q) be finite.Then on each half-
branch of %Z (that is, connected component of V - {p}) a defines an orientation. Given a, we
say that a half-branch is inbound or outbound accordingly as the orientation is towards or
away from the base point p. We have from Definition (1.2) two real artinian &-modules
R*(a) with non-degenerate quadratic forms JI,‘.
Y t
a2 = xdx l ydy a2=xdx*ydy
Fig. 1. Examples of orientations induced by l-forms on singular curves. The curves are y3 - x2 = 0 and xy = 0.
In each case 01, is a Gorenstein generator.
THEOREM (2.1). Let W be a reduced real analytic curve germ and a be afinite meromorphic
l-form on V. Then:
# {outbound half-branches} - # (inbound half-branches} = 2 Sig(+:) + 2 Sig($,),
where Sig denotes the signature of a quadratic form, that is the diflerence between the number
of positive and the number of negative eigenvalues.
COROLLARY (2.2). For a = CXidXi we have R-(a) = 0 and every half-branch is outbound,
so the number of half-branches of 59 is equal to 2Sig($:). E4
Before proving this theorem we show that in the case of an isolated complete inter-
section curve it reduces to the theorem of Aoki, Fukuda and Nishimura [l]. Let
f = (fil * * . ,f,-l):R”;o~R”-‘,o
508 James Montaldi and Duco van Straten
define an isolated complete intersection germ % =f-‘(0) and let gEmrg define a finite
mapping. From (1.4)(iv), R+(dg) = Sy/Jac(g), which is the local algebra of the finite map
F = U-1, * * . ,f.- 1, Jac(g)) and R-(dg) = 0. The Jacobian of F is then Jac(Jac(g)).
THEOREM (Aoki, Fukuda, Nishimura). Let f be as above and put g = xx:. If
cp : cO,/juc(g) + (w is any linearfunctional with cp(Jac(Jac(g))) > 0 and if B, is the symmetric
bilinear form defined by B,(a. b) = cp(a . b) then
2 Sig B, = # {half-branches of U> q
We show that our quadratic form $,‘, c1 = dg is of the form B, for a suitable cp. Define
q: Cn,/Jac(g) + Iw by
where o is a Gorenstein generator as in (1.4) (iv). Then
q(Jac(Jac(g))) = Res
Remark (2.3). In fact, the method of Aoki et al. generalizes to the following: Let ferns
define a finite mapping, then 2 Sig B, is the difference between the number of half-branches
with g > 0 and the number of half-branches on which g < 0. Note further that the signature
of the quadratic form as B, can be computed as the dimension of &/Jac(g) minus twice the
dimension of a maximal square-zero ideal, comparing Theorems 1.1 and 1.2 of Eisenbud
and Levine [7].
Proof of Theorem (2.1). Let n: d -+ %’ be the normalization of % and let p = n* z be the
pull back of the l-form c1 to the normalization.
LEMMA (2.4). Sig$: + Sig$; = Sig$; + Sig$i.
Proof Consider the subspace P? = OV + Q&I of fig(*) = Q,(e). On V there is the
quadratic form Y = ‘I-“, = YP defined in (1.3), from which the Ic/‘s are induced. With respect
to Y one has 0% = (O&’ and ~3 = (Sg./?)‘. So the statement follows from the following
lemma:
LEMMA (2.5). Let Y be a quadratic form on a vector space Y whose null space has finite
codimension in Y. Then for any subspace 4p of V one has:
Sig Y,, + Sig YlyL = SigY q IXI
The above two lemmas show that our sum of signatures does not change under
normalization of g. We have reduced the proof of Theorem (2.1) to the special case where V
is smooth: since the normalization map is an isomorphism away from the singular point, the
numbers of outbound and of inbound half-branches are the same on V? and 3. So we can
assume:
OcP = ,+i w{ti> jil R{sj9 uj>/(“f + l)*
Here a is the number of real branches and b is the number of complex conjugate pairs, so
ONE-FORMS ON SINGULAR CURVES 509
r = a + 2b is the total number of branches of %. Similarly, since ?? is smooth:
0’~ = ~ iW{ti}.dti ~ iw{Sj, Uj}.dsj/(uf + 1).
i=l j= 1
The modules R* split into corresponding direct sums, the summands being pairwise
orthogonal with respect to $*, so we reduce to the case where cw has only one summand.
LEMMA (2.6). Let I?% = [w {f>, cc)% = [w {t}.dt. Consider a jinire I-form z = c ajrj.dt of
j2 N
order N (so aN # 0). Then
if N is even
if N is odd ’
Here$=$+ifN>Oand$=$-ifN50.
ProoJ Suppose N L 0. Then R; = 0 and dim RL = N. We can choose as a basis for
R+:(tmN.a, tmN+l.a,. . . , t-’ .a}, with respect to which the matrix of II/’ has a very simple
form with the number uN on and zeros below the anti-diagonal. From this one reads off the
signature immediately. For N I 0 the proof is of course similar. Ix)
LEMMA (2.7). Let &W = rW{s, u}/(u’ + l), 0% = L?w.ds. Consider a finite I-form r on %.
Then Sig $h = 0.
Proof: On the @%-modules R * we now have a transformation U: w H u.o, whose square
is - 1. Thus $*(uo,, uwI) = $*(u*.w,, 02) = - $*(wi, w,), so II/* is under the auto-
morphism U equivalent to - I/ *. Consequently Sig +.f = 0. EJ
We have now proved Theorem (2.1), since by Lemma (2.7) the complex conjugate
branches of the complexification of ‘3 give no contribution to the signature, while by
Lemma (2.6) the real half-branches give a contribution in agreement with the orientation: if
the order N of a l-form a on a branch is even, then this branch has one inbound and one
outbound half-branch, while if N is odd both half-branches are outbound or inbound
accordingly as the sign of uN is positive or negative. LZJ
COROLLARY (2.8). Let $5’ be a real Gorenstein curve and let w be a Gorenstein generator.
Then the number of inbound half-branches with respect to o is the same as the number of
outbound half-branches.
Proof In this case R+ = R- = 0 by (1.4) (ii). Hence the result follows from
Theorem (2.1). q
Remarks (2.9). Corollary (2.8) should be seen as an expression of a “geometric sym-
metry” for Gorenstein curves. In the case of an isolated comp!ete intersection case (2.8) can
be proved more directly as follows: Let f: OX”, 0 + UP-l, 0 define Q and consider a small
sphere S around 0 in R”, transverse to %‘. The map& has 0 as a regular value and for
x Ef,; ‘(0) the jacobian off;, at x is positive or negative accordingly as x lies on an outbound
or inbound half-branch. (This can be seen from the formula for o given in (1.4)(iv).) Since the
degree of any map S + W-r is zero, the result follows. More generally, for a smoothable
Gorenstein curve singularity (2.8) is clear,
510 James Montaldi and Duco van Straten
Acknowledgement-This research was undertaken at the University of Utrecht. James Montaldi was partially
supported by the N.W.O. (the Dutch Organization for Scientific Research). Duco van Straten was supported by the
Dutch Ministry of Social Security.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
REFERENCES
K. AOKI, T. FUKUDA and T. NISHIMURA: On the number of branches of the zero locus of a map germ
(KY’, 0) + (DB”-i, 0). Topology and Computer Science. (1987), 347-363.
V. I. ARNOLD: Indices of singular points of l-forms on a manifold with boundary, the convolution of
invariants of groups generated by reflections, and the singular projection of smooth surfaces. Russian Math.
Surveys 34(2) (1979), l-42. (Transl. from Uspekhi Mat. Nauk. 34(2) (1979), 3-38.)
R. BASSEIN: On smooothable curve singularities: Local methods. Math. Ann. 230 (1977). 273-277.
J. W. BRUCE and R. M. ROBERTS: Critical points of functions on analytic varieties. Topology 27 (1988). 57-90.
R.-O. BUCHWEITZ: Thesis, Universite de Paris VII, (1981).
R.-O. BUCHWEITZ and G.-M. GREUEL: The Milnor number and deformations of complex curve singularities.
Iwent. Math. 58 (1980), 241-281.
D. EISENBUD and H. I. LEVINE: An algebraic formula for the degree of a C” map germ. Ann. Mnth. 106 (1977).
19-44.
J. HERZOG and R. WALDI: Cotangent functor of curve singularities. Manuscr. Math. 55 (1986) 307-341.
E. J. N. LOOIJENGA: Isolated singular points of complete intersections. Lecture Notes of the London Math. Sot.
77, Cambridge Univ. Press (1984).
J. MILNOR: Singular points of complex hypersurfaces. Ann. Math. Stud. 61, Princeton Univ. Press. Princeton
(1968).
J.-P. SERRE: Groupes algebriques et Corps de Classes. Hermann, Paris (1969).
H. M. WEBER, Lehrbuch der Algebra, band I, (Zc Auflage), F. VIEWEG, Braunschweig (1912).
Mathematics Institute
University of Warwick
Coventry CV4 7AL
U.K.
F. B. Mathematik Universitiit Kaiserslautern
Erwin Schriidingerstrape
6750 Kaiserslautern
West Germany.
