On the legacy of free divisors: discriminants and Morse-type

On the legacy of free divisors: discriminants and Morse-typesingularities

James Damon

American Journal of Mathematics, Volume 120, Number 3, June 1998,pp. 453-492 (Article)

Published by The Johns Hopkins University PressDOI: 10.1353/ajm.1998.0017

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ON THE LEGACY OF FREE DIVISORS: DISCRIMINANTS ANDMORSE-TYPE SINGULARITIES

By JAMES DAMON

Abstract. We investigate when the freeness of a divisor (V, 0) is inherited by the discriminantsfor the versal deformations of nonlinear sections of V. We introduce Morse-type singularities forsections and give a criterion for freeness of the discriminant in terms of (V, 0) generically havingMorse-type singularities. This criterion is applied to determine when the bifurcation sets of mappingsand smoothings of space curves and complete intersections are free. It also explains the failure offreeness for discriminantal arrangements of hyperplanes.

Introduction. Since the introduction of the notion of free divisors by Saito[Sa], we have gradually become aware of just how commonplace they are. Saito[Sa] and Looijenga [L2, Chapter 6] showed that the discriminants of the versalunfoldings of isolated hypersurface and complete intersection singularities arealways free divisors. Terao [To1] and Bruce [Br] proved that the bifurcation setsassociated to the versal unfoldings of isolated hypersurface singularities are alsofree divisors. Also, Terao [To2] proved that Coxeter arrangements (hyperplanearrangements consisting of the reflecting hyperplanes of a Coxeter group) arelikewise free divisors when viewed as germs of hypersurfaces. Very recently,Van Straten [VS] showed that the discriminant of the versal deformation of aspace curve singularity is a free divisor.

While free divisors are generally rigid spaces canonically associated to uni-versal objects such as the versal unfoldings, they give rise, via the operationsof pull-back and transverse intersection, to large rich families of “almost freedivisors and complete intersections.” These families not only include isolatedcomplete intersection singularities, but also discriminants of finitely determinedmap germs, bifurcation sets for unfoldings of isolated hypersurface singularities,generic hyperplane arrangements based on free arrangements, etc. (see [DM],[D8], and [D9]).

Importantly, although such almost free singularities are in general highlynonisolated, they have the singular analog of the Milnor fiber with the same ho-motopy properties as for isolated singularities. There is also an algebraic formula

Manuscript received September 13, 1997; revised February 25, 1997.Research supported in part by the National Science Foundation.American Journal of Mathematics 120 (1998), 453–492.

453

454 JAMES DAMON

for the associated singular Milnor number (again see [DM], [D8], and [D9]).These formulas allow us to make explicit calculations of higher multiplicities fornonisolated singularities as Teissier [Te] originally did for isolated hypersurfacesingularities.

In this paper we examine how the freeness of a divisor V , 0 � Cp, 0 leaves a

further legacy of this freeness via the discriminant for the deformation theory ofnonlinear sections of V , 0. Such a theory is given by KV -equivalence (see [D4]and [D6]). Let F be a versal unfolding of f0: C

n, 0 ! Cp, 0 for KV -equivalence.

We define the KV-discriminant DV (F) to be the subset of parameter values whereF fails to be (algebraically) transverse to V .

If V , 0 is a free divisor, our first theorem gives a sufficient condition that thisfreeness is inherited by the discriminant DV (F). The sufficient condition is thatthe discriminant is reduced using a 0th Fitting ideal structure. Then, the moduleof logarithmic derivations Derlog(DV(F)) is shown to be exactly the module ofKV-liftable vector fields.

Second, we establish a sufficient condition that this inheritance of freenessholds for all finite KV -codimension germs f0: C

n, 0 ! Cp, 0. This condition

involves V generically having “Morse-type singularities” in dimension n wheren < hn(V), the holonomic codimension of V . Such Morse-type singularitieshave only one singular vanishing cycle in their singular Milnor fibers (so thatPicard-Lefschetz theory can be applied for KV-equivalence). For example, theMorse-type singularities of dimension two which occur as V varies among thestable discriminants in C

3 are the (complex analogs) of the generic bifurcationsfor Legendrian curves [A1], which are the Legendrian analogues of those usedby Arnold [A2] in his construction of invariants for plane curve singularities (seealso x6).

We classify “Morse-type singularities” and deduce a number of properties.As a consequence we establish that the property of generically having Morse-type singularities is possessed by discriminants of versal deformations of bothsimple hypersurface singularities and sufficiently low codimension ICIS singular-ities and their multigerms, by Boolean arrangements, as well as other examples.An important fact which emerges is that the freeness of classical discriminants,bifurcation sets, discriminants for arrangements, etc. are all part of this singleresult for KV -discriminants.

We deduce several key results. First, we extend the results of Bruce [Br]and Terao [To2]. They proved bifurcation sets are free divisors for hypersurfacegerms. We extend this to bifurcation sets for finitely determined map germsf0: C

n, 0 ! Cp, 0 with n � p provided (n, p) satisfy certain conditions. For

example, it follows that the freeness of bifurcation sets holds independent of nfor corank 1 singularities when p < 5 and for arbitrary complete intersectionswhen p < 4. Mond and DuPlessis have indicated to the author that they have alsoobtained a similar version of this result in terms of the “semi-nice dimensions.”

ON THE LEGACY OF FREE DIVISORS 455

Furthermore, the dimension restrictions we obtain are exact. Also, in x9 we showquite generally that the bifurcation sets of smoothings of space curve singularitiesare free, as well as bifurcation sets for functions on ICIS.

Next, we consider the Manin-Schechtman discriminantal arrangements [MS].Orlik-Terao [OT] show that in general they are not free, although Bayer-Brandt[By] [BB] have found special configurations which do yield free discriminantalarrangements. We show that the discriminantal arrangements are not the fulldiscriminants, but are only linear sections of the KV-discriminant. Furthermore,while the discriminantal arrangements may not be free, we show that the fullKV-discriminant is always a free divisor.

Just as the Manin-Schechtman discriminantal arrangements are generaliza-tions of braid arrangements, there is another direction in which braid arrange-ments generalize for “multitype particle systems.” This can be done in such away as to simultaneously include both colored and uncolored braids. We showthat quite generally all configuration spaces for such systems arise by consideringnonlinear Boolean arrangements. Specifically the configuration spaces arise as thecomplement of the bifurcation sets for nonlinear Boolean arrangements. Hence,just as in the case of both colored and uncolored braids, the configuration spacesare complements of free divisors.

The failure of V to generically have Morse-type singularities is due to highermultiplicities of V at points near 0. This is a common situation which leads toa notion of a free* divisor, which is a weaker form of free divisor. In part twoof this paper we consider this situation and prove that the module of KV-liftablevector fields is still free of the correct rank and belongs to Derlog(DV(F)). Nowthe KV-discriminant is no longer reduced using the 0th Fitting-ideal structure.We show that the KV-discriminant is always a free* divisor provided we remainbelow the holonomic codimension of V .

This weakened version appears when we also consider the KV -discriminantfor the versal deformations of almost free complete intersections. Now we mustuse a more restrictive notion of liftable vector field, which only agrees with theusual notion for the case of ICIS. However, the module of such liftable vectorfields is again a free submodule of Derlog(DV(F)). We again prove Derlog(DV(F))is a free* divisor with a nonreduced structure for the discriminant.

The author is grateful for a number of comments by David Mond.

1. Free divisors, holonomic codimension, and KV-equivalence. We be-gin by recalling Saito’s notion of a free divisor V , 0, which is defined by thefreeness of the module of logarithmic vector fields. Related is the module ofvector fields annihilating a good defining equation h for V . These modules allowus to define “logarithmic tangent spaces” used for transversality conditions. First,we introduce some notation. For a holomorphic germ f0: C

n, 0 ! Cp, 0, the

tangent space to the space of germs C(n, p) at f0 consists of germs of vector fields�: C

n, 0 ! TCp such that � � � = f0 (for �: TC

p, 0 ! Cp the projection), and

456 JAMES DAMON

is denoted by �( f0). Thus,

�( f0) ��! OC n ,0

(@

@y1, : : : ,

@

@yp

)= OC n ,0

�@

@yi

�

(we shall denote the R-module generated by '1, : : : ,'k by Rf'1, : : : ,'kg, orRf'ig if k is understood). Also, we let

�n = ��idC n ,0

� �! OC n ,0

�@

@xi

�

and similarly for �p. We also denote the maximal ideal of OC n ,0 by mn.If (V , 0) � C

p, 0 is a germ of a variety, then we consider the module ofvector fields tangent to V . Let I(V) denote the ideal of germs vanishing on V .Then (following Saito [Sa]) we let

Derlog(V) = f� 2 �n: �(I(V)) � I(V)g.

It extends to a sheaf of vector fields tangent to V , Derlog(V) which is easily seento be coherent [Sa]. Then, V , 0 is called a Free Divisor by Saito if Derlog(V) isa free OC p ,0-module. Its rank is then necessarily p. We have already mentioneda number of examples of free divisors due to Saito [Sa], Looijenga [L2], Terao[To1] [To2], and Bruce [Br] and Van Straten [VS].

Likewise, let H be an equation defining V . We also define

Derlog(H) = f� 2 �p: �(H) = 0g.

The associated sheaf Derlog(H) is also coherent. We recall (see [DM]) that for Hto be a good defining equation for the free divisor V , 0 � C

p, 0 means that V isdefined locally by a germ H: C

n, 0 ! C , 0 such that there is a germ of an “Euler-like” vector field e with e(H) = H. This follows if V is weighted homogeneous,and H can be chosen weighted homogeneous of non-zero weight. However, asobserved in [DM], there is always a good defining equation for V � C . Also, ifV is a free divisor then so is V � C , and by results in [D6], for the considerationof equivalence of sections of germs of varieties, we can replace V by V � C

without changing the deformation theory. Then, by [DM, Lemma 3.3]

Derlog(V) = Derlog(H)�OC p ,0feg

Hence, if V is a free divisor with good defining equation, then Derlog(H) is afree OC p ,0-module of rank p� 1.

Logarithmic tangent spaces. If M � �n is an OC n ,0-module, let f�ig de-note a set of generators for M, and let hMi(x) denote the subspace of TxC

n


spanned by f�i (x)g (by coherence this is well-defined for x in some sufficientlysmall neighborhood of 0). Using it, we define the logarithmic tangent space to Vat x 2 V by

Tlog(V)(x) = hDerlog(V)i(x).

This agrees with the usual tangent space at the smooth points of V . As observedin [DM, Proposition 3.11], the elements of Derlog(V) are tangent to the strata ofthe canonical Whitney stratification of V (defined e.g. in [M2] or [Gb]). Thus, ifSi is the canonical Whitney stratum containing x then Tlog(V)(x) � TxSi. Just asfor Derlog(V), we can define

Tlog(H)(x) = hDerlog(H)i(x).

It follows from Lemma 2.1 of [D8] that whether Tlog(H)(x) = Tlog(V)(x) is inde-pendent of the good defining equation H.

If Si is the canonical Whitney stratum containing x, there are the inclusions

Tlog(H)(x) � Tlog(V)(x) � TxSi.(1.1)

Holonomic and other codimensions. We recall certain codimensions whichmeasure when we have equality between the tangent spaces in (1.1). For thecanonical Whitney stratification S of V , a stratum Si 2 S is a holonomic stratumif for all x 2 Si,

Tlog(V)(x) = TxSi.(1.2)

It is a H-holonomic stratum if (1.2) holds instead for Tlog(H)(x); and it is aweighted homogeneous stratum if near each x 2 Si there are local coordinatesabout x so that (V , x) is weighted homogeneous near x. We observe that if V isweighted homogeneous near x, then there is a H and an “Euler vector field” e sothat e(H) = H and e(x) = 0. Hence,

Tlog(H)(x) = Tlog(V)(x).(1.3)

Then, we recall the various codimensions:

h(V)def= maxfk: all strata Wi of codimension < k are H-holonomicg,

i.e., h(V) is the codimension of the largest stratum which is not H-holonomic.We similarly define holonomic codimension hn(V) and the weighted homogeneouscodimension wh(V). By (1.2) and (1.3) we have

minfhn(V), wh(V)g � h(V) � hn(V).(1.4)

458 JAMES DAMON

If all strata are holonomic then we say that V is holonomic and write hn(V) = 1(i.e. the empty stratum has infinite codimension). We analogously have the notionof being H-holonomic.

KV-equivalence. Next we recall the notion of KV equivalence (see [D4],[D6], and [DM] for details). It is defined using a subgroup of the contact groupK, introduced by Mather [M-III], which consists of germs of diffeomorphismsof C

p+n, 0 of the form Ψ(x, y) = (Ψ1(x), Ψ2(x, y)) such that Ψ(C n � f0g) �C

n � f0g. It defines K-equivalence for germs f0: Cn, 0 ! C

p, 0 via the action:graph(Ψ � f0) = Ψ(graph( f0))).

If (V , 0) � Cp, 0 is a germ of a variety, then we define

KV = fΨ 2 K: Ψ(C n � V) � Cn � Vg.

This yields KV -equivalence by the restriction of the above action. Just as K-equivalence captures the equivalence of the germs of varieties f�1

0 (0), so too theequivalence KV captures the equivalence of the germs of varieties f�1

0 (V).We can extend KV to the group of unfolding-equivalences KV,un for unfold-

ings on parameters u 2 Cq, consisting of Ψ(x, y, u) = (Ψ1(x, u), Ψ2(x, y, u), u)

with Ψ(C n � V � Cq) � C

n � V � Cq. It similarly acts on unfoldings. These

groups (together with the associated unfolding groups) are “geometric subgroupsof A or K” which satisfy the basic theorems of singularity theory by [D4]. Theirassociated extended tangent spaces are given by

TKV,e � f0 = OC n ,0

�@f0@x1

, : : : ,@f0@xn

, �0 � f0, : : : , �r � f0

�;

and for an unfolding F(x, u) = (F(x, u), u),

TKV,un,e � F = OC n+q ,0

(@F@x1

, : : : ,@F@xn

, �0 � F, : : : , �r � F

)

where Derlog(V) is generated by �0, : : : , �r. These are the “deformation theoretic”tangent spaces and the quotients

NKV,e � f0 = �( f0)=TKV,e � f0 and NKV,un,e � F = �(F)=TKV,un,e � F

are the analogues of T1. Also, we say that f0 has finite KV-codimension if

dimC (NKV,e � f0) <1.

Remark. There is another equivalence, KH-equivalence, an analog of KV-equivalence for which Ψ preserves instead the level sets of the defining equation


H. It is the KH,e-codimension of a germ which equals its singular Milnor number(see [DM] and [D8]).

In addition to the tangent and normal spaces, we also consider the associatedsheaves. Let T KV � f0,T KV � F,NKV � f0, and NKV � F denote the associatedsheaves to the tangent and normal spaces TKV,e � f0, TKV,un,e � F, NKV,e � f0, andNKV,un,e � F.

2. Critical sets and discriminants for KV -equivalence. In this section wedefine both the critical set and discriminant for KV-equivalence. We shall see thatfor the case of V a free divisor, the critical sets and discriminants have manycommon features with the usual critical sets and discriminants for germs definingisolated complete intersection singularities. Specifically we use Teissier’s method[Te2] for associating a nonreduced structure to the discriminant via the 0th Fittingideal.

Given V , 0 � Cp, 0 and a germ f0: C

n, 0 ! Cp, 0 which has finite KV-

codimension, let F: Cn+q, 0 ! C

p+q, 0 be a KV-versal unfolding of f0. We writeF(x, u) = (F(x, u), u), again using local coordinates u for C

q. Also, we denote by� the projection C

n+q, 0 ! Cq, 0.

Definition 2.1. We define the KV -critical set of F to be

CV(F) = supp(NKV,un,e � F) =�supp(NKV � F)

�(where NKV � F is the normal sheaf as defined in x2). The KV-discriminant ofF is defined to be DV (F) = �(CV(F)).

LEMMA 2.2. Let F: Cn+q, 0 ! C

p+q, 0 be aKV-versal unfolding of f0. Then (1)theKV-critical set and discriminant of F are analytic subsets of the same dimension;and (2) CV(F) consists of points (x0, u0), with y0 = F(x0, u0), such that the germF(�, u0): C

n, x0 ! Cp, y0 is not algebraically transverse to V at x0.

Proof. First for (2), Let I(V) and Θ(F) denote the sheaves associated to I(V)and �(F). The sheaf inclusion

I(V) �Θ(F) � T KV � F

implies that if (x, u) 2 supp(NKV �F) then F(x, u) 2 V . Next, if F(x, u) 2 V , then(x, u) =2 supp(NKV � F) is equivalent to

TlogV(F(x,u)) + DF(x, u)(T(x,u)Cn) = TF(x,u)C

p.(2.3)

However, (2.3) is exactly the statement that F is algebraically transverse to V at(x, u) (see e.g. [D8, x2]).

For (1), the definition of CV (F) as supp(NKV �F) implies that it is an analyticgerm. Next, f0: C

n, 0 ! Cp, 0 having finite KV-codimension implies by [D4,

460 JAMES DAMON

x2] (or see [D6, x1]) that f0 is algebraically transverse to V in a puncturedneighborhood of 0. Hence, by the characterization of CV (F) in terms of algebraictransversality, it follows that CV (F) \ (C n � f0g) = f0g. Thus, dim CV (F) � qand � j CV(F) is finite to one. Hence, by Grauert’s theorem DV (F)) = �(CV(F))is the image of an analytic subset CV (F) under a finite map, hence is also ananalytic germ of the same dimension as CV (F).

Remark. If V is a free divisor with good defining equation H, then it canbe shown that an equivalent description of points (x0, u0) 2 CV(F) is that therestriction of the projection �: C

n+q, 0 ! Cq, 0 to X = F�1(V) is not the germ

of a submersion at (x0, u0).Second, in the case that V is a free divisor we have more specific information

about CV (F) and DV (F).

PROPOSITION 2.4. Suppose that V , 0 � Cp, 0 is a free divisor, and that F: C

n+q,0 ! C

p+q, 0 is a KV-versal unfolding of f0: Cn, 0 ! C

p, 0, with n < hn(V). Then,both CV(F) and DV(F) are Cohen-Macaulay of dimension q� 1.

Proof. As V is a free divisor, Derlog(V) is freely generated by p elements,�0, : : : , �p�1. By Lemma 2.2, CV(F) consists of those points (x, u) 2 C

n+q forwhich

dimC

*@F@x1

(x, u), : : : ,@F@xn

(x, u), �0 � F(x, u), : : : , �p�1 � F(x, u)

+< p.

By results of Macaulay-Northcott [Mc] [No], this defines a subvariety of Cn+q

of codimension � n + p � (p � 1) = n + 1; so its dimension � q � 1. Also,if the codimension is exactly n + 1 (i.e. dimension exactly q � 1), then it is adeterminantal variety, and so Cohen-Macaulay.

If dim CV (F) = q, then so would dim�(CV(F)) = q, implying �(CV(F)) islocally onto near 0. However, as F is versal, the map F is algebraically trans-verse to V . As n < hn(V), geometric and algebraic transversality agree (see [D8,x3]). Thus, F is geometrically transverse to V (i.e., to the strata of the canon-ical Whitney stratification) in a neighborhood of 0. Thus, by the parametrizedtransversality theorem, there are points u0 2 C

q arbitrarily close to 0 for which(a representative of) F(�, u0) is geometrically transverse to V . Again it is alge-braically transverse. Thus, by (2) of Lemma 2.3, u0 =2 DV (F) = �(CV(F)). Thus,neither can have dimension q; hence, their common dimension is at most q� 1.

By the above discussion, since the dimension � q � 1, it must be exactlyq � 1. Thus, the critical set CV(F) is Cohen-Macaulay of dimension q � 1 and� j CV (F) is finite to one. Now we can apply a result from commutative algebra(see [Se, Chapter 4]) which implies that DV (F) = �(CV(F)) is Cohen-Macaulayof dimension q� 1.


We observe that the only two properties of F which were used in Proposi-tion 2.4 were: that F: C

n+q, 0 ! Cp, 0, viewed as a map, is algebraically trans-

verse to V , and that n < hn(V). Together these imply that generically F(�, u0) istranverse to V . Thus, we obtain as a corollary

COROLLARY 2.5. Suppose V , 0 � Cp, 0 is a free divisor, and that F: C

n+q, 0 !C

p+q, 0 is any unfolding of a KV-finite codimension germ f0: Cn, 0 ! C

p, 0 suchthat the map germ F is transverse to V. If n < hn(V), then both CV (F) and DV (F)are Cohen-Macaulay of dimension q� 1.

3. Freeness of the module of liftable vector fields. We again let V , 0 �C

p, 0 be a free divisor and suppose f0: Cn, 0 ! C

p, 0 has finite KV-codimension.We also let f'1, : : : ,'qg project to a basis for NKV,e � f0 so that

F(x, u) = (F(x, u), u) with F(x, u) = f0(x) +qX

j=1

uj'j(3.1)

is theKV-versal unfolding of f0. Then, it follows from the preparation theorem thatNKV,un,e �F is a finitely generated OC q ,0-module on the generators f'1, : : : ,'qg.Hence, we have an exact sequence

0 ��! L�

��! OC q ,0

n@@ui

o ��! NKV,un,e � F ��! 0(3.2)

where the map � sends @@ui

7! 'i and L denotes the kernel of �.

PROPOSITION 3.3. Suppose that V , 0 � Cp, 0 is a free divisor, and that f0: C n, 0 !

Cp, 0 has finite KV-codimension with n < hn(V). Let F: C

n+q, 0 ! Cp+q, 0 be a

KV-versal unfolding of f0. Then, L has the following properties:

(i) L is the Lie algebra of liftable vector fields (see Definition 3.5);

(ii) L is a free OC q ,0-module; and

(iii) L is of rank q so � is given by a q � q-matrix whose determinant is thegenerator for the 0th Fitting ideal of NKV,un,e � F and defines the KV discriminantof F (possibly with nonreduced structure).

It will then follow by Saito’s criterion [Sa, x2], that provided the 0th Fittingideal structure for DV (F) is reduced, then DV (F) is a free divisor. We shallestablish these properties, concluding with a proof that the KV discriminant isa free divisor provided that the 0th Fitting ideal structure for DV (F) is reduced,and then establish sufficient conditions for this.

The proof will be carried out in several steps. First, (ii) follows immediatelyby the direct analogue of an argument of Looijenga [L2, Corollary 6.13].

LEMMA 3.4. For the free divisor V , 0 � Cp, 0, with F a KV-versal unfolding

of f0, L is a free OC q ,0-module of rank q.

462 JAMES DAMON

Proof of Lemma 3.4. Again we can first apply results of Macaulay [Ma] andNorthcott [No] to NKV,un,e �F. It is the quotient of a free OC n+q ,0-module of rankp by a submodule on n + p generators. Thus, supp(NKV,un,e � F) has dimension� n + q � (n + 1) = q � 1 and if there is equality, then NKV,un,e � F is a Cohen-Macaulay module. Since CV (F) = supp(NKV,un,e � F) and has dimension q � 1,we conclude that NKV,un,e �F is a Cohen-Macaulay OC n+q ,0-module with supportCV(F). Thus, it is a Cohen-MacaulayOCV (F),0-module. As � j CV (F) is finite withDV(F) = �(CV(F)), it follows that NKV,un,e �F is Cohen-Macaulay as a ODV (F),0-module, and hence as a OC q ,0 -module. Then, the Auslander-Buchsbaum formulaimplies that NKV,un,e �F has projective dimension 1. This implies that in (3.2), Lis a free OC q ,0-module.

Definition 3.5. We say a vector field � 2 �q is a KV-liftable vector field (ormore briefly a “liftable vector field” ) if there are germs of vector fields

� 2 OC n+q ,0

�@

@x1, : : : ,

@

@xn

�and � 2 OC n+q ,0f�0, : : : , �p�1g

satisfying:

(� + �) (F) = � � F.(3.6)

Then, as

��(F) + � � F 2 TKV,un,e � F and �(F) = �(�)

(3.6) is equivalent to � 2 L. Thus, L is the vector space of liftable vector fields.To check that it is a Lie algebra, suppose that �i 2 �q are liftable for i = 1, 2.Thus, there are �i, and �i satisfying (3.6) for each �i. If we write �0 = � + � , then(3.6) may be written in the form

�0(F) = � � F.(3.7)

Then, the Lie algebra structure follows from the following lemma.

LEMMA 3.8.

(1) If � 2 OC n+q ,0f@@xig and � 2 �q, then the Lie bracket [�, �] 2 OC n+q ,0f

@@xig.

(2) Both OC n+q ,0f@@xig and OC p+q ,0f�0, : : : , �p�1g are Lie algebras for the

usual Lie bracket of vector fields.

Proof. Both of these follow by straighforward calculation using in (2) thatDerlog(V) is a Lie algebra. For example, by linearity, its enough to check (1) for� = g(u) @

@uiand � = h(x, u) @

@xj, and then [�, �] = g @h

@ui

@@xj

.


Note for (2) both are the Lie algebras for the groups of unfoldings of diffeo-morphisms, Dun and DV,un.

To finally establish (i) of (3.4), we observe that

[�01, �02] = [�1, �2] + [�1, �2] + [�1, �2] + [�1, �2].(3.9)

The first three terms on the RHS are in OC n+q ,0f@@xig by (1) of Lemma 3.8. Denote

the sum of these terms by �00. Then, (3.7) implies that �0i and �i are F-related.Hence, so are [�01, �02] and [�1, �2]. Thus, [�1, �2] is liftable using �00, [�1, �2]. Thisyields (i).

Furthermore, by (ii) of (3.3), we know that L is a free OC q ,0-module. As �is an inclusion, its rank is no greater than q, however, it cannot be less than qas supp(NKV,un,e � F) (as an OC q ,0-module) has dimension q � 1. Thus, L hasrank q, and supp(NKV,un,e �F) is defined by the vanishing of det (�), which is thegenerator of the 0th Fitting ideal of NKV,un,e � F. This completes (iii).

Lastly, we are in a position to apply Saito’s criterion, which we recall.

3.10. Saito’s criterion. Let �1, : : : , �n be germs of holomorphic vector fieldson C

n given by

�i =nX

j=1

aij@

@xj.

Suppose that theOC n ,0-module generated by the �i is a Lie algebra. If H = det (aij)is reduced, then the hypersurface germ X defined by H is a free divisor withgenerators f�ig for Derlog(X).

Applying this to L allows us to conclude the freeness of DV (F).

THEOREM 1. Suppose that V , 0 � Cp, 0 is a free divisor and that F is a KV-

versal unfolding of f0. If the KV-discriminant DV (F) has reduced structure via the0th Fitting ideal, then DV(F) is a free divisor with

Derlog(DV(F)) = module of KV-liftable vector fields.

Proof. By (i) of (3.3), L is a Lie algebra generated by the q free OC q ,0-modulegenerators. Futhermore, the coordinate functions of these generators are exactlythe entries of the matrix representation of �. Thus, provided det (�) defines DV(F)with a reduced structure, it follows by Saito’s criterion that DV(F) is free and thegenerators of Derlog(DV(F)) are the generators for the module of liftable vectorfields L. Thus, we have obtained the first form of our main result.

464 JAMES DAMON

4. Morse-type singularities. By Theorem 1, we have a sufficient conditionfor the freeness of the KV-discriminant DV (F) of a versal unfolding F, providedDV(F) endowed with the structure determined by the 0th Fitting ideal is reduced.We define in this section the notion of a Morse-type singularity for a sectionof a germ V . Geometrically these singularities are similar to those of Morsesingularities for isolated hypersurface singularities in that provided n < h(V)they have a single singular vanishing cycle. We derive the basic properties forsuch singularities, including sufficient conditions for existence, normal forms, andgenericity. Then, in the next section we verify that V having generic Morse-typesingularities implies that DV (F) is a free divisor for any KV-versal unfolding F.

Definition 4.1. Given V , 0 � Cp, 0 and an integer n > 0, then a germ

g: Cn, 0 ! C

p, 0 is a Morse-type singularity in dimension n if g is KV -equivalentto a germ f0 which has KV,e–codim = 1 and for a common choice of lo-cal coordinates, both f0 and V are weighted homogeneous. Furthermore, wesay V has a Morse-type singularity in dimension n at x if there is a germg: C

n, 0 ! Cp, x which is a Morse-type singularity in dimension n (for this,

we use K(V,x)-equivalence).

Remark 4.2. If g: Cn, 0 ! C

p, 0 is a Morse-type singularity in dimen-sion n for V , then first the weighted homogeneity implies KH,e-codim(g) =KV,e-codim(g) = 1. Hence, provided n < h(V) by [DM], there is exactly onesingular vanishing cycle in the singular Milnor fiber of g. Thus, this is analogousto a Morse singularity for isolated hypersurface singularities.

We give several examples of Morse-type singularities. The verification willfollow from the results of this section.

Example 4.3. Morse-type singularities of dimension 2 for discriminants ofstable multigerms in C

3.

These are the discriminants for A3, the swallowtail, the multigerm A2A1, andCusp �C in Figure 4.4, and the multigerms A2

1 = A1A1, A31, and Fold �C

2

in Figure 4.5. We show the pull-backs of the discriminants by the Morse-typesingularities together with their stabilizations exhibiting the singular vanishingcycles.

We note that all but one of the generic bifurcations for Legendrian curvesclassified by Arnold [A2] [A1] appear here, except for a second real form of A2

1which is the “bec–a–bec” bifurcation and does not have a real singular vanishingcycle.

A second example is for the Boolean arrangement A � C4. The Morse-type

singularity of dimension 3 is obtained as the inclusion of a generic hyperplanesection. The pull-back together with the singular Milnor fiber, showing the sin-gular vanishing cycle, are given in Figure 4.6.


Figure 4.4.

Figure 4.5.

Finally an example of a free divisor without a Morse-type singularity is givenby the free hyperplane arrangement A � C

3 defined by Q = xyz(x� y). A Morse-type singularity of dimension 2, if it existed, would be obtained as the inclusion ofa generic hyperplane section. However, we see that it would give an arrangementof 4 lines in C

2 with singular Milnor fiber having 2 singular cycles as shown inFigure 4.7. This corresponds to the section having KA,e-codimension 2.

That these examples have the indicated properties will follow from the resultsfrom the remainder of this section and x6 and x7. We shall be fairly thoroughin describing these germs of Morse-type at arbitrary points of V . Before doingso, we first show that for such a singularity the versal unfolding has a reducedKV-discriminant.

Let f0: Cn, 0 ! C

p, 0 be a Morse-type singularity for V , 0 � Cp, 0. Since two

KV-equivalent germs have KV-equivalent versal unfoldings, we may assume localcoordinates are already chosen so both f0 and V , 0 are weighted homogeneous.As KV,e-codim( f0) = 1, we may assume the space NKV,e � f0 is spanned by avector @

@yp= (0, : : : , 0, 1). Thus,

df0(0)(TCn) + Tlog(V)0 + h(0, : : : , 0, 1)i = TC

p.(4.8)

Hence we may assume by a linear change of coordinates that Tlog(V)0 is the sub-space y1, : : : , ys and df0(0)(TC

n) is the subspace ys+1, : : : , yp�1. If df0(0)(TCn)\

Tlog(V)0 6= f0g then a KV -equivalence will allow us to assume the intersection isf0g.

Next, by the versality theorem for KV-equivalence [D4], the KV-versal un-

466 JAMES DAMON

Figure 4.6.

Figure 4.7.

folding is given by

F(x, u) = (F(x, u), u) with F(x, u) = f0(x) + (0, : : : , 0, u).(4.9)

LEMMA 4.10. For the KV-versal unfolding F in (4.9) with wt(yp) 6= 0, thediscriminant DV (F) (defined by u = 0) is reduced and u @

@u is a liftable vector field.In particular, DV (F) is a free divisor.

Proof. Let e0 be the Euler vector field on Cp and e the Euler vector field on

Cp so that e0( f0) = e � f0. If wt(yp) = bp, then

e0(F) = e0( f0) and e � F = e � f0 + (0, : : : , 0, bpu).(4.11)

Hence, by 4.9 and 4.11

�e0 + bpu

@

@u

�(F) = e � F

which says exactly that bpu @@u is KV-liftable, and generates the module of liftable

vector fields (as @@u is not liftable). Thus, the 0th Fitting ideal of NKV,un,e � F as

an OC ,0 -module is generated by u, so DV (F) is reduced and free.

Second, we derive the normal form for a Morse-type singularity. For a germV , 0 � C

p, 0, we let r = dim (TlogV(0)).


LEMMA 4.12. (Local Normal Form) Let f0: Cn, 0 ! C

p, 0 be a Morse-typesingularity for V , 0 � C

p, 0. Then, up to KV-equivalence, we may assume V, 0 =C

r � V0, 0 for V0, 0 � Cp0 , 0, and with respect to coordinates for which V0, 0 is

weighted homogeneous, f0 has the form

f0(x1, : : : , xn) =

0@0, : : : , 0, x1, : : : , xp0�1,

nXj=p0

x2j

1A .

Proof. Suppose we have already applied a KV-equivalence so that both V andf0 are weighted homogeneous. We make a further change of coordinates so thatV , 0 = C

r � V0, 0. We prove this is possible by induction on r. It is trivially truefor r = 0. Suppose it is true for r0 < r. Let �1, : : : , �r 2 Derlog(V) be weightedhomogeneous vector fields which span TlogV(0). We may make a linear weightedhomogeneous change of coordinates so that �i(0) = @

@yi. Let V1 = V\(f0g�C

p�1).

Then, by the versality theorem for KV-equivalence, the inclusion i: Cp�1 ,! C

p

on the last p�1 coordinates is KV -versal, so the deformation by translation in they1 direction is KV -trivial. Furthermore, the trivialization is given by integratingthe weighted homogeneous vector field �1. Hence, by a weighted homogeneouschange of coordinates, V , 0 = C � V1, 0; and by induction V1, 0 = C

r�1 � V0, 0,yielding the desired form for V , 0.

Composing with this weighted homogeneous change of coordinates, f0 is stillweighted homogeneous, and now V , 0 = C

r � V0, 0 so Derlog(V) has generatorsf @@y1

, : : : , @@yr

, �0, : : : , �p0�1g where �i 2 Derlog(V0) and �i(0) = 0. Also, �0 is theEuler vector field.

Next, we change coordinates for f0. If �p0 denotes projection onto Cp0 along

Cr, then �p0 � f0 is KV-equivalent to f0 and hence has KV,e-codimension 1. Thus,

we may assume f0 = (0, f1) with f1 weighted homogeneous and by [D6, x1]

KV0,e � codim( f1) = KV,ecodim( f0) = 1.

Thus, f1 has the same properties as f0 so we may as well assume r = 0, p0 = p,V = V0, f1 = f0, and so TlogV(0) = (0). Then, with these assumptions

KV,e � codim( f0) � dim�TC

p=(df0(0)(TCn) + TlogV(0))

�.

Thus, dim (df0(0)(TCn)) � p � 1. If this latter dimension equals p, then f0 is

algebraically transverse to V and KV,e � codim( f0) = 0. Thus, f0 is a corank onemap. We may suppose that df0(0)(TC

n) is the subspace (y1, : : : , yp�1).Then, we may make a change of coordinates in C

n so that f0 has the formof an unfolding

f0(x1, : : : , xn) = (x1, : : : , xp�1, f0(x1, : : : , xn))(4.13)

468 JAMES DAMON

with df0(0) = 0. This is obtained in a standard way by composing with '�1 where

'(x1, : : : , xn) = (y1 � f0, : : : , yp�1 � f0, xp0 , : : : , xn).

Hence, it is obtained by a weighted homogenous change of coordinates.Finally we must show that by a final KV-equivalence, f0 may be put in the

desired form. It is only here that particular properties of V come into play. Wesee from the form of f0 in (4.13) that

@

@yi2 TKV,e � f0 for i = 1, : : : , p � 1.(4.14)

Hence, since KV,e � codim( f0) = 1, (4.14) implies that the image of TKV,e � f0under the projection onto OC n ,0f

@@ypg ' OC n ,0 has codimension one. Thus, the

image must be mnf@@ypg. Hence, mn must be generated by the n germs

(�i(yp) � f0, i = 1, : : : , p � 1;

@f0@xj

, j = p, : : : , n

).(4.15)

This follows as �0 is the Euler vector field, so by the Euler relation and (4.13),if bp = wt(yp)

�0(yp) � f0 = bpf0 2 m2n.

Also, by the form of f0 in (4.13), for i = 1, : : : , p� 1

�i(yp) � f0 2 mp�1OC n ,0 mod�

m2n

�

and hence project to generators of this ideal. Thus,

f�i(yp), i = 1, : : : , p � 1g generate mp�1OC p ,0 mod�

(yp) + m2p

�.(4.16)

Hence, (4.15) and (4.16) imply

(@f0@xj

=@ f0@xj

, j = p, : : : , n

)spans mn=

�(x1, : : : , xp�1) + m2

n

�.(4.17)


Thus,

TKV � f0 = mn

(@

@y1, : : : ,

@

@yp�1

)(4.18)

+(x1, : : : , xp0�1)

(@

@yp

)+ m2

n

(@

@yp

).

Then, by (4.18) together with Mather’s Geometric Lemma, all such f0 of theform (4.13) and satisfying (4.16) are KV -equivalent. In particular, we may choosef0 =

Pnj=p0 x2

j .Conversely, if we choose f0 as in (4.13) with f0 as above, then provided V

satisfies (4.16), f0 is a Morse-type singularity.

Remark 4.19. It must be noted in our use of Mather’s geometric lemma thatit has not been established that the induced action of KV on ` � jets is a Liegroup action. In the addendum x10, we establish this for the case where V0 hasall positive weights. This takes care of all of the free divisors which interest usas the condition n < hn(V) which appears in the main theorems excludes othercases. However, the lemma is true as stated. For the full result we again give inthe addendum a direct proof that the conditions in (4.18) or (4.27) for a familyimply that the family is KV-trivial. For this proof, we use the reduction lemma forgeometrically defined subgroups of A or K [D3]. With it we prove that trivialityholds for families for KV -equivalence.

Also, using the reasoning of the lemma we may analogously obtain normalforms for Morse-type singularities in the real case using the same proof exceptwe must replace f0 =

Pnj=p0 "jx2

j , where "j = �1.Given one Morse-type singularity, we can deduce the existence of Morse-type

singularities in all other allowable dimensions.

COROLLARY 4.20. If V , 0 � Cp, 0 has a Morse-type singularity in dimension

n, then it has Morse-type singularities in dimensions m � p0 � 1 where p0 = p� rand r = dim (Tlog(V)0).

Hence, we shall speak of V having Morse-type singularities (in the allowabledimensions) without specifying a particular dimension.

Proof. We note that the only condition which is needed is (4.16) which isindependent of m � p0 � 1.

Remark 4.21. We further note that condition (4.16) can be given a geometricinterpretation. This relates to a question raised by Arnold relating to the role ofthe group of linearized automorphisms of V , 0 (see e.g. [A1]). Having expressedV , 0 = C

r�V0, 0 with V0, 0 � Cp0 , 0, then the linear parts of f�i, i = 0, : : : , p0�1g

470 JAMES DAMON

generate the Lie algebra of the Lie group Aut1(V0) of linearized automorphisms ofV0, 0 (they are not really automorphisms, but the linear parts of automorphisms).

Also, �0, as the Euler vector field, will be tangent to any weighted subspace.Hence, we may state the sufficient condition (4.16) geometrically in terms of theLie group Aut1(V0).

COROLLARY 4.22. Suppose V , 0 = Cr � V0, 0 is weighted homogeneous with

V0, 0 � Cp0 , 0 and (Tlog(V0)(0)) = (0). Then V has Morse-type singularities in all

allowable dimensions if and only if there is a weighted hyperplane in Cp0 which is

transverse to the orbits of Aut1(V0) in a punctured neighborhood of 0.

Proof. If the weighted hyperplane is denoted Cp0�1, then �0 is tangent to it so

the condition that C p0�1 is transverse to the orbits of Aut1(V0) is the condition thatthe tangent space of the orbit at y 2 C

p0�1, spanned by f�i(y), i = 0, : : : , p0 � 1g,is complementary to C

p0�1. Keeping in mind that �0 is tangent, we see that thiscondition becomes:

f�i(yp)(y), i = 1, : : : , p0 � 1g only vanish at 0.(4.23)

By the Nullstellsatz, this is equivalent to (4.16).

A third consequence concerns the genericity of Morse-type singularities. LetΣ0 denote the 2-jets of germs f0: C

n, 0 ! Cp, 0 which are algebraically trans-

verse to V at 0. Its complement, denoted Σ1, consists of germs which fail to bealgebraically transverse.

COROLLARY 4.24. If V , 0 � Cp, 0 has Morse-type singularities, then there is a

Zariski open dense subset of Σ1 consisting of jets of Morse-type singularities.

Proof. We use the local coordinates from (4.13). Then, there is a Zariski opensubset in the Grassmanian Gp0�1(C p0) of p0 � 1 planes Π � C

p0 which does notcontain @

@yp. Furthermore, there is a smaller Zariski open dense subset U0 having

the additional property that

fyp, �i(yp), i = 1, : : : , p0 � 1g spans mp0=m2p0 .(4.25)

Then, for �p0 : Cp, 0 ! C

p0 , 0 denoting projection along Cr, consider a germ f0

with

�p0 � df0(0)(TCn) 2 U0.(4.26)

Let K = ker(�p0 � df0(0)). Then, we consider the Zariski open dense subset U of2-jets f0 for which (4.26) holds and d2(yp � f0)(0)jK is nonsingular.

We claim that all f0 with j2( f0)(0) 2 U are Morse-type singularities. It isenough to show that for any family of germs ft 2 U are KV-equivalent for then


U is connected and contains one Morse-type singularity. By applying the firstpart of the reduction in the Normal Form Lemma 4.12 to the family ft, we mayassume that the family has the form (4.13) where f0 is replaced by f0t, except nowfor t 6= 0 f0t may have linear terms in (x1, : : : , xp0�1). Now, applying Nakayama’slemma

TKV � ft = mnfWtg + (x1, : : : , xp0�1)

(@

@yp

)+ m2

n

�@

@yi

�(4.27)

where Wt is the subspace spanned by TCr and dft(0)(TC

n). Thus, dimC (TKV � ft)is constant independent of t, and by (4.27)

@ft@t

2 TKV � ft for each t.

Thus, by Mather’s Geometric Lemma (again see the Addendum), the family ft isKV-trivial, completing the proof.

For a discriminant V , the coordinate yp in the normal form in (4.12) andthe condition (4.16) will play a distinguished role. Although we can multiply allweights by the same positive constant, the sign of the weight wt(yp) is intrinsic.

Definition 4.28. We say that a free divisor V has exceptional weight of pos-itive, negative, or zero type if V satisfies (4.16) with -wt(yp) having the corre-sponding positive or negative sign or = 0.

Lastly, we establish a final result which allows us to extend the Morse-type singularities for discriminants of stable germs to discriminants of stablemultigerms. We recall from [D8] that given germs Vi, 0 � C

pi for i = 1, 2, wecan form the product union

V1 �[V2 = (V1 � Cp2 ) [ (C p1 � V2).

Inductively, we can repeat this construction for a finite number of Vi. By Mather’smultitransversality characterization of stability [M-V], the discriminant of a stablemultigerm is the product union of the discriminants of the individual germs.Hence, results for the product union allow us to relate properties of germs andmultigerms.

PROPOSITION 4.29. If Vi, 0 � Cpi , 0 for i = 1, 2 have Morse-type singularities

of the same nonzero exceptional weight type, then the product union V1 �[V2 hasMorse-type singularities of the same exceptional weight type.

Proof. We let V , 0 � Cp, 0 denote the product union V1 �[V2 where p = p1+p2.

We may independently make weighted homogeneous change coordinates to factor

472 JAMES DAMON

Vi, 0 = Cri � Vi0, 0. Then, the product union factors

V1 �[V2 = Cr1+r2 � (V10 �[V20) .

Hence, we may suppose that TlogVi(0) = (0). Furthermore, we suppose that C pi�1

are the weighted subspaces given by Corollary 4.22. Then, as each Vi has thesame exceptional weight type, we may multiply weights if necessary so thatwt(y(1)

p1) = wt(y(2)

p2) where each Vj has coordinates (y( j)

i ) . Then, we consider theweighted homogeneous subspace

M = (C p1�1 � f0g) + (f0g � Cp2�1) +

* @

@y(1)p1

,@

@y(2)p2

!+.

Let f�(i)j g denote the generators for Derlog(Vi) with �(i)

0 the Euler vectorfield.Then, by Proposition 3.1 of [D8], Derlog(V) has generators

f�j: j = 1, : : : , pg = f�(i)j : i = 1, 2, j > 0; �(1)

0 � �(2)0 , �(1)

0 + �(2)0 g

with (1=2)(�(1)0 + �(2)

0 ) the Euler vectorfield. Likewise, we use coordinates for C p

(y1, : : : , yp) = (y(1)1 , : : : , y(1)

p1�1, y(2)1 , : : : , y(2)

p2�1, y(1)p1

+ y(2)p2

, y(1)p1� y(2)

p2).

Then, since f�(i)j (yp) = ��(i)

j (y(i)pi

): j > 0g

f�(i)j (yp): j > 0g generate OC p ,0fy(i)

1 , : : : , y(i)pi�1g mod OC p ,0fy(i)

pig.(4.29)

Also,

(�(1)0 � �(2)

0 )(yp) = (�(1)0 � �(2)

0 )�

y(1)p1� y(2)

p2

�(4.30)

= �(1)0 (y(1)

p1) + �(2)

0 (y(2)p2

)

= b(y(1)p1

+ y(2)p2

) = byp�1

where b denotes the common (nonzero) weight of y(1)p1

and y(2)p2

. Then (4.29) fori = 1, 2 and (4.30) together imply that the product union V satisfies (4.16) for thesubspace M, yielding the result.

5. Reduced KV-discriminants. Now, we are in a position to deduce that,provided V has Morse-type singularities of dimension n < hn(V) at all points(near 0) on strata of codimension� n+1, then DV (F) is reduced for anyKV -versalunfolding F of a germ f0: C

n, 0 ! Cp, 0.


Definition 5.1. We say that a free divisor V , 0 � Cp, 0, generically has Morse-

type singularities in dimension n if all points on canonical stata of V of codimen-sion � n+1 have Morse singularities of nonzero exceptional weight type, and anystratum of codimension > n + 1 lies in the closure of a stratum of codimension= n + 1.

PROPOSITION 5.2. Let V , 0 � Cp, 0 be a free divisor and let n < hn(V). Sup-

pose V generically has Morse-type singularities in dimension n. Then the KV-discriminant of a versal unfolding for any f0: C

n, 0 ! Cp, 0 is reduced (with the

0th Fitting ideal structure).

Proof. If G is a generator for the 0th Fitting ideal for NKV,un,e � F, then thegerm of G at u0 also defines the 0th Fitting ideal for NKV,un,e � (F, (x, u0)), wherenow potentially F is a multigerm with singularities at several points. However,by the parametrized transversality theorem we may deform u so only one singu-larity remains (i.e., by the finite determinacy at each point, we may add termsof sufficiently high order so that one singularity remains unchanged up to equiv-alence while others are perturbed in general directions so that the parametrizedtransversality theorem applies). Thus, on a Zariski open subset W of DV (F), wewill have only single KV singularities of F occurring.

Furthermore, we claim that any singularity deforms to a Morse-type singu-larity. This follows from the following lemma.

LEMMA 5.3. Let V , 0 � Cp, 0 be a free divisor which has a Morse-type singu-

larity in dimension n at 0. Then, any finitelyKV-determined germ f0: C n, 0 ! Cp, 0

which is not algebraically transverse to V at 0 can be deformed to a Morse-typesingularity.

First using the lemma, we argue as follows. By the openness of versality forKV-equivalence, if we can deform a singularity F(�, u0) to a Morse-type singu-larity (and then if necessary deform away singular points at other points), thenthis occurs within the versal unfolding F so the stratum of u0 is contained in theclosure of W.

First, consider a point F(x0, u0) = y0 on a stratum of codimension > n +1. Then, by assumption on V , we may deform the image y0 to a stratum ofcodimension of = n + 1. Thus, we may assume this for our original y0.

Second, consider a point F(x0, u0) = y0 on a stratum of codimension � n + 1.By the lemma and assumption on V , we may deform F(�, u0) to a Morse-typesingularity. Hence, this occurs in F, so as above, the stratum of u0 is containedin the closure of W.

Thus, every singularity occurring in F deforms to a single Morse-type singu-larity; hence, every irreducible component of DV(F) intersects W. By assumption,G is reduced at every point of the Zariski open subset of W consisting of Morse-type singularities. Hence, G is itself reduced.

474 JAMES DAMON

Proof of Lemma 5.3. If f0 has a singularity at 0, then in the notation ofCorollary 4.24, j2( f0)(0) 2 Σ1. However, that corollary implies that there is aZariski open set of 2-jets U � Σ1 consisting of jets of Morse-type singularities.Hence, we may construct a deformation ft of f0 so that j2( ft)(0) 2 U for t 6= 0.This is the desired deformation.

This proposition together with Theorem 1 yields

THEOREM 2. Let V , 0 � Cp, 0 be a free divisor which generically has Morse-

type singularities in dimension n where n < hn(V). Then, the KV-discriminant ofthe versal unfolding for any f0: C

n, 0 ! Cp, 0 is a free divisor. Moreover,

Derlog(DV(F)) = module of KV-liftable vector fields.

Remark. The conclusion of the theorem is actually false without the conditionn < hn(V). See Example 6.5.

6. Bifurcation sets for finitely determined germs. In this section we provean extension to higher dimensional target spaces of a result of Bruce and Teraoconcerning the freeness of bifurcation sets.

Let f0: Cn, 0 ! C

p, 0 be a finitely (A-)determined germ, and let F: Cn+q, 0 !

Cp+q, 0 be its (A-)versal unfolding. More generally we can consider a multigerm

f0: Cn, S ! C

p, 0 and its (A-)versal unfolding F. Then, F, viewed as a map germ,is stable. We may choose a stable representative F: U ! C

p+q with U = U1�U2.Let C(F) denote the critical set of F. We may suppose that F�1(0)\C(F) = f0gand F j C(F) \ U is a finite-to-one map onto D(F), the ordinary discriminant ofF. We write F in the form F(x, u) = (F(x, u), u). Then, we consider the bifurcationset

B(F) = fu 2 Cq: F(�, u): U1 ! C

p is not (A-)stableg.

This gives a well-defined germ of an analytic set at 0. We mentioned that in thecase p = 1, Bruce and Terao have proven that B(F) is a free divisor. We considerthe general case p � 1. To describe this extension we identify distinguishedclasses of (multi)germs.

Definition 6.1. We say that a finitely A-determined (multi)germ f0: Cn, S !

Cp, 0 with n � p belongs to the “distinguished bifurcation class of (multi)germs”

if it satisfies one of the following:

(1) “general case”: n 6= p + 1 and p � 4;

(2) “worse case”: n = p + 1 and p � 3; and

(3) “best cases”:

(i) corank 1 (multi)germs such that:


(a) n = p + 1 and p � 6 or

(b) n > p + 1 and p � 5

(ii) Σn�p+1 and Σ2,(1) (multi)germs without restriction on n � p.

Remark. The Σn�p+1 germs are germs in the K-equivalence class of the Ak

germs for n � p; and the Σ2,(1) germs are those in the K-equivalence class of theI2,b germs for n = p (see below in (6.6)). Then, the Terao-Bruce result extendsas follows.

THEOREM 3. Let f0: Cn, S ! C

p, 0 be a finitely A-determined germ whichbelongs to the distinguished bifurcation class. Then, the bifurcation set of its A-versal unfolding is a free divisor.

Remark. David Mond and Andrew DuPlessis have indicated that they havealso obtained a theorem similar to this.

The proof of this theorem will occupy the rest of this section. We shall deduceit as a consequence of the preceding results on the freeness of KV-discriminants.To do this, we show that theKD(F)-discriminant can be identified with B(F). Then,it is sufficient to determine when D(F) generically has Morse-type singularitiesso that Theorem 2 applies.

We begin by recalling results from [D6]. We use the notation preceding thetheorem. If g0: C

p, 0 ! Cp+q, 0 denotes inclusion, then g0 is transverse to F and

f0 is the pull-back of F via g0.

Cn+q, S

F��! C

p+q, 0x?? g0

x??C

n, Sf0

��! Cp, 0

Then, the properties of f0 involving A-equivalence are related to the propertiesof g0 involving KV -equivalence for V = D(F).

6.2. Relation between A and KV-equivalence. There are the followingrelations between f0 and g0:

(1) f0 is finitely A-determined if and only if g0 is finitely KV-determined;

(2) KV,e-codim(g0) = Ae-codim( f0);

(3) f0 is a A-stable multigerm if and only if g0 is a KV -stable germ; and

(4) if G is an unfolding of g0, with pull-back denoted by F1, then G isKV-versal if and only if F1 is A-versal.

476 JAMES DAMON

Thus, by (3) and (4), B(F) is the KD(F)-discriminant of the KD(F)-versalunfolding of g0. By Theorem 2, the freeness of the KD(F)-discriminant followsprovided D(F) has generic Morse-type singularities in dimension p.

To determine at which points Morse-type singularities occur, we identify themby the properties of the corresponding pull-back germs. First, we note deforma-tions of g0 will still be transverse to F and hence induce a deformation of f0. Also,if f0 is weighted homogeneous, then F may be chosen weighted homogeneousso that both g0 and D(F) are weighted homogeneous (and conversely F and g0

weighted homogeneous implies the pull-back f0 is). Thus, a Morse-type singu-larity for D(F) with F weighted homogeneous induces by pull-back a weightedhomogeneous (multi)germ of Ae-codimension 1, and conversely.

Furthermore, as already mentioned, it follows from Mather’s multitransversal-ity characterization of stability [M-V] that at a point corresponding to a multigerm,the discriminant is the product union (as defined in x4) of the discriminants ofthe individual stable germs. Thus, to verify that points of discriminants of stablegerms have Morse-type singularities it is sufficient by Proposition 4.28 to verifythat the discriminants of the individual stable germs have Morse-type singular-ities and that each of them is of the same exceptional weight type. Moreover,by (3) of (6.2) above, it is enough for an individual germ to establish that theK-equivalence class of the germ has a weighted homogeneous unfolding whichviewed as a germ has Ae-codimension 1.

Now we can invoke two different results concerning the existence of germsof Ae-codimension 1. First, consider a weighted homogeneous germ f0: C

n, 0 !C , 0 defining either an isolated hypersurface singularity or a germ f0: C

4, 0 !C

2, 0. Weights are assigned to weighted homogeneous terms in NKe � f0 so thatterms with the same weights as f0 are assigned weight 0 (the unfolding parameterfor a term is then assigned negative the weight of the term so that all of the termsof the unfolding have the same weight). Each of the preceding two families ofgerms have in NKe � f0 a maximal weight summand having dimension 1 (inthe hypersurface case it is generated by the Hessian). Let F be the unfoldingof f0 versal except for the maximal weight term. Then, by the results of [D1]which uses the calculations of Looijenga [L1] and Wirthmuller [Wi] together withduality results in [D2] and [Wa], we conclude:

PROPOSITION 6.3. Let F denote the unfolding versal in nonmaximal weight ofa weighted homogeneous germ f0 defining an isolated hypersurface singularity ora surface singularity in C

4. If either the weight 0 part of NKe � f0 = (0) or it is themaximal weight part, then F viewed as a germ has Ae-codimension 1.

Remark 6.4. Together with the Normal Form Lemma 4.12, Proposition 4.28and the results described in (6.2), this proposition allows us to classify the Ae-codimension 1 germs and multigerms appearing in the K-equivalence classes of(6.3).


For the purpose of considering multigerms, we note that the sections of thediscriminant corresponding to these unfoldings have exceptional weight type asfollows: for simple singularities it is positive, for simple elliptic singularities itis zero, and for the exceptional unimodal singularities it is negative.

Example (6.5). Bifurcation sets and KV-discriminants which are not hypersur-faces. For any of the simple elliptic singularities, the unfolding in nonmaximalweight F, viewed as a germ of a mapping, is weighted homogeneous and hasAe-codimension 1. Thus, it is a Morse-type singularity; however, its target codi-mension equals the holonomic codimension so Theorem 2 doesn’t apply. In fact,the one-parameter versal unfolding just varies the modulus parameter for theparticular simple elliptic family. Hence, the bifurcation set (which is also a KV-discriminant for a free divisor) consists of all of the unfolding space! Thus, it isnot even a hypersurface, let alone free. This same phenomena arises wheneverp � hn(D(F)).

The duality results underlying the results in Proposition 6.3 are not valid moregenerally. However, as pointed out to the author by David Mond and AndrewDuPlessis, Goryunov [Go] has independently by different methods classified allA-simple germs. The relevant parts of his results for us are as follows.

PROPOSITION 6.6. For the following K-equivalence classes of complete inter-sections, there are unfoldings which are weighted homogeneous and are of Ae-codimension 1 when viewed as germs:

(1) simple hypersurface germs;

(2) the 0-dimensional complete intersections Ia,b in Mather’s notation [M-VI](or Fa,b in Giusti’s [Gi]): (x, y) 7! (xy, x a + y b) with a = 2;

(3) in particular, there are noAe-codimension 1 germs obtained as unfoldingsof curve singularities C

3, 0 ! C2, 0 nor of the germ Ia,b with a, b � 3; and

(4) other K classes of germs of corank > 1 (not already listed) for whichn � p = 0, 1 deform to one of the germs in (3), hence, cannot have Ae-codimen-sion 1.

Remark 6.7. Goryunov gives explicit germs. In the case of simple isolatedhypersurface singularities, they are the same as those obtained by Proposition 6.3.

Proof of Theorem 3. By the preceding discussion, it is sufficient to identifywhen the discriminant of a stable (multi) germ D(F) � C

p0 generically hasMorse-type singularities in dimension p and also p < hn(D(F)).

This gives three conditions which must be met :

(1) p < hn(D(F));

(2) germs corresponding to points in nearby strata of codimension � p + 1must have Morse-type singularities; and

478 JAMES DAMON

(3) nearby strata of codimension > p + 1 must lie in the closures of strataof codimension p + 1.

For (1), we must avoid the dimensions where moduli appear in the classifi-cation of stable germs. This is referred to by Mather as the “nice dimensions”[M-VI]. Then, the classification of simple germs in the nice dimensions showsthat (3) holds.

For (2), by the preceding propositions, p < s � 1 where s denotes the Ke-codimension of canonical strata whose K-class(es) does not contain an unfoldingof Ae-codimension 1.

Thus, if the classification of singularities precludes (2), then only the conditionfor the nice dimensions comes into play; this includes case (3ii) in (6.1).

If we exclude the case n = p+1, then there are two key strata. The first stratumdefined is by the collection of K-classes of germs C

4, 0 ! C2, 0 defined by pairs

of generic quadrics. This family has a single modulus. Note the individual germshave discriminants which can have Morse-type-singularities in dimension 6 byProposition 6.3, but these are in the holonomic stratum. However, the germs inthis stratum have Ke-codimension 7; they do not have Morse-type singularities indimension 5 (in the source space). Thus, we must have p < 6� 1 = 5. For zero-dimensional complete intersections, the revelant stratum is for I3,3 which hascodimension 6, and by Proposition 6.6 does not have Morse-type singularitiesin dimension 5. Thus, we must have p < 6 � 1 = 5. By the K-classificationof singularities, these are the lowest codimensional strata without Morse-typesingularities; hence, we do have free divisors by Theorem 2 if p < 6 � 1 = 5(provided n 6= p + 1).

For the case of complete intersections with n = p + 1, the relevant stratum isthat of the generic curve singularity (xy, x2 + y2 + z2) which has codimension 5.Again by Proposition 6.6, it does not have Morse-type singularities in dimension4. Thus, we require p < 5 � 1 = 4. By the classification of singularities, this isagain sufficient.

Next if we have corank 1 singularities, then the preceding strata will notappear so the relevant strata are the E6 and E7 strata defined by the collectionof K-classes of the simple elliptic singularities x3 + y3 + z3 + txyz, respectivelyx4 + y4 + tx2y2, for varying modulus values t. The E6-stratum has codimension7 (in the source space) and the E7-stratum has codimension 8. Thus, for corank1 singularities, if n = p + 1 then the E7-stratum is the relevant stratum (E6 can’toccur). An individual E7 germ can’t have a Morse-type singularity of dimension7 so we must have p < 8 � 1 = 7, while if n � p + 2 then the E6-stratum canappear and is the relevant stratum so we must have p < 7� 1 = 6. Again in eachof these cases, the numerical conditions and Propositions 6.3 and 6.6 imply thatthe strata which occur correspond to simple multigerms so they generically haveMorse-type singularities. This completes the proof.


7. Discriminantal arrangements. Manin and Schechtman [MS] defined adiscriminantal arrangement which naturally extends the braid arrangement. LetA = [n

i=1Hi � Ck be a central arrangment of n hyperplanes Hi which are in general

position off the origin. Then, each hyperplane can be independently translated byai in a normal direction. The set of new affine arrangements can be parametrizedby points a = (a1, : : : , an) 2 C

n. Let U(n, k) denote the subset of such affinearrangements which are in general position in all of C

k. It is the complement ofan arrangement B(n, k), which is called a discriminantal arrangement. Althoughit is an arrangement in C

n, uniform translation by a point in Ck acts on U(n, k)

and B(n, k); hence, B(n, k) has a trivial factor of C k and the quotient by the actiondefines a reduced form of it in C

n�k.A special case is B(n, 1) which is just the braid arrangement Bn. However,

unlike the braid arrangement, B(n, k) is in general not a free arrangement. Orlik-Terao [OT, Proposition 5.6.6] have shown that B(k + 3, k) is not free when k �2. On the other hand, Falk [F] has shown that the discriminantal arrangementdepends upon the particular initial form of the general position arrangement. Also,Bayer-Brandt [By] [B], adapting examples of Falk [F], have shown that there arespecial general position arrangements of 6 planes A � C

3 whose discriminantalarrangements are free arrangements even though the general such is not.

To clarify this situation, we take the perspective of singularity theory, espe-cially [D8] that such arrangments are almost free arrangements A � C

k obtainedas the pull-back of the Boolean arrangement An � C

n by a linear inclusion': C

k ! Cn (recall the Boolean arrangement An is the union of coordinate

hyperplanes in Cn). The fact that A is a central general position arrangement is

equivalent to ' being transverse to An in a punctured neighborhood of 0 (i.e.to the coordinate planes and their intersections, except 0). Hence, ' has finiteKAn-codimension.

Definition 7.1. A hypersurface singularity X, 0 � Ck, 0 is a nonlinear Boolean

arrangement if X = '�1(An) for a map germ ': Ck, 0 ! C

n, 0 which is transverseto the An in a punctured neighborhood of 0.

This is a special case of a general nonlinear arrangement based on a freearrangement defined in [D8]. Then the main result we obtain does not evendepend upon ' being linear although it has consequences for linear inclusions '.

THEOREM 4. If X, 0 � Ck, 0 is a nonlinear Boolean arrangement defined by

': Ck, 0 ! C

n, 0, then, the KAn-discriminant of the versal unfolding of ' is a freedivisor.

Before proving the theorem we draw some consequences. First is the corol-lary.

480 JAMES DAMON

Figure 7.5.

COROLLARY 7.2. Let A � Ck be a central general position arrangement defined

by ': Ck ! C

n. Then, the KAn-discriminant of the versal unfolding of ' is a freedivisor.

The relation between this KAn-discriminant and the discriminantal arrange-ment is as follows. We may view the reduced form of the discriminantal ar-rangement as an arrangement in C

n�k. Then, the KAn-versal unfolding Φ(x, u) =(Φ(x, u), u) will contain n � k constant terms corresponding to the n � k normaldirections to '(C k) � C

n. The points u of the KAn-discriminant in this subspaceare exactly those for which there is an x where Φ(�, u) fails to be transverse toAn at Φ(x, u). This says exactly that the pull-back Φ(�, u)�1(An) is not in generalposition, or u 2 B(n, k). Hence, we obtain

COROLLARY 7.3. Let A � Ck be a central general position arrangement de-

fined by ' with B(n, k) the associated discriminantal arrangement. Then, B(n, k)is obtained as an n � k dimensional section of the KAn-discriminant of the versalunfolding of ' (by the parameter space for the constant term deformations).

Example 7.4. Thus, B(n, k) is the linear part of the full KAn-discriminant. Tosee the relationship between these two discriminants , we consider the simpleexample of an arrangement of five lines in C

2 defined by a linear inclusion': C

2 ! C5. Then, the KAn,e-codimension of ' is also equal to the number of

singular vanishing cycles. We can see in Figure 7.5 the 6 singular cycles in thesingular Milnor fiber.

Then, a computation of NKA5,e � ' shows that the versal unfolding has sixparameters: three of which are for constant terms corresponding to the normaldirections to '(C 2); two of which are linear terms corresponding to nonuniformrotation of just some of the hyperplanes; and one quadratic term. The constantterms correspond to the translations giving rise to the discriminantal arrange-ment. The linear terms appear because not all general position arrangements areequivalent via the action of GL(C k). The most surprising term from the point ofview of arrangements is the presence of the quadratic term. Specifically, in orderto describe all deformations of a linear general position arrangement one mustallow nonlinear bending! Furthermore, a calculation shows that in this case ifone only takes the part of the unfolding coming from the constant terms then this


unfolding has finite KA5-codimension as an unfolding. Then one can apply thetopological versality theorem of [D7, Theorem 4] to conclude that the KA5-versalunfolding is topologically KA5-equivalent to a trivial extension of the unfoldingby constant terms. In particular, the KA5-discriminant is topologically equivalentto B(5, 2)� C

3.

Remark 7.6. In general one should expect that for “very generic arrangements”the unfolding by the constant terms will have finite KAn-codimension as an un-folding so the topological versality theorem will apply and the KAn-discriminantwill be topologically equivalent to a trivial factor times B(n, k). Otherwise therewill be a nontrivial change occurring in the KAn-discriminant corresponding tothe special arrangements identified by Falk and Bayer.

Proof of Theorem 4. We wish to apply Theorem 2. Then, the Boolean arrange-ment An is a free arrangement for all n, and hn(An) = 1. Thus, all we need toverify is that An generically has Morse-type singularities in all dimensions � 1.This follows from the following lemma.

LEMMA 7.7. Boolean arrangements An generically have Morse-type singular-ities in all dimensions � 1.

Proof of Lemma 7.7. For this we observe that at any point, the arrangementAn is locally equivalent to As � C

n�s for some s � 0. Thus, it is sufficient toshow that for any n > 0, An has Morse-type singularities. However, if we choosea generic hyperplane section of An (through the origin), then it is homogeneousand by [D8, x5], it has KAn,e-codimension 1. Thus, it is a Morse-type singularity.Hence, by Corollary 4.22, for any n � 0, An has Morse-type singularities in allallowable dimensions. Thus, An generically has Morse-type singularities.

8. Multiple type braid configurations and nonlinear arrangements. Inaddition to discriminantal arrangements, there is another way in which braid ar-rangements naturally extend. The braid arrangement arises by considering config-urations of n distinct points in C . If the distinct identity of the points is retained,we have the colored braids described by the complement of the braid arrange-ment. If just the set of points is considered then, we instead have the complementof the stable discriminant of the An�1-singularity.

Suppose now we consider the configuration of several distinct sets of pointswith the ith set containing ni distinct points in C . For example, for two sets,we can think of the points as either black or white points, without the individualpoints being distinguished, except by their color. Then, we want there to remain ni

points of each color and not to have points of different colors occupying the sameposition. What can we say about the space of such configurations? In particular,do they still share properties in common with the complements of discriminantsof the An�1-singularities or of braid arrangements?

482 JAMES DAMON

In general, we will refer to configurations consisting of k distinct types ofpoints, with sets of ni points of the ith type, which do not occupy commonpositions as k-type Braid configurations. For given fixed k and n = (n1, : : : , nk),we denote the space of allowable configurations of points by Bk(n). The collectionof such spaces include Bk(1, : : : , 1), which is the configuration of colored braidsand is the complement of the braid arrangement. Also included is B1(n) whichdescribes the uncolored n-braids and is the complement of the stable discriminantof an An�1-singularity.

In the definition of Bk(n), there are N =P

i ni total points, so the configurationspace will be a subset of C

N (note that a uniform translation of all the pointsacts freely so Bk(n) has a one-dimensional trivial factor, and we could considerinstead the reduced space in C

N�1). Then, we claim that we can describe Bk(n)as the complement of the KV -discriminant for a nonlinear Boolean arrangement.Hence, by Theorem 3, it will follow that Bk(n) is the complement of a freedivisor. Define f0: C , 0 ! C

k, 0 by f0(x) = (xn1 , : : : , xnk ). Let Ak � Ck denote the

Boolean arrangement. Then, f0, viewed as a nonlinear section of Ak, just mapsC to a curve in C

k which only intersects Ak at 0. We construct a KAk -versal

unfolding F: C1+N , 0 ! C

k+N , 0 of f0 as follows: Let u( j) = (u( j)1 , : : : , u( j)

nj ),u = (u(1), : : : , u(k)) , and define

Pj(x, u) = xnj + u( j)1 xnj�1 + � � � + u( j)

nj�1x1 + u( j)nj

, j = 1, : : : , k.

Also, define

F(x, u) = (F(x, u), u) where F(x, u) = (P1(x, u), : : : , Pk(x, u)).(8.1)

Note here N =P

i ni.

THEOREM 8.2. The space Bk(n) of k-type braid configurations is the comple-ment of the KAk-discriminant DAk (F), which is a free divisor in C

N.

Proof. Derlog(Ak) is generated by fzi@@zig, where the zi are local coordinates

for Ck. It is straightforward to verify using the infinitesimal characterization of

versality that F is the KAk -versal unfolding of f0. However, F is not the miniversalunfolding, which requires one fewer parameter. In fact, the miniversal unfoldingcan be obtained by replacing the k parameters u( j)

1 , j = 1, : : : , k, by generic linearcombinations of k � 1 parameters (w1, : : : , wk�1). This should not be surprisingin light of the theorem as we have seen that Bk(n) is trivial in one direction.

Then, by Theorem 4 we know that for the Boolean arrangement Ak, DAk (F)is a free divisor.

Finally, we can use the geometric characterization of the KAk -discriminant toidentify Bk(n) with the complement of DAk (F). Let the coordinate hyperplaneszi = 0 be denoted by Hi. Then, F(�, u, w): C , x ! C

k, y, with y 2 Ak will fail to


be transverse to Ak unless y 2 Hi for exactly one i and

@Pi

@x jx=0=

@

@x(zi � F(�, u))jx=0 6= 0.(8.3)

In particular, u 2 DAk (F) if and only if either there is an x such that F(x, u) isin more than one Hi or (by (8.3)) some Pi(�, u) will have a multiple root at x.However, if we let the roots of Pi represent the points for the ith type, then forF(x, u) to be in more than one Hi is equivalent to points of distinct types being atthe position x. Also, a multiple root of Pi(�, u) at x implies there are fewer thanni points of type i.

Hence, DAk (F) is exactly the complement of Bk(n).

Example 8.4. Consider the special case of two types, each consisting of npoints. Now, we describe the reduced configuration space of B2(n, n) as the com-plement of DA2(F) � C

2n�1 for the miniversalKA2-unfolding F of f0(x) = (xn, xn).Then, F is defined by

F(x, u, v, w) = (F(x, u, w), u, v, w)

where

F(x, u, v, w) = (P1(x, u, w), P2(x, v, w))

with u = (u1, : : : , un�1), v = (v1, : : : , vn�1), and

P1(x, u, w) = xn + a1wxn�1 + u1xn�2 + : : : un�2x1 + un�1

and

P2(x, v, w) = xn + a2wxn�1 + v1xn�2 + : : : vn�2x1 + vn�1.

Here (a1, a2) is a vector which is not a multiple of (1, 1).Then, DA2(F) has three components corresponding to the curve F(�, u, v, w)

being tangent to the z1-axis, the z2-axis, or passing through the origin (see Figure8.5).

The component corresponding to the curve passing through the origin is givenby the set

f(u, v, w) 2 C2n�1: there exists x 2 C such that P1(x, u, w) = P2(x, v, w) = 0g.

However, this is exactly the image of the stable corank 1 map germ G: C2n�2, 0 !

C2n�1, 0 defined by

G(x, u0, v 0, w) = (P01(x, u0, w), P02(x, v 0, w), u0, v 0, w)

484 JAMES DAMON

Figure 8.5.

where u0 = (u1, : : : , un�2), v 0 = (v1, : : : , vn�2) and

P01(x, u0, w) = P1(x, u, w)� un�1 and P02(x, v 0, w) = P1(x, v, w)� vn�1.

Each of the remaining two components which correspond to tangency to one ofthe axes can be described as the image of the multiple root set of one of the P2.

Thus, the image of a corank 1 stable map germ G: Cm, 0 ! C

m+1, 0 is nota free divisor (for it already fails for the Whitney umbrella, see e.g. [D6, x1]).However, David Mond had discovered that by adding the adjoint divisor to theimage it becomes a free divisor. Via the 2-type braid configuration, we obtain analternate way to add naturally a divisor to the image of the stable corank 1 germto obtain a free divisor.

9. Smoothings of space curves and complete intersections. In this sectionwe consider various consequences of the main theorem for smoothings of curvesingularities, functions on complete intersections, higher order cusp singularities,and examples where nonreduced structure fails.

Smoothings of space curve singularities. Let C, 0 � Cn, 0 be a reduced

space curve singularity. A smoothing of C, 0 is a germ ': X0, 0 ! C , 0 which isa flat deformation of C, 0. We consider finite parameter deformations Φ: X , 0 !C

q+1, 0 of smoothings. We let Xu = Φ�1(C � fug). For finite S � X withΦ(S) = (y, u), consider the germ ΦS = Φ j Xu: Xu, S ! C , y. We define thebifurcation set B(Φ) for the deformation of smoothings Φ by

fu 2 Cq: for some finite S � Xu, ΦS is not the versal deformation of (Xu, S)g.

Because C, 0 is a reduced space curve singularity, it has an unobstructed versaldeformation Ψ: Y , 0 ! C

p, 0. Let ∆ denote the discriminant of Ψ.By versality, any other flat deformation ': X0, 0 ! C

r, 0 of C, 0 is (up toisomorphism) the pull-back of Ψ by a map germ f : C

r, 0 ! Cp, 0. In particular,

a smoothing ': X0, 0 ! C , 0 is induced by a map f : C , 0 ! Cp, 0 and because


' is a smoothing, f (C )\∆ = f0g. Thus, f has finite K∆-codimension, and hencehas a K∆-versal unfolding F: C

1+q, 0 ! Cp+q, 0 which pulls-back Ψ to give a

versal deformation Φ of the smoothing '.

THEOREM 9.1. For any reduced space curve C, 0, the bifurcation set B(Φ) ofthe (versal deformation of the) smoothing ' is a free divisor.

Proof. We observe that the bifurcation set consists of u 2 Cq such that ΦS

fails to be versal at some finite S � Xu. However, this corresponds to either F(S, u)not being a point v of the discriminant ∆ for which Yv has a single singular pointwith a one-parameter versal deformation, or F(�, u) failing to versally unfold thesingular point.

A basic fact we need is the following.

LEMMA 9.2. For any reduced space curve C, 0, the bifurcation set B(Φ) is theK∆-discriminant of F.

Proof of the lemma. Let

∆1 = fv 2 ∆: Yv is not a complete intersection curveg.

Then ∆1 has codimension 3 in Cp (see e.g. [VS, x3]).

Thus, if F(S, u) = v 2 ∆1 then by the openness of versality, Xu cannot bethe versal deformation of Yv . Second, if F(S, u) = v =2 ∆1, then Yv is a completeintersection curve singularity so by (6.2) ΦS j Xu, (S, u) is the versal deformationat the singular point if and only if F is transverse to ∆ at v.

Thus, to prove the theorem, it is sufficient to show that ∆ is a free divisorwhich generically has Morse-type singularities in dimension 1, and 1 < hn(∆).However, by the result of Van Straten [VS], ∆ is a free divisor. Also, as we havestated, off ∆1, which has codimension 3, the strata of ∆ correspond to ordinarydouble points. Thus, hn(∆) � codim(∆1) � 3. Also, ordinary double points havecorank 1, hence by Propositions 6.3 and 6.6, they have Morse-type singularitiesin all dimensions � 1.

Smoothings and functions on isolated complete intersections. Supposewe wish to consider functions on complete intersections with isolated singularitiesf0: X, 0 ! C

p, where we allow both the function f0 and complete intersection X, 0to deform. In [D5, x3] we showed that it is possible to describe the equivalence ofsuch germs, allowing both germs and varieties to vary, by a geometrically definedsubgroup of A or K so that all of the basic theorems of singularity theory apply.

Mond and Montaldi [MM] showed that, just as for A-equivalence in (6.2),such an equivalence could be described by KV-equivalence for the sections tothe stable discriminant V = D(G) for the stable germ G defining X0 = f�1

0 (0).The key difference is that now the immersion g0: C

p, 0 ,! Cp0 , 0 may not be

486 JAMES DAMON

transverse to the map germ G so the pull-back X of g0(C p) by G will not be asmooth space.

Now, suppose F: X , 0 ! Cp+q, 0 is the versal deformation defined by

F(x, u) = (F(x, u), u) where dim (X, 0) = n so dim (X , 0) = n + q. Then the bi-furcation set B(F) consists of parameter values u such that there is a finite subsetS � Xu = F�1(C p � fug) such that the germ F(x, u): Xu, S ! C

p is not stablefor the equivalence allowing both source space and germ to deform. This impliesthat (Xu, S) is smooth and that F(x, u) j (Xu, S) is a stable multigerm. Again, bythe theorem of Mond-Montaldi, the bifurcation set B(F) of the versal deformationwill be exactly the KD(G)-discriminant of g0.

Thus, provided D(G) has generic Morse-type singularities in dimension pand p < hn(D(G)), then by Theorem 2, the KD(G)-discriminant for the versaldeformation of g0 is a free divisor. Hence, we can conclude:

THEOREM 9.3. Let f0: X, 0 ! Cp, 0 be a complete intersection germ defining

an isolated singularity X0, 0 on an isolated complete intersection singularity X, 0.Also, let F: X , 0 ! C

p+q, 0 be the versal deformation for A-equivalence whichallows both f0 and X to vary. Then, the bifurcation set B(F) is a free divisor providedeither:

(1) X0, 0 is a curve singularity and p � 3, or

(2) dim (X0, 0) 6= 1 and p � 4.

In particular, as a corollary we obtain the analog of Theorem 9.1

COROLLARY 9.4. Let': X, 0 ! C , 0 be a smoothing of an isolated complete in-tersection singularity X0, 0. Then, the bifurcation set B(Φ) of the versal deformationΦ of the smoothing ' is a free divisor.

Proof of Theorem 9.3. Just as in the proof of Theorem 3 in x6, we must avoidthe problem strata in the discriminant of the stable germ G. By the same criteria,if X0, 0 is a curve singularity then p < 4, otherwise p < 5.

The corollary is an immediate consequence of the theorem as p = 1 is alwaysallowable.

Higher order cusp singularities. The Zariski example

g(x, y, z) = (x2 + y2)3 � (x3 + z3)2 = 0

is a plane curve singularity in C P2 which has six singular points which are cuspslying on a quadric. This remarkable example is well-known for the propertythat its fundamental group differs from that of other curve singularities with sixcusps not lying on a quadric. From the point of view of local singularity theory,g: C

3, 0 ! C , 0 provides an example of a nonisolated surface singularity, with


cusps along its zero set. It is obtained as the pull-back of the cusp singularityh(u, v) = u2 � v3 = 0 by a complete intersection germ f : C

3, 0 ! C2, 0 given by

f (x, y, z) = (x2 + y2, x3 + z3).Since the singularity is not isolated it does not have a versal unfolding in

the usual sense. However, as a nonlinear section of cusp C, 0 � C2, 0, f has

finite KC-codimension and so by the KV-versality theorem, f has a KC-versalunfolding F: C

3+q, 0 ! C2+q, 0. For example, in [D8,x10] it is shown that the

Zariski example has a versal unfolding with seven parameters.More generally we consider kth order cusps Ck, 0 � C

2, 0 defined by hk(u, v) =u2 � vk = 0 for k � 3. All of these higher order cusp singularities share the samebasic freeness of discriminants.

THEOREM 9.5. LetCk, 0 � C2, 0 be a kth order cusp singularity and let f0: C n, 0 !

C2, 0 be (algebraically) transverse to Ck in a punctured neighborhood of 0. Then,

theKCk -discriminant of the versal unfolding F: Cn+q, 0 ! C

2+q, 0 is a free divisor.

Proof of Theorem 9.5. First, all isolated plane curve singularities C, 0 � C2, 0

are free divisors by Saito [Sa]. Since hn(Ck) = 1, it is sufficient to show that kthorder cusps generically have Morse-type singularities in all dimensions n � 1.

Since the smooth strata clearly do, it is sufficient to consider the origin.However, with Ck defined by hk(u, v) = u2 � vk = 0, Derlog(Ck) is generated by

�0 = ku@

@u+ 2v

@

@vand �1 = kvk�1 @

@u+ 2u

@

@v.

Then, for the weighted subspace spanned by u we see that 2u = �1(v) spans mu;thus Ck satisfies (4.16) and hence by Corollary 4.22 has Morse singularities in alldimensions � 1. Hence, we can apply Theorem 2 to conclude the result.

Nonreduced KV-discriminants for free divisors. We earlier consideredan example of a free arrangement A � C

3 defined by Q = xyz(x � y). Thisarrangement is the product union of the free arrangement of three lines A0 � C

2

defined by xy(x�y) = 0 with the trivial free divisor f0g � C . We let A00 � C2 be

the arrangement obtained by pulling back out A via a generic linear embedding': C

2 ! C3. Such an arrangement of four lines has for its singular Milnor fiber

the affine arrangement with two singular cycles as shown in Figure 4.7. Thiscorresponds to the section having KA,e-codimension 2.

In fact, the KA-versal unfolding has the form

Φ(x, u, v) = ('(x) + v� + u`(x), u, v)

where � is a constant term and `(x) is linear. Again a calulation shows thatthe restricted unfolding Φ1(x, v) = ('(x) + v�, v) has finite KA-codimension asan unfolding. Thus, again by the topological versality theorem in [D7], Φ1 is

488 JAMES DAMON

topologically KA-versal and Φ is topologically KA-trivial along u. Thus, the KA-discriminant is the u-axis. Moreover, the proof shows that the vector field v @

@u isKA-liftable. Thus, although the module of KA-liftable vector fields is free, it isgenerated by v @

@u and the Euler vector field on C2, which is e = v @

@v as wt(u) = 0.Thus, the 0th Fitting ideal is generated by v2 and defines the KA-discriminant asthe u-axis with nonreduced structure.

Thus, while any reduced curve singularity in C2 is free by Saito [Sa], the

key point is the nonreduced structure from the 0th Fitting ideal. We shall see inpart two of this paper that this is typical when Morse-type singularities are notgeneric, and we have to settle for a weakened form of free divisor.

10. Addendum. In order to obtain the local normal form for Morse-typesingularities in Lemma 4.12, we applied Mather’s Geometric Lemma for KV-equivalence. To be able to do so, we must know that the KV -action descendsto a Lie group action on `-jets. Here we justify this assertion in the case thatthe weights for V0 are all positive. Second, we show in general that we mayobtain the conclusion of Mather’s Geometric Lemma by directly proving localKV-triviality for families satisfying (4.18) or more generally (4.27).

First suppose that we consider a germ of a weighted homogeneous varietyV , 0 � C

p such that all weights wt(yi) = bi > 0. Let I(V) have homogeneousgenerators fh1, : : : , hrg with wt(hi) = di. Also, we assign weights wt(xi) = 1 forC

n. Then, we consider the set Kw(`)(n, p) of weighted � `-jets (i.e., invertiblepolynomial germs of weight � `) of the form Φ(x, y) = ('1(x),'2(x, y)). Herewe choose ` > max (di). Then, this is a Lie group under composition (weighted� `-jets of the uinversal gauge group). We consider the subgroup

Kw(`)V = fΦ(x, y) 2 Kw(`)(n, p): '�2(hi) 2 I(V) � OC n+p ,0g.

PROPOSITION 10.1. Let ` > max (di), then the action KV factors through Kw(`)V

which is an algebraic Lie group.

Proof. Suppose we write each hi =P

a�,iy�. For any weighted space W welet Wk denote the weight k part, so that

�I(V) � OC n+p ,0

�k =

XI(V)j � (OC n+p ,0)k�j.(10.2)

For each k we choose a linear map

Lk: (OC n+p ,0)k ! Cmk

with

ker(Lk) =�I(V) � OC n+p ,0

�k .


Then, if we write '2(x, y) = ('21, : : : ,'2p) with '2i(x, y) =P

ci�rxry�. Then,we may expand '�2(hi) =

Pgij where wt(gij) = j and gij =

Psij�rxry� where

each sij�r is a fixed polynomial function of the coefficients ci�r depending on theweights and the coefficients a�,i. Thus the condition that

'�2(hi) 2 I(V) � OC n+p ,0 for i = 1, : : : , r

is that

(Lj)(gij) = 0 for i = 1, : : : , r, j = 1, : : : , `(10.3)

However, the equations in (10.3) are linear equations in the coefficients sij�r, andhence polynomial equations in the coefficients ci�r. Thus, Kw(`)

V is a Zariski closedsubgroup of the algebraic Lie group Kw(`) defined by polynomial equations, andhence, an algebraic Lie group (by e.g. [B, Chapter 1]).

Second, we consider more general V , 0 where we suppose that some of theweights may be zero or negative. An alternate method for proving that the fam-ilies satisfying (4.18) or (4.27) are KV -trivial is to use directly the results forgeometrically defined subgroups of A or K. Since (4.18) is a special version of(4.27), it is enough to prove it for the latter.

PROPOSITION 10.4. Let ft: Cn ! C

p be a family of germs of the form (4.13)such that for V , 0 � C

p, the KV-tangent spaces satisfy (4.27) with

@ft@t

2 TKV � ft for each t in a small neighborhood of 0.

Then, the family ft is KV-trivial.

Proof. We recall that (4.27) states

TKV � ft = mnfWtg + (x1, : : : , xp0�1)

(@

@yp

)+ m2

n

�@

@yi

�

where Wt is the subspace spanned by TCr and dft(0)(TC

n). We assume we havemade a parametrized change of coordinates so that ker(dft(0)) is the subspace(x1, : : : , xp�1) for all t. Then

(xi

@

@yj, xi

@ft@xk

; 1 � j � r, 1 � k � p� 1, 1 � i � n; �l � ft, 1 � l � p� 1

)

490 JAMES DAMON

is a smoothly varying basis for

mnfWtg + (x1, : : : , xp0�1)

(@

@yp

)mod m2

n

�@

@yi

�.

Thus, since

@ft@t

2 TKV � ft for each t,

we may represent @ft@t smoothly in this basis. This allows us to find (�1, �1) 2

TKV,un so that

@ft@t

� ��1( ft) + �1 � ft mod m2nOC n+1 ,0

�@

@yi

�.(10.5)

Let

M = m2nOC n+1 ,0

(@

@yj, 1 � j � r;

@ft@xk

; 1 � k � p� 1

)

+ mnOC n+1 ,0

(�i � ft, 1 � l � p� 1;

@ft@xj

, p � j � n

).

The infinitesimal orbit map sends

M �! m2nOC n+1 ,0

�@

@yi

�.(10.6)

Furthermore, if we divide by (t)OC n+1 ,0, we obtain the induced map from theinfinitesimal orbit map for f0, and this is surjective. Thus, by the preparationtheorem, (10.6) is surjective. Thus, by (10.5) and (10.6), there are (�2, �2) 2TKV,un so that

@ft@t

+ �1( ft)� �1 � ft = ��2( ft) + �2 � ft.(10.7)

However, rearranging the terms of (10.7), we may solve

@ft@t

= ��( ft) + � � ft

for (�, �) 2 TKV,un. Lastly we use the reduction Lemma 9.1 in [D3], but asit applies to triviality of families ([D3, 10.9]) to conclude that the family ft isKV-trivial.


DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NORTH CAROLINA, CHAPEL HILL, NC,27599

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