Fitting Ideals and Multiple Points of Analytic Mappings

David Mond Ruud Pellikaan

Introdu~don

Let f:X ~ Y be a finite analytic map. The purpose of this paper is to find a good

choice of analytic structure for the set Mk(f) of points in Y whose preimage consists of k

or more points, counting multiplicity. We want this structure to predict the behaviour of

deformations of the mapping, and thus for example to tell us how many ordinary

n+l-tuple points will emerge from each "degenerate" point in the image, on deforming

an unstable (non-generic) mapping from the n-dimensional space X into a smooth

n+l-fold Y. Such mappings and deformations occur naturally in the family of

projections into IP n+l of an n-dimensional variety X c_ ~N.

Our interest in the scheme structure of Mk(f) contrasts with the approach in

enumerative geometry, where Mk(f) is only defined modulo rational equivalence, c.f. [21],

[32], [33], [34], t38].

We define the analytic structure on Mk(f) by means of the Fitting ideals of the

coherent Oy-module f,O)~. This is discussed, with further motivation, in §1. The rest

of the paper is concerned with ghowing that when X is Cohen-Macaulay and Y is

smooth, with dim Y = dim )( + 1, this analytic structure is the right one. There are four

requirements here: the structure so defined should be calculable, it should be reduced at

ordinary k-tuple points, it should commute with base change, and it should behave well

under deformation (more precisely, if F is a deformation of f over base S, Mk(F) should

be flat over S). In §1 it is shown that the second and third requirements hold, and in §2

we give an algorithm for constructing a presentation of the Oy-module f.O)~, from

which all of the Fitting ideals may be read off.

108

The fourth requirement is not so easily dealt with. In §3 we show that the

assumption that :~ is Cohen-Macaulay is enough to guarantee that M l(f) and M2(f) are

themselves Cohen-Macaulay, provided the codimension of M2(f) is 2 (of course for

Ml(f) this is well known), and from this, flatness follows. However, it is necessary to

assume also that )~ is Gorenstein (see §4 and Example 7.5 below), in order to conclude

that M3(f) is Cohen-Macaulay.

Our results on M 3 in §4 lead to a necessary condition (4.7, below) for a

hypersurface X to have Gorenstein normalisation, entirely in terms of the singular locus

of X. In §5 we show that for a certain class of maps (those which exhibit O~ as a cyclic

extension of O X, where X = f(X)), all of the M k are Cohen-Macaulay, for 1 < k < n+l

(provided certain natural requirements on codimension are met), and hence flat over the

base of a deformation.

In §6 we apply our results on M 2 to show that if F:(• n+l × Cd,0) ~ (¢ × cd,0) is

an unfolding of the germ f:(~n+l,0) ~ (C,0), then the singular subspace of the

discriminant D F is Cohen-Macaulay, and we use this to give the bifurcation set

B c_ (cd,0) a canonical hypersurface structure.

Section 7 consists of examples. It is followed by an Appendix containing a brief

discussion of some of the results from commutative algebra that we use.

The authors thank Andrew du Plessis, Miles Reid, Joseph Steenbrink and Duco

van Straten for helpful discussions and remarks, and Jan Stevens for his example (7.6

below).

This paper is in final form and no similar version is to be submitted elsewhere.

109

§1. Fitting Ideals.

1.1 Definition. Let M be a finitely presented R-module (R a commutative ring with unit) and let

RP ~--~ Rq. , M ~ 0

be a presentation (i.e. exact). The kth Fitting ideal of M, Fk(M), is defined to be the

ideal in R generated by all (q-k) x(q-k) minors of the matrix ~,, for q > k > q-p. Fk(M)

is defined to be equal to R, for k _> q, and 0 for k < q-p.

In [20] (but see alternatively [63] Chapter 1 or [41] 4.D) it is shown that the Fitting ideals are indeed well defined - that is, they do not depend upon the choice of a

presentation. Of course, they depend upon the ring R over which M is a module; however, they "commute with base change":

1.2 Lemma. Let M be an R-module of finite presentation and let cp:R~S be a (unitary)

ring homomorphism. Then the k-th Fitting ideal, Fk(M®RS), of the S-module M®RS,

is equal to the ideal generated in S by q~(Fk(M)). In consequence,

S/Fk(M®RS) -~ R/Fk(M ) ®R S.

Proof. This follows from the definition by the right exactness of tensor product, n

If S is a coherent sheaf on the analytic space X, the kth Fitting ideal sheaf Fk(S) is defined to be the sheaf associated to the presheaf which assigns to the open set U, the

k th Fitting ideal of the F(U,0x) module F(U,S). The coherence of S implies that Fk(S) is

coherent also, and that Fk(S)x is the k th Fitting ideal of the OX,x-module Sx.

The usefulness to us of these Fitting ideal sheaves begins with

1.3 proposition. For a coherent sheaf 5 on the analytic space X,

V(Fk(S)) = {xeXISx requires more than k generators over OX, x}

Proof (c.f. [41]). Take a presentation of S over some Stein open set U in X,

0X OXl q ~ SIU ~ 0

associated with sections s 1 ..... Sq eF(U,S) which generate it over Ox(U). Then the

columns of X generate all relations among the s i over 0 X. If for some xeU Sx is

generated by k elements over OX, x, then it will be generated by some k of the germs at x

of the sections s 1 ..... Sq. Let us suppose these are s 1 ..... s k. Then there are relations

110

k k

Sk+l = E i . i ~Jk+lSl . . . . . Sq = E [3qSi, inSx. 1 1

As these relations lie in the 0X, x linear span of the columns of the matrix X, this matrix

must have rank at least q-k at x, and hence

x¢ V(Fk.(s)).

The converse holds by a similar argument. []

1.4 Corollary. V(F0(S)) = supp (S). []

Of course the Fitting ideals are not in general radical ideals.

Now let f:)~ . Y be a finite morphism of analytic spaces; then f.O)~ is a coherent sheaf of

Oy-modules, by e.g. [23] Chapter 1, and to it we can associate the Fitting ideal sheaves

Fk(f.O~). From 1.3, we have

1.5 Proposition.

V(Fk(f ,O~))={yeY1 E c l i m ~ g X ' x >k} xe f-l(y) f *Ta,y

Proof. By Nakayama's lemma, the sum on the right hand side is the number of elements in a minimal generating set for ( f , 0 ~ ) y over Oy,y. n

Thus, for example, V(FI(f .0)~)) is the set of points in Y which have at least two

preimages in )~, or lie in the image of the ramification locus of f.

For mappings in codimension two or more, Fitting ideals do not usually give a

good structure: for example if f:(¢,0) -* (C3,0) is given by t ~ (t3,t4,t 5) then 0¢ , 0

requires three generators over 0¢3,0, so that F0(f .Oc) is generated at 0 by polynomials

of order at least three, whereas the ideal I of all functions vanishing on the image of f is

equal to

(X1X3_X 2, 3 2 2 X2X3-X 1' X 3 - X 1X2)"

In fact (one calculates) F0(f ,0c ,0) = (X1,X2,X3)I, so F0 is not a Cohen-Macaulay

ideal, as it contains an (X 1,X2,X3)-primary embedded component. However, for

mappings in codimension one into smooth spaces, the Fitting ideals do give, under favourable circumstances, a reasonable structure to the image and to the multiple point sets, as we shall see in the succeeding sections.

111

sets, as we shall see in the succeeding sections.

Definition. Let f:X -, ~;n+l be a finite map, with dim X=n. An ordinary k-tuple point o f / i s a point y ~ Cn+l such that f4(y) consists of k distinct points x 1 .... x k, at each of

which J~ is smooth and f is locally an embedding, such that the tangent spaces at y to the

images fCX,xi) are in general position in Ty~: n+l .

Remark. By explicitly constructing a presentation of ( f . 0~)y over 0~;n+ 1 ,Y when y is an

ordinary k-tuple point (see the example discussed after 5.3 below) one checks easily that the ideals 5q(f.OJ~)y are radical. Note that the germ at y of k smooth hypersurfaces

whose tangent spaces are in general position in Ty~; n+l is isomorphic to the germ in

Ty~;n+l ~ ~;n+l defined by the union of their tangent spaces.

We expect that for reasonable mappings, most of the points in V(Fk(f,O~)) are indeed

k+l-tuple points, i.e. have k+l distinct preimages, at least when the dimension of

V(Fk(f,0x)) is greater than 0. When X is smooth, a more precise formulation of this is

possible:

Coniecture. Let f:(¢n,0 ~ (~n+l,0) be a finitely determined map-germ, and suppose dim V(Fk(f.0cn)0) > 0. Then for any representative of f, there are neighbourhoods U 1 of 0

in ~;n and U 2 of 0 in ~;n+l, with f(U1) c_ U2 , such that the set of points y in

V(Fk(f ,0u1)) which are not ordinary k+l-tuple points is a proper subvariety of

V(Fk(f, Ou1)).

This conjecture is proved for map-germs of corank 1 in [43], section 2; it is also easy to check that it is true for low values of k and n.

For a finite mapping f: JX-,Y, let us denote Yk(f,0j~) by Fk(f), or just Fk where there is

no risk of confusion. We denote the variety of zeros of yk(f), with its structure sheaf

0Y/Fk(f ), by Mk+ 1 (f), or simply by Mk+ 1 .

Definition. Let f:)( ~ Y be a map of analytic spaces. An unfolding of f is a diagram of analytic spaces

112

.~ F

\ ,..-. 10}

which is commutative and such that each quadrilateral face is a fibre square and i and j

are embeddings. In short, F :~ ' . 9"is an unfolding of f:J~ ~ Y over S. An unfolding of a map f is called f/at or a deformation if both p and q are flat maps, i.e. p: .~". S and q:9% S are deformations of X and Y respectively.

Notation. Let J~s = p"(s) and Ys = q'l(s); then we have a map fs:)(s -~ Ys-

1.6 Proposition. Let F : . ~ y b e a (not necessarily flat) unfolding of f:)~-,Y. Then

Mk(f) ~ Mk(F)

0 c, S is also a fibre square.

Proof. This follows from 1.2, since Oj~ = O y~OSO{0 } and O y = 09~9sO{0} : for any

yEff;, tensofing a presentation of F.(OX~)y over Oy, y with 0{0 } over 0 S, one obtains a

presentation of (f.O)~)y over Oy,y. (Of course, ID{0 } = I~.) []

1.7. Corollary. Let fs:)~s ~ Ys be the fibre of F:.~', y o v e r s e S. Suppose

2 dim~; 0y,y/Srk(f)y = m < oo. yeY

Then there is a neighbourhood of U of 0 in S such that for s ~ U,

E dim¢ OY,Y/yk(fs)y _< m. yEY s

Proof. By 1.6, the hypothesis implies that the projection ~ from Mk(F ) to the base S of

the unfolding is finite. It follows that zC.OMk(F ) is coherent, and (by Nakayama's

Lemma) that m is the minimal number of generators of the stalk of this sheaf at 0 a S, as OS,0-module. The minimal number of generators of a coherent sheaf of 0 S modules is upper semicontinuous, and so the corollary follows. D

113

We leave the reader to formulate the corresponding local version.

It is natural to ask when equality holds in 1.7, and, more generally, to ask when the projection Mk(F) ~ S is flat. These questions are most easily answered with the help of

the Cohen-Macaulay property. For suppose that S is smooth of dimension d. Then equality in 1.7 holds precisely when Mk(F ) is Cohen-Macaulay of dimension d (see

A.lb), and, more generally, if Mk(F) is Cohen-Macaulay of dimension d' then flatness

over S holds at O a S if and only if the dimension of the fibre Mk(f) is d' - d, (A.la).

In this paper we are concerned with showing that for unfoldings F of finite maps f from

n-dimensional Cohen-Macaulay space X into ¢ n+l, the schemes Mk(F) are Cohen-

Macaulay. We do this by showing that they are detenninantal varieties in the sense of Eagon-Northcott [18]. This notion, and the definition and properties of Cohen-Macaulay schemes, are described briefly in the Appendix.

Although the Fitting ideals Fk(f) are generated by the minors of the matrix ~. of a

presentation, in general it does not follow immediately that the Mk+ 1 are determinantal

varieties, since for k>0 they will not satisfy the condition on codimension (see the def'mition before Theorem A.2.) In order to prove that Mk+ 1 is a determinantal variety it

is necessary to identify Fk(f') as the ideal generated by the minors of a different,

appropriately chosen, matrix, and the construction of such matrices (which depends heavily on the fact that 0 X is a sheaf of rings ) occupies a substantial part of this paper.

§2.

114

Cons~action of a Presentation.

Throughout this section (and indeed, throughout the rest of this paper) (X,,~) is the

multi-germ of a Cohen-Macaulay variety of dimension n, and f:O(,K)~(¢ n+1,0) is a finite analytic map. By the Weierstrass preparation theorem (see e.g. [23] Chapter I), ~ , ~ is a finite On+ 1 -module (where On+ 1 = O¢n+1,0 ), and if the classes of g 1,.-,gh in

0)'~,~/f*m0 generate it as vector space over • = On+l/m0, then g 1,...gh generate O~,~ as

On+ 1 -module. Evidently we may take g 1 = 1, and we shall do so in all o f what follows.

A presentation of O~,~ over On+ 1 is an exact sequence

ot

of On+ 1 -modules. If gl ..... gh generate O)~,~ then one may take q=h and o~(ei) -- gi (ei

= i th member of usual basis).

2.1 Lemma. There is a presentation (1) in which ~. is injective. In any presentation, ~. is injecfive if and only if p=q.

Proof. (c.f. [41], 4.8) As O~,~ is a Cohen-Macaulay ring, it is Cohen-Macaulay as an

On+l module, i.e.

depth0n+l (0)'~,~) = dimOn+l (0)~,~) = n

The syzygies theorem for finite modules over regular local rings ([41, [57] Chapter IV),

gives

depth R M+dim horn R(M) = dim R

dim horn On+l (0~,~) = 1

and so if (1) is a minimal presentation of O)~,~ over On+ 1 , ~. is injective.

If ~. is injective then clearly p_<q. As O~,~ is a torsion On+ 1 -module, tensoring (1) with

Jfn+l • the field of fractions of On+ 1 , we obtain

and so p>q also. Conversely if p=q and ~ is not injective, we must have det ~. = 0 identically. This is absurd, as det ~. generates FO(O~,~), whose variety of zeros is the

115

hypersurface f0~). n

2.2 Algorithm for constructing a presentation.

Choose a projection ;~:¢n+1 ~ s n such that f = roof is still finite (this is "Noether normalisation" and may always be done). After a coordinate change we may suppose that

f(x) = (f(x),fn+l(X)). As 0:~,,~ is Cohen-Macaulay, it becomes a free On-module v i a l*

(see A.1). Let 1 = gl ,g2 ..... gh be generators.

. i Then there exist umque a j e O n, l<_i,j<_h, such that

gj fn+l = ~ (a ] °f) gi

Of course the gi also generate OX',~ over On+ 1 via f* . Now observe that

fn+l = Xn+l ° f; letting

~,!-- a ! * zc for iej, X! = a ! o zc - Xn+ 1, J 3 t x

the preceding equations may be viewed as relations among the gi over 0n+ 1 ,

kj 1" gl + "'" + Xh" ej) j gh = 0

We claim that the Fj generate the module Ker ct of relations among the gi, and do so

freely. To see this it is convenient to embed (X,~) as a divisor in (Xx¢,(~,0)) and define a

map-ge rm F:()~xtg,(~,0)) ~ ( c n + l , 0 ) by F(x,t) = ( f (x) , fn+l (X)+t ) . Then

~,~xC,(~,0)/F*'m,0 ~ 0 2 ( , ~ / f * ~ 0 ' so that the gi (considered as functions on )~xC

independent of the t-variable) generate 0~xC,(~,0 ) over On+ 1 . Moreover, they form a

free On+ 1 - basis, since (XxC,(~,,0)) is Cohen-Macaulay. Since (as one checks easily)

h -i_~l Xi

tgj = jgj, the matrix ~. with entries ~.! we have obtained, is in fact the matrix, with J

respect to the free basis g 1,..,gh, of the On+ 1 -linear map to : 0)~xt£,(~,0 ) ~ O)~x$,(~,0) defined by multiplication by -t. Thus, we have a commutative diagram

h X h On+ 1 --~ On+ 1 -~ Coker X -* 0

II H

to 0;~xC,(~,0) ---) 6)~xC,(~,0) -) Coker t o = 6)~,~ -, 0

of 0n+ 1 -modules, showing that k is injective and Coker X = 0~ ,~ .

116

2.4 The case where X is Oorenstein.

Here we make use of local duality for a finite mapping, in order to give a somewhat more elegant construction of a presentation, in which the matrix X is symmetric. For a discussion of the theory of dualising sheaves, the most appropriate reference is probably [40], where Lipman, following Kunz [36], def'mes the dualising sheaf c0)~ by means of

Noether normalisation. In this approach, the isomorphism (1) below, which is all we use, becomes in effect the definition of the dualising sheaf.

Let O~cn+l , o))~ , O ~ x C be dualising sheaves on their respective spaces. Each is

unique up to canonical isomorphism. The finite morphism F:(Xx¢,(~,0)) -, (~n+1,0)

induces an isomorphism of O)~x$,(~,0)- modules

(1) Homon+ 1 (O)~x~,(~,0), C0$n+1,0 ) -= O)~x¢,(~,0 ).

For a Gorenstein space the dualising sheaf is isomorphic to the structure sheaf, and so

when CX,~) (and therefore )~,xC,(~,0))) is Gorenstein, (1) becomes

(2) Horn On+ 1 (O~x~,(~,0) , On+l) ~ O~x¢~,(~,0)

(for s n + l is certainly Gorenstein).

In [55], Scheja and Storch construct such an isomorphism explicitly in the case where

F:(V,0) --, (~n+1,0) is a finite mapping and (V,0) is a complete intersection of dimension n+l. It is given as follows: let the regular sequence H 1 ..... H d define the germ (V,0) in

($n+d+l,0), and let F1 ..... Fn+l be the map ( cn+d+ l ,0 )o ($n+l,0) whose restriction to V

is F. Then a generator of Hom(0n+ 1 (OV, 0 , (0n+ 1 ) as O V ,0 module is given by the

homomorphism yI:(0V, 0 -, On+ 1 defined by

n(r) = Tr(r/J)

where Tr: KV,0 -~ Xn+l is the trace homomorphism, KV,0 , Kn+l are the fields of

fractions of OV, 0 and On+ 1 respectively, and J is the restriction to V of the Jacobian

determinant of the mapping (¢n+d+l ,0) ~ (¢n+d+l ,0 ) with components F1 ..... Fn+l,

H 1 ..... H d .

In general, an isomorphism of B-modules HomA(B,A) ~ B (where the ring B is an A-

module) gives a perfect A-bilinear pairing < , > :BxB~A; for if q e Horn A (B,A)

generates it over B, then the pairing <b,c> = Tl(bc ) is clearly perfect, since we recover the

117

isomorphism B-,HomA(B,A) by sending b to <b, >.

The key to the usefulness to us of this pairing is the fact that with respect to it, the homomorphism of A-modules ¢Pr:B~B defined by multiplication by the element rEB, is

self-adjoint.

Let us return to the situation that interests us. If gl,.. gh is simultaneously a free basis of

t~x , , (~ ,O ) over On+ 1 via F, and of O~,~ via f, choose a dua/ basis g'l ..... g'h of

~x~;,(~,O) with respect to the pairing < , > defined by (2); that is, a basis such that

<gi,g'j> = 8ij. Then the restriction of the g'i to (X,~) gives an On-basis for O~,~ via f*.

We now modify the presentation of O~,~, over 0n+ 1 obtained in 2.2, by composing k

h h G' with the isomorphism 0n+l~0n+ 1 induced by multiplying by the (invertible) matrix [ I ] G

(= the identity written with respect to the bases G' and G). So we have a presentation

G' h x[I]G

O-*On+ 1 ---> oh1 4 0~,~-* 0

GG ~ G' Since ~. = [q0] , ~. [I] G [¢P] and so writing k[I] G' = ~t, we have = G )

cp(g'j) = tg'j = ~.~tt!g' i, and i j

k <Xkt!g i , g'k> = <-tg'j , g'k> }.tj= i J

i 1 ° = <g ' j , - tg ' j> = <g ' j , X l . tkg i> = l.t J . i

That is, we have proved

2.5 Proposition. Let CX,~) be the germ of a Gorenstein variety of dimension n, and let

f:0(,R) ~ (cn+1,0) be a finite mapping. Then there exists a presentation

h cx o-, oh1 -, o

of O)~,~ over 0n+ 1 , in which p. is a symmetric matrix and a (e l ) = 1. a

2.6 Example.

Let f:(C2,0) ~ (•3,0) be defined by

118

fix,y) = (x,xy+y3, xy2+cy 4)

(this is a member of the unimodal family P3 of [46]). Writing f(x,y) = (x,xy+y 3) we find that 0 2 (source) is generated over 0 2 (target) viaff by 1,y and y2. Solving the equations

2 yJ(xy2+cy4) = ~ a] (x,xy+y3)y i O< j < 2

as in 2.2 we obtain the (asymmetric) presentation matrix

-X 3 (l--c) X1X 2 cX~

cX2 (c -1)~-x 3 (1-2c)x~x 2

(1-c)xt cXa (c-l) ~ - x 3

By adding X 1 (column 1) to column 3 and then interchanging columns 1 and 3 we obtain the symmetric matrix ~.

"cX _XtX 3 (1-c)X X2 -x 3

(l-c) X1X 2 (c-l) X~I-X 3 cX2

-X 3 cX 2 (l-c) X 1

corresponding to a presentation taken with respect to dual bases 1, y, y2 and y2-x, y, 1.

119

§3. Frequently Fitting Spaee, s are DetermlnRntal Varieties.

First we record the well known fact that under favourable circumstances Y0 defines the

reduced image.

3.1 Proposition. Let (X,~) be an n-dimensional irreducible Cohen-Macaulay variety

germ and let f:(:K,~) --~ (~:n+1,0) be a finite map-germ. Let X be the (reduced) image of

f, let I X be the ideal in On+ 1 of functions vanishing on X, and let d be the degree of the

map (X,~) ~ (X,0). Then 5r0(f) = IX d.

Proof. Any representative of f becomes a branched cover over a sufficiently small Stein neighbourhood U of 0 in X. At a regular point xeU which is a regular value of f, let h x

generate IX, x (= ideal in Ocn+I ,x of functions vanishing on (X,x)). As f is unramified

over x, (f.0)~) x is a free OX, x module of rank d, and hence can be presented over

0cn+l ,x by

hxI d (9 -~ 0 21 "* (f*0X)x "~ 0 ¢ ,x ~ O~:n+l,x

where I is the identity matrix of rank d. Thus we have 70(f)x = IX,x d.

As 5r0(f)0 = F0(O)~,~) is principal (by 2.1) and def'mes the same locus as Ix, 0, by 1.4, the

result follows by the coherence of Y0(f). []

3.2 Proposition. Let (X,~) be as in 3.1, but no longer irreducible, let (XI,~) ..... 0(m,~) be

its distinct irreducible components, let X i be the reduced image of )~i under f, let d i =

degree f:(Xi,~)~(Xi,0), and let Ixi,0 be the ideal of functions vanishing on (Xi,0). Then

m

F0(f) = Y I I x i , 0 di %1

Proof. This is proved in the same way as 3.1. o

We shall refer to a map f as in 3.2 as having degree 1 onto its image if each d i is equal to

1, and X i ¢ Xj for iej.

Our next result establishes the structure of determinantal variety for M2(f) in case f is of

degree 1 onto its image. First we need a lemma:

3.3 Lemma. Let

hoh.+, o

120

i be a presentation, with ct(ei) = gi, and let mj be the minor determinant obtained from ~. by

_k :!:m i deleting i and column j. For all i,j,k with 1 <id&-~ h, we have uJjg i = jgk. r o w

Proof. Fixing j, the remaining columns of ~. give h-1 relations among the gi:

h Z i (1) ~..~gi = 0 i=1

and thus

h k = -~L,g k

1 <$<h, *#j.

i=l

iCk

The matrix M k of coefficients of the gi on the left hand side of the system of equations

(2), has determinant mk. Let N k be the matrix obtained from M k by replacing the i th J 3 J

column of M by the column on the right hand side of (2). Then det N = ~:mjg k. Since

by Cramer's rule

det Mjkgi = det N~j,

the result follows, n

3.4 Theorem. Let CX,~) be an n-dimensional Cohen-Macaulay germ, and let f:0],R)

(~;n+1,0) be a finite map, of degree 1 onto its image. If

h cc

is a presentation, with cz(ei) = 1, then 5rl(O)~,~) is equal to the ideal 5rl ' generated by the

maximal minors of the matrix ~ 1 obtained by deleting the first row of X. In particular,

M2(f) is a determinantal variety.

121

Proof. Let C c_ 19X, 0 be the conductor ideal of the map (X,R) -* (X,0) induced by f

(where (X,0) is the image of f): that is, C is the ideal of elements r a0x , 0 such that

r.0~,R c_ 0X, 0. It is in fact the largest ideal of ¢9X, 0 which is also an ideal in 0)~,R. Let

J c_ 0n+ 1 be the preimage of Cunder the projection On+ 1 ~ 0X, 0 -~ 0. Then we have F1

c_ J; to see this, it's only necessary to show that for each i,j,k, m]g k a 0X, 0. As gl,-.,gk

Z k generate 0~,R over 0 n+l, there exist akt ~ 0X, 0 such that gsgt = as t gk.

k

3.3,

s 1 Z k 1 Z mjk mj gt = mj gsgt = as t gkmj = akt ~ OX,0 k k

and hence Yl c_ C.

Now by

Observe now that

On+ 1 -~ 0~,,~/~X, 0 ~ 0

is exact, so that FI' = F0(OX,R/0X,0) (as On+ 1 -module). Since f is of degree 1 onto its image, the support of 0)~,~/OX, 0 is of codimension at least 1 in X, and hence at least 2

in cn+l . The t h e o r e m of [9] t h e r e f o r e i m p l i e s tha t

F0(Cg~,~/0X,0) = Ann 0n+l (0~,~/19X,0), and since this annihilator is J, we have

F1c_ J = FI' C_ F1,

whence F1 = FI'. Since F1 is generated by the maximal minors of an (h-l) x h matrix, its grade is less than or equal to 2; the hypothesis that f be of degree 1 onto its image implies that grade (F1) = 2, and hence F1 is a determinantal ideal and M 2 a determinantal

variety. []

Another proof of 3.4 may be found in [29].

It will be useful to record:

3.5 P ropos i t i on . Let (X,~) be Cohen-Macaulay of dimension n, and let

f:(X,R) ~ (~n+1,0) be finite and of degree 1 onto its image, (X,0). Let CO_ OX, 0 be the

conductor ideal of the induced map (X,R) ~ (X,0).

Then

122

i) z l ( f ) O x , o = c

ii) C 0~,~ ~" o~,~ (as 01~,~-modules)

CO~,~ is principal if and only if (X,i) is Gorenstein. iii)

Proof.

(i) is already shown in the proof of 3.4.

(ii) Since X is a hypersurface, it is Gorenstein. Thus, in the isomorphism of dualising modules

Homox,o (0~,~, oX,O) -z,_ co~,~,

COX, 0 may be replaced by 0X, 0. As f is of degree t onto its image,

Hom0x,0 (0)~,~, 0X,0) embeds in OX, 0, via the map sending the homomorphism o to

o(1). Seen in this way, HOm0X,0 (03~,2,0X, 0) is none other than C 03~,~.

(iii) This is immediate from (ii). []

The following consequence of 3.4 does not seem to be in the literature:

3.6 Theorem. Let (C,2) be a reduced curve-germ and let f:(C,~) ~ (~2,0) be finite and of

degree 1 onto its image (C,0). Writing 8 = 8(C,0), 8 = 5(C,~), 0 ~ = 0~,~ and

0C = 0C,0 we have

5 = 6 + dime 02/FI(f )

Proof. Consider the following diagram of modules and morphisms:

dO C c+ dO(~

m co C <... co~

By each inclusion we have written the length of its cokemel. Here tl and ti are the Milnor numbers of these curves, (C,0) and (C,2) respectively, as defined by Buchweitz and Greuel ([12]). There is an inclusion of o ~ in co C, defined as follows:

co(~ -~ Homoc(O~, O3C) ~- Homoc(O~, OC) = C,

123

where Cis the conductor ideal in 0 C, and then C ~ • C ~' co C. With respect to this

inclusion we have

m = d£m¢ COciCO ~ = dira¢ OcfC.

Buchweitz and Greuel showed that Milnor's formula I.t=28-r+l continues to hold with their extended definition, where r is the number of branches. Composing the inclusions, we have

I.t = d i m mc/d(~ C = 8-g+ I~+m.

However by 3.2, d ime OC/C = dime (~2/Yl(f);

hence

25-r+1 = 8-8+2~-r+1 + dirn~ (92/71(0

from which the desired equality follows. Note that the number of branches of (C,~) and (C,0) must be the same. []

We are grateful to J. Steenbrink for suggesting the lines of the proof of 3.6, and also for pointing out to us the following more geometric proof, valid for smoothable curves:

Let (X,~) -~ (¢,0) be a smoothing of (C,~), with fibre Ct, and let F:(X,~) ~ (¢2,0) extend

f, let C t = F(Ct), and suppose that for t~0, C t has only ordinary crossings. For the level

preserving map F:(X,~) -+ (¢2 x (~,0) defined by F(x) = (F(x),rc(x)) we have

(9¢2,0/9"1(t") - 0¢2 x ¢ ,0/Yl( ~ @0¢2 x ¢,0 (9¢2,0

(see 1.6) and since, by 3.4, OG2 x ¢,0/Srl(r: ) is Cohen-Macaulay, the number of points

in the fibre of M2(r:) over ¢ is constant. Thus, C t has d :=d im¢O2/71( f ) ordinary

crossings. Each of these may be smoothed, to give a curve Ct ° which is, of course, a

Milnor fibre for C O = (C,o). In doing so, we raise the genus by d. Thus we have

%(Ct' ) -_ x(Ct)-d = z(Ct)-2d

and hence, as C t' and Ct are Milnor fibres for (C,0) and (C,~) respectively we have

1-~t(C,0) = 1 -~ (~ , ~)-2d,

124

Now using Milnor's formula I.t = 2~-r+l, we obtain the result.

3.7 Example. Let (C,0) = V(xy) c_ (~2,0) and let f:(C,0) ~ ((£2,0) be defined by (x,y)

(x2+y3,y2+x3). Calculating a presentation, we find that d im~92/71( f ) = 5. The image

of the deformation ft(x,y) --- (x2+y3+ty,y2+x3+tx) (to0) has 6 nodes, but of course the

node at (0,0) does not lie in M2(ft) since the germ of ft at (0,0) defines an embedding.

3.8 Equations defining an analytic space.

In order to be able to give examples, we describe a method for finding the equations of an

n-dimensional Cohen-Macaulay analytic space )(, starting from the image and double-

point locus of a finite map f:C{,i) ~ (~:n+1,0). The idea goes back to Petri [49] for projective curves, and has been used several times since, e.g. by Arbarello and Semesi [2], and by Catanese [14] for projective surfaces.

3.9 Proposition. Let (:K,~) be a multi-germ of a Cohen-Macaulay analytic space of

dimension n and f:(X,K) ~ (•n+1,0) be a finite map of degree 1 onto its image (X,0). Let

~. be an (h × h) presentation matrix of f , (0 )~ ,~ as 0n+ 1 -module with respect to

h generators g 1 = 1, g2,.-,gh. Suppose that there exist elements ~i h , {x i j E 0n+ 1 such that

i X k k m l f o r a l l l < i j < h (i) mj = k

(ii) ~ ~.jkl Xilk = Xtxik~.k£1 in OX,O for all 1 <i,j,£ < h k k

Then gigj= X a k j g k . k

Proof. Observe that there always exist elements ~-ij k ~ 0n+l such that mj = X by k

Theorem (3.9). In 0X, 0 we have that

m , -- g i m l = . . m k = k k

xjklxilkm~ k I m ~ = ~ a i j m l , = = OtiJ ~'k'l k

125

, Z k by (3.3), (i), (3.3), (i), (ii) and (i) respectively. Dividing by m 1 gives gigj = °tij gk. o

k

3.10 Corollary. If moreover CX,~) is Gorcnstein then we may take k to be symmetric, and

k. i Z k i Z k there exist elements ~-ij m On+ 1 such that mj = ~. m k and gigj = ~. gk- k k

Proof. By Proposition (2.5) we may assume that k is symmetric; by Theorem 3.4 there

k i rn~. k 1 exist elements ~-ij such that mj = ~ i k By the symmetry of ~. we have m I = mk, and

we may assume ~.i k = ~i- Thus a = ~-ij satisfies (ii), hence gigj = ~- gk- o k

3.11 Proposition. Let (X,~) be a mult i -germ of a Cohen-Macaulay analytic space of

dimension n and let f:(X,.,2) ~ (¢n+1,0) be a finite map. Let k be an (h x h) presentation matrix of f , (O~,~) as On+ 1 -module with respect to generators gl = 1,g 2 ..... gh. Suppose

gigj = Z k

txij gk. Then the multi-germ (X,~) can be embedded in (C n+l x Ch-1 , {0} x k

~ h - 1 ) with equations

h 1 j

( * ) X i + ~ ' i v j=0 l ~ i < h , andViV j = c t + j=2

Proof. See Catanese [14] Theorem 4.3. o

h Z k O~ij V k ; 2 < i,j _< h. k=2

3.12 Definition. Let k be an (h × h)-matrix and ~1 the matrix obtained from ~. by

deleting the first row. If Ih_ 1 (~1) = Ih_ 1 (~.) then we say that ~. satisfies the rank

condition, R. C.

3.13 Remark. Ih_ 1 (~1) = Ih_ 1 (L) if and only if there exist ~i k such that

i k 1 mj = ~..~jm k

126

3.14 Proposition. Suppose X is an (h x h)-matrix with entries in 0n+ 1 satisfying R.C.

Let I = Ih_ 1 ( t) and g = det k and let (X,0) be the germ in (¢n+1,0) defined by g. Let C

= IOX, 0 and suppose V(I) has codimension 2 in (¢n+l,0). If R = Coker(~.) then R ~,

HOm0X,0(C,O, R is a semi-local ring and R is generated by g 1,..,gh as 0n+ 1 -module,

i m I

where gi is multiplication by "-i" on C.

m 1

Proof. In case k is a symmetric matrix Catanese [14] Theorem 4.7 relates the result to a

- - = ~ k - - i 1 theorem of Rouchf-Capelli. He def'mes a product eie j ~ - - i j e k, in case mj = ~%i k m k

k

h and ~ is the image of the i th standard basis element of 0n+ 1 in Coker (k), which he shows

to be well defined, commutative and associative.

The general result is proved by de Jong and van Straten [29] as follows.

The sequence

0 h-1 (~-~)t 0 h l ~ I ~ 0 n+l

is exact, since I = I h_ 1 (~') and V(I) has codimension 2, see A.2.

The sequence

1 1 is since det exact g = = 2.,Z.km k.

Thus

0-) Homox,o(C, Ox,o)~ Oh.o ~-~ 4 ,0 is exact, since Hom(-,OX,O) is right-exact and C = IOx, 0 by definition.

But the sequence

h ~" h adO~) ~ ~ h .... -~ 0 X,O - ) 0 X,O "-> 0 xho x.o

127

is exact and periodic with period 2 [191. Thus Coker (~.) ~ ker (k) ~ Horn OX,0(C, Ox,0).

The isomorphism from R to HOmOx,o(C, OX,0) is given by

i m 1

~ gi, where gi is multiplication by 1 " Furthermore,

m 1

1,i 1 i 1 i ml, j g = mlm j - mj ml in On+ 1

1 i (see 4.1 below), and thus gimj = mj in 0 X, which is an element of C, since ~ satisfies

R.C. Hence giCc_ C, so R ~ Hom0x,0(C,c"). This makes R into a commutative ring with

a unit element. R is a finitely generated module over the local ring On+ 1 . Hence R is a semi-local ring. t3

3.15 Corollary. Suppose moreover g , 0. Let (J~,~) be the multi-germ of an analytic

space with semi-local ring R. Then O(,x) is Cohen-Macaulay. If (X,~) is embedded in (C n+l × ¢ h- I , {0} × igh-1 ) with equations (,) and f:(?(,~) ~ (¢n+1,0) is the restriction of

the projection onto (¢n+1,0) then f.(O)~,~) has presentation matrix X. If moreover X is

symmetric then (X,~) is Gorenstein.

Proof. The sequence

0n+l ~ R ~ 0

is exact since det X = g , 0, and by definition of R. Hence pd~)n+lR = 1 and dim R = n.

Therefore depth R = n and R is a Cohen-Macaulay ring and (X,~) a Cohen-Macaulay analytic space. Furthermore Coker ~. = R = f.(Oj~,~) as On+ 1 -modules. Hence ~. is a

presentation matrix of f*(O~,~). Now I = 5rl(f) is generated by {m~ I i= 1,..,h} since X

satisfies R.C., and IOj~,~ = (mll) 0)~,~ since

1 i i 1 rni = ml = ml gl = ml giin O~,~,

by the symmetry of ~., as g 1 = 1 and by 3.3 respectively. But I O ~ , i m ¢0j~,~, by 3.5 (ii).

Hence the dualising module of (X,~) is generated by one element, and (J~,~) is Gorenstein, by 3.5 (iii).

128

~4. Triple Points.

Throughout this section f:(X,xo)-,(~n+1,0) will be a finite mapping of an n-dimensional

Gorenstein variety germ of degree 1 onto its image, except where explicit mention to the

contrary is made. We may thus assume given a symmetric presentation matrix I.t = [9]G'

as in 2.4, where G and G' are dual bases of 0~x~, (x0) over 0n+ 1 and gl = 1. Let F2' be

the ideal in 0n+ 1 generated by the (h-2) × (h-2) minors of the matrix till obtained by

deleting the first row and first column of It; and let miil"~ J2 the minor determinant of ~t

obtained by deleting rows i 1 and i 2, and columns Jl and J2. Thus 72' -- /{ml,j I " 1,i

2<i,j~h}).

4.1. Proposition. F2' = Y2 = (F12:F0)

Proof: By [161 or [151, Section 6,we have the Laplace identity

il,i 2 il i 2 il i2 (1) mji , j2g = + m j . m j +mj mj

- 1 2 - 2 1

where g = det(kt) generates F0. Thus F2 c_ (Z12:F0). We will prove the proposition by

showing that (F12:F0) F ' = 2 . To do so, we obtain first a presentation of the ideal

I' = (g)+F12, and then use this to calculate (FI2:F0). We need

4.2. Lemma. F0 + yl 2 = F 0 + m l F 1

Proof. From (1), we obtain

1,i 1 i 1 j i 1 1 1 ml, j g = + m l m j + m j m 1 = + mjm 1 + mi r a j . Since F1 is generated by

1 1 m i , i = 1 ..... h, this shows that F12 is contained in FoF2 + mlF1. D

1 1 Proof of 4.1 (continued). We obtain relations among the generators g, mlm i 1 < i < h of

F0+F12 as follows: first, from

h • 1 1

g = 2..,(-1)1-11~i m i we obtain i=l

129

h 1 ~ i _ l l 1 1

mlg = ~ ( - 1 ) gi mlmi l--I

(r0

Second, all relations among the

have

h

~ ( - 1 ) i - l l x k m : - - 0 i= l

1 m i come from rows of the matrix ~t: for k--2,...,h we

(rk)

1 1 Multiplying these by mll we obtain relations among the m lm i . We thus obtain a

complex K"

h ~ h + l ~ 0-*0n+ 1 0n+ 1 F0+F12 0

1 1 in which 13(e0) -- g, 13(ei) -- m l m i for 1 < i < h, and ~ is the matrix whose columns are the

coefficients of the relations r 1,...r k obtained above. Thus ~ is equal to [1] 1" m 1 0...0

h+l t

,~ ~t

h

The h × h minors of~ , however, generate Y0 + 5r12' and it follows by A.3 that K is exact,

and hence that it gives in particular a presentation for S O + 9-12 over 0n+ 1 . Suppose now

that

sends thei,j thgeneratOrei, j to m ~ m ] , l < i < j < h . DefineS: 0!+h~ h+ 1) h+l

- - - "* ~)n+l by

1,i lifting in 7 (see the diagram below): 8(ei,j) = + m 1 . e 0 + multiples of e,..,e h (using (1)).

,J By diagram chasing in the diagram

130

0 $

,I.8 ,l.i

0 h 1 ~ 0n+h+ll ~ Y0 + Fa 2-~0

j~NA ,],j

yO+ Srl 2

yl 2

$

0

we obtain a presentation

0~h(h+2) ~,8] h+l j~ YO+F12 n+l "~ On+ 1 "~ 2 -)0

Evidently the elements of the first row of ~,8] generate AnnOn+l(F 0 + Yl2 /Y ' I 2) =

1 1 (Sr12:Sr0); and these are just m 1, together with the generators of 5r2 '. As m 1 ~ Y2', we

have

(Y12:Sr0) = Y2'

and this completes the proof. []

One can obtain a geometrical interpretation of the matrix bi] by observing that by 4.2 and

the Hilbert-Burch theorem, (bil) t is the matrix of a presentation of 5ri over On+ 1 . Of

course, by symmetry (lil) t = P-1 (delete the first column instead of the first row) and now,

.1 obtaining bt I by deleting the first row of b11, we recognise it as the matrix of a

presentation of 9"1/(m I )" Thus,

131

that is, it is the first Fitting ideal of the On+ 1 module obtained as the quotient of FI(O/~,~)

by its subideal (ml) Thus, ~ e zeros of F2' (which by 4.1 are of course just the zeros of

F2) are just those points where FI(O~,~) needs at least two generators besides m I. If one

defines a generic element of F1 to be one whose preimage in 0~ ,~ generates the

conductor ideal C, then one can show that the first Fitting ideal of fl(o: S)/(ml ) is

1 unchanged if one replaces m 1 by any generic element o~ of FI(O~,2). Any set of

generators of F1 over On+ 1 must contain at least one genetic element, and thus we can

view F2', and hence F2, as measuring those points where F1 is not a complete intersection

ideal. In this connection it is an instructive exercise to show that at an ordinary triple

point XYZ=0 in 033, where the first Fitting ideal F1 of the normalisation is generated by

YZ, XZ and XY, the first Fitting ideal of F1/(aYZ+bXZ+cXY) (a,b,c, ~ 03) is equal to

the maximal ideal if and only if at least two of a,b and c are units, while aYZ+bXZ+cXY is genetic if and only if all of a,b and c are units.

The hypersurface defined by a genetic element of f l is referred to in the literature

as an adjointhypersuneace - see e.g. [141, [51.

As an important corollary of 4.1 we have

4.3. Theorem. Let f:(X,~) ~ (03n+1 0) be a finite mapping from the n-dimensional

Gorenstein space (X,2) and suppose that f is of degree 1 onto its image, and that codim V(F2) -> 3. Then F2 is a symmetric determinantal ideal, and in particular is Cohen-

Macaulay. If n=2 and there exist deformations of f:(J~,~) ~ (033,0) exhibiting only

ordinary triple points, then the number of these is equal to dim03 03/F2.

Proof. This follows from 4.1 by A.5 and A.1.6. t3

The next results reveals the somewhat surprising fact that the number of virtual triple

points in a mapping of a Gorenstein surface 0(,2) to 033, for which F1 is radical, depends

only on F1 itself. Recall that for any ideal I c_ On+l, the primitive idealfI is defined to

be {hEI I J h c I}, where Jh is the Jacobian ideal of h. Thus, if I is radical, f I consists of

the defining equations of all hypersurfaces singular along V(I). Chapter 1 of [481 (and [49]) contain other characterisations o f f I . If I is a complete intersection ideal, then f I =

I2; otherwise, 12 is strictly contained infI, as may be seen for example in the case where I = (YZ, ZX, XY) defines the singular locus of the standard triple point XYZ = 0. Here,

132

fI = 12 + (xYz). That f I / I 2 be cyclic is in fact quite typical, for we have

4.4. Theorem. Let (X,~) be Gorenstein of dimension n, let f:(X,~) ~ (¢n+1,0) be a finite

mapping, and suppose that F1 is radical and codim (M2(f)) = 2. ThenfY 1 -- F12 + F0.

Proof. It is clear that fYl _~ F12 + :TO, since f(X) is singular along V(F1), and that

V(fF1) = V(F12 + F0) = M2 as sets. To prove the theorem, observe that from the

presentation of F12 + F0 given in the proof of 4.1 we see that dim hom0n+l (F12 + Y0) =

1, and hence dim horn0n+l(On+l/F12 + :TO) = 2. Thus dep th0n+l (0n+l /F12 + F0) =

n - l , which coincides with its dimension, so that 0 n + l / F 1 2 + F 0 is a Cohen-Macaulay

On+ 1 -module. In particular, 0n+l/9:12 + F 0 has no embedded primes. We now use the

following elementary lemma of primary decomposition:

4.5. Lemma. Let I c_ J be two ideals in the ring R, which define the same variety in Spec R. (i.e. are contained in the same primes of R). If I is unmixed (that is, if R / I has no

embedded primes) and if I and J are generically equal (i.e. IRp = JRp for each minimal

overprime p of I) then I= J.

Proof. Let M = J / I . Then M c, R / I , so ASsR(M) c_ ASsR(R/I) . As SuppRM _D

ASSR(M ) ([45] Theorem 6.5) we have

ASSR(M ) -__ ASSR(M ) ~ SuppRM c_ ASsR(R/I) c3 SuppRM

Now because R / I has no embedded components, ASsR(R/I) = {minimal overprimes of

I}, and so ASSR(M) = ~, since Mp = 0 for every minimal overprime p of I. Hence

M = O . o

Proof of 4.4 (continued). To apply 4.5 we need to check only that FI 2 + F 0 and fY l are

generically equal. But this is evident, for at a smooth point of V(F1), F1 is generated by

a regular sequence and sofF1 is generically equal to F12 since F1 is radical, n

4.6. Corollary, With the hypotheses of 4.3 and 4.4, suppose also that n=2. Then if there

exist deformations of f:(X,~) -, (C3,0) having only ordinary triple points, the number T of

these is equal to d ims fF1/912 .

Proof. By 4.3, T = d i m • 0 3 / F 2. Now combining 4.1 and 4.4 we have F2 =

Annon+l (~F1/F12), and sincefF1/F12 is a cyclic 0n+1 -module (generated by g = det.

l.t),f F1/ F12 =- On+l / AnnOn+l(f F1/ F12) = On+l/F2. o

133

As a corollary to 4.4, we have a necessary condition that a weakly normal surface singularity has Gorenstein normalisation.

4.7. Corol l~. Let (X,o) be the germ of a weakly normal surface in (¢3,0) with defining equation g=0 and with singular locus (~,o) defined by the ideal I in 0 3. Let

n:CX,~)~(X,o) be the normalisation map; if (X,~) is Gorenstein thenfI = I2+(g) and I~0"I) = I-tg)+l, where ~ = minimum number of generators.

Proof. The hypothesis of weak normality (see, e.g. [1] for the original definition and discussion) means merely that X is reduced and that away from 0 its only singularities are points of transverse crossing of two smooth sheets (Am, in Siersma's terminology, c.f.

[48], [59] [58]). This guarantees that Yl(n.0)~) is radical (and therefore equal to I) since it

is clearly radical at Am points, and is a Cohen-Macaulay ideal, by 3.4 (so that 4.5

applies, with I = 71(n,0~), J = ~,/Srl(n,O~) ). Thus, if (X,~) is Gorenstein, 4.4 applies,

and It(J'I) = It(I) + 1 by 4.2. []

Huneke, in [27], gives examples of prime ideals I in 0 3 def'ming a curve in (¢3,0)

such thatfI/I 2 is cyclic but p.(~I) > It(I)+l. See 7.4 below for further discussion of this point.

The role of f I / I 2 in predicting the number of triple points in a generic deformation of a

map f:(i{,~) ~ (¢3,0) (J{ Gorenstein) is mimicked in the following formula, which depends upon a result of R. Piene's [51].

4.8. Proposition. Let f:(¢2,0) ~ (¢3,0) be of degree 1 with weakly normal image (X,0). Then a generic deformation of f has C pinch points ( = cross-caps = Whitney umbrellas, etc.) where

c-- fZl/hJ+Z0

(here the Fitting ideals are those of f , OG2, and J is the Jacobian ideal of the generator of

Fo).

Proof. Weak normality implies that f has isolated non-immersive point, and hence that C

= dim¢02/R, where R is the ramification ideal, generated by the 2 x 2 minors of the

matrix of df; for then R is a Cohen-Macaulay ideal, and so dime 02/R is just the

intersection number of df:(¢2,o) -, L(¢2,¢ 3) with the variety Y. 1 c_ L(¢2,G3) of matrices

of rank <2. Now in [51] Piene proves that CR = f*J.

134

and

1 Let m 1 ~ 0~2,o be some fixed generator of Cand let X --- f((g2). Then

1 (m~) 2 (0(g2,o) = mlCOX, o .x,_ F12 + Fo/Fo

1 2 1 F12+Fo (m 1) R = (ml)(f*J) = F1JOx, 0. Hence 02/R ~ =fF1/F1J+Fo._ o

F1J+Fo

135

§5. Cyclic Extensions.

Although our results so far apply only to the first and second Fitting ideals, we expect that they can be generalised to the higher Fitting ideals. In this section we provide evidence to support this view by proving that when O~,K is cyclic over On+ 1 (that is,

when O~,~/f*mn+l ~ ~[t]/(th ) for some finite heN, and when certain genericity

hypotheses hold, a / /of the ideals Yk(f), for l<_k.~_n, are determinantal.

If f:(cn,0) ~ (~n+l,0) is a finite mapping with dim Key df 0 = 1 then O n is cyclic

over On+l. This case has been examined in [43] and [47], where multiple points are

studied from a different point of view, via the construction of "source multiple-point-

~k schemes" Df c_ (cn)k, consisting essentially of k-tuples of points in C n sharing the same

image under f. If r~:I) k ~ Cn is the map induced by projecting (~n)k onto any one of its

factors, then our M k is equal (set-theoretically)to fon(I)k).

Suppose therefore that (9~,2 is cyclic over On+ 1 , and let y ~ 07~,2 be some element

projecting to the generator t of ¢[t]/(th ). By the preparation theorem, the elements

1,y ..... yh-1 generate 0~ ,~ over On+ 1 . Fix a presentation P

h 0 ~ (9n+ 1 On+ 1 -~ 0)~,~ -4 0

1 h t with respect to this generating set; each column (~.j ..... ~.j ) of ~. is thus a relation

i~l~.] = be the matrix obtained from by deleting the first k rows, and let yi- 1 0. Let X "=

5rk ' be the ideal generated by the maximal minors of ~k. Thus 7k' c_ 7k. We will show

that V(5-k') = V(Tk), and if moreover codim V(Srk ') = k + 1 (so that 5-k' is determinantal)

and codim V(Srk+l ') = k + 2, then 5rk ' = 7k. In order to deal with 7n, we will need an

extra hypothesis, that there exist a deformation F:(.~,2) ~ (cn+2,0) of the pair ((X,~),f) for which the previous hypothesis holds for the ideals 5rn(F) and 5rn+l (F).

5.1. Lemma. Let O~,xo be cyclic over On+ l, and let 5rk, Yk' be as defined in the

preceding paragraph. Then V(Srk) = V(Y-k').

Proof. For any representation f of the map-germ f:(O~,~) -, (C n+1,0), we may choose a

136

Stein neighbourhood U of 0 ~ ~;n+l such that, restricting f to f'l(U), the sections,

1,y ..... yh-1 generate (f.0j~)(U), and such that the presentation P of 0~,Xo extends to a

presentation

h ~" oh _, f . 0 ~ I U 040U~ ~ 0

of f*OJ~ I U"

Recall from 1.3 that Mk+ 1 = {z~ U I (f*Oj~) z requires at least k+l generators over OU,z}.

We claim that if zCMk+ 1, then in fact ( f ,O~) z is generated by 1,y,...y k-1. This is

nothing more than Nakayama's lemma: let s be the least integer such that some relation yS

S-I

= Z o t i u i (o~fi¢) holds in M = (f*Oj~)Z/mz.(f,O~)z. Multiplying this relation by i=0

powers of y, one sees that 1,y ..... yS-1 generate M over ¢, and hence that 1,y ..... yS-1

generate ( f ,O~) z over OU, z. Clearly 1,y ..... yS-1 are a basis for M, and hence a minimal

generating set for ( f ,O~) z, and thus s _< k. This proves the claim.

It follows that as sets

V(Fk) = supp (f, Oj~)/Ou<l,y ..... yk-1 >).

Now the matrix X k is actually the matrix of a presentation of f * O ~ / 0 U < l , y ..... yk-1 >,

and since for a coherent sheafS, V(fo(S) ) = supp (S) as sets, we have

V(Fk) = V ( F o ( f * O ~ ) / O U < l , y ..... yk- l>) ) = V(Fk') as sets. D

5.2. Theorem. Let tgj~,~ be cyclic over 0n+ 1 , and suppose further that codim Mk+ 1 --

k+l, codim Mk+ 2 = k+2. Then Fk = Fk' and is therefore determinantal.

Proof. By 5.1, Mk+ 1 = V(Fk') as sets, and now the hypothesis on codimension of V(Fk)

implies that Fk' is determinantal and hence unmixed (A.2). The theorem will follow by

4.5, provided wecan show that Fk and Fk' are generically equal. Now by the assumption

on codimension Mk+2, a generic point of each component of Mk+ 1 does not lie in Mk+ 2.

Let z by such a point. By the proof of 5.1, (f*0j~) z = 0U,z{1,y ..... yk}, so there exist relations

Rj yk+j 0 k = ~3j +...+~3j yk 1 < j < h-k

It follows that by means of column operations we can bring the matrix ~. of the

137

presentation to the form

Fk = ({a} 1 <i,j _ k+l}),

Fk' = ({ak+11 1 <_j _<k+l}).

We can also characterise Fk' as the ideal generated by the coefficients ~k in all relations

R oq+a2-y+---+aky k = 0

among the generators 1,y ..... yk over 0U,z: this is clear, as any such relation is a linear

combination over 0U, z of the column of A. We complete the proof by showing that in

any relation R, ~ of the coefficients lie in Fk'. This may be shown inductively: suppose

that for somej with 1 _<j < k, we have ~j+l,Otj+2 ..... ~k ~ Fk'. Multiplying the relation R

by yk-j and using the relations Rj, we obtain a relation in which the coefficient of yk is

otj + 0~j+ 1 [31 +...+ o~ k _j.

It follows that ~j ~ Fk'.

Thus, we have proved that when z ~ Mk+ 1 - Mk+2, (Fk)z = (Fk')z, and hence

that Fk and Fk' are generically equal. We conclude from 4.5 that Yk = Fk'. n

5.3. Theorem. Let 0)~,~ be cyclic over On+ 1 , and suppose that

i) codim Mk+ 1 = n+l ii) there exists a deformation

(2,~)

0 1"

(¢n+l ,o)

c_~ (x,~) ~,p

C.~ ¢,o F

1"~ 2

(¢:n+l × ¢,o)

of the map f:(X,~) --~ (¢:n+1,0), such that

codim Mn+ 1 (F) = n+l

138

codim Mn+2(F) = n+2

Then (in the terminology of 5.2), Fn(f) = Fn'.

Proof. We have 0X, i /F .mn+2 ~ 0)~,~/f , mn+ 1 and so our presentation P of 0~ ,~ over

0n+ 1 lifts to a presentation P of 0X, 2 over On+ 2. Thus

(*) 0n+l / Fk(f) = 0n+2/(t)+ Yk(F)

where t is the parameter in the base C of the deformation; a similar isomorphism holds for

Fk'. By theorem 5.2 applied to the Fitting ideals of F, Ox, ~, we have

Fnff') = Fn' (F)

and now by (,) we conclude that Fn(f) = Fn' (f)- []

As an application of 5.2 and 5.3 we consider the case where X is the union of

hypersurfaces X 1 ..... X m in (C n+1,0), each with isolated singularity at 0, such that each

n+l among the X i meet only at 0. Let h 1 ..... h m be their defining equations. Then let

m (X,~) be the disjoint union of the (Xi,0), so that 0)~,~ = i=el On+l/(hi)" Let f be the

obvious map (X,2) ~ (C n+l ,0) with image (X,0). Then there is a natural choice of

presentation of f ,(O~,~) over On+ 1 , namely

m m 0 -~ 0n+ 1 On+ 1 ~ f.0)~,2 ~ 0

where X is the diagonal matrix with entries h 1 ..... h m.

Thus Fk(f) = ({hil...him_k I t < i 1 <..-<im_ k -< m}). However, 0~,y, is cyclic over 0n+ l ,

for it is generated by the element (1 ..... 1) and the first m-1 powers of the element (1,2 ..... m). Thus, since the cc~mension requirement is met, by the assumption that each

n+l of the h i form a regular sequence, from 5.2 it follows that all of the F k , for

0 < k < n - l , are Cohen-Macaulay. As there exists a l-parameter deformation of X, in which all of the X i are moved into general position with respect to one another, and as

this deformation lifts to a deformation of f:()~,~) -~ (~n+1,0), 5.3 also applies, and we conclude from A.lb that during such a deformation dim C 0n+l /Fn(f) ordinary n + l -

tuple points emerge.

The matrix of a presentation of f , O ~ with respect to the cyclic generating set may

139

be obtained by multiplying ~. on the left by the inverse of the van der Monde matrix M on (1,2 ..... m). If instead we left multiply ~. by M, we obtain the matrix ~ of another

presentation, for which it is immediately evident that Fk(f) = Im_k~k), and this gives

another (direct) proof that 5rk(f) is determinantal.

After this paper was written we learned that the results of this section can be found in "Courbes de l'espace projectif: vari~t~s de s~cantes" by L. Gruson and C. Peskine, in Enumerative Geometry and Classical Algebraic Geometry) Le Barz, Hervier, Eds., Birkh~user, 1982.

140

§6. The Structure of Bifurcation Sets.

As an application of the results of §3 we indicate how to put a canonical analytic

structure on the level bifurcation set of a deformation of an isolated hypersurface

singularity.

Let f:(¢n+1,0) ~ (¢,0) have finite Milnor number (and hence isolated singularity at 0)

and let F:(¢ n+l x ed,0) -~ (¢: x ed,0) be an unfolding of f, with F(x,u) = (fu(x),u). Set

C F = critical set of F = {(x,u I dF(x,u) is not an epimorphism}

D F = discriminant of F = F(C F)

B I = {u ~ ed 13x E ¢:n+1 s.t. fu has degenerate critical point at x}

B 2 = {u e ~d 13x 1 ~ x 2 e C n+l s.t. fu has critical points at x 1 and

x 2, with fu(Xl) = fu(X2)}.

Then B 1 is the "source bifurcation

Consider the diagram

en+l× cd F ~ ~×C d

t -t F CF

set" and B 2 the "level bifurcation set".

S in which ~ and p are projections, and Crc is the critical set of ~ :CF~¢d (that is, Cr~ is

defined in ¢ n+l x ¢ d by the n+2 equations

afu ax i 0 i = 1,...n+1, det t~xi3xjJ 0)

and ]~ is the singular locus of the discriminant D F. Then as sets

B l = ~(C~)

U B2 = p(1).

The first of these equations is evident: the second follows from the fact that ]~ consists of

two kinds of points: those with two or more preimages (such points project down to ed to give B2) and those which have a "degenerate" preimage - that is, which have a preimage

141

in Cm At such points D is, typically, cuspidal. In [61t, Teissier suggested the now

accepted view that the "correct" structure for D F is that provided by 0¢, × ¢,d/Sro(F*OCF ). When F is Thom-transversal then this gives D F its reduced

structure ([41] 4.11). By the same token, we give B 1 the structure O~d/Fo(rC,OCrc), and

B1UB 2 the structure O~d/Fo(P ,O~) , where Y~ has structure sheaf

0]~ = 0¢ x ¢,d/Yl(F, OCF). The key point here is that when the codimension of Cr~ is CF

is 1 (it can be no greater), then C~ is a complete intersection of dimension d- l , and hence

Cohen-Macaulay. It follows from 3.1 that Fo(~,OC~) is a principal ideal. Similarly,

when FK2 F is of degree 1 onto its image, FI(F , OCF) is determinantal (for C F is also a

complete intersection) by 3.4, and so ]~ is Cohen-Macaulay of dimension d-1. Another application of 3.1 shows that Fo(P,O~) is principal also. Now by the commutativity of

the diagram above, Fo(P,O]~) c_ 5-o(r~,OCr~), and as both are principal, we obtain a

principal ideal I L = yo(p,O]~). Yo(r~,OCF)- 1, which defines B 2. We leave to the reader

the straightforward proof of

6.1. Proposition. If F is an Re-versal unfolding of f, then rc:]~ ~ cd is of degree 1 onto its

image, so by 3.1 Fo(P,O~) and Fo(rC,OCrc) are both radical, and hence I L is radical.

6.2. Example. We take the unfolding

F(x,y,a,b,c,) = (x3+y3+ax+by+cxy,a,b,c)

of the D 4 singularity. A calculation shows that with respect to basis 1,x,y,xy of OCF over

01+ d, there is a presentation of OCF with matrix

-Y l b c 2 - ~ " a 2 2 7 l a c 2 - ~ b 2 5 a b c

2 1 2 2 l c 3 - y - _-~-bc ~b2~ ~ ac ga 27 3

1 1 3 2 1 3~b - ~ac ---.c -Y - ~ - a 2 + ~ bc 2 27

2 1..~c3_ Y 1 2 b - a -~c ~- 3 27

(Here Y is the target variable in •.) Write Y = 1--~-c3-y. 27

generated by the four minors

Then by 3.4, FI(F, OCF) is

142

~3+ 2~{4a3+4b3-7abc2} ~ + 8-~a2b2c - #a3c3+b3c3)

2 2 2 1 2 4 8 1 8 4+ .~ . a be -2-~-b? c --~-ab 3 2a~2+(98-b2c - -Gac3)2 +

2b~2+(98--a2c-~'bc3)~ + ~ 4 + 2~ ab2c2 - 2~ a2c4 - 2~ a3b 27

8 -- ~c(a3+b3 ) 1 3 c 7 2 _ 3 by - -

Since O]~,o/p,m 3 =-- ¢~] / (~3) , 0]~,o is generated over 03 by 1,~ and ~2. The second,

third and fourth generators of F1 (F*Oc F) then generate all relations among 1,~/and~ 2 as

elements of 0]~, so that we immediately obtain a presentation

in which • is the transpose of the matrix

- ~-abc 3 4 - ~c(a3+b 3) - 38--ab c

_ 2~b4 2 1 8 9~bc3 8 2 - ~-ab2c 2 + .~-ffa2c 4 + ~-a3b - ~ a c -2b

_ 2~a4 _ _~a2bc2 + 2_~b7 2c4 + 2_~ab 3 1 8 ~ac 3 - -2a ~b2c

One then calculates that

8 256 2 2 ~ F0(P,0]~) = ((a3_b3){9 - .~abc4 - 2569 (a3+b3)c2- -~--a b })

A similar (easier) calculation shows F0(rc,OCr~) equal to the second factor of the

generator of Y0(p,(Oy), so we conclude that

I L = (a3-b3).

The method used by L~, Teissier and others to find the bifurcation set (c.f. [61]), by regarding it as the discriminant A(~) of the projection rc:D F ~ Cd, gives it a non-reduced structure. In fact, in our terminology,

A(rQ = 3B 1 + 2B2

143

([61], Chapter m, 5.11).

Another way of finding the equations of the discriminant and the bifurcation set of a versal deformation is described by Bruce [6], Saito [53] and Temo [62] following the work of Arnold [3], Goryunov [22], Lyashko [42] and Zakalyukin [64]. Their method involves the construction of vector fields tangent to the discfiminant resp. bifurcation set, but the calculations that they do turn out to be exactly the same as ours. Our method however is not restricted to deformations which are versa], and indeed commutes with base change. G. Schneider, in [56], gives the singular locus of the discfiminant an analytic structure

using F0(F.OCF/ODF), and shows that it is Cohen-Macaulay. See also [24].

6.3. Remark. The Fitting spaces V(Yk(F,OCF)), for F as in 6.1, are the closures of the x-

constant strata (where x = Tjurina number). This is evident from 1.5, for

Z (x~)~(x ,u) dime OCF,(X,U)/F, m(Y,u)

Z dime

(x,u)~Cr:C-~(x,u)

Ocn+l+d, (x,u)

F* m(y,u ) + (~1 /)Xn+t

'K" o¢n+l ,x = ,7., d ime

(x,u)~CFC'a~t(x,u) (fu,Jfu)

Z x(f u ; x). (x,u~ c~--'(x,u)

Thus, 5.2 and 5.3 can be interpreted in this case as saying that in a versal unfolding of an A k singularity, all of the closures of the x-constant strata are determinantal varieties (for

the hypotheses of 5.2 and 5.3 certainly hold).

144

§7. Examples.

7.1 Example. In 2.6 we gave as an example a symmetric presentation matrix of

f.(O~2,0), where f:0I~2,0) ~ ((£3,0) is the map f(x,y) = (x,xy+y3, xy2+cy4). Observe that

fore ~ 0,1, Y2 = (X1,X2,X3), so by 4.3 a generic deformation of f will exhibit 1 triple

point. For the exceptional values c = 0,1, d im~03/72 is infinite. The family P3

1 3 described in [46] has two other exceptional values, c = ~- and c = ~-. That they are

1 3 exceptional (in fact weak normality of the image fails for all four values c = 0, ~, 1, ~ ) is

not immediately evident from the matrix ~., but can be detected by considering 5rl.0•2 0.

1 By 3.3, this is the principal ideal generated by m 1.f

= (1-c)2x 3 + (1-c+c2)x2y 2 + (c+c2)xy 4 + c2y 6

1 3 1 1 When c = ~-, ~this is equal to -~- (x+y2) 3, -, (x+3y2)2(x+y 2) respectively, and the fact

that there is a repeated factor in each case shows that f is not finitely determined (see [43], 3.4).

This example highlights a weakness of this approach to multiple points of mappings, namely, that we cannot distinguish between say a curve of ordinary double points and a

3 cuspidal edge (as in the case e = .~, where the image of f is the swallowtail surface) by

examining 71 alone. A more subtle understanding of the geometry can be obtained by

using s o u r c e double-point schemes, on which there is a natural :~2 action which does

distinguish between ordinary double points and non-immersive points, as in [47] and [43], but there the higher multiple point schemes (corresponding to k-tuple points for k > 2) are constructed only for corank 1 map-germs.

7.2 Example. Let it, Jt for t = 1,2,3 be positive integers such that i t -<Jt and consider the following 3 × 3 symmetric matrix,

il X~2-i2 ~3-i3 J2 J3" X 1 X 2 X 3

• i3 i 1 X~2 X3 X1

J3 il X 3 X 1 X~

145

1 One checks easily that ~ satisfies R.C., and indeed, writing m t = mt, t = 1,2,3, one has

that

adj(~.) =

"m 1 m 2 m 3

.,)3-i3 J2-i2 J3-i3 m2 "&3 m3 X 2 X 3 m

J2-i2_ J3-i3 x2-J2-i2m2 m 3 X2 X 3 m 1

k s Thec°eff ic ients~ 's t inmt E k = , ~-s tmk (see 3.9 above) can be seen in this last matrix.

k

il J2 J3 3i 1 j2-i2 j3-i3 i,2+2J2 - _ i3+J3 Let g = dot ~. = 3X/X,2-X 3- - X 1 "X~ - X 3 - - X ~ x 3 . ~ n g fI, and g

defines a surface (X,O) in (q;3,0) with singular locus ~g. I f J g is the Jacobi ideal of g,

X has along ~-{0} a transversal self intersection (i.e. transverse type A 1). Furthermore,

the Xl -ax is is a singular line of X if and only if j2 > i2+2 or j3 > i3+2, or j2 > i 2 and

J3 > i3-

Let (X,~) be the multi-germ of analytic space in (~13 × 11;2, 0 × C 2) defined by the seven equations

V I = I

3

S=I

t = •,2,3

Z k ~.~t Vk -- VsV t

146

2 < s < t < 3 k=l

Let V = V 2 and W = V 3. We can view (X,~) in (~3 x ¢2, {0} x C 2) with coordinates

X 1,X2,X3, V,W, as defined by the 5 equations

• . i 1

• , °

4+x';v+4w--0 XJ3-i3 V2= W

J2-L2 J3-13 VW -- X 2 X 3

W 2 = 4 - i 2 v

_ i l J2-i2 J3-i3 J2__ _ J 3 - i 3 . since X 1 X 2 X 3 + X2W + X 3 w = 0 is a consequence of the other five equations.

These 5 equations come from the 4 x 4 Pfaffians of the following anti-symmetric 5 × 5

matrix:

_X j3-i3 0 V 0 -W

v 0 w

v 0 -w 0 x~ 2-~

il i3 _X~ 2-i2 0 _X1 W X3

Hence, by A.5, (X,~) is a Gorenstein surface. If f: (X,~) -+ (X,0) is the projection map then f . (0 ~,~) has 7V as symmetric presentation matrix.

Denote the embedding dimension of O~,R) by e. Then

147

e < 4 e , i 2 = j2or i3 = J3

e < 3 *:~ J2 -- 1 or j3 --- 1 or i 2 = J2 and i 3 = J3

e = 2¢:*j2-- 1 and j3 = i3+1 or j3 = 1 a n d j 2 = i2+1

In particular e = 2 if and only if (J~,~) --- (~:2,0), in which case f is the normalisation map

i 3i~+2 (¢2,0) ~ (X,0) given by f(X 1,W) - (X 1, -XI 1W - W ~ ,W3). If moreover i 1 = 1, we

get the map Hi3+l described in [4611, [47].

The surface (X,0) is reducible if i 2 - J2 (mod 3) and i 3 - J3 (mod 3).

7.3 Example. Now consider the matrix

X=

i j -" j -i J2 bX~2-i2 J3 " "" _ 1 212 3 3 +cXJ3 13 t,x l+a) X2 X 3 x 2+ X 3

• . _. i 3 i 1 X~2 + bX~ 2v2 X 3 + c X l + a

• " " i l + a X ~ + b XJ3 + cXJ3 13 X 1

and let • be the matrix obtained by deleting the first row. Then I2 (~ ) defines a

4-dimensional germ ({,0) in (C 3 × C3,0) whose projection to (~3,0) (second copy) is a deformation of the curve germ ~ = V(I2(q~)) of the previous example.

Let (.~,~) be the germ of the analytic space in (~3 × tI~3 × fI~2, 0 x 1122) defined by the equations

O~+b) X22-i2+ (xi3+ c) V + (xi~+ a) W = 0

(xi33+ c) xJ33-i3 + ~ + a)V + 0 ~ 2 b ) W = 0

V 2 = X j3-i3 W

VW = xJ2-i2 xJ3-i3

w2 = xJ2 v

which come from the 4 × 4 Pfaffians of the 5 x 5 antisymmetric matrix

148

J3-i3 I. 2 • - V i l X 2- + b X I + 0 X3

-J3-i3 0 V 0 - W -X 3

X• J3 - - b - V 0 W X 3 + c

V 0 -W 0 4 2-i2

il J3 4 2-i2 0 -X 1 - a W -X 3 - c -

Specialising to a=b=c=0 gives the antisymmetric matrix of the previous section.

is a Gorenstein analytic space and

X ,-- X w -

1' l' X ~ -----~ X

/ / {0} ~ ¢ 3

is a (flat) deformation of f:X -* X (here X = V (det A ) , see [29]).

HenceX

Let F be the map (.~,~) --) (¢3 × ¢3,0 ) defined by projection (forget V and W). The image

of F is x, and F , O ~ h a s / ~ as presentation matrix.

For every s = (a,b,c) we have a map fs'.Xs-, X s, (Xs and X s are the fibres of ~" and X

over s). The only triple points of fs are ordinary in case abc ~ 0, and occur at the

solutions of

i 1 i2 i 3 X 1 + a = X~ + b = X 3 + c = 0

There are thus i li2i3 ordinary triple points if abc ~ 0. One checks easily that this is equal

to dim¢O3/9~2(f), as stated in 4.3.

Although (X,0) is not weakly normal in general, and its singular locus in general strictly contains ~, we can find a weakly normal surface with singular locus ~ and with Gorenstein normalisation as follows: consider the family parametrised by ~. = (~.1,~.2,~.3),

149

i 1 0 xJ33-i3 X ~ + b -V X l + a

J3-i3 0 V 0 - W -X 3

x~ J3 - - b - V 0 W X 3 + c

v o -w 0 <2-h

il J3 < 2-i2 0 -X 1 - a W -X 3 - c -

Specialising to a=b=c=0 gives the antisymmetric matrix of the previous section. Hence~" is a Gorenstein analytic space and

X ~- X v

1'l' X ~ - - - ~ X

{o}

is a (flat) deformation of f:X ~ X (here X = V (det A ) , see [de Jong, van Straten]).

Let F be the map (.~,~) ~ (~3 x C3,0) def'med by projection (forget V and W). The image of F is X, and F , O . ~ h a s / k as presentation matrix.

For every s = (a,b,c) we have a map fs:Xs ~ X s, (Xs a n d X s are the fibres of ~'and X

over s). The only triple points of fs are ordinary in case abe ~ 0, and occur at the

solutions of

i I i2 i 3 X l + a = X 2 + b - - X 3 + c = 0

There are thus ili2i 3 ordinary triple points if abe ~ 0. One checks easily that this is equal

to dimcO3/72(f) , as stated in 4.3.

Although (X,0) is not weakly normal in general, and its singular locus in general strictly contains ~, we can find a weakly normal surface with singular locus ~ and with Gorenstein normalisation as follows: consider the family parametrised by t = (k 1,~.2,k3),

150

2 2 g)~ = g + ~.lm~ + 9~2m2 + ~.3m3 .

This defines a deformation of (X,~) with fixed singular locus (~,0), see [291. The surface

X)~ defined by g~ has for general values of 9~ a singular locus contained in V(m 2, m 2,

) by Bertini's theorem. Hence X)~ is weakly normal with singular locus (~,0), for

some values of ~.. The deformation of (X,~) gives a deformation of X -* X ([29]). Hence

we get a family of maps f~.:?~. ~ X)~ with :~)~ Gorenstein and normal for general values

of ~.

7.4 Example, Following [27] and by means of our 4.7 we give an example of a reduced

curve (~,0) c (C3,0), with defining ideal I, such that no surface singular along ~ can have Gorenstein normalisation. Let

i ix; 1 X 3 X 1 cp=

J3 Jl X 3 X 1

where i t , Jt are positive integers with i t < j t, and let I = I2(cp). (Note the differences

between q~ and the matrix 21 of 7.2). By [27],fi/i2 is cyclic, and indeed if we obtain the

3 × 3 matrix )~ by adjoining to q0 the top row ~X 1 X~ X 3 , , , we find

that 9~ satisfies R.C., and that f I = (det)~) + 12 in the special case i l= i2= i 3 = 1. The

argument here is that det ~. ~ f I (since)~ satisfies R.C.), but det ~. ¢ 12, while by [27] 2.11, d i m ~ I / I 2 = 1. However i f j l > 1, no surface V(g), with g ~ f I , can have Gorenstein

normalisation. For let L det ~. be the images of I, det ~. in O3 / (X I) -~ ¢~X2'X3}" We

have (d'~ ~.) + r e

_- + _ 2 _ 2 _ _ 2 + _ 2 + (X2X3, X 2 X3, X2X 3 " ,X 2 2J2)

and hence

5 = ~t((d'~ )~) + r 2 ) < g((det ).) + 12) = g(j'I).

Since ~(J'i) > g(i) + 1, by 4.7 no surface V(g) with g c f I can have Gorenstein normalisation.

151

7.5 Example. Let X be the matrix

"X2X 3 -2X1X 3 XIX 2"

X 1 X 2 X 3

X I 2X 2 3X 3

Then X satisfies R.C. Let I = I2(~,), g = det X, and (X,0) the germ in (q;3,0) defined by g.

Also let C= IOX, 0. Let C~,0) be the germ with local ring 0j~,0 = Homox,0 (C, 63 (see

3.15). Then f:()~,0) . (X,0) is the normalisation of (X,0). Le t / k 1 be the matrix

"X2X3+tX 1 -2X1X 3 X1X2 ]

x 2 x 3 [

1 X 1 2X 2 3X 3

Then /~1 satisfies R.C. Let G = det A 1" Then G defines a hypersufface (Xl,0) in

(a;3 × C,0). We have a map F1:(.~1,0) ~ (Xl,O), see 3.15, which is a fiat unfolding of f

over ~. The surface J~l,t is non-singular and the map fl , t : Xl , t ~ Xl,t (the fibre of F 1 over t) has one ordinary triple point, for small t # 0.

Let A 2 be the matrix

"x2x 3 + s3x 2 - s2x 3

X 1

XI+S 1

-2X1X 3 + S1X 3 - S3X 1

X2

2X 2 +S 2

XlX 2 + S2X 1 - S1X2]

1 X 3

3X 3 + S 3

This also satisfies R.C. Again by 3.15 we get a map F2:(X2,0 ) . (X2,0) of analytic spaces

which is a flat unfolding of f over C 3. The surface X2,s is non-singular and f2,s~2,s 4

J~2,s has no triple point, for general (s 1,s2,s3) ~ ~;3. Hence the triple point locus of F2 is

not flat over ~;3.

De Jong and Schreyer remarked that ()(,0) is isomorphic to the rational normal surface in

(C5,0) given by the vanishing of the 2 × 2 minors of

152

iY Y2 Y3 [Y Y2 Y3] equivalently Y2 Y3 Y4

Y2 Y3 Y4 Y5 of Y3 Y4 Y5

which one encounters in [52]. The base of the mini-versal deformation of (X,0) consists of two components (~,0) and (~3,0) which correspond to the unfoldings F 1 and F 2 of f above. This example also occurs in [48]. There it is shown that the number of cross-caps need not be constant in a deformation. A remarkable result of de Jong [28] asserts that

n cross-caps - 2n ordinary triple points

is constant under deformation. Indeed, in our example, f l , t has 6 cross-caps and 1 ordinary triple point, while f2,t has 4 cross-caps and no triple point.

7.6 Example. (J. Stevens). If (~,0) is the union of 6 lines in (~3,0) in general position (not all are contained in a quadric cone), and if I is the ideal defining (~,0), then dim C fI / I2 is strictly~greater than the maximum number of triple points in a deformation

of (~_,,0). Moreover, JI/i2 is not cyclic. Hence by 4.7 there is no weakly normal surface

(X,0) in (C3,0) with (~,0) as its singular locus, and which has Gorenstein normalisafion.

153

Appendix

Analytic Spaces Def'med by Determinants

Let (X,o) be a germ of an analytic space in (¢N,o) defined by the ideal I in 0 = O N. We

say that (X,o) in a hypersurface if I is generated by one element which is not a unit. The

germ (X,o) is called a complete intersection if I is generated by r elements where r is the

codimension of (X,o) in (¢N,o), or equivalently: I is generated by an O N-sequence

f l ..... fr, with fi ~ raN. The germ (X,o) is called Cohen-Macaulay if 0 X is a Cohen

Macaulay local ring, i.e. depth OX, 0 = dim (X,0), i.e. there is a OX,0-regular sequence

X 1 ..... X n in rnX, 0, where n=dim(X,0).

Every finitely generated O-module M has a Finite O-free resolution, since O is a

regular ring [26] [57], i.e. there exists an exact sequence of O-modules

O O ¢obr-1 ..... _. oul obo M_.O

The resolution is called minimal if all the entries of the matrices of all the cpi are

elements of m. For a minimal resolution we have that b i = dim• Tor~(M,C). The

homological or projective dimension of M is the length of a minimal resolution, i.e.

pdoM = max{i I Tor~(M,C) ~ 0}.

The following identity ("the syzygies theorem") holds c.f. [57], [5]

pd0M + depth0M = N.

A.1. Theorem. Let x:V -¢ S be an analytic map and suppose that S is smooth. Then

a) if V is Cohen-Macaulay at v 0 then x is flat at v 0 if and only if the dimension

of x'~(Tr(v0)) at v 0 is equal to dim V -dim S.

154

b) if ~ is finite, dim V = dim S and rc~(~(v0)) = {vo}, then the following are

equivalent

i) (ib

(2)

V is Cohen-Macaulay at v 0

n , 0 V is flee over 0 S at v 0.

n is flat at v 0

Proof a) [45] Theorem 15.1, page 116 and 23.1, page 179.

b) (i) e:~ (ii) follows from the syzygies theorem. (ii) ¢:~ (iii) is proved for example

in the corollary to Theorem 7.11, page 53, of [45]. n

In particular, if X is Cohen-Macaulay, the local ring 0 X has a minimal O-free resolution

of length r = codimension of X in cN. In this case b r = d ime TOr0r(0X,0 ) is called the

Gorenstein type of (X,o).

Another way of defining the Gorenstein type of (X,o) is by means of the minimal number

of generators of the dualising module COX, o. If we take Grothendieck's definition of the

r f2Nc~N,o), dualising module, COx, o = Ext 0 (0X, c.f. [25], then we see that COx, o ~- Coker

(~pr) which has minimal number of generators equal to br, in case the resolution is

minimal.

For instance, if CX,o) is a complete intersection def'med by the regular sequence (f 1,...fr)

then the Koszul complex of the sequence (fl,...fr)

0 ~ Arc r -~ A r-1 ~r .~ .... -~ A20r~ 0 -~ 0 x -~ 0

gives a minimal resolution of 0 X and therefore (X,o) is Cohen-Macaulay with

Gorenstein type 1.

A Cohen-Macaulay analytic space is called Gorenste/n if its Gorenstein type is 1.

155

Let ¢p be a (r x s)-matrix with entries in 0 and r > s, We denote by Ik(Cp) the ideal

generated by the (k x k)-minors of cp, in case 0 < k < s.

A.2. Theorem. Let ¢p be an (r x s)-matrix with entries in 0. If 0 < k < s -< r then codim

VOk(Cp)) < ( r -k+l) (s-k+l) . If equality holds then Ik(Cp) defines a Cohen-Macaulay

space which is Gorenstein if and only if r = s or k = 1.

Proof. See [18], [1(3], [11], [17] for the statement about the Cohen-Macaulay property, and

[7], [60] for the Gorenstein property.

A germ of an analytic space X is called deterrninantal if it is defined by the ideal Ik(C p)

where 9 is a (r × s)-matrix and 0 < k < s <_ r and codim X = ( r -k+l) (s-k+1).

In particular, if ¢p is a (t+l) × t-matrix then It(q~), defines a Cohen-Macaulay space if its

codimension is 2. A converse also holds.

A.3. Theorem. If (X,o) is a Cohen-Macaulay analytic space of codlmension 2 in (C N,0)

and of Gorenstein type t, then X is determinantal defined by It(cp), where q) is a (t+l) x t -

matrix. Moreover the following sequence is exact

¢p ,5 0-* Ot--.> ot+l .-> O~ O X -, 0,

where A = (A l , . ,~t+l) and A i = (-1) i det ¢pi and cp i is the (t x 0-matr ix obtained from ¢p

by deleting the ith row of ¢p.

Proof. See [26], [13], [54]. n

The following corollary is due to Serre.

A.4..Corollary. If (X,o) is a Gorenstein analytic space of codimension 2 then (X,o) is a

complete intersection. B

The case that cp is a symmetric or anti-symmetric matrix is also treated.

156

A.5. Theorem. Let q~ be a symmetric (s × s)-matrix with entries in 0. If 0 < k < s then

codim V(Ik(q~) < ~(s-k+l)(s-k+2). If equality holds then Ik(q~) defines a Cohen-

Macaulay analytic space with Gorenstein type I s (s-1) (if s>2).

Proof. See [30], [37]. []

Let q~ be an anti-symmetric (s × s)-matrix. If s is even then det cp = A2 for some

A e 0. A 2p-Pfaffian of (p is an e!ement A such that A2 = det cp, where ~0 is a (2p x 2p)

ant i -symmetr ic submatrix of cp, obtained by deleting s -2p columns and the

corresponding s-2p rows. Let Pf2p(Cp) be the ideal generated by the 2p-Pfaffians of cp,

where 0 < 2p < s.

A.6. Theorem. Let q~ be an anti-symmetric (s × s)-matrix with entries in 0. If 0 < 2p < s

then codim V(Pf2p(Cp) ) < I (s-2p+l)(s-2p+2). If equality holds then Pf2p(~0) def'mes a

Cohen-Macaulay analytic space which is Gorenstein if s = 2p+l or s = 2p+2.

Proof. See [31], [35], [39]. n

In particular Pf2p(q0) defines a Gorenstein analytic space if ~0 is an anti-

symmetric (2p+l) × (2p+l)-matrix and codim V(Pf2p(Cp) ) = 3. The converse also holds.

A.7. Theorem. Let (X,o) be a germ of a Gorenstein analytic space in (¢N,0) of

codimension 3. If the minimal number of generators of the ideal I defining X is ~t then

I.t = 2p+l and I = Pf2p(q0), where q~ is an anti-symmetric (2p+l) × (2p+l)-matrix.

Moreover the following sequence is exact

A* q~ A 0 ~ 0 ~ 02P +1 --~ O2p+l ...) 0 ~ 0 X ~ 0,

2 where A = (A 1,..,A2p+l ) and A i = det ~pi, and q~i is the matrix obtained from q~ by

deleting the i th row and column from cp.

Proof. See [81. n

157

Remark: Although the above mentioned results hold in greater generality, we

prefer to state them over 0, since we need them only in this context.

158

R e f e r e n c e s

.

.

.

.

.

.

.

.

.

10.

11.

12.

13.

14.

15.

Andreotti, A. & P. Holm. Quasianalytic and parametric spaces, in Real and complex singularities, Osto 1976, P. Holm, ed., Sijthoff & Noordhoff Int. Publishers, 1977, 13-98.

Arbarello, E. & E. Sernesi. Petti's approach to the study of the ideal associated to a special divisor, Inventiones Math. 49 (1978), 99-119.

Arnold, V.I. Wavefront evolution and the equivariant Morse Lemma, Comm. Pure Appl. Math. 29 (1976), 557-582.

Auslander, M. & D.A. Buchsbaum. Homological dimension in local tings, Trans. Amer. Math. Soc. 85 (1957), 390-405.

Barth, W. Counting singularities of quadratic forms on vectorfields, in Vector bundles and differential equations, Proceedings, Nice 1979, Hirschowitz, ed., Birkhauser, Progress in Math. 7, 1980.

Bruce, J.W. Vector fields on discrirninants and bifurcation varieties, Bull. London Math. Soc. 17 (1985), 257-262.

Bruns, W. The canonical module of a determinantal ring, in Commutative Algebra, Durham 1981, R.Y. Sharp, ed., London Math. Soc., LNS. 72.

Buchsbaum, D.A. & D. Eisenbud. Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math. 99 (1974), 447-485.

Buchsbaum, D . A . & D . Eisenbud. What annihilates a module? Journ. of Alg. 47 (1977), 231-243.

Buchsbaum, D.A. & D. Eisenbud. Genetic free resolutions and a family of generically perfect ideals, Adv. Math. 18 (1975), 245-301.

Buchsbaum, D.A. & D.S. Rim. A generalised Koszul complex II, depth and multiplicity, Trans. Amer. Math. Soc. II/(1964), 197-225.

Buchweitz, R.-O. & G.-M. Greuel. The Milnor number and deformations of complex curve singularities, Invent. Math. 58 (1980), 241-281.

Burch, L. On ideals of finite homological dimension in local tings, Math. Proc. Camb. Phil. Soc. 64 (1968), 941-946.

Catanese, F. Commutative algebra methods and equations of regular surfaces, Lecture Notes in Math. 1056, 68-111, Berlin, Springer-Verlag, 1984.

Concini, C. de, D.Eisenbud, & D. Procesi. Young diagrams and determinantal varieties, Inventiones Math. 56 (1980), 129-165.

159

16. Doubillet, P., G.-C. Rota & J. Stein. On the foundations of combinatorial theory IX, Combinatorial methods in invariant theory, Studies in Applied Math. 53 (1974), 105-216.

17. Eagon, J.A. & M. Hochster. Cohen-Macaulay rings, invariant theory, and the generic perfection of determinental loci, Amer. J. Math. 93 (1971), 1020-1059.

18. Eagon, J.A. & D.G. Northcott. Ideals defined by matrices and a certain complex associated to them. Proc. RoyalSoc. A 269 (1962), 188-204.

19. Eisenbud, D. Homological algebra on a complete intersection, with applications to group representations, Trans. Amer. Math. Soc. 260 (1980), 35-64.

20. Fitting, H. Die Determinantenideale einer Moduls Jahresber, Deutcher Math. Verein, 46 (1936), 195-220.

21. Fulton, W. Intersection theory, Ergeb. Math. Grensgebiete, 3 Folge, Band 2, Berlin, Springer-Verlag, 1984.

22. Goryunov, V.V. Bifurcation diagrams of some simple and quasihomogeneous singularities, Funct. Anal. Appl. 17 (1983), 97-108.

23. Grauert, M. & R. Remmert. Theory of Stein spaces, Grundlehren der Math. Wissenschaffen, 236, Berlin, Spring-Verlag, 1979.

24. Greuel, G.-M. & D.T. L~. Spitzen, Doppelpunllte und vertikale Tangenten in der Discriminante verseller Deformationen yon vollst~ndigen Durchschnitten, Math. Ann. 227 (1976), 71-88.

25. Hartshorne, R. Residues and duality, Lecture notes in Math. no. 20, Heidelberg, Springer-Verlag, 1966.

26. Hilbert, D. 0ber die Theorie der algebraischen Formen, Math. Ann. 36 (1890), 473-534.

27. Huneke, C. The primary components of and integral closures of ideals in 3- dimensional regular local rings, Math. Ann. 275 (1986), 617-635.

28. Jong, T. de. The virtual number of Doo points, Preprint, Universit~it Kaiserslautem, 1987.

29. Jong, T. de & D. van Straten. Deformations of surface singularities, in preparation.

30. Joztfiak, T. Ideals generated by minors of a symmetric matrix, Comment. Math. Helv. 53 (1978), 594-607.

31. Joztfiak, T. & P. Pragacz. Ideals generated by Pfaffians, Journ. of Alg. 61 (1979), 189-198.

32. Kleiman, S.L. Multiple-point formulas h iteration, Acta. Math. 147 (1981), 13-49.

33. Kleiman, S.L. The enumerative theory of singularities, in Real and complex singularities, Oslo 1976, P. Holm, ed., Sijthoff and Noordhoff, 1977, 297-396.

160

34. Kleiman, S.L. Multiple-point formulas for maps, in Enumerative geometry and classical algebraic geometry, P. Le Barz, ed., Progress in Math. vol. 24, Birkhauser, 1982, 237-252.

35. Kleppe, H. & D. Laksov. The algebraic structure and deformation of Pfaffian schemes, Journ. of Alg. 64 (1980), 167-189.

36. Kunz, E. Holomorphe Differentialformen auf algebraischen Variet~ten mit Singularit~ten I, Manuscripta Math. 15 (1975), 91-108.

37. Kutz, R.E. Cohen-Macaulay rings and ideal theory in rings of invariants of algebraic groups, Trans. Amer. Math. Soc. 194 (1974), 115-129.

38. Laksov, D. Residual intersections and Todd's formula for the double locus of a morphism, Acta Math. 140 (1978), 75-92.

39. Lascoux, A. Syzygies des vari6t6s d6terminantales, Adv. Math. 30 (1978), 202-237.

40. Lipman, J. Dualising sheaves differentials and residues on algebraic varieties, Astdrisque, 117 (1984), 1-139.

41. Looijenga, E.J.N. Isolated singular points on complete intersections, London Math. Soc. LNS 77, Cambridge University Press, 1984.

42. Lyashko, O.V. The geometry of bifurcation diagrams, Russian Math. Surveys, 34:3 (1979), 209-210.

43. Marar, W.L. & D. Mond. Multiple-point schemes for corank 1 maps, Preprint, University of Warwick, 1988.

44. Matsumura, H. Commutative algebra, 2nd ed., Math. Lecture Note Series, Benjamin/Cummings Publ. Co., 1980.

45. Matsumura, H. Commutative ring theory, Cambridge Studies in Advanced Math., 8, Cambridge University Press, 1986.

46. Mond, D. On the classification of germs of maps from IR 2 to Itl 3, Proc. London. Math. Soc. 3:50 (1985), 333-369.

47. Mond, D. Some remarks on the geometry and classification of germs of maps from surfaces to 3-space, Topology, 26:3 (1987), 361-383.

48. Pellikaan, R. Hypersurface singularities and resolutions of Jacobi modules, thesis, Rijksuniversiteit Utrecht, 1985.

49. Pellikaan, R. Finite determinancy of functions with non-isolated singularities, to appear in Proc. London Math. Soc. (1988).

50. Petri, K. 0ber Spezialkurven I, Math. Ann. 93 (1924), 182-209.

5t. Piene, R. Ideals associated to a desingularisation, Proc. Summer Meeting Copenhagen 1978, Lecture Notes in Math., 732, 503-517, Berlin, Springer-Verlag, 1979.

161

52. Pinkham, H. Deformations of algebraic varities with IB m action, Astdrisque, 20 (1974), 2-131.

53. Saito, K. Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo, Sec. 1A, 27 (1980), 265-291.

54.

55.

Schaps, M. Deformations of Cohen-Macaulay schemes of codimension 2 and nonsingular deformations of space curves, Amer. J. Math. 99 (1977), 669-685.

Scheja, G. & U. Storch. tJber Spurfunktionen bei Vollst~ndingen Durschnitten, J. Reine u Angew. Math. 278/279 (1975), 174-189.

56. Schneider, G. Monodromie und Schnittform, zu ein Vermutung yon E. Brieskom, thesis, Universit~t Kaiserslautem, 1987.

57. Serre, J.-P. Alg~bre locale, multiplicitds, Lect. Notes in Math. 11, Berlin, Springer- Verlag, 1965.

58. Siersma, D. Isolated line singularities, in Singularities atArcata 1981, Proc. Symp. Pure Math. 40, P. Orlik, ed., 405-496.

59. Straten, D. van. Weakly normal surface singularities and their improvements, thesis, Leiden, Rijksuniversiteit, 1987.

60. Svanes, T. Coherent cohomology on Schubert subschemes of flag manifolds and applications, Adv. Math. t4 (1974), 369-453.

61. Teissier, B. The hunting of invariants in the geometry of discriminants, in Real and complex singularities, P. Holm, ed., Oslo, Sijthoff & Noordhoff Int. Publishers, 1977, 565-678.

62. Terao, H. The bifurcation set and logarithmic vector fields, Math. Ann. 263 (1983), 313-321.

63. Tougeron, J.C. Ideaux de fonctions diff~rentiables, Ergeb. Math. Grenzgebiete, 2 Folge, 71, BerLin, Springer-Verlag, 1972.

64. Zakalyukin, V.M. Bifurcations of wavefronts depending on one parameter, Funct. Anal. Appl. 10 (1976), 139-140.

David Mond Mathematics Institute University of Warwick

Ruud Pellikaan Technische Universiteit

Eindhoven