Local Embeddings of Lines in Singular Hypersurfaces

Local Embeddings of Lines in Singular Hypersurfaces Jiang Guangfeng

Dirk Siersma

Abstract

Lines on hypersurfaces with isolated singularities are classi
ed. New normal forms of simple singularities with respect to lines are obtained. Several invariants are introduced.

Key Words and Phrases: classication, line, singular hypersurface, embedding. Mathematics Subject Classi
cation: 14J17, 32S25, 58C27

Introduction

Simple surface singularities have been known for long time since they occur in several di
erent situations. They have at least fteen characterizations 4]. One often describes them by the normal forms of their dening equations in certain coordinate system. This has been done by Arnol'd 1] and Siersma 8, 9]. The idea is to classify all the surfaces that contain the origin 0 under the right equivalence dened by the group R, consisting of all the coordinate transformations preserving 0. The equivalence classes without parameters in their normal forms are called the simple surface singularities, or the A ; D ; E singularities. The observation by Gonzalez-Sprinberg and Lejeune-Jalabert 5] that on a singular surface X in C 3 one can not always nd a smooth line passing through the singular point was our starting point for classifying functions f , where the corresponding hypersurface X = f ;1 (O) contains a smooth curve. We take the following set-up. Take a line L as the x-axis of the coordinate system in C n+1 , classify all the hypersurfaces that contain L under the equivalence dened by the subgroup RL of R, consisting of all the coordinate transformations preserving L. If we compare this with the usual R-classication of surfaces, the result classies the lines contained in X . Our work is in fact about the pair (X
L). We obtain the following results:  The RL -classication is ner than the R-classication, i.e. there are di
erent `positions' of lines on the same hypersurface. 1

1 LINES CONTAINED IN SINGULAR SPACES

2

 There exist an RL invariant  , which counts the maximal number of Morse

points on the line over all Morsications.   is invariant under sections with generic hyperplanes containing L.  Another proof of the non-existence of lines on E8 surface. We also introduce a sequence of higher order RL invariants which gives more information about lines on hypersurfaces. It turns out that our work tells how lines are embedded in a hypersurface. What brought this problem to our attention was the study of functions with non-isolated singularities on singular spaces 7]. Natural questions are: given a singular hypersurface X with isolated singularity, does there exist smooth curves on X passing through the singular point of X ? If there exist smooth curves on X , how many of them? etc.

Acknowledgements This work was done during the visiting of the rst author

to the department of mathematics at Utrecht University, supported by the host department, Education Commission, Science and Technology Commission of Liaoning Province of P. R. China. We thank all of them. 1 Lines contained in singular spaces

1.1 Let L be a smooth curve in C n+1 . Since L is locally biholomorphic to a line, we often say that L is a line. Let X be a singular space embedded in C n+1 . If for a singular point O 2 Xsing, there exists a line (smooth curve) L in C n+1 such that O 2 L and L n fOg  Xreg, we say that X contains (or has ) smooth curve passing through O. Very often, we say that X contains (or has) a line passing through O. 1.2 On a singular surface X in C 3 one can not always nd a line passing through (not contained in) the singular locus of X . Gonzalez-Sprinberg and Lejeune-Jalabert 5], by using resolutions of singularities, have proved a criterion for the existence of lines on any (two dimensional) surface, and given a natural partition of the set of smooth curves on (X
0) into families. Especially, on surfaces with Ak, Dk (k  4), E6 or E7 type singularity, there exist lines passing through the singular point. In fact one can nd easily lines on this kind of surfaces simply by looking at the dening equations. But it is not easy to tell how many di
erent lines there are on a surface, and what kind of invariants can be used to tell the di
erences. The hardest thing is to prove the non-existence of smooth curve on certain surfaces (e.g. E8 type surface loc.cit.). There exists similar question for hypersurfaces with other dimensions.

2 RL -EQUIVALENCE

3

1.3 Isolated hypersurface singularities Let X be a hypersurface germ with

isolated singularity at the origin. The singularity of X is simple if and only if X can be dened in a neighborhood of the origin in C n+2 by the following equations (see 1] and 8, 9]):

Ak : Dk : E6 : E7 : E8 :

xk+1 + y2 + z12 +    + zn2 = 0 (k  1)
x2y + yk;1 + z12 +    + zn2 = 0 (k  4)
x4 + y3 + z12 +    + zn2 = 0
x3y + y3 + z12 +    + zn2 = 0
x5 + y3 + z12 +    + zn2 = 0:

The left hand sides of the equations are called the normal forms of the simple (or A-D-E) singularities. The following equations dene the simple elliptic hypersurface singularities (see 2] and 8, 9]): E~6 : x2y + z13 + y2z1 + y3 + z22 +    + zn2 = 0
43 + 27 2 6= 0 E~7 : x4 + y4 + x2y2 + z12 +    + zn2 = 0
 6= 2 E~8 : x3 + xy4 + y6 + z12 +    + zn2 = 0
43 + 27 2 6= 0: 2

RL-equivalence

2.1 Let On+1 (or O) be the stalk at 0 of the holomorphic structure sheaf OC n+1 , m be the maximal ideal of O. Let R be the group of all the holomorphic automorphisms of C n+1 that preserve 0. Take L as the x-axis in C n+1 , then the dening ideal of L is g := (y1
: : :
yn ), and RL := f 2 R j g  gg: It has an action on mg from the right hand side. This denes an equivalence relation on mg. We recall some notations from 10]. Two germs f
g 2 mg are called RL-equivalent if there exists a  2 RL such that f = g . A germ f 2 mg is called k ; RLdetermined in mg if for each g 2 mg with f ; g 2 mk+1 \ g = mk g, f and g are RL-equivalent. 2.2 Theorem If mk g  mg (f ) + mk+1 g


2 RL -EQUIVALENCE

4

then f is k ; RL-determined, where


!




@f
: : :
@f + g g (f ) := m @f @x @y1 @yn is the tangent space at f of the RL-orbit RL (f ) of f .

!

Proof Sincce the proof is a standard argument in singularity theory and very similar to 10], we give a brief proof. For any function g 2 mk g, dene a one parameter family of functions F := f + t(g ; f ). Let ~g (F ) be the collection of all the di
erentials of F by the one-parameter families of vector elds preserving 0 and L. By the assumption and Nakayama lemma, one nds that there exists a vector eld of the above mentioned type, such that g ; f = (F ). The ow of gives an RL-equivalence between f and g by continuous induction.
2.3 Denitions For a germ h 2 mg, the RL -codimension of h is dened by

cL = cL(h) := dimC gm (h)

Moreover there exists the next RL-invariant

g

 = (h) := dimC J (hO) + g


where J (h) is the Jacobian ideal of h. Note that

1  (h)  (h)
where (h) is the Milnor number of h. The number  is equal to the torsion number dened in 7], see also x2.4. One can always write h in the following form

h=

X

X

Aj xyj + Bjk yj yk
where Aj are functions of one variable x: Aj 2 O1, and Bjk 2 O. With this notation we have the following simple formula

 = dimC x(A
O: :1:
A ) 1

n

2.4 Torsion number We recall the denition of the torsion number from 7]. This

section plays no role in the rest of the paper. Let L and X be analytic spaces in C n+1 dened by the ideal g and h respectively.

2 RL -EQUIVALENCE

5

If L  X , then h  g. As a subspace, L can be dened by the ideal g, the image of g in OX := O=h. The OL := O=g-module g M := gg2 = g2 + h is called the conormal module of g. Denote by T (M ) the torsion submodule of M , and by N := M=T (M ) the torsionless factored module. We have the following exact sequence 0 ;! T (M ) ;! M ;! N ;! 0: In case L is a line on an isolated complete intersection singularity X , this sequence splits. The dimension of T (M ), as a C vector space, is called the torsion number of L and X , denoted by %(L
X ) := dimC T (M ) When L is a line on a hypersurface X with isolated singuarity, then  = %(L
X ). In fact, the dening equation h of X can be written in the form h  xA1(x)y1 mod g2 by a coordinate transformation preserving L, which has been taken to be the x-axis. A calculation shows (see 7] for details) that M = (xA (x)gy ) + g2
T (M ) = (xA (x)y ) + (y(2y)1)+ y (y
: : :
y ) : 1 1 1 1 1 2 n 1 Hence dim T (M ) = dim (xAO(1x)) = : 1

2.5 Semi-continuity of (h) For h 2 gm with isolated singularity at 0, there exist

deformations of h in gm such that for generic parameters, the deformed function has only A1 type critical points. Is (h) a constant with respect to the deformation? The answer is negative in general, as easy examples show. In order to prove a semi-continuity result we consider a map H : (C n+1  C

0) ;! (C  C

0) dened by H (z
s) = (hs(z)
s), where hs is a deformation of h in mg. The critical locus of H is the germ at 0 of (

CH := (z
s) 2 C

n+1

C




s j @h @zj = 0
j = 0
: : : n

)

Let O~ be the stalk at 0 of the structure sheaf of C n+1  C
. We consider O as a subring of O~ . Denote
! @h @h s s J (hs) := @z
: : :
@z O~ : 0

n

3 CLASSIFICATION OF LINES CONTAINED IN HYPERSURFACES

6

Obviously, CH \ (L  C
) can be dened by the ideal J (hs) + gO~ . Let O~
: (C n+1  C

0) ;! (C

0)
(z
s) = s: ~ := J (hs) + gO~ Denote p := jCH \(LC
) : Theorem Let h dene an isolated singularity at 0. 1) p is nite analytic map 2) there exist representatives of all the germs considered such that for all s 2 C


(h) 

X

(Ps)2p;1(s)

(hsP )

where hsP denotes the germ of hs at point P . Proof By 6] theorem 5(d) and (h) < 1, p is nite. Next we prove 2). Still we denote by ~ the sheaf dened by ~ . Since p is nite, p(~ ) is again coherent. For any s 2 C  , we have M ~ (p(~ ))s P : = (Ps)2p;1 (s)

Note that p;1 (0) = f0g and the minimal generating set of ~ 0 consists of (h) elements. For s near 0, let s be the number of the elements in the minimal generating set of (p(~ ))s. Then s  (h). Hence
!
! ~0 ~P X X   (h) = dim   dim = (h ): p (m(C
0)~ 0 (Ps)2p;1(s) p(m(C
s)~ P (Ps)2p;1(s) sP




Remark that there always exist deformations of h in mg with only A1 points, called Morsications. An easy calculation shows that there exist Morsications hs of h such that the number of A1 points of hs sitting on L is exact (h). Conclusion (h) is semi-continuous under deformations and equal to the maximal number of A1 points of hs sitting on L, where hs is a Morsication of h in gm:




3 Classication of lines contained in hypersurfaces

The aim of this section is to classify lines on surfaces in C 3 (see x3.3), but rst (for completeness) we classify the lines contained in curves in C 2 . The equivalence is dened by RL-action.

3 CLASSIFICATION OF LINES CONTAINED IN HYPERSURFACES

7

3.1 Theorem A curve X  C 2 with an isolated simple or elliptic simple singularity contains a line passing through its singular point if and only if it is of type A2k;1 (k  1)
Dk (k  4)
E7
E~7 or E~8:

If we choose the singular point as the origin and the line as the x-axis of the local coordinate system, the dening equation of X is RL-equivalent to one of the equations in table 1.

Table 1 : Curves containing x-axis Type of X Equations A2k;1 xk y + y2 = 0 (k  1) (k  1) Dk x2y + yk;1 = 0 (k  4) (k  4) xly + xy2 = 0 (k = 2l
l  3) E7 x3y + y3 = 0 xy(x + y)(x + ty) = 0 (t 6= 0
1) E~7 ~E8 x2y2 + x4y + y3 = 0 ( 6= 0
14 )

 Name cL k A2k;1k k ; 1 2 Dk2 k ; 2 l Dkl l 3 E73 4 ~ 3 E73 6 ~ 4 E84 6

Sketch of the proof One considers the 2-jet j 2h of a function h 2 (xy
y 2). Let r be the rank of the coecient matrix of j 2h. If r = 2, then h is RL-equivalent to A11. If r = 1, one nds that h is RL-equivalent to A2k;1k (k > 1) or y2. If r = 0, one needs to consider j 3h = ax2y + bxy 2 + cy3. In case a 6= 0, j 3h is RL-equivalent to D42 if b2 ;4ac 6= 0 and to x2y if b2 ;4ac = 0. In the latter case, one considers the higher jets and nds the Dk2 singularities. In case a = 0
b 6= 0, one nds the D2ll's by considering the higher jets of h. In case a = b = 0
c 6= 0, one nds that j 4h is RL-equivalent to E73 or y3 + RL RL x2y2 or y3. If j 4h ! y3 + x2y2 (where ! means \to be RL-equivalent to "), R R L L then j 5h ! y3 + x2y2 + x4y ! E~84 ( 6= 0
41 ). One stops here since the RL RL singularity of h is neither simple nor elliptic simple if j 4h ! y3 or j 5h ! y3+ x2y2 + x4y ( = 0
14 ). It remains to study the case j 3h = 0. In the generic case, j 4h is RL-equivalent to E~73. In the non-generic case, the singularity of h is neither simple nor elliptic simple.
Proposition If h 2 mg denes an isolated singularity and n = 1, then

cL = ; :

3 CLASSIFICATION OF LINES CONTAINED IN HYPERSURFACES

8

Proof It is an easy exercise in algebra to show that for any n  1

cL = ;  + : where  = dim (mJ(gh())h)\mg : And if n = 1, we have  = 0.
3.2 A class of singularities S (with respect to an equivalence) is adjacent to a class T (notation: T ; S ) if every function f 2 S can be deformed into a function of T by an arbitarily small perturbation (cf. 3]). For example, since x2y + y6 + cy5 is RL-equivalent to D6:2, hence D6:2 ; D7:2. In the following diagram, we record some adjacencies from the proof of the theorem above, called the classication tree. Note that it does not contain all the adjacencies. Classication Tree 1

cL 0

A11 6

1 2

A32 6 I @ @

A53 6

3 4 5 6

@

D42 6 I @@

A74

D52

D63

6

6

6

A95

D62

D84

6

6

6

A116

D72

D105

6

6

6

A137

D82

D105

6

...

6

...

6

...

H Y HH

HH

E73 6A K A A

E~84

A

A A

E~73

3 CLASSIFICATION OF LINES CONTAINED IN HYPERSURFACES

9

3.3 Theorem A surface X  C 3 with a simple isolated singularity contains a line passing through its singular point if and only if it is of type: Ak (k  1)
Dk (k  4)
E6
or E7: Moreover, if we choose the singular point of X as the origin and the line L on X as the x-axis in C 3 , that is, L is dened by the ideal g = (y
z), then the dening equation of X is RL -equivalent to one of the equations in table 2.

Table 2 : Simple Surface Singularities Passing Through x-axis

Type of X Equations xy + zk+1 = 0 Ak xly + xsz2 + yz = 0 (k  1) (k = 2l + s ; 1
l  2
s  0) x2y + yk;1 + z2 = 0 x2y + xz2 + y2 = 0 Dk xly + xy2 + z2 = 0 (k = 2l
l  3)
(k  4) xly + xz2 + y2 = 0 (k = 2l + 1
l  3) E6 x2z + y3 + z2 = 0 E7 x3y + y3 + z2 = 0

 Name 1 Ak1 l Akl 2 Dk2 2 D52 l Dkl l Dkl 2 E62 3 E73

cL

k;1 k;1 k;1 4

k;1 k;1 5 6

Sketch of the proof It is just classifying the functions in (y
z )m by choosing all the coordinate transformations in RL. One starts from the 2-jet of h 2 (y
z)m: j 2h = 2a12xy + 2a13xz + a22y2 + 2a23yz + a33z2
aij = aji Let r be the rank of the coecient matrix (aij ) of j 2h. If r = 3, then (a12
a13) 6= 0. We can assume a12 6= 0. Then RL RL R A j 2h ! xy + az2 ! xy + z2 ! 1 where we still denote by x
y
z the new coordinates by abusing of notations. RL Suppose r = 2. If (a12
a13) 6= 0, then j 2h ! xy. The higher order jet RL k +1 k +1 j h ! xy + az which, by theorem 3.2, is (k + 1)-RL -determined in (y
z)m, and RL-equivalent to xy + zk+1 if a 6= 0. RL RL If (a12
a13) = 0, j 2h ! yz. For l  2, j l+1h ! yz + axly + bxlz RL  ab 6= 0: j l+1h ! yz + xly + z2, which is A2l;1l. RL  ab = 0
a 6= 0: j l+1h ! yz + xly which is not nitely RL-determined in RL (y
z)m. But j l+s+1 h ! yz + xly + xsz which, if  6= 0, is (l + s + 1)-RL determined in (y
z)m, RL-equivalent to yz + xly + xsz.

3 CLASSIFICATION OF LINES CONTAINED IN HYPERSURFACES

10

Suppose r = 1. We may assume j 2h = z2. A calculation shows that RL j 3h ! z2 + axy2 + bx2y + cx2z + ey3 which is RL-equivalent to z2 + x2y + ey3 if b 6= 0. Further study of this case, gives Dk2 (k  4). If b = 0, we have  ac 6= 0: h is RL-equivalent to D52.  a 6= 0
c = 0: by studying the higher jet of h, we obtain D2ll and D2l+1l (l  3).  a = 0
ce 6= 0: we obtain E62. RL z2 + y3 + x3y +  a = c = 0
e 6= 0: we can write j 3 = z2 + y3, and j 4h ! x3z + x2y2. A calculation shows that h is RL -equivalent to E73 if  6= 0. And in case  = 0, there are no simple germs with 4-jet RL-equivalent to

z2 + y3 + x3z + x2y2.  a = e = 0: there are no simple germs with 3-jet RL -equivalent to z2 + cx2z. If r = 0, also there are no simple germs with 2-jet RL-equivalent to 0. E8 does not appear in the list. An intuitive reason for this is that one can bring the 4-jet of the Ek (k  7) singularities into the normal form: z2 + y3 + ax3y: In case a 6= 0, one has E7, otherwise, there is no term x4, so one can never get E8.

The list of R-simple singularities is now exhausted. h i k+1 One can easily calculate h k i the (h): (h) = 1
2
up to 2 when h denes an Ak surface, (h) = 2 or 2 when h denes a Dk surface, and so on.
3.4 Corollary 1) There does not exist smooth curves on surface with E8 type isolated singularity 2) All the normal forms of the germs in table 2 are also RL -simple. And all the RL simple germs of (y
z)m are contained in table 2. Proof 1) follows from the proof of the theorem in x3.3. 2) A germ h 2 (y
z)m is RL-simple if cL(h) < 1 and any small deformation h~ of h in (y
z)m cuts only nite number of the RL-orbits of (y
z)m. It follows from the classication procedure that all the germs in table 2 are RL-simple and that they are the only ones. All RL -simple germs in (y
z)m are also R-simple germs, which are all contained in table 2 except E8.
3.5 Remarks For a hypersurface X dened by h = 0, two lines L1 and L2 in X are in one class if there exists a ' 2 R such that '(L1) = L2 and h ' = h. The results of 5] say that there exist k, 3, 2 and 1 families of smooth curves on Ak (k  1), Dk (k  4), E6 and E7 surfaces respectively. In our classication of

3 CLASSIFICATION OF LINES CONTAINED IN HYPERSURFACES

11

h

i

smooth curves on A ; D ; E6 ; E7 surfaces above, there exist k+1 2 , 2, 1 and 1 class(es) smooth curves on Ak (k  1), Dk (k  4), E6 and E7 surfaces respectively. The reason for this is the symmetry property of the simple singularities. Especially, there is only one class of smooth curves on A1, A2, D4, E6 and E7 surface. There exist two di
erent classes of smooth curves on D5 surface with the same  = 2. On Ak (k  3) and Dk (k  6) surfaces the number  tells the di
erences of the smooth curves on these surfaces. 3.6 The classication tree We record some adjacencies in the proof of the theorem in x3.3 in the classication tree 2. Note that this table contains only some of the adjacencies. (cf. x3.2) The Classication Tree 2

cL 0 1 2 3

A11 6

A21

6@ I @

A31 A32 X y XX X 6

XX XX

6

A41 A42 6

6@ I @

6

6

6

6

6

6@ I @

6

6

6

6

6

6

6

6@ I @

...6

...6

...6

...6

XXX X

XX

D42

6 I @ @  52 52 6 6@ I @

4

A51 A52 A53

D

5

A61 A62 A63

D62 D63 E62

6

A71 A72 A73 A74

D72 D73 E73

A81 A82 A83 A84

D82 D84

A91 A92 A93 A94 A95

D92 D94

7 8 ...

...6

D

6

6

6

6

6

6

...6

...6

6

3.7 Lines on surfaces with simple elliptic singularities If we continue the calculations for the case r = 1
0 in the proof of theorem 3.3 and record the adjancencies

3 CLASSIFICATION OF LINES CONTAINED IN HYPERSURFACES

12

appeared, we obtain the following theorem and classifcation tree. Theorem A surface X with simple elliptic singularity at O has a line L passing through O. And if one chooses L as the x;axis of the local coordinate system, the dening equation of X is RL-equivalent to one of the equations in table 3.

Table 3 : Simple Elliptic Surface Singularities Containing x-axis Type of X E~6

E~7 E~8

 Name cL

Equations x2y + d0xz2 + d1y3 + d2y2z + d3yz2 + d4z3 = 0 (For generic di (0  i  4)
especially "1 6= 0) x2z + x3y + bx2y2 + cxy3 + dy4 + z2 = 0 ( 6= 0
1
"2 6= 0) xy(x ; y)(x ; y) + z2 = 0 ( 6= 0
1) x3z + x2y2 + x4y + y3 + z2 = 0 (D 6= 0) x4y + x2y2 + y3 + z2 = 0 ( 6= 0
14 )

We give the explanation of the notations in table 3.

22d 0 4 66 c 0 66 0 ;12d1 1 := det 6 0 0 64 0 d2 0

0

0

7

2 3 3 4

E~72 E~73 E~83 E~84

8 8 9 9

3

0

3 77 77 77 : c 5

0 0 0 0

3 + 2b 2b + 3c c + 4d

 is the cross ratio of the four lines de
ned by and

E~62

0 ;4d0 d1 ;3d0 d2 ;2d0 d3 ;6d1 ;4d2 ;2d3 0 7 7 ;8d2 ;4d3 0 ;c2 77 : 4d 1 3d2 2d3 d4 77 2d3 3d4 ;c2 0 5 d2 2d3 3d4 ;c2

22 3 2b c 0 66c ;1 3 2b c 660 0 3 + 2b 2b + 3c c + 4d 2 := det 60 3 + 2b 2b + 3c c + 4d 0 64 0 0 ;1 3 2b 0

2

; 14 x4 + x3 y + bx2 y2 + cxy3 + dy4 = 0: D := 94  6 ; 34  3 + 163 3 ; 43  2 2 :

3 CLASSIFICATION OF LINES CONTAINED IN HYPERSURFACES

13

Classication Tree 3 D52  E~62 @ I @

E62 E~72 E~73 @ I @

E73  E~83  E~84

3.8 Remark For E~62, there are two RL moduli. E~72, E~73, E~83 and E~84 are one RL modular. 3.9 Invariance of  under sections with hyperplanes through L Let L be a line contained in a hypersurface X  C n+1 with isolated singularity, H a generic hyperplane in the pencil of hyperplanes passing through L. Then X \ H has also isolated singularity and contains L. A straightforward computation shows: Proposition (X \ H ) = (X ). In fact we can continue cutting until we are in the curve case. If we apply this on the surfaces dened by the equation in table 2, we have

X

Akl Dkl D52 E62 E73 X \ H A2l;1l A2l;1l A32 A32 A53 Obviously we have in these examples (X \ H ) = 2 ; 1. This reminds very much to Milnor's double point formula

= 2 + 1 ; r
where is the maximal number of double points on a deformed curve, and r is the number of the branches of the curve. But this is a coincidence, which holds since X \ H is of type Aodd. Accordining to the above proposition

 = (X ) = (X \ H )  (X \ H ) Inspection shows (Aodd) = (Aodd). However for X with E~62 type singularity, this is not the case. X \ H is of D42 type singularity and = 3,  = 2 and r = 3. 3.10 Remark For all the germs in table 2 and 3, we have cL = ; 1, especially cL(h) does not depend on . This means that for those surfaces we can not move from one position of a line into another position. This was a surprise to us in other dimensions this is di
erent (see also question 4.4).

4 LINES ON HYPERSURFACES IN C 4

14

4 Lines on hypersurfaces in C 4

About the existence of lines on hypersurfaces in C 4 , we mention rst that on corank 2 singularities there always exist lines due to the suspension xy + g(z
w): This gives lines on, for example, the E8 hypersurface in C 4 . And  = 1 all the time. For corank 3 singularities one can consider the suspension g(x
y
z) + w2. If g contains the x-axis, so does this suspension with the same . But there are examples of corank 3 singularities which do not contain any lines. Here is one of them:

h = w2 + x11 + y13 + z15 = 0: It is natural to consider the above classications with four variables. We give here the beginning of the classication for simple singularities. The proof is very similar to the cases considered in x3. Also here we record some of adjancencies in the classication tree. We distinguish between  = 1 and  > 1. 4.1 Theorem Let X  C 4 be hypersurfaces with isolated simple singularity. If X contains x-axis and  = 1, then the dening equation of X is RL -equivalent to one of the equations in table 4.


Table 4 Type of X

Equations

 Name

Ak

xy + zk+1 + w2 = 0

1

Dk

xy + z2w + wk;1 = 0 xy + z4 + w3 = 0 xy + z3w + w3 = 0 xy + z5 + w3 = 0

1 Dk1 k ; 1 1 E61 5 1 E71 6 1 E81 7

(k  1) (k  4)

E6 E7 E8

cL

Ak1 k ; 1

4.2 Theorem Let X  C 4 be hypersurfaces with simple isolated singularity. If X contains x-axis,  > 1 and cL(X )  8, then the dening equation of X is RL equivalent to one of the equations in table 5.


4 LINES ON HYPERSURFACES IN C 4

15

Table 5 Type of X

Equations xly + xsz2 + yz + w2 = 0 (k = 2l + s ; 1
l  2
s  0) x2y + yk;1 + z2 + w2 = 0 x2z + xl;3w + xy2 + wz = 0 (l  5) xly + xy2 + z2 + w2 = 0 (l  3) xlz + xy2 + z2 + w2 = 0 (l  3) x2 y + y 2 + z 3 + w 2 = 0 x2z + y3 + xyw + zw = 0 x3y + y3 + z2 + w2 = 0 x2z + y3 + xw2 + zw = 0

Ak

(k  1)

Dk

(k  4)

E6 E7 E8

 l

Name

cL

Akl

3l + s ; 3

2 2

Dk2 Dl2 D2ll D2l+1l E62 E72 E73 E82

k l

l l

2 2 3 2

3l ; 2 3l ; 1 6 7 8 8

The Classication Tree 4

A11

 6  21   6@ I @   31 32 X y XX XXX  XXX 6 6  XXX XX 41 41 42  6  6 6@ I @   51 52 53 51  6 6 6 6

A A

A

D

A

A

D

A

A

A



D42

6 I @ @  52 52 6 6

D

D

6

i P PP Y H HH P @ I @ H PPP  63 62 72 62 6 6 6 6

6

6

E61 D61 A61 A62 A63

D62 D

E71 D71 A71 A72 A73 A74

D72 D72 D73 E73 E82

6

6

6

6

6@ I @

6

6

6

6

6

6

E81 D81 A81 A82 A83 A84 6

6

6

6

6@ I @

D91 A91 A92 A93 A94 A95 ...6

...6

...6

...6

...6

...6

D

6

D82

D

D84

D92

D

D94

6

...6

82

6

92

...6

6

...6

E

E

4 LINES ON HYPERSURFACES IN C 4

16

4.3 Remark 1) One may compare table 4 with the list of simple hypersurface singularities in x1.3 2) If we compare with the surface case, we nd that on E8 hypersurface there exist two classes of lines. Also on other hypersufaces, there exists more lines than in the case of surfaces in C 3 . 4.4 Question As claimed by the proposition in x3.1, all the known examples show the following remarkable formulae:

cL = + (n ; 2) ; n + 1: Is it true for any hypersurfaces with isolated singularities in C n+1 ? 4.5 Higher order invariants For h 2 mg dening an isolated singularity, there exists the following ltration: J (h) + g  J (h) + g2      J (h) + gk      J (h): Dene

k = k (h) := J (h)O+ gk
k = 1
2
   :

and the higher order invariants: k = k (h) := dimC k
k = 1
2
   : Note that there exists a k0 such that  = 1  2      k0 = : Denote

 = (h) = 1
2
: : :
k0 = ]
where k0 is the minimal k such that k = , called the length of the sequence (h). The (h) give more information about how a line is embedded into a singular hypersurface. Recall that in table 2 and 5, we have Dl2 and Dl2 (l  5) with the same . Easy calculations show the following: For surfaces in C 3 , (D52) = 2
3
4
5] with length 4, and (D52) = 2
4
5] with length 3. For the Dk type hypersurfaces in C 4 we have (Dl2) = 2
: : :
l] (l  5) with length l ; 1, (D52) = 2
4
5] with length 3. For l  3, (D2l2) = 2
5
7
: : :
2l ; 1
2l] with length l, and (D2l+12) = 2
5
7
: : :
2l ; 1
2l + 1
] with length l. So, higher order invariants k 's distinguish Dl2 and Dl2:

REFERENCES

17

References 1] Arnol'd, V. I.: Normal forms of functions near degenerate critical points, Weyl groups Ak  Dk  Ek and Lagrange singularities, Functional Anal. and its Appl., Vol.6, No.4 (1972), 254272. 2] Arnol'd, V. I.: Classi
cation of 1-modular critical points of functions, Functional Anal. and its Appl., Vol.7, No.3 (1972). 230-231. 3] Arnold, V. I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of dierentiable Maps, Vol. I, II, Birkhauser, 1985. 4] Durfee, A. H.: Fifteen characterizations of rational double points and simple critical points, Ens. Math., 25 (1979), 131-163. 5] Gonzalez-Sprinberg, G., Lejeune-Jalabert, M.: Families of smooth curves on surface singularities and wedges, Ann. Polonici Math., LXVII.2 (1997), 179-190. 6] Gunning, R. : Lectures on complex analytic varieties II. Finite analytic mappings, Princeton University Press, 1970. 7] Jiang, G: Functions with non-isolated singularities on singular spaces, Thesis, Universiteit Utrecht, 1998. 8] Siersma, D.: The singularities of C 1 -functions of right codimension smaller or equal than eight, Indag. Math., Vol.35, No. 1 (1973), 31-37. 9] Siersma, D.: Classication and deformation of singularities, Thesis, Amsterdam University, 1974. 10] Siersma, D.: Isolated line singularities, Proc. of Symp. in Pure Math., Vol.40 Part 2 (1983), 485-496.

Jiang, G. Department of Mathematics Jinzhou Normal University Jinzhou, Liaoning, 121000 P. R. China Siersma, D. Department of Mathematics Utrecht University P.O.Box 80.010, 3508 TA Utrecht The Netherlands email:[email protected]