Abstract. Simple surface singularities have been known for long time since they occur in several

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1 ocal Embeddings of ines in Singular Hypersurfaces Jiang Guangfeng Dirk Siersma Abstract ines on hypersurfaces with isolated singularities are classied 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 Classication: 14J17, 32S25, 58C27 Introduction Simple surface singularities have been known for long time since they occur in several dierent 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 ejeune-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 as the x-axis of the coordinate system in C n+1, classify all the hypersurfaces that contain under the equivalence dened by the subgroup R of R, consisting of all the coordinate transformations preserving 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; ) We obtain the following results: The R -classication is ner than the R-classication, ie there are dierent `positions' of lines on the same hypersurface 1

2 1 INES CONTAINED IN SINGUAR SPACES 2 There exist an R invariant, which counts the maximal number of Morse points on the line over all Morsications is invariant under sections with generic hyperplanes containing Another proof of the non-existence of lines on E 8 surface We also introduce a sequence of higher order R 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 iaoning Province of P R China We thank all of them 1 ines contained in singular spaces 11 et be a smooth curve in C n+1 Since is locally biholomorphic to a line, we often say that is a line et X be a singular space embedded in C n+1 If for a singular point O 2 X sing, there exists a line (smooth curve) in C n+1 such that O 2 and n fog X reg, 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 12 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 ejeune-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 A k, D k (k 4), E or E 7 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 dierent lines there are on a surface, and what kind of invariants can be used to tell the dierences The hardest thing is to prove the non-existence of smooth curve on certain surfaces (eg E 8 type surface loccit) There exists similar question for hypersurfaces with other dimensions

3 2 R -EQUIVAENCE 3 13 Isolated hypersurface singularities et 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]): A k : x k+1 + y 2 + z z 2 n = 0 (k 1); D k : x 2 y + y k?1 + z z2 n = 0 (k 4); E : x 4 + y 3 + z z 2 n = 0; E 7 : x 3 y + y 3 + z z 2 n = 0; E 8 : x 5 + y 3 + z z2 = 0: n 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 : x 2 y + z y 2 z 1 + y 3 + z z 2 n = 0; = 0; ~E 7 : x 4 + y 4 + x 2 y 2 + z z2 n = 0; = 2; ~E 8 : x 3 + xy 4 + y + z z 2 n = 0; = 0: 2 R -equivalence 21 et O n+1 (or O) be the stalk at 0 of the holomorphic structure sheaf O C n+1, m be the maximal ideal of O et R be the group of all the holomorphic automorphisms of C n+1 that preserve 0 Take as the x-axis in C n+1, then the dening ideal of is g := (y 1 ; : : : ; y n ), and R := 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 R -equivalent if there exists a 2 R such that f = g A germ f 2 mg is called k? R - determined in mg if for each g 2 mg with f? g 2 m k+1 \ g = m k g, f and g are R -equivalent 22 Theorem If m k g m g (f) + m k+1 g;

4 2 R -EQUIVAENCE 4 then f is k? R -determined, where g (f) 1 ; : : n! is the tangent space at f of the R -orbit R (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 m k g, dene a one parameter family of functions F := f + t(g? f) et ~ g (F ) be the collection of all the dierentials of F by the one-parameter families of vector elds preserving 0 and 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 R -equivalence between f and g by continuous induction 23 Denitions For a germ h 2 mg, the R -codimension of h is dened by c = c (h) := dim C Moreover there exists the next R -invariant = (h) := dim C where J(h) is the Jacobian ideal of h Note that 1 (h) (h); gm g (h) O J(h) + g ; where (h) is the Milnor number of h The number is equal to the torsion number dened in [7], see also x24 One can always write h in the following form h = X A j xy j + X B jk y j y k ; where A j are functions of one variable x: A j 2 O 1, and B jk 2 O With this notation we have the following simple formula = dim C O 1 x(a 1 ; : : : ; A n ) 24 Torsion number We recall the denition of the torsion number from [7] This section plays no role in the rest of the paper et and X be analytic spaces in C n+1 dened by the ideal g and h respectively

5 2 R -EQUIVAENCE 5 If X, then h g As a subspace, can be dened by the ideal g, the image of g in O X := O=h The O := O=g-module M := g g 2 = g g 2 + 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 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 and X, denoted by %(; X) := dim C T (M) When is a line on a hypersurface X with isolated singuarity, then = %(; X) In fact, the dening equation h of X can be written in the form h xa 1 (x)y 1 mod g 2 by a coordinate transformation preserving, which has been taken to be the x-axis A calculation shows (see [7] for details) that Hence M = g (xa 1 (x)y 1 ) + g 2 ; T (M) = (y 1 ) (xa 1 (x)y 1 ) + (y 2 1) + y 1 (y 2 ; : : : ; y n ) : dim T (M) = dim O 1 (xa 1 (x)) = : 25 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 A 1 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) = (h s (z); s), where h s is a deformation of h in mg The critical locus of H is the germ at 0 of ( ) C H := (z; s) 2 C n+1 C s = 0; j = 0; : : : j et O ~ be the stalk at 0 of the structure sheaf of C n+1 C We consider O as a subring of O ~ Denote! J(h s ) s ; : : : s n

6 3 CASSIFICATION OF INES CONTAINED IN HYPERSURFACES Obviously, C H \ ( C ) can be dened by the ideal J(h s ) + g ~ O et ~ := Denote p := j CH \(C ) : ~O J(h s ) + g ~O ; : (C n+1 C ; 0)?! (C ; 0); (z; s) = s: Theorem et 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 (P;s)2p?1 (s) (h s;p ) where h s;p denotes the germ of h s at point P Proof By [] 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 (p ( )) ~ M s = ~ P : (P;s)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 (h) = dim! ~ 0 p (m (C ;0) ~ 0 X (P;s)2p?1 (s) dim! ~ P p (m (C ;s) ~ = P X (P;s)2p?1 (s) Remark that there always exist deformations of h in mg with only A 1 (h s;p ): points, called Morsications An easy calculation shows that there exist Morsications h s of h such that the number of A 1 points of h s sitting on is exact (h) Conclusion (h) is semi-continuous under deformations and equal to the maximal number of A 1 points of h s sitting on, where h s 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 x33), but rst (for completeness) we classify the lines contained in curves in C 2 The equivalence is dened by R -action

7 3 CASSIFICATION OF INES CONTAINED IN HYPERSURFACES 7 31 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 A 2k?1 (k 1); D k (k 4); E 7 ; ~ E7 or ~ E8 : 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 R -equivalent to one of the equations in table 1 Table 1 : Curves containing x-axis Type of X Equations Name c A 2k?1 (k 1) x k y + y 2 = 0 (k 1) k A 2k?1;k k? 1 D k x 2 y + y k?1 = 0 (k 4) 2 D k;2 k? 2 (k 4) x l y + xy 2 = 0 (k = 2l; l 3) l D k;l l E 7 x 3 y + y 3 = 0 3 E 7;3 4 ~E 7 xy(x + y)(x + ty) = 0 (t = 0; 1) 3 E7;3 ~ ~E 8 x 2 y 2 + x 4 y + y 3 = 0 ( = 0; 1) 4 4 E8;4 ~ Sketch of the proof One considers the 2-jet j 2 h of a function h 2 (xy; y 2 ) et r be the rank of the coecient matrix of j 2 h If r = 2, then h is R -equivalent to A 1;1 If r = 1, one nds that h is R -equivalent to A 2k?1;k (k > 1) or y 2 If r = 0, one needs to consider j 3 h = ax 2 y + bxy 2 + cy 3 In case a = 0, j 3 h is R -equivalent to D 4;2 if b 2?4ac = 0 and to x 2 y if b 2?4ac = 0 In the latter case, one considers the higher jets and nds the D k;2 singularities In case a = 0; b = 0, one nds the D 2l;l 's by considering the higher jets of h In case a = b = 0; c = 0, one nds that j 4 h is R -equivalent to E 7;3 or y 3 + x 2 y 2 or y 3 If j 4 R y 3 + x 2 y 2 (where R! means \to be R -equivalent to "), then j 5 R y 3 + x 2 y 2 + x 4 R y! E ~ 8;4 ( = 0; 1 ) One stops here since the 4 singularity of h is neither simple nor elliptic simple if j 4 R y 3 or j 5 R y 3 + x 2 y 2 + x 4 y ( = 0; 1 ) 4 It remains to study the case j 3 h = 0 In the generic case, j 4 h is R -equivalent to E ~ 7;3 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 c =? :

8 3 CASSIFICATION OF INES CONTAINED IN HYPERSURFACES 8 Proof It is an easy exercise in algebra to show that for any n 1 where = dim (mj(h))\mg g(h) : And if n = 1, we have = 0 c =? + : 32 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 x 2 y + y + cy 5 is R -equivalent to D :2, hence D :2? D 7:2 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 c 0 A 1; A 5;3 A 7;4 A 9;5 A 11; A 13;7 D 4;2 D 5;2 D ;2 D 7;2 D ;3 D 8;4 D 10;5 D 10;5 HY H H H H E 7;3 AK A A A A A ~E 8;4 E7;3 ~

9 3 CASSIFICATION OF INES CONTAINED IN HYPERSURFACES 9 33 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: A k (k 1); D k (k 4); E ; or E 7 : Moreover, if we choose the singular point of X as the origin and the line on X as the x-axis in C 3, that is, is dened by the ideal g = (y; z), then the dening equation of X is R -equivalent to one of the equations in table 2 Table 2 : Simple Surface Singularities Passing Through x-axis Type of X Equations Name c xy + z k+1 = 0 A k 1 A x (k 1) l y + x s z 2 k;1 k? 1 + yz = 0 l A k;l k? 1 (k = 2l + s? 1; l 2; s 0) D k (k 4) x 2 y + y k?1 + z 2 = 0 x 2 y + xz 2 + y 2 = 0 x l y + xy 2 + z 2 = 0 (k = 2l; l 3); x l y + xz 2 + y 2 = 0 (k = 2l + 1; l 3) 2 2 l l D k;2 D 5;2 D k;l D k;l k? 1 4 k? 1 k? 1 E x 2 z + y 3 + z 2 = 0 2 E ;2 5 E 7 x 3 y + y 3 + z 2 = 0 3 E 7;3 Sketch of the proof It is just classifying the functions in (y; z)m by choosing all the coordinate transformations in R One starts from the 2-jet of h 2 (y; z)m: j 2 h = 2a 12 xy + 2a 13 xz + a 22 y 2 + 2a 23 yz + a 33 z 2 ; a ij = a ji et r be the rank of the coecient matrix (a ij ) of j 2 h If r = 3, then (a 12 ; a 13 ) = 0 We can assume a 12 = 0 Then j 2 R xy + az 2 R! xy + z R! 2 A 1 where we still denote by x; y; z the new coordinates by abusing of notations Suppose r = 2 If (a 12 ; a 13 ) = 0, then j 2 R xy The higher order jet j k+1 R xy + az k+1 which, by theorem 32, is (k + 1)-R -determined in (y; z)m, and R -equivalent to xy + z k+1 if a = 0 If (a 12 ; a 13 ) = 0, j 2 R yz For l 2, j l+1 R yz + ax l y + bx l z ab = 0: j l+1 R yz + x l y + z 2, which is A 2l?1;l ab = 0; a = 0: j l+1 R h yz + x l y which is not nitely R -determined in (y; z)m But j l+s+1 R yz + x l y + x s z which, if = 0, is (l + s + 1)-R - determined in (y; z)m, R -equivalent to yz + x l y + x s z

10 3 CASSIFICATION OF INES CONTAINED IN HYPERSURFACES 10 Suppose r = 1 We may assume j 2 h = z 2 A calculation shows that j 3 h R! z 2 + axy 2 + bx 2 y + cx 2 z + ey 3 which is R -equivalent to z 2 + x 2 y + ey 3 if b = 0 Further study of this case, gives D k;2 (k 4) If b = 0, we have ac = 0: h is R -equivalent to D 5;2 a = 0; c = 0: by studying the higher jet of h, we obtain D 2l;l and D 2l+1;l (l 3) a = 0; ce = 0: we obtain E ;2 a = c = 0; e = 0: we can write j 3 = z 2 + y 3, and j 4 h R! z 2 + y 3 + x 3 y + x 3 z + x 2 y 2 A calculation shows that h is R -equivalent to E 7;3 if = 0 And in case = 0, there are no simple germs with 4-jet R -equivalent to z 2 + y 3 + x 3 z + x 2 y 2 a = e = 0: there are no simple germs with 3-jet R -equivalent to z 2 + cx 2 z If r = 0, also there are no simple germs with 2-jet R -equivalent to 0 E 8 does not appear in the list An intuitive reason for this is that one can bring the 4-jet of the E k (k 7) singularities into the normal form: z 2 + y 3 + ax 3 y: In case a = 0, one has E 7, otherwise, there is no term x 4, so one can never get E 8 The list of R-simple singularities is now exhausted h i One can easily calculate h k+1 the (h): (h) = 1; 2; up to 2 when h denes an Ak k surface, (h) = 2 or 2i when h denes a Dk surface, and so on 34 Corollary 1) There does not exist smooth curves on surface with E 8 type isolated singularity; 2) All the normal forms of the germs in table 2 are also R -simple And all the R simple germs of (y; z)m are contained in table 2 Proof 1) follows from the proof of the theorem in x33 2) A germ h 2 (y; z)m is R -simple if c (h) < 1 and any small deformation ~ h of h in (y; z)m cuts only nite number of the R -orbits of (y; z)m It follows from the classication procedure that all the germs in table 2 are R -simple and that they are the only ones All R -simple germs in (y; z)m are also R-simple germs, which are all contained in table 2 except E 8 35 Remarks For a hypersurface X dened by h = 0, two lines 1 and 2 in X are in one class if there exists a ' 2 R such that '( 1 ) = 2 and h ' = h The results of [5] say that there exist k, 3, 2 and 1 families of smooth curves on A k (k 1), D k (k 4), E and E 7 surfaces respectively In our classication of

11 3 CASSIFICATION OF INES CONTAINED IN HYPERSURFACES 11 h i k+1 smooth curves on A? D? E? E 7 surfaces above, there exist 2, 2, 1 and 1 class(es) smooth curves on A k (k 1), D k (k 4), E and E 7 surfaces respectively The reason for this is the symmetry property of the simple singularities Especially, there is only one class of smooth curves on A 1, A 2, D 4, E and E 7 surface There exist two dierent classes of smooth curves on D 5 surface with the same = 2 On A k (k 3) and D k (k ) surfaces the number tells the dierences of the smooth curves on these surfaces 3 The classication tree We record some adjacencies in the proof of the theorem in x33 in the classication tree 2 Note that this table contains only some of the adjacencies (cf x32) The Classication Tree 2 c A 1;1 A A 3;1 A 4;1 A 5;1 A ;1 A 7;1 A 8;1 A 9;1 A 3;2 A A 5;2 A ;2 A 7;2 A 8;2 A 9;2 Xy X XX X XXXXXXXX A 5;3 A A 7;3 A 8;3 A 9;3 A 7;4 A A 9;4 A 9;5 D 4;2 D 5;2 D ;2 D 7;2 D 8;2 D D D ;3 D 7;3 D 8;4 D 9;4 E ;2 E 7;3 37 ines on surfaces with simple elliptic singularities If we continue the calculations for the case r = 1; 0 in the proof of theorem 33 and record the adjancencies

12 3 CASSIFICATION OF INES 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 passing through O And if one chooses as the x?axis of the local coordinate system, the dening equation of X is R -equivalent to one of the equations in table 3 Table 3 : Simple Elliptic Surface Singularities Containing x-axis Type of X Equations Name c x ~E 2 y + d 0 xz 2 + d 1 y 3 + d 2 y 2 z + d 3 yz 2 + d 4 z 3 = 0 (For generic d i (0 i 4); especially 1 = 0) x 2 z + x 3 y + bx 2 y 2 + cxy 3 + dy 4 + z 2 = 0 ~E 7 ( = 0; 1; 2 = 0) xy(x? y)(x? y) + z 2 = 0 ( = 0; 1) ~E 8 x 3 z + x 2 y 2 + x 4 y + y 3 + z 2 = 0 (D = 0) x 4 y + x 2 y 2 + y 3 + z 2 = 0 ( = 0; 1 4 ) ~ E; ~E 7;2 8 ~E 7;3 8 ~E 8;3 9 ~E 8;4 9 We give the explanation of the notations in table 3 1 := det 2 := det 2 4 2d 4 0 0?4d 0d 1?3d 0d 2?2d 0d 3 c 0?d 1?4d 2?2d 3 0 0?12d 1?8d 2?4d 3 0?c is the cross ratio of the four lines dened by 0 0 4d 1 3d 2 2d 3 d 4 0 d 2 2d 3 3d 4?c d 2 2d 3 3d 4?c b c 0 0 c?1 3 2b c b 2b + 3c c + 4d b 2b + 3c c + 4d ?1 3 2b c b 2b + 3c c + 4d 3 : : 7 5 and? 1 4 x4 + x 3 y + bx 2 y 2 + cxy 3 + dy 4 = 0: D := 9 4? ? :

13 3 CASSIFICATION OF INES CONTAINED IN HYPERSURFACES 13 Classication Tree 3 D 5;2 ~ E ;2 E7;2 ~ E 7;3 ~ E8;3 ~ E8;4 38 Remark For ~ E;2, there are two R moduli ~ E7;2, ~ E7;3, ~ E8;3 and ~ E8;4 are one R modular 39 Invariance of under sections with hyperplanes through et 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 Then X \ H has also isolated singularity and contains 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 A k;l D k;l D 5;2 E ;2 E 7;3 X \ H A 2l?1;l A 2l?1;l A 3;2 A 3;2 A 5;3 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 A odd Accordining to the above proposition = (X) = (X \ H) (X \ H) Inspection shows (A odd ) = (A odd ) However for X with ~ E;2 type singularity, this is not the case X \ H is of D 4;2 type singularity and = 3, = 2 and r = Remark For all the germs in table 2 and 3, we have c =? 1, especially c (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 dierent (see also question 44)

14 4 INES ON HYPERSURFACES IN C ines 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 E 8 hypersurface in C 4 And = 1 all the time For corank 3 singularities one can consider the suspension g(x; y; z) + w 2 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 = w 2 + x 11 + y 13 + z 15 = 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 41 Theorem et X C 4 be hypersurfaces with isolated simple singularity If X contains x-axis and = 1, then the dening equation of X is R -equivalent to one of the equations in table 4 Table 4 Type of X A k (k 1) Equations Name c xy + z k+1 + w 2 = 0 1 A k;1 k? 1 D k (k 4) xy + z 2 w + w k?1 = 0 1 D k;1 k? 1 E xy + z 4 + w 3 = 0 1 E ;1 5 E 7 xy + z 3 w + w 3 = 0 1 E 7;1 E 8 xy + z 5 + w 3 = 0 1 E 8; Theorem et X C 4 be hypersurfaces with simple isolated singularity If X contains x-axis, > 1 and c (X) 8, then the dening equation of X is R equivalent to one of the equations in table 5

15 4 INES ON HYPERSURFACES IN C 4 15 Type of X A k (k 1) Table 5 Equations Name c x l y + x s z 2 + yz + w 2 = 0 (k = 2l + s? 1; l 2; s 0) l A k;l 3l + s? 3 x 2 y + y k?1 + z 2 + w 2 = 0 2 D k;2 k x 2 z + x l?3 w + xy 2 + wz = 0 (l 5) 2 D l;2 l x l y + xy 2 + z 2 + w 2 = 0 (l 3) l D 2l;l 3l? 2 x l z + xy 2 + z 2 + w 2 = 0 (l 3) l D 2l+1;l 3l? 1 E x 2 y + y 2 + z 3 + w 2 = 0 2 E ;2 x 2 z + y 3 + xyw + zw = 0 2 E 7;2 7 E 7 x 3 y + y 3 + z 2 + w 2 = 0 3 E 7;3 8 E 8 x 2 z + y 3 + xw 2 + zw = 0 2 E 8;2 8 D k (k 4) The Classication Tree 4 E ;1 E 7;1 E 8;1 D 4;1 D 5;1 D ;1 D 7;1 D 8;1 D 9;1 A 1;1 A A 3;1 A 4;1 A 5;1 A ;1 A 7;1 A 8;1 A 9;1 A 3;2 A A 5;2 A ;2 A 7;2 A 8;2 A 9;2 Xy X XX X XXXXXXXX A 5;3 A A 7;3 A 8;3 A 9;3 A 7;4 A A 9;4 A 9;5 D 4;2 D 5;2 D ;2 D 7;2 D 8;2 D D 5;2 D ;2 D 7;2 D 8;2 D 9;2 PPi HY H PPPP H D ;3 E ;2 D 7;3 D 8;4 D 9;4 E 7;3 E 7;2 E 8;2

16 4 INES ON HYPERSURFACES IN C Remark 1) One may compare table 4 with the list of simple hypersurface singularities in x13; 2) If we compare with the surface case, we nd that on E 8 hypersurface there exist two classes of lines Also on other hypersufaces, there exists more lines than in the case of surfaces in C 3 44 Question As claimed by the proposition in x31, all the known examples show the following remarkable formulae: c = + (n? 2)? n + 1: Is it true for any hypersurfaces with isolated singularities in C n+1? 45 Higher order invariants For h 2 mg dening an isolated singularity, there exists the following ltration: Dene J(h) + g J(h) + g 2 J(h) + g k J(h): k = k (h) := and the higher order invariants: Note that there exists a k 0 such that Denote O ; k = 1; 2; : k J(h) + g k = k (h) := dim C k ; k = 1; 2; : = 1 2 k0 = : = (h) = [ 1 ; 2 ; : : : ; k0 = ]; where k 0 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 D l;2 and D l;2 (l 5) with the same Easy calculations show the following: For surfaces in C 3, (D 5;2 ) = [2; 3; 4; 5] with length 4, and (D5;2) = [2; 4; 5] with length 3 For the D k type hypersurfaces in C 4 we have (D l;2 ) = [2; : : : ; l] (l 5) with length l? 1, (D5;2) = [2; 4; 5] with length 3 For l 3, (D2l;2) = [2; 5; 7; : : : ; 2l? 1; 2l] with length l, and (D2l+1;2) = [2; 5; 7; : : : ; 2l? 1; 2l + 1; ] with length l So, higher order invariants k 's distinguish D l;2 and Dl;2:

17 REFERENCES 17 References [1] Arnol'd, V I: Normal forms of functions near degenerate critical points, Weyl groups A k; D k; E k and agrange singularities, Functional Anal and its Appl, Vol, No4 (1972), [2] Arnol'd, V I: Classication of 1-modular critical points of functions, Functional Anal and its Appl, Vol7, No3 (1972) [3] Arnold, V I, Gusein-Zade, SM, Varchenko, AN: 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), [5] Gonzalez-Sprinberg, G, ejeune-jalabert, M: Families of smooth curves on surface singularities and wedges, Ann Polonici Math, XVII2 (1997), [] Gunning, R : ectures 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, Vol35, No 1 (1973), [9] Siersma, D: Classication and deformation of singularities, Thesis, Amsterdam University, 1974 [10] Siersma, D: Isolated line singularities, Proc of Symp in Pure Math, Vol40 Part 2 (1983), Jiang, G Department of Mathematics Jinzhou Normal University Jinzhou, iaoning, P R China Siersma, D Department of Mathematics Utrecht University POBox 80010, 3508 TA Utrecht The Netherlands siersma@mathuunl