Fully simple singularities of plane and space curves

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1 Proc. London Math. Soc. Page1of21 C 2008 London Mathematcal Socety do: /plms/pdn001 Fully smple sngulartes of plane and space curves M. Zhtomrsk Abstract In ths work we ntroduce the defnton of fully smple sngulartes of parameterzed curves and explan that ths defnton s more natural than the defnton of smple sngulartes. The set of fully smple sngulartes s much smaller than the set of smple ones. We determne and classfy all fully smple sngulartes of plane and space curves, wth any number of components. Our classfcaton results mply that any fully smple sngularty of a plane or a space curve s quas-homogeneous (whereas there s a number of non-quas-homogeneous smple sngulartes). Another outcome of our classfcaton results s a one-to-one correspondence between the fully smple sngulartes of plane curves and the classcal A-D-E sngulartes of functons. 1. Introducton and man results 1.1. Sngulartes of parameterzed curves: smple and fully smple sngulartes We deal wth parameterzed curves. In what follows by a curve n R n we mean a map Γ:(a, b) R n. All objects are assumed to be of the class C. The purpose of ths work s to determne and classfy sngulartes of Γ at a pont p of ts mage satsfyng the followng condton: C: All sngulartes of curves suffcently close to Γ at ponts of ther mages suffcently close to p are exhausted by a fnte number of sngulartes. Sngulartes satsfyng C wll be called fully smple. Precse defnton s gven below. The reason for ths termnology s as follows: a smple sngularty of a curve does not need to be fully smple. In order to explan ths clam one should start wth a defnton of the sngularty of a curve Γ:(a, b) R n at a pont p n the mage of Γ. Let t 0 1,...,t 0 d (a, b) be the nverse mages of the pont p. Here d 1. The local structure of Γ at p can be descrbed by a multgerm wth d components. Defnton. A multgerm of a curve n R n s a collecton γ =(γ 1,...,γ d ), where γ : (R, 0) (R n, 0) are map germs; they are called the components of γ. The collecton s defned up to the order of the components. To descrbe the local structure of Γ at p by a multgerm γ =(γ 1,...,γ d ) take local coordnates t centered at t 0 and a local coordnate system x centered at p. Then γ s the germ of Γ at the pont t 0 expressed n the local coordnates t and x, =1,...,d. Changng the local coordnates t and x we obtan another multgerm γ =( γ 1,..., γ d )whch s RL-equvalent to γ. Ths means that there exst local dffeomorphsms φ :(R, 0) (R, 0) (parameterzatons of the components) and a local dffeomorphsm Φ : (R n, 0) (R n, 0) Receved 20 July 2005; revsed 16 Aprl Mathematcs Subject Classfcaton 14B05, 14H50, 58K60. The work on ths paper was supported by the Israel Scence Foundaton, grant 1356/04.

2 Page 2 of 21 M. ZHITOMIRSKII such that Φ γ = γ φ, =1,...,d, up to numeraton of the components of one of the mult-germs. Defnton. The sngularty of a curve Γ n R n at a pont p n the mage of Γ s the RL-equvalence class of some (and then any) multgerm γ descrbng the local structure of Γ at p n the space of all multgerms wth d components, where d 1 s the number of the nverse mages of the pont p. In most classfcaton problems the frst task s to determne and classfy smple sngulartes. The smple sngulartes wth one component of curves n R n were determned and classfed by Bruce and Gaffney n [3] forn = 2, by Gbbson and Hobbs n [4] forn = 3 and by Arnol d n [1] for an arbtrary n. These results were contnued n [5, 6], where Kolgushkn and Sadykov classfed all smple sngulartes of curves n R n, wth any number of components and for any n. The works [5, 6] contan almost 150 normal forms. The defnton of smple germ or multgerm used n all cted above works s as follows. Defnton. A multgerm γ =(γ 1,...,γ d ) (and a sngularty defned by γ) s called smple f there exsts k < such that the sngulartes defned by multgerms wth d components and the k-jet suffcently close to the k-jet of γ are exhausted by a fnte number of sngulartes. The startng pont for the present work s the observaton that a bg part of smple sngulartes do not satsfy condton C above, see Subsecton 1.2. Therefore sngulartes satsfyng C wll be called fully smple. A precse defnton of fully smple sngulartes requres arcs maps F :[a, b] R n. Defnton. We wll say that an arc F :[a, b] R n represents a multgerm γ f the mage of F contans 0 R n, the ponts F (a)andf(b) are dfferent from 0, and γ defnes the sngularty of F at 0 R n. (The sngularty of an arc F :[a, b] R n at a pont p of ts mage, dfferent from F (a) andf (b), s the sngularty at p of the curve F (a,b).) Man Defnton. Let γ be a multgerm of a parameterzed curve n R n and let F :[a, b] R n be an arc representng γ. The multgerm γ (and the sngularty defned by γ) s called fully smple f there exsts k< such that the sngulartes of all arcs F :[a, b] R n suffcently C k -close to the arc F at all ponts of ther mages suffcently close to 0 R n are exhausted by a fnte number of sngulartes. One can easly prove that the defnton s correct (the choce of an arc F representng γ s rrelevant) and that any fully smple multgerm s smple. The defnton remans the same f we restrct ourselves to the sngulartes at 0 R n of those arcs representng γ, the mage of whch contans 0 R n. In the present work we determne and classfy all fully smple sngulartes of plane and space curves. We show that the set of such sngulartes s much smaller than the set of smple sngulartes and ther classfcaton s much more natural. In partcular, any fully smple sngularty s quas-homogeneous and for the fully smple sngulartes of plane curves there s a natural bjecton wth the classcal A, D, E 6,E 7,E 8 smple sngulartes of functons. The reason why a smple sngularty mght be not fully smple s the adjacency of a smple sngularty descrbed by a multgerm wth d components to a sngularty class consstng of non-smple multgerms wth more than d components. By defnton, a sngularty class of multgerms s any set of multgerms whch s closed wth respect to the RL-equvalence, that s the unon of a fnte or nfnte number of sngulartes. (A sngle sngularty s also a sngularty class.)

3 FULLY SIMPLE SINGULARITIES Page 3 of 21 Defnton. Let Q be a class of a multgerms of curves n R n and let γ be a fxed multgerm represented by an arc F :[a, b] R n. The multgerm γ (and the sngularty defned by γ) adjons the class Q (notaton Q γ) f for any k< then there exsts a sequence of arcs F :[a, b] R n tendng to the arc F n the C k -topology, wth mages that contan 0 R n,and wth sngulartes at 0 R n defned by multgerms of the class Q. A sngularty class Q adjons a sngularty class Q (notaton: Q Q) f any sngularty n Q adjons Q. It s easy to check that the defnton s correct, that s the choce of the arc Γ representng γ s rrelevant. It s also easy to check that f a sngularty adjons a class contanng no fully smple multgerms (n partcular, no smple multgerms) then ths sngularty s not fully smple Examples of smple but not fully smple sngulartes Let us gve few obvous examples of smple sngulartes wth d 1 components whch adjon certan sngularty classes consstng of multgerms wth d 1 >d components and contanng no smple multgerms. Such smple sngulartes are not fully smple. We need the followng defnton. Defnton. The multplcty of a curve germ γ :(R, 0) (R n, 0) s the mnmal p such that j p γ 0 (the multplcty s f γ has zero Taylor seres). The multplcty of a multgerm (γ 1,...,γ d ) s the sum of the multplctes of ts components. For example, the multplcty of the plane curve multgerm wth two components (t 2 1,t 2k+1 1 ), (t 4 2,t 3 2) s equal to = 5. Proposton 1.1. There are no fully smple multgerms of plane curves of multplcty 4 or more. There are no fully smple multgerms of space curves of multplcty 5 or more. On the other hand there s a bg number of smple plane curve sngulartes of multplcty 4 and smple space curve sngulartes of multplcty 5, see [3 6]. Some examples: the plane curve sngulartes wth one component (t 4,t 5 ± t 7 ), (t 4,t 5 ), (t 4,t 7 ± t 9 ); the plane curve sngulartes wth two components ( (t 1, 0), (t 4 2,t 3 2) ), ( (t 2 1,t 2k+1 1 ), (t 2s+1 2,t 2 2) ). To prove Proposton 1.1 ntroduce the followng notaton. By (I,I,...,I) R n (or smply (I,I,...,I)) wth I repeated r tmes we denote the class of multgerms of curves n R n consstng of r mmersed components. Proposton 1.1 s a drect corollary of the followng two clams. Proposton 1.2. Any multgerm of multplcty at least r adjons the class (I,I,...,I) wth I repeated r tmes. Proposton 1.3. The classes (I,I,I,I)) R 2 and (I,I,I,I,I)) R 3 contan no smple multgerms. Proof of Proposton 1.2. It s easy to prove that f a multgerm γ adjons a class Q and a multgerm γ adjons a class Q then the multgerm (γ, γ) wth d + d components adjons the class (Q, Q) ={(ψ, ψ) :ψ Q, ψ Q}. Ths statement reduces Proposton 1.2 to the case of one component. It suffces to consder a curve germ of multplcty r,thats,agermγ :(R, 0) (R n, 0) of the form γ(t) =t r f(t), where f :(R, 0) R n s a map germ such that f(0) 0.Let F (t) be a non-vanshng functon defned on the nterval t [ 1, 1] wth the germ f(t) atthe

4 Page 4 of 21 M. ZHITOMIRSKII pont t = 0. Consder the arc Γ ɛ =(t ɛ 1 )... (t ɛ r )F (t),t [ 1, 1]. The arc Γ 0 represents the germ γ. Ifɛ 1,...,ɛ r ( 1, 1) are dstnct numbers then the sngularty at 0 R n of the arc Γ ɛ conssts of r mmersed curves. Proposton 1.3 s a well-known statement. In the RL-classfcaton of multgerms of the class (I,I,I,I) R 2 (respectvely (I,I,I,I,I) R 3) a modulus, that s, parameter dstngushng close non-equvalent multgerms occurs already n the classfcaton of 1-jets t s the crossrato nvarant n the classfcaton of tuples consstng of four (respectvely fve) 1-dmensonal subspaces of R 2 (respectvely R 3 ) wth respect to the group of lnear transformatons. Let us gve one more example of smple but not fully smple germ γ :(R, 0) (R 2, 0). Introduce the followng class of multgerms. (I I... I) R n s the subclass of (I,I,...,I) R n consstng of multgerms such that the mages of all components have the same tangent lne at 0. Proposton 1.4. The class (I I I) R 2 contans no smple multgerms. Ths statement s also well known: there s a modulus n the RL-classfcaton of the 2-jets of multgerms of ths class. A generc 2-jet can be descrbed by the normal form (t 1, 0), (t 2,t 2 2), (t 3,at 2 3),a {0, 1}, and t s easy to prove that the parameter a s a modulus (t s a modulus n the classfcaton of a tuple consstng of three parabolas whch are tangent at 0 wth respect to the group of 2-jets of local dffeomorphsms; the same modulus as n the classfcaton of the sngularty class J 10 of functon germs, see [2]). Consder now a plane curve sngularty defned by a germ of the form Consder the deformaton γ : x(t) =t 3,y(t) =t 7 f(t). (1.1) γ ɛ : x =(t ɛ 1 )(t ɛ 2 )(t ɛ 3 ), y = ( (t ɛ 1 )(t ɛ 2 )(t ɛ 3 ) ) 2 t f(t). If ɛ 1,ɛ 2,ɛ 3 are dstnct numbers then the sngularty at 0 R 2 of the curve γ ɛ conssts of three mmersed curves tangent to the x-axes. Therefore the germ γ adjons the class (I I I) R 2 and consequently t s not fully smple. On the other hand, f f(0) 0 then γ s smple, see [3] Fully smple sngulartes wth one component Our results for the case of one component are as follows: all fully smple plane curve sngulartes wth one component are exhausted by the sngulartes A 2k, (3, 4), (3, 5) and all fully smple space curve sngulartes wth one component are exhausted by the sngulartes A 2k, (3, 4, 5), (3, 4, 0), (3, 5, 7), (3, 5, 0), (3, 7, 8), (1.2) (4, 5, 6), (4, 5, 7), (4, 6, 7). (1.3) Here we use the followng usual notaton. A 2k denotes the sngularty defned by the germ (t 2,t 2k+1 ) n the case of plane curves and (t 2,t 2k+1, 0) n the case of space curves; (q, p) denotes the plane curve sngularty defned by the germ (t q,t p ); by (q, p, r) and (q, p, 0) we denote the space curve sngularty defned by the germ (t q,t p,t r ) and (t q,t p, 0), respectvely.

5 FULLY SIMPLE SINGULARITIES Page 5 of Fencng sngularty classes A tuple of sngularty classes wll be called fencng (for fully smple sngulartes of curves n R n ) f any of these classes contans no fully smple multgerms and any sngularty whch s not fully smple adjons one of these classes, wth a possble excepton of nfntely-degenerate sngulartes (a certan class of sngulartes of nfnte codmenson). It turns out that for fully smple sngulartes of plane and space curves there s a tuple of fencng classes wth multgerms that consst of mmersed curves only. To present these classes we use the followng notaton. By (I I I) # R we denote the subclass of the class (I I I) 3 R 3 consstng of multgerms of space curves wth planar 2-jet. Here we use the followng defnton. Defnton. A multgerm γ of a space curve s called planar f ts mage (that s, the unon of the mages of the components) s contaned n a non-sngular surface. The r-jet of γ s called planar f there exsts a planar multgerm γ such that j r γ = j r γ. By (I,I,I,I) # R we denote the subclass of the class (I,I,I,I) 3 R 3 consstng of multgerms of space curves such that the tangent lnes at 0 R 3 to the mages of the four components do not span T 0 R 3. Proposton 1.5. The classes (I I I) # R 3 and (I,I,I,I) # R 3 contan no smple multgerms. Proof. The defntons of these classes and Propostons 1.3 and 1.4 mply that there s a modulus n the RL-classfcaton of the 1-jets of multgerms of the class (I,I,I,I) # R and of the 3 2-jets of multgerms of the class (I I I) # R. 3 Defne also the followng two classes of nfnte codmenson. By (A, ) we denote the class of multgerms of curves n R n wth any number 1 or more of components such that the Taylor seres of one of the components s RL-equvalent to (t 2, 0,...,0); By ((I,I), ) we denote the class of multgerms of curves n R n wth any number 2 or more of components such that two of them, say γ 1 and γ 2, are mmersed curves wth tangency of nfnte order, that s, n sutable coordnates the mage of γ 1 s the x 1 -axes and γ 2 : x = a (t) where the functon germs a 2 (t),...,a n (t) have zero Taylor seres. Theorem A. A sngularty of a parameterzed curve n R n, n =2, 3 s fully smple f and only f t does not belong to the classes (A, ), ((I,I), ) and does not adjon any of the followng classes: n =2: (I I I) R 2, (I,I,I,I) R 2; (1.4) n =3: (I I I) # R, (I,I,I,I) # 3 R, (I,I,I,I,I) 3 R 3. (1.5) By Theorem A the followng conjecture holds for n =2, 3. Conjecture A1. Denote by F n the class of non-smple multgerms of curves n R n wth at most (n + 2) mmersed components. A sngularty of a curve n R n s fully smple f and only f t does not belong to the classes (A, ), ((I,I), ) and does not adjon the class F n.

6 Page 6 of 21 M. ZHITOMIRSKII 1.5. Full smplcty and quas-homogenety The quas-homogenety of a sngularty of a curve n R n s the followng property: a multgerm defnng ths sngularty s RL-equvalent to a multgerm wth components of the form x = a t rλ, =1,...,n, where λ 1,...,λ n > 0 are postve numbers (weghts) whch are the same for all components, whereas the coeffcents a 1,...,a n R mght vary from a component to a component. A number of smple sngulartes of plane and space curves, ncludng sngulartes wth one components (that s smple germs) are not quas-homogeneous, see [3] and [4]. The smplest examples are the plane curve germs (t 3,t 7 + t 8 ) and (t 4,t 5 + t 7 ) and the space curve germs (t 3,t 7 + t 8,t 11 ) and (t 4,t 5 + t 7,t 11 ). On the other hand, the classfcaton results of the present work mply the followng. Theorem B. Any fully smple sngularty of a parameterzed plane or space curve, wth any number of components, s quas-homogeneous. Conjecture B1. any n. Theorem B holds for sngulartes of parameterzed curves n R n for 1.6. Plan for the paper In Secton 2 we present results on determnaton and classfcaton of fully smple sngulartes of plane curves (Theorem C) and establsh the bjecton between such sngulartes and the classcal A, D, E 6,E 7,E 8 smple sngulartes of functons. We also compare the lst of fully smple sngulartes of plane curves wth a much more nvolved lst of smple sngulartes. (In ths purpose we explan n canoncal terms some of the normal forms obtaned n [5, 6].) In Secton 3 we determne and classfy all fully smple space curves sngulartes (Theorem D). The proof of Theorems C and D and smultaneously the proof of Theorem A s contaned n Sectons 4 7. Theorem B s a drect corollary of the obtaned classfcaton results Analytc curves (C, 0) (C n, 0) The defnton of fully smple sngulartes can be extended, n a natural way, to such curves. All results reman the same up to the followng obvous changes: A ± 2+1 A 2+1, D ± 2+4 D 2+4, 0, (Subsecton 2.2), and consequently the second column of Table 1 contans all smple V -sngulartes (not only those whch have the property of zeros); c ± c and b ± b (Tables 3 and 4, Subsecton 3.3) Notatons for sngularty classes We wll use the followng notatons for certan sngularty classes, contnung the notatons n Subsecton 1.4. By (I,I) we denote the class of multgerms consstng of two mmersed components γ 1,γ 2 wth tangency of fnte order 0. Here and n what follows the order of tangency between mmersed components γ 1,γ 2 s the order of tangency between ther mages. It wll be denoted ord(γ 1,γ 2 ). Fx a local coordnate system x 1,...,x n such that the mage of γ 1 s the x 1 -axes. Let γ 2 : x = a (t). Then ord(γ 1,γ 2 )sthe mnmal such that at least one of the functon germs a 2 (t),...,a n (t) has non-zero ( + 1)-jet. The zero order of tangency means that the mages of γ 1 and γ 2 are not tangent. By (I,I,...,I),0 wth I repeated r 3 tmes we denote the class of multgerms consstng of r mmersed components such that, up to numeraton of the components, no two of the

7 FULLY SIMPLE SINGULARITIES Page 7 of 21 frst (r 1) components are tangent and one of them has tangency of order wth the last component. The case = 0 means that no two of the r components are tangent. Gven a class Q of sngulartes wth one component we denote by (I,Q) the class of multgerms (μ, ψ) consstng of an mmersed component μ and a sngular component ψ Q. Smlarly, by ((I,I),Q) we denote the class of multgerms (μ 1,μ 2,ψ) consstng of two mmersed components μ 1,μ 2 such that (μ 1,μ 2 ) (I,I) and a sngular component ψ Q. Gven two classes Q 1,Q 2 of sngulartes wth one component we denote by (Q 1,Q 2 )the class of multgerms (ψ 1,ψ 2 ) such that ψ 1 Q 1,ψ 2 Q Fully smple sngulartes of plane curves 2.1. Determnaton and classfcaton of fully smple sngulartes Recall the defnton of the tangent lne to a curve germ γ :(R, 0) (R n, 0) wth non-zero Taylor seres. It s the lmt as t 0 of the 1-dmensonal subspaces span( γ(t)) T γ(t) R n.let j p 1 γ =0andj p γ 0. Then γ(t) =t p 1 γ(t), where γ s an mmersed curve germ, and the tangent lne to γ s the tangent lne at 0 to the mage of γ (f γ s mmersed then p =1and γ = γ). Notaton. The tangent lne to a curve germ γ :(R, 0) (R n, 0) wth non-zero Taylor seres wll be denoted l(γ). Gven a class Q of plane curve sngulartes wth one component, we denote by I Q and I Q the subclass of the class (I,Q) consstng of multgerms wth an mmersed component μ and a sngular component ψ Q such that l(μ) l(ψ)andl(μ) =l(ψ), respectvely. Theorem C. A plane curve sngularty s fully smple f and only f t belongs to one of the classes n the frst column of Table 1. Each of these classes s a sngularty, that s all ts multgerms are RL-equvalent (to the normal form gven n the frst column) Fully smple plane curve sngulartes and smple sngulartes of functons Table 1 mples that there s a natural bjecton between fully smple plane curve sngulartes and the classcal smple V -sngulartes of functons of two varables havng the property of zeros. Recall from [2] thatthev-equvalence of functon germs f,g :(R 2, 0) (R, 0) means that g can be obtaned from f by a local dffeomorphsm (change of coordnates) and multplcaton by a non-vanshng functon. The V -equvalence class of a functon germ s called the V -sngularty. A V -sngularty defned by a functon germ f s called smple f there exsts k< such that the V -sngulartes defned by functon germs wth the k-jet suffcently close to the k-jet of f are exhausted by a fnte number of V -sngulartes. It s classcally known (see [2]) that the smple V -sngulartes are exhausted by the seres A ± 2k 1 : y2 x 2k,A 2k : y 2 x 2k+1,D ± 2k+2 : xy 2 ± x 2k+1,D 2k+3 : xy 2 x 2k+2, (k 1) and E 6 : y 3 x 4,E 7 : y 3 x 3 y, E 8 : y 3 x 5. We wll say that a V -sngularty defned by a functon germ f has the property of zeros f any functon germ vanshng on the set {f =0} belongs to the deal generated by f. It s clear that the sngulartes A + 2k 1 and D+ 2k+2 do not have the property of zeros, and t s easy to prove that all other smple V -sngulartes have ths property. Thus the second column of Table 1 contans all smple V -sngulartes havng the property of zeros. The correspondence between the sngulartes n the frst and the second column of Table 1 s as follows. Gven a multgerm γ, we assocate to t the deal I γ consstng of functon germs vanshng on the mage of each of the components of γ. Ifγ does not belong to a certan class of

8 Page 8 of 21 M. ZHITOMIRSKII nfnte codmenson (contanng no smple multgerms) then the deal I γ s 1-generated. Let f γ be one of the generators. The V -sngularty defned by f γ s nvarantly related to the sngularty of a parameterzed plane curve defned by the multgerm γ: a multgerm γ s RL-equvalent to γ f and only f the functon germ f γ s V -equvalent to f γ. We wll say that the V -sngularty defned by f γ s the functonal realzaton of the plane curve sngularty defned by γ. It s easy to check that every sngularty n the second column of Table 1 s the functonal realzaton of the sngularty n the frst column and the same row. Therefore Theorem C mples the followng corollary. Table 1. Fully smple plane curve sngulartes and smple V -sngulartes of functons. Indexes: k 1, 0. Fully smple sngulartes of plane curves Smple V-sngulartes of functons A 2k :(t 2,t 2k+1 ) A 2k : y 2 x 2k+1 (3, 4) : (t 3,t 4 ) E 6 : y 3 x 4 (3, 5) : (t 3,t 5 ) E 8 : y 3 x 5 (I,I) :(t 1, 0), (t 2,t +1 2 ) A 2+1 : y2 x 2+2 I A 2k :(t 2 1,t2k+1 1 ), (0,t 2 ) D 2k+3 : xy 2 x 2k+2 I A 2 :(t 2 1,t3 1 ), (t 2, 0) E 7 : y 3 x 3 y (I,I,I) 0, :(t 1, 0), (0,t 2 ), (t +1 3,t 3 ) D 2+4 : xy2 x 2+3 Theorem C1. A sngularty of a parameterzed plane curve s fully smple f and only f ts functonal realzaton s smple. Is t possble to prove Theorem C1 wthout usng Theorem C? It s not hard to prove that the adjacency of two plane curve sngulartes mples the adjacency of ther functonal realzatons, that s, the part f n Theorem C1. On the other hand, the only f part of Theorem C1 s non-trval and maybe surprsng because the adjacency of the functonal realzatons of two plane curve sngulartes does not mply the adjacency of these sngulartes. Trval examples can be found already wthn Table 1: A 2k (I,I) k (3, 4) I A 2 A 2k A 2k+1 E 6 E Fully smple versus smple plane curve sngulartes Accordng to results n [3] for the case of one component and [5, 6] for the case of 2 or more components components, a plane curve sngularty s smple f and only f t belongs to one of the classes n Table 2 below. Table 2. Smple and fully smple plane curve sngulartes. Indexes: k, s 1, 0 Class of smple sngulartes Fully smple? Class of smple sngulartes Fully smple? A 2k Yes I A 2k, k 1 Only f k =1 E 6k, E 6k+2 Only f k =1 I E 6k, I E 6k+2 No (t 4,t 6 + t 7+2 ) No I (3, 4), I (3, 5) No (t 4,t 5 ± t 7 ), (t 4,t 5 ), (t 4,t 7 ± t 9 ) No A 2k A 2s No (I,I) Yes (I,I,I),0 Yes I A 2k Yes ((I,I) 0,A 2k ) No

9 FULLY SIMPLE SINGULARITIES Page 9 of 21 In ths table the usual notaton E 6k and E 6k+2 are used for the class of germs whch are RL-equvalent to germs of the form (t 3,t 3k+1 + hgher order terms) and (t 3,t 3k+2 + hgher order terms), respectvely. Note that E 6 =(3, 4) and E 8 =(3, 5). The class A 2k A 2s conssts of multgerms wth two sngular components ψ 1 A 2k,ψ 2 A 2s such that l(ψ 1 ) l(ψ 2 ). The table shows whch of the smple sngulartes are fully smple. Each of the classes n Table 2 conssts of a fnte number of sngulartes. For example, the class I A 2k conssts of (2k 1) sngulartes (t 2 1,t1 2k 1 ), (t 2,t j+1 2 ),j =1,...,2k Fully smple sngulartes of space curves To formulate the results on determnaton and classfcaton of such sngulartes we need, except the tangent lne l(γ) to an mmersed or sngular germ γ :(R, 0) (R 3, 0), (see the begnnng of Secton 2) the defnton of the tangent plane to a multgerm of a space curve (Subsecton 3.1) and the defnton of the order of tangency between a non-sngular curve germ μ :(R, 0) (R 3, 0) and a space cusp ψ A 2k (Subsecton 3.2). A theorem on determnaton and classfcaton of fully smple space curve sngulartes s formulated n Subsecton 4.3. Though the lst of such sngulartes s much bgger than that of fully smple plane curve sngulartes, t s far from beng as nvolved as the classfcaton of smple space curve sngulartes obtaned n [5, 6] Tangent plane to a space curve multgerm One of the equvalent defntons s as follows. At frst we defne the order of tangency between any space curve multgerm γ wth d components γ : x = a (t),y = b (t),z = c (t) and a germ of a non-sngular surface S R 3.LetS = {H(x, y, z) =0}. Consder the functons R (t) =H(a (t),b (t),c (t)). Defnton and Notaton. The order of tangency between γ and S s the mnmal nteger r such that at least one of the functons R (t),=1,...,d has non-zero (r + 1)-jet. The order of tangency wll be denoted ord(γ,s). If the mage of γ s contaned n S then ord(γ,s) =. If γ conssts of one mmersed component transversal to S then ord(γ,s) = 0. Defnton and Notaton. The tangent plane L(γ) to a multgerm γ of a space curve s well defned f all non-sngular surfaces S R 3 for whch ord(γ,s) takes maxmal possble value have the same tangent plane at 0. In ths case L(γ) =T 0 S. Examples. We need two examples when the tangent plane s well defned. (1) It s well known that f γ conssts of one sngular component then the tangent plane L(γ) s well defned, provded that the Taylor seres of γ s not RL-equvalent to a seres of the form (at r, 0, 0), where r 2,a R. Under ths assumpton, excludng a class of nfnte codmenson, γ s RL-equvalent to a germ of the form (x, y, z) =(t q,t p, 0) + o(t p ), where 2 q<p and p 0modq. In the coordnates of ths normal form one has L(γ) =span( / x, / y). (In the same coordnates the tangent lne l(γ), whch s always a subspace of the tangent plane, s spanned by the vector / x). If γ s planar (for example, γ A 2k (3, 4, 0) (3, 5, 0)) then L(γ) s the tangent plane at 0 to some (and then any) non-sngular surface contanng the mage of γ. (2) If γ =(μ 1,μ 2 ) s a multgerm of the class (I,I) then L(γ) s the tangent plane at 0 to some (and then any) non-sngular surface contanng the mages of the mmersed curves μ 1 and

10 Page 10 of 21 M. ZHITOMIRSKII μ 2. Any multgerm of the class (I,I) s RL-equvalent to the multgerm (t 1, 0, 0), (t 2,t +1 2, 0); n these coordnates L(γ) =span( / x, / y), where x and y are the frst two coordnates The order of tangency between an mmersed curve and a cusp n R 3 Let μ be an mmersed space curve germ and let ψ A 2k. Defnton and Notaton. The order of tangency between μ and ψ s the number ord(μ, ψ) = mn ord(μ, S), where the mnmum s taken over all non-sngular surfaces S contanng the mage of ψ. Lemma 3.1. Let μ be an mmersed space curve germ and let ψ A 2k.Ifl(μ) L(ψ) then ord(μ, ψ) =0.Ifl(μ) L(ψ), but l(μ) l(ψ) then ord(μ, ψ) =1.Ifl(μ) =l(ψ) then 1 ord(μ, ψ) 2k and for generc couple (μ, ψ) wthn ths case one has ord(μ, ψ) =1. Proof. Take local coordnates n whch ψ : x = t 2,y = t 2k+1,z =0, μ : x = a(t),y = b(t),z = c(t). Any non-sngular surface S contanng the mage of ψ s descrbed by equaton z (y 2 x 2k+1 )g(x, y) = 0, wth an arbtrary functon g(x, y). Let R(t) =c(t) (b 2 (t) a 2k+1 (t)) g(a(t),b(t)). The lemma follows from the followng observatons. (1) The condton l(μ) L(ψ) means that c (0) 0. In ths case R (0) 0. (2) The condton l(μ) L(ψ),l(μ) l(ψ) means that c (0) = 0, b (0) 0. In ths case R (0) = 0 and f g(0, 0) s a generc number then R (0) 0. (3) The condton l(μ) =l(ψ) means that c (0) = b (0) = 0. If c (0) 0 then R (0) 0. Snce μ s mmersed then a (0) 0 and t s easy to see that f g(x, y) r wth a generc r R then j 2k+1 R Determnaton and classfcaton of fully smple sngulartes In Tables 3 and 4 below we denote mmersed components by μ, μ 1,μ 2,..., and sngular components by ψ, ψ 1,ψ 2,... Theorem D. Any fully smple space curve sngularty has not more than four components. A space curve sngularty wth one component s fully smple f and only f t s one of the sngulartes (1.2), (1.3). A space curve sngularty wth two components (respectvely three or four components) s fully smple f and only f t belongs to one of the classes gven n the frst column of Table 3(respectvely Table 4) and satsfes the restrctons gven n the second column. Any such sngularty s RL-equvalent to one and only one of the normal forms gven n the thrd column. The normal forms are dstngushed n coordnate-free terms n the part Cases of the second column of Tables 3 and 4. Here by coordnate-free terms we mean the mutual poston of the tangent lnes and the tangent planes to the components and/or the orders of tangency between the components. It s easy to see that the second column of Tables 3 and 4 contans all possble cases wthn the gven restrctons. For example, wthn the restrcton L(ψ 1 ) L(ψ 2 ) for multgerms of the class (A 2,A 2 ), see Table 3, the case l(ψ 1 ) L(ψ 2 ),l(ψ 2 ) L(ψ 1 ),l(ψ 1 ) l(ψ 2 ) s mpossble.

11 FULLY SIMPLE SINGULARITIES Page 11 of ProofofTheoremsC, D and A (outlne) Throughout the proof we wll use the followng notaton. Notaton. By W we denote the unon of the classes n Table 1. We wll use the same notaton W for the unon of space curve sngulartes (1.2) and (1.3), and the subclasses of the classes n Tables 3 and 4 consstng of multgerms satsfyng the restrctons gven n the second column of these tables. Table 3. Fully smple space curve sngulartes wth two components. A sngularty s fully smple f and only f Class Cases Sngulartes (I,I), 0 No restrctons ((t 1, 0, 0), (t 2,t +1 2, 0) (I,A 2 ) No restrctons a : l(μ) L(ψ) a :(t 2 1,t3 1, 0), (0, 0,t 2) b : l(μ) L(ψ); l(μ) l(ψ) b :(t 2 1,t3 1, 0), (0,t 2, 0) c : l(μ) =l(ψ); ord(μ, ψ) =1 c :(t 2 1,t3 1, 0), (t 2, 0,t 2 2 ) d : l(μ) =l(ψ); ord(μ, ψ) =2 d :(t 2 1,t3 1, 0), (t 2, 0, 0) (I,A 4 ) f l(μ) =l(ψ) then ord(μ, ψ) =1 a : l(μ) L(ψ) a :(t 2 1,t5 1, 0), (0, 0,t 2) b : l(μ) L(ψ); l(μ) l(ψ) b :(t 2 1,t5 1, 0), (0,t 2, 0) c : l(μ) =l(ψ); ord(μ, ψ) =1 c :(t 2 1,t5 1, 0), (t 2, 0,t 2 2 ) (I,A 2k ),k 3 l(μ) l(ψ) a : l(μ) L(ψ) b : l(μ) L(ψ) a :(t 2 1,t2k+1 1, 0), (0, 0,t 2 ) b :(t 2 1,t2k+1 1, 0), (0,t 2, 0) (I,(3, 4, 5)) No restrctons a : l(μ) L(ψ) a :(t 3 1,t4 1,t5 1 ), (0, 0,t 2) b : l(μ) L(ψ); l(μ) l(ψ) b :(t 3 1,t4 1,t5 1 ), (0,t 2, 0) c : l(μ) =l(ψ); c :(t 3 1,t4 1,t5 1 ), (t 2, 0, 0) (I,(3, 4, 0)) l(μ) L(ψ) (t 3 1,t4 1, 0), (0, 0,t 2) (I,(3, 5, 7)) (t 3 1,t5 1,t7 1 ), (0, 0,t 2) (I,(3, 5, 0)) (t 3 1,t5 1, 0), (0, 0,t 2) (A 2,A 2 ) L(ψ 1 ) L(ψ 2 ) a : l(ψ 1 ) L(ψ 2 ),l(ψ 2 ) L(ψ 1 ) a :(t 2 1,t3 1, 0), (0,t3 2,t2 2 ) b : l(ψ 1 ) L(ψ 2 ),l(ψ 2 ) L(ψ 1 ) b :(t 2 1,t3 1, 0), (t3 2, 0,t2 2 ) (up to numeraton of ψ 1,ψ 2 ) c : l(ψ 1 )=l(ψ 2 ) c ± :(t 2 1,t3 1, 0), (±t2 2, 0,t3 2 ) l(ψ 2 ) L(ψ 1 ) (A 2,A 2k ),k 2 (assumng ψ 1 A 2,ψ 2 A 2k ) a : l(ψ 1 ) L(ψ 2 ) b : l(ψ 1 ) L(ψ 2 ) a :(t 2 1,t3 1, 0), (0,t2k+12,t 2 2 ) b :(t 2 1,t3 1, 0), (t2k+12, 0,t 2 2 )

12 Page 12 of 21 M. ZHITOMIRSKII Usng ths notaton, Theorems C and D can be joned as follows. Theorem 4.1. () A plane or space curve sngularty s fully smple f and only f t belongs to the class W. () Any multgerm wth at least two components of a plane or space curve of the class W s RL-equvalent to the normal form gven n Tables 1, 3 and 4. Table 4. Fully smple space curve sngulartes wth 3 and 4 components. A sngularty s fully smple f and only f Class Cases Sngulartes μ 1 =(t 1, 0, 0) (I,I,I),0 No restrctons μ 2 =(t 2,t +1 2, 0) a : l(μ 3 ) L(μ 1,μ 2 ) a : μ 3 =(0, 0,t 3 ) b : l(μ 3 ) L(μ 1,μ 2 ) b : μ 3 =(0,t 3, 0) (assumng ord(μ 1,μ 2 )=) I I I the 2-jet of the multgerm μ 1 =(t 1, 0, 0) s not planar μ 2 =(t 2,t 2 2, 0) μ 3 =(t 3, 0,t 2 3 ) ψ =(t 2 1,t3 1, 0) ((I,I) 0,A 2 ) span(l(μ 1 ),l(μ 2 )) L(ψ) μ 1 =(0, 0,t 2 ), a : l(μ 1 ),l(μ 2 ) L(ψ) a : μ 2 =(0,t 3,t 3 ) b : l(μ 2 ) L(ψ),l(μ 2 ) l(ψ) b : μ 2 =(0,t 3, 0) c 1 : l(μ 2 )=l(ψ), ord(μ 2,ψ)=1 c 1 : μ 2 =(t 3, 0,t 2 3 ) c 2 : l(μ 2 )=l(ψ), ord(μ 2,ψ)=2 c 2 : μ 2 =(t 3, 0, 0) (uptonumeratonofμ 1,μ 2 ) ψ =(t 2 1,t2k+1 1, 0) ((I,I) 0,A 2k ),k 2 l(μ 1 ),l(μ 2 ),l(ψ) spant 0 R 3 μ 1 =(0, 0,t 2 ) a : l(μ 1 ),l(μ 2 ) L(ψ) a : μ 2 =(0,t 3,t 3 ) b : l(μ 1 ) L(ψ),l(μ 2 ) L(ψ) b : μ 2 =(0,t 3, 0) (uptonumeratonofμ 1,μ 2 ) ψ =(t 2 1,t3 1, 0), ((I,I),A 2 ), 1 l(μ 1 )=l(μ 2 ) L(ψ) μ 1 =(0, 0,t 2 ), a : l(ψ) L(μ 1,μ 2 ) a : μ 2 =(0,t +1 3,t 3 ) b : l(ψ) L(μ 1,μ 2 ) b ± : μ 2 =(±t +1 3, 0,t 3 ) (± +ff s even) μ 1 =(t 1, 0, 0), μ 2 =(0,t 2, 0), (I,I,I,I),0, 0 l(μ 1 ),l(μ 2 ),l(μ 3 ),l(μ 4 )spant 0 R 3 μ 3 =(0, 0,t 3 ), a : l(μ 2 ),l(μ 3 ) L(μ 1,μ 4 ) a : μ 4 =(t 4,t +1 4,t +1 4 ) b : l(μ 2 ) L(μ 1,μ 4 )or b : μ 4 =(t 4,t +1 4, 0) l(μ 3 ) L(μ 1,μ 4 ) (assumng that l(μ 1 ),l(μ 2 ),l(μ 3 ) span T 0 R 3 and ord(μ 1,μ 4 )=)

13 FULLY SIMPLE SINGULARITIES Page 13 of 21 The second statement of ths theorem can be proved by the standard normalzaton technques. It also follows from a straghtforward analyss of the lst of normal forms obtaned n [5, 6]. Let us show that Theorem 4.1, () follows from Propostons 4.2 and 4.3 below. In what follows by the fencng classes we mean the fencng sngularty classes defned n Subsecton 1.4: the classes (1.4) n the case of plane curves and the classes (1.5) n the case of space curves. Proposton 4.2. Any plane or space curve sngularty beyond the class W and the classes (A, ) and ((I,I), ) adjons one of the fencng classes. Proposton 4.3. of the fencng classes. Any plane or space curve sngularty of the class W does not adjon any Theorem 4.1() from Propostons 4.2 and 4.3. Snce, as we showed n Subsecton 1.4, the fencng classes contan no smple multgerms then Proposton 4.2 mples the only f part of Theorem 4.1(). The proof of the f part s as follows. Assume, to obtan a contradcton, that a multgerm γ W s not fully smple. Represent γ by an arc Γ defned on the segment [ 1, 1]. Then for any k there exsts a sequence of arcs Γ 1, Γ 2,... defned on [ 1, 1] and tendng to Γ n the C k topology such that the sngulartes at the orgn of the arcs Γ 1 and Γ 2 are dfferent for any 1 2. The class W conssts of a countable number of sngulartes, see Tables 1, 3 and 4. It s easy to check (usng the noton of the codmenson of sngulartes) that any fxed sngularty of the class W adjons not more than a fnte number of sngulartes n ths class. Therefore the sequence Γ can be chosen n such a way that the sngulartes at the orgn of Γ do not belong to the tuple W. By Proposton 4.2 these sngulartes adjon one of the fencng classes. Ths means that for any k and any fxed there exsts a sequence of arcs Γ,1, Γ,2,... defned on [ 1, 1], tendng to Γ n the C k -topology, and such that the sngulartes at the orgn of Γ,1, Γ,2,...belong to one of the fencng classes. Consder the sequence of arcs Γ 1,1, Γ 2,2,... It tends to the arc Γ n the C k topology and the sngularty at the orgn of Γ, belongs to one of the fencng classes. Therefore γ adjons one of the fencng classes whch contradcts to Proposton 4.3. Theorem A s a drect corollary of Theorem 4.1() and Propostons 4.2 and 4.3. Therefore to prove Theorems C, D and A t suffces to prove Propostons 4.2 and 4.3. Proposton 4.2 s proved n Secton 5. Proposton 4.3 s proved n Sectons 6 and Proof of Proposton 4.2 Throughout the proof we use the followng deformaton of t r : 5.1. The case of one component P r,ɛ (t) =(t ɛ 1 ) (t ɛ 2 )... (t ɛ r ). The classfcaton results n [3] mply that any plane curve germ γ :(R, 0) (R 2, 0) beyond the sngulartes A 2k, (3, 4), (3, 5) and the class A ether has multplcty 4 or more, or s RLequvalent to a germs of form (1.1). As we showed n Subsecton 1.2, n the frst case γ adjons the fencng class (I,I,I,I) R 2 and n the second case t adjons the fencng class (I I I) R 2. The classfcaton results n [4] mply that any space curve germ γ :(R, 0) (R 3, 0) beyond the sngulartes (1.2), (1.3) and the class A ether has multplcty 5 or more, (and then t adjons the fencng class (I,I,I,I,I) R 3, see Proposton 1.2) or s RL-equvalent to a germ of

14 Page 14 of 21 M. ZHITOMIRSKII one of the followng forms: x(t) =t 3, y(t) =t 7 b(t), z(t) =t 10 c(t); (5.1) x(t) =t 4, y(t) =t 5 b(t), z(t) =t 9 c(t). (5.2) Let us show that germ (5.1) adjons the fencng class (I I I) # R and germ (5.2) adjons the 3 fencng class (I,I,I,I) # R. To prove the adjacency (I I I) # 3 R (5.1) t suffces to consder 3 the deformaton x = P 3,ɛ (t), y = P 2 3,ɛ(t) t b(t), z = P 3 3,ɛ(t) t c(t). (5.3) In fact, f ɛ 1,ɛ 2,ɛ 3 are dstnct then the sngularty at 0 R 3 of the curve (5.3) conssts of three mmersed curves tangent to the x-axes and havng tangency of order 2 or more wth the plane z =0. The adjacency (I,I,I,I) # R (5.2) can be realzed by the deformaton 3 x = P 4,ɛ (t), y = P 4,ɛ (t) t b(t), z = P 2 4,ɛ(t) t c(t). (5.4) In fact, the sngularty at 0 R 3 of the curve (5.4) conssts of four mmersed curves tangent to the plane z = Plane curve sngulartes wth at least two components It s easy to check that any plane curve sngularty wth at least two components beyond the sngulartes n Table 1 and the classes (A, ) and((i,i), ) ether has multplcty 4 or more ((and then t adjons the fencng class (I,I,I,I) R 2 by Proposton 1.2), or belongs to the fencng class (I I I) R 2, or belongs to the class I A 2k 4. The latter class adjons the fencng class (I I I) R 2 by the followng lemma. Lemma 5.1. If k 2 then any plane curve germ ψ A 2k adjons the class of multgerms wth two mmersed components μ 1,μ 2 tangent to the lne l(ψ) and such that the order of tangency between μ 1,μ 2 s equal to (k 1). Proof. Take local coordnates n whch ψ has the form x = t 2 1,y = t 2k+1 1. Then l(ψ) = span( / x). The requred adjacency s realzed by the deformaton γ ɛ : x = P 2,ɛ (t),y = P k 2,ɛ(t) t. In fact, f ɛ 1 ɛ 2 then the sngularty at 0 R 2 of the curve γ ɛ conssts of two mmersed curves tangent to the x-axes; t s easy to see that the order of tangency s equal to (k 1) Space curve sngulartes wth at least two components The proof s based on the followng lemmas. Lemma 5.2. Any space curve germ ψ A 2k adjons the class of multgerms wth two mmersed components μ 1,μ 2 such that l(μ 1 ),l(μ 2 ) L(ψ).Ifk 2 then ψ adjons the subclass of ths class consstng of multgerms (μ 1,μ 2 ) such that μ 1 and μ 2 are tangent to the lne l(ψ) and ord(μ 1,μ 2 )=(k 1). Proof. Follows from Lemma 5.1 and the fact that any space curve germ ψ A 2k s planar: ts mage s contaned n a non-sngular surface tangent to the plane L(ψ) whch contans the lne l(ψ).

15 FULLY SIMPLE SINGULARITIES Page 15 of 21 Lemma 5.3. Any germ ψ :(R, 0) (R 3, 0) satsfyng the condton j 2 ψ =0, j 3 ψ 0, ψ (3, 4, 5) (3, 4, 0) (3, 5, 7) (3, 5, 0) (5.5) adjons the class of multgerms consstng of three components whose mages have the same tangent lne at 0 R 3. Proof. In sutable coordnates ψ has the form x = t 3, y = t 7 b(t), z = t 7 c(t), see [4]. The requred adjacency s realzed by the deformaton x = P 3,ɛ (t), y= P3,ɛ(t) 2 t b(t), z= P3,ɛ(t) 2 t c(t). In fact, f ɛ 1,ɛ 2,ɛ 3 are dstnct then the sngularty at 0 R 3 of ths curve conssts of three mmersed components tangent to the x-axes. Lemma 5.4. Any germ ψ (3, 4, 0) (3, 5, 7) (3, 5, 0) adjons the class of multgerms wth three mmersed components tangent to the plane L(ψ). Proof. In sutable coordnates ψ has the form x = t 3, y = t 4 b(t), z = t 7 c(t) andl(ψ) = span( / x, / y). The requred adjacency s realzed by the deformaton x = P 3,ɛ (t),y = P 3,ɛ (t) t b(t),z = P 2 3,ɛ(t) t c(t). In fact, f ɛ 1,ɛ 2,ɛ 3 are dstnct then the sngularty at 0 R 3 of ths curve conssts of three mmersed components tangent to the plane z =0. Let us show that these lemmas mply that any space curve sngularty wth at least two components beyond the class W adjons one of the fencng classes. We start wth space curve sngulartes wth at least two components whch do not belong to any of the classes gven n the frst column of Tables 3 and 4, nether to the classes (A, ) and((i,i), ). It s easy to check that any such sngularty γ ether belongs to one of the the fencng classes (I I I) # R, 3 (I,I,I,I) # R 3 or satsfes one of the followng condtons: (a) γ has multplcty 5 or more; (b) γ (A 2k,A 2s ), where k, s 2orγ ((I,I),A 2k ) where 1andk 2; (c) γ =(μ, ψ), where the sngular component ψ satsfes (5.5). In case (a) the sngularty adjons the fencng class (I,I,I,I,I) R 3 by Proposton 1.2. In case (b) Lemma 5.2 mples that γ adjons the fencng class (I,I,I,I) # R. The same holds n case 3 (c) by Lemma 5.3. Now we analyze space curve sngulartes γ wth at least two components whch belong to one of the classes n the frst column of Tables 3 and 4, but do not satsfy the restrcton gven n the second column of these tables. In the case of classes I I I and (I,I,I,I),0 such sngulartes belong to one of the fencng classes (I I I) # R, (I,I,I,I) # 3 R. In the case of 3 other classes n Tables 3 and 4, such that the restrctons n the second column are not empty, γ satsfes one of the followng condtons: (d) γ =(μ, ψ) (I,A 4 ), l(μ) =l(ψ), ord(μ, ψ) 2; (e) γ =(μ, ψ) (I,A 2k 6 ), l(μ) =l(ψ); (f) γ =(μ, ψ), ψ (3, 4, 0) (3, 5, 7) (3, 5, 0), l(μ) L(ψ); (g) γ =(ψ 1,ψ 2 ) (A 2,A 2 ), L(ψ 1 )=L(ψ 2 ); (h) γ =(ψ 1,ψ 2 ) (A 2,A 2k 4 ), l(ψ 2 ) L(ψ 1 ); () γ =(μ 1,μ 2,ψ) ((I,I),A 2 ), 0, l(μ 1 ),l(μ 2 ) L(ψ); (j) γ =(μ 1,μ 2,ψ) ((I,I),A 2k ), 0, k 2, the lnes l(μ 1 ),l(μ 2 ),l(ψ) do not span T 0 R 3. Lemma 5.2 mples that n the cases (g), (h), (), (j) the sngularty γ adjons the fencng class (I,I,I,I) # R and n the cases (d) and (e) t adjons the fencng class (I I I) # 3 R.Inthe 3 remanng case (f) the sngularty adjons the fencng class (I,I,I,I) # R by Lemma

16 Page 16 of 21 M. ZHITOMIRSKII 6. Proof of Proposton 4.3 For sngulartes wth mmersed components only Proposton 4.3 s trval. In what follows we consder sngulartes wth at least one sngular component. For such sngulartes, n ths secton we prove Proposton 4.3 modulo several lemmas whch are proved n the next secton. Lemma 6.1. A sngularty of a curve n R n of multplcty less than r does not adjon the class consstng of multgerms wth r mmersed components. It s easy to check that ths lemma mples the followng corollary. Corollary 6.2. Plane and space curve sngulartes of the class W do not adjon the fencng class (I,I,I,I) R 2 and (I,I,I,I,I) R 3, respectvely. A plane or space curve sngularty of the class A 2k does not adjon any of the fencng classes. The next lemma excludes certan adjacences of sngulartes consstng of mmersed curve(s) and a cusp. Lemma 6.3. Let n =2or n =3. Consder the arc F : x = t 2, y = t 2k+1 (n =2), F : x = t 2, y = t 2k+1, z =0 (n =3) defned on [ 1, 1]. Assume that a sequence of arcs F :[ 1, 1] R n tends to the arc F n the C 3 -topology and the sngularty of F at 0 R n conssts of two mmersed components μ (1),μ (2). Then the followng holds: (1) the sequences of tangent lnes l(μ (1) ), l(μ (2) ) tend to the lne span( / x); (2) f k =1then l(μ (1) ) l(μ (2) ) for suffcently large ; (3) f k =1and n =3then the sequence of planes spanned by l(μ (1) ) and l(μ (2) ) tends to the plane span( / x, / y). The frst two statements of Lemma 6.3 mply the absence of the followng adjacences. Corollary 6.4. (I I I) R 2. The plane curve sngulartes I A 2k and I A 2 do not adjon the class Corollares 6.2 and 6.4 and the obvous adjacency (3, 4) (3, 5) reduce Proposton 4.3 for plane curve sngulartes to the followng lemma. Lemma 6.5. The sngularty (3, 5) does not adjon the class (I I I) R 2. Now we consder space curve sngulartes. Lemmas 6.1 and 6.3 mply the followng statement. Corollary 6.6. Let γ be a space curve multgerm of one of the classes (I,A 2k ),k 2, (A 2,A 2k ), ((I,I) 0,A 2k ), ((I,I),A 2 ). If γ satsfes the restrctons n the second column of Tables 3 and 4 then t does not adjon any of the classes (I,I,I,I) # R 3, I I I.

17 FULLY SIMPLE SINGULARITIES Page 17 of 21 Corollares 6.2 and 6.6 and the obvous adjacences (3, 4, 5) (3, 4, 0) (3, 5, 7) (3, 5, 0) (3, 7, 8); (4, 5, 6) (4, 5, 7) (4, 6, 7) reduce Proposton 4.3 for space curve sngulartes to the followng statement: (*) none of the sngulartes (3, 7, 8), (4, 6, 7) and none of the sngulartes of the classes W (I,A 4 ),W (I,(3, 4, 5)),W (I,(3, 5, 0)) adjons any of the fencng classes (I I I) # R 3, (I,I,I,I) # R 3. Remnd that the ntersecton of W wth a class n Tables 3 and 4 s the subclass consstng of multgerms satsfyng the restrctons n the second column. By Table 3, each of the classes W (I,A 4 ), W (I,(3, 4, 5)) conssts of three sngulartes a, b, c and ther coordnate free defnton n the second column mples the adjacences a b c. The class W (I,(3, 5, 0)) conssts of a sngle sngularty. Therefore (*) reduces to the followng statement. (**) none of the followng space curve sngulartes adjons any of the fencng classes (I I I) # R 3, (I,I,I,I) # R 3 : (3, 7, 8); (4, 6, 7); c (I,A 4 ); c (I,(3, 4, 5)); W (I,(3, 5, 0)). (6.1) The sngularty W (I,(3, 5, 0)) conssts of multgerms (μ, ψ) (I,(3, 5, 0)) such that l(μ) L(ψ). Such multgerms do not adjon the class (I,I,I,I) # R by Lemma 6.5. They do not adjon 3 the class I I I and consequently the class (I I I) # R by the followng lemma. 3 Lemma 6.7. Let F :[ 1, 1] R 3 be a sequence of arcs tendng to the arc x = t 3,y = t 5,z =0 n the C r -topology wth suffcently large r and such that the sngularty of F at 0 R 3 conssts of two mmersed components μ (1),μ (2) tangent to the same lne l. Then l l(t 3,t 5, 0) = span( / x). Now, let us prove the adjacences (I,A 4 ) c (3, 7, 8) (4, 6, 7); (I,(3, 4, 5)) c (4, 6, 7). (6.2) The adjacency (3, 7, 8) (4, 6, 7) s realzed by the deformaton (ɛt 3 + t 4,t 6,t 7 ) (ths germ belongs to the class (3, 7, 8) f ɛ 0). The other two adjacences are realzed by the followng deformatons of the curves (t 3,t 7,t 8 ) and (t 4,t 6,t 7 ): x =(t ɛ 1 ) 2 (t ɛ 2 ), y =(t ɛ 1 ) 5 (t ɛ 2 ) 2, z =(t ɛ 1 ) 6 (t ɛ 2 ) 2 ; (6.3) x =(t ɛ 1 ) 3 (t ɛ 2 ), y =(t ɛ 1 ) 4 (t ɛ 2 ) 2, z =(t ɛ 1 ) 5 (t ɛ 2 ) 2. (6.4) In fact, the sngularty at 0 R 3 of the curve (6.3) (respectvely (6.4)) conssts of an mmersed component μ and a sngular component ψ A 4 (respectvely ψ (3, 4, 5)) such that l(μ) = l(ψ) sthex-axes. Adjacences (6.2) and Lemma 6.7 reduce the clam (**) to the followng statement. Lemma 6.8. The sngularty (4, 6, 7) does not adjon any of the fencng classes (I I I) # R, (I,I,I,I) # 3 R Proof of the lemmas n Secton Auxlary statements Throughout the proof the followng statements wll be used.

18 Page 18 of 21 M. ZHITOMIRSKII Proposton 7.1. Let P (t) =a 0 + a 1 t a k t k, where k 0 and a k 0.Letf (t),t [ 1, 1] be a sequence of functons tendng to P (t) n the C k -topology. Then for suffcently large the set f 1 (0) conssts of at most k ponts. Proposton 7.2. Let a (1),...,a (p) sequence of functons be sequences of ponts n [ 1, 1]. Letq, r p. Ifthe F (t) =(t a (1) )... (t a (p) ) G (t), t [ 1, 1] tends to the functon t q n the C r -topology then the sequences a (1),...,a (p) sequence G (t) tends to the functon t q p n the C r p topology. tend to 0 and the To prove Proposton 7.1 note that the sequence of kth dervatves f (k) (t) tends to a nonzero constant. Therefore for suffcently large one has f (k) (t) 0,t [ 1, 1]. Ths mples Proposton 7.1 by the classcal Rolle theorem. The frst statement of Proposton 7.2 s also obvous: f one of the sequence a (1),...,a (p) does not converge to 0 then t has a lmt pont a 0 whch contradcts to the condton F (t) t q. To prove the second statement t suffces to consder the case p =1.Thuswehave to prove the followng asserton. (*) f the sequence of ponts a tends to 0 and the sequence of functons of the form f (t) =(t a ) φ (t),t [ 1, 1] tends to 0 n the C r 1 -topology then the sequence of functons φ (t),t [ 1, 1], tends to 0 n the C r 1 -topology. Fx δ (0, 1) and a number m such that a < 1 δ for any >m. Consder the functons φ (t) =φ +m (t + a ) defned for t [ δ, δ]. Then the sequence of functons t φ (t),t [ δ, δ] tends to 0 n the C r -topology and t follows that the sequence φ (t) tends to 0 n the C r 1 - topology (the latter mplcaton s a known fact; we leave the proof to a reader). Therefore the sequence φ (t), t [ δ/2,δ/2] tends to 0 n the C r 1 -topology and (*) follows ProofofLemma6.1 Ths lemma s almost straghtforward corollary of Proposton ProofofLemma6.3 We wll consder the case n = 3 (the proof for n = 2 s the same). The arcs F have the form The lnes l ( μ (1) F : x(t) =R (t)a (t), y(t) =R (t)b (t), z(t) =R (t)c (t), (7.1) R (t) = ( t a (1) ) ( (2)) (1) t a, a a (2). (7.2) ) are spanned by the vectors ), l ( μ (2) v (j) = A(a (j) ) x + B(a(j) ) y + C(a(j) ), j =1, 2. (7.3) z By Proposton 7.2 the sequences a (1),a (2) tend to 0 and the sequences of functons A (t),b (t),c (t) tend to 1,t 2k 1, 0, respectvely, n the C 1 -topology. Therefore the sequences of vectors v (1) and v (2) Express now the vector v (2) tend to the vector / x. Ths mples the frst statement of the lemma. n the form v (2) = v (1) + v (a (2) a (1) ), (7.4) v = A (t,1 ) x + B (t,2 ) y + C (t,3 ) z, (7.5)

19 FULLY SIMPLE SINGULARITIES Page 19 of 21 where t,1,t,2,t,3 [a (1),a (2) ]. In the case k = 1 the sequences A (t) andc (t) tend to 0 and the sequence B (t) tends to 1 n the C0 -topology. Therefore v / y. Ths mples the second and the thrd statements of the lemma Proof of Lemma 6.5 Let F :[ 1, 1] R 2 be a sequence of arcs tendng to the arc x = t 3,y = t 5 n the C r - topology wth suffcently large r and such that the sngularty of F at 0 R 2 conssts of three mmersed components μ (1),μ (2),μ (3). We have to prove that for suffcently large ether l ( μ (1) ) ( (2)) ( (1)) ( (3)) l μ or l μ l μ. We wll prove ths statement wth r = 5. Assume, to get contradcton, that l(μ (1) )= l(μ (2) )=l(μ (3) )=l for suffcently large. Then we may assume that ths holds for all. Any sequence of 1-dmensonal subspaces of T 0 R 2 has a convergent subsequence. Therefore there s no loss of generalty to assume that l = l s a fxed 1-dmensonal subspace of T 0 R 2. Let l = span(a / x + b / y), (a, b) (0, 0). The arcs F have the form F : x =(t a (1) )(t a (2) )(t a (3) )A (t), y =(t a (1) )(t a (2) )(t a (3) )B (t), where a (1),a (2),a (3) are dstnct ponts n ( 1, 1). Introduce the sequence G (t) =b A (t) a B (t). The lnes l(μ (j) one has ) are spanned by the vectors A (a (j) ) / x + B (a (j) ) / y, j =1, 2, 3, therefore G (a (1) )=G (a (2) )=G (a (3) )=0. (7.6) The assumpton that F (t 3,t 5 )nthec 5 -topology mples by Proposton 7.2 that A 1, B t 2 n the C 2 -topology. Consequently the sequence G tends to b at 2 n the C 2 -topology. Proposton 7.1 and (7.6) wth a (1),a (2),a (3) mply that a = b = 0 and we get contradcton Proof of Lemma 6.7 We wll prove ths lemma wth r = 3. The arcs F have the form (7.1), (7.2) and the lnes l(μ (1) )andl(μ (2) ) are spanned by the vectors (7.3). Express the vector v (2) n the form (7.4), (7.5). By Proposton 7.2 the assumpton that F (t 3,t 5, 0) n the C 3 -topology mples that the sequences of ponts a (1),a (2) tend to 0 and the sequences of functons A,B,C tend to t, t 3, 0, respectvely, n the C 1 -topology. It follows that v (1),v (2) 0and v / x. Snce the vectors v (1) and v (2) are proportonal ths mples the lemma Proof of Lemma 6.8 We have to prove the followng statements for a sequence of arcs F :[ 1, 1] R 3 tendng to the arc x = t 4,y = t 6,z = t 7 n the C r -topology wth suffcently large r: () f the sngularty of F at 0 R 3 conssts of four mmersed components μ (1),...,μ (4) then for suffcently large the lnes l(μ (1) ),...,l(μ (4) ) span T 0 R 3 ; () f the sngularty of F at 0 R 3 conssts of three mmersed components μ (1),μ (2),μ (3) tangent to the same lne l then for suffcently large the 2-jet of the multgerm (μ (1),μ (2),μ (3) ) s not planar.