ON SINGULARITIES OF SMOOTH MAPS TO A SPACE WITH A FIXED CONE B. Z. Shapiro
Department of Mathematics, University of Stockholm, S-10691, Sweden,
[email protected] Abstract. We consider germs of smooth maps f : (Rl ; 0) ! (Rn; c); c 2 C where C is the standard nondegenerate cone of some signature and classify their singularities under the actions of two natural groups of di eomorphisms preserving C . Occuring singularities are subdivided into 3 classes: regular, semiregular and irregular. In the regular case the classi cation of singularities is reduced to the classi cation of the usual singularities of germs of functions. We present the list of simple semiregular singularities and also analyze some irregular singularities.
x0.
Preliminaries and results The singularities of maps f : (Rl ; 0) ! (Rn ; s); s 2 S to a target space with some xed
strati ed variety S were considered by several authors, see e.g. [AVG],[A1], [L],[Si]. In the case when S is a hyperplane they are called boundary singularities and were investigated in details in [A1]. In particular, it was shown that the simple boundary singularities correspond to the A-, B -, C -, D-, E -series and F4 in the classi cation of simple Lie algebras. A much more general class of actions (including all actions on source and target with a strati ed S used in this note) was studied by J. Damon [Da1-2]. He proved the existence of the versal unfoldings and nite determinacy of germs of maps in this situation. (Therefore, the normal forms we present can be achieved in formal, analytic and smooth categories.) In what follows we will always assume that the dimension of the source is less than the dimension of the target. Main Notation.
Let s 2 S be a point on a strati ed variety S and let AS denote the product of the group of local di eomorphisms (Rl ; 0) ! (Rl ; 0) of the source by the group of local di eomorphisms (Rn ; s) ! (Rn; s) of the target preserving S . By K we denote the group of contact transformations , i.e. elements of K are di eomorphisms of (Rl Rn ; 0) preserving a) projection on Rl (inducing di eomorphisms of Rl ) and b) the subspace (Rl 0). Thus two germs f1 and f2 : (Rl ; 0) ! (Rn ; 0) are K-equivalent if there exists a di eomorphism of the source and a germ M : (Rl ; 0) ! GL(Rn ) such that f1(x) = M (x)f2(g(x)). (Another standard notation for K-equivalence introduced by J. Martinet and used in [AVG] is V -equivalence.) Let KS denote the subgroup of K consisting of all di eomorphisms H 2 K such that H (Rl S ) Rl S . 1991 Mathematics Subject Classi cation. Primary 58C27, Secondary 14B10. Key words and phrases. Singularities of smooth maps, versal unfoldings, bifurcation diagrams. Typeset by AMS-TEX 1
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B. Z. SHAPIRO
Let Ol denote the ring of germs of analytic functions on (Rl ; 0) and ml Ol denote the minimal ideal of functions vanishing at the origin. The space of germs of maps (Rl ; 0) ! Rn is denoted by Ol(n) and the space of germs of maps (Rl ; 0) ! (Rn ; 0) sending the origin of the source to the origin of the target is denoted by Mnl . Obviously, Ol(n) = Ol Ol (n factors) and Mnl = ml ml . Some standard notions of the singularity theory. An arbitrary germ (x; ) = ( (x); ) : (Rl Rr ; 0 0) ! (Rn Rr ; s 0) such that (x; 0) = f (x) is called an unfolding of the map f : (Rl ; 0) ! (Rn ; s). The additional space of parameters Rr is called the base of the unfolding. If additionally, for all one has
(0; ) = s then such a is called an origin-preserving unfolding of f . Let us consider the action of a group G (G equals AS or KS ) on the space of germs f : (Rl ; 0) ! (Rn ; s). Two unfoldings of f with the same base 0 : (Rl Rr ; 0 0) ! (Rn Rr ; s 0) and 00 : (Rl Rr ; 0 0) ! (Rn Rr ; s 0) are called G-equivalent if the exists a germ of smooth map g : (Rr ; 0) ! (G; e) where e is the identical di eomorphism such that 0 (x; ) = g() 00 (x; ). Consider a smooth map : (Rr1 ; 0) ! (Rr2 ; 0). The deformation induced by from 2 : (Rl Rr2 ; 0 0) ! (Rn Rr2 ; s 0) is de ned as 2 = 2 : (Rl Rr1 ; 0 0) ! (Rn Rr1 ; s 0). A G-versal unfolding of a map f : (Rl ; 0) ! (Rn ; s) is an unfolding such that any other unfolding is G-equivalent to some unfolding induced from by an appropriate map of bases, see [AVG],p.142. If the dimension of the base of is the minimal possible then is called a miniversal unfolding. (We will always work with miniversal unfoldings and omit the pre x 'mini'.) A bifurcation diagram is a subset Bif of the base such that for any 2 Bif the corresponding map f 2 is not in general position w.r.t. S , i.e. either f is not an immersion or the image f (Rl ) is nontransversal to S . Two unfoldings of a germ f : (Rl ; 0) ! (Rn ; S ) are called equivalent if each of them can be induced from the other by an appropriate map of their bases. Bifurcation diagrams Bif1 1 and Bif2 2 of two unfoldings with the bases 1 and 2 resp. are called coinciding if there exists a di eomorphism of the pairs (1; Bif1 ) and (2; Bif2 ). Exactly the same de nitions as above work if we restrict our considerations to the class of origin-preserving unfoldings. Corresponding versal unfoldings will be called origin-preserving or restricted versal unfoldings. Recall that two germs of functions on the same number of variables are called equivalent if there exists a local di eomorphism sending one to the other. Two functions on a di erent number of variables are called stably equivalent if they become equivalent after addition of nondegenerate quadratic forms of extra variables. By the modality of a singularity of a germ f : (Rl ; 0) ! (Rn ; s) (under the action of a chosen group G) we call the minimal number m of parameters such that a suciently small neighborhood of the orbit of f can be covered by a nite number of m-parameter families of orbits. A singularity of a germ f : (Rl ; 0) ! (Rn ; C ) is called simple (under the action of the chosen group) if its modality is zero. A germ f : (Rl ; 0) ! (Rn ; C ) is called stable if for any germ f~ close to f there exists a point (x; y) close to (0; 0) such that f~ considered as f~ : (Rl ; x) ! (Rn ; y) is A-equivalent to f. The main definition. A germ f : (Rl ; 0) ! (Rn ; s); s 2 S is called regular if the following 2 conditions are sati ed a) f is an embedding; b) s belongs to one of the top-dimensional strata of S . (In this case S can be replaced by a germ of a smooth submanifold.) If only a) is satis ed the germ f is called semiregular and nally if a) is violated such an
MAPS IN A SPACE WITH A CONE
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f is called irregular.
0.1.
Regular singularities. Remark. If f is (semi)regular then its AS - and KS -versal unfoldings are equivalent (see
lemma 1.1) and in this case we just call either of them a versal unfolding. Analogously, its restricted AS - and KS -versal unfoldings are equivalent and we call either of them a restricted versal unfolding. Let S Rn be a germ of a smooth manifold of codimension k and s 2 S be some point. Let us x k functions h1 ; : : :; hk de ning S as a germ of complete intersection S : fhi = 0; i = 1; : : :; kg in some neighborhood of s. For any germ f : (Rl ; 0) ! (Rn ; s); s 2 S we call by the induced germ fS the germ h f : (Rl ; 0) ! Rn ; s), i.e. the pullback of the functions h1 ; : : :; hk by f (Rl ) in the neighborhood of s. Proposition A. A versal unfolding and the bifurcation diagram of a regular germ f : (Rl ; 0) ! (Rn ; S ) is equivalent to a K-versal unfolding and bifurcation diagram of its induced germ fS . Remark. As was pointed put by the referee Proposition A also holds for KS -unfoldings without the requirement that f is an embedding, see [Da 3] but is virtually always false if s is not a regular point of S . If S : fQ = 0g is a germ of hypersurface in Rn and f : (Rl ; 0) ! (Rn ; s) is a map then Qf : (Rl ; 0) ! (R; 0) will denote the germ of the induced function Q f = f Q. Corollary. If S : fQ = 0g is a germ of a hypersurface then a versal deformation of a germ of a regular map f : (Rl ; 0) ! (Rn ; s) is equivalent to a K-versal deformation of the induced fuction Qf : (Rl ; 0) ! (R; 0). Therefore, if S is a hypersurface then the classi cation of singularities of regular maps coincides with the K-classi cation of germs of functions. In particular, they have the same list of simple singularities. Corollary B. If C is a nondegenerate cone then for any xed k the list of bifurcation diagrams in versal unfoldings of regular maps f : (Rl ; 0) ! (Rn ; C ) occuring in generic kparameter families of maps stabilize as soon as n > k + l. This means that for any such map regular f1 to the space of dimension n1 > k + l there exists a regular map f2 to the space of dimension n2 k + l with the coinciding bifurcation diagram.
0.2.
Semiregular singularities. Proposition C. A versal unfolding of a semiregular germ f : (Rl ; 0) ! (Rn ; c); c 2 C
can be obtained by extending its reduced versal unfolding by a (n ? l)-dimensional space of parallel shifts of f in the directions transversal to the image f (Rl) at the origin of the target. Proposition D. Taking the induced functions one de nes a mapping from origin-preserving unfoldings of f to unfoldings of Qf in m2l with the following properties. Let (x; ) = ( (x); ) be some origin-preserving unfolding then a) If 1 (x) and 2 (x) lie in a single AC -orbit then Q1 and Q2 lie in a single K-orbit. b) any unfolding of Qf in m2l can be induced from an origin-preserving unfolding of f . In particular, the deformation of Qf induced from any reduced versal unfolding red (x; ) of a semiregular germ f : (Rl ; 0) ! (Rn ; C ) is equivalent to reduced K-versal unfolding of QfP . (Recall that in its turn any reduced K-versal unfolding of Qf is equivalent to Qf + i ei where ei 2 m2l are representatives of any basis of the quotient module @Qf f m2l =ml ( @Q @x1 ; : : :; @xl ; Q).) Corollary E. The modality of a semiregular germ is not less than the K-modality of the induced function Qf . Therefore, the necessary condition for a semiregular germ f : (Rl ; 0) !
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B. Z. SHAPIRO
(Rn ; C ) to be simple is that the induced Qf is stably equivalent to one of the germs of the K-simple singularities, i.e. belongs to one of the A-, D- or E -series. Remark F. A (n ? l)-dimensional family of parallel shifts of a semiregular f which is included in its versal unfolding in addition to its reduced versal unfolding (see Proposition C) induces the subdeformation of Qf which can be normalized as
Qf +
cr X i=1
i x i +
n?l X j =cr+1
2j ;
where xi ; i = 1; : : :; cr are coordinates in the kernel of the quadratic part of Qf . Therefore, bifurcation diagrams of a versal unfolding and of a reduced versal unfolding are given as the zero sets of the stably equivalent functions. Theorem G. A germ of a semiregular map f : (Rl ; 0) ! (Rn ; c); c 2 C ; l < n such that the quadratic part of Qf has corank 1 and Qf = xk + : : : ; 6= 0 along the kernel can be reduced by to the normal form fk : (x1; xk1 ; x2; : : : ; xl; 0; : : :; 0); where Q = (y1 y2 + PntheAy2C)-action i=3 i and C : fQ = 0g: Its reduced versal unfolding depends on k ? 1 parameters and can be chosen as red (x1; : : :; xl ; 1; : : :; k?1 ) : fy1 = x1; y2 = xk1 + 1 xk1?1 + + k?1 x1 ; y3 = x2; : : :; yl+1 = xl ; yl+2 = 0; : : :; yn = 0g: The bifurcation diagram of this unfolding has two irreducible components. The rst component consists of all sets such that the induced function Qf at the origin has a more complicated singularity than Morse singularity. The second component consists of all sets such that Qf has only singular points di erent from the origin. Thus the considered bifurcation diagram coincides with the bifurcation diagram of the singularity Bk?1, i.e. consists of all (k ? 1)-tuples (1 ; : : :; k?1) such that xk1?1 + 1xk1 + + k?1 as a polynomial in x1 has a multiple or zero root. Recall that Bk is a boundary singularity described for the rst time in [A1]. Corollary H. Under the same assumptions as in Theorem G, a versal unfolding of fk depends on k + n ? l ? 1 parameters and can be chosen in the form (x1; : : :; xl ; 1; : : :; k+n?l?1 ) : fy1 = x1; y2 = xk1 + 1xk1?1 + + k ; y3 = x2 ; : : :; yl+1 = xl ; yl+2 = k+1 ; : : :; yn = k+n?l?1 g: The bifurcation diagram of this unfolding consists of all (k +n?l?1)-tuples (1; : : :; k+n?l?1 ) such that the hypersurface xk1+1 + 1 xk1 + + k x1 x22 x2l 2k+1 2k+n?l?1 = 0 is singular. If l = k ? 1 then the bifurcation diagram coincides with that of the singularity Bk and if l = k ? 2 with that of Dk+1 . By a trivial (r; s)-extension of a germ f : (Rp; 0) ! (Rq ; 0) we mean the map f : (Rp Rr Rs; 0) ! (Rq Rr Rs; 0) which is equal to (f; id; 0). Theorem I. The normal forms of simple the trivial (r; s)-extensions of the following Pn f yare 2 normal forms. (Below Q = ( y y + ) for the A-series and Q = (y1 y2 + y3y4 + 1 2 j =3 j Pn y2) in the rest of the cases.) j =5 j 1) Qf = Ak ; f = (x1; xk1 ); see its reduced versal unfolding in theorem G; 2) Qf = Dk+1 ; ( forms are di erent only if k is odd)
MAPS IN A SPACE WITH A CONE
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the normal form: f = (x1; xk1?1; x2; x1 x2); its reduced versal unfolding is red (x1; x2; 1; : : :; k ) = fx1 ; xk1?1 + 1xk1?2 + + k?2 x1 ; x2; x1x2 + k?1 x1 + k x2 g; 3) Qf = E6; f = (x1 ; x21; x2; x32); its reduced versal unfolding is red (x1; x2; 1; : : :; 5) = fx1; x21 + 1 x1; x2; x32 + 2 x22 + 3x1 x2 + 4 x1 + 5x2 g; 4) Qf = E7 ; the normal form: f = (x1; x21; x2; x1x22), its reduced versal unfolding is red (x1 ; x2; 1; : : :; 6) = fx1; x21 + 1x1 ; x2; x1x22 + 2 x21 + 3x22 + 4 x1x2 + 5x1 + 6 x2g; 5) Qf = E8; f = (x1; x21; x2; x42), its reduced versal unfolding is red (x1 ; x2; 1; : : :; 7) = fx1 ; x21 +1 x1 ; x2; x42 +2 x32 +3 x22 +4 x2 +5 x1x22 +6 x1 x2 +7 x1g: The adjacency of simple semiregular singularities coincides with the usual adjacency of the A ? D ? E -series, see Fig.1, compare [A3]. Remark. In all the cases 1)-5) the number of parameters in a reduced versal unfolding of f equals ? 1, where is the Milnor number of Qf and the induced deformation of Qf P is equivalent to Qf + i ei , where ei 2 m2l are representatives of any basis of the quotient @Qf @Qf @Qf f 2 module m2l =ml ( @Q @x1 ; : : :; @xl ) = ml =ml ( @x1 ; : : :; @xl ; Qf ), compare [A3]. The last identity holds in all these cases since Qf is quasihomogeneous. The bifurcation diagram is reducible and consists of two components one of which is a cylinder over the usual cone in R3 in the cases 2-5 and is a smooth in the case 1. This component correponds to the case when Qf has at the origin a more complicated singularity than just Morse singularity. Remark. When l = n ? 1 the only type of singularities which occurs is Qf = Ak . Other simple singularities are realized only if n ? l is at least 2 while in the regular case (i.e. when f (0) is a smooth point of the cone) one can realize all the simple boundary singularities. Yet another di erence is that in the semiregular case not all the simple boundary singularities are realized due to the fact that Q is a nondegenerate quadratic form in the ambient space. A
1
A
2
A
3
D
4
A4
A5
A6
D5
D6
D
E
E
E8
6
7
A7
7
Fig.1. Adjacency of complex simple semiregular singularities. Proposition B0 . Consider the space Mnl of all germs of maps sending the origin to the
vertex of the cone. For any xed k the list of versal unfoldings and bifurcation diagrams of a semiregular germs f : (Rl ; 0) ! (Rn ; C ) occuring in generic k-parameter families of germs from Mnl stabilize as soon as n k + l, compare B.
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0.3.
B. Z. SHAPIRO
Irregular singularities.
y3
2
2
3
1 2
y=yy handle
tangent plane
vertex y
y
2
1
Fig. 2. The standard Whitney umbrella.
The standard Whitney umbrella is a hypersurface in R3 given by y3 = x1 x2; y2 = x2 ; y1 = x21 in the parametric form or by the equation y32 = y1y22. A Whitney umbrella is a hypersurface di eomorphic to the standard one. A Whitney umbrella de nes a cross in the tangent plane at its vertex consisting of the tangent line to its handle and the tangent space to a Whitney umbrella considered as the image of a map. (On Fig.2. these lines are y1- and y2 -axis resp.) The rst nontrivial irregular germ f : (R2 ; 0) ! (R3; C ) occurs when f (R2) is a standard Whitney umbrella and f (0) is a smooth point of the cone C . In this case we can substitute C by R2 . Moreover instead of normalizing a Whitney umbrella in a space with a hyperplane R2 one can normalize an imbedding of a smooth hypersurface in a space with the xed Whitney umbrella, see lemma 3.1. Theorem J. The above nontrivial irregular case leads to the following singularities. a) If C is transversal to the tangent plane at the vertex of the Whitney umbrella then the Whitney umbrella and a germ of C in the neighborhood of f (0) can be normalized as fy32 = y1y22; y1 = y2 g. b) There are 2 cases of codimension 1 when the tangent plane to C is given by either y1 = 0 or y2 = 0, see Fig.2. In both cases one gets families of singularities with the bifurcation diagrams coincides with that of the singularity Bk for some k 2. c) in the most complicated case y3 = 0 any map is equivalent to one of the following k : (x2 ; x1; xk2 ) which has a versal deformation k (x1; x2; 0; : : :; k ) = fy1 = x2; y2 =
MAPS IN A SPACE WITH A CONE
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x1 ; y3 = 0 x1 + xk2 + k xk2?1 + + 1 g and its bifurcation diagram consists of three irreducible components, namely, i) 1 = 0; ii) p(x2 ) = xk2 + k xk2?1 + + 1 has a multiple zero and iii) p(20 ) = 0: These components correspond to nontransversality of a smooth hypersurface to the vertex, the handle and to the smooth part of the Whitney umbrella resp. Recently D. Mond has informed the author that the same problem in the case when the smooth germ and the germ of Whitney umbrella meet transversally was studied by him in [Mo]. We consider the real case which is more complicated than the complex one. All the results hold in the complex case as well if one drops the signs in theorem I. Some preliminary results in this direction were obtained by the author in 1989. The original motivation to consider the special case of the cone came from the hyperbolic systems with variable coecients, compare [A2]. Sincere thanks are due to V. I. Arnold for his constant support and encouragement and to V. V. Goryunov for the assistance and explanation of the papers [Da1-2]. The considered problem is closely related to the more complicated results by V. V. Goryunov on simple projections, see [G1-2] although (to the best of the author's knowledge) the above results do not follow from [G1-2]. The author is very obliged to the referee for the constructive criticism which enabled to improve substantially the quality of the original version. The author is very greatful to IHES for the hospitality in January 94 which allowed him not only to accomplish the present article but also to enjoy the beauty of Paris.
x1.
Regular case 1.1. Lemma, see [Da2]. Let f : (Rl; 0) ! (Rn; S ) be a germ of a map then:n a) a AS -versal unfolding (x; ) of f is equivalent to f + i ei where ei 2 Ol are represen-
tatives of any basis of the quotient module
(1)
Oln =fOl (@f=@x1; : : :; @f=@xl) + f On (v1 f; : : :; vr f )g:
Here vi ; (i = 1; : : :r) is a basis of the module of vector elds tangent to S , i.e. preserving the ideal of S ; vi f is the restriction of the vector eld vi to the image f (Rl ). The denominator in the r.h.s. presents the so-called extended tangent space Te AS (f ) to the action of the group of AS -unfoldings on f calculated at the 0 values of parameters, see [Da2]. b) analogously, a KS -versal unfolding e (x; ) of f is equivalent to f + i ei where ei 2 Oln are representatives of any basis of the quotient module (2)
Oln =fOl (@f=@x1; : : :; @f=@xl; v1 f; : : : ; vr f )g;
Analogously, the denominator in the r.h.s. presents the extended tangent space Te KS (f ) to the action of the group of KS -unfoldings on f calculated at the 0 values of parameters. c) a reduced AS -versal unfolding red (x; ) of the map f is equivalent to f + i ei where ei 2 Mnl are representatives of any basis of the quotient module (3)
Mnl =fml (@f=@x1; : : :; @f=@xl) + f On (~v1 f; : : :; v~p f )g;
where v~1 ; : : :; v~p is a basis of the module of vector elds tangent to S and preserving the origin in the target;
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B. Z. SHAPIRO
The denominator in the r.h.s. presents the usual tangent space T AS (f ) to the AS -action on f . d) analogously, a reduced KS -versal unfolding ~ red (x; ) is equivalent to f + i ei where ei 2 Mnl are representatives of any basis of the quotient module (4)
Mnl =fml (@f=@x1; : : : ; @f=@xl) + Ol (~v1 f; : : :; v~p f )g:
The denominator in the r.h.s. presents the usual tangent space T KS (f ) to the KS -action on f . The proof of this statement is standard and analogous to the proofs of the corresponding statements about the R-, RL- and K-equivalences, see [AVG], p.122. 1.2.l Proof nof nProposition A. We start with the following reformulation. Let f : (R ; 0) ! (R ; R ?k ) be a germ of a regular map. (Recall that in the regular case any strati ed S can be substituted by Rn?k .) Denote by f a complement to the extended tangent space Te ARn?k (f ); denote by ~ f a complement to the extended tangent space Te KRn?k (f ) and, nally, denote by ~ f a complement to the extended tangent space Te K( f ), where is the projection of Rn along Rn?k on Rk and the composition f maps (Rl ; 0) to (Rk ; 0). These complements f , ~ f and ~ f are versal unfoldings of the germs f and f resp. relative to the corresponding group of di eomorphisms ARn?k , KRn?k and K, see [Da2]. Proposition A0 . Versal unfoldings f ; ~ f and ~ f are equivalent, i.e. each of them can be induced from any other, see [AVG], p.147 and above. Proof. The equivalence of f and ~ f follows directly from the formulas (1) ? (2) and the fact that f On is isomorphic to Ol since f is an embedding. Let us show the equivalence of ~ f and ~ f , i.e. that they can be induced from each other. Note that any map from ~ f is de ned by the inverse image of Rn?k in the space Rl w.r.t. the group of all di eomorphisms of Rl (under the assumption that the inverse image of Rn?k in Rl has positive codimension). Analogously, any map from ~ f is de ned by the inverse image of the origin in Rl w.r.t. the same group. Thus we must show that for any map from ~ f there exists a map from ~ f with the di eomorphic inverse image of the origin and, conversely, for any map from ~ f there exists a map from ~ f with the di eomorphic inverse image of Rn?k . The last statement is obvious in one direction. And conversely, let us for any h 2 ~ f construct a map g the inverse image of Rn?k of which coincides with the inverse image of the origin for h. Let f1 ; : : :; fn?k be coordinate functions of the map f which are 'forgotten' by the projection . Then, obviously, one can take the map g = ff1; : : : ; fn?k ; hg. Remark. As was pointed out by the referee a more general 'invariance of KV -equivalence under suspension' was proved in [Da3]. Corollary. Versal unfoldings f and ~ f of a regular germ f : (Rl ; 0) ! (Rn ; C ) are equivalent to a K-versal unfolding of the function Qf , where Q is the quadratic form de ning the cone C and Qf is the pullback of Q in the space Rl . Proof. Apply the previous proposition in the neigborhhood of a smooth (by the de nition of regularity) point on the cone. Remark. If f is the germ of a smooth curve with the order k of tangency to a germ of a smooth hypersurface then its KS -versal unfolding of the is equivalent to a K-versal unfolding of the map xk : (R; 0) ! (R; 0). (In this case K-versal and R-versal unfoldings are equivalent.)
MAPS IN A SPACE WITH A CONE
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1.3. Proof of Corollary B. Corollary B follows directly from the above corollary and the following count of parameters. If f is a regular germ f : (Rl ; 0) ! (Rn ; C ) occuring in a generic k-parameter family then the corank cr of the quadratic part of Qf satis es the inequality cr(cr ? 1)=2 k and thus p k) ] variables. Indeed the Qf is stably equivalent to a function on at most (k) = [ 1+ (1+8 2 codimension of the stratum of all quadratic forms on Rl of corank cr equals to cr(cr2?1) and by weak transversality a generic (k ? 1)-parameter family of quadratic parts of the zero levels of functions can intersect such a stratum only if k > cr(cr2?1) . The parametrized Morse lemma provides that such a germ Qpf is stably equivalent to a germ of function on cr variables. The assumption cr < (k) = 1+ 21+8k completes the proof. x2.
Semiregular singularities 2.1. Proof of Proposition C. The tangent space T AC of a semiregular germ f lies in the extended tangent space Te AC (f ).
Moreover, the latter is obtained from the former by adding the l-dimensional vector space of parallel shifts of coordinates in the source since all vector elds preserving C vanish at the vertex of the cone and the linear part of f is nondegenerate, compare denominators in the formulas (1)-(3). Thus the AC -versal unfolding of f di ers from its reduced AC -versal unfolding only by some subspace of parallel shifts of the target. The deformation induced by the l-dimensional space of those shifts of the target which preserve the tangent space to f (Rl ) at the origin is cancelled by the l-dimensional space of the parallel shifts of the source. Therefore, the (n ? l)-dimensional quotient space belongs to the versal unfolding. 2.2. Proof of proposition D. We start with the following simple statement. Consider the group Om;n of all linear transformations preserving some nondegenerate quadratic form Q in Rn the index of which (i.e. the number of negative squares) is equal to m. The group Om;n acts in the obvious way on the Grassmanian Gl;n . 2.2.1. Lemma. An orbit of Om;n-action on the Grassmanian Gl;n consists of all l-dimensional subspaces L with a given corank and index of the restriction of Q L. Moreover, any such subspace L Rn of dimension l with the corank cr and the indexP there always exists a P P cr m basis (y1; : : :; yn ) such that Q = i=1 y2i?1 y2i ? j=2+crcr+1 yj2 + nk=m+cr+1 yk2 where the subspace L is spanned by the coordinates y1; y3; : : :; y2cr?1; y2cr+1; y2cr+2; : : :; y2cr+ and ym+cr+1; : : :; ym+l? . Proof. Use the reduction of the degenerate quadratic form QL to its normal form. 2.2.2. Theorem. Let f : (Rl; 0) ! (Rn; 0) (0 is the vertex of C ) be a semiregular map such that the corank of the quadratic part of Qf equals cr. Then the map f can be reduced to the form (x1; 1 (x1; : : :; xl ); x2; 2 (x1; : : :; xl ); : : :; xcr ; cr (x1 : : : ; xl ); xcr+1; : : :; xl ; 0; : : :; 0), where 1 ; : : :; cr 2 m2l in an appropriate system of coordinates P (x1 ; : : :; xl) and (y1; : : :; yn ) in which the cone C is given by Q = y1 y2 + + y2cr?1y2cr + nj=2cr+1 yj2 = 0. Moreover, a reduced versal unfolding of f can be chosen as (x1 ; 1(x1; : : :; xl ; ); x2; 2 (x1; : : :xl ; ); : : :; xcr ; cr (x1; : : :; xl ; ); xcr+1; : : :; xl ; 0; : : :; 0), where 1 ; : : :; cr 2 m for all values of parameters . Proof of theorem 2.2.2 is based on Mather's homotopy method in its usual form, i.e. if ft ; t 2 [0; 1] is a family of maps such that @f@tt 2 T AC (ft ) for all t 2 [0; 1] then f0 is AC equivalent to f1 . In order to present the tangent spaces T AC (ft) more explicitly, using lemma 1.1 we need the following proposition.
10
B. Z. SHAPIRO
Proposition. A basis of the module of vector elds tangent to a homogeneous surface H with an isolated singularity at the origin can be chosen in the following way, see [L]. One of the generators is the standard linear (y1 @y@1 ; : : :; yn @y@n ) while the rest are all (2 2) @ eld @ determinants of the (2 n)-matrix P@y1 ;; :: :: :: ;; P@yn ; where P is the polynomial de ning y1 yn @P the hypersurface and Pyi = @yi . P Remark.Applying this statement to C given by Q = y1 y2 + + y2cr?1 y2cr +P n2cr+1 yj2 one gets the basis of vector elds tangent to C consisting of the Euler vector eld: ni=1 yi @=@yi and 3 groups of generators: a) y~j @=@yi ? y~i @=@yj ; i < j 2cr where ~i = i ? 1 if i is even and ~i = i + 1 if i is odd (the same rule applies to ~j and j ); b) 2yj @=@yi ? y~i @=@yj ; i 2cr; 2cr < j n; c) yj @=@yi yi @=@yj ; 2cr < i < j n. Proof. Applying lemma 2.2.1. to the image of the nondegenerate linear part of f one reduces it by a linear transformation to (x1; 0; x2; 0; : : :; xcr ; 0; xcr+1; xcr+2; : : :; xl ; 0; : : :; 0) and, therefore, reduces f itself to (x1 ; 1; x2; 2; : : :; xcr ; cr ; xcr+1; xcr+2; : : :; xl ; cr+1; : : : ; n?l ), where all j 2 m2l . Let us get rid of cr+1 ; : : :; n?l by a C -preserving di eomorphism. First we remove all terms in cr+1 ; : : :; n?l ) divisible by any xj where j cr. Namely, for any cr + 1 k n?l and j cr there exists a vector eld of the form (0; 0; : : :; k ; 0; : : :; 0; xj ; 0; : : :; 0) with k standing at the position 2j and xj standing at the position cr + k. According to [AVG] using the homotopy method with this vector eld we can remove all terms in k divisible by xj . In such a way we reduce f to the form (x1; 1; x2; 2; : : : ; xcr ; cr ; xcr+1; xcr+2; : : : ; xl; ~cr+1 ; : : : ; ~n?l ) where all ~k depend only on xcr+1 ; : : :; xl. Here we use Mather's homotopy method again. Denote by ft ; t 2 [0; 1] the family (x1; 1 ; x2; 2; : : :; xcr ; cr ; xcr+1; xcr+2; : : :; xl ; t ~cr+1 ; : : :; t ~n?l ). Then @f@tt = (0; : : :; 0; ~cr+1; : : :; ~n?l ). Denote by Ml?cr the Ol?cr submodule consisting of all maps of the form (0; : : :; 0; cr+1; : : :; n?l ), i 2 m2l?cr (xcr+1; : : :; xl ). Let us show using the above basis that Ml?cr belongs to T AC (ft) for all t and thus f0 is AC -equivalent to f1. By lemma 1.1 the tangent space to T AC (ft) is the Ol -module fml (@ft=@x1; : : :; @ft=@xl ) + Ol (~v1 ft ; : : :; v~p ft )g where v~i belongs to the above basis. As above among the vector elds of the basis there exist all vector elds of the form (0; 0; : : :; ~k ; 0; : : :; 0; xj ; 0; : : :; 0), cr + 1 k n ? l and cr + 1 j l where ~k stands at the position cr + j and xj stands at the position cr + k. This means that Ml?cr is contained in the tangent space to all ft and the homotopy method gives the necessary reduction. Using lemma 2.2.2 one gets Proposition D directly. Indeed, the reduced versal unfolding of a semiregular f consists of the semiregular maps and, therefore, their induced functions belong to m2l . Taking the induced functions one maps the whole orbit AC f onto the whole orbit KQf . Indeed, one can cover the whole RQf -orbit by changing coordinates in the source and one can multiply the induced function Qf by an arbitrary nonvanishing function by using the di eomorphisms of the target which preserve the cone C . Since the tangent space T KQf @Qf f to the orbit KQf at Qf is identi ed with ml f @Q @x1 ; : : :; @xn ; Qg. We now show that any unfolding Q = Qf + (x1; : : : ; xl; 1; : : : ; k ) in the class m2l can be covered by an appropriate origin-preserving unfolding = f P (x1; : : : ; xl) + (x1; : : :; xl; 1 ; : : :; P k ). By lemma 2.2.2 cr one has Q can be reduced to Q = i=1 xi i (x1 ; : : :; xl ; 1; : : :; k )+ lj=cr+1 x2j ; i 2 ml for all values of parameters 1 ; : : :; k . The di erence Q ? Qf = (x1 ; : : :; xl; 1 ; : : :; k ) can P be expanded as 1ijl xi xj i;j (x1; : : : ; xl; 1; : : : ; k ). To obtain (x1; : : : ; xl; 1; : : : ; k )
MAPS IN A SPACE WITH A CONE
11
we the original map f = (x1 ; 1; : : :; xcrP ; cr ; xcr+1; : : :; xl ; 0 : : :; 0) by adding P rstx deform to for all i cr . It is left to compensate ~ = i;j j i cr+1ij l xi xj i;j (x1 ; : : :; xl ; 1 ; : : :; k ) by deformingP(xcr+1; : : :; xl ; 0; : : :; 0). Since ~ is a small deformation of a nondegenerate quadratic form lcr+1 x2j then the parametrized Morse lemma provides the existence of the necessary deformation. Proof of Corollary E. It follows immediately from the fact that taking induced functions one maps the orbit AC f submersively onto KQf , proposition D and the fact that the K-modality of any function with a singularity at the origin equals the K-modality of in the space m2l . The last statement is proved along the same lines as the corresponding statement for the R-equivalence, see [Ga]. Proof of Remark F. By the proposition C and lemma 2.2.1 the family of parallel shifts can be written in the form (x1 ; 1 + 1 ; x2; 2 + 2 ; : : :; xcr ; cr + cr ; xcr+1; : : :; xl ; cr+1; : : :; n?l ); where cr is the corank of Qf . This directly gives the necessary answer. 2.3. Proof of theorem G. We start with the easiest case f : (R; 0) ! (Rn; C ), where f is the germ of a smooth curve passing through the origin tangent to some ruling of the cone C : fQ = y1y2 y32 yn2 = 0g ; By lemma 2.2.2 it suces to consider the case of plane curves images of which lie in the plane spanned by y1 and y2. Let us denote by fk (x) the parameterized curve (y1 = x; y2 = xk ; yi = 0; i 3) and show that any other germ of a smooth curve passing through the origin is di emorphic to one of those. The extended tangent space Te AC (fk ) of fk is presented as
80 x 1 0 x 1 0 0 1 0 0 1 9 0 1 1 > > > BB kxk?1 CC BB xk CC BB ?xk CC BB 0 CC BB 0 CC > > > > 0 0 BB 0. CC B B C B C B C C 0 C B 0 C > > > B < B C B C = . . . . . . . B B C B C B C B C C . . . C + Ox B . C ; B . C ; : : : ; B . C ; B . C ; : : : Te AC (fk ) = Ox B BB ... CC BB ... CC BB ... CC BB ?xk CC BB ?x CC > > > > B@ .. CA B > B C B C B C > C . . 0 0 > > @ @ A @ A @ A A . . . . . > > . . : ; 0
0
0
..
..
for all 3 i n. It is convenient to choose the reduced versal unfolding red (x; 1; : : :; k?1 ) : fy1 = x; y2 = xk + 1xk?1 + + k?1 x; y3 = 0; : : :; yn = 0g using (3)-(4). The inverse image of the cone in the extended space of parameters (x; 1; : : :; k?1 ) has the form xk+1 +1 xk + +k?1 x2 = 0: The bifurcation diagram in the space of parameters (1 ; : : :; k?1) is the hypersurface of the singularities of projection or, in other words, the set of all (k ? 1)-tuples (1 ; : : :; k?1) for which xk?1 + 1xk + + k?1 as a polynomial in the variable x has a multiple or zero root. Since for any h consisting of terms of degree > k the tangent space T AC (fk + h) contains all monomials of degree k it follows that the AC -orbit of fk contains all fk + h (probably after multiplying Q by ?1) according to [Da2]. Therefore, these cases do not require separate consideration. Let us now mention the necessary changes to adjust the above proof to the case of a semiregular map f : (Rl ; 0) ! (Rn ; C ) such that the quadratic part of the restriction Qf of Q = y1y2 y32 yn2 de ning C has a one-dimensional kernel. One can assume without loss of generality that the 1-jet of f is (x1 ; 0; x2; : : :; xl ; 0; : : :; 0). The quadratic part of Qf has the form x22 x2l by lemma 2.2.1. After analogous considerations of the 1-jets of the vector
12
B. Z. SHAPIRO
elds one gets that the only family of germs to be considered is (x1; xk1 ; x2; : : :; xl ; 0; : : :; 0). Now the consideration of the corresponding vector elds gives that its reduced versal unfolding can be chosen in the form (x1; xk1 + 1 xk1?1 + + k?1 x1; x2; : : :; xl ; 0; : : :; 0). The restriction of the cone C has the form x1 (xk1 + 1 xk1?1 + + k?1 x1 ) x22 x2l = 0, i.e. is stably equivalent to the same restriction as in the previous case. Proof of Corollary H. The AC -versal unfolding of the map (x1; xk1 ; x2; : : : ; xl ; 0; : : :; 0) can be chosen as (x1; xk1 + 1 xk1?1 + + k ; x2; : : :; xl ; k+1; : : :; n+k?l?1 ) by proposition C. The rest follows immediately. On Fig.3 one can see the monomials included in this versal unfolding of fk ; the arrow shows competing monomials, i.e. either of them but not both must be included in the versal unfolding. When l = n ? 1 then, rather obviously, the above versal unfolding and its bifurcation diagram are equivalent to that of the singularity Bk . Let us show that when l = n ? 2 the bifurcation diagram coincides with the bifurcation diagram for the singularity Dk+1 . This follows from the form of the standard versal unfolding of Dk+1 , see [AVG]. Namely, taking the standard versal unfolding of Dk+1 as (x1; x2; 1; : : :; k+1 = xk1 + x1 x22 + 1 xk1?1 + + k + k+1 x one gets that its bifurcation diagram is de ned by the system 8=0 > < @ 2 k?1 = x2 + kx1 + (k ? 1)1xk1?2 + + k?1 = 0 @x 1 > : @ = 2x1x2 + 2k+1 = 0: @x2 Using the expression x1 = ?xk2+1 from the last equation one gets that the rst two equation are equivalent to the condition that the polynomial xk1+1 + 1xk1 + + k x1 ? 2k+1 has a multiple zero. Unfortunately the direct relation between the above semiregular singularity and Dk+1 is still unclear. y
1
y
2
k-1 y
3
. .
y
k
Fig.3. Circled monomials are included in the versal unfolding of f
k
MAPS IN A SPACE WITH A CONE
2.4.
Proof of theorem I.
13
Using corollary E one gets that the corank of the induced function Qf of a simple semiregular f is at most 2 since all simple singularities of functions have the corank 2. The case of corank 1 is covered by theorem G. In order to classify the simple singularities of corank 2 it suces to consider only the case f : (R2; 0) ! (R4 ; C ) where C given by Q = y1y2 + y3 y4 and the 1-jet of f equal to (x1; 0; x2; 0) by lemmas 2.2.1 and 2.2.2. Any semiregular map satisfying the above conditions will be called an adjusted 2 ! 4-map. Consideration of simple adjusted 2 ! 4-maps splits into a series a lemmas. Since we are only interested in the cone C we allow to multiply its de ning form Q by ?1 while nding the normal forms of f . 2.4.1. Lemma. Any adjusted 2 ! 4-map can be reduced to one of the following two forms: i) (x1 ; xk1 ; x2; x2g(x1; x2)), k 2 and g 2 m2 ; ii) (x1; 0; x2; h(x1; x2)), h 2 m22 . Proof. Obviously, any adjusted 2 ! 4-map f reduces to (x1; 1(x1 ; x2); x2; 2 (x1; x2)) where 1 ; 2 2 m. The vector eld V = (0; x2; 0; ?x1) is tangent to the AC -orbit of any such f , see 2.3. Therefore one can remove either of all the terms of 1 which are divisible by x2 or of all the terms of 2 divisible by x1 . Namely, we apply the homotopy method in the form: if ft satis es @f@tt 2 T AC (ft ) for all 0 t 1 then f0 is AC -equivalent to f1 . Since v 2 T AC (ft ) all ft = f + t(0; hx2 ; 0; ?hx1) then f is equivalent to f + (x1 ; 1 + hx2 ; x2; 2 ? hx1 ). Thus, if either 1 itself is divisible by x2 or 2 is divisible by x1 one can remove of the corresponding function completely and obtain the case ii) up to renumbering of components. Let xk11 be the smallest power of x1 in 1 (x1; x2) and xl11 be the smallest power of x1 in 2 (x1; x2). Let us show that using the homotopy method and renumbering of components f can be reduced to (x1; xk1 ; x2; ~2 (x1; x2)), where k = min(k1; l1). After some straightforward transformations of the basis of the tangent vector elds one gets only two elds which preserve the rst and the third components and a ect powers of x1 of the second component, namely, (0; 1; 0; 2) and 2 (0; 2; 0; x1 @ @x2 ). Using them we can remove all powers of x1 of degree greater than min(k1 ; l1 ) and multiplying the rst component by a constant and dividing the second component by the same constant we get that the only power of x1 included in 1 is xk1 (possibly after multiplication of Q by ?1). Finally, we remove of all powers of x2 in 1 as described before. Now we arrive at the form (x1 ; xk1 ; x2; ~2(x1 ; x2)), where ~2 (x1; x2) is not divisible by x1 . Let us assume that xl1, l k is the smallest power of x1 in ~2 (x1; x2). In this situation considering the basis of tangent vector elds one gets the vector eld (0; 0; 0; (k +1)xk1 +x2 @@x ~12 ). We again apply the homotopy method using the last vector eld and remove all powers of x1 in ~2 and get that the fourth component is divisible by x2 . The statement is proved. 2.4.2. Proposition. In the case i) of lemma 2.4.1 an adjusted 2 ! 4-map is simple if and only if one of the following possibilities holds. If k 4 then in order to be simple g must have a nondegenerate linear part x1 + x2 ; 2 + 2 6= 0. If 6= 0 then g reduces either to g(x1; x2) = x1 which gives 1) Qf = Dk+2 ; k 3; f = (x1; xk1 ; x2; x1 x2). If k is even then the forms coincide and one can drop signs of the last component. If = 0 then g reduces to g(x1; x2) = x2 which after renumeration of components coincides with the case k = 2 below. If k = 2 and g has a nonvanishing linear part then g reduces to g(x1; x2) = x1 which leads to 2) Qf = D4 ; f = (x1 ; x21; x2; x1x2 ).
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If k = 2 and g is a simple singularity with vanishing linear part and nonvanishing quadratic part x22 + x1 x2 + x21; 2 + 2 + 2 6= 0 then one of the following options is possible. If 6= 0 then g reduces to x22 which gives 3) Qf = E6; f = (x1 ; x21; x2; x32). If = 0; 6= 0 then g reduces to x1 x2 which gives 4) Qf = E7; f = (x1; x21; x2; x1x22 ). If = = 0 then using the homotopy method with the vector eld V from lemma 2.4.1 one can get rid of the whole quadratic part of g and obtain either the case 5) of this lemma or the case ii) of lemma 2.4.1. If k = 2 and g 2 m32 then in order to be simple g must have a nontrivial cubic part containing x32 ; 6= 0 and in this case it reduces to g = x32 which gives 5) Qf = E8; f = (x1; x21; x2; x42). Proof. Case 1). If k 3 then the induced function Qf = xk1+1 +x22 g; g 2 m2 . Since any simple germ of function has a nontrivial cubic form, (see e.g. [A3]) then g has a nonvanishing linear part x1 + x2 ; 2 + 2 6= 0. Let us present g = x1 + x2 + g2, where g2 2 m22 . Working with the basis one gets 4 vector elds with the rst three components vanishing and the following nontrivial quadratic parts of the last component, (0; 0; 0; k x1x2 +(k +1) x22 + : : : ); (0; 0; 0; x22 + : : : ); (0; 0; 0; 2 x1x2 + 3 x22 + : : : ) and (0; 0; 0; 2 x21 + 3 x1 x2 + : : : ), where : : : denotes all terms of order at least 3. If 6= 0 then using the second eld we can remove of x22 in the fourth component of f , i.e. of x2 in g. Finally, by multiplying the third and the fourth components of f by an appropriate constant and its inverse we force = 1. If k is even then changing the sign of x1 one can force the last component to be x1x2 . If = 0 then we can normalize the coecient by making = 1. In this case one gets f = (x1 ; xk1 ; x2; x22 + : : : ) which coincides with the case 2) below up to renumbering of components. Case 2) is similar to the case 1). Case 3). If k = 2 one gets Qf = x31 + x22g: According to the classi cation of simple germs of functions we conclude that g has at least a nontrivial 3-jet. Let us rst assume that its 2{jet is nonvanishing, i.e. g = x22 + x1 x2 + x21 + g3; 2 + 2 + 2 6= 0: Working with the basis of vector elds one gets the following 4 elds a ecting only the fourth component of f : (0; 0; 0; x21x2 + x1 x22 + x32 + : : : ); (0; 0; 0; 3x21 + 2 x1x22 + x32 + : : : ); (0; 0; 0; 2 x21x2 + 3 x1 x22 +4 x32 + : : : ) and (0; 0; 0; 2 x31 +3 x21 x2 +4 x1 x22 + : : : ): We denote them by v1 ; : : :; v4 resp. Since v2 has the term 3x21 one can always get rid of the term x21 in g. Assuming that
= 0 one gets v4 ? v2 = 4 x1 x22 + : : : , where : : : denotes all terms of order at least 4. If 6= 0 one can get rid of the term x1 x2 in g and moreover normalize it to g = x22 + g3. Case 4). If = 0 and 6= 0 then g reduces to g = x1 x2 + g3 . Multiplying x2 by an appropriate constant one gets g = x1 x2 + g3. Case 5). Let us assume that f = (x1; x21; x2; x2(P3 + g4)), where P3 denotes the cubic part. One concludes that if Qf is a simple germ of function with the 3-jet equal to x31 then its 5-jet must contain x31 + x52 + : : : ; 6= 0 and therefore P3 = x32 + : : : , see [A3], p.13. Let us check that in this case P3 reduces to x32. At rst we force = 1 by the usual trick. Let P3 = x32 + x1 x22 + x21x2 + x32 . The appropriate basis of vector elds is (0; 0; 0; 3x42+2 x1 x32 + x21x22 +: : : ); (0; 0; 0; 3x21+ x42 +2 x1x32 +3x21x22 +: : : ); (0; 0; 0; 5x1x32 + 4 x21 x22 + 3 x31x2 + 2x41 + : : : ). Using v2 one removes x1x22 and x31 in g. Taking = = 0 one gets v4 ? 34 x22 v2 = 5x1x32 + : : : and removes the term x1 x22 in g. Thus, g = x32 + g4:
MAPS IN A SPACE WITH A CONE
15
In order to nish the proof of simplicity it suces to check that all the jets of f presented in the formulation of lemma 2.4.2 are sucient and describe their adjacency. Suciency will be discussed in the separate statement 2.4.5 and adjacency is postponed until 2.4.6. 2.4.3. Lemma. In the case ii) of lemma 2.4.1 an adjusted 2 ! 4-map is simple if and only if the function h has a nontrivial quadratic part h = x21 + x1 x2 + x22 + h3 ; 2 + 2 + 2 6= 0 and one of the following possibilities holds. If 6= 0 then h reduces either to x21 x22 which gives a) Qf = D4 ; f = (x1 ; 0; x2; x21 x22 ) or h reduces to x21 which gives b) Qf = Dk+2 ; f = (x1; 0; x2; x21 xk2 ); k 3. Here the -forms are di erent only if k is even. If = 0 then h reduces to x1 x2 which gives c) Qf = D2k ; k 3; f = (x1 ; 0; x2; x1x2 + xk1 ). If = = 0 then h reduces to x22 which gives d) Qf = E7; f = (x1; 0; x2; x22 + x31). Proof. Considering the basis of the module of tangent vector elds one gets the following 4-tuple of vector elds a ecting only the fourth component (0; 0; 0; x1x2 + 2 x22 + : : : ); (0; 0; 0; 2 x21 + x1 x2 + : : : ); (0; 0; 0; 2 x1x2 + x22 + : : : ) and (0; 0; 0; x21 + x1 x2 +
x22 + : : : ). After some obvious linear transformations one gets (0; 0; 0; : : : ); (0; 0; 0; 2 x21 + x1 x2 + : : : ); (0; 0; 0; 2 x1x2 + x22 + : : : ) and (0; 0; 0; (2 ? 2 )x22 + : : : ) denoted by v1 ; : : :; v4 resp. Case a). If 6= 0 then one normalizes it by making = 1 (possibly after multiplying Q by ?1). Then using v3 one cancels x1 x2 in h. We arrive at h = x21 + x22 + h3 : If 6= 0 then using the last eld we can normalize it by making = 1. Thus, h = x21 x22 + h3 . In 2.4.5. we will discuss the suciency of the presented 2-jet. Case b). If = 0 then h = x21 + h3 . Using the elds v2 and v3 one can remove of all terms in h3 divisible by x1 x2. Therefore h = x21 + xk2?1 + : : : . Analogous arguments show that we can remove of all the terms denoted by : : : and obtain h = x21 xk2?1 . Case c). If = 0 and 6= 0 then using v3 one removes x22 in h and normalizes = 1. Now using v2 and v3 one removes all the terms in h3 divisible by x1 x2 and by x22 . Thus, h = x1 x2 + xk1 : : : . The same arguments show that h normalizes to h = x1 x2 + xk1 . Case d). If = = 0 then h = x22 + h3 . The constant normalizes to 1 and the arguments analogous to the case 4 of lemma 2.4.2 shows that if f is simple then h3 contains x31 and reduces to h3 = x31 + h4 .
2.4.4.
Different normal forms.
Lemmata 2.4.1-2.4.3 give us the unique normal forms for the singularities Ak , E6 and E8 and provide the following list of normal forms for Dk+1 and E7. Qf = Dk+1 ; ( forms are di erent only if k is odd) a) the normal form: f = (x1 ; xk1?1; x2; x1x2 ); its reduced versal unfolding is red (x1; x2; 1; : : :; k ) = fx1 ; xk1?1 + 1xk1?2 + + k?2 x1 ; x2; x1x2 + k?1 x1 + k x2 g; b) the second normal form: f = (x1 ; 0; x2; x21 xk2?1); its reduced versal unfolding is ~ red (x1; x2; 1; : : :; k ) = fx1 ; 1x1; x2; x21 xk2?1 + 2x1 + 3 x2 + + k xk2?2 g; c) if k 3 then there exists the third normal form: D2+k : f = (x1; 0; x2; x1x2 + xk1 ), its reduced versal unfolding is ~ red (x1; x2; 1; : : :; 2k?1 ) = (x1 ; 1x1 + 2 x21 + + k?1 xk1?1 ; x2; x1x2 + xk1 + k x2 + k+1 x1 + k+2 x21 + + 2k?1 xk1?1 ):
16
B. Z. SHAPIRO
Qf = E7 ; a) the normal form: f = (x1 ; x21; x2; x1x22 ), its reduced versal unfolding is red (x1 ; x2; 1; : : :; 6) = fx1; x21 + 1x1 ; x2; x1x22 + 2 x21 + 3x22 + 4 x1x2 + 5x1 + 6 x2g; b) the second normal form: f = (x1 ; 0; x2; x31 + x22), its reduced versal unfolding is ~ red (x1 ; x2; 1 ; : : :; 6) = fx1 ; 1x1 + 2 x21 ; x2; x22 + x31 + 3 x1 + 4 x21 + 5 x2 + 6 x1 x2g; Proposition. In each of the above subcases all the normal forms present 1 orbit.
Proof. In order to show that the rst and the second normal forms belong to the same orbit in all the subcases one should notice the following. Any change of coordinates of the form y~1 = y1; y~2 = y2 + y3h; y~3 = y3; y~4 = y4 ? y1h, where h is an arbitrary smooth function preserves the cone C : fQ = y1y2 + y3 y4 = 0g. For an adjusted 2 ! 4-map f that means that if one adds x2 h to the second coordinate and simultaneously subtracts x1h from the fourth coordinate then one obtains another map belonging to the orbit of f . This explains why the rst and the second normal forms belong to the same orbit. It is left to show that the third normal form belongs to the same orbit as the second normal form of D2+k , i.e. why (x1; 0; x2; x21 ? x22k?2) = (x1 ; 0; x2; x1x2 + xk1 ). The second normal form is equivalent to (x1; x22 + x2 xk1?1 ; x2; 0) or after renumbering of coordinates to (x1; 0; x2; x21 + x1 xk2?1 ). Making the coordinate change x~1 = x1 ? xk2?1 one transforms the last map into (x1 + xk2?1 ; 0; x2; x1 + 2x1xk2?1 ). Multiplying x2 by a constant one gets (x1 + xk2?1 ; 0; x2; x1 + x1 xk2?1). Finally, any change of coordinates of the form y~1 = y1 + hy3 ; y~2 = y2 ; y~3 = y3 ; y~4 = y4 ? hy2 preserves the cone. This transformation applied to (x1 + xk2?1 ; 0; x2; x1 + x1xk2?1 ) with h = ? xk2?2 gives (x1; 0; x2; x1 + x1 xk2?1 ).
2.4.5.
Criterion of sufficiency of a given jet.
Recall that the k-jet is called sucient under the action of a chosen group if any perturbation of f by any terms of degree greater than k belongs to the orbit of f . Let us denote by Mnl (j ) the Ol -module of all germs (Rl ; 0) ! (Rn ; 0) all components of which have degree j . By general results of [Da2] one can apply Mazer's homotopy method to check stability of a given jet. Thus suciency of the k-jet of a germ f (under the action of AC ) is equivalent to the fact that the AC -tangent space to any f + , 2 Mnl (j ) contains the whole module Mnl (j ). The last condition for semiregular germs f is equivalent to the following statement. Criterion of sufficiency. The k-jet ofa semiregular germ f : (Rl ; 0) ! (Rn ; 0); (0 is the vertex of the cone C ) is AC -sucient if and only if the Ol -module fm2l (@f=@x1; : : :; @f=@xl)+ ml (~v1 f; : : :; v~p f )g contains the submodule Mnl (k + 1), see notation in lemma 1.1. Proof. Let us sketch the proof of suciency of the above condition (the necessity is almost obvious). Indeed, let us assume that fm2l (@f=@x1; : : :; @f=@xl)+ml (~v1 f; : : :; v~p f )g contains the submodule Mnl(k + 1). Then since the vector elds v1 ; : : :; vp are linear the same is true for all f + ; 2 Mnl (k + 1). Therefore for all the tangent space T AC (f + ) contains Mnl (k + 1). Thus, for the family ft = f + t , t 2 [0; 1] the velocity vector @f@tt belongs to T AC (ft ) and therefore f is AC -equivalent to f + . Using this criterion one can check that all the normal forms in Theorem I and section 2.4.4 are sucient. Let us illustrate this in the most complicated case Qf = D2+k : (x1; 0; x2; x1x2 + xk1 ); k 3, see 2.4.4. We must show that the k-jet is sucient. The corresponding vector elds are (1; 0; 0; x2 + kxk1?1 ); (0; 0; 1; x1); (x1; 0; x2; x1x2 + xk1 ); (x1; 0; 0; 0); (x1x2 + xk1 ; 0; 0; 0);
MAPS IN A SPACE WITH A CONE
17
(x2; 0; 0; 0); (0; x2 + kxk1?1 ; ?x1; 0); (0; x2; 0; ?x1) and (0; 0; ?x2; x1x2 + xk1 ). After some simpli cations one gets (1; 0; 0; x2 + kxk1?1 ); (0; 0; 1; x1); (0; 0; 0; xk1 ); (0; 0; 0; x1x2 ); (0; xk1 ; 0; 2x21); (0; x2; 0; ?x1) and (0; 0; 0; x22). We must represent any (k + 1)-jet, k 3 by m22 f(1; 0; 0; x2 + kxk1?1 ); (0; 0; 1; x1)g +m2 f(0; 0; 0; xk1 ); (0; 0; 0; x1x2); (0; xk1 ; 0; 2x21); (0; x2; 0; ?x1); (0; 0; 0; x22)g. Any perturbation of degree k + 1 of the rst and the third components is cancelled by m22 f(1; 0; 0; x2 + kxk1 ?1 ); (0; 0; 1; x1)g. Any perturbation of degree k + 1 of the second coordinate is removed by m2 f(0; 0; 0; xk1 ); (0; 0; 0; x1x2)g. Finally, any perturbation of degree k + 1 of the fourth coordinate is contained in m22 f(0; xk1 ; 0; 2x21); (0; x2; 0; ?x1); (0; 0; 0; x22)g.
2.4.6.
Adjacency of simple semiregular germs. Proposition. The adjacency of the simple semiregular germs coincides with the adjacency
of the corresponding induced singularities (and, therefore, is presented on Fig.1). Proof. The statement follows from the analysis of the reduced versal unfoldings given in Theorem I together with lemmas 2.4.2-2.4.3. For Qf = Ak the statement is obvious. For Qf = Dk+1 we will analyze the reduced versal unfolding of the rst normal form. Nonvanishing k?1 or k give nontrivial quadratic part of Qf and thus lead to Al ; l k. Nonvanishing 1 ; : : :; k?2 lead to Dl ; l k. In the case 3) Qf = E6 nonvanishing 1 ; 4 or 5 lead to Al . Nonvanishing 2 or 3 give D4 or D5 according to lemma 2.4.2. Analogous arguments for E7 and E8 nish the proof. Recall that reduced versalPunfoldings of the simple semiregular singularities are equivalent as versal unfoldings to Qf + i ei , where ei 2 m2l are representa@Qf f tives of a basis of m2l =ml ( @Q @x1 ; : : :; @xl ) and thus reduced versal unfoldings are di erent from the usual versal unfoldings of the induced function Qf . In particular, the bifucation diagram of the former contains two irreducible components while the bifurcation diagram of the latter is irreducible.
2.5. Proposition B0. 2.5.1. Lemma. Fixing a nondegenerate quadratic form Q in Rn with the number of negative
squares equal to m let us consider the strati cation of Grassmanian Gl;n into strata Scr , where Scr consists of all l-planes such that the corank of the restriction of Q to any of these planes equals cr min(l; n; m ? n). Then codim Scr = cr(cr ? 1)=2. This lemma is proved by the same argument as the analogous statement for the quadratic forms. Proof of B 0 . Any semiregular f : (Rl ; 0) ! (Rn ; C ) occuring in a family of semiregular maps with k parameters has the corank cr of the quadratic part of Qf satisfying cr (k) = p1+8 1+ [ 2 k ] by lemma 2.5.1. Therefore, any such f can be reduced to (x1 ; 1; x2; 2; : : :; x(k) ; (k) ; x(k)+1 ; : : :; xl ; 0; : : :; 0) and its reduced AC -unfolding deforms only 1 ; : : :; (k) . Thus all such semiregular singularities are eqivalent to semiregular singularities on at most (k) variables. If the number of variables is restricted then the stabilization is obvious.
x3.
Irregular singularities
Let us start proof of theorem J with the following statement, see [Da3]. 3.1. Lemma. Let f : Rl ! Rn be a germ of embedding and : Rk ! Rn be a germs of a stable map (see the de nition of the stability of germs in [AVG], page 115). Let im(f ) ( ) and im( ) (f ) be an Aim(f ) -versal unfolding of and an Aim( )-versal unfolding of f resp. Then im(f ) ( ) is equivalent to im( ) (f ) as versal unfoldings. Remark. Apparently the same statement holds if both f and g are stable germs.
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3.2.
B. Z. SHAPIRO
Proof of theorem J.
Proof. Now let f : (R2; 0) ! (R3; C ) be a germ of Whitney umbrella tangent to the cone C at some point di erent from the origin. In this case one can obviously substitute C by a germ of a smooth hypersurface. Thus we can normalize a germ of smooth hypersurface in the presence of the standard Whitney umbrella fy32 = y1 y22g using the lemma 3.1. The natural basis of vector elds tangent to the standard Whitney umbrella is v1 = (0; y2; y3); v2 = (2y1; 0; y3); v3 = (2y3; 0; y12); v4 = (0; y3; y1y2): At rst we enumerate all cases of nontransversality of the tangent plane to a smooth germ w.r.t. the standard Whitney umbrella, i.e. all orbits of positive codimension of the action of 1-jets of vector elds preserving the Whitney umbrella on the 1-jets of germs of smooth hypersurfaces. One has to consider the following 3 types of 1-jets of f : a) ( x1 + x2 ; x2; x1), b) (x2; x1; x1) and c) (x2 ; x1; 0). Case a). The 1-jets of vector elds have the form ( ; 0; 1); ( ; 1; 0); (0; x2; x1); (2 x1 + 2 x2 ; 0; x1); (x1; 0; 0); (0; x1; 0); or after reduction ( ; 0; 1); ( ; 1; 0); ( x1 + x2 ; 0; 0); ( x2 ; 0; 0); (x1; 0; 0); ( x1 ; 0; 0): It splits into 2 subcases: a0) The typical case of codimension 0 when 6= 0; in this case the tangent plane can be reduced to the form y1 = y2 = x2; y3 = x1 . a00 ) The special case of codimension 1 when = 0; in this case the tangent hyperplane is y1 = 0 and the 1-jet of f is reduced to the form y1 = 0; y2 = x2; y3 = x1. Case b). For the 1-jet (x2; x1; x1) one gets (after some obvious simpli cations) the following 1-jets of vector elds (0; ; 1); (1; 0; 0); (0; 0; x1); (0; x1; 0): Therefore the initial jet of the map f can be always reduced to the form (x2 ; 0; x1): Let us now consider the subcases of positive codimension. The subcase a0 ) is generic. In the subcase a00) the 1-jet of f is (0; x2; x1) and the 1-jets of vector elds have the form (0; 1; 0); (0; 0; 1); (x1; 0; 0): This means that one can restrict consideration to the germs of maps of the form (g(x2); x2; x1) according to results of [AVG] p.180. Let us consider the series of maps Fk : (xk2 ; x2; x1). The corresponding basis of vector elds has the form (0; 0; 1); (kxk2?1 ; 1; 0); (0; x2; x1); (2xk2 ; 0; x1); (2x1; 0; 0); (0; x1; 0): The versal unfolding of Fk is
k (x1; x2; 1; : : :; k ) : fy1 = xk2 + 1 xk2?1 + + 1 ; y2 = x2 ; y3 = x1g: The inverse image of the Whitney umbrella is given by the formula x21 ? x22 (xk2 + k xk2?1 + + 1 )2 = x21 ? x22 p(x2 ) = 0: For a generic set of the inverse image of Whitney umbrella has a point of transversal sel ntersection at the origin and is smooth at other points. Violation of genericity occurs when p(x2) has the zero or multiple root. Thus one gets the bifurcation diagram of the singularity Bk . Case b). The 1-jet of the map f is (x2 ; 0x1). The corresponding 1-jets of the vector elds are (1; 0; 0); (0; 0; 1); (0; 0; x2); (2x1; 0; x2); (2x2; 0; 0); (0; x2; 0) or after reduction (1; 0; 0); (0; 0; 1); (0; x2; 0): According to the general technique it suces to consider the family of germs k : (x2 ; xk1 ; x1). The corresponding 1-jets of the vector elds are (1; kxk1?1; 0); (0; 0; 1); (0; xk1 ; x2); (2x1; 0; x2); (2x2; 0; x21k ); (0; x2; xk1+1 ) or after reduction (1; kxk1?1; 0); (0; 0; 1); (0; xk1 ; 0); (0; x2; 0):
MAPS IN A SPACE WITH A CONE
19
The versal unfolding is k (x1; x2; 1; : : :; k ) = fy1 = x1 ; y2 = xk1 + k xk1?1 + + k ; y3 = x2 g. The inverse image of the Whitney umbrella is the curve given by x22 ? x1 (xk1 + k xk1?1 + + 1 )2 = x22 ? x1 p2 (x1) = 0. This curve has transversal sel ntersections which lie on the x1 -axis which correspond to the simple zeros of the polynomial p and is tangent to the x2-axis at the origin. Its degeneracies are caused either by a multiple root of p or if p vanishes at the origin. Thus the bifurcation diagram is the same as for the singularity Bk . Case c). One has to consider the series of maps k : (x2 ; x1; xk2 ) and the corresponding initial jet of the vector elds are (0; 1; 0); (1; 0; kxk2?1); (0; x1; xk2 ); (2x2 ; 0; xk2 ); (2xk2 ; 0; x21); (0; xk2 ; x1 x2) or after reduction (0; 1; 0); (1; 0; kxk2?1); (0; 0; xk2 ); (0; 0; x21); (0; 0; x1x2 ): The same argument as above shows that in the case c) considerations can be restricted only to the cases k . The versal unfolding of k is given by the formula k (x1; x2; 0; : : :; k ) : fy1 = x2 ; y2 = x; y3 = 0x1 + xk2 + k xk2?1 + + 1 g: The inverse image of the Whitney umbrella is given by (0 x1 + p(x2 ))2 ? x21x2 = 0; where p(x2) = xk2 + k xk2?1 + + 1 . Let us describe the cases of nongeneric position. The natural strati cation of Whitney umbrella consists of its vertex, its handle and its smooth open 2-dimensional part, see Fig.1. If R denotes 0 x1 + p(x2 ) then the nontransversality to the vertex implies R = x1 = x2 = 0. Therefore, it gives 1 = 0. The nontransversality to the handle implies R = x1 = 0 and therefore it gives that p has a multiple root. Finally, it is easy to check that the nontransversality to the smooth part is equivalent to p(20 ) = 0.
x4.
Final remarks.
The following questions are quite natural from the point of view of the singularity theory. 1) Extend the theory of vanishing cycles and the technique of Dynkin diagrams to the considered case. 2) Compare modalities of the versal unfolding of f and its induced function Qf . 3) Develop some method to calculate the dimension of the reduced versal unfolding of a semiregular f (at least in the quasihomogeneous case) and compare it with the Milnor number of Qf . 4) Study the semiregular singularities in the case when C is a generic (quasi)homogeneous polynomial of some (multi)degree. References
V. I. Arnold, Critical points of functions on a manifold with boundary, the simple Lie groups Bk ,Ck and F4 and singularities of evolutes, Russ. Math. Surv. 33 (1978), no. 5, 99{116. [A2] V. I. Arnold, On the surfaces de ned by hyperbolic equations, Mat.Zametki (1988), no. 1, 3{18. [A3] V. I. Arnold, Normal forms of functions near degenerate critical points, Weyl groups Ak , Dk ,Ek and Lagrange singularities, Funct. Anal. Appl. 6 (1972), no. 4, 3{25. [AGLV] V. I. Arnold, V. V. Goryunov, O. V. Lyashko, V. A. Vassiliev, Singularity theory I, Encyclopedia Math. Sciences, Dynamical Systems VI, Springer, 1993. [AVG] V. I. Arnold, A. N. Varchenko and S. M. Gusein-Zade, Singularities of di erentiable maps, Birkhauser, 1985. [Da1] J. Damon, Unfoldings of sections of singularities and Gorenstein surface singularities, Columbia Univ. (1982), preprint. [Da2] J. Damon, The unfolding and determinacy theorems for subgroups of A and K, Memoires of the AMS 50 (1984), no. 306. [Da3] J. Damon, A-equivalence and the equivalence of sections of images and discriminants, Warwick proceedings, Lecture Notes in Mathematics 1462 1 (1989), no. 306, 93{121. [A1]
20 [G1] [G2] [Ga] [L] [Ma] [Mo] [Si]
B. Z. SHAPIRO V. V. Goryunov, Singularities of projections of complete intersections, Itogi nauki i tehniki, Sovr. problemy matematiki 22 (1983), 167{206. V. V. Goryunov, Simple projections, Sibirian Math. Jour. 25 (1984), no. 1, 61{68. A. M. Gabrielov, Bifurcations, Dynkin diagrams and modality of isolated singularities, Funct. Anal. Appl. 8 (1974), no. 2, 7{12. O. V. Lyashko, Classi cation of critical points of functions on varities with singular boundary, Funct. Anal. and Appl. 17 (1983), no. 3, 28{36. J. Mather, Finitely determined map germs, Publ. Math. IHES 36 (1968), 127{156. D. Mond, On the classi cation of germs from R2 to R3 , Proc. of London Math. Soc. 50 (1985), 333{369. D. Siersma, Singularities of functions on boundaries, corners etc, Quart. J. Math, Oxford Ser 32 (1981), no. 125, 119{127.
