Real And Complex Singularities

An exact sequence of the form

0 — V -^ U, means that ?/> is injective, while a sequence of the type

W -^ V —-> 0, means that (f> is surjective. Suppose that W is a subspace of the vector space V. Then

is a vector space. Moreover, if dim# V < oo, then w

„ V - dim*: W.

(4.5)

The above equality follows from the fact that, if v\,... vm, vm+i,..., vn, is a basis for V, where ? , > i , . . . ?;m, form a basis for W, then vm+i,..., t7n is a basis for ^.

Irreducible Plane Curve Singularities

59

PROPOSITION 4.7 Suppose that we have an exact sequence of vector spaces

Then V is finite dimensional if and only if U and W are finite dimensional. In this case, V = dim# W + dim^ U. Proof:

Since the sequence is exact we have that dim# W = dim^ t/)(W) = dim^- ker (<£>),

and from the isomorphism theorem, it follows that

,

ker(
dim/f V = dimx ker(
PROPOSITION 4.8 Let W C V C U be a chain of vector spaces such that dimK % < oo. Then,

v

.

u

Proof: This follows from the Proposition (4.8) and from the exact sequence:

o —> W *1— > W - ^ - + V^ — > o D

PROPOSITION 4.9 (Noether Isomorphism Theorem) Let V and W be subspaces of a vector space U. Then

V

^ V+W

vnw ~ w ' Proof:

The map:

V +W V +W

V

vnw

I—>

V

is well denned, surjective and its kernel is W.

D

THEOREM 4.10 Let V be a vector space and let L : V —>• V be a linear injective map. Let W be a vector subspace ofV such that L(W) C W. We have that (i) —

w ~ L(wy

(ii)

// ^ and LYV, are finite dimensional, then

W

V

L(vy

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Proof:

(i) Consider the sequence,

where j is the inclusion homomorphism, and L is the homomorphism induced by L. Since L is inject! ve, we have that ker(I) - {v e V] L(v) e L(W}} = W, which implies that the above sequence is exact. (ii) Consider the following chains of inclusions: WC W + L(V] C K and

L(W) CW r\L(V) CL(V). Since from (i) and from the hypothesis of (ii), we have that

we get, from the Proposition (4.9), the following equalities: r V dim K -fa II

=

iW+L(V) dim K —^-^II

-t-,

r dim*

V

W+L(V)

WC\L(V)

(4-6) >

where the equality

W + L(V)_^ W

L(V)

follows from the Proposition (4.10). Consequently, we get V

,

WClL(V)

Since from hypothesis, dim/<- jjy^ < oo, from the chain

L(V) CW + L(V) C V, we get

V

V

W + L(V]

W + L(V) This implies, in presence of the Proposition (4.10) and of (4.7), that w

,-

W + L(V)

Irreducible Plane Curve Singularities

61

From (4.7), (4.9), and from the chain below,

L(w] cwn L(V) c w, we get that

w

w

,.

wr\L(v)

The result then follows from this last equality together with (4.7), (4.9) and (4.8). D

4.3 INTERSECTION INDICES In this section we will introduce a method to express numerically the order or the degree of contact of two algebroid plane curves. This will be done through the fundamental notion of intersection index. Let K be an arbitrary field. We will denote by M , as usual, the maximal ideal of PROPOSITION 4.11 Let /, g G M.. The following conditions are equivalent. i) / and g are relatively prime.

K\\X Y]] ii) The dimension of —^ , as a K -vector space, is finite. Proof: The above conditions are preserved by automorphisms of /f [[X, Y}] and by multiplication of / and g by units in -ftT[[A", Y]]. Therefore, we may assume that / and g are pseudo-polynomials. (i) =>• (ii) Since / and g are relatively prime in /f [[A", y]], they are relatively prime in X[[X]][y], and therefore RY(f,g) ^ 0. So RY(f,g) = Xru, where r > 0 and w(0) ^ 0. Since R Y ( f , g ] € (f,g) (Proposition (2.20)), it foUows that RY(f,g)(Q) = 0, hence r > 1 and we have Xr £ (/, g) Suppose that degy (/) = n, then from the Division Theorem, for all h € A"[[X, Y]], there exist elements q E ^[[^, Y]] and ooPO,...,a n _ipO G K[[X]}, such that h — fq + O,Q(X] + - • • + an_i (X)Y n~1

From this it follows that the image h of h in K[[X,Y]]/(f,g) is in the K-vector space generated by~XlY3,Q (i) Suppose that / and g are not relatively prime in .&T[[A", y]]. There exists h G K[[X]][Y] non-invertible and pseudo-polynomial such that / = hfi

e g = hg2.

Hence {/,#} C (h). We have that l,X,X , . . . are elements linearly independent over K in ^[[X, Y]]/(h) , because otherwise, there would exist a polynomial P(X] G

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K[X] such that P(X) = hih, h\ € K[[X, Y}}, which is not possible because h is a pseudo-polynomial. The result now follows because dimK

K[[X,Y]} L . D

DEFINITION 4.12 Let f , g G M.. The intersection index of f and g is the integer (including oo) K{[X,Y}} l JJ . (/. 9)

In what follows we will use the notation z' to represent the image of z e ^[[X, Y}] in Of. So, it is immediate to verify that

O

.

DEFINITION 4.13 We will say that two algebroid curves (/) and (g] are transversal, if (/) and (g) are regular and their tangent lines are distinct. THEOREM 4.14 Let f,g,h e M, * an automorphism of K[[X,Y}] and u and v units in K[[X,Y]}. The intersection index has the following properties: i)

!(/, g) < oo if and only if f and g are relatively prime in K^X, Y]}.

ii)

!(/,. 9) = % , / ) .

iii)

I(*(f),$(g))=I(uf,vg)=I(f,g).

iv)

I(/,s/i)=I(/,«/)+I(M).

v)

!(/, g} = 1 if and only if (/) and (g} are transversal.

vi)

I(f,g-hf)

=!(/,,g}.

Proof: The first assertion follows immediately from Proposition (4.12). The assertions (ii), (iii) and (vi) follow immediately from the definition of intersection index. We are now going to prove (iv). It is sufficient to prove that the following sequence of ^-vector spaces is exact, n

_ ^ Of (h')

1>

Of (g'h'}

9

Of _ (g'}

where the map tp is induced by the projection (p: n Of

Notice that

M^ a

J^L ®f

n

o

Irreducible Plane Curve Singularities

63

The map 4> is defined by

V>(F) - JZ, where the bar represents the residual class modulo (hr), whereas the double bar represents the residual class modulo (h'g'). The map tp is clearly a homomorphism of K-vector spaces. To prove that V is injective, suppose that V;(2/) = 0 and notice that

Hence there exists a' e Of such that g'z' = a'g'h', which in turn implies that there exists (3 6 K[[X, Y}] such that g(z - ah) = gz- agh = (3f. Since / and g have no common factors, it follows that / divides z — ah, which shows that z' = 0. To conclude the proof of (iv), observe that

Finally, we will prove property (v) . Suppose that / and g are regular with distinct tangents. Then after a linear change of coordinates we may assume that / = X + f\ and g = Y +gi, where fljgl € M2 . We may then write e

=

where u and v are units and /2 , 52 We then have Y(v - IT W2) = Yv- u-lYgih =g- u~lg2f e {/, g). Since (v — M~ 1 ^2/2)(0,0) = v (0, 0) ^ 0, it follows that v — u~1g2/2 is a unit and therefore Y G {/,#). In a similar way one shows that X e {/, g ) . It then follows that (f,g) = (X, y), and therefore,

...

.

..

K\[X,Y\]

,.

K[[X,Y]]

.

K/,.) = dim^ ^-^- = dim, -—- = 1. Conversely, if mult(/) > 2 or mult(cjr) > 2, or also if (/) and (g) are regular and have the same tangent line we may, after a linear change of coordinates, suppose that f = Yfi + h and g = Y9l + g^ with / 2 ,# 2 € M2 and flt9l e K[[X,Y]]. Hence {/, g) c (Y) + M2 and therefore,

dimK

-

,Y}}

-

^ dimK

,

, = dimK D

The properties listed in the Theorem (4.15) determine uniquely the intersection index of algebroid plane curves. This is the content of the next result.

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THEOREM 4.15 Let I' be a map I' :

M x M

—>

IN U {00}

(/,.g)

>-»

!'(/,. 9)

having properties (i) £o (vi), o/ t/?,e Theorem (4-15). Then I' = I. Proof: The proof will be done by induction on the value of !'(/, g). We may assume that f , g £ M\ {0} are relatively prime in /f[[X, F]], because otherwise we would have, from (i), that !'(/,
Let f , g £ M\ {0} such that !'(/, g) = r. Let $ be an automorphism of K[[X, Y}} such that 3>(/) and <J>(
Suppose that degy(p) = n and degy(g) = r^i. From property (ii) we may assume that n < m. Define where g2 € A'[pf,F]l. If 92 is a unit, then from properties (iii),(iv) and (vi), we have that

In the same way, one shows that !(/>, q] = n, and therefore, V(p, q) = I(p, q}. Suppose that q% is not a unit, then from properties (vi) and (iv), and by the inductive assumption we have that

since I'(p, X) < r - 1 and r(p, q2) < r - 1. This implies that

The above theorem is an executable algorithm to compute !(/, g) for all pair of power series / and g in M which are pseudo-polynomials. EXAMPLE 4.16 We will find the intersection index of Y7 - X2 and Y5 - X3. l(Y7 - X2,Y5 - X3) = l(Y7 -X2 - Y2(Y5 - X3},Y5 - X3) = l(X2(XY2

- 1), Y5 - X3) = I ( X 2 , Y5 - X3) = 2 • l(X, Y5 - X3} = 2 - I(X, Y5) = 1 Q - I ( X , Y ) = 10.

Irreducible Plane Curve Singularities

65

THEOREM 4.17 Let f and g be pseudo-polynomials in K[[X]][Y].

Let

be the decomposition of f into irreducible factors, which we may suppose that are pseudo-polynomials. Then

Proof: If / and g have common components, we have nothing to prove since the above three terms are infinite. Suppose now that / and g have no common factors. Since from property (iv) of Theorem (4.15), and from Problem 2.4.2, we have that

and

It will be sufficient to prove that if / is irreducible, then

Consider the ^-linear map L:

h(T]

~

n

where n — degr(/) and (T ,
(g') ~ g(T^^(T})Rlp ~ L (W) On the other hand, V = K[(T\] L(V] (g(T From the Corollary (4.6) we have that — < oo,

and since

it follows from the Theorem (4.11) that

!(/, .9) = di

66

Hefez On the other hand, if C is a primitive nth root of unity, from the Proposition

(2.22).

Hence mult(fly(/,0)) = n.nm\i(g(X,(p(X±)))

= mult (0(7™, D

THEOREM 4.18 Let f , g e At. We /iowe tf

with equality if and only if (/) and (g) have no common tangents. Proof: Let f = fi • • • fr and g = g\ • • • gs, be respectively the decompositions of / and g into irreducible factors. From the Theorem (4.15) (ii) and (iv), we have that I(/,<7)=Tl(/?,.9,).

(4-10)

On the other hand, since mult (/?< 1/12) = mult (hi) 4- mult(/7,2), we have that mult (/) - mult (g) -

JT mult (fl ) • mult (gj ) .

(4.11)

Therefore, from (4.10) and (4.11). it follows that it is sufficient to prove the theorem for / and g irreducible. Suppose that coordinates have been chosen in such a way that / and g are associated to pseudo-polynomials in /C[[X]][F] and that (/) has Y as its tangent line. Let (Tn, <£>(T)), where n = mult(/), be a Puiseux parametrization of (/). Since Y is the tangent line of (/), recall from (3.6) that mult( n. Suppose now that g(X,Y) = (aX + bY)m + gm+1(X,Y) 4- • • • , where gm+i(X, Y) H ---- e Mm+l . Then we have that !(/, g) = mult((«T" + with equality if and only if a ^ 0; that is, the tangent line of (g) is not Y, or equivalently, if the tangent lines of (/) and g are distinct. n The intersection index of two curves is easily and effectively computable, either by means of the Theorem (4.15), or by means of the Theorem (4.18), when either both curves are given in cartesian form or one of them in cartesian form and the other in parametric form. In Chapter 8 we will show how to compute the intersection index when both curves are given in parametric form.

Irreducible Plane Curve Singularities

67

PROBLEMS 3.1) Compute the intersection indices of the following pairs of curves: a) (Y2-X3) and (Y2 - X5}. b) (Y2-X3}and(X2-Y3). c) ((X2 + Y2)2 - (X2 - r 2 )) and ((X2 + Y2}2 4- (X2 - Y2)}. d) Compute the intersection indices of the curves of Section 3.3, taken by pairs. 3.2) Let / = Y4 - 2X3Y2 - 4X5Y -X6-X7. Denote by /£> the partial derivative of order i of / with respect to the indeterminate Y. Compute !(/, fy '), !(/, f y ] and

68

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Resolution of singularities

In this chapter we will be concerned with the process of resolution of singularities of irreducible plane algebroid curves. In the analytic context, this process consists in transforming a germ of an analytic branch into a non-singular germ, by means of a finite sequence of certain birational transformations. This technique was introduced by Max Noether, the father of Algebraic Geometry, in the nineteenth century. In order to motivate our study, we will introduce the subject in the more geometric context of germs of analytic plane curves, and then develop it in the context of formal geometry.

5.1

QUADRATIC TRANSFORMS IN
DEFINITION 5.1 A quadratic transformation or a blowing-up centered at the origin of (E is a map which in .some coordinate system of (E2 is of the form,:

T :

(E2 (X 1} yi)

—> ^

(E2 (X,Y) = (X1,X1Y1)

The line E : X\ — 0 is called the exceptional divisor of the blowing-up and it is equal to T~l (0,0). Let L : Y - aX = 0 be a line through the origin of (E2, then T~l(L] is the union of the exceptional divisor E and the line YI = a, which cuts E in the point of coordinates (0, a). Hence, the points of the exceptional divisor correspond to the directions through the origin of (E2, excepting the direction of the Y axis. The transformation T induces an analytic automorphism of (E2 \ E. Let Cf be a germ of an analytic curve denned by an element / 6 M. C${X. Y}. Let us see what is the set T~~l(Cf ). Suppose that the expansion of / in homogeneous polynomials is given by )= l

Fn(X,Y)+Fn+1(X,Y)

So, the equation (5.1) of T~ (C/) is given by the series

69

Irreducible Plane Curve Singularities

which will be called the total transform of /, whose associated curve T 1(Cf) will be called the total transform of Cf. The curve Cw, determined by the equation f ( l ) ( X i , Y i ) = 0, where

is called the strict transform of Cf. Since T~ 1 (0,0) is the exceptional divisor, with equation E : X\ = 0, we have that

EXAMPLE 5.2 Consider the curve Cf where / = Y2 - a2X2 - X3 = 0, with a ^ 0. Since the curve is denned by a polynomial, we will not have convergence problems and therefore it is globally defined in all (E2. The total transform of Cf is T-l(Cf) : X2Y2 - a2X2 - X* = X2(Y2 - a2 - XJ = 0, whereas the strict transform of Cf is given by the tangent lines of Cf at the origin, given by - a2X2 =

: Y2

— 0. Notice that

aX}(Y + aX] = 0,

have as strict transforms two horizontal lines which pass through the points PI = (0, a) and P% — (0, —a), which are precisely the points of intersection of C/u) with E (see the figure below).

Figure 1

Notice that after one blowing-up, the curve Cf is transformed into a smooth curve Cf(i) and its two branches through the origin are transformed into portions of the curve passing through the points PI and P-Z. EXAMPLE 5.3 Consider the curve C/, where / = Y'2 — X3 = 0. Here again we have a curve defined globally in (E2. The total transform of Cf is T~l(Cf) : X2Y2 - X3 = X2(Y2 - *i) = 0,

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whereas the strict transform of Cf is given by Cf vertical hne X\ — 0 as a tangent line in the origin.

:Y?

= 0, which has the

Figure 2 Observe that also here after one blowing-up we succeeded to transform Cf into a smooth curve C/o;, and since Cf has only one branch through the origin, it is transformed into the unique branch of Cr(\) at the origin. EXAMPLE 5.4 Consider the curve Cf where / = Y2 — Xr>. Here again we have a curve defined globally in (C". The total transform of Cf is 'T~l (f1

/

whereas the strict a tangent line the previous example, parabola F22 - X2

Y"2v"2 V'2/"V"2 n (Cf)\ .. X -XV"5 - Av3\ lYl l=X-i(Yl 1)=0,

transform of Cf is given by C^D : Y-,2 — Xf = 0, which has as line YI = 0. Notice that the curve Cf(i) is the curve Cf of the and therefore with one more blowing-up we transform it into the = 0.

Figure 3 Some of the facts pointed out in the above examples always happen. For example, we will show that after a finite number of blowing-ups, it is possible to transform any algebroid irreducible plane curve into an algebroid regular curve. This procedure will be called the resolution of the singularities of the curve, which will allow to study a singularity transforming it into several simpler singularities.

Irreducible Plane Curve Singularities

71

5.2 RESOLUTION OF SINGULARITIES OF PLANE CURVES From now on, K will be an arbitrary algebraically closed field.

DEFINITION 5.5 The quadratic transformation a from the ring K[[X,Y^\ into the ring K^_X\,Y\^ is the homomorphism of K-algebras defined by a: K\\X,Y}} X Y

—> .-» i->

K^X^Y^} Xl XiYi

r:

—>

#[[*!, Yi]]

or by K[[X,Y}} v

v y

Y

i-»

Y!

These transformations are not invertible, but they are birational; that is, they define isomorphisms between K((X, Y)) and K((Xi, YI)), respectively the fields of fractions The homomorphism a transforms the ideal (X, Y) into the ideal

When we are working with the ring (K{X,Y}, this geometrically corresponds to the condition T -1 (0,0) = E of the previous section. The transform by a of f(X,Y)

= Fn(X,Y} + Fn+l(X,Y)

+ •••

is the element of /f[p£"i, YI]] defined by

a(f)=f(Xl,X1Yl)= =

Fn(X1,XiY1)+Fn+1(X1,X1Y1 X^(Fn(l,Y1)+X1Fn+l(l,Y1}

The series

where n = mult(/), will be called the strict transform by a of (/), and will be denoted also by

PROPOSITION 5.6 Let f,ge

K[[X,Y}}.

0 a *(/) is invertible in K[[Xi,Yi]] if, and only if, f is regular in X; that is, the initial form of f is of the form Fn — (cXn H ---- ), where c e K \ {0} and n is a non-negative integer (the case n = 0 corresponds to f invertible in K[[X,Y]]}. < iii) mult(<7*(/)) < mult(/).

72

Hefez iv) / / / is- a Weierstrass polynomial in K[[X}}[Y] of degree n with tangent cone. (Yn), then
Proof:

If we write / = Fn(X, Y)+Fn+1(X, Y)+- • -, then we have a* (/) - Fn (1, Y^+

XiF n+1 (i,y 1 ) + . . - .

(i) So, cr*(/) is a unit if and only if F n (l,0) is a non-zero constant. This in turn is equivalent to say that Fn(X, Y) — (cXn + • • • ) > where c 6 K*. (ii) This follows easily from the definition of a*, (iii) In F n (l, YI) there are monomials of degree less or equal than n = mult(/). The result follows by observing that these monomials cannot cancel with other monomials of cr*(/), because the latter are all multiple of X\. (iv) Let / = Yn + ai(X}Yn~l -\ \-an(X] be a Weierstrass polynomial with tangent cone (Yn)- that is m u l t ( a t ( X ) ) > i. Since, '•

w y

'1

'

j^

*i

'

'

xn

and

X\ the result follows. (v) Since / is irreducible, from the Unitarigent Lemma we must have either / regular in X or regular in Y . When / is regular in Y but not regular in X , then the tangent cone of (/) is of the form (Yn). If / is regular in X, then from (i) we get that cr*(/) is a unit. So it is sufficient to show that if we assume / irreducible with tangent cone (Yn) and &*(f) reducible, then we get a contradiction. Let's assume that / is so. From (i) and (ii) we have that if / and g are associated, then cr*(f) and cr*(g) are associated. Hence we may assume that / is a Weierstrass polynomial in Y. From (iv) we have that a*(f] is a pseudo-polynomial of degree n. Since cr*(/) is reducible in K[[X, Y}}, from the Lemmas (2.10) and (2.11), it factors into a product of irreducible pseudo- polynomials. Let h be an irreducible pseudo-polynomial of degree 0 < r < n that divides a*(/), hence Xrh(X,Y/X] would be a non-trivial factor of /, a contradiction. D Remark that if / <£ K[[X, Y}} is irreducible and such that
+ •••+ an(X) + Yn+lh(X, Y},

with ar(X) e K[[X}}, mult(a i (A')) > i, a 0 (0) ^ 0 and h E K[[X, Y}}. PROPOSITION 5.7 Let f G K{[X, Y}} be an irreducible power series with tangent cone(Yn}. Suppose that I ( f , Y ) =m. Then (i) I^/),^) = m - n and I( n, then mult( n, then (a*(/)) has tangent cone (Y"}; and if m — n — n, then neither (X\) nor (Yi} are the tangent lines of (a*(f}},

Irreducible Plane Curve Singularities

73

iii) Ifm-n
has tangent cone

( vm—n\ 1A1 )•

Proof: Since / is regular of order n in Y, we may write it as follows: / = aQ(X}Yn 4- ax (X)Yn~l + • • • + an(X] + Yn+lh(XJ F), where a0(0) ^ 0, h(X,Y] € K[(X,Y}}, and mult(a n (X)) = I(/,F) = m. From the Lemma (3.17) we also have that mult(aipO) >?: — . Now,

where So, we have that &o(^i) = a o(^i) and i)) = mult(ai(Xi}) -i>i

ffl

n

(5.1)

(i) From (3.6) we have that m > n because the tangent cone of (/) is (Yn}. Since b0(Xi) is a unit and i1—1 > 0, for i > 1, it follows from (5.1) that

!(**(/), tt) =I(fer»(Xi),yi) =mult(6 n (Xi)) =mult(a n (Xi))-n = m - n . (ii) If m — n > n, then from (5.1) it follows that mult(6j(Xi)) > i and since 6o(0) 7^ 0, we have that mult(cr*(/)) = n = mult(/). Now, if m — n > n, then from (5.1), nm\i(bi(Xi)) > i, showing that (F/1) is the tangent cone of cr*(/). On the other hand, if m — n = n, then the tangent line of cr*(/) has as summands at least the two terms of lower order in bQ(X\)Yn and in bn(X\). This shows that neither (Yi) nor (Xi) are the tangent lines of (
.m~2n n

m — 2n n

we have that mult(a*(/)) =m-n and that (X?~n) is the tangent cone of (
y.

The following result will be applied often.

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LEMMA 5.8 Let f e K[[X,Y}} be irreducible of multiplicity n and regular in Y. There exists an automorphism
Since / is irreducible and regular of order n in Y , we may write 1

where a 0 (0) ^ 0, h(X,Y) (3.17) we also have that

+ • • • +an(X)

+Yn+lh(X,Y),

G K [ \ X , Y } } , and I(/,Y) - mult(a n (X)). From Lemma

n

If n doesn't divide !(/, Y), then just take $ = Id. Suppose now that !(/, y) = nr, for some integer r > 1. Let c be a parameter in K and define We then have (Y +c.Xr)n+lh(X, Y + c.Xr), and since mult(a;pO) > ?: ——- = zr, n it follows that we ma write

with6o(c,0) =

> ?'r, if 1 < i < n — 1 > nr, if i = n, and P(c) is a polynomial of degree n in c. Let c0 e /C be such that F(c0) = 0, then

Yn+lh"(x, y), where ^(^T) = ^(c 0 , X), for 0 = 1, . . . , n, and /j/^X, Y) = h'(co,X, Y). So, $! (/) is irreducible, regular in Y and such that irmlt(bn(X)) > nr. If n doesn't divide mult(6 n (X)), we are done, since I($i(/), Y) is equal to mult(bn(X)). Otherwise, we repeat the above process obtaining an automorphism $2 such that

* 2 (/) = c*(X)Yn + cl(X)Yn~l + ••• + cn(X) + Yn+lh"(X1Y), with c0(0) / 0 and mult(c n (X)) > mu\t(bn(X)). This process must stop at some stage because otherwise we would get an automorphism $ such that = a'0(X)Yn + a'l(X}Yn-1 +•••+ a'n_1(X)Y + Yn+lh(X, Y), which is reducible; a contradiction.

Irreducible Plane Curve Singularities

75

PROPOSITION 5.9 Let (/) be irreducible with tangent cone (Yn}. Let fM = = o - * ( f ( i ~ l ) ) . I f n doesn't divide m := I ( f , Y ) , then [-1 = minjz; mult(/<*>) ^ mult(/)| Ln J «. > Proof: Since n doesn't divide m, by the euclidian algorithm we have m = nq + r, with 0 < r < n and o = — . Now, from the Proposition (5.7) we have that Ln J )

) = mult(/).

n We are now ready to describe the process of resolution of singularities for an irreducible algebroid plane curve. Assume that / e /f[[A", F]] is irreducible of multiplicity n, regular in Y and such that n doesn't divide ra = !(/, Y). We use the notation f^ — cr*(/) and /« = ^(f^-V), for z = 1, . . . ,q = [m/n]. Put n' = mult(/^), then we have that n' < n. If n' = 1 we stop. Otherwise, from Proposition (5.7.iii) we have that /^ is power series regular of order n' > 1 in the indeterminate Xq. Let m' — I(f(q\Xq) and put q' = [m! /n'}. If ra' is a multiple of n', then by the Lemma (5.8) we may change coordinates by an automorphism <£ of -^[[A", Y]] in order that ra' is no more divisible by n' . We still denote $(/ (
and /<*>=r*(/«-i>),

» = 9 + 2 , . . . , g + 9/.

Now, either n" = mult(/^ 9+9/ )) = 1, and we stop; or from the analog of Proposition (5.7.iii) we have that /te+9 ) js a regular power series of order n" > 1 in the indeterminate Yq+q', to which we continue to apply the above process, using a this time. Since the multiplicities are decreasing at some point we get a regular curve The sequence /, /^, . . . , /^ is called the canonical resolution of (/). We have proved the following result: THEOREM 5.10 The canonical resolution of an irreducible algebroid plane curve leads after finitely many steps to a non-singular algebroid plane curve. The canonical resolution of (/) determines the numerical sequence mult(/) > multt/W) > • • • > mult(/ w ) > - • • > mult(fW) = 1, of great importance in this theory, called the multiplicity sequence of (/). We say that two plane branches are equiresoluble if their multiplicity sequences are equal. EXAMPLE 5.11 We will determine the canonical resolution of / = Y4 - X7 . In the present case, we have that /W = <7*(/) = Yf - Xf. Therefore, applying r* to /(1\ we get /( 2 ) = F2 — X$, which is not singular. The multiplicity sequence in this case is (4,3, 1) (see figure below).

76

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c ra

]1 *"

f h

X

c

f

2

\

y

/ \ '

Figure 4

PROBLEMS 2.1) Resolve the singularities of the curves given below and determine their multiplicity sequences. a) Y5-X8. b) Y4 - 2X3Y2 - 4X5Y - X6 - X7.

5.3

NOETHER'S FORMULA

In this section we will present a useful formula due to Max Noether that relates the intersection index of two curves with that of their strict transforms. Let / £ K[[X, Y}} be irreducible of multiplicity n and regular in Y. We know from Section 3.4 that / has a Puiseux parametrization (T n , f ( T ) ) with m — mult(<^(T)) > n. According to the Lemma (5.8), performing a change of coordinates, if necessary, we may assume that m is not a multiple of n. Consider Since f ( l ^ ( X i , YI) is regular in YI of order n, and

it follows that

is a Puiseux parametrization of have

= &*(/}. Hence, for g e K[[Xi, YI]], we always

Recall now that the local ring Of injects into K[[T]} by means of the homomorphism HV : Of -^ A^ C K[[T]}. g ^

Irreducible Plane Curve Singularities

For the same reason, the local ring 0/(i) injects into homomorphism 9

•-*

We will see in the next proposition that the local ring of an irreducible algebroid plane curve injects naturally into the local ring of its blow-up. PROPOSITION 5.12 Let f e A"[[A",y]] be irreducible with tangent cone (Yn}. Then there is a natural injective homomorphism $ which makes the diagram below commutative

of ± o.w A, V

"

Proof: Let 0:

Of g(X,Y)

->

g(Xl,X1Yl)

where the overlines have the obvious meaning. Suppose that the multiplicity of / is n. The map is well denned and injective because, from the Proposition (4.3), it may be viewed as a homomorphism of K[[X]]-module • ••0

->

g(x,xyi)

which is clearly well defined and injective.

On the other hand, which proves the assertion.

D

PROPOSITION 5.13 Let (/) and (g) be two algebroid plane curves with f irreducible. Then we have I(f,g) = mult(/) Proof: Denote by n and n' respectively the multiplicities of / and g. We may change coordinates in such a way that / becomes regular in Y. Let (Tn, p(T}) be a Puiseux parametrization of (/), then (Tn, ^pp-) is a primitive parametrization of f^. So we have =-n-n'+vf(g} = which proves the result.

D

Let / and g be two algebroid irreducible plane curves. With a finite sequence, say of length N, of blowing-ups of either type cr, or T, we are led necessarily to the

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situation in which /( ' and q^ ' have distinct tangents. This observation, together with the Proposition (5.13) and the Theorem (4.19), proves a classical formula due to Max Noether, whose statement we give below. THEOREM 5.14 (Xoether's formula) Let f and g be two irreducible algebroid plane curves. We have that N

!(/, g} = /^n i-O

where f ( ° ' = f and g(°' = g.

Irreducible Plane Curve Singularities

6

79

Semigroups of plane branches

In this chapter we will introduce the semigroup of values of an algebroid irreducible plane curve. This is an invariant under the equivalence of curves and was classically recognized as the classifying topological invariant in the case of germs of complex analytic irreducible plane curves (see for example [5]). We will relate this semigroup to the characteristic integers, to the Puiseux pairs and to other numerical invariant of the curve. This will give us tools to compute such semigroups.

6.1

SEMIGROUPS OF THE NATURALS

Let G 7^ {0} be a subset of ]N, containing the element 0. We say that G is a semigroup in IN if it is closed under addition. In this section we will study semigroups under a strictly arithmetical point of view. The element min(G \ {0}) will be called the multiplicity of G, and will be denoted by mul If XQ, . . . , xr € IN, then the set (xo, . . . , Xr) = {XQXQ -\ ----- h A r X r ;

A I , . . . , Xr G IN}

is a semigroup in IN, called the semigroup generated by x 0 , . - . , x r . The elements XQ, . . . ,x r are called generators for G. For example, we have (3) = {0,3, 6, 9,...},

and

(3, 5} ={0,3, 5, 6, 8, 9, 10, 11,...}. PROPOSITION 6.1 Given any semigroup G in IN, there exists a unique finite set of elements vo,...,vg in G such that

i) v o < • • • < vg, and Vi ^ Vj mod VQ, for i ^ j, ii) G= (v0,...,vg), iii) {VQ, . . . , vg} is contained in any set of generators of G.

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Proof: The elements VQ, . . . , vg are defined inductively as follows. Put VQ = mult ((7) and define

vi = min(G\ (VQ)) . It is clear that v0 ^ v^ mod VQ, because otherwise v\ would be in (?;0), which is a contradiction. Define for i > 2, Vj = min (G \ (v0, . . . , ^-i}) . It is clear that ?;7 ^ Vj mod^o, for j < i, because otherwise v^ would be in (VQ, . . . ,Vi_i), which is a contradiction. Since Vi ^ Vj mod VQ, for * 7^ j, then for some g < VQ, we must have

It is now easy to check that VQ, . . . ,vg satisfy conditions (i) and (iii) in the statement of the proposition. D The set {VQ, . . . ,vg} of the Proposition (5.6) will be called the minimal system of generators of G, and the integer g will be called the genus of the semigroup G (notice that g < mult(G) - 1). Given a semigroup in IX, the elements of IN \ G are called the gaps of G. A semigroup may have finitely or infinitely many gaps. For example the semigroup (3) has infinitely many gaps, while the gaps of (3, 5} are the integers 1, 2,4 and 7. When the number of gaps of G is finite, there exists a unique element c € G, called the conductor of G, such that a) c-l& G b) if z 6 IX and z > c, then z e G. If n > 1, then obviously the semigroup (77) has no conductor, since no element m G IX with ra ^ 0 mod n is in (n). PROPOSITION 6.2 Let G he a semigroup in IX. The following assertions are equivalent i) G has a conductor, ii) The elements of G have GCD equal to one, iii) There exist two consecutive integers in G. Proof: (i) => (ii) Since G C (GCD(G)), it is clear that, if G has a conductor, then GCD(G) = 1. (ii) => (iii) Let VQ, . . . , vg be the minimal system of generators of G. Then clearly GCD(G] — 1 implies GCD(?;0, . - - , v g ] = 1. So, in this case, there exist integers AQ, . . . , Xg such that XQVQ H ----- \-XgVg — 1.

Transferring to the second member the terms with the A?:'s negative, the result follows immediately. (iii) => (i) Let a and a + I be two elements in G, then the set {0, a + 1, 2(a + 1), . . . , (a - l)(a + 1)}

Irreducible Plane Curve Singularities

81

is a complete residue system, modulo a. So, any integer n > (a — l)(a + 1) may be written as n — i(a + 1) + j'a, i = 0 , . . . , (a - 1), j > 0, which implies that G has a conductor c < (a — l)(a + 1).

D

Notice that in the proof of the proposition we got the estimate c < (a — l)(a + 1) for the conductor of G, when a and a + 1 are elements of G. Let G = (#0, • • • i xr} be a semigroup in IN with conductor c. We define below two sequences of natural numbers associated to the set of generators XQ, .. - , xr of G. Put e0 = .TO and n0 = 1. For i = 1,... ,r, define Cj = GCD(xo,... ,Xi)

and

ni — ——. e i

From the definition of the e,;'s it is clear that e,: e 7 ;_ 1; for all i = 1,... ,r, and also er = GCD(x0, - . . , xr] = 1. PROBLEMS 1.1) Let XQ,XI, • • • , xr be a set of generators of a semigroup with conductor. Let the rii's and e^'s be as defined in this section. Show that XQ = n\n
6.2

SEMIGROUPS OF VALUES

From now on, / will denote an irreducible element in M C /f[[A", K]]. Hence, the ring Of = K[[X, Y]]/(f) will be a domain. Let v = Vf the valuation associated to /, as defined in Chapter 4. To simplify notation, we will denote the residual class of Y modulo {/) by y and that of X by x or also by X, since this will not cause any confusion. DEFINITION 6.3 The semigroup of values associated to the curve (/) is the set S(f) = {!(/,
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The semigroup of values of a plane branch has a conductor. In fact, since equivalent branches have same semigroup, we may assume that the branch is represented by a Weierstrass polynomial /, so from the Theorem (4.5) we have that any element in /^[[T]] of multiplicity greater than mult'j-^Dy (/)(T n )) is already in Av, hence its multiplicity is in S(f}. Alternatively, this also follows from the observation we made in Section 4.1, before the Theorem (4.5), that the field of fractions of A^ is K((T)). Indeed, there exist hi,h-2 € A^ such that T — h\jhi, so mult (hi) — mult(h 2 ) + 1, which in view of the Proposition (5.7), implies that S(f) has a conductor. EXAMPLE 6.4 If / = Y2 - X3, we have that

Indeed, v(x] — I(f,X] = 2 and v(y) = !(/, Y) — 3. Since any natural number I > 2 may be written as I = r-2+.s-3, with r, .s e IX, it follows that I = v(xry8} e S(f). The conductor of £(/) is 2. EXAMPLE 6.5 If

= Y2-X5, we have that

Indeed, ii(x} — I ( f , X ) — 2, v(y] = !(/, Y) = 5. Now, it is easy to write any natural number / > 4 as / = r-2+.s-5, with r, s e IN, and therefore / = v(xrys) e S(f}. The conductor in this case is 4. EXAMPLE 6.6 Let / = F4 - X7 . Since X = T4 and Y = T7 is a Puiseux parametrization of (/), then given g = ]T\ aitjXlY:i 6 K[[X, Y}}, we have that T4j+7-? - 4r + 7.s, for some natural numbers r and .s. This shows that S(f) is generated by 4 and 7. Therefore,

5(/) = (0, 4, 7, 8, 11, 12, 14, 15, 16, 18, 19 . . .}, with conductor 18. How one can, in general, determine the semigroup <S(/)? This is not a priori an easy question to answer and an important part of this chapter will be devoted to it. The least element of S(f) \ {0} is the integer mult(/) since from the Theorem (4.19), we have that 7; (19) = I(/,.g) > mult(/), for all g € M C K [ \ X , Y } } , and if (L) is a line not belonging to the tangent cone of (/), we have that v(L) = mult(/). To study the semigroup of values of a plane branch (/), we will assume that char(K) — 0 and that coordinates have been chosen in order that / is regular in Y and the tangent cone of (/) is (Yn). So, / is of the form / - a0(X)Yn +al(X)Yn~l

+ •• • +an(X)

+Yn+lh(X,Y),

with ao(0) / 0 and mult(o ? ) > i. In particular-, we have that a n ( X ) — cXm + • • •, with m > n. We also will choose coordinates such that m is not a multiple of n (see Lemma (5.8)).

83

Irreducible Plane Curve Singularities

Let

X = Tn

with ra and n as above, be a Puiseux parametrization of (/). REMARK 6.7 In the above situation we have that ra = min(5(/) \ {n}). Indeed, ra e £(/), because v(y) = ra. On the other hand, ii K[[X,Y}}, then a,/3

From this, it is clear that min(S'(/) \ (n)) > ra, and since I(/,y) = ra, the equahty follows. So, if t>o, vi, . . . , t>s represent the minimal system of generators of 5(/), we have, in this case, that VQ = n and t>i — m. If we denote by /?o,/2i, . . ./37 the characteristic exponents of the above Puiseux parametrization of (/), we have that /?0 = n and /^ = m. We will still denote by rjj and £-, the integers corresponding to the characteristic exponents as defined in Section 3.4. Recall that we denoted the group of the rth roots of 1 in K by Ur. For j — 0, . . . 7, we put hence LEMMA 6.8 J/C € G^ \ G^+i, ^en C^'+1 7^ 1. Proof: Indeed, if the we had C^J+1 = 1> then we would have (^ = £#>+i = 1, and since €j+i — GCD(^, /3J+1), we would have (£J+l = 1, a contradiction, since

LEMMA 6.9 Let C € G fc _i \ Gk. Ifl>k, then (Gt C G f c _i \ Gk (see figure below).

Figure 1

84

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Proof: Indeed, from the hypothesis, we have that C5""1 — 1? but (£k ^ 1. Let p be any element of G\. Since £/ divides e^-i, it follows that ((p)5*"1 = 1, and therefore, Cp € Gfc-i • We now want to prove that £p ^ Gfc. Suppose by absurdity that £p e G^. Since e/ divides c^, we have that

which is a contradiction.

D

For every j = 2 , . . . , 7, we define

P-(T) = 'V^ 6 T'. i=0\

For commodity, we define /iv+i = oo. So,

Consider now, for j = 1,...,7, P j ( X » ) € AT((X«)). We want to study the following tower of extensions:

LEMMA 6.10 Let C, p 6 G0 = f/ n and j e {2, . . . , 7}. We have, that P 7 (CT) = R(PT) ^^ CP-1 e G,_!. Proof: Suppose that

Hence, C,1 — p1 for i = n , f i i , . . . , f t j _ \ and therefore, (C,p~1}1 = 1, for i — GCD(n,/?i, . . . , / ^ _ i ) = £j_i. This implies that Cp"1 e GJ-I. Conversely, suppose that f^- 1 = //j- 1 . From Section 3.4 we know that e 7 -_! divides all the exponents i such that /^_i < ?' < /^, for which bx ^ 0 and since £j-i divides e? for all / < j - 1, it follows that £ 7 -_i divides all the exponents i with /Ji < i < ftj for which /;, 7^ 0. Therefore,

proving thus that P 7 (CT) = Pj(pT). The above result gives immediately the following corollary.

Irreducible Plane Curve Singularities

85

COROLLARY6.il The Galois group of the extension

K((Xn))/K((X))(Pj(Xn))

Since the Galois group of the extension K((X"))/K((X)) is GO = Un, hence abelian, it follows readily from the above corollary that the Galois group of K((XD(Pj(X±})IK((X)) is Go/G^i. Denoting by [("] the residual class of (" in GO = C/n, modulo the subgroup G-/-I, we define which makes sense by the Lemma (6.10). It then follows that the conjugates of P j ( X n ] in K((X})* over K ( ( X ) ) , are the P,-([C]Xn), where [(] € G0/Gj-i. Therefore, Y

( - MC]*-)) e K[[x]]\y\.

The polynomial f j ( X , Y ) is of degree n/£j_i = 771772 • • -^-i in V (see Problem 3.4.1) and it is the minimal polynomial of P j ( X * } over K ( ( X } } . Consequently, we have that

We define

( & = n, if j = 0 V? = { A, if j = 1 [ «(/,-), if j = 2 , . . . , 7 Therefore, the ?;,,'s are elements in S ( f ) . The following theorem, due to Zariski [15] will give us important information about these integers, relating them to the characteristic exponents /3o, . • • , /37 of (/). To state the Theorem we will need the following notation. Define for k — 0, . . . , n, Mk = K[[X}} + K[\X}}y + K[[X]}y2 + ••• + K[[X]}yk

C Of,

where n = mult(/) and y is the residual class of Y in Of. From the Division Theorem it follows easily that M n _i = Of. THEOREM 6.12 i)

For j = 2, . . . , 7, we have that V

J~l „

V~^ £fc— 1

£fc r,

,

r>

J = 2^ -— fa+ PJ £j 1

k=l

-

//^ i \

(6.1)

ii) If h € Mfc w^/i k < i j i - - - r j j , then v(h] 6 ^=0^-^- ^n Particular,

iii) The GCD o/ej-i anrf 7jj is e^, /or j = 1, . . . ,7. Moreover, Vj is the smallest non-zero element of S(f) which is not divisible by £j-\. iv) The genus of S(f) is 7 and the integers v0, ...,v 7 are a minimal system of generators.

86

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Proof:

(i) From the Lemma (6.9) it follows that if j > fc, then each set Gk-i \Gk is

partitioned into the disjoint union of —

— cosets of GJ-I. Moreover, using the

definitions and the Lemma (6.10), we have that v(y — Pj([(,]T}} = /4, VC G Gk-i \ GkIt then follows that .7-1

(ii) We will prove this result by induction over j. Case j = 1: let // = 0,0(3:) + yai(x) + • • • + ykak(x), with k < r/i. For every ?' = 0 , . . . , r/! — 1, we have that ?)(?/a,;(.r)) = ifti

mod n.

If i and / are two distinct non-negative integers less than r\\, we have that (i - /)/?! ^ 0 mod H,

because otherwise.

which, since — and r/] = ^~ are relatively prime, would imply that r/i divides i — I. This is a contradiction because 0 < i — I < r/i. Therefore, i)(y'ai(x)) ^ v(yla,i(x}),

if ?', / < rji with i ^ I.

l

It then follows that ;>(/?) = v(y a,i(x)) for some i = 0 , . . . , k. This shows that v(h] G nlN + /^iIN = t'oIN' + vilN. Suppose now that the result is true for j — 1. Let h G M^ C O/, and suppose that k < 771 • • • r/r Suppose that h = ~h(X,Y), where h(X, Y] is a polynomial in A"[ of degree < k. Expanding h(X,Y) with respect to the powers of f j ( X , Y ) , we may write

where s < r/,, and AQ(X,Y},...,A8(X,Y] y /j(A",y) = ^ i - - - 7 / j _ i . Therefore,

G X[[JsiT]][y] are of degrees less than

//. = .40(.r, ?y) + Al (or, ?y)/7 + • • • ^(

where Ai(x,y) e A//f, with A/ < r/i • • -T/ 7 -_i, for i = 0, . . . , ,s. From the inductive hypothesis this implies, for ?' = 0, . . . , .s, that

.7-1 »;(Ai(.T,y))67;I/]\.

(6.2)

Irreducible Plane Curve Singularities

87

In particular, from (6.2), it follows that v(Ai(x,y)) = 0 mod e 7 -_i. From this and again from (6.2), it follows that v(Ai(x,y)ffi

= ivj = i p j , mod £j_i.

If i,/ < T)J and i ^ /, we have that (i — l)(3j ^ mod £j_i; because otherwise, we

which implies that r\j divides i — I, because — and r\j = —— are relatively prime. £ £ j j This is a contradiction because 0 < \i — l\ < rjj. This shows that the v(Ai(x, y)/j) are distinct for i = 0 , . . . , s. Therefore, using (6.2), we see that there exists i, with i = 0 , . . . , s, such that 3

v(h] = v ( A i ( x , y ) f j )

= v(Ai(x,y))+ivj

G J^IX, i/=0

which concludes the proof of (ii). (iii) Observe that from (6.1) we clearly have that GCD(uj, £j-\) = £j. It also follows easily from (6.1) that VQ < v\ < • • • < v7. Since, from (ii), VQ < v\ < ••• < v 7 , generate S(f), and that £ 7 ;_i divides VQ, ..., Vi^i, we have that v^ is the least non-zero element of S(f) which is not divisible byei-i. (iv) It follows from (ii) and (iii) above that VQ,VI, ... ,« 7 , is the minimal system of generators of S(f), because VQ = n = min(5(/) \ {0}), vi = f3\ = mm(S(f) \ nIN), and for alH = 2 , . . . , 7,

In particular this shows that 7 is equal to the genus g of S(f).

D

From this we see that for all i = 0 , . . . , g one has Cj = £i and

rji = n^,

where the e^s and the n^s are the integers we associated in Section 1 to any sequence of integers VQ, ..., vg. In particular, since 77, > 1 for i = 1,... ,9 (see Problem 3.4.2), it follows that n{ > 1, for i = 1,... ,g. By the above observation, and using the relations in Problem 1.1, formula (6.1) may be written as follows: Vj — ("i - 1)^2 • • • nj-iPi + («-2 - 1)^3 • • • nj-iPz H ) a• (n' i — -\\a 1) /? • i -p, k)'

1-

,„ 0, (6-3)

From this formula we get easily the following formula: vui — =n TV-ji—i1 —HI—i 3- i 4-i t^ii 3"'i—i<

i* — = ^i• 2 • • ->yn

4-} v(ftu'^*7

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Because the /i 7 's are increasing, it follows immediately from (6.4) that vi+1>niVi,

i = 0,...,.g-l.

(6.5)

A semigroup S with a minimal system of generators VQ, vi,..., vg satisfying condition (6.5) will be called a strongly increasing semigroup. So, we have shown that the semigroup of values of a plane branch is a strongly increasing semigroup. THEOREM 6.13 Let f be an irreducible power series in K[[X,Y}] with a given Puiseux parametrization. Then S(f] and the characteristic integers of (/) determine each other. Proof: Indeed, given the characteristic integers (fa, fti,..., flg} of (/), we may determine by the formulas (6.4) the integers VQ, ..., vg and therefore the semigroup S(f) that they generate. Conversely, given S ( f ) , then its minimal system of generators VQ, ... ,vg is well determined so also are the integers n = VQ, and the integers n,;, denned inductively by ^, e7- = GCD(e l -_ 1 ,^.), e0 = n. Now, the A's may be determined recursively by the formulas (6.4).

D

Next we show that the condition on VQ, . . . , vg to be strongly increasing, together with the condition n?; > 1, for all i — 1, . . . ,g are sufficient to guarantee that there exists a curve (/) such that S(f) has VQ, . . . , vg as its minimal system of generators. THEOREM 6.14 Let S be a given semigroup with a conductor and with minimal system of generators VQ, . . . . vg such that HJ > I and Vj > n ? -_i?',:_i for all i = 1. . . . ,g. Then there exists a plane algebroid irreducible curve (/) such that S(f) = S. Proof: Put n = /io = VQ, arid fi\ = VQ. By the equations (6.4) we get an increasing sequence of integers A),/:?!,/^ • • • ,fig, because of the conditions ?;7 > n,-_it',;_i. The conditions n, > 1, i = 1, . . . , g, imply that GCD(/^o, • - - , fti~i] ^ GCD(/io, . . . , A). Now, let /3i +T'6* + ... + T /3 », (p(r) = T and take

where (," is an nth primitive root of 1. It is easy to see that f ( X , Y ) G /('[[.Xjjfy] determines an irreducible algebroid plane curve (/) with characteristic exponents /io, • • • , ftg, and therefore with semigroup S. n

PROBLEMS 2.1) Determine the minimal system of generators of the semigroups of the branches given in Problem 3.4.3.

Irreducible Plane Curve Singularities

2.2) a) / b) / c) f 2.3)

89

Determine S(f] in the following cases: = Y5 - X8. = Yn -Xn+l. = Y7-X9. Show that one has, for i = 1,... ,#, the formulas: A = V{ - (rii_i - l)Vi_i

(n2 - 1)V2 - ("'1 - ]>!•

[Hint: Use formulas (6.4)] 2.4) Let (/) be an irreducible algebroid plane curve and let t>o, • • • , vg be the minimal set of generators of 5(/). Suppose that /i 6 A'[[A", Y]} is such that !(/, ti) — Vi, for some i = 0 , . . . , g, show that h is an irreducible power series.

6.3 SEMIGROUPS AND CARTESIAN EQUATIONS Up to now we have shown how to determine the semigroup S(f) through a NewtonPuiseux parametrization of (/) . In this section we will show how to determine from / itself the characteristic integers of the curve (/), and therefore, by formulas (6.4) we will be able to recover the generators of S(f). Let (/) be an irreducible algebroid plane curve regular in Y and parametrized by

LEMMA 6.15 //C e Gk \ Gk+1, then mult(p(T) -
Since

and when C € Gk \ Gk+i we have /9fc + l-l

0IH-1-

= E Since ("fc+1 7^ 1, the result follows immediately.

D

O-J /*

We will use the notation:

fy

:— ——r.

THEOREM 6.16 Let f e -K"[[A", Y]] be irreducible of multiplicity n and regular in Y. Let j, k be integers such that 1 < j < n and Q < k < g — 1. If e^+i < j' < e^, then

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90

Proof:

If M is a unit in K[[X, Y}} then

!(«/, (ufM

= I ( u f , uYf + ufY) = I(uf, w/y) = !(/, / y ).

So, we may assume that / is a Weierstrass polynomial. If (T n ,
Differentiating the above expression, we get

E

E We know by the Lemma (6.15) that the multiplicity of H depends upon the set G ? \ GJ.+ I , where C is located, and it will be smaller as i is smaller. Since e^+i < j < e-k, it follows that n — e.^ < n — j < n —fifc+i>arid therefore smallest multiplicities in the above expression will occur when A = (G'o \ Gk) U J, where J C Gk \ G^+\ (seefigurebelow).

Figure 2 Therefore, we must evaluate the multiplicity of the following sum:

E

Irreducible Plane Curve Singularities

91

From the Lemma (6.15), the multiplicity of the first factor is (n - ei)/?i + (ei - e2)#> H ----- h (ek-i ~ e k } f t k . On the other hand, each summand in the second factor has multiplicity

and leading coefficient

To finish the proof, we must show that

E

\T767

/

#J=ek-j

is non-zero. But, since 1 - rjf3k+l = 0, for 77 G Gk+i, because e fc+1 divides /4+i, it follows that (6.6) is equal to

E Observe that when 77 varies in Gk the expression r//3fe+1 assumes nk+\ =

p,

distinct values, each one repeated e^+i times. With this, we may conclude that, modulo the sign, the expression (6.7) is the coefficient of ZJ of the polynomial

which is also, modulo the sign, equal to V^

(nk^L j

, --K-,-1 , ^ Q

o COROLLARY 6.17 If ek+i <j<ek, then I(fJ$-1})-I(fjP).

(6-8)

Remark that if / is regular in X, then all results in this section are valid interchanging the roles of X and Y. The above corollary allows to compute the characteristic integers of an irreducible algebroid plane curve, given in cartesian form. For this, if / is regular in Y then it is sufficient to compute the differences (6.8). More precisely, put in (6.8) j — 2 > eg = 1. Then we have

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Next, compute the difference (6.8) for j = 3,..., until the last value of j for which it remains unchanged. This value of j is precisely e.g-\. Next, compute the difference (6.8) for j — eg~\ + 1, getting fig-\. Next, compute the difference (6.8) for j = e g _i + 2 , . . . , until the last value of j for which this difference remains unchanged. This new value of j is cg_^. Proceeding in this way we get all the characteristic integers of (/). If instead / is not regular in Y then it is necessarily regular in X. Hence we may use the same method as above interchanging the roles of X and Y. The results of this section were essentially taken from [7]. PROBLEMS 3.1)

Determine the characteristic integers of the curve (/), where / = y 4 - 2X3Y2 - 4X5Y -f X6 - X7.

[Hint: Use the computations you did in Problem 4.3.2 and then apply the above procedure.]

6.4

APERY SEQUENCE OF THE SEMIGROUP OF VALUES

In his 1947 note [2], M. Apery introduced the important notion of Apery sequence associated to a semigroup of the naturals. As we will see, several notions and invariants may be expressed in terms of these sequences. Let S be a semigroup with conductor c and let p G S \ {0}. We define the Apery sequence O,Q, . . . , op-i of S, with respect to p, inductively by the conditions: (IQ — 0 and « 9 — mm

where «, -f- pIS = {o,7; + Xp \ A € IN}.

If a £ IN. we will denote by [a] the residual class of a, modulo p, in IN. It is easy to verify from the above definition that the following properties hold: (a) a0 = 0 < ai < •• • < a T ,._]. (b) ai ^ a 7 -, mod p,

0 < i < j < p — 1.

(c) o,7 = min ([o,-| fl S}.

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93

The property (d) implies that {p, ai, . . . , a p _i} generates S over IN. Property (e) follows on one hand, from the fact that c—1— a p _i — p doesn't belong to S, because of (c). On the other hand, for all r > 0, we have that c + r = a p _i — (p — 1) + r G 5", because otherwise we would have for some i = 0, . . . , p — 1 that c -f r — a^ + Xp, with A < 0. This would imply that a p _! — ctj = Xp + p — 1 < 0, a contradiction. According to (c) and (d) above, the elements of S are of the form a^ + Xp for some i = 0, . . . ,p — 1 and A > 0; while the gaps of S are of the form a; + Xp for some i = 0, . . . , p - 1 and A < 0. When we take p = n = min (S \ {0}), then the Apery sequence with respect to n will be called simply the Apery sequence of S and the set A = {oo,. .. ,a n -i} will be called the Apery set of S. Let / 6 jFf [[X, F]j be an irreducible power series, regular of order n in Y. Recall from the Proposition (4.3) that Of = K[[X}} © K[[X]]y © • • • © K[\X}]yn-1 . Recall also that in Chapter 3 we defined, for k = 0, 1, . . . , n — 1, Mk = K\[X\] © K[[X]]y © • • • © K[[[X}}yk.

So, AT[[X]] = Mo C Mi C M2 C • • • C M n _i = Of. The next result, concerning the spaces M^, will play an important role in what follows. PROPOSITION 6.18 (Apery) Let f 6 K[[X,Y}} be irreducible, regular of order n in Y and such that n doesn't divide m = !(/, V). Put M_i = {0} amd yo = 1. Then for every k = 0,1,..., n — I, there exists an element y^ £ yk + MI--I such that

Proof: Let (Tn, yj(T)) be a Puiseux parametrization of (/) , where
Since ra is not a multiple of n, defining 7/1 = y, we have that

v ( y i ) = v(y] =m& v(Mo) = nIN. Notice that Consider the element <^0(-X')+v?i(-X')yi G -^i such that ?/2 = y 2 —
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and consequently, y2 E MI, which is absurd. We then have that y2 E y1 + MI and v(y2) & v(M\). Notice that M2 = K[[X}} + K[[X}}yi

+K[[X]}y2.

Consider the element (pa(X} + (pi(X}y\ + (f>2(X)y2 E M2 such that y3 — y5 — \ (X}y\ — (M2). Notice that A/3 = #[[*]] + K[[X]]?/1 + K[[X]]y2 + K[[X}}y3. We continue this way until we prove the result.

D

Remark that during the proof of the above proposition we have shown that Of = K[[X}} 4- K[[X]]yi

+••• + K[[X]}yn^.

(6.9)

We then have that the y^., k = 0, 1, . . . , n — 1, found in the above proposition generate Of as a K [{X}} -module. LEMMA 6.19 (Azevedo) Suppose that, for k = 0, . . . , n — 1, we hove elements y^ G ;t/fc 4- M f e _i un^/?, ^(?//c) ^ ^(M f c _i), where ?y0 = 1 «ri^ M-i = {0}. Then /or all i,j with 0 < i,j; < n — 1. and ?] -j- j < n — 17 we ha?;e that

Proof:

Write ?/,; = a + y7 and j/7 = fe 4- ?/ with a 6 M7;_i and 6 6 Mj_i . Then

y^j = 0.6 4- ay-7 4- by1 4- ?/?/ = c 4- y' +J , where c = ah 4- ay-7 4- by1 e M 7 ; +7 _i. Since ?y?+-7 = ?/j + j — d, for some d 6 MJ+J_I, it follows that 7/j?/7- = c — d + y1+j. Define e = c — d E M,; +J _I. If v ( y i + j ) < v ( y i y j ) , then v(yi+j] = v(e) € ^(A-/ i + 7 -_i), which is a contradiction because v(yi+i) £ v(Mi+j-i). D

REMARK 6.20 We have that v(y,} < v ( y j ) , whenever 0 < i < j < 77. - 1. Indeed, since for / > 1, ('(?/,;) ^ ?;(A/j_i), we have that ??(?/,;) 7^ 0, hence it follows that v(yj) > '(;(?/,.) 4- i ' ( y j - i ) > t'(?/i)PROPOSITION 6.21 Let f E /f[[X, Y]] be irreducible and regular in Y of order n. Put yo — I and let ?/i, . . . , ?y n _i be elements of Of such that y^ E ;(/A" + Mk-i, and v ( y k ) & v(Mk-i)- Denoting by [r] the residual class of the integer r, 'modulo n, then for all k — 0, . . . , n — 1, we have that (i) v(Mk)=U*=0{v(yi)} (ii) v(yi) ^ v(yj)

+ n}K;

mod n, for all i,j = 0, . . . , n — 1, w?'£h ?' ^ j.

(iii) v(yk] = min ([v(yk)\ n 5(/)), /or a// fc = 0, . . . , n - 1.

Irreducible Plane Curve Singularities

95

Proof: If n = 1, we have nothing to prove since in this case, S(f) — IN, and assertions (i), (ii) e (iii) are trivially satisfied. We then may assume that n > 1. (i) We are going to prove this part by induction on k. For k — 0, the assertion is easily proved since every element of X [[X]] is the product of a power of x by a unit in this ring, hence its value will be a positive multiple of n. Suppose then that for some k such that 1 < k < n— 1, we have v(Mk-i) = [v(yi) + An; 0 < i < k - 1, A > 0}. We will prove the assertion for k. Since Mk = Mk_\ + K[[X]]yk, we may write any element (3 of Mk in the form ft = a + a ( X } y k , with a E M fc _i and a(X) G A"[[Jf]]. We initially prove that v(a) ^ v(a(X)yk}. Indeed, if the contrary was true, using the inductive hypothesis, we would have, for some i < k — 1, a relation of the following type and therefore, in view of the Remark (6.20), v(yk] = v(yj 4- (A — //)n, with A > //. Hence, again from the inductive hypothesis, v(yk} £E v(Mk-i), which is a contradiction. Hence, from the inductive hypothesis, v(a) ^ v ( a ( X ) y k ) . It follows from this that either, v(/3) — v(a) e v(Mfc-i), or v(/3] = v(a(X}yk] — v(yk] +//n, which proves the result. (ii) Suppose by absurdity that v(yi] = v(yj) mod n, for some pair of integers ( i , j ) with i ^ j. Suppose that v(yj) = v(yi) + Xn, and without loss of generality that j > i. It then follows from the Remark (6.20) that A > 0, hence from (i), v(yj) e •y(Mj), which is a contradiction. (iii) It follows from (ii) that each residual class of integers modulo n contains exactly one of the integers v(yi\ i = 0, . . . ,n — 1. On the other hand, the only elements in the residual class of v(yi) are from (i) the integers of the form v(yi] + An, with A > 0. Therefore, v(yk] = min ({v(yk)} H S(f)) ; Vfc, k = 0, . . . , n - 1. D

Conditions (ii) and (iii) in the above proposition say that aj = v (?/j), i — 0, . . . , n — 1, is the Apery sequence of S(f). COROLLARY 6.22 Let i and j be two distinct integers, i, j — 0, . . . , n — 1, and let ai(X),aj(X)eK[[X]]\{0}. Then v(ai(x)yi) ^ v ( a j ( x ) y j ) mod n. Proof: We have already observed before that v(oti(X)) = Ajn for some natural number A^. If we had v(oti(X}yi) = v(aj(X}yj), then assuming j > i, we would have 0 < v(yj) — v(yi) = (Aj — Aj)n. It would follow that A, > A^, hence v(yj] e v(Mj) C f(Mj_i), a contradiction. D COROLLARY 6.23 We have Of = K[(X}} 0 K[[X]]yi 0 • • • 0 Proof: In view of (6.9), it is sufficient to prove that yo,...,yn-i are independent over /f[[A"]]. In fact, if we had a non-trivial relation ao(X) + ai(X)yi -\ ---- +

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a n _i(X)7/ n _i = 0, from the Corollary (6.11) above, it would exist i e {0,..., n — 1} such that Qii 0 ami oo = v(ty — "(rto(^0 4- f*i(X}yi -f- • which is a contradiction.

D

COROLLARY 6.24 Letl = z0,zi,..., z n _i e O f 6e suc/i tfia* (i) zk£yk + M f c _i, Vfc = 0 , . . . , n - 1, and (ii) ^(-£o)> ^(21)1 • • • i "(^n-i) are pairwise non-congruent modulo n. Then v(zk] — v ( y k ) , for all k = 0 , . . . , n — 1. Proof: From Proposition (6.21.iii), it is sufficient to show that v ( z k ) ^ v(Mk_i] for all AT = 1,..., n — 1, since (i) arid this condition determine uniquely the v ( y k ) . Since from the Proposition (6.21.i), i;(Mfc_i) intersects only k residual classes modulo n, and since v(zi) € '('(M^-i), for all i = 0 , . . . , k — 1, it follows that v ( z k ) ^ v(Mk_i), which proves the assertion. n Since the Apery set of a semigroup is a set of generators, it is clear that the Apery set and the minimal set of generators determine each other. In the next chapter we will see how, in the case of semigroups of values of branches, we can obtain the Apery sequence from the minimal set of generators of the semigroup and how to recognize which elements of the Apery sequence form the minimal system of generators of the semigroup. PROBLEMS 4.1) Determine the Apery sequences for the following semigroups: a) (5,8). b) ( n , n + l ) . c) (6,9,19).

6.5

SEMIGROUPS AND BLOWING-UPS

In this section we will relate the semigroup of a singularity of a plane branch with the multiplicity sequence obtained from the desingularization process that we discussed in Chapter 5. Let / 6 K[[X, Y]] be an irreducible power series of multiplicity n and regular in Y. Then from the Proposition (5.12) it follows that if /W = a*(/), then S(f) C PROPOSITION 6.25 Let f e K[[X, Y}} be an irreducible power seri.es regular in Y and with tangent cone (Yn). Let f^ =
Irreducible Plane Curve Singularities

97

Proof: In the above conditions, from the Proposition (5.7), we have that cr*(/) is regular of order n in Y\ and is such that ^(D (x] = n. Since a'Q, a'1? . . . , a'n_l form a complete residue system modulo n, the set {a^, a( + n, . . . , a'n_1 + (n— l)n} is also a complete residue system modulo n. From the Corollary (6.24), it will be sufficient to show that for every k = l , . . . , n — 1, there exists yk E Mk_i + yk, such that v(yk] = a'fc -f fcn. For every A; = 0, . . . , n — 1, define M^_ : in an analogous way as we did for Mk with / (1) in the place of /. Let y'k G (|-)fc + Mk_lt be such that v(j/£) = a'k, whose existence in Of(i) is secured by the Propositions (6.18) and (6.21) and put y'0 = 1. Multiplying y'k by X fc, for A: = 0, . . . , n - 1, we get yk = Xky'k € M fc _i + yk . Hence v(yk) — v(y'k] + v ( X k ] — a'k + kn, finishing the proof. D When the power series is regular in X, we may apply a similar argument using T instead of a, showing that the result may applied to the sequence of semigroups occurring in the canonical resolution of (/) . COROLLARY 6.26 Let c and c^ be respectively the conductors ofS(f) Then c = c^ + n(n - 1).

and S(/(1)).

Proof: We have only to use the above relations among the Apery sequences of S(f) and S(f^} with respect to n and the relations c = a n _i — (n — 1) and c^ =

Notice that the equality c^ — c — n(n — 1), we have just proved, gives us another proof that after finitely many blowing-ups we get at the end a non-singular algebroid curve. COROLLARY 6.27 S^/^) and mult(/) determine S(f]

and conversely.

COROLLARY 6.28 Two plane branches are equisingular if, and only if, they are equiresoluble. Proof:

Consider the canonical resolution of a branch (/):

Since / ^ is non-singular, its associated semigroup is IN. Since the Apery sequence of IV is given by a\ = i, then from Proposition (6.25) we have that the Apery sequence of S ( f ( N ~ 1 ^ ) is determined if we know the multiplicity of y^"1) and vice-versa. Using an inductive reasoning, we see that the multiplicity sequence n^ of the /W determines and is determined by the semigroup S(f}. D Corollary (6.11), used repeatedly gives us the following formula for the conductor: r

c = ^nw(n(i)-l), i=0

where n^ = mult(/).

(6.10)

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EXAMPLE 6.29 In the case of the curve h = Y4 - X7, we saw in Chapter 5 that the multiplicity sequence in the canonical resolution is (4,3,1), which gives c — 4(4 — 1) 4- 3(3 — 1) = 18, in accordance with the result in the Example (6.6). PROBLEMS 5.1) Determine the sequence of semigroups and their respective conductors in the canonical resolution of the curves in Problem 2.1, Chapter 5.

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7

99

Semigroups of the naturals

In this chapter we will study semigroups of natural numbers from a purely arithmetical point of view. This will show that most of the properties of semigroups of values of plane branches are consequence of the very basic inequality (6.5) we proved in Section 6.2 involving their minimal system of generators.

7.1

SEMI GROUPS WITH CONDUCTORS

Given a semigroup G = (x0, . . . , x r ) in IN, then any element in G may be represented in several ways in the form H ----- 1- A r x r ; A 0 , . . . , A r G IN. For example, if G = (x 0 ,xi) = (3,5), then we have 45 = 15xo = 9xi = 5xo + 6x1 .

However, when G = (XQ, . . . , x r ) is a semigroup with a conductor, we will show in the Proposition (7.2) below that the elements of G may be represented uniquely as a combination of special type of the elements XQ, . . . , x r . Let Hi and e^, i = 0, . . . , g, be the integers associated to XQ, . . . , xg as defined in the Section 6.1. The following lemma will be fundamental. LEMMA 7.1 Let XQ,XI, . . . ,x r G IN with GCD(xo,xi, . . . ,x r ) = 1 and let ei and HI, i = 0, . . . , r, be their associated integers. For every m G IN there is a unique solution for the congruence r

m = 2_^ sixi mod XQ; with 0 < Sj < n i5 i — 1, . . . , r. i—l

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Proof: By induction on r. Case r — 1: Since e\ — GCD(xo,xi) = 1, it follows that HI = — = — = .TO, and A.XQ + iix\ = 1 for some integers A and /x Then el 1 m = mAxQ + m//,.xi. Dividing rn/y, by XQ, we get q and ,si with 0 < si < XQ = ni, such that /.tm = qx.Q + .s'i-

Putting together the above two displayed equalities, we get m = ,siXi mod XQ, with 0 < .s'i < HI. Let us now suppose the result true for r > 1. Let XQ, . . . , xr+\ be positive integers satisfying the hypothesis of the Lemma. Consider the sequence x0 = —, x\ — er

— , . . . x' = —, which satisfies the inductive assumption; that is, for every integer er er m' there exist .s, and A such that r

in! — V^ .s-jx'j + Ax 0 , with 0 < si < ri-, ?' = !,...,r, where

By hypothesis, there exist AQ, . . . , A r+ i such that

and therefore, for every integer m, we have AJ.X! -\ ----- (- X'rxr + A / r+1 x r+ i. Dividing A'r+1 by e r , we get integers q and .sr+i, with 0 < A>+I < er = nr+i, such that A^.+1 = qer -+- ,s r +i- Therefore, 7,,, = AO.XO + AI-TI H ----- h \'rxr + (qer + sr+i)xr+i. Since er divides .TO, - - • , x r , there exists an integer m' such that

Now, the result follows by writing in! in terms of the the x'-, as above, and using the fact that x, = x'e r , i = 1, . . . , r. Uniqueness Since we have proved the existence of the solution of the congruences, the uniqueness will follow from the fact that

D

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PROPOSITION 7.2 Let XQ,XI, ... ,xr be natural relatively prime numbers. i) Every natural number m has a unique representation as r *•

ri ^

.

•

_ rn

-i

ii) If m > Y^i,=-i(ni ~ l) x i ~ x o> then in the above representation SQ > 0. Proof: Part (i) is a consequence of the Lemma (7.1). To prove (ii), suppose that m > Y^i=i(ni ~~ tyxi ~~ xo- From (i) we have a unique representation m —

SiXi, 0 < Si < n^ i = 1, . . . , r, SQ € TL. i=0

So, from the hypothesis we get r

r

r

]P SiXi =m> y^(nj - l)xi - XQ > ^ SiXi - XQ, i=0

i=l

i=l

which implies that SOXQ > —XQ, giving SQ > 0.

n

COROLLARY 7.3 Let G be a semigroup in IN, with conductor c, and let XQ, . . . ,xr be any set of generators of G. Then we have

- l)Xj - x0 -f 1. Notice that in the representation of the Proposition (7.2), we have that if SQ > 0, then m £ G. The converse is not always true; that is, we may have m G G with SQ < 0, as one can see in the following example. EXAMPLE 7.4 Let G = (8, 10, 11) = {0, 8, 10, 11, 16, 18, 19, 20, 21, 22, 24, 26, 27, . . .}.

We have that the conductor of G is c = 26. Also, EQ = 8, e\ = 2, 62 = 1, n\ = 4 and n2 = 2. The element 22 e G, in the above representation, is written as 22 = 3 • X\ + 0 • X2 — XQ.

There is an important class of semigroups in IN, where this converse, we mentioned above, is true. To describe this class of semigroups, let us define the following notion. Let XQ, • • • , xr be a sequence of relatively prime integers, and let the n^'s be the associated integers. We say that this sequence is nice if, for alH = 1, . . . , r, we have

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PROPOSITION 7.5 Let .TO, .. - , xr be a nice sequence of integers. IfG = {.TO, . . . , xr), then i) every element m 6 G is uniquely representable in the form m = 2^SiXi, 0 ?;=o ii) the conductor c. of G is given by

Proof: (i) Let m — XQXQ -f • • • + \rxr G G, with A7; G IN. Write A r = ^ r n r + ,sy with 0 < «sy < nr. Then we have m = .syxy + qrnrxr + AQ.TQ + • • • + A r _ix r _!, 0 < «sy < n r . Since the sequence is nice, we get that qrnrxr + AQ.TO H

h A r _ i x r _ i G {.TO, - - - ^r-i}-

Now, repeat this procedure to the coefficient of x r _! of the above element written as a combination of .TO, • • • , -T r _i with natural coefficients, and so on. (ii) From the Corollary (7.3), we know that

Now, since from (i) we have that Y^l=\(ni ~~ ~^)xi ~ xo & G, the result follows. D In reality, it can be proved that assertions (i) and (ii) in the Proposition (7.5) are equivalent to each other, and also equivalent to the fact that .TO, . . . , .xy is a nice sequence (see for example [1]). A semigroup G in Es with conductor r will be called symmetric if,

Mz 6 TL,

z£G<=>c-l-z& G.

PROPOSITION 7.6 Let G be a semigroup in IN with conductor c. The following assertions are equivalent, i) G is symmetric. ii) 2 | G n [ 0 , r ) | = r ,

iii) 2 1 N \ G =c. iv) 1N\G = |G'n[0,c-) .

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Proof: Since (IN \ G) U (G n [0,c)) = [0,c), it follows that (ii),(iii) and (iv) are equivalent . Consider now the bijection ¥>:[0,c-l] 2;

— i—»

[0,c-l] c—1—z

Since G is a semigroup and c— 1 £ G, we have that <^(G n [0, c)) C IN \ G. Now, G symmetric is equivalent to the fact that (/?(G n [0, c — 1] ) = IX \ G. This proves that (i) is equivalent to (iv). D So, if G is symmetric, c is even and there as many gaps as non-gaps of G in the interval [0, c). PROPOSITION 7.7 Every semigroup generated by a nice sequence is symmetric. Proof: Let XQ,. . . ,x r be a nice sequence. We know from the Proposition (7.5.ii), that the conductor c of G = (XQ, . . • , xr) is equal to Y^i=i(ni ~ ^-}xi ~~ xo + 1. Now from the Proposition (7.2.1), we also know that any z 6 IN may be written uniquely as

and, from the Proposition (7.5.1), 2 € G if and only if SQ > 0. Now, consider r

c- 1- z = y^(nj - 1 - Si)xj - (s0 4Hence, we have that s0 > 0 if and only if — (SQ + 1} < — 1, which proves that G is symmetric. n

PROBLEMS 1.1) Let 1 < XQ < x\ be two relatively prime integers. Show that XQ, x\ forms a nice sequence. Conclude that the semigroup they generate is symmetric and its conductor iS C- (VQ - l}(vi - 1).

7.2

STRONGLY INCREASING SEMIGROUPS

In this section we will explore some properties of strongly increasing semigroups, which will be used intensively in what follows. Let XQ, a?i, • • • , xr be a sequence of natural relatively prime numbers and let 6j and nj, i = 0,... ,r, be their associated numbers. We say that the sequence is strongly increasing if Xi > n;_iZi_i, Vz = 1, ...,r. It follows immediately from the above definition that a strongly increasing sequence is an increasing sequence.

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LEMMA 7.8 Let XQ < x\ < • • • < xr be a strongly increasing sequence of natural numbers. Then n

J - iH?' ?' = 0, . . . , r - 1.

.7=0 Proof: By induction on i. For i = 0, this is obvious. Suppose the inequality true for x,;, then i-l

i

.7=0

3 =0

D

PROPOSITION 7.9 Every strongly increasing sequence of natural numbers is nice. In particular, it generates a symmetric semigroup. Proof: Let XQ < x± < • • • < xr be a strongly increasing sequence of natural numbers. We will show that this sequence is nice, which in particular, by the Proposition (7.7), will imply that the semigroup generated by it is symmetric. Consider the sequence x'Q = — , x( = — , . . . ,.x' = — . Then it is easy to verify

From the Lemma (7.8), we have that

.7=0

which implies that

4^>EK-iK.7=0

and therefore, from the corollary (7.3) it follows that

which implies clearly that riiXj € ( i c o , ^ , . . . ,

Recall from Section 6.2 that a semigroup G is called strongly increasing if its minimal system of generators is strongly increasing. Let x.y e IX r . We will say that x is smaller than y in the reverse lexicographical order, writing x -< y, if the last non-zero coordinate of y — x is positive. This establishes a total order relation in IN r .

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Let XQ, ... ,x r be a sequence of positive relatively prime integers. Consider the integers n o , . . . , nr associated to XQ, . . . , xr and define the set 7—1 /

\

f /

\

-- TXTr

f\

^

-.

-1

m1

,— TXTT

E(XQ, ... ,x r ) — {(.Si,..., sr) € JM ; U < Si < n i? ?, = 1,..., r) C LM . We will always consider E(XQ, . . . , x r ) equipped with the reverse lexicographical order and IN with its usual order. LEMMA 7.10 7/x 0 ,..., xr is a strongly increasing sequence, then the map ..,xr)

—>

IN,

preserves orders. Proof: Clearly, it is enough to prove, for all j = 1,..., r — 1, and all ,^ with Sj < n^, i = 1, . . . , j , that Z^

The above inequality is proved as follows: y^J

s.x.

<

(n, .

j j

PROBLEMS 2.1) Give an example of a nice sequence which is not strongly increasing. 2.2) Determine the conductors of the semigroups of the branches in Problem 3.4.3. [Hint: Use the computations you did in Problem 6.2.1.]

7.3

APERY SEQUENCES

Recall from Section 6.4 the definitions of the Apery sequence and the Apery set of a semigroup S. PROPOSITION 7.11 Let S = (XQ, . . . ,xr), where XQ, . . . ,x r is a nice sequence. We have that

where n\, . . . , nr are the integers associated to XQ, . . . , xr and A is the Apery set of S.

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Proof: We know from the Proposition (7.5) that every m G S may be written in a unique way as r

$iXi -f .SO-TO, 0 < .s,: < HJ,

i = 1 , . . . , r and .s'o G IN.

This implies that each sum X^=1 .s,;.T?: is minimal in its congruence class modulo n — XQ. so we have
0 < -s, < n?:, % — 1,.. ., 1 r > C X. J

Now, since / "

•

V 1— J-

\ x

the result follows.

D

LEMMA 7.12 Let xo,...,xr be a sequence of positive integers such that GCD(x0, ... ,xr) = I. The map

is an order preserving map that is a bisection between E(XQ, . . . , xr] and {0,1,..., n — I } , where n = XQ = n-\_ ... iir. Proof:

The inequalities

. . HJ - n0

_

show that A preserves orders and that its image is contained in the set {0, 1, . . . , n — 1}. Since A is order preserving, it follows that it is injective and because both sets E(XQ, . . . , x>) and (0, 1, . . . , n — 1} have the same cardinality n, it follows that A is a bijection onto this last set. D The next result will relate the Apery sequence and the minimal set of generators of a strongly increasing semigroup. More precisely, since we know that

but not necessary in the same order, we want to identify which Oj corresponds to a given sum ^1=1 sixii an(l which element of the Apery sequence corresponds to a given generator ?;7. This will be easy to answer when the semigroup is strongly increasing.

Irreducible Plane Curve Singularities

107

THEOREM 7.13 Let DO, • • • , vg be a strongly increasing sequence of integers and let a o , . . - , a n _ i , where n = VQ, be the Apery sequence of G = ( v Q j . . . , v g ) . Then one has: i) //(«!,..., sg) G E(v0,..., vg), then r ;•_, Sin 0 ...ni_i

=: \

«.?).•

ii) For i = 1,..., g, Vj = a » . e

i—1

Proof: The proof of (i) follows immediately from the facts we proved in the Lemmas (7.10) and (7.12) that £ and A are maps preserving orders. To prove (ii) just observe that ( S I , . . . , S P ) = ( 0 , . . . , 1,... ,0), where 1 is in position z, gives us that Vi = ano...ni_l = a-^-

a

7.4 APPLICATION TO SEMIGROUPS OF VALUES Since the semigroup of values S(f) of a plane branch (/) is strongly increasing, from the Proposition (7.9) its minimal system of generators VQ, ... ,vg forms a nice sequence. Hence from the Proposition (7.5) the conductor of S(f) is given by 9

c = y~](nj - l)vi - v0 + 1, i=i

(7.1)

and from the Proposition (7.7), it follows that S(f) is a symmetric semigroup, and hence from the Proposition (7.6), c is an even integer. PROPOSITION 7.14 Let (/) be a plane branch and let ?;Q, . . . , vg be the minimal system of generators of S(f). If c is the conductor of S ( f ) , then C = ngVg - (3g - VQ + 1.

Proof:

Proposition (7.5) combined with Problem 6.2.3 imply that

c - ngVg - [vg - (rig-i - I)v 9 _i

(HI - 1)^] - v0 + 1 - ngvg - flg - v0 -f 1. D

COROLLARY 7.15 Let (/) be an algebroid irreducible plane curve. I f v Q , . . . , v g is the minimal system of generators of S(f) and c its conductor, then

Proof:

From (6.3) we have that

vg = (n! - I)n 2 • • • ng-i0i + (n2 - I)n3 • - - ng_i/32 -\

h (ng-i - l)/3 s _i + Pg,

which replaced in the formula of the Proposition (7.11), gives the result, taking into account that (3o = v0 and that gj = ni+i • • • ng (see Problem 6.1.1). D

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Hefez

COROLLARY 7.16 Let f e K[[X, Y}} be an irreducible power series. Let n = !(/, X] and rn — !(/, Y}. If c is the conductor of S ( f ) , then

!(/, /Y) = <' + n - 1,

ami !(/, f x } = <' + rn - 1.

Proof: Suppose that / is regular- in Y (the case / regular in X is analogous). From the Theorem (6.16), corresponding to the case k — g — 1 and j — 1, we have that

which is equal, from the Corollary (7.15), to c + n — 1. Let (;r(T),7/(T)) be a primitive parametrization of (/). From the chain rule we have that fx (x(T},y(T}}x'(T] + fY ( x ( T ) , y(T}}y'(T] = 0, hence !(/,/*)

= rnulta Y (.r(T),;y(T))) = mult(/y (:/:(7 ), y ( T ) ) y ( I ) ) — mult(.7: (./ )) = c 4- n — I + rn — 1 — (n — 1) = c + rn - 1. 1 J_ / J*

/

/ T1 \

/ rr~i \\

/ / f~T~i \\

1L. {

I IT^ \ \

D

In [10], Milnor introduced the following number: //./ :=dim R '—;—'—-. U X ? JYI

The theorem below, proved by Milnor by topological methods, was reproved algebraically by Risler in [12]. First we need a proposition. PROPOSITION 7.17 Let f e K\{X,Y}} be irreducible and regular in Y. Then

Proof: Suppose that (/^)y 1 = p"1 . • .p" r , where each p7 is an irreducible power series. So,

1=1 and

So it is sufficient to prove, for (p) any branch of ((/( ^yj, that

Suppose (p) parametrized by power series (<^, V>) in T. Then (1) JU" • / ( 1 ) (^,V') = V"-1 [n/ (v, VO 4-^(/ ( 1 ) )x 1 (^V')]j ^t/J L (LJ-

(7-3)

Irreducible Plane Curve Singularities

109

Since mult ( -^-^ • / (1) 0/>, V-') J = « • mult (yj) + mult (/(1)(v^')) - 1, and \dT J \ /

(n- l)mult(^) -Hmult we get (7.2) from (7.3).

D

THEOREM 7.18 Let (/) fee a plane branch. If ^f is Milnor's number of (/) and c the conductor of S ( f ) , then ^/ = c. Proof: Suppose that / is irreducible of multiplicity n and regular in Y (the other case is analogous). From the definition of strict transform we have XlYl).

(7.4)

Differentiating expression (7.4) in YI we get

hence

Differentiating (7.4) in Xi we get n/(1) +Xl(fM)Xl

= (/x) (1) +*i(/y) (1) -

(7.6)

So, from (7.6) we get

hence from the Proposition (7.17) and (7.5) we get

I(/ (1) ,(/y) (1) ) = I((/x) (1) ,(/y) (1) ),

(7.7)

Now, using the Proposition (5.13) we get I(/ (1) , 0V ) (1) ) = ~n(n - 1) + !(/, / y ) = -n(n - 1) + c + n - 1.

(7.8)

and = -(n - I)2 + I(/x, /y).

Putting (7.7), (7.8) and (7.9) together we get the result,

(7-9)

D

Theorem (7.18) implies in particular that Milnor's number is invariant under the relation of equisingularity of curves which, in turn, implies that it is invariant under equivalence of curves. We now will relate the semigroup of a plane branch (/) with that of its strict transform

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THEOREM 7.19 Suppose that (/) is a plane branch with tangent cone (Yn) and that !(/, y) is not divisible by n. Lei r>o, - . - ,vg is the minimal set of generators of S(f). Then the minimal system, of generators of S(a*(f}} is given by

where

i)

VQ < v{ < • • • < v'g

if

ii) iii)

v( < v0 < v'2 < • • • < v'g v{ < v'2 < • • - < v'g

if VQ > v(, and v( J(VQ, if VQ > v{, and v( VQ,

7/o < v{ ,

2

, . . . , uj), j = 1, • • • , g-

The proof of the above theorem is a consequence of the Theorem (7.13) and will be left as an exercise.

Irreducible Plane Curve Singularities

8

111

Contact among branches

In this chapter we will study the contact among branches. This will provide us the last and till now missing way to compute the intersection index of two branches when both are given in parametric form. The contact formula is classical and has many applications. It was used by O. Zariski in [14] to study the saturation of the local ring of a singular curve and by M. Merle in [M] to describe the behaviour of the generic polar of a plane branch.

8.1

THE CONTACT FORMULA

Let (/) and (g) be two branches denned by two power series regular in Y, both with tangent line (Y) arid Puiseux parametrizations:

x = Tn (8.1)

= Tn' with characteristic integers (/30, . . . , /3g) and (/%, . . . , /3'g,) respectively. Let £ and £ denote respectively an nth and n'th roots of unity. We will say that the branches (/) and (h) have contact order a € Q U {oo}, if nn' By changing, if necessary, the above Puiseux parametrizations of (/) and (h) by equivalent ones, we may suppose that

-

nn If (/) and (h) have distinct tangent lines then the contact order is not denned, and if they have same tangent line but distinct from (Y), then we change coordinates in order that (Y) becomes the common tangent line.

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From the above definition it follows that if, (/) and (h) have contact order a, then the power series (p(Tn ) and 4J(Tn] coincide up to order arm' — 1. If S(f) = (vQ,...,vg} and S(h) — (v'0,. . . , v ' g l ) , we will use, as previously, the notation n, , e; , n'- and e' for the associated integers to the Vi and to the v'- respectively (or which is the same to the fij and the /^, respectively). REMARK 8.1 Observe that if mult(<^(r n ' ) - 4'(Tn}} is realized by a power of a term in <^(Tn ), then nn'a = in' for some integer i. Alternatively, if the multiplicity is realized by a power of a term in i{>(Tn), then nn'a = jn for some integer j. So, either na or n'a is an integer and if, for example, na is an integer with fiq < i = na < ftq+i, for some q (where we put fig+i = oo), then eq divides na. Analogous result is valid in the other case. PROPOSITION 8.2 Let (/) and [h] be, two branches with contact order a such that ftq ^

^, M?+l

—
-. ,

i

t ••• jor some q > 1. rrLnen

0<*<
Proof:

if a =

-,

Let us write

and with 6j0 • • • 6,r + 1 7^ 0, b'- • • • b'^

i

7^ 0 and n'ir < ann' < n'ir+i.

Suppose that ann' = rwn{n'ir+i,njs+i}, then r = s and n'if = iijt,

bit=b'jt,

^ = 0,...,r.

On the other hand, 6 V + 1 7^ 7/7 if n'ir+i = njr+\. Now, since n'/^q < ann' < n'fiq+1 and GCD(r/n, n ' y : 0 , . . . , n/v^) = n'GCD(n, i0,...,«/) - nGCD(n', jo, - - - , jf), the result follows from the definitions of the e ? ,
D

COROLLARY 8.3 Let (/) and (h) be two plane branches with characteristic integers respectively (/ig, ft\,..., tf g } and (ji'Q, / 1 [ , . . . , /^'/). Suppose S(f) — (t'o, • • • , '';g) and S ( f ' ) = (v'0,.. , , v ' g , } and that (/) and (h) have contact order a with ^ < a < ^^, where q > 1, then

'or

<

I o < ?; < o,

•/:/ a >

Irreducible Plane Curve Singularities

Proof:

113

From relations (6.4) we get the following:

n _ 0o _

_ A

^~^~""~^

4 = > n_ _

v0 _

_ Vi

^~^~"'~^r

The result now follows from the Proposition (8.2).

D

Remark that if n'e^ = ne\ for i = 0 , . . . , r, then n?: = n\ in the same range. The above Corollary motivates the following definition: p = max (j € IN; -, = ^, Vi, 3

"I

I

0 < i < j} .

V'Q

(8.2)

J

If S(f) = S(h), then p = g = g'. On the other hand, if 5(/) ^ S(h), then ' ^ fl'p+in and the contact order a among (/) and (h) satisfies the inequality:

Thus, we have n

(8.3)

n

which is also valid when S(f) = S(h) if we set, as we already did, /?p+i = oo. Recall the following definition:

G, = {CeK; C e i =l}THEOREM 8.4 Let (/) and (h) be branches defined by power series regular in Y, with tangent line (Y), parametrized as in (8.1). Suppose that (/) and (h} have contact order a and let S(f) = (VQ, . . . ,vg). If a < — , then I ( f , h ) = ann' . In addition, if we put no = 1, then the assertions below are equivalent. i)

— < a < —— , for some q = 1 . . . , g. ,

mult(h) Proof:

no...n9_i

n\...nq (n — 1}. Since / is associated to the F-polynomial

Let GQ = {( e K;

(Y ' (P(^) ' it; follows that I(/,/i) = mult/(r"',^(T)) = mult

mult

-E n ^—' E -—' 2

»=1 C e G i _ i \ G j

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where (?g+i = 0. From the above formula it follows that if a. < —, then !(/, ti) — ann'.

i) => ii)

Suppose now that — < <x < ———, for some q = ! , . . . , # . If we put n n G__i = 0, then from the Lemma (6.8), for ( <E GV_i \ G ? , i = 1,... ,q — 1, we have that mult U>(Tn) - ^Tn']\ = rifa. (8.4) At this point we divide our proof into two cases. Case 1)

a = —. In this case, when C 6 G g _i, we have that

mult V'(^ n ) - vKOf 1 ")

= mult ( ( T n ) - (^(T n )

= ri fi q = arm'.

(8.5)

The above equation together with (8.4) imply g 1 1 ( ~ \ !(/,//,) = - e.q-iannf + V(e ? _i - e^ri fa , ri \ ^—' / \

7=1

/

which from formula (6.1) gives

— < a < q+ . In this case, for £ 6 G a , we have that (8.5) holds. This n n together with (8.4) imply that Case 2)

From this last equality together with (6.1) and (6.4) we may conclude that ^— = eqa + -eq(vq+i - (3q+i) = eqa + -eq(nqvq -fiq]— n' * ri " ' — , n no. -

n0...nq-i ii) => i)

ni...nq

Suppose that, for some q = 1, . . . , g, l(fjl)

mult(/i)

Vq

ri0 . . . nq_i

-r

HI ... nq

If (/) and (h) have contact of order a, with — < a < '• , we have to prove n n that g = g and o- = a. From the first part of the proof, we have that q > 0 and

Irreducible Plane Curve Singularities

115

The proof will be complete if we show that if q > 0, q' > 0 and q' ^ q"', then Va',a" with ?*- < a' < ^1±L and ^ < a" < n

n

n

n

we have --^ '

- rq /3q>

Vq" -q

no" - j3q

no...?V/_i

n\...nq»

,

n\...nq>

Indeed, if q" = q' -+- 1, then from (6.4) we have ,

HO ... nqi =

n\... nq'+i Vq'

"i

ftq'

n0...7V_i

(- -*

n\... nq>

+l -

fiq'

:<- >

ni...nq>

n\ ... nq< Vq>

2

n0...nq'_i

1

not' - /3q>

1-

n\ .. .nq>

where the last inequality follows from the condition — < a' < n now follows from these considerations.

q+

n

. The result D

The Theorem (8.4) will allow us to compute the intersection index of two branches when both are given in parametric form.

PROBLEMS 1.1) Determine the contact order and the intersection index for the following pairs of branches: a) (T 4 ,T 6 + T8 + T9) and (T2, T3 + T4 + T5}. b) (T 4 ,T 6 + T8 + T11) and (T2, T3 + T4 + T5). 1.2) Prove in details the Corollary (8.3).

8.2

PAIRS OF BRANCHES WITH GIVEN SEMIGROUPS

Let F = (VQ, ... ,vg) and H = (v'Q,..., v'g,} be two strongly increasing semigroups. We will put vg+i = oo. Our objective in this section will be to determine the set F(F, H) = {!(/, /i); S(f) = F and S(h) = H} .

This will be done here in characteristic zero, since we use Puiseux parametrization. In arbitrary characteristic this was done by V. Bayer in [4]. Motivated by the Corollary (8.3), we define the integer p(F, H} as being the integer p in (8.2) THEOREM 8.5 With notation as above, we have T(F,H) = {e'^Vi + jcjci, 0 < i < p, 0 < j < 7ri+1} U {loo} where e_i = e'_^ = 1,

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and

Proof: We will prove a stronger result. Namely, for (/) any fixed branch with 5(/) = F and when h varies in such a way that S(h] = H, then we have |I(M); heK[\X,Y}},

S(h} = H}=T(F\H).

The intersection index !(/, /;,) will be computed using the formula in the Theorem (8.4). {-}'.

ftr

Recall from the Corollary (8.3) that —7 = — for i = 0, . . . , p. n n If for some i = 0, . . . p — 1 we have n

it'

n

(8.6)

n'

then from the Theorem (8.4) and the remark after the Corollary (8.3) we have

Since e'^na—fa] — ei(n'a—/?'), it follows from the Remark (8.1) that e'^na—fa] = je'jCi for some integer j. Since a GQ varies as in (8.6) then we have j varying in the interval _ na - fa fa+i - fa _fi'i+i- fa 0 <. j _ < •— . Observe that for i = 0 , . . . , p — 1,

and we get part of T(F. H). Now, if nun

n

rr

n

n

then from the Theorem (8.4),

where j is an integer such that n 0

nn ^ ~ ftp ^ - •< mm < i. = -

y

Pp+\

p^p

Putting mm

ft

, — f'pft (V ' P+l

I'P

I

•

^ — "—

yI

l/

P+l

"'pl'p

Irreducible Plane Curve Singularities

117

we get the rest of F(F, H ), with the exception of 1^, if F =£ H. The remaining possibility, in view of (8.3), is mm <

•• n

*-

—— > ?r

a < max

n

n

We will split our analysis into two cases. p+

< p . Observe that this condition is equivalent to p+ < p+ . n n n n' In such case we must have a = —-— , because, if otherwise, a > —— , we would n n Case a)

a

QI

have —-— = —-— , contradicting the definition of p. n n Hence, from the Theorem (8.4), we have !(/, h] = n

'

— = e'pvp+l = loo.

Tl\ . . . Tip

0' -p-- > -p-^-. The proof is analogous to that of case a). n n

Case b)

D

Put p = p(F,H) and let TTJ+I be as in the Theorem (8.5). Consider the following map e:

{(*J); 0 < * < p , 0 < j < 7 r i + 1 } u { o o } (*».?) 00

—-> ^

e'i-iVi + je'iCi,

l~>

IOQ.

Concerning the map e we have the following result. PROPOSITION 8.6 If we give the lexicographical order to the domain of e and consider oo os its biggest element, then e is an order preserving map. Proof:

Since the v j form a strongly increasing sequence we have eiVi+i > eiHiVi = ei_iVi, Vz = 0, . . . < / .

(8.7)

The above inequality with the fact that -4 = — for all i = 0, . . . , p imply that e i v0 e'i-iVi > e / i _ 2 v i _i, i = 1, . . . , p .

(8.8)

Let i and £ be integers such that 0 < I < i < p. To prove that 'i > E'^VI + keee'f, for all j and k such that 0 < j < Vi+l ~ UiVi and 0 < k < ^+1 ~ niV* , it is enough e» ee to show that

118

Hefez

which is true because of (8.8). On the other hand, if i = £, it is clear that for j > k, we have f'-'i-iVi + Jed > ?'i-\ve, + keie'f. To conclude the proof we have to show that loo > ep-ivp + (irp+i - l}e'pep. If F = H, this is obvious since I^ = oo. Suppose now that F ^ H. We will prove the above inequality only in the case VQV'+I < vp+iv'0, since the other case is similar. v' + l Remark that in the present case we have that ~np+\ = —

7?/

V+i

8.3

H ?/ V — ?/

P PJ

MAXIMAL CONTACT

Let (/) be a fixed branch with semigroup of values G of genus g. If (h) is any branch, we will denote the genus of S(h) by gen(/i) . We want to introduce a notion of maximal contact among (/) and all branches (/?,), under the restriction that gen(/?,) is less or equal to a given integer 7, where 7 < g. There is no hope to define this maximal contact by taking ); gen(» < 7}, since this last set is not bounded. Indeed, let G — (vo,vi,v<2, . . •) of genus at least 2. If we take Hn — (n,n + 1), we have p(G,Hn] = 0, and since VQ(H + 1) < nv\, for sufficiently large n, then the element 1^ of F(G, Hn) is VQ(U + 1), for sufficiently large n. This shows that sup{I(/, h); geu(h) < 1} > sup{v0(n + 1); n <E IN} = oo. In contrast we have the following result. PROPOSITION 8.7 Let (/) be a given branch with semigroup of values G — (VQ, . . . , vg}. Let 7 be an integer less than g. Then M7(/):=max( ^ h\ ; gen(/») < 7} = ^ [mult(/)mult(/?,) J ?;

Irreducible Plane Curve Singularities

119

Proof: We clearly have

Suppose that H = (v'0,...,vj) with i < 7. From the Theorem (8.5), for F(£,H), we have •r 1 ^ 1 ., if v0v'p+l > v'0vp+l. Since in any case we have Ln

with equality holding if vov'p+1 > v'0vp+i, and since from (8.7) the sequence is increasing, it follows that

D

We will say that a branch (h) such that gen(/i) < 7 has maximal contact of order 7 with (/) if we have nmlt(/)mult(/i) Define

r7~/( G which is strongly increasing semigroup. Since p(G, G 7 ) = 7 and ^7+1 = oo, it follows that in F(G, G 7 ) we have loo = e 7 ^ 7+ i = v 7+ i. Hence the maximal contact of order 7 with (/) occurs for a branch (h) such that

Hefez

[1] G. Angermiiller. Die Wertehalbgruppe einer ebenen irreduziblen algebroiden Kurve. Math. Zeitschr. 153:267-282, 1977. [2] R. Apery. Sur les Branches Superlineaires des Courbes Algebriques. CRAS, 222:1198-1200. 1946. [3] A. Azevedo. The Jacobian Ideal of a Plane Algebroid Curve. PhD. Thesis, Purdue, 1967. [4] V. Bayer. Semigroup of Two irreducible Algebroid Plane Curves. Manuscripta Math. 39: 207-241. 1985. [5] E. Brieskorn, H. Knorrer. Plane Algebraic Curves. Birkhauser Verlag, 1986. [6] A. Campillo. Algebroid Curves in Positive Characteristic. Springer Lecture Notes in Mathematics 813, 1980. [7] A. Dickenstein, C. Sessa. An Integral Criterion for the Equivalence of Plane Curves. Manuscripta Math. 37:1-9, 1982. [8] A. Hefez, M.E. Hernarides. Computational Methods in the Local Theory of Curves. Publicagoes Matematicas, IMPA, Rio de Janeiro, 2001. [9] M. Merle. Invariants Polaires des Courbes Planes. Inventiones Math. 41:103-111 1977. [10] J. Milnor. Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies, 61, Princeton Univ. Press, 1968. [11] M. Puiseux. Recherches sur les Fonctions Algebriques. Journal de Math. 1. 15:365-480, 1850. [12] J.J. Risler. Sur L'ideal Jacobien D'une Courbe Plane. Bull. Soc. Math. France, 99:305-311, 1971. [13] R. Walker. Algebraic Curves. Springer-Verlag, 1978. [14] O. Zariski. General Theory of Saturation and of Saturated Local Rings II. Am. J. Math. 93:872-964, 1971. [15] O. Zariski. Le Probleme des Modules pour les Branches Planes. Hermann, Paris, 1986. [16] O. Zariski, P. Samuel. Commutative Algebra, Volume 2, Van Nostrand, 1960.

Openness and Multitransversality C.T.C. WALL Department of Mathematical Sciences The University of Liverpool, Liverpool L69 7ZL, United Kingdom. E-mail: [email protected]

The transversality theorems of Thorn and Mather have been used in a wide variety of situations to show that the family of maps satisfying certain families of conditions is dense (even, a dense G^-set) in the space of all maps. However although the original result of Thorn showed that the set of maps was also open, this is not the case for later extensions of the result, and in particular theorems showing that a set of maps defined by multitransversality conditions is open are very few. We begin this paper by a fuller discussion of the problem, and then a review of previous work in this area, concluding with a simple criterion due to du Plessis. In the rest of the paper we demonstrate the effectiveness of this criterion by applying it to a number of problems showing the openness of families of curves in a plane defined by collections of multitransversality conditions. Similar applications to curves in 3-space will be discussed elsewhere.

1

THE PROBLEM OF OPENNESS

The main thrust of Thorn's original transversality theorem, and of its numerous later variants and extensions, is to approximate a map / : X —» Y by one transverse to one or more submanifolds of Y; in other words, to show that the set of maps satisfying the prescribed transversality condition is dense in the space C°° (X, Y) of all maps, provided with some appropriate topology. In the case of a single submanifold W closed in Y it is immediate that the set of maps transverse to W is also open in C°° (X, Y), with any of the usual topologies. 121

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The case of a collection of submanifolds is less trivial. It is natural to suppose that these stratify a subset W of Y, and then Thorn suggested a condition for the set of transverse maps to be open. The final result appears in a paper [18] of Trot man which includes also a rather full list of references to previous papers on the topic, most of which contain errors; thus illustrating the rather delicate nature of the question. THEOREM 1.1 LetS be a locally finite stratification of a closed subset W of a Cl manifold Y. Then the following conditions are equivalent: S is A-regular in the sense of Whitney; For every C1-manifold X, {z e Jl(X,Y) : z rtl S} is open in Jl(X,Y}; For every C1-manifold X, {/ e Cl(X,Y) : f rh S} is open in Cl(X,Y] with the strong Cl-topology. For the non-specialist, the result can be summarised (inaccurately) by the slogan: 'Transversality to a stratification is an open condition if and only if the stratification is A-regular'. The common error to which Trot man draws attention is as follows. Say that S satisfies condition (t) if every Cl map of a manifold to Y which is transverse to S at a point x £ X is also transverse to S in a neighbourhood of x. It was thought that (t) was sufficient to imply that the set of maps transverse to S was open. The situation was clarified in [18] where it was shown that although condition (A) implies (t), the converse is false in general; whereas, by Theorem 1.1, (A) is the condition ensuring openness of transversality. This is the more annoying as attempts to prove openness lead naturally to looking at condition (t). The two conditions are indeed equivalent for subanalytic stratifications S by [17|, and this is sufficient for many purposes. For the multitransversality theorem of Mather [13] one first defines X^ to be the subset of Xr consisting of r-tuples of distinct points, and r , J k ( X , Y ] to be the preimage of X1-^ under the natural projection (Jk (X, Y}}r —> Xr. Then a Ck map / : X —> Y induces by restriction of (jkf)r a section rjkf to the projection rJk(X, Y} —> X^. If W is a submanifold of rJk(X,Y), then the theorem states that the set of maps / such that r j k f is transverse to W is dense in Ck(X,Y}. To establish openness of the set of transverse maps in this case is much less straightforward. What one can see easily is that if W is compact (we allow W to have boundary, so it may be a submanifold of a larger manifold W with boundary) then multitransversality to W is indeed an open condition. However, the submanifolds of k k r always contain r , J ( X , Y ) of interest are never compact: their closures in ( J ( X , Y ) r points which project to the fat diagonal in X . We seek to develop methods of proving that certain sets defined by collections of multitransversality conditions are open. In the next section we discuss general methods. This excludes special devices, such as the ingenious use in [2] of an auxiliary map to deduce openness of the set of maps there considered from openness of stable maps. We suppose throughout that the source manifold X is compact. Then the ordinary and strong C°° topologies on C°°(X, Y} coincide, and we may use this topology from now on without further mention. Most positive results may be extended without difficulty to proper maps if X is not compact, using the strong C°° topology; we refer the reader to [15, Chapter 4] for a discussion of some of the issues that arise in further situations.

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123

REVIEW OF KNOWN METHODS

The idea of using condition (t) may be rescued to some extent as follows. We deal with the case r = 2 of 'bigerms'; essentially the same argument will deal with jets of any multiplicity. LEMMA 2.1 Suppose X is compact and we are given A-regular stratifications S of a closed subset A of Jk(X, Y} and T of a closed subset B of 2 Y transverse to S there exist a neighbourhood U of f and a neighbourhood V of the diagonal in X x X such that for any g 6 U, jkg is transverse to S and the image of ijkg is disjoint from B D K~1V. Then the set of f with j k f ( \ ] S and 2Jkf rti T is open in C°°(X, Y). (ii) The same follows if we suppose merely that for any f : X —>• Y transverse to S and any x G X there exist a neighbourhood Ux of f and a neighbourhood Vx of "X. in X such that for any g £ Ux, jkg \vx is transverse to S and the image of ijkg \vm is disjoint from B. Proof: (i) Let V\ be a compact neighbourhood of the diagonal in X x X with V\ C V, and V<2 a compact subset of X x X, disjoint from the diagonal, with V\ U V^, = X x X. Then the set of g with 2Jk9 \v2 transverse to T is open, and its intersection with U is a neighbourhood of / consisting of maps transverse everywhere to T'. (ii) We may suppose the Vx open. Since X is compact, it is covered by finitely many of them. The union of the corresponding Vx x Vx is then a neighbourhood V of the diagonal in X x X, and the intersection of the Ux is an open neighbourhood U of f. These satisfy the hypotheses of (i), so the conclusion follows from the first part. D It is not enough to suppose that the image of 2Jkf is disjoint from BC\7r~lV: it seems certain that examples similar to Trotman's, pointing out the error noted above, can be constructed. An approach using stratifications of function spaces is given in Looijenga's thesis [12]. We recall some of his definitions. A subspace T of a Frechet space J- is a weak codimension k submanifold if, for any / £ T, there exist a neighbourhood Uf of / in f and a C^-submersion p/ : Uf —> R fc such that pjl(0) = T fl Uf. There is an obvious notion of transversality to such a submanifold, and for any C^-manifold t/, the T-transversal maps are open and dense 'mC°°(U,F). If U is an open subset of F, and S a locally finite partition of U into weak submanifolds of T of finite codimension, then we call (U,S) a weak Whitney stratification if, for any smooth mapping G of a (finite dimensional) manifold U into U which is transverse to the members of 5, G~l(U,S] defines a Whitney stratification ofU. (In fact, A-regularity is enough for our purposes.) The following can be proved. PROPOSITION 2.2 // (U, S) is a weak Whitney stratification, and M is a compact manifold, then the set of maps transverse to S is open and dense in C°°(M,U}. Looijenga establishes a weak Whitney stratification for the space of maps from a manifold into R by combining basic results on .A-unfoldings of map-germs into R with some technical theorems of Sergeraert [16] which yield local product structures

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in function space. This approach has two disadvantages: it makes heavy use of the apparatus of Frechet spaces, which does not appear very relevant to the problem, and it uses several results specific to the case when the target has dimension 1. However, our function spaces are indeed Frechet, and not Banach spaces: the results of [16] are applicable here, while those of Abraham [1] are not. To make further progress using this method, it would be necessary to give direct constructions of local product structures in function spaces: a likely method would be to develop the theory of ,4-unfoldings far enough to give continuous maps of function spaces. This appears to be an interesting question. The major known result concerning openness of multitransversality concerns the specific, but very important, example of stability. A map is said to be Cr'-stable if it has a neighbourhood such that all maps in that neighbourhood are equivalent to it under Cr-diffeomorphisms of source and target. It thus follows from the definition that the set of Cr-stable maps is open. The following, however, is far from trivial. If we write MT-stable to denote maps which are C°-stable in the sense of Thorn and Mather - we recall that this class of maps is defined by a family of multitransversality conditions - then the result states that MT-stable maps form an open set in C°°(X,Y}. There are two proofs of this result. The argument of Mather [14] can be summarised as follows. Write 17 for the set of MT-stable maps. The multitransversality conditions imposed are strong enough to imply finite singularity type, so any / €E 17 has a proper stable unfolding {/' : X' —->• Y';i, j}, where 'unfolding' means that X

-^

i I

X'

Y 3 i

-C

Y'

is a pullback diagram. It is shown in the theory that any proper stable map has a canonical stratification, and this gives in particular a regular stratification S? of Y'. Now by the theory of versal unfoldings there exist a neighbourhood U of / and a continuous map ($,#) : U —> C°°(X,X'} x C°°(Y,Y'} such that for all g e U, (F, (#), ^(.9)) is an unfolding of g. Now transversality to <52 is an open condition, so there is an open neighbourhood W of j in C°°(Y,Y'} such that any V> 6 W is transverse to 8-2. Then fy~~l (W) is a neighbourhood of / contained in 17. This proof has the merit of reducing multitransversality of / to ordinary transversality of j where we can apply the previous result. However to do this it requires the existence of a common unfolding F of all maps near /, and a regular stratification of F. Moreover while for this particular purpose it suffices to use /C-versal unfoldings, for other problems one tends to need .-4-versal unfoldings and level-preserving families, so this leads back to the questions mentioned above. The second proof is due in principle to Looijenga [9]. First, if 17 is not a neighbourhood of /, then there exists a sequence {/7} of maps not in 17 and converging to / (this is justified in [15, 3.4.11] despite the fact that the topology does not have countable neighbourhood bases). Passing to a subsequence, we may embed

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these in a 1-parameter family forming a C°° map F : XxI-^YxIof the form F(x,t) = (ft(x),t) with /o = / and fi/n the sequence. A contradiction is now obtained by showing that all ft for t small enough satisfy the multitransversality condition. The above mentioned justification uses the following criterion, which is due to du Plessis. PROPOSITION 2.3 [15, Corollary 3.4.12] Let N and P be C°°-manifolds, with N compact. Then a subset U. is open in C°°(N,P) if and only if for all 1-parameter families {ft : N —> P \ t 6 U} (for U a neighbourhood of 0 in R) defining a smooth map F | N x U -> P x U, the set {t € U \ ft € U} is open in R. In the remainder of this paper we will show how to use this criterion to obtain specific results. However, the question of obtaining a full analogue of Theorem 1.1 remains both open and challenging. To apply this result to a particular problem, we take U as the set of maps satisfying a given set of multitransversality conditions. To show that U is open we consider a 1parameter family {ft} of maps with /o satisfying the conditions, and need to deduce that they hold for ft for all sufficiently small t. This does not require us to go further into function space technicalities, and is close in spirit to the inadequate but simple argument using condition (t). We shall spend the remainder of this paper in illustrating this method for proving openness. We confine ourselves here to problems concerning plane curves: applications to families of space curves will be considered elsewhere.

3

CONTACT WITH LINES

We consider plane curves as given by parametrisations / : Sl —>• M, where M is some model of the plane, with /(51) = F. The stable maps form the subset consisting of immersions whose only self-intersections are normal crossings (thus with only 2 branches at each). The openness of this set thus follows. However we now show this directly - i.e. without using the characterisation of stable maps. LEMMA 3.1 Let Co denote the set of immersions f : Sl —>• M whose only singularities are normal crossings. Then CQ is open in C°C(S1,M}. Proof: We will use Proposition 2.3, so consider a smooth 1-parameter family F = {ft '• Sl —> M 1 1 G U}, with /o € CQ, and suppose there exist arbitrarily small values of t for which ft ^ Co', we seek a contradiction. We may choose a sequence of values t i - > 0 with/ t i <£C0. First suppose, for infinitely many values of i, ft. not an immersion: then it has a critical point Pi. Since S1 is compact we may suppose (passing to a subsequence of the ti if necessary) that Pi converges to a limit P. Since F is smooth, it follows by continuity of first derivatives that P is a critical point of /Q: a contradiction. Next suppose there exist infinitely many values of i for which ft. has a nontransverse double point given, say, by fti(Pi) = fti(Qi)- By compactness, we may again suppose that Pt converges to a limit PQ and Qt to a limit Q0. If PQ ^ Qo then f o ( S l ) has a non-transverse double point, contrary to hypothesis. But if P0 = Qo

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then since F : Sl x U —>• P2 x U is an immersion, it embeds some neighbourhood of (Po, 0), and this also is a contradiction. Similarly if there are infinitely many values of i giving a triple point /^(Pi) = f t , ( Q i ) = f t i ( R i ) , then we may suppose that all three sequences converge: Pj —» PO etc. If the limits Po,Qo and RQ are all distinct, then /0 has a triple point: a contradiction. But if they are not, e.g. PQ = QQ, we obtain the same contradiction as in the previous case. D There are numerous extensions to this result: here is the simplest, where we give a different style of argument which clearly extends to many other cases. LEMMA 3.2 Let C'Q denote the set of maps f : Sl —>• M whose only singularities are normal crossings and simple cusps. Then C'Q is open in C°°(Sl,M). Proof: Again consider a smooth 1-parameter family F — {ft : Sl —> M t € 17}, with /Q G C'Q, and suppose there exist arbitrarily small values of t for which ft 0 CQ, and seek a contradiction. If there is a sequence £,• —>• 0 such that ft. ( S 1 } has a singular point fti(Pi] — Xi which is not a node or cusp, we may suppose as before that P± converges to a point P, and hence X^ converges, to X, say. Now if TO has a node at X, then as in the previous proof we see that no singularities worse than nodes can appear nearby. If we have a cusp, then the family F in a neighbourhood of P is induced from the versa! unfolding (for right-left equivalence) of a cusp singularity, and since no nearby curve in the versal unfolding has a singularity other than an ordinary node or cusp, the same must hold here. D We next take M to be the projective plane P2 = P2(1R), and impose a genericity condition on the family of intersections of F with lines. Let us first suppose that F is already stable. Each of the following conditions defines a residual, hence dense, set by Mather's multitransversality theorem: there are no hyperflexes, there are no fiecnodes, an inflexional tangent cannot be a bitangent, an inflexional tangent may not pass through a node, there are no tritarigents, a bitangent may not pass through a node, a nodal tangent may not touch the curve elsewhere, a nodal tangent may not pass through another node, the line joining two nodes may not touch the curve, no line passes through three nodes. We may give a unified statement of these conditions as follows. If the local intersection number of curves F and C at a point P is fc, define the contact number of C with F at P to be k— 1, and define the total contact number K,(C, F) of C with F to be the sum of the contact numbers at all their common points, or equivalently, the total intersection number C.T diminished by the number #(C ft F) of points of intersection. Then the above conditions on F are equivalent to requiring that the total contact number with any line is at most 2. Here we use 'intersection number' in the real sense: if C is given by an equation f ( x , y ) = 0 and F by a parametrisation ( x , y ) = (a(i),b(t}} then the intersection number at t = 0 is the multiplicity of 0 as root of /(«(<), b(t)) = 0, i.e. the order

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of (f)(t) = f ( a ( t ) , b(t)) at t = 0. If 4>(t) has order k at 0, and we have a 1-parameter family, leading to 0 U (£), then the sum of orders of vanishing of P2 \ t € U} be a smooth 1-parameter family, with /o € C\. Suppose there exist arbitrarily small values of t for which there is a line Lt whose total contact number with Tt is at least 3. Since we are working in project! ve space, the set of all lines forms the dual projective space P2v and is, in particular, compact. Thus we may pick a sequence of values of t converging to 0 for which Lt converges to a line LQ. For each point P G LQ fl FQ, with intersection number a 4- 1, say, choose a neighbourhood U of P not containing any other point of LQ fl FQ. Then the sum of the intersection numbers of Ft with Lt at points of U is upper semicontinous. For small t either the number of points of Lt H U fl Tt is 0, so the intersection number and contact number are also 0, or it is > 1, so as the total intersection number is < a + 1, the contact number is < a. So the total contact number also is upper semicontinuous. Hence the total contact number of LQ with FQ is > 3, a contradiction. D In this proof we did not make any assumptions on the singularities of F: assuming in addition that / is stable, or that it is an embedding, defines the intersection of two open subsets of function space, which is then also open. However, the condition on contact numbers itself imposes a sharp condition on the possible singularities. We can contemplate various notions of 'singularity' of a plane curve, but the classifications most familiar to the author refer to singularities of a defining equation of the curve. We thus indicate briefly (adapting [6, p. 56]) how to obtain a defining equation from a parametrisation. The multiplicity of F at a point is the minimum of its local intersection numbers with a line through the point. Suppose this finite: explicitly, suppose F has a single branch having intersection number k < oo at the origin with the x-axis. In the local parametrisation x = a(i] and y =. b(t) with b of order k at 0. We can thus write b(t) = uk and choose u as a new local parameter: say x = A(u),y = uk. Now by the Malgrange preparation theorem we may find C°° functions Ai(y] such that A(u) — ^0~l ulAi(y}. For 0 < j < k — 1 we may multiply by u* to obtain the system fc-l

k-j-l

which we regard as simultaneous linear equations in {ul 1 0 < ? < A; — 1}. Equating the determinant to 0 gives a monic equation of degree k in x whose coefficients are C°° functions of y. Provided the local intersection number of F with y = 0 at the origin is finite, there are only a finite number of branches. We may use this procedure for each branch, and then multiply the equations together. We thus have an equation, and may use

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the standard (Arnol'd) notations of singularity theory to describe the cases that will arise. For example, we have LEMMA 3.4 I f T e C i then any singularity of T has type AI or A2. Thus C\ CC'Q. Proof: Suppose F 6 C\. If T had a singular point P of order > 3 then choosing L to be tangent to some branch at P would give (L.F)p > 4, so /f,(L,F)p > 3, a contradiction. If P has multiplicity 2, and does not have type AI, take a local equation /(.x, y] = 0. We may suppose / has 2-jet y2 . Now if the coefficient c of x3 in / is non-zero, we have an AI singularity. But if c, = 0, the local intersection number of F with y = 0 is at least 4, and again we have a contradiction. D We have introduced the use of the total contact number as a convenient device for formulating a collection of multitransversality conditions. Essentially the same argument could be given in more detail, and would then show that several subsets of the above collection of conditions also define open sets in the space of curves. The argument yields also the following generalisation, which is useful in certain situations. PROPOSITION 3.5 Suppose A a (perhaps reducible) curve such that for all lines L we have «;(!/, A) < 2. Then the set of curves F such that for all lines L we have K(L, F U A) < 2 is open.

4

REMARKS ON DUALITY

We have just shown that any curve F € C\ has only node and cusp singularities. We next show that the same holds for the dual. Recall that a curve F is said to have ordinary singularities in the sense of Pliicker if neither F nor its dual curve has singularities other than ordinary nodes and cusps. Here, of course, the dual curve Fv is the family of tangents to F, considered as a curve in the dual projective plane P v . LEMMA 4.1 If T € C\, then Fv has only node and. cusp singularities, so F has ordinary singularities. Proof: We recall some results from [20] . If G denotes a curve-germ at P with welldefined tangent L, (7V its dual (with centre L and unique tangent P) write m — m(G} for the multiplicity of G and i = i(G) for the local intersection number of G with L; similarly for m v and ?"v. Then, according to Zeuthen [21], iv = i = m + rav '. Let L be a (singular) point of F v , let the tangents there be PI and the corresponding germs F^. Thus the line L is tangent to F at the points P 7 , and the corresponding germs are F^. Applying Zeuthen's formula to each of these germs (with an obvious notation) and summing, we obtain

n r) which is no less than LT-

T

Thus Fv has no points of multiplicity above 2. Now further results of [20] show that if Fv has an A^ singularity with k > 3, then F either has the same, or has a singular point of higher multiplicity. But neither of these is the case. D

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By a result of Bruce [4], the local intersection number of two touching curves is equal to that of their duals at the corresponding point, provided that at least one of the curves has non-zero curvature. It is thus natural to seek a self-dual theory. Define C± to consist of curves F such that Fv e C\. Were the duality map continuous on (e.g.) the set of maps with ordinary singularities, we could deduce the openness of GI from Proposition 3.3. However, the duality map is far from continuous, as we see by considering, for example, the family fu(t) = (u + t2, ut + t3) of cubic curves Tu. For u ^ 0 we have a nodal cubic, whose dual has degree 4; for u = 0 a cuspidal cubic, with dual of degree 3. We thus have to treat the dual cases separately. We review which further situations are excluded by the C\ hypothesis. A line with «(L, F) = 3 would have intersections with F of multiplicities 4, 3 and 2, or 2 and 2 and 2. In the first case we either have a hyperflex tangent (giving an EQ on the dual) or a flecnode, with the tangent as a flexional tangent to one branch. For the remaining cases introduce the notations A, 5, C for the 3 types of local intersection number 3: inflexional tangent, nodal tangent, cuspidal tangent respectively; and [7, V, W for the types of local intersection number 2: ordinary tangent, nodal chord, cuspidal chord. Eight of the 19 cases correspond to similar configurations on the dual: Figure an T: AU Figure on Fv : D5

AV UUW

AW AW

BU BU

CU flecnode

UUU UUV D4 UUV

UUW AV

Of the remaining 11 cases, 4 configurations on Fv correspond to a bitangent (BV, CV] or inflexional tangent (BW, CW) of F passing through a point of contact of a bitangent (BV, BW] or inflexional tangent (CV,CW). The next 3 correspond to 2 bitangents (t/VV), a bitangent and a flex tangent (UVW) or 2 flex tangents (UWW) meeting on F, and the final 4 cases (VVV, VVW, VWW, WWW] to the concurrence of 3 lines, each a bitangent or flex tangent. We do not prove openness of C± , but establish LEMMA 4.2 The set C{ := C\ D C{ is open. Proof: By Proposition 3.3, C\ is open. The additional explicit conditions defining C* are the exclusions of the 11 situations just listed. We thus consider a 1-parameter family Fu in Ci, with FQ G C%. Suppose, if possible, that for arbitrarily small w, Tu presents one of these 11 situations. Each of these refers to the concurrence of 3 curves, each a line or F. We may thus also suppose that each of the lines, and the point of concurrence, also converge, while of course the curves Tu converge to F = FQ. A sequence of inflexional tangents converges to the tangent at a flex or hyperflex; and a sequence of bitangents converges to a bitangent or hyperflex tangent. Since FQ € C*, it does not present any of the configurations in question. It follows that two (at least) of the limiting lines must coincide. But also F has no hyperflex. As the number of flex tangents and bitangents can only vary when the family of curves passes through a hyperflex, we have our contradiction. D

5

CONTACT WITH CIRCLES

Our next example considers the conditions studied in a series of papers by Bruce, Giblin and Gibson starting with [5]; see also [3]. The symmetry set of the plane

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curve F is defined to be the locus of centres of circles C whose total contact number /c((7, F) > 2. For several results they impose the condition that this number is at most 3 for any circle. Certainly this defines a dense set of curves: we now seek to prove it open. Take our model M of the plane to be the inversive plane S*2, which may be identified with P1 (C). The group of the geometry is PGL2(C), which preserves the family of circles but does not respect distances. Coordinates may be obtained by stereographic projection of S2 to a plane: this maps circles to circles and straight lines. We will study the contact of F with the family of all circles in the plane. We show THEOREM 5.1 Let C<2 be the family of smooth curves f : Sl —> S2 such that for each circle C in S2, the total contact number of C with F is at most 3. Then C<2 is an open set. All the arguments used in the proof of Proposition 3.3 apply here with one exception: we need to know that the family of all circles is compact, so that a sequence of circles contains a convergent subsequence. If we identify S2 with the unit sphere in R 3 , then the circles are the intersections of S2 with the planes ax + by + cz — d where we may normalise a 2 + b2 + c2 = 1 and then require \d < 1 for the intersection to be a circle. We compactify the space of circles by allowing \d\ < 1, thus including 'point' circles. It is then necessary to consider the total contact number of F with a sequence of circles shrinking to a point. Thus we need to study the structure of F near a point. We recall from elementary geometry that if F is a smooth point of a plane curve A then among the circles touching A at P there is a unique one with local intersection number > 3: the 'circle of curvature' at P. LEMMA 5.2 J/F is a curve in €2, then F has no singularities other than (ordinary) nodes and cusps: we have €2 C C'0. Proof: We argue by contradiction. If P is a singular point of F of multiplicity > 4, we take a circle C touching one of the branches of F at P: then the local intersection number (C.T)P > 5, so K(C, F) > 4: a contradiction. If P is a singular point of multiplicity 3 such that one of the branches of F at P is smooth, then the circle of curvature C of that branch again has (C.F)p > 5, giving a contradiction. If F has multiplicity 3 and just one branch at P, we can make a linear change of coordinates to put its equation in the form y3 + ax4 + higher terms = 0. If a ^ 0 here, we have a singularity of type EQ: we defer consideration of this case. If a = 0 take C as the circle y = 0, which again gives a contradiction since (C.F)p > 5. Now suppose P is a singular point of multiplicity 2. If there are 2 branches BI, B-2 at P, then both are smooth and if the singularity is riot a simple node, the two are tangent. Take C as the circle of curvature of B\: this has local intersection number > 3 with BI and > 2 with BI-, giymg at least 5 in all: a contradiction. If there is a single branch at P, with a singular point of type A%k, the equation may be written in the form (y — 0(x)) 2 — x2k+l, and if k > 2 then the circle of curvature of y — 0(x) has local intersection number at least 5 with F at P. Thus only singularities of types AI and AI remain. We must now return to the case when F has a singular point P of type EQ. For any circle C having the same tangent as F at P, the local intersection number is 4,

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which does not of itself yield a contradiction. Consider the family of all such curves (7, which are the intersections of S2 with a variable plane through a fixed line tangent to the sphere: begin with the plane tangent to S2 (so having intersection a point circle) and rotate continuously through an angle TT till we return to the original plane. We may suppose the local equation of F to be y3 = x4 added to higher order terms. So a small circle below the x-axis meets F only at P. A small circle x2 + y2 = 2ay just above the axis (i.e. with a > 0 small) meets it where (substituting for x2 and ignoring the higher order terms) y3 = (y2 — lay}2. Apart from the root y = 0 this has one root near y = I and one with y = 4a2 + . . . . Such circles thus cross F at least twice. During the rotation of the plane there must be a first point where the circle C of intersection with S2 meets — and hence is tangent to — F. For this circle we have an additional contribution to K and now have the desired contradiction. D This final argument is an essential point in our proof: it can be shown by direct calculation that for any curve FQ with a singular point of type EQ there exist 1parameter families Ft of curves and Ct of circles shrinking to a point circle Co at the singularity such that «(Ct, F^) = 4. The proof of Theorem 5.1 is completed by the following lemma. LEMMA 5.3 Let F = {fu : S1 —» S2} be a smooth 1-parameter family of maps with f = f0; write F := f ( S l ) and let P G S1, X = f ( P ) . (i) Let X be a smooth point ofT. Then there exist neighbourhoods V of X in S2 and U o/O in R such that if C is a circle contained in V and u e U, then K , ( C , f u ( S 1 ) ) < 1. (ii) Let X be a simple node ofT. Then there exist neighbourhoods V of X in S2 and U o/O in R such that if C is a circle contained in V and u 6 U, then K , ( C , f u ( S 1 ) ) < 2. (Hi) Let X be a simple cusp o f T . Then there exist neighbourhoods V of X in S2 and U o/O in R such that ifC is a circle contained in V and u e U, then K(C, fu(S1)) < 3. Proof: (i) We take local coordinates with respect to which / takes the form f ( t ) = (t + 0(£), ?/>(£)), where 0 and ip denote functions of higher order (i.e. divisible by t2). The deformed curve fu may be written in the same form, but now the error terms 0(t, u),^(t, u) belong to the ideal generated by t2 and u. We consider intersections with the circle (x — a(u}}2 + (y — b(u})2 = r(u}2 where all of a(w), b(u) and r(u) must tend to 0 with u. Intersection points are given by (t + (t,u] - a(u})2 + (ip(t,u) - b(u)}2 - r(u}2 = 0. When u becomes zero, the left hand side of the equation reduces to t2 added to higher order terms. The local intersection number is thus at most 2 in the limit. It follows that for all small enough w, K < 1. (ii) We have just seen that for a smooth branch, small circles cannot be either circles of curvature or bitangent. Thus when we have 2 smooth branches, the only possibilities for contributions to K are for circles simply touching one or other branch, or passing through the point of intersection. Since a circle cannot do all three (as the branches are not tangent at the point of intersection), we must have K < 2. (iii) We follow the same approach as in (i). We have a family of curves given by ( x , y ) = (t2 + (j>(t,u},t3 + ^(t,u)}, with (j) <E (£ 3 ,w), V G (£ 4 ,w), and consider the contact with the same family of circles as before. This leads to the equation (t2 + <j)(t, u) - a(u}}2 + (t3 + i{>(t, u) - b(u))2 - r(u}2 = 0.

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To state the results let us first define, for any A = ^q aqAq in HQ(X$>\{f a closed n-cycle 6 (A) in Hn(F, C) where F is the Milnor fiber of fa at 0. DEFINITION 3.2 Assume that F = f^fa) where is chosen in ]0, rj[. For Aq C {/ > 0} let 8(Aq) = Aq H F (with the For Aq C {/ < 0} let us consider the closed C°° C°° -trivialisation of Milnor's fibration along the half details see [2]). Then define 6(Aq) := -&*(Aq n fa1

= 0}, C)

the base point t0 in D(Q, r,) \ {0} orientation chosen before). embedding induced by following a circle toe16 , 0 e [TT, 2?r] (for more (-to)) and 6(A) := £g o,q6(Aq).

We remark that the definition of 6 (A) is not the convention of [8, 17] because the choice in 2.1 is not the same that in [2]. First we consider the case where A 7^ 1. THEOREM 3.3 Assume that the complexified germ fc : (Cn+1, 0) —> (C, 0) of f has an almost isolated singularity at 0 for the eigenvalue A = exp (— 2i7ru) where u e]0, l[n. Then we have equivalence between: (i) the distribution fA fs D has a pole of order > ko at —u — v for v G N, v large enough, (U) (T - XId)k°~l(8(A)x) ^ 0 in Hn(F, C) A , where 8(A)X is the X-component of 8 (A) in the spectral decomposition of the monodromy T acting on Hn(F, C). Of course in condition (ii) we mean that 8(A)X = 0 4=^ ko = 0. The proof of (i) => (ii) is given in [8] and its converse in [17]. Basically, five key points are used . (1) Each vector e € H£(F, C) is the class of a semi-meromorphic n-form uj with poles along {fa = 0}, /c-proper support and relatively closed (du A dfa = 0 ). (2) Each Jordan block of size k for the monodromy T acting on H£(F, C) x can be induced by semi-meromorphic n-forms, with /-proper supports, o>i,..., ujk as in (1) satisfying the relation : Vj G [1, k] , duj = (m + u)(dfa/fa) A Uj + (dfa/ fa) A u^_i, UJQ = 0, for some m G N. (3) By integration of such representative of a Jordan block on An {/ = £}, we obtain fiber-integral of the form t \m uP(\og \t \) where P is a polynomial of degree < k — 1. Choosing now a Jordan block which is not orthogonal to (T — Xld) °~l(8(A))x for the Poincare duality on F, allows to produce a polynomial of degree > ko — 1. We easily conclude by Mellin's transform. The converse needs also the following key points. (4) Each distribution associated to a polar part at —u — v, v E N, for fA fs D is supported by {0}. (5) The canonical map Can : H™(F, C) A —>• Hn(F, C)A is an isomorphism. As a consequence of this strategy it is not very hard to obtain, using that all our homological constructions are defined over R (see [15]): COROLLARY 3.4 In the same setting of the previous theorem, for v & N large enough, the order of the pole for JA fs D at —u — v and u — v are the. same. Remark that by theorem 3.1 they do not depend on v large enough; for some examples see [16]. Now we have a result analogous to theorem 3.1 for the case A = 1 (see [2]).

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THEOREM 3.5 Assume that the singularity of the complexified genn of f has an almost isolated singularity relatively to the eigenvalue 1 of the monodrom,y, then we have equivalence between : (i) o(A) has a nan zero component S(A)l on Hn(F, C)l , (ii) the meromo'rphic extension to the complex plane of the holomorphic distribution ,s, -"ft.s > 0 i—> IA fs D admits a pole of order > 2 at s = — (n + 1). Moreover, the order of the pole at v € N, v > n + 1, of

A

is exactly the nilpotency order of T ~ 1 acting on We just give here the key point in the proof. The main difficulty of the A = 1 case compared to the previous case (A ^ 1) is that, because the canonical map Can : H£(F, C)l —> Hn(F, C) x is no longer an isomorphism, the intersection form coming from the Poincare duality H£(F. C) x Hn(F, C) —> C has to be replaced by the canonical (non degenerated) hermitian form (see [3])

h : Hn(F, C) x Hn(F, C) —> C. This hermitian form is related to the variation map Var : Hn(F, C): —> H£(F, C)1 which has been denned and computed in term of differential forms in [3] under the present assumption.

4

OSCILLATING INTEGRALS

Now apply Fubini's theorem to the Fourier transform of the fiber-integral, we obtain

r

F

I exp(irt)( (f(x)dx/df)dt J-r, JAn{f=t]

f

= I exp (ir f)ip(x)dx . JA

Using the asymptotic expansion of the fiber-integral when t —> 0± leads to an asymptotic expansion for the oscillating integral

when T •—>• ±00 in the scale (±r)~ a (log±r) where a 6 Bf and k 6 [0, n}. In [21] Palamodov introduced the real asymptotic spectrum of the previous oscillating integral ( in the case A = ^q Aq = XR \ {f = 0}). This is the subset 5R(/, 0) in Bf of ft satisfying : (a) there exists a non zero term of the type (±r)~^(log±r) , k £ iO, n], in the asymptotics when r —» ±CXD for some (p e C^(X^}, (b) each spectral element is endowed with a order kp which is the largest k such that (±T) (log ±r) ~ appears, (c) the C[[l/±r]] -module of asymptotics is generated by the finite family (of leading terms) (±r)~ /3 (log±r) 9 , ft e S R ( f , 0), q < kp. Of course ft E SR(/, 0) has order kp > 1 if, and only if, JA fs D has a pole of order kp > 1 at —ft — rn for some integer m € N. Palamodov has conjectured that for any germ / : (En+1, 0) —> (R, 0)

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143

singular at 0 (i.e df(Q) = 0) the real asymptotic spectrum of / at 0 is non empty. It means that there exists some (p G C^(X^) such that Jx exp (irf}(p(x]dx is not, as a function of r, in the Schwartz space «S(R). The second author proved this conjecture in [14] and we shall give an outline the proof of it. Remark that, by complex conjugacy, we can restrict ourselves to the asymptotics for r —> +00.

4.1

Real isolated singularity

By a real isolated singularity we mean that for x G XR, df(x) = 0 ^=^ x = 0. We remark that if the singularity is isolated in the complex, the answer to the conjecture would be positive if the cycle 8(A) ^ 0 for A = ]Tq aqAq = X$> \ {f = 0}. But even in this restrictive case, we don't know how to check whether that cycle is trivial or not. In fact, the proof is based on Malgrange idea [20] which consists to relate the oscillating integral to the saddle point method. For 0 « 277' « 77 « 1, we denote here by fa : Xc —> D, D = £>(0, 77), a Milnor representative of the complexified germ of /. For a (n 4- l)-chain F C Xc such that dF C Xlc, Xlc := {z G Xc, &/c(z) > 0}» and a (n + l)-holomorphic form 17, the integral Jr exp (ir/)17 admits, for r —> +00, the same type of asymptotic expansion as before. This expansion depends only, for fixed 17, on [F] the class of F in the relative homology Hn+i(Xc, X^, C), because for every compact K C X£, the function r i—> JK exp (rr/)17 is flat at +00. The following result precises and extends to real isolated singularity case the proposition 7 ([20]). THEOREM 4.1 Assume that the germ has a real isolated singularity at 0 and let p be any function in C^°(X^) such that p = 1 near 0. Then there exists a (n + l)-chain 8^ (called real) satisfying : (i) d8^_ c Xfc, 8m n R n+1 is a real neighbourhood ofO, (ii) the class of 8^ in Hn+i(Xc, X£, C) is not trivial, and (Hi) for any (n + I)-holomorphic form 17 the two oscillating integrals r

r

have the same asymptotic expansion for r —> +00. We first construct a 8n satisfying the condition (i), by pushing a real small sphere (around 0) toward Xlc with the help of the gradient of the imaginary part of /c. This is possible because S/c and /c have the same critical set which can supposed to be in the complex singular fiber {/c =0}. As each coefficient of r~ a (logr) is a distribution supported by the origin, the condition (iii) is automatically fulfilled for £R. But it is more difficult to see whether the condition (ii) is fulfilled. Secondly, we construct a smooth vector field S°° (resp. T°°) on an open complex neighbourhood of {/ = 0} \ {0} in such a way that its scalar product in Cn+1 with the gradient of the real part (resp. gradient of the imaginary part) is equal 1 and [S°°, T°°] = 0 outside a small complex ball. By using the integral curves of such fields, we build a smooth manifold A(to) C Xlc (0 < tQ « 1) of real dimension n + 2, and with quasi-regular boundary U^ =1 A^(t 0 ) such that : (a) A1 = M n+1 n A (to) is a fixed closed neighbourhood of 0 in R n+1 , independent of £0, with /(A 1 ) = [—77', 77'],

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(b) the volume of A J (t 0 ), j = 2; 3, tends to 0 for t0 -»• 0+, (c) A 4 (t 0 ) C {z, -2r/' < Furthemore, by using the fibration given by IK/c and Fubini's formula on A 4 (to), we can see that for all v 6 [—2//, 2r/], Jv+it0 '•= A 4 (t 0 ) n {3£/c = v} is a n-cycle in the fiber over v + HQ. And particularly, there is an homotopy in A(to) between the cycle 7it0 and c^A1 which is a real n- sphere. Let now 0 be a (n + l)-holomorphic form such that J A1 £1 ^ 0 ( take for instance Q ^ 0 and positive on R n+1 ). Then using the property (b), Stokes' formula on A(t 0 ) and Gauss-Manin theory we get : (1) f fi/d/c ^ 0 for all v 6 [-2r/, 2r/]. Let h(w, .), w e [0, 1] be the previous homotopy. We set by definition 6^ := A (to) U {h(w, x), (to, x) 6 [0, 1] x c)A 1 (to)}. The intersection <$R fl <$R is a neighbourhood of 0 in R"-4"1; then, by the saddle point method, one can easily conclude that the two chains have the same class in the relative cohomology. Recall that, as X<£ is Stein and contractible, by Milnor's fibration we have an isomorphism between Hn+i(Xc, X^, C) and the homology in degree n of the fiber of /c at it0, and it is clear that the cycle 7,-fo is the image of the class [<*>R] by this isomorphism. The condition (b) implies that this class is not trivial and, for fi satisfying (1), the integral J6 exp (?'r/)f2 is not flat at +00. Finally by (Hi), we get an oscillating integral r i—> I exp (irf)pO, which is not in <S(R). 1

4.2

General case

The procedure is to reduce the question to the isolated singularity case, by considering the restriction of the distribution T^, wich are the coefficient of r~ a (logr) , (a:, fc) 6 Bf x [0, n]. By the Fourier and Mellin inverse transforms, these distributions are in the vector space spanned by the distributions U% ±,(fi, q) € Bf x [1, n + 1} corresponding to the poles of /|, < /J., := f±f>0 (±f}a(x)(p(x)dx for 3?.s- > 0 and

(R, 0) given by the restriction of / to H. THEOREM 4.2 Assume that the real singularity is non isolated. Then (i) there exists a generic, real hyperplan H, through a generic point of (df) (0) \ {0}, which is non characteristic for all distributions /J. and Up ±, (ii) each restriction on H: (f±},H (f/S ± ) is well defined, and we have (f±)
The two following crucial facts are used in the proof. (a) For a fixed complex number A, the characteristic variety of the holonomic T>xcmodule T>xcfc — {^xc/P ^ 'Dx?, Pfc = 0, at a generic point } is contained in the variety VJ, n fc (ty x (Cn+1 \ 0) where V^c := \JSjt0T* 1 Xc is the characteristic variety of the DjYc-rnodule ^Xr[ s ]/c> ^T-i/ s ^c being the conormal of the Milnor /c ( ) fiber over s ([19], proposition 6.1 ). It is interesting to note that if for all j £ N*, A — j is not a root of the fr-function of fc then we have

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145

we also refer to [10] (theorem 7.6.1) for further results on the regular holonomic distributions. (b) Locally at 0, the complex singular fiber /c-1(0) can be endowed with a stratification Yl = (Vj) • > so called good stratification which particularly satisfyies the condition (see [11]): if £ is non zero cotangent vector to the stratum Vj at 2, then

(1) Firstly, by using the definition of the distribution /J. one can show that all wave front sets WF00(f^), and then WF00(t/'^±) (resp. WF^T*}}, are contained in the intersection V*fc fl /-^O) x (En+1 \ 0) (resp. V;c n (d/T^O) x (Rn+1 \ 0)). (2) Secondly, as T>xcfc '1S holonomic, there exists a generic non characteristic complex hyperplan HC through a generic point a € (dfc)~ (0) \ {0} and by (b) we can choose it in such a way that a 6 (rf/)~ 1 (0) \ {0} and H = Hc n R n+1 is a real hyperplan. Then H is non characteristic for all the distributions above. (3) Accordingly, the restriction of each distribution to H is well defined by the limit (in the space of distributions on //) of the restriction of its regularizing sequence (see [18]). The fact that for /£ this convergence is uniform on any compact subset of C \ Bf and the Bernstein identity imply the remainder of the statement. COROLLARY 4.3 The real asymptotic spectrum is not empty. We remark that the case of dim^((df)~ (0)) = n reduces to the one variable case, for which the proof is trivial. So we can assume dim^((df}~ (0) < n. In this later case, by induction on the dimension of the singular locus, the non characteristic real hyperplan given by the previous theorem at a generic point of the biggest stratum gives the reduction to the real isolated singularity case.

5 EXAMPLES In the previous sections, we exactly deal with the three approachs (a) embedding criterion, (b) almost isolated singularity and (c) saddle point method. This allows in principle to determine ( as far as the computation of the complex Gauss-Manin system is known ) poles of the concerned distributions or singular terms of the oscillating integral corresponding to a fixed eigenvalue of the monodromy. The method (b) complements the method (a). The last method don't need any hypothesis on the singularity but we still can't (for the moment) use it for a connected component. All these methods determine spectral elements up to shift by an integer. For this raison, in the following examples, we will essentially compute exp(—2i7rS^(f, 0)). A

In this section we illustrate each method by an example. Let dz^ 0 < j < n denotes the form of degree n: (dzo A .../\dzn)/dzj where ZQ, •••, zn are the coordinates in Cn+1. The first main difficulty which arises is to compute Milnor numbers and Jordan bases of the monodromy on the vector spaces H*(F, C), which is not easy in general. Nevertheless, by using the Leray exact sequence for the pair (Xc, F} and some properties of the construtible sheaf given by the cohomology of degree 1, we succeed in some quasi-homogeneous cases (see [16]).

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EXAMPLE 5.1 Compact fiber On C 4 , the monodromy of the quasi-homogeneous polynomial fc '• \

/

9

9

9X2

,

9

is semi-simple with eigenvalues — 1; 1 . The restriction / of /c to E4 has a compact fiber and an isolated singularity at 0. Using the same arguments as in the example 4.1 of [16], we have dimcH3(F, C) = 2, F being the Milnor fiber. Consider now the A

A

A

A

holomorphic 3-forrn a; = ZQ dzo —z\ dz\ +2z2 dz-2 — 1z3 dz3. Then it satisfies on C4 the relations : du = ^ A „,

d[(zl

+ *2)a;] = 2^C

^ JC

A (Z2 + Z2}uj_

JC

Denote respectively by .?;, //, z and w the real part of ZQ, z\i ^2 and 23. Then by a change of variables and Stokes' formula, we obtain for t > 0 : dx A dy A dz A dw .t 3 / 2 . and

(jj

'{f=t} r

6 /

J

{f=t}

L -M/
(x2 + y2)dx A dy A dz A dw .t2.

As the integrals on {/ < 1} are positive, we conclude : (i) the vectors e\ = [ u / f c ] \ F an(l e2 = [(zo + •^iV/'/cW are eigenvectors corresponding respectively to —1 and 1 and provide a base of the cohomology in degree 3 of Milnor's fiber, (ii) the C[[£]]-module of asymptotic expansions of j't,p, for t tending to 0 + , (f> test form of degree 3 in C^°(E 4 ), is generated by t2 and £ 2 . Finally, we get exp ( — 2 i 7 r S ^ ( f , 0)) = {1; —1} and each spectral element has order one. EXAMPLE 5.2 Almost isolated singularity On C3, the monodromy of the homogeneous polynomial fc :

is semi-simple with one eigenvalues — 1. This is a complex isolated singularity (then almost isolated); it is clear that the cohomology in degree 2 of Milnor's fiber is 3

A

A

A

generated by the vector [u/f£\,F where u; = ZQ dz$ —z\ dz\ 4-22 dz?. Set/(.T, y, z) = x2 + y2 - z2 the restriction of fc on E3. Then the semi-analytic set E3 \ {/ = 0} admits three connected components A\ — {/ > 0}, Ao — [f < 0, z > 0} and A3 = {f < 0, z < 0}. We have 6(Ai) = 0 and 6(A2) = $(A3) ^ 0. Computation done in [8] shows that for A = a-\A\ -f a 2 A 2 + 0,3AS, the distribution J'A fs D admits a simple pole located in ~N — ^ if, and only if, a^ + 03 7^ 0.

Real Asymptotic Spectrum

147

EXAMPLE 5.3 OsciUating integral On C3 , the monodromy of the quasi-homogeneous polynomial fc : fc(zo, zi, z2) = ZQZ-L - z2 A

is semi-simple.

A

Consider now the holomorphic 2-forms u\ = 3zo dzo — 3zi dz\

A

+4z2 dz2, uz = Z2&i, 6^3 = ZQZ\UJ\ and u;4 = ZQZ\z-2^i. Then they satisfy the relations :

where a\ — |, a% = |, a^ = | and a4 = |. On the other hand, by the same arguments as in example 4.1 in [16], one can show that dimcH2(F, C) = 4. For 1 < J < 4, we set GJ — [ujj/ fcj].F. The Milnor fiber of fc is biholomorphic to the Milnor fiber of the function gc defined by gc(zo, z\, z 2 ) = (z$ + z\} - z\. Thus, by considering on M3 (see example 4.3 in [16]) the distributions JAgs d, g(x, y-, z] = (x2 + y2) — z 3 , we can see that the cohomology in degree 2 of Milnor's fiber of fc is generated by the eigenvectors ei, 62, 63 and e^ corresponding respectively to the eigenvalue AI = exp(— 5«7r/3), A2 = AI, AS = exp(— 2z?r/3) and A4 = A 3 . The singularity of fc is almost isolated at 0 relatively to \% and A 4 ; the transverse singularity along the critical locus is a cusp for which eigenvalues of the monodromy are precisely AI and A2 . A direct computation on the oscillating integral shows that exp(— 2i7rSyt(f, 0)) = {Ai, A2}, each element having order 2. This corresponds to the phenomenon described in [4] in the complex setting. Remark that as in the proof of corollary 1, we can show that for any singularity, the set of complex numbers exp(— 2z7rS]R(/, 0)) is self-conjugate. EXAMPLE 5.4 OsciUating integral In [22] Scherk considered the non quasi-homogeneous polynomial fc '•

He computed Milnor number (/z = 11) and gave a base of the local algebra. This is an example where the monodromy is not semi-simple and non compact real fiber. We remark that in [1] the first author used the theory of (a, fr)-modules to show the existence of a (2, 2)- Jordan block of the monodromy for the family ZQ + zf By considering the chain 6^ of the theorem 4. 1 and the form dz0 A dz\ one can see that \ € £R(/, 0), but our method here doesn't give any information about its multiplicity. It is to be expected that Brieskorn's formula (D) (in [22], page 30) permits to get more spectral elements. The authors would like to thank the referee for his remarks which permit to improve this paper.

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REFERENCES [1] D. Barlet. Un critere d'existence de bloc de Jordan pour un (a, 6)—module. Prepublications de FInstitut Elie Cartan no 10 (2000). [2] D. Barlet. Multiple Poles at Negative Integers for \A f x D in the Case of an Almost isolated Singularity. Publ. RIMS, Kyoto Univ. 35:571-584, 1999. [3] D. Barlet. La variation pour une hypersurface ayant une singularite presque isolee relativement a la valeur propre 1. Revue de 1'Inst. Cartan 15:5-29, 1997. [4] D. Barlet. Emmelernerits de strates consecutives pour les cycles evanescent. Ann. Scient. Ecole Norm. Sup. 4ieme serie, t. 24:401-506, 1991. [5] D. Barlet. Contribution effective dans le cas reel. Compositio Mathematica 56:351-359, 1985. [6] D. Barlet. Contribution du cup-produit de la fibre de Milnor aux poles de / Annales de FInstitut Fourier, t. 34, Fasc. 4:75-107, 1984. [7] D. Barlet. Contribution effective de la monodromie aux developpements asymptotiques. Ann. Scient. Ecole Norm. Sup. 17:293-315, 1984. [8] D. Barlet, A. Mardhy. Un critere topologique d'existence des poles pour le prolongement merornorphe de JA f x D. Ann. Inst. Fourier (Grenoble) 43 3:743-750, 1993 (et son Erratum, Ann. Inst. Fourier(Grenoble), 44 2:629-630, 1994.) [9] I.N. Bernstein. The analytic continuation of generalized functions with respect to a parameter. FAP 6 4:26-40, 1972. [10] J.E. Bjork. Analytic P-module and Applications. Kluwer Academic Publishers. Netherlands, 1993. [11] H.A. Hamm, L.D. Trang. Un theoreme de Zariski du type Lefschetz. Ann. scient. EC. Norm. Sup., 4ierne serie, t. 6:317-366, 1973. [12] M.E. Herrera. Integration on semi-analytic set, Bull. Soc. Math. France 94:141180, 1966. [13] P. Jeanquartier. Developpement asymptotique de la distribution de Dirac attachee a une fonction analytique. C. R. Acad. Sc. Paris 271:1159-1161, 1970. [14] A. Jeddi. Preuve d'une conjecture de V. P. Palamodov, to appear: Topology 41 2. [15] A. Jeddi. Poles de fA / s « et conjugaison complexe. C.R. Acad. Sci. Paris, t. 320:165-168, Serie I, 1995. [16] A. Jeddi. Singularite reelle isolee. Ann. Ins. Fourier, Grenoble 41:1:87-116, 1991. [17] A. Jeddi, A Mardhy. Poles de JA fs» pour une singularite presque isolee. Manuscripta Math. 97:435 452, 1998.

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[18] L. Hormander. The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis. Springer-Verlag, Berlin-Heidelberg-New York, 1983. [19] M. Kashiwara. B-functions and holonomic systems, Invent. Math. 38:33-53, 1977. [20] B. Malgrange. Integrates asymtotiques et monodromie. Ann. Scient. EC. Norm. Sup. 4ieme serie, t, 7:405-430, 1974. [21] V.P. Palamodov. Asymptotic expansions of integrals in complex and real regions. Math USSR, vol. 55:1:207-236, 1986. [22] J. Scherk. On the Gauss-Manin Connection of an Isolated Hypersurface Singularity. Math. Ann. 238:23-32, 1978.

Deformations of Boundary Singularities and Non-Crystallographic Coxeter Groups INNA SCHERBAK School of Mathematical Sciences, Tel Aviv University, Ramat Aviv 69978, Israel E-mail: [email protected] Abstract

Singularities connected with Coxeter groups h(k], HS and GI are studied. At first, such singularities were found by O. Lyashko in the classification of critical points of non-singular functions on a singular hypersurface. We establish a link between these critical points and boundary singularities. We describe a class of deformations of boundary singularities which provides miniversal deformations of critical points of non-singular functions on a singular hypersurface. In particular, Coxeter groups I^k) and HS turn out to be connected with unimodal boundary singularities B^_^ and F£, respectively, and group G
INTRODUCTION Since 1972 it has been known that singularities of holomorphic functions are closely related to the geometry of Coxeter groups. To describe this relation, we recall briefly some data about singularities and about Coxeter groups. BIFURCATION DIAGRAMS OF SINGULARITIES Let / be a holomorphic function germ at a critical point. Its deformation is a holomorphic family germ F(-,A) such that .F(-, O) = /, A € C M . For an equivalence relation on the set of the holomorphic germs, deformation F is called versal with respect to the relation, if F contains (for values of parameter A close to O) representatives of all equivalence classes close enough to /. A versal deformation with the minimal possible dimension of the parameter space, C M , is called miniversal. The values of A's such that the corresponding germ in a miniversal deformation has zero as its critical value form a hypersurface in CM called a bifurcation diagram. A classical example is provided by an ordinary 151

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singularity, which is a class of stable equivalency. It is well known that for a singularity of finite multiplicity, miniversal deformations exist, the bifurcation diagram is unique up to a diffeomorphism. and its structure contains a lot of information about the singularity ([5]). MANIFOLDS OF NON-REGULAR ORBITS OF COXETER GROUPS A finite group generated by reflections in /^-dimensional euclidean space, i.e. a Coxeter group, has a basis of invariants, that is, the manifold of the orbits of the complexified action of the group is furnished with a natural structure of a smooth algebraic variety C M . The number of points in the orbit of a typical point is equal to the number of elements in this group. However, some orbits are smaller. These non-regular orbits form an algebraic hypersurface in the orbit space called a manifold of non-regular orbits. The list of the crystallograph!c Coxeter groups (or of the Weyl groups of the simple Lie groups) contains groups A^.D^ (//, > 4), EG, £7, Eg having root systems with only roots of equal length, and groups J3M (p, > 2),C M (// > 3),F4 and G*2 with inhomogeneous root systems. Besides the Weyl groups, the list of Coxeter groups contains non-crystallographic groups ^(p) (p > 5) - the symmetry groups of regular />gons, HS - the symmetry group of an icosahedron, and H^ - the symmetry group of a hyper-icosahedron In [1] it was established that the bifurcation diagrams of the simple ordinary singularities A^, D^ (// > 4), EQ, E7, Eg are diffeomorphic to the manifolds of the non-regular orbits of the corresponding reflection groups acting on the complex space. The extension of this connection to include other reflection groups has been a problem stimulating a deep research in the singularity theory. Deformations which are miniversal with respect to other equivalence relations appear naturally by considering singularities with additional structures as boundaries, obstacles, symmetries etc. Among the corresponding bifurcation diagrams one can recognize the manifolds of the non-regular orbits of other reflection groups. After papers [3], [12], [7] this is a "standard" way of finding reflection groups in the singularity theory. Namely, miniversal deformations of the simple singularities on manifolds with boundary are to be identified with deformations of the simple singularities A<2^-\, .D/x+i, EQ which are miniversal in the class of Z2-invariant functions ([3], [5]). The corresponding bifurcation diagrams are diffeomorphic to the manifolds of the nonregular orbits of reflection groups B M , C M , F±. A similar example is given by singularities of the distance function in the problem of avoiding an obstacle [12]. The manifolds of the non-regular orbits of Coxeter groups /2(5), //a, //4 are realized as the bifurcation diagrams of deformations of the simple singularities Aj, DQ, Eg, respectively, which are miniversal in the class of the singularities of even multiplicity. Recently, the link between unitary reflection groups and singularities of functions with cyclic simmetry was established by Goryunov [7]. Unlike the previous cases, singularities connected with Coxeter groups I^p] (p > 5), #3, G"2, have appeared in [8] in a different way. These are singularities of critical points of functions on a singular hypersurface, and the corresponding critical points are not simple but unimodal. In the present paper we propose a construction that allows us to include groups /2(p), #3 and G-2 into the generic framework of the singularity theory described above.

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We establish a link between critical points of non-singular functions on a singular hypersurface and boundary singularities. It turns out that there is one-to-one correspondence between simple boundary singularities and simple critical points on a singular surface. We prove that the bifurcation diagram of a critical point of a non-singular function on a singular surface is the bifurcation diagram of a certain deformation of the restriction of the function to the surface which is miniversal with respect to the stable equivalence of functions on a singular surface. This deformation is related to the corresponding boundary singularity. In particular, unimodal singularities I^p), p > 5, and HS are stable equivalence classes of critical points of non-singular functions on hypersurfaces of types A p _i and AS, respectively. These critical points correspond to the unimodal boundary singularities B^_1 and F*, respectively. The bifurcation diagrams of these critical points are the bifurcation diagrams of certain deformations of simple singularities A% and A4 related to the boundary singularities B^,_l and F£, respectively. The simple critical point G-2 on a hypersurface of type A% corresponds to the simple boundary singularity F± = F$, and the manifold of the non-regular orbits G-2 appears as the bifurcation diagram of a deformation of A2 related to F4. Main results of this paper are announced in [11]. The structure of the paper is as follows. In Sec. 1 we recall the theory of critical points of functions on a singular hypersurface ([8], [4]). Sec. 2 is devoted to the case of non-singular functions on a singular hypersurface. In Sec. 3 we recall some facts from the boundary singularity theory ([5], [13]). In Sec. 4 a connection between non-singular functions on a singular hypersurface and boundary singularities is described. In Sec. 5, the case when the hypersurface has a singularity of type Ak is considered. We prove that the origin is a non-critical point for a function germ on a singular hypersurface if and only if the corresponding boundary germ is of type Bk- In Sec. 6, for a wide class of critical points on a singular hypersurface, we prove that the local ring is isomorphic to the local ring of the ordinary singularity given by the restriction of the corresponding boundary germ to the boundary. In Sec. 7 we describe miniversal deformations of a critical point on a singular hypersurface in terms of certain deformations of the restriction of the corresponding boundary germ to the boundary. As a corollary, we get (Sec. 8) that the manifolds of the non-regular orbits of reflection groups /2(p) and HS are diffeomorphic to generic hyperplane sections of the corresponding bifurcation diagrams. In Sec. 9 we consider the critical point of type G 2 .

1 PRELIMINARIES 1.1

Critical points of functions on a singular hypersurface

In this section we recall the theory of critical points of functions on a manifold with singular boundary of [8]. We use slightly different terminology and call these critical points "critical points of functions on a singular hypersurface" in order not to mess them up with critical points on a manifold with boundary participating in our considerations. A function germ f on a singular hypersurface V is a triple (/, V, C n ) where • / : (Cn, O) —» (C, 0) is a germ of a holomorphic function; • V = {z € Cn | h(z) = 0} is a germ of a hypersurface with an isolated singular point at O.

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Germs of diffeomorphisms (C n ,O) —> (C n ,O) act on the set of the triples, and two triples are equivalent if they lie in the same orbit of this action. For ra > n, denote by TT the natural projection TT : Cm = Cn x C m ~ n -»• C n ,

In Cm , we define hypersurface V — \h(z\ , . . . , zn) + z£+1 H ----- h z^ =0} and function germ / = TT*/. The triple (/, V,Cm) is called t/ie stabilization of (/, I/, C n ). Functions on singular hypersurfaces are stable equivalent if they have equivalent stabilizations. Stable equivalent triples have the same singularity of hypersurfaces. We say that O is a -non- critical point of (/, V, C n ) if VQ = /~ 1 (O) is a germ of a smooth hypersurface (i.e. / is non-singular at O) transversal to V at O. In the opposite case we say that O is a critical point of the triple (/, V, C n ). The transversality at a singular point of V means the following. Consider PT*Cn, the projectivization of the cotangent bundle of C n , and the canonical projection p : PT*Cn -v C n ,

P (Z!,

. . . , z n , P l : - - - : p n ) = (*i, . . . , z n ).

For any subvariety M = {^(2) = 0} in C n , denote by PM the image of M under the embedding in PT*Cn: (2 € M) i—>• (2, tangent plane to M at z). In other words, PM is the following subvariety of PT*Cn: PM = {(z} = 0,

Pi|*

(J Z"j

-Pj|£ = 0, 1 < i < j < n}. (s^i

In particular, if V has an isolated critical point at O, then PV is reducible and consists of PT*{O} and Vl - P1/\PT*{O}, the closure of PV \ PT*{O}. We say that VQ = f ~ l ( O ] is transversal to V at O, if V0 is smooth and Vi^PVQ^p-l(U} = 0, where U is a small neighborhood of the origin in C n . This means that the tangent plane to Vb at the origin, considered as a point in PT*C", does not belong to Vi. In the case when hypersurface V has an isolated simple singularity, the classification of critical points on V reduces to the description of the orbits of the Lie group of diffeomorphisms preserving V. The Lie algebra of this group, TV, consists of the vector fields preserving V . Vector field v preserves hypersurface V, v 6 TV, if the directional derivative L^h belongs to the principal ideal (/?,), i.e. < £>,grad/i >= gh for some smooth germ g. We define the ideal

and the local algebra Qf\V

=

where On is the ring of the holomorphic germs (C n ,O) —* (C,0). The multiplicity /z(/, V] of the critical point O of (/, V, C n ) is defined as ti(f,V)

= dimcQf\v - 1.

For any stabilization (/, V , Cn+k) of (/, V, C n ), the local algebras Qf\v antl Qf\v are isomorphic, and thus the multiplicity /i(/,F,C n )) = /z(/,F,C n + & ) = /*(/, K) does not depend on dimension.

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By the usual way, the notions of modality, versal deformation, bifurcation diagram can be denned for this situation. In particular, the modality of (/, V, C n ) is the minimal number ra such that a small neighborhood of the orbit of this germ (under the action of diffeomorphisms preserving V) is covered by a finite number of raparameter families of orbits. When ra is equal to 0 or 1, a critical point is called simple or unimodal, respectively. In the case // < oo, one can take a versal deformation of (/, V, C n ) in the form (F, V, C n ), where F is the family of functions F(z, A) = f ( z ) + \0e0 + ••• + A M e M , here e o , . . . , e M are representatives of a basis of the local algebra Q/\v °ver C, and A = ( A o , . . . , A M ) € C M+1 is the parameter of versal deformation. The bifurcation diagram E(/, V) of the critical point (/, V, C n ) is a hypersurface in the base of versal deformation, C M+1 , formed by the parameter values A £ CM+1 such that 0 is a critical value of (F(-, A), V, C n ). It turns out that simple and unimodal critical points appear only on a hypersurface with a simple singularity of type Ak. In [8], it was proven that for a critical point of modality 1, the bifurcation diagram is analytically trivial along the strata p, = const, and the classification of the simple and unimodal critical points on a hypersurface of type Ak was obtained. A part of the classification is connected with reflection groups H% and /2(p)Namely, a critical point of type hip), P > 4, is given by the germ x + ey + z 2 , e ^ 0, on the hypersurface xy = zp of type -A p _i, and a critical point of type H% is given by the germ x + y + ez3 on the hypersurface xy = z5 of type A^. These critical points are unimodal and e is a parameter along the strata fj, = const. The main result of [8] is the following theorem. THEOREM 1.1 The intersection of the bifurcation diagram of a critical point of type Ii(p}i p > 4, or HS, with a hyperplane in the base of versal deformation which is transversal to the stratum // = const, is biholomorphic equivalent to the manifold of the non-regular orbits of the corresponding group generated by reflections, acting on the complexification of the Euclidean space C M .

1.2

Non-singular functions on a singular hypersurface

As it was pointed out in [8], if the number of variables is greater then two, then simple and unimodal critical points on a singular hypersurface can appear only for function germs with non-zero 1-jet. Moreover, simple critical points appear only on a hypersurface of type A^. On a boundary of type D^ or E^, there are only unimodal non-critical points. For that reason, in the paper, we study triples (/, V,C n ) where / : (C n ,(9) —> (C,0) is a non-singular germ, i.e. Vb = f ~ 1 ( O ) is a germ of smooth hypersurface. Let z\,..., zn be coordinates in Cn such that df/dzi does not vanishes at the origin. The change of variables x = /, t/i = 22, • • • , yn-i = zn, gives an equivalent triple (x,V = {g(x,yl,...,yn_l)=0},Cn).

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We denote y = (?;i , . . . , yn-\ ), 0o(?/) = #(0, y\, • • • , y n -i)- In this case Vb = {a: = 0} and thus PVb = {x = 0, ? ! = . . . =?„_!=()}. The tangent plane to V0 at the origin is point (0, . . . , 0; 1 : 0 : • • • : 0) in PT*Cn with coordinates (.x, T/J, . . . , yn-\]pQ : p\ : • • • : pn~i)- Further, PV is given by PV = {9(,, «,) = 0,Po^ =w ^, P]|U Pi ^, ! < , < , < „}. diji dx cw/i c% Therefore the intersection PVo fl PV is P V b n P V = { 0 ( a : , j / i , . . . , y n - i ) = 0 , .T = 0, P o ~ = 0 ,

Pi

= 0, i = l , . . . , n - l }

^?/i

and thus PV0r\V1={g(x,y1,...,yn-l)=Q,

3 = 0, |^- = 0, Pi = 0, i = 1, .. . , n - 1}.

^?y?'

EXAMPLE 1.2 Let y - {xk+l = y2} c C2, fc > 2. Then for the triple (.7;, V, C 2 ), the origin is a non-critical point, whereas for the triple (?/, V. C 2 ), the origin is a critical point. Indeed, PV = {xk+l = y2, (k + l)xkPl = 2yp0} - {y = x^ , (k + l)xkPl = 2x^Po}. If x ^ 0, we get

fc + 1 Po = —

fc-i

For / = x, we have PFT = {x = pi = 0} and the corresponding point in PT*C2 is (0,0; 1 : 0) £ Vi = P1/\PT*{O}, whereas for / = y, PFM = {?/ = p0 = 0} and the corresponding point in PT*C2 is (0,0; 0 : 1) e Vi.

1.3

Boundary singularities

Here we recall some basic facts of boundary singularities theory [13]. A boundary germ is a triple (#, F,C n ), where Y is a germ at O of a smooth hypersurface, called a boundary, and g : (C n ,O) —> (C,0) is a holomorphic germ such that both g and g0 = g\Y : (Y,O) —* (C,0) have isolated critical points. In appropriate local coordinates (x, yi, . . . ,yn-i) °f C", the boundary is Y = {x — 0}, and 0 0 (y) =0(0, x / i , . . . , 2/n_i). A stabilization of (_g, F, C n ) is a boundary germ ( n, where

g(%, yi,--, ym-i) = g(x, y\, • • • , y n -i) + 2/J H ----- 1- 2/m-n ^ = i-T = °l-

A boundary singularity is a boundary germ considered up to germs of diffeomorphisms preserving the boundary arid up to stabilizations. The multiplicity //(#, Y} of the critical point of the boundary germ (g, Y, C n ) is the dimension over C of the local ring Q(g,Y) = On/I(g,Y], where I ( g , Y ] is the ideal generated by xdg/dx, dg/dy\, . . . , dg/dyn-\ :

The multiplicity of a boundary germ and of any its stabilization is the same.

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In a natural sense, boundary singularity (g, Y, C n ) is an extension of two ordinary singularities given by germs g and go. These two ordinary singularities are called the decomposition of (#, y, C n ). Recall that for an ordinary singularity given by a holomorphic germ f ( z i , . . . , zn) at the critical point O, the multiplicity is the dimension over C of the local ring Q(f] = O n / I ( f ) , where ideal /(/) is generated by the partial derivatives df/dzi , . . . , df/dzn:

For boundary singularity (#,y, C n ) with decomposition (g,go)'.

A versal deformation of boundary singularity (g, y, C n ) one can take in the form G(x, y, A) = g(x, y} + EAjeJ + where {e'^ 1 < i < ^(go)} represent a basis of Q(<7o)> and {e'L 1 < j < /^(s1)} represent a basis of Q(g). The bifurcation diagram of the boundary singularity (.g,Y, C n ) has two irreducible components, which are bifurcation diagrams of g and go respectively multiplying by complex spaces of appropriate dimensions. In generic case, the decomposition does not define a boundary singularity (see example 4), but boundary singularities with decomposition of type (A^, A{) are welldefined by their decomposition ( [10]). In particular, if go is a Morse function (i.e. of type AI) and g is of type Ak, then (#, y, C n ) is of type Bk+i- It can be given by the function germ xk+l + Q(y), where Q(y] = Q(y\, • • • , 2/n-i) is a Morse function, the boundary is x = 0.

2

LINK BETWEEN NON-SINGULAR FUNCTIONS NON SINGULAR HYPERSUPERFACES AND BOUNDARY SINGULARITIES

If (f(z),V = {g(z) = 0},C n ) is a germ of non-singular function / on a singular hypersurface V, such that //(/, V] < oo, then triple (g(z),Vo = [f(z) = 0},Cn) is a boundary germ. Conversely, any boundary germ (g(x, y ) , Y — {x = 0}, C n ) defines a germ (x, V = {g(x,y} = 0},C n ) of a non-singular function on a singular hypersurface. PROPOSITION 2.1 Equivalent germs of non-singular functions on a singular hypersurface define the same boundary singularity Proof: If (/,, Fj,C n ), i = 1,2, are equivalent triples and /i,/2 are germs of nonsingular functions, then there exist germs of diffeomorphisms • (C n ,O), such that 0 sends (A,Vi,C n ) to (/ 2 ,V2,C n ) and ^ sends (/;,!/;, C n ) to (ar,{^(a;,y) = 0},C"),» = l,2. The corresponding boundary germs are (^(x, y), y, C71) (i = 1, 2) with the boundary y = {x = 0}, and one goes to another by the diffeomorphism 02 ° 4> °
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Simple critical points

Comparing the lists of the simple critical points on singular surfaces [8] and of the simple boundary singularities [3] , we get the following PROPOSITION 2.2 The simple, critical points on a singular hypersurface correspond to the simple boundary singularities Ck, k> 2, F±. N on- critical points on a singular hypersurface correspond to the simple boundary singularities B^, k > 2. EXAMPLE 2.3 • Simple critical point (7o appeal's in [8] for non-singular germ x + y on a singular surface xy = z3. It is easy to check that the corresponding (simple) boundary singularity is F$. It can be given by germ x2 + y3 and boundary {x = 0}. • Critical points /2 (/>),/> > 4, appear in the Lyashko classification on a hypersurface of type Ap-\. These critical points are unimodal. The normal form of the function germ is / = x + ey + z 2 , where f ^ 0 is a parameter along the strata // — const. The hypersurface V is given by V = {xy = zp + Q}, where Q is a Morse function in additional variables. The corresponding boundary singularity has decomposition (Ap-\, AS}. Indeed, Ap_i is the type of the hypersurface. Consider f~l(O) = VQ = {x+fy+z2 = 0}. The intersection with the boundary, V n VQ, is given by VnV0 = {zp + (fy + z2)y = 0}. We have: zp-r-(ey + z2)y = zp -az4 Jr(^/a.z2 + ^y}2, where a = l/4f, therefore this is a singularity of type A3. Thus the boundary singularity corresponding to the critical point of type I^p] is the unimodal boundary singularity of type Bp^ (in notations of [10]). HS -singii lari ty. In the Lyashko list, a critical point of type H$ is given by germ x + y + ez3 on a hypersurface xy — z° of type A$ (e is a parameter along the strata // = const). After the change of variables

we get germ z° — (X — y — f.z3)y on the boundary X = 0. It is easy to see that this boundary singularity has decomposition (A^.A^). This is unimodal boundary singularity F% (in notations of [10]).

2.2

Singular hypersurface of type A^

Here we consider triples (/, V, Cn) such that / is a non-singular germ, and hypersurface V has a simple singularity of type Ak at the origin. We call such triples L-germs. Note that the critical points on a singular hypersurface, which are connected with reflection groups 6?2, Ii(p)-,P > 4, and //s, are L-germs. For a L-germ, we can choose coordinates such that the triple is of the form (/(x,y), (xk+l = Q(y)}, C n ), where Q(y) - Q(yi,... ,y n -i) is a Morse function.

Deformations of Boundary Singularities

THEOREM 2.4 The origin is a non- critical point of the triple (f(x,y), Q(y}}, Cn), if and only ifdf/dx 0 ^ 0.

159

V = {xn+l =

Proof: As it follows from [2], [9], if df /dx does not vanish at the origin, then / can be reduced to a: by a diffeomorphism preserving {xn+1 = Q(y}}, and we get a triple which is stable equivalent to the triple (x, V, C2) of the example 1. If df/dx o = 0 , then df/dyi\o ^ 0 for some 1 < i < n - 1 (recall that / is non- singular at the origin). We can assume

Indeed, if / = a\y\ + • • • + a n _iy n _i + (terms of order > 2), and a\ required change of variables is ( z , y i , . . . , y n _ i ) ^ (x,Y\ = a\y\ H ----- h a n -i:?/n-i,2/2, • • • ,2/n-i)The tangent plane to {/ = 0} at the origin corresponds to point (0, . . . , 0; 0 : 1 : 0 : • • • : 0). This point is obviously in V\ (the intersection with plane {y% = • • • = yn-i = 0} reduce this case to the case of two variables considered in example 1). D THEOREM 2.5 The origin is a non-critical point for a L-germ, if and only if the corresponding boundary germ is of type

Proof: Consider L-germ (/(x,?/), {xn+l = Q(y)}, C n ), where Q is a Morse function. The condition that the origin is a non-critical point for the L-germ means that df/dx\o 7^ 0, i.e. the equation / = 0 can be solved with respect to x: x = 0(y), where <j)(y) is a holomorphic function germ. Then the restriction of the function xk+l — Q(y) to the boundary, (0(y)) fc+1 ~ Q(y]i has a non-degenerate quadratic part and therefore defines an ordinary singularity of type A\ . Therefore boundary singularity (xfc+1 — Q(y},f = 0, C n ) has decomposition ( A k , A i ] , i.e. this is a boundary singularity of type B^+i • D If the hypersurface, V, has another simple singularity, i.e. a singularity of type Dk (k > 4) or Ek (k = 6, 7,8), then one can prove that a germ (x, V = {g(x, y] = 0},C n+1 ) does not have the origin as a critical point if the germ go(y) = g(0, y] defines a singularity of type A^ . EXAMPLE 2.6 The following germs on the boundary x = 0 define unimodal noncritical points (x, V = {g(x,y) = 0},C 2 ) on a hypersurface of type D^, k > 4, and Ek, k = 6, 7, 8, respectively: (Dk,A2): (E6,A2) : (E7,A2): (E8,A2}:

g(x,y) = xy2 + y3 + axk~l , a ^ 0, g(x, y} = x4 + y3 + ax3y, g(x,y) =x3y + y3+ax5, g(x,y}=x5+y3+ax4.

Consider hypersurface V — {xa +yb = 0}, a > b. Direct calculations show that O is a non-critical point for triple (x, V, C 2 ) and O is a critical point for triple (y, V, C 2 ). More generically, we have the following proposition.

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PROPOSITION 2.7 Let hyper surface V — {g(xi,... ,x n ) = 0} be given by a quasihomogeneous junction g with deg.7:? = al, 1 < i < n — 1, a\ > 0.3 > • • • > an. Then the origin is a critical point of (x\, V, C n ). Proof: Note first of all, that if g is a quasi-homogeneous function with deg 0. Point (O; 1 : 0 : • • • : 0) corresponds to the function x\.

2.3

D

Local rings

It appears that in many cases, the local ring Q/\v — On/lf\v °f the triple (/, V — [h(z] = 0}, C n ) is isomorphic to the local ring Q(ho) of the ordinary singularity given by ho = h\f=Q. First we establish this result in the quasi-homogeneous case. THEOREM 2.8 Let (
oyn-i

)

and by the Hamilton! an vector fields Ox dyk dg

0

-yi vyi

,

Oyk dx1 dg d

^r~'^r~' £7/7 uyi

1

. .

<^J
Applying these vector fields to the function ,x, we get the generators of Ix\v which are x, dg/dyk,k — 1, . . . , n — 1. Therefore QX]V

= On/ < X,

^-, . . . , -^- >= On

COROLLARY 2.9 // (g(x, y),Y — {x — 0}, C n ) is a boundary germ given by a quasihomogeneous function g, then /-/(.x, V) = p<(go) — 1 The similar statement is hold for L- germs of modality 0 or 1.

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THEOREM 2.10 // (f,V = {h(z} = 0},Cn) is a L-germ of modality 0 or 1, then: (i) Qf\v=Q(h\f=0); (ii) The proof of the theorem can be obtained by direct calculations using the list of normal forms of simple and unimodal critical poins and their versal deformations given in [8]. We say that (/, V = {h(z] = 0},C n ) is a good triple if it satisfies condition (i) (and, hence, (ii)}. In particular, triples of types H% and I<2,(p} are good. Using the parametrized Morse lemma, one can prove the similar result for triples (x, V = {g(x, y) = 0}, C n ) such that the corresponding boundary germs (g(x, y ) , Y = {y = 0}, C n ) have decomposition (Ak, At). Unfortunately we do not know a proof working for all these cases. It would be interesting to get such a proof and to understand general conditions for 'triple to be good. Note that for L-triples, the equivalence class of a critical point (/, V = {h(z} = 0}, C n ) does not in general defined by the equivalence class of the germ ^|/=o, as the following example shows. EXAMPLE 2.11 Consider two germs on C3 with the boundary {x - 0}: 7 7 h\—xz — z 4 — zy 2 — y 4 , h? = xz — z 4 — y 4 — zy 3 .

As it follows from [10], they define non-equivalent boundary singularities with the same decomposition (A3,D$). This means that the critical points (x,Vi = {hi = 0},C3) (i = 1,2) on a singular hypersurface of type A3 have the same singularity, namely D5, of the restrictions to the bundary, hi\x=o, but they are non-equivalent L- germs. For holomorphic functions, an isolated critical point always has a finite multiplicity. Next example provides a non-singular function having a critical point of infinite multiplicity at an isolated singular point of a hypersurface. EXAMPLE 2.12 Consider the following critical point on a singular hypersurface: (x, V = {x3 + xz2 + y2z = 0}, C n ).

It is clear that hypersurface V has an isolated singularity at O, but the multiplicity of the critical point //(x, V} = oo. Indeed, V is given by a homogeneous function, therefore theorem 3 gives Qx\y = On-i/I(go), where go — y2z. Function go has a non-isolated critical point at 0 (in fact, the line y = 0 is the line of critical points), that means that dime Qx\v — °°-

2.4

Versal deformations and bifurcation diagrams of good triples

Let L = (,r, V = {g(x, y) =0}, C n ) be a good triple. A versal deformation of L one can take in the form Lx = (x + X0eo + •••+ A^e M , V, C n ),

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where {ei = e ? ;(y), 0 < i < //,} represent a basis of the local ring Q(go). We define a deformation GL(X, y. A) of boundary germ (g(x, y), {x = 0}, C") which corresponds to the versal deformation of the germ L as the family GL(X, ?/, A) = g(x + A 0 e 0 H

(- A^e M , y).

The restriction of this family to the boundary, G°L(y, A) = Gr,(0, y, A) = g(\oe.0 H

h AMeM, y ) ,

is a deformation of go = g(0, y) related to the boundary germ (g(x, y), {x — 0}, C n ). We call this deformation boundary deformation of go and the corresponding bifurcation diagram the boundary diagram. One can see that the boundary diagrams of equivalent boundary germs are diffeomorphic. It turns out that the bifurcation diagram of this deformation is the bifurcation diagram of L. THEOREM 2.13 The bifurcation diagram of L i.s E(L) = (A 6 C^ +] | 0 is a critical value of G°L(-,\)} Proof:

Change of variables (x, y) K-). (X = x - A 0 e 0

A^e^, y)

gives an equivalent family Z A = (A", VA = {g(X + AQCO + • • • + A M e M , ? / ) = 0},C"). The bifurcation diagram of this family is given by S(L) = (A € C^+1 | 0 is a critical value of Z A }. Consider FT*C" with coordinates (x, y i , . . . , yn-\\pQ : p\ : • • • : pn-i}- Variety PV\ of tangent planes to V\ is given by

„ (v

,•,

GL(X,y,*)

n

VGL

= 0, PQ—

ay,:

OGL

OGL

C/A

ay?

=/>i-rrrr, jpiT:

OGL

=Pj—^—5

a?y7

. .

?, J = 1, - • • , n - 1.

The intersection with PVo = {-^C = p\ = • • • — pn-i = 0} gives PVxnPV0 = {X=Pl = - - - = p n _! = 0, G L ( X , y , A ) = 0, PodGL/dyi = 0}. Thus the intersection V\ PI PVo is given by V r 1 nPVo = {G L (X, ? / ,A)-0, X = 0 , dGL/dyi = Q,

Pi

- 0, i = 1,... ,n - 1} =

= {Gj(y, A) - 0, 0Gj/0W = 0, Pi = 0, i = 1 , . . . , n - 1}. It is non-empty if and only if 0 is the critical value of G°L(•, A).

D

Deformations of Boundary Singularities

2.5

163

Critical points G 2 ? ^3 and II(P]

The simple critical point G% appears in the Lyashko classification for non-singular germ x + y on a singular surface xy = z3. The corresponding simple boundary singularity F4 is given by germ x2 + y3 and boundary {x = 0}. The boundary deformation is given by GQG(y, A) = (Ai 4- X2y}2 4- y3. Simple calculations show THEOREM 2.14 The bifurcation diagram of the deformation G^ is diffeomorphic the manifold of the non-regular orbits of the group G<2

to

Consider function f = x having a critical point of type H3 at the origin on a hypersurface V = {x2 + y5 = 0}. Its versal deformation is x 4- X3y3 4- A 2 y 2 4- Xy + XQ. Thus the corresponding deformation of the unimodal boundary singularity F% is GH(x, y, X } = y ' ° + (x + X3y3 + X2y2 4- Xy + A 0 ) 2 . As it is proven in[8], the bifurcation diagram of this critical point is analytically trivial along the stratum JJL = const and AS is a parameter along this stratum. The bifurcation diagram of this critical point is the bifurcation diagram of the deformation G°H(y,X)=y5 + (X3y3+X2y2+Xy

+ X0}2

of simple singularity A4. A critical point of type / 2 (p), p > 5, can be given by germ / = x on a hypersurface V = {(x 4- y 2 ) 2 4-y p = 0}. Its versal deformation is x + y2 4- X2y2 4- Xiy 4- A0. Again, as it follows from [8], the bifurcation digram is analytically trivial along the stratum // = const and A 2 is a parameter along this stratum. The corresponding deformation of the unimodal boundary singularity B3_1 is Gr(x, y, A) = yp + (x + y2 + X2y2 + Xiy + A 0 ) 2 . The bifurcation diagram of this critical point is the bifurcation diagram of the deformation £j(j/> A ) = yP + (y2 + A22/2 4- Ait/ + A 0 ) 2 of simple singularity A%. Triples Hs and /2(p) are good, therefore A3 (resp. A 2 ) is a parameter along the stratum /.j, — const for the deformation GH (resp. Gj} of the unimodal boundary singularity F£ (resp. B3^ as well. The bifurcation diagram of critical point HS (resp. /2(p)) is the component of the bifurcation diagram of GH (resp. G/) which corresponds to the restriction to the boundary. Thus get the following result. THEOREM 2.15 The bifurcation diagram of the deformation G® (G°H resp.) of a simple singularity A$ (A± resp.) is diffeomorphic to the manifold of the non-regular orbits of the group /2(j>) (H^ resp.) multiplying by a complex line Acknowledgment: A part of this paper was written at the Isaac Newton Institute for Mathematical Sciences, Cambridge. Author thanks the Institute and the organizers of the Programme on Singularity Theory for their hospitality and support.

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REFERENCES [I] V.I. Arnol'd. Normal forms of functions near degenerate critical points, the Weyl groups yifc, Dfc, E/,. and Lagrangian singularities. Functional Anal. Appl. 6:254272, 1972. [2] V.I. Arnol'd. Wave front evolution and equivariarit Morse lemma. Comm. Pure Appl. Math. 29:557 582, 1976. [3] V.I. Arnol'd. Critical points of function on a manifold with a boundary, the simple Lie groups Bk,Ck, F4, and singularities of evolutes. Russian Math. Surveys 33:91105, 1978. [4] V.I. Arnol'd, V.V. Goryunov, O.V. Lyashko, V.A. Vassiliev. Singularities II. Encycl. Math. Sci. 39, Springer Verlag, 1993. [5] V.I. Arnol'd, S.M. Gusein-Zade, A.X. Varcheriko. Singularities of Differentiable Maps, Vol. I. Birkhauser, Basel, 1985. [6] N. Bourbaki. Groupes et algebres de Lie, Chapters IV, V and VI. Hermann, Paris, 1968. [7] V. Goryunov. Unitary reflection groups associated with singularities of functions with cyclic symmetry. Russian Math. Surveys 54 5:873-893, 1999. [8] O. Lyashko. Classification of critical points of functions on a manifold with singular boundary. Functional Anal. Appl. 17 3:187-193, 1983. [9] O. Lyashko. Geometry of bifurcation diagrams. Journal of Soviet Math. 27:25362759, 1984. [10] I. Scherbak. Boundary singularities with simple decomposition. Journal of Soviet Math. 60:1681-1693, 1992. [II] I. Scherbak. Xoncristallographic Coxeter Groups arid Boundary Singularities. Functional Anal. Appl. 35:2, 2001. [12] O. Scherbak. Wave fronts and reflection groups. Russian Math. Surveys 43 3:149J 194, 1988. [13] I. Scherbak, ASzpirglas. Boundary singularities: topology and duality. Advances in Soviet Math. 21:213-223, 1994.

Transversal Whitney Topology and Singularities of Haefliger Foliations SHYUICHIIZUMIYA Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan E-mail: [email protected] KUNIHIDE MARUYAMA* Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan E-mail: [email protected]

Dedicated to the memory of Professor Luiz A. Favaro Abstract

In order to study singularities of Haefliger foliation, we define the notion of transversally Whitney C°° -topology modulo a regular foliation on the set of C°°mappings into a foliated manifold. We prove a kind of transversality theorem with respect to this new topology. All arguments we use here are analogous to those of the theory for the ordinary Whitney C°°-topology. However, this is the first attempt for the study of generic properties of Haefliger foliations

1

INTRODUCTION

In his paper [2] Prof. Luiz A. Favaro studied ^-stability of (7°°-mappings into a foliated manifold. The definition of .F-st ability is given as follows: Let M, N be C°°manifolds and N is regularly foliated by F. We consider the space of C°°-mappings C°°(M, N) with the Whitney C°° topology. We say that / G C°°(M, TV) is f-stable * Currrent address: NEC nogawaryou 211, miyamae-ku nogawa 3139, kawasaki 2160001, Japan. 165

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Izumiya and Maruyama

if there exists a neighbourhood Vf of / in C°°(A1, N) such that for any g <E Vf. there exist a diffeomorphism /?. : M —» M and a diffeomorphism k : N —> N taking leaves of f to leaves of T such that g — k o / o /r~ ] . The infinitesimal version of the above stability is denned as follows: / E C°° (M, N} is injinitKtiimally f-stable if for any w e r°°'(f*TN), there exist w e r°°(TA/), ;, e r°°(7W) with 7r(v/) is locally constant along the leaves and w — df o w -f ?; o /, where TT : TN —> TN/TF is the canonical projection. One of the purpose in his paper is to consider that the above two definitions are equivalent or not. He has shown that it is true in the local sense. Of course his assertion is rather clear in the present time, because there are a lot of tools and results in singularity theory now (c.f., Damon [1]). Global ^"-stability is, however, still an open question now. Unfortunately, we can riot contribute to the global jF-stability in this paper. Let us consider the meaning of the above stability of mappings. We say that a C°°-niapping F : M —> AT is .F^-nonsingular at x <E M if / is transversal to JF at x. Otherwise, we say that / is jF^-singular at x. The above notion of stability corresponds to the stability of ^-"^-singularities. One of the motivations to study J-^singularities is given in the study of singular Haefliger foliations. It is clear that if / is .F^-nonsingular at any point, f ~ l J - is a regular foliation. If / has jF^-singularities, then f~lF is a singular Haefliger foliation. We now review the definition of Haefliger foliations. Suppose that there exist an open covering M = Uig/V, and a local C°°mappiiig 07; : Vj —> R9 on each V, with the following properties: (1) If Vi n Vj ^ 0, there exist open neighbourhoods Ut of faty n V 7 -), Uj of 0? (Vi f"1 Vj) anc^ a (liffeomorphisrn V'?;? : Uj —> Ur such that T/'^- o j(x] for xeViCiVj. (2) If Vi n Vj n Vk ^ 0, V'?.7 ? V'7fc, V'ifc satisfy the cocycle condition: V'ifc(^) = V'ij ° ^jk(x] for any x 6 Vr n V,- n Vk. In this case, H — {(0,;, Vi) \ i e /} is called a Haefliger foliation and q is called the codimension o f H . We can extend each level set 0~1(c;) C V,; to Vj if 0"1 (c) fl Vj ^ 0. The maximal connected component consisting of such level sets is called a leaf ofH. We say that x G M is a singular point of 7i if x is a singular point of 0j for a? 6 Vi. It has been known (and easy to prove) the following proposition. PROPOSITION 1.1 Let H be a Haefliger foliation on M. Then there exist a smooth manifold AT, a regular foliation J- and a C°° -mapping f : M —> TV such that f~~lJ- =

n.

This proposition gives a strong motivation for the study of JF-1 -singularities of C°°mapping / : M —> TV. In the definition of ^F-stability and infinitesimal ^-"-stability, the directions for perturbations of / E C' 00 (M,TV) contains too many information for the study of jF^-singularities. It can be perturbed into the tangent direction of leaves of T. In this paper, we study the stability of singularities of the Haefliger foliation f~lF. For the purpose, we do not need the perturbation corresponding to the tangent direction of leaves. Our main purpose is summarised that we introduce a certain topology on C°°(M, TV) describing the perturbation of / e C°°(M, TV) with respect to only the transversal direction of J-'. All arguments in this note are analogous to those for the jet-transversality theory on the ordinary Whitney C°°-topology. Nevertheless, we write down the results because these contain some new concepts arid these are the fundamental results for the study of singularities of Haefliger foliations.

Transversal Whitney Topology

2

167

TRANSVERSAL WHITNEY TOPOLOGY MODULO f

In this section we introduce a new topology on C°° (M, AT) along the line of the description of Whitney C*°°-topology in [3]. First, we borrow the notion of jet spaces modulo T from Ikegami [4] for our purpose. Let /, g : (M, x) —> (N, y] be C°°-map germs. Since f is a regular foliation on N, there exists a local chart 771 x 772 : U —>• DqxDr C R9 x Rr around y e A7 such that UftF = {(r?i xr/ 2 )~ 1 ({c}xD r ) | c € D9}. For any k > 1, we say that / and g have k-th order contact at x modulo f if 771 o/ and r/i o g have k-th order contact at x in the ordinary sense (cf., [3]). We can easily show that the above relation is an equivalence relation among the set of all C^-map germs C°° (M, N)(x^. We denote the set of equivalence classes by J fc (M, N; F)(x,y) and we call it a k-jet space modulo jF. For k = 0, we consider that J°(M, N; f ) ( x , y ) = MxN. The element in J fc (M, A7";/")^^) represented by a map-germ / : (M,x) —> (N,y) is denoted jkpf(x}. We can define the k-jet bundle modulo f by

The topology on Jk(M,N;F] is defined like as the ordinary k-jet bundle. We also define a mapping a x ft : Jk(M,N;F) —> M x N by a x ft(jkFf(x}} = (xj(x}}. We can summarise basic properties of J fe (M, N; F}. For the detailed descriptions, see Ikegami [4]. PROPOSITION 2.1 Let M,N be smooth manifolds such that N is regularly foliated by F with codiiruF = q. a) Jk(M,N;J-') is a smooth manifold. b) For any smooth mapping f : M —>• AT, we can define a smooth mapping j^f : M —>• J fe (M, N;J-") by (jkpf}(x] = j^fix). We call j^f a k-jet extension modulo f'. c) a x ft : J fc (M, A/"; J") —> MxN is a smooth fibre bundle with the fibre Bkmq = {P : (R m ,0) —> (R 9 ^)

P : polynomial map with degree P < k }.

We now define the notion of Whitney C°°-topology modulo F. The idea is very simple. We adopt Jk (M, N; J") for the new topology instead of Jk (M, Ar) for the ordinary Whitney C°°-topology. For any non-negative integer k and a subset U C Jk(M, N- /"), we denote that M>([7) = {/ 6 C°°(M, AT) | jkFf(M)

C U }.

It is clear that M£(t7) n M^(V] = M^(U n V). It follows from these facts that W£ = {Mj,(U) | U : open subset} form a basis of a topology on C°°(M,N). We can induce a topology on C°° (M, AT) such that the basis of the topology is given by W^ = U^L0Wf. We denote C°°(M,N-Jr) the space of smooth mappings with this new topology. We call this topology Whitney C°° -topology modulo F. We can prove several basic facts on Whitney C°°-topology modulo f like as the ordinary Whitney C°°-topology. PROPOSITION 2.2 1) C°°(M,N]F} 2) The mapping

is a Baire space.

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defined by jjr(/) — jjrf is continuous. Here we adopt the ordinary Whitney C°° -topology on C°°(M, J fc (M, N; f}). Proof: Let d^ be a complete metric on J f c (M, N;F}. Let U\, Uo, . . . be count ably many open dense subsets of (7°°(M, TV; F}. For any open subset V C C°°(M, N\ F ] , we. have to show that V n n^G^ ^ 0. Since V C C°°(Af, N;f) is an open subset with respect to Whitney C°° topology modulo JF, there exist a natural number k.Q and an open subset W C ./fc°_(Af, N; .F) such that M^(W] C V and M^(W] ^ 0. It is enough to show that Mf(W] n H^f/, ^ 0. For the purpose, we choose functions / i , / 2 , - - - £ CY°°(M, A^; JF), non- negative integers fcl5 fco, • • • and open sets H^ C
(Ai) /,- e Mjr(WO n n - A ^ ( w ; ? ) n [/,. (fli) M,r(W) C Ui and /?. € Mf(Wi), (Ci] d^f^J^f^x)}

< I for 7 > 1 and 1 < ,s < i.

(Dj) For any x e A/, there exists a local chart 77! x 772 : V —> Dq x Dr of Ar around /i(x) such that I/ n JF = {^"^c') <" € £>9}, /i(x) e V" and r;2 o /,: = r/2 o /: The condition D7 is independent on the choice of local charts (r/i x 772, ^) and it is the essential difference from the arguments on the ordinary Whitney C00- topology. We can choose the above subjects by the almost same arguments as those on the ordinary Whitney C00- topology. By the condition (C,;), jjr/i(.x¥) (?' = 1 , 2 , . . . ) is a Cauchy sequence in ,7s (Af, ./V;JF), so that we have the limit ys(x) = lim^oo jjr/i(.r). Since J°(M,/V;.F) = M x TV, we have j ^ f r ( x ) - (x,/j(x)). Therefore, we define a mapping # : A-f —> A^ by .90 (-^) = (XTQ(X))- In order to prove that g is a smooth mapping, we need the condition (-D7:). By exactly the same arguments as those on the ordinary Whitney C°° -topology, we can show that rji o g is smooth for a local chart rii x r/2 : V —> Dq x Dr of N around g(x). By the condition (I),;), r;2 o g — r/2 o fl. This means that ^ is smooth. By the conditions (-Ai), (^) and exactly the same arguments as those on the ordinary Whitney C°°- topology, we can show that g G Mjr(W) n n^f/j. This completes the proof of the assertion 1). Let U C J*(M, Jk(M,N;F} be an open subset. Since M(U] = {/ G C°°(M,Jk(M,N',f))

| //(A/) C C7 }

is an element of the open basis for the ordinary Whitney C00- topology, it is enough to show that (j$:}~l(M(U)) is an open set in (7°°(M, AT; J^). We define a map

by «fc,£(o") = jf'(jkpf}(x}, where a = j^ f'f(x). It is clear that 0;^^ is a well-defined smooth map. Therefore, fx^lf(U) is an open submanifold of Jk+e(M,N]P). Since <*k,i°£+lf = j*°£f, we have Mr(a^t(U}) = (j^)-l(M(U}), so that it is open. This completes the proof. D

Transversal Whitney Topology

3

169

A JET TRANSVERSALITY THEOREM MODULO T

In this section we formulate the transversality theorem modulo T and give a proof. The proof of the theorem is a direct analogy of the proof of the original jet transversality theorem [3], so that we only give the sketch of the proof here. THEOREM 3.1 Let M, TV be smooth manifolds and F be a regular foliation on N. For any submanifold W C J fc (M, TV ;.?•"), the set T$f = {/ E C°°(M,N]f}

fef

: transversal to W }

is a residual subset ofC°°(M, TV; .F). If W is a closed set, then T^ is an open set. Proof: The proof is analogous to the proof of the ordinary jet transversality theorem, so that we only describe the different part from the ordinary one. We show that TW is a intersection of countable open dense subsets of C'00(M,./V;.7r). Let VFi,W"2 ,• • • CW be an open covering of W with the following properties: 1) Wi C W.

2) Wi is compact. 3) There exist coordinate neighbourhoods Ui of M and Vj of TV such that a x ft(Wi) cU,x Vi. 4) Ui is compact. Since F is a regular foliation on TV, we can choose the coordinate system 771 x 772 : Vi —> DI x Dr C Rq x Rr with Vt r\ F = {(rji x r)2)~l({c} x Dr) c E D* }. We define a set T£. = {/ G C°°(M, N] f ]

j^f

: transversal to WonWt}.

Since T^ = n^1T^., it is enough to show that T^. is an open dense subset. We denote by Ti = {g E C°°(M, J fc (M, TV; /")) | g : transversal to W on Wi }. Since we adopt the ordinary Whitney C^-topology on C°°(M, J fc (M, TV; J")), it is known that T^ is an open set ( [3], page 57). It follows from the fact that fy is continuous, we have an open subset TV, = (j^)~l(Ti). In order to prove the density, let V> :Ui —> R n and ^i x r/2 : V^ —» D^ x Dr C Eg x R r be coordinate charts. Let p:Rn—>[0,1],

p':Rr—^[0,1]

be continuous functions respectively defined by 1 0

on a neighbourhood of ^ o a(W,;) off

1

on a neighbourhood of ryi o a(Wi)

0

Off 77!

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Izumiya and Maruyama

Let B(in,r : k} be a set of polynomial maps p : R m —> R r with degreep < k and p(0) — 0. Let / : A/ —> N be a smooth mapping. For any b € B(ni,r : fc), we define a smooth mapping gb : M —> A7 by

^ru/k'^oye^ We now define a smooth mapping G : MxB(ni, r : k) —> N by G(x, b}=gb(x). We also define a mapping <£ : AtxB(in,r : A:) —> Jk(M, N] JF) by $(.x%/>)=j£-(fc(x). By the similar arguments as those of in the proof of ordinary jet transversality theorem, we can easily show that there exists an open neighbourhood B of 0 £ B(in,r : k) such that <£ Af x /? is a local diffeomorphism at the point (j-, b) e Af x B with <£(x,/}) £ M7,;, so that $ A:/ x L? is transversal to W on Wi. By the fundamental transversality theorem of Thorn (cf., [3!), we have a sequence {bj}^^ C B which is uniformly convergent to 0 £ B such that jkpCf\,j is transversal to W on Wj. Since go — f and gb = f outside of L/ 7 , we have lim ? yoo gb. = f in C°°(M, A"; T\ On the other hand, suppose that W is a closed set. Since we adopt the ordinary Whitney C°°-topology on C°°(M, J fc (M, A7; J 7 )), it has been known that the set Tfv ={F e C°°(Al,Jk(M,N;^))

F : transversal to W }

is an open set (cf., [3]). By the assertion 2) of Proposition 2.2, j£ : (^(Af, A^; f} —> C°°(AL ./ fc (M, is continuous. Hence, T£ = ( j ^ ) ~ l ( T f v )

is an open subset of C°° (M, N; ^).

D

We remark that the multi-jet version of the above theorem also holds. Since we do not consider global situation here, we omit to describe the assertion and the proof.

4

APPLICATIONS

In this section we give some applications of the jet transversality theorem modulo 3F. 1) Thom-Boardman Haefliger foliations Let M, N be smooth manifolds and f be a regular foliation on N with codimj- = q. We now consider the (ordinary] Thom-Boardman singular set S 7 (m,^) C ,Jk(m,q). Since Jk(in,q) is considered to be the fibre of Jk(M,N;F], we can define a Thom-Boardman subbundle modulo T £ / (M,./V;.F) C Jk(M,N\T} as usual. By Theorem 3.1, the set TE/(M,N ; :F) = {/ e C°°(M,N;F)

jkfj

: transversal to S 7 (M,A^;^) }

is a residual subset of C°°(M, N-J="). For any / £ T E / (MJV .^- ) , we call H = f"1^ a Thom-Boardman Haefliger foliation with /. Suppose that the Boardman symbol is given by / = (^1,^2, • • • , ifc). For any / £ T-£,i(M we can choose local covering [Uj}j£j of M and local C*00 -mappings 07- : Uj —> Dq C E9 which define the Haefliger foliation 7i = /~1Jr. By the construction of the Thom-Boardman singular set , the set S11 (H) = {x £ M | rankd(0j)x = (7 - *i for x £ Uj}

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is a smooth submanifold of M. Moreover, = {x e Zil(h] | rank d(<^- 1 E^ft)^ = min(dimE l l (W), 9 - z 2 ) f or x e Uj} is also a smooth submanifold of M. This procedure can be continued to k times. As a special case, we assume that q = 1, k = 1 and / = (1). For any / 6 T^(M,N;F)I 7i — f~lF is called a Morse type Haefliger foliation. We have the following problem associated to Morse type Haefliger foliations: PROBLEM: Suppose that F is a regular foliation on N with codimF = 1 and 7Y = f~lF is a Morse type Haefliger foliation. Can we have a generalised Morse inequality? It might be related to the topology of the leaf space N/F. 2) Stability of mappings modulo F For any f , g e C°°(M, N;F), we say that /, g are Ayr-equivalent if there exist a diffeomorphism h : M —> M and a smooth mapping k : N —> N taking leaves of F to leaves such that the induced mapping k : N/F —>• N/F is a homeomorphism and g = k o f o h~l, where / = TT o / and TT : N —> N IT is the canonical projection. The local version of the above definition is as follows: For smooth map-germs / : (M, x] —>• (TV, y] and g : (M, x'} —> (A7', ?/), f , g are Ayr-equivalent if there exist diffeomorphism-germs h : (M, x) —> (M, x') and fci : (Dq , 0) —>• (Dq, 0) such that k\ o TTI o (771 x 772) o / = TT\ o (rj{ x 772) o g o h, where rji x 7/2 : (N,y) —> (Dq x £T,0),r/i x r?2 : (TV,?/') —> (£>9 x L> r ,0) are local charts and TTi : (Dq x £) r ,0) —>• (Dq,Q] is the canonical projection. We remark that the above definition does not depend on the choice of local charts 771 x 772, f]'\ x ^2We now define the notion of stability corresponding to each definition. We say that / G C°°(M, N; F) is Ayr-stable modulo F if there exists a neighbourhood V) of f in C°°(M, N;F] such that / and g are .^-equivalent for any g e V/. We also say that a smooth map-germ / : (M, x) —> (N,y) is Ayr-stable modulo T if for any representative / : t/ —> F of / with a local chart 77! x 772 : V —> DqxDr, there exists a neighbourhood Vj of / in C°°(U, V\F\V} such that for any g 6 V? there exists a point x' € U with / and the germ g : (M, x') —> (N,g(x'}) are ^^-equivalent . The notion of the Ajr-st ability modulo F for smooth map-germs is rather trivial because it is almost the same as the ordinary definition of the stability of smooth map-germs. However, the local stability of global mappings is important for the study of Haefliger foliations. We say that / € C°°(M,N\J:} is locally Ayr-stable modulo F if the mapgerm / : (M, x) —> (N,f(x)} at any point x 6 M represented by / is ,4jr-stable modulo J- as a smooth map-germ. We now define the infinitesimal version of the above notions of stability. We have the normal bundle TT^ = TN/TF of the foliation F. We say that / e C°°(M, N; F} is infinitesimally Ayr-stable modulo F if for any w € r oo (/*^ r - L ), there exist u G r°°(TM), v € r°°(r^r±) such that w = TTjr o df o u + v o /, where TTJP : TN —> TFL is the canonical projection. We can also define the germ version of the infinitesimally *4jr-stability modulo F as usual. A smooth map-germ / : (M, x) —» (N, y} is infinitesimally Ayr-stable modulo F as a germ if for any w E F00 (f*F±)x, there exist u € F°°(rM):c, v € r°°(TF-L)f(x) such that w = IT? odf ou + v o f for any x € M, where we denote that T00(f*F±)x,u € r oo (rM) x ,r oo (T^) /(x) are sets of germs of smooth sections of each vector bundle. Since the local ^4^r-group is the geometric subgroup of A and 1C in the sense of Damon [1], the infinitesimally ^4^-st ability modulo F implies the local ^4jr-stability modulo F. The converse is also true by the jet trans versality theorem modulo F.

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PROPOSITION 4.1 For f e C°°(M,N;F), f is locally A?-stable modulo T if and only if f is infinitesimally Ay?-stable modulo J- in the. local sense. Moreover, we have the following characterisation of local stability: Since the fibre of the jet bundle modulo T Jk(M, TV; F} can be identified with the ordinary jet space J fc (m, g), we can consider the /C-orbit in J f c (m, q). Here 1C is the group corresponding to the contact equivalence in the sense of Mather (c.f., [3, 5]). For any z = fyf(x] £ J k ( M , N ] f } , we denote the /C-orbit through z by K.k(z). By the characterisation theorem of Mather [5], we have the following proposition: PROPOSITION 4.2 Let f : (M,.r) —> ( N , y ) be a smooth map-germ. Then f is Ap-stable modulo f if and only if j^f is transversal to lCk(z] for k > q + 1. It follows from the transversality theorem modulo T and the celebrated results of Mather [5] that we have the following theorem. THEOREM 4.3 //(dim/if, codimJF) is in the nice dimensions in the sense of Mather [5], the set of locally Ap-stable mappings modulo JF is open and dense in C°°(M,N\F}. FINAL REMARK: We remark that Prof. J. P. Dufour pointed out that if we consider the foliation T given by the irrational flow on the torus TV = T 2 , there might be no global .Ajr-stable mappings modulo T from any compact manifold into N. This means that we need strong assumption on the foliation for the existence of global ^4jr-stable mappings modulo J-'. Apologies and Acknowledgments It was 1994 August, the first author proposed collaboration to Prof. L. A. Favaro on the study of smooth mappings along the above direction. We promised to proceed this project in that time. However, we have not been able to realise it because of his sorrowful death. The first author really would like to apologise to him that we did not start the project earlier. On the other hand, the authors would like to thank the organisers of "The 6th Workshop On Real and Complex Singularities" for giving us an opportunity to talk about this incomplete project. The first author is also grateful to people at USP and UFSCAR for their kind and warm-hearted hospitality during his stay in Sao Carlos.

REFERENCES [1] J. Damon. The unfolding arid determinacy theorems for subgroups of A and 1C. Memoirs Amer. Math. Soc. 50 306. 1984. [2] L.A. Favaro. Differentiable mappings between foliated manifolds. Bol. Soc. Brasil Mat. 8:39-46, 1977. [3] M Golubitsky, V. Guillemin. Stable Mappings and Their Singularities. Graduate Texts in Math. 14 Springer-Verlag, 1973. [4] G. Ikegami. Vector fields tangent to foliations. Japan. J. Math. 112:95-120, 1986. [5] J.N. Mather. LStability of C00-mappings LThe division theorem. Ann. of Math.. 87:89-104, 1968.

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II: Infmitesimally stability implies stability. Ann. of Math. 89:254-

291, 1969. Ill: Finitely determined map-germs. Publi. Math. I.H.E.S. 35:127-

156, 1968. IV: Classification of stable germs by R algebras. Pub. Math. I.H.E.S. 37:223-248, 1970. VrTransversality. Adv. Math., 4:301-336, 1970. VLThe nice dimensions. Lecture Notes in Math. 192:207-253, 1972.

On a Conjecture of Chisini for Coverings of the Plane with A-D-E-Singularities VALENTINE S. KULIKOV Moscow State University of Printing, Department of mathematics 2A, Prjanishnikova str. Moscow 127550, E-mail:[email protected]

1

INTRODUCTION

This is a report about our joint paper with my brother, to be published in Izvestiya RAN: Ser.Mat. 64:6 (2000) (see also math.AG/00021-74). We are interested in generic coverings / : X —> P2 of projective plane by surfaces X with A-D-Esingularities (in other terminology, rational double points, Du Val singularities and etc.). A classical result is that a generic projection of a non-singular surface X C Pr to P2 is a finite covering with at most folds and pleats as singularities, and the discriminant (= branch) curve B C P2 is cuspidal. A fold (resp., a pleat) is a singularity of a map locally equivalent to a projection of a surface, defined by equation x = z2 (resp. y = z3 -f- xz ), to the x, y-plane. A cuspidal curve is a curve, which has at most ordinary nodes and cusps (locally denned by equation xy — 0 and y2 = x 3 ). Over a node b £ B there are two folds, and over a cusp — one pleat. A generic (or simple) covering is a covering, which possesses the same properties as a generic projection. First of all, I explain why it is important to consider coverings by surfaces with A-D-E-singularities. A presentation of an algebraic variety as a finite covering of the projective space is one of the effective ways of studying projective varieties as well as their moduli. To compare we recall what such an approach gives in the case of curves. For a curve C of genus g a generic covering / : C —> P1 is a covering such that in every fibre there is at most one ramification point which is a double point (or a singular point of / of This work was partially supported by grants RFBR 99-01-01133, NWO-RFBR 047008-005 and INTAS-00-0259. 175

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type AI }. Let B C P1 be the set of branch points, and d = deg 73, i.e. d = ij(#). Then according to the Hurwitz formula d — 2 TV + 2g — 2, where N = deg f . If N > y + 1, then any curve of genus g can be presented as a simple covering of P1 of degree N. The set of all simple coverings (up to equivalence) / : C —> P1 of degree N with d branch points is parametrized by the Hurwitz variety H = HN'd. Let ¥d \ A (A is the discriminant) be the protective space parametrizing the sets of d different points of P 1 , and let M g be the moduli space of curves of genus g. There are two maps: a map fi : iff —> Pd \ A sending / to the set of branch points B C P 1 , and a map // : H —> Mg, sending / to the class of curves isomorphic to C. Hurwitz introduced and investigated the variety // in 1891. He proved that the variety // is connected, and h is a finite unrainified covering. In modern functorial language H was studied also by W.Fulton (1969). The map /;, is surjective and has fibres of dimension TV -f- (N — g + 1) (/ : C --> P1 is a rational function on C\ the first summand TV is the number of points on C. chosen to be f ~ l ( o o ) or the divisor of poles of function /; the second summand is the number of / with fixed poles defined by Riemann-Roch theorem). We obtain the number of moduli 2/V + 2g — 2 — (27V — g + l] = 3g — 3 and the irreducibility of the moduli space Mg. In the case of surfaces we also can consider an analog of Hiirwitz variety H of all generic coverings (up to equivalence) / : S —» P2 of degree TV and with discriminant curves B of degree d witli given number n of nodes and given number c of cusps. Let P17, // = d( * -. be a protective space parametrizing curves of degree d, and h : PI —» P" be a mapping sending a covering / to its discriminant curve B. A so called Chisini conjecture concerns the fibres of h. It claims that if B is the discriminant curve of a generic covering / of degree TV > 5, then / is defined by the curve B uniquely, up to equivalence. In other words, it means that the mapping h is injective (and, besides, TV = deg f is determined by B). In [6] it is proved that the Chisini conjecture is true for almost all generic coverings. In particular, it is true for generic coverings defined by a multiple canonical class. A construction of the moduli space of surfaces of general type uses pluricanonical maps. As it is known [3] , if S is a minimal surface of general type, then for m > 5 the linear system \rnKs blows down only (-2)-curves and gives a birational map of S to a surface X C Pr (the canonical model) with at most A-D-E-singularities. This requires a generalization of the notion of generic covering to the case of surfaces with A-D-E-singularities. In our paper we: 1) generalize a classical result on singularities of generic projections of non-singular surfaces to the case of surfaces with A-D-E-sirigularities; this gives a notion of a generic covering; 2) compute numerical invariants of a generic covering in terms of intersection numbers on a minimal resolution of the surface; 3) obtain an inequality (the main inequality), which is the main tool in proving the Chisini conjecture; this inequality bounds the degree of a generic covering in the case of existence of another nonequivalent generic covering; our proof of the main inequality requires some investigations in singularity theory; 4) study local structure of fibre products of generic coverings with A-D-E-singularities and 5) investigate the canonical cycle of an A-D-E-singularity, with the help of which we compute numerical invariants of generic coverings; finally, 6) we apply the main inequality to prove the Chisini conjecture for generic m-canonical

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177

coverings of surfaces of general type.

2

GENERIC PROJECTIONS OF SURFACES WITH A-D-E-SINGULARITIES

We consider a slightly more general situation, when a surface X may possess isolated singularities of multiplicity two, locally denned by equations z2 — h(x,y], which are called double planes. All A-D-E-singularities are double planes. THEOREM 2.1 Let X c Pr be a surface with at most isolated singularities of the form z2 = h(x,y) (= 'double planes'), X —> P2 be the restriction to X of a generic projection Pr\L —> P 2 , where L is a generic linear subspace o/P 2 of dimension r — 3. Then (i) f is a finite covering; (ii) at non-singular points of X the covering f has as singularities at most either double points (folds), or singular points of cuspidal type (pleats); in a neighbourhood of these points f is equivalent to the projection of a surface x = z 2 , respectively y = z3 -f xz, to the x, y-plane; (Hi) in a neighbourhood of a point s G Sing X the covering f is analytically equivalent to a projection of a surface z'2 = h(x,y] to the x, y-plane; from (ii) and (Hi) it follows that the ramification divisor is reduced, i.e. f*(B) = 2R + C, where B = f(R), and R and C are reduced curves; (iv) except singular points f(SingX}

the discriminant curve B is cuspidal;

(v) the restriction of f to R is of degree 1. We prove a weakened version of Theorem 2.1, in which the projection goes after a Veronese embedding of the initial embedding. This is quite enough for the purposes described above. Actually, the main difficulty in the proof of this theorem lies in the classical case, when the surface X is non-singular. Unfortunately the authors do not know a complete (and mordern) proof of this theorem, and it seems that such a proof does not exist. Thus, its proof, even in the case of a non-singular surface, take interest. We begin with a projection TTL of X onto a surface Y in P3 with ordinary singularities (double and triple points and pinches ('Whitney umbrellas')). Then we prove that for a generic point £ 6 P3 all lines / 9 £ are at most simple (i.e. (Y-l)y = 2) bitangents and simple stationary tangents (i.e. (Y • l}y = 3) with respect to Y, and I is transversal to tangent cones at the singular points. The projection / = TT^ OTT L I : X —> P2 at singular points is a covering of degree 2 or 3. To obtain normal forms of the projection at the singular points, we use a simple lemma [1] LEMMA 2.2 Let (A",0) C (C 3 ,0) be a non-singular surface, and (C 3 ,0) -> (C 2 ,0) be a smooth morphism, the restriction of which f : X —> C2 is a finite covering of degree //. Then one can choose local coordinates x,y in C2 and x,y,z in C3 such that the restriction to X is defined by the equation y = z» + Ai(x)z"- 2 + • • • + A M _ 2 (x) 2 ,

Kulikov

178 and f is the projection along z arris.

In the proof, the covering / is considered as a 2-parameter family of 0-dimensional hypersurface singularities of multiplicity //. and, consequently, / is induced by a miniversal deformation of the singularity of type A M _I. From this lemma it follows that for a non-singular X a covering (^",0) —* (C 2 ,0) of degree 2 is a fold, and a covering of degree 3 locally is a projection to the ,7:,?y-plane of one of the surfaces y — z3 4- xkz, k — 1,2,.. ., or y = z'A (k = oo). For a generic point £ e P3 the exponent k = 1. i.e. we have a pleat.

?^ = 0

Figure 1

3

GENERIC COVERINGS

Let X be a surface with A-D-E-singularities, an singularities of type An, n > 1, dn singularities of type £)„, n > 4, and en singularities of type En, n = 6,7,8. A finite morphism / : X —>• P2 is called generic (or simple] covering, if it possesses the properties of a generic projection. The ramification divisor R of / is reduced, /*(B) = 2R + C. The discriminant curve B has, firstly, 'the same' singularities as the surface X (and as the curve /£), which are locally defined by the equation h(x,y] = 0, where z2 = h(x,y) is an equation of the corresponding singular- point. These singularities on D we call s-singularities, in particular, s-nodes and s-cusps. Besides, there are nodes and cusps on B originated from singularities of the map /, which we call p-nodes and p-cusps. Set N = deg/. Let deg/J = d and let B has n nodes and c cusps, na = ai and c.s = a-2 of which originates from Sing X, and np and cp are p-nodes and p-cusps. The covering / is unbranched in P2 \ B. Over a point 6 6 B \ Sing B the covering / has one fold x = z2 and N — 2 'etale sheets'. Over an s-point b e Sing B f has one singular point of X and N — 2 etale sheets; over a p-cusp / has one pleat and N — 3 etale sheets; over p-node / has two folds and TV — 4 etale sheets. Let TT : S —* X be a minimal resolution of X, and / = / o TT : S —> P2. Let L C X be the preimage of a generic line I C P2. Denote by R and L the proper transforms of R and L on S. Then R is a normalization of the curve R, and L ~ L. The singularities of X being Gorenstein, the divisor R is locally principal, and

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179

where Z = ^ZzgSm x ^x ls ^e canomcal cycle of S1, Zx are the canonical cycles of singularities x € SingX. We'll discuss the cycles Zx later and compute the numbers 6X = — |(Z|), which we call defects of A-D-E-singularities. Set <Sx = We compute intersection numbers (Z 2 ) = N,L-R = d,L-Z = Q,R-Z = 28x, (Z2) = -28X ,

(R2) = 3d +g - i - 6 x , (R + Z)2 = 3d + g-l+Sx =3d + pa(R)-l, where d — |d, g is the geometric genus of B (or R), and pa(R} = 9 + ±-4j-- — np — cp is the arithmetical genus of R. Besides, we compute the self- intersection number of the canonical class (K2S) = 9N- 9d+pa(R) - 1 = 97V + ]-d(d - 12) - np - cp. Zi

and the topological Euler characteristic e(S) = 3N + 2g - 2 + 26X - cp.

From Noether's formula (Kg} -f e(S} = I2pa, we have d(d — 3) l

nv

cv

and as in the case of a non-singular surface X, we obtain that np = 0 (mod 4) , cp = 0 (mod 3).

4

MAIN INEQUALITY

Let /i : A"i —> P2 and f^'.X^^ P2 be two generic coverings of surfaces with A-D-Esingularities and with the same discriminant curve B. Let f * ( B } = 2Ri -fCj, i = 1, 2. With respect to a pair of coverings f\ and /2 nodes and cusps of B are subdivided into four types: ss-, sp-, ps- and pp-nodes and cusps. For example, a sp-node b € B is a node, which is a s-node for f i and a p-node for /2. The number of sp- nodes is denoted by nsp. Then Tl — Tlss

i T^sp

i ^ps

i Tlpp 5

C — Css ~p CSp -p f'pg

i

C-ppi

where n^ and c^ are numbers of bft-nodes and bjj-cusps of . In particular, nss +nsp = tti is the number of singularities of type AI, and ss + sp — a<2 is the number of singularities of type A% on the surface X\ . THEOREM 4.1 If f i and f% are nonequivalent generic coverings of degrees NI and, N2, then 2

4(3J+ g l -l) - 2(3d +9 l - l ) - i ! '

where g\ = pa(R\} is the arithmetic genus of the curve RI, and i\ — 2nsp + 2csp

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180

We briefly sketch the proof of this theorem. To compare coverings /i and f2, we consider a normalization X of the fiber product Xx — X\ XP2X2 and the, corresponding commutative diagram

CXi

13 c The surface X is a A^A^ -sheeted covering of P2 and it has at most A-D-Esingularities, which lie over Sing B. LEMMA 4.2 // covering* j\ and f2 are non equivalent, then the surface X is irreducible. For the proof it is shown that the group TTJ = Tr^P2 \ D) acts transitively on a fibre of / : X —> P 2 . It is used that the monodromy homomorphisms (/?,; : ?TI —> SNU i — I} 2, are epimorphisms. The main idea in the proof of Theorem 4.1 is the same as in [6] in the case of non-singular surfaces: we partition the preimage g^1 (R^} of the curve R\ into two parts where R is the curve mapped by g2 onto R2, and C is the curve mapped by g2 onto C*2, and apply the Hodge index theorem to the pair of divisors R and C on X. More precisely, the surface X has A-D-E-singularitis, and so to use intersection theory we have to resolve the singularities. Let TT : S —> X be a minimal resolution of singularities of X, and denote by R and ~ the proper transforms of R and C on S. Let 7 r * ( g ~ l ( R l ) ) = R} be the total transform of 7^ on S. Then

R} = R +

where

ZR, C = C + Z

where ZR and Zc are the sums ^Zx, where Zx is the canonical cycle of a point x G Sm_g X, and the summation runs over x 6 R, respectively x G C. We apply the Hodge index theorem to divisors R and C. Recall that this theorem states that for a non-singular projective surface S the cup product form on H^' (S) is non-degenerate of type (1, /i 1 - 1 — 1). This gives that if R and C are divisors on S and (R2) > 0, then

R-C

(C2)

This is the desired main inequality. So our goal is to compute the mentioned intersection numbers and to show that (R2) > 0.

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One more remark. To compute intersection numbers we need to make additional a-processes on S (and on X\} to disjoint the curves R and C. For then we obtain a covering X —> X\ and in neighbourhoods of proper transforms R, C and R± of R, C and .Ri, the restrictions R —»• R\ and C —> R\ are finite coverings of degrees 2 and 7V2 - 2 respectively. Therefore, (R2) = 2 (R*) , (C2) - (7V2 - 2) (R*) , R • C = 0. If we know exactly the performed a-processes, we can compute the desired intersection numbers. Thus, we have to examine carefully the normalization X of the fibre product, and the divisors R and C on X. This problem is local in its nature. Let U C P2 be a sufficiently small neighbourhood (in complex topology) of a point b e Sing B. The preimage /-f1 (U) is a disjoint union of two parts, /f 1 ([/) = V\ U V/, where Vi is a part containing the ramification curve R\, and V{ is a part not containing R\ and etale mapped to U. The map V/ —» t/ is an unramified covering of degree 7V"i — fc(fc = 2,3 or 4 depending on the type of the singular point 6). Analogously, /2~1(C7) = V2 U V^'. Then f ~ l ( U ) is a disjoint union of four open sets - of normalizations of fibre products

And only W and W meet the curve g^1(Ri) and are essential over g^l(Ri) = R + C through the windows W and W. W and it is studied in detail for different types of singular points W is auxiliary. It does not meet R. W consists of N% — k isomorphically mapped to V\. Thus, to finish the proof we have to study the windows W.

5

for us. We can look is the main window b 6 B. The window disjoint components

LOCAL STRUCTURE OF FIBRE PRODUCTS

It is not difficult to describe the window W, when b is a non-singular or an ss-point of B. In this case V\ and V-2 are double planes z\ = h(x,y) ,z\ = /i(x, y), where h ( x , y ) = 0 locally defines B. Then Wx = V\ X(/ V-z is defined in C4 3 (x,y,z\,z<2) by two equations z\ = h(x,y), z\ = h(x,y}. Hence, z\ = z|, and we see that Wx consists of two components, W* — W£ U W 2 X , lying in subspaces z\ — z^ and z\ — —Z2. W£ and W£ are isomorphic to V\ and V?,, and they meet along a curve z\ = Z2 = 0, h(x, y) = 0. We obtain that W = W^ U W£ is a disjoint union of two components isomorphic to V\ and V^. A visually-schematic picture is as follows

Figure 2

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Kulikov

It remains to describe the window W for sp- and pp-points of type A\ and A? of B. The most interesting is the case of an sp-point of type A^. So let /2 : X? —+ C2 be a two-sheeted covering branched along a cusp D : y2 — x 3 , where /2 is a projection of a surface w'2 — x3 — y2 onto x, y-plane. Here 0 E B is a s-point of type A^. Let /3 : X% —> C2 be a projection of a surface y = z 3 + xz onto x,y-plane. We can consider it as a ininiversal deformation z3 + a^z + «3 = 0 (0,2 = x - ,a 3 = —y) of a zero-dimensional singularity of type A^. Its discriminant curve J3 is denned by equation 4o| -4- 27a| = 0. To bring the equation of 1? to the form y2 = x3, we make a substitution

We obtain a 3-sheeted covering /3 : X5 —* C 2 , where /3 is a projection of a surface X3, defined by equation —1y — z3 — 3xz, to the x,y-plane. Or we can assume that X% ~ C2 3 (x,z), arid /3 is defined by formulae x — x, y = -^(z3 — 3xz) (see Fig. 1). Then the ramification curve is denned by equation x = z1', and f$(B) = 2R + C, where C is defined by equation x — ~z2. Thus, we have a p-point 0 G D of type A^. To obtain X — X^^Xz we must call to mind that in singularity theory there is a Viete map /6 : C2 —> C2 of degree 6, branched along a cusp. It appears to be the desired normalization of J^2 x c 2 ^3- We recall that the Viete map sends the roots of a polinomial to its coefficients,

where (z — z\)(z — z
ai = -(zi 4- z2 + z 3 ), a 2 = ziz 2 + z2z3 4- z 3 zi, a,3 = The map v is ramified along diagonals A = U^ 7 {zi = z 7 }, and ^(A) = D is defined by the discriminant of a polynomial of degree three. The map /g is obtained by restricting v to the invariant subspace C2 = {zi + z^ + z3 = 0 } (i.e. ai — 0). In coordinates x, y, where n^ = —3.x, 0,3 = 2y, this map C2 - (zi + z2 + 23 = 0} = X6 -^ C2 3 (x,y)

is defined b formulae

The ramification divisor R — LI + L2 + L3 consists of three lines Li : Zj = zk, zi + z2 + z3 = 0 , {?;, j, k} = (1, 2, 3} ,

and the discriminant curve B has equation y 2 = x 3 , and f * ( B ) = 2Li + 1L<2 + 2L 3 . We obtain a commutative diagram

183

Coverings of the Plane with A-D-E-Singularities = {Zl + 22 + Z3 = 0}

+ 2y = 0} ,

(1)

(x, ?/) 6 C2 D B : y2 = x

describing a normalization of the fibre product W = Vi *uVi in a neighbourhood of a sp-point of type A^. It is easy to see that g$ is a factorization under the action of a cyclic group Za = AS C $3, X% = XQ/AS, and #2 is a factorization under the action of a cyclic group of order two Z2 ^ S2 = {(1), (2, 3)} C £3. The diagram (1) can be visually-schematic presented as follows

Figure 3 Direct computations show that x — z2 — \(zi — z\](z\ — 23) , and x — |z2 = Yf (-^s — 22) > i- e -

g2(R)=L2+L3,g*2(C)=2Ll. It is interesting to note (Catanese F., [4] ) that the constructed coverings /2,/s and /e are characterized as the only coverings / : (X,0) —» (C 2 ,0) by a normal irreducible surface X, branched along an ordinary cusp Z? C C 2 , and the ramification curve of which is reduced, i.e. f * ( B ) = 2R + C. To finish the consideration of the case of sp-points of type ^2, we recall that for computing the intersection numbers we have to resolve the singular point of X2 and to disjoint the curves LI + Ls (= R) and LI (= C) on XQ. For this purpose we resolve the singularity of the cusp B by a-processes,

Kulikov

184

8

-3

-* 3

Figure 4 and obtain • C2 . rr = 03 o ri2 00-^ Then we lift (1) to C2 and obtain a commutative diagram

where Xi is a normalization of Xi x C2 C 2 . We obtain a detailed description of the right square of this diagram. Replacing the varieties at its vertices by the total transforms of the curve B, we obtain the following picture

Coverings of the Plane with A-D-E-Singularities

185

-6

-f

-1

-f

-f

-1

•v.

-Z

-3

-3

-3

-1

-f

Figure 5 The rule of notation is as follows. The exceptional curves E\,E<±, ES on C2 are already denoted. Under double indexing E^- the first index indicates the variety X$, where -E?;,j lies, and the second index indicates to what curve Ej the curve Eij is mapped onC 2 .

6

CANONICAL CYCLE OF A DU VAL SINGULARITY

Let (X, x] be a 2-dimensional A-D-E-singularity. Let TT : X —-> X be a minimal resolution, L = TT~ I (X) be the exceptional curve. As is known, the canonical class KX is trivial in a neighbourhood of L, that is we can choose a divisor in K^ with a support not intersecting L. In other words, there is a differetial form a; on X, which has neither poles nor zeroes in a neighbourhood of L, i.e. (a;) = 0. On the other hand, (X, x) can be considered as a double plane, that is as a 2-sheeted covering X —>• Y of the plane Y = C2 (locally). Let z2 — h(x, y) be an equation of (X,x), B : /i(x, y) = 0 be the discriminant curve, f ~ l ( B ) = /?, defined by the equation 2 = 0, be the ramification curve. We can consider the differential form u; = /* (dx A dy) lifted from Y. Then on X the divisor (o>) = (2) = R + Z, where R C X is the proper transform of R , Z = ^^jiLi is a cycle on L. We shall say that Z is the canonical cycle of a 2-dimensional A-D-E-singularity. Thus, — Z is a cycle on the exceptional curve L, which is equivalent to the ramification curve .R in a neighbourhood of L. Let us calculate the canonical cycle for all A-D-E-singularities.

Kulikov

186

As for any double plane, a resolution of an A-D-E-singularity can be obtained by means of a resolution of the discriminant curve B C Y — C 2 , D : h ( x , y ) — 0. Let a : Y —> Y be a composition of a-processes, such that the total transform of D is a divisor with normal crossings. Let (?*(B) = B + Y^i=i <'M?i where B is the proper transform of B, / ? ~ P1 r, are the exceptional curves, as well as their proper transforms, glued by rr-processes. Let X be the normalization of Y x Y X, and / and TT be induced by projections,

X D R 3 x, R : z = 0

\J...\JLrCX

Set / 1 (/j) = Li. The curve L, is either irreducible or consists of two components Li •= L': + L", where L' ~ P1 , L" ~ P1. The mapping / is a 2-sheeted covering branched along the curve B 4- ^ -odd ^ - ^° ^e more graphic we denote the curves /i, for which « z are odd, also by /",, and L?; - respectively by Lj. The surface X has singularities of type A\ over nodes of the branch curve B + Y^k- If this curve is non-singular, that is, a disconnected union of components (one can reach this by performing one additional a-processes for each node), then X is non-singular and is a resolution of the singularity ( X . x ) . Let R be the proper transform of R w.r.t. TT (— the proper transform of B w.r.t. / ). We have f * ( l i ) — 2L7;, if a,- is odd, and f * ( l i ) — Li, if on is even. We have

= 22 - 2R

((

a, -odd

«* -even

and, consequently, (z) = /? 4- Z, where

Z= a,- -odd

«, -even

For example, for the singularity Eg : .x3 4- y° , these computations give

Z = 3Li 4- 5L2 4- 9L3 4- 15L4 4- 10L5 + 8L6 + 12L7 4- 6L

Coverings of the Plane with A-D-E-Singularities

/

•*"

*

*

— fr

20

187

-f

A

•*"

1

'*

I

**-* 3 -1 V

-4 *2 -*

Figure 6 Define a defect 6 of an A-D-E-singularity by the formula

Computing the canonical cycle for all A-D-E-singularities, we obtain the following LEMMA 6.1 For different

6=

types of A-D-E-singularities the defect equals /or type Anf j + 1 /or iype Dn^1 for types ^n, n = 6,7,8.

In particular, for the type A\ (nodes) and A<2 (cusps) the defect 6 = 1. Actually the defect 6 is equal to the <5-invariant (genus) of the one-dimensional A-D-E-sirigularity.

7

APPLICATION OF THE MAIN INEQUALITY FOR THE PROOF OF THE CHISINI CONJECTURE FOR GENERIC m-CANONICAL COVERINGS

Let S be a minimal model of a surface of general type with numerical invariants (Kg] = k and e(S] = e. Let X be a canonical model of the surface S, and TT : S —> X be the blowing down of (-2)-curves. Let / : X —> P2 be a generic mcanonical covering, i.e. a generic projection onto P2 of the surface X = pm(S), where <£>m : S —> FP"1""1 is a m-canonical map, defined by the complete linear system ™>Ks\, pm = |m(m - l)fc + x(S}. As is well known [3] , by a theorem of Bombieri m(S) ~ X for m > 5, and (f>m gives the blowing down TT.

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Kulikov

For a m-canonical covering we can express d = (leg73, N = (leg/ and pa(R} by rn and k,

d = m(3rn + 1)A;, N = m2k, pa(R} - 1 = i(3m -f l)(3m -t- 2}k. THEOREM 7.1 Let Si and 82 be minimal models of surface* of general type with the same (A'|) and x(£), and let f\ : X\ —> P 2 , /2 : X? —* P2 be generic, m-canonical coverings with the same discriminant curve. Then for m > 5 the coverings f i and f? are equivalent. So let Si and So be two surfaces of general type with numerical invariants k and e. Let /,- : X,- —> P2. i = 1,2, be their w,?-canonical coverings having the same discriminant curve B C P 2 . In this situation we have mi = m-2 = rn and deg/i = deg/2 = N. By the main inequality, for the proof that f\ and /2 are equivalent, it is sufficient to show that an inequality N (2(3d+pa(R) - 1) - ,.) > 4(3d + pa(R) - 1) holds (here R stands for R-[). Using a Hirzebruch-Miyaoka inequality ([3], p.215), •s' < | (3e(S) — (Kg}}, for the number .s- of disjoint (-2)-curves on S (to estimate i ) , we obtain that it is enough to show that an equality of the form 3m 3 (2m + 1)A: - 4(3m + I) 2 - (~k + 12)w2 > 0 holds. It is easv to show that it is indeed so for rn > 3 and for all k.

REFERENCES [1] V.I. Arnol'd. Indices of singular points of 1-forms on a manifold with a boundary, convolution of invariants of groups generated by reflections, and singular projections of smooth surfaces. Usp. Math. Xauk. 34:2:3-38, 1979. (Engl. translation in Russ. Math. Surv., 34 2:1-42, 1979.) [2] V.I. Arnol'd, S.M. Gusein-Zade, A.N. Varchenko. Singularities of differentiate maps, Vol. I, II. Birkhauser, 1985. [3] W. Barth. C. Peters, A. Van de Ven. Compact complex surfaces. Springer, 1984. [4] F. Catanese. On a Problem of Chisini. Duke Math. J. 53 1:33-42 1986. [5] Ph. Griffiths, J. Harris. Principles of algebraic geometry. John Wiley & Sons, New York, 1978. [6] Vic. S. Kulikov. On a Chisini Conjecture. Izvestiya RAN: Ser.Mat., 63:683-116, 1999. [7] B. Moishezon. Complex Surfaces and Connected Sums of Complex Projective Planes. LXM 603, Springer, 1977. (8] O. Zariski. Algebraic surfaces. Berlin.Verlag von Julius Springer, 1935 SpringerVerlag, 1971.

Not AH Codimension 1 Germs Have Good Real Pictures DAVID MONO Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdon E-mail: [email protected] ROBERTA G. WIK ATIQUE Departamento de Matematica, Institute de Ciencias Matematicas e de Computagao, Universidade de Sao Paulo - Campus de Sao Carlos, Caixa Postal 668, 15560-970, Sao Carlos SP, Brazil E-mail: rwikCiicmc.se.usp.br

1

INTRODUCTION

Some time ago the first author conjectured (in [9]) that every «4e-codimension 1 equivalence class of map-germs (• (C p ,0), with n > p — 1 and (n,p] nice dimensions, should have a real form with a "good real perturbation" — that is, the ^-equivalence class should contain a real germ (one whose power-series expansion has purely real coefficients) which moreover should have a real perturbation whose real discriminant (if n > p) or real image (if n = p — 1) carries the vanishing homology of its complexification. The purpose of this note is to give an example for which this does not hold — thus proving the conjecture false. The example is not hard to describe: it is the simplest ^4e-codimension 1 class of map-germ / : (C n ,0) —> (C n+1 ,0) of corank 2 (i.e. such that ker(rfo/) has dimension 2). Since the codimension in jet space Jl(n, n + 1) of the set S2 of matrices of kernel rank 2 is equal to 6, such a singularity cannot occur if n < 5. In fact one can easily construct map-germs (C 5 ,0) —» (C 6 ,0) of .Ae-codimension 1 and corank 2. From the point of view of this paper, the most straightforward procedure is by transverse pull-back from the stable map-germ F : (C 6 ,0) —» (C 7 ,0) defined by

F(A, B, C, D, x, y) = (A, B, C, D, x2 + Ay, xy + Bx + Cy, y2 + Dx}. 189

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Mond and \Vik Atique

That is, if i : C6 —>• C' is transverse to F. then the fibre square (C 7 ,0)

T* (C 6 ,0) gives us a map-germ ?'*(F) from the smooth 5-dimensional complex manifold germ (C6 X(p 7 C 6 ,0) to (C 6 ,0). If i is suitably generic (in a sense we will make precise below) then z*(F) has yte-codirriension 1. Every map-germ (C°,0) —>• (C ,0) with local algebra isomorphic to that of F (which is the simplest local algebra of type S 2 , isomorphic to (9^2 /(ni^ ) 2 ), can be obtained by transverse pull-back from F. To give a concrete example, one can compute that the germ /('(/,, v, u1,;/:, //) = ('(I,.'(', w, x~ + uy, xy + vx -+- wy, y — v/,.7,1) has y^-codimension 1. This germ is obtained as i*(F] by taking

Of course there is no canonical choice of coordinates on the fibre product (C6 x ^7 C .0). and we could equally well have written ?'*(F) in a different form. In fact the ^-equivalence class of i*(F) is determined by the image L of i. and so a more economical way of studying the codirnension 1 map-germs obtained in this way is to consider the space of smooth hypersurface-germs transverse to F. For the same reason we also adopt the notation F\L in place of ?'*(F), although the reader will of course appreciate that since L is contained in the target of F, this denotes base-change rather than restriction. In Section 2 we show first that to obtain every ^-equivalence class we need consider only hyperplanes, and among hyperplanes only those of the form kA -{-IB + rnC + nD = 0. We then show that within the space C of these hyperplanes there is a discriminant A, itself a hypersurface, such that if L £ A then F\L has .Ae-codimension 1. In Section 3 we study the topology of the complement in £j£ of A^; we do not determine its homology, but we do locate at least one point in each connected component. If LI and L<2 are in the same connected component of £jg> \ A^, the two germs F|L] and F]/ 2 are ^-equivalent (over R) (Remark 2.5 below) arid thus we do not need to examine both of them. In Section 4, we compute the homotopy type of the image of a stable perturbation of map-germs F|L, for each hyperplane L in the list that we determine in Section 3. Up to homeomorphism each real codimension 1 map-germ has two real stable perturbations, corresponding to the sign of the deformation parameter; however here it turns out that the two are isomorphic.This implies a rather strong version of triviality of the monodromy of the family of complex stable perturbations, over C*. Using Morse theory, we show that each of these complex images has the homotopy type of a 5-sphere; in consequence, the algebraic closure of the real image of each stable perturbation has the homotopy type of an n-sphere for some n between 0 and 5. For none of the examples in our list does n equal 5. It turns out that A C £ is a free divisor, with some curious properties; in Section 5 we record some of these.

Not All Codimension 1 Germs Have Good Real Pictures

191

We are grateful to Jan Stevens for a geometrical observation concerning A, and to the organisers of the Vlth Workshop for running such an enjoyable meeting.

2

A-CODIMENSION OF TYPE E2

1 GERMS (c5,0) -» (c 6 ,0)

LEMMA 2.1 Let i: (C6,0) —»(C 7 ,0) be a map germ such that i*(F] has Ae-codimension 1. Then 1. i is an immersion, and 2. i*(F] is A-equivalent to i*^(F}, where if, is the linear part of i. Proof: Write f = i* (F). (1) By a theorem of J. Damon ([6], but see also [10], Section

'e '

tt(0C6)+i*(Der(log£7))

where E = F(C6}. As F is not a trivial unfolding of any lower-dimensional germ, Der(logE') C m

C7,oGC7> hence fit '\

(T*^

1 = *4e-codimension J f > dini(p —77 = dimp ( L —;-r^:—7;-rr-rrr C ~ * ' It follows that the inequality in the centre is an equality, and i is an immersion. (2) is essentially Proposition 2.6 of [1], and can also be found, in this formulation, as of [13]. For completeness we outline the proof. Since i*(F} has *4e-codimension 1, we must have

by Nakayama's lemma this holds if and only if

hence it depends only on the linear part of i. Let ix = ie + X(i — if] be the linear interpolation between if and i. For each value of A, TK-E^x = m(P7 O ^( ? 'A)> and a standard argument using Mather's Lemma (of [7]) and finite /C^-determinacy now shows that the family i\ is /C^-trivial (over E). It follows that the family i\(F} is ,/4-trivial, and i*(F) is ,4-equivalent to i*((F}. D

LEMMA 2.2 If the Ae-codimension of i*((F] is equal to 1 then if is K,E~ equivalent to a linear immersion whose image has equation of the form kA + lB + rnC + nD = 0.

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Proof: Let p define the image of if. By the preceding lemma we may assume that p is linear. We can express the /C£,e-normal space of if (and hence the ^4e-normal space of if(F}) in terms of p:

For if we write C' = L x C and identify p with projection to C. then ^LxC

*p(Der(log E)} + (p)O(p)

0 L x C / C + Der(logE)

Xow

and within

so

If '/£(-F) has *4e-codimension 1, then fp(Der(logE)) + (p)Q(p) must be equal to •m.£7 Q. Xow the module tp(Der(logE}) + (p)0(p) is the tangent space of p under the action of the group 1C(E], the semi-direct product of the group HE of diffeomorphisms of (C ,0) preserving E, with the group C. This is a geometric group in the sense of Damon [5]; it follows that p is finitely /C(E)-determined. There is a 7-dimensional family of linear maps ;;,

d := {p = kA + IB + mC + nD + pU + qV + rW : A:, /, m, n, p, q, r 6 C}, and within £j a discriminant, A1; in principle possibly equal to all of C\, or reduced to the zero map, consisting of linear maps for which £p(Der(log E)} + (p)0(p] fails to be equal to Wp? A calculation with Macaulay shows that the anriihilator in C[A, . . ., W, k, . . ., r] of T)C(E)p is generated by /(A;, /, m, n] := 8kl3 + 3Z 2 m 2 4- 24klmn + 8rn3n - 16A;2ri2 together with polynomials in the ideal (/I, . . ., I47). Thus, the equation of the discriminant is / = 0. and in particular is independent of the coefficients p,q,r. Suppose L € £1 \ AI has equation £, and let P.Q be obtained from f. by setting to zero the coefficients of U, V and W . Then for each t, the hyperplane Lt = {(1 — *Xo + ^ = 0} also lies outside AI, and it follows from Mather's Lemma (and finite determinacy) that the linear

Not All Codimension 1 Germs Have Good Real Pictures

193

interpolation between i and the map IQ is /C(J5)-trivial. Hence the corresponding family of immersions C6 —>• C7 is /C^-trivial. This completes the proof. D The Lemma tells us we can forget the family C\ of all linear maps C7 —> C and consider only those of the form kA + IB + rnC + nD; we denote the family of these by C. REMARK 2.3 The equation for the discriminant A may also be obtained by the following alternative method, which also relies on Macaulay, but which is primarily topological rather than deformation-theoretic. Suppose that L is a hyperplane in C 7 , with defining equation hL(A,...,W) = UL • (A,..., W} = 0, and let G be a reduced defining equation for the image E of F. If the ^4e-codimension of F\L is 1, then the family of normal translations of L, Lt := {£UL} + L, t E (C,0), induces an «4e-versal deformation of F\L. This follows from the fact that for a deformation of a codimension 1 germ, versality is equivalent to infinitesimal non-triviality. By a Morse-theoretic argument of Dirk Siersma ([12]), the image of F\Lt, i.e. E fl Lt, has the homotopy type of a wedge of spheres, whose number, the image Milnor number of F|£, ^i(F^)), is equal to the sum of the Milnor numbers of the (isolated) singular points of G\i which move off the zero locus Er\Lt as £ moves off 0. Let us call these points indicator points. For each value of t, the number of indicator points is finite; as t varies, they form a curve. In fact this curve is simply the polar curve of G with respect to the linear form hi; its equations can easily be obtained as the generators of the transporter ideal JL : G, where JL is the ideal of maximal minors of the 2 x 7 jacobian matrix of the map (G, hi). It is worth noting that V(Ji : G) evidently contains all the indicator points. However, a priori it is not clear that every zero of JL : G (off {t — 0}) is an indicator point. Nevertheless, this is the case. For provided t /= 0, each germ of F\Lt is rightleft stable. In Mather's nice dimensions (which contain (5,6), cf [8]), every stable germ is weighted homogeneous in appropriate coordinates, and so its image has a weighted homogeneous defining equation. It follows that E n Lt is locally weighted homogeneous at each point, and thus G\Lt € J\it at any point where G = 0. Since, for t 7^ 0, the deformation induced by varying t is right-left trivial, it follows that G G JL at any point where G = 0 and t ^ 0. Hence the set of indicator points coincides with V(JL : G). All of this can be done simultaneously for the family of all hyperplanes L — {kA + IB + rnC + nD = 0}. We work in C[A,..., W, fc," .., n}. Let h(A,..., n) = kA -+- IB + mC + nD, let J be the ideal of maximal minors of the jacobian of (G, h] with respect to the variables A,..., W, and let Jo := J : G.

Evidently ZQ has a (fc, /, ra, n)-primary component, corresponding to the absent hyperplane for which k = l = m = n = Q, which we remove by using a transporter ideal. We find that Z = ZQ : (&, /, ra, n) has no (A;, /, ra, n)-primary component, and is thus the ideal we require. It is equal to

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(Ak - 1/3BI + l/3Cm - Dn, 4Bm - 4Ck - 3DL 3 Am + 4Bn - 4C7, B2 - 3W. C2 We claim that V(T] —> £ restricts to a locally trivial fibration over C \ A, with fibre over L £ II \ A equal to the set of indicator points of the family F\fjt, where Lt = {/UL} -f L is the normal translation of L. In fact this follows immediately from what we showed in the preceding paragraph: for t ^ 0 and L ^ A. the deformation induced by varying L is locally trivial, and so the fibrewise equality {indicator points} = V(-!L '• G) gives a relative equality {relative indicator points} = V(J : G) = V(I). Xow F\L has image Milnor number 1 if and only if the curve of indicator points in the deformation f] L f is non-singular and transverse to L. We find the ideal of the set of points where this fails, by taking the determinant of the jacobian matrix of the pair (gi , . . . ,
1 then it is 2- determined for A-

Proof: Since / := F\L has ^le-codimension 1, and there is no stable germ in the same /C-orbit, its .4-orbit (in any jet space) must be open in its /C-orbit, and the tangent spaces to the two orbits coincide at each point of the former. The latter tangent space is easily shown to contain mjL5 #(/). The result now follows by Mather's Lemma. D

Not All Codimension 1 Germs Have Good Real Pictures

3

195

THE COMPLEMENT OF THE REAL DISCRIMINANT AR

To determine a list of normal forms for ^4e-codimension 1 germs (R°,0) —> (E 6 ,0) of type S 2 , we have to select one real point from each component of P(£ \ A). This can be done as follows: within each component of S3 \ Aj^, the function /| S3 achieves at least one local maximum or local minimum. The critical points of f\ 53 are the zeros of the ideal / of 2 x 2 minors of the jacobian matrix of (/, r 2 ), where r2 is the usual distance-squared function. To eliminate the components lying in A we calculate the transporter ideal / : /. Once again by a Macaulay calculation, we find that this is equal to (km - 4/7/m, Im - 7/4/n, k2 - 1/3/2 + l/3m2 - n2). This ideal defines a curve consisting of the eight lines through (0,0,0,0), shown in the left hand column of table 1. The second column shows the signs of the eigenvalues of the Hessian of /| 53 at the points where these lines meet the sphere. The third column shows information obtained in the next section: it is the dimension of the sphere to which the real section Lt PI E D R6 is homotopy-equivalent for * ^ 0. By a Morse-theoretic argument given in the next section, this is one less than the index displayed in the third column of Table 2. Line

Eigenvalues

Homotopy dimen-

Sp{(l,>/3,0,0)}nS 3 Sp{(l,-v / 3,0,0)}nS' 3

+ + -

4 3

+ + -

3

+ + + + + +

3 3

+ + - -

4 4

sion of Lt n E n E6

,^3,l)}nS 3

,-\/3,l)}nS 3

s p {(i,o,o,i)}ns 33 s p (i,o,o,-i)}ns

Sp{(4,7,7,4)}n,S 3 Sp{(4,7,-7,-4)}n,S 3

4

Table 1: Critical points of /| S3 To obtain a representative of each A- equivalence class we need take only those F|L for which L corresponds to one of the lines at which /| 53 has a local maximum or minimum: L = {A + V/3S = 0}, L = {v/3Cr + D = 0}, L = {A + D = 0} and L — {A — D — 0}. In fact F has an obvious symmetry permuting x and y in the source and U and W in the target, and permuting also A and D, and B and C in both source and target. This symmetry interchanges the first and second of the hyperplanes just listed, and therefore we do not need to consider both.

4

THE TOPOLOGY OF REAL PERTURBATIONS Q C M7 be a hyperplane (containing 0), let UL ^ 0 be orthogonal to LJJ, and 1 1 be the affine translate of L-^ by £UL- If F\L has «Ae-codimension 1, then

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the family FUJI,, sK.L]n>

l&.f

is an Ace- versa! deformation of Fm . IK.]TL|n> 1KL

From 2.4 it follows (cf [3]) that L-^ t C\ K^ is homotopy-equivalent to a d-sphere, for some d between 0 and 5. For by Siersma's result, F\Lt has just one indicator point; this must therefore be real (otherwise its complex conjugate would also be an indicator point), and now LTO fl Em is hornotopy-eqiiivalent to the boundary of a A'-cell, where k is the index of G p l r at the indicator point. ' IB?

The indicator point can be determined by substituting the values of k. /, m and n into the expressions for the generators of the ideal X listed in Remark 2.3. The Hessian of G\ it at the indicator point can be computed as P* HcP, where HC is the Hessian of G and P is a 7 x 6 matrix whose columns form an orthonormal basis for L. The results of this computation are displayed in Table 2. In principle we should display the signs of the eigenvalues of the Hessian of G| Lt at the indicator point for two values of /. one positive arid one negative. However, there is an obvious symmetry which renders this unnecessary: if (A, B,C,D,U,V,W) is the indicator point for Lt then ( — A —B, —C, —D, U, V, W} is the indicator point for L_,, and the Hessian matrices of G at these two indicator points are the same. Equation of L

Ideal of indicator point

Index

A + x/3B

(V3A - J3, C, D, (7, V, 3W - D2)

5

A - V3B

(\/3^4 + B, C\ D, ?/, V, 3W - B2)

4

V3C + D

(V3D - C\ A,B,W, K, 3f7 - C 2 )

5

X/3C-D

(v^O + ^A/J^^^fJ-C 2 )

4

A+D

(B,C,A- D, D2 - 4K VT, [/)

4

A-D

(B,(7,A + J D,D 2 +41/,MU/)

4

4^+75+7(7+41)

(B-lD,A-D,C-B,D2-$W,U-W,V+JIW)

5

), C7+C, D 2 -

M^, f/-M/, K-

W)

5

Table 2: Indicator points and Morse indices

5

CONCLUDING REMARKS

This counterexample reveals a sharp difference between singularities of corank 1 and of higher corank. The main theorem of [4] states that every .Ae-codimension 1 multigerm (Cn, S) —> (C;\ 0) (n > p — 1, (n,p] nice dimensions), in which each component has corank < 1, has a real form with a good real perturbation, and thus a "good real picture" .

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197

It is clear from our example here that good real pictures cannot be expected for germs of corank greater than 1.

6

APPENDIX: GEOMETRY OF THE DISCRIMINANT A

Jan Stevens has pointed out to us that the discriminant A of Section 3 is the affine cone over the tangent developable surface of the rational normal curve (a twisted cubic) in P3 parametrised by (u, v) ^ (u3, -2m'2, 2u2v, -v3). For it is easy to check that the singular locus of A is the cone over this curve, and then to check that A coincides with the tangent developable. The tangent developable has a cuspidal edge along the curve, and is otherwise smooth. The map RP1 x RP1 -> RP3 parametrising the real part of the tangent developable is therefore a homeomorphism onto its image, and from this it follows by a straightforward duality argument that the complement of A^ has just two connected components. Thus, it is no surprise that the eight lines in the complement of Aj£ that we listed in Section 3, and the eight hyperplanes in L C R7 that they correspond to, should give rise to just two inequivalent real codimension 1 map-germs It also turns out that A is a free divisor. This is easy to check, using Saito's criterion (c/ [11]): a Macaulay calculation shows that the module Der(log A) of vector fields on L which are tangent to A is generated by the vector fields Xi? • • ••> X4 whose coefficients in the basis d/dk, d/dl, 3/dm, d/dn form the columns of the matrix / k M _

"

-3/4A: j

v

-/^

™ —1/4™, \ n 3/4n

0

-3/4™

m

n

k 3/41

I 0

and the determinant of this matrix is a reduced defining equation of A. We note for future reference that xi • / = 4f and Xi ' f — 0 f°r ?' 7^ 1The free divisor A is unusual in that all of the generating vector fields are linear. This makes it particularly easy to calculate the cohomology of the complement: from its geometrical description, A is clearly locally quasihomogeneous, and thus we can use the theorem of [2]: THEOREM 6.1 If D C Cn is a locally quasihomogenenous free divisor, then integration of forms along cycles defines an isomorphism

/? 9 (r(C n ,ft'(logA))) ~Hq(Cn\ A;C). D

Via a contracting homotopy defined by the Lie derivative with respect to the Euler vector field, the complex £7*(logA)o is quasi-isomorphic to its subcomplex of weight zero. As all the generating vector fields are linear, they have weight zero, and

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so dually the generators uj\ , o>2, ^3, <^4 of £ll (log A) (determined by the property that ujj(Xj) = <*>?;, 7) also have weight zero. Thus the weight zero subalgebra of 17* (log A) is the free exterior algebra over C on generators wi, . . ..(^4. As soon as we determine the coefficients F' ,. in = \

F7;

which must be simply complex numbers, determination of H*(C4 \ A) is reduced to linear- algebra. In fact we have

dui

= 0

and thus

~UJ\ A

= = =

A UJ4

0 0 0

and =

0

for all ?', j,k. Hence

PROPOSITION 6.2

//9(£\A;C) = C, generated by uj\ A ^2 A 0^3 A 1^4

if

q

=4

The equality do;2 = — u/'s A ^4 shows that the family / : C4 —> C is a logarithmic contact bundle. For, as with any free divisor £) = {/ = 0}, the wedge product of all the generators of £7 L (logD) is equal to the generator of the top-dimensional module of logarithmic differential forms. Since ui\ = df / / can be taken as one of these generators, this shows that the wedge product of the remaining generators of 171(logD) generates the relative dualising module ujf (for it can be written as the "wedge division" dz\ A • • • A d z n / d f } and thus its restriction to the regular fibres is a holomorphic volume form. Since in our example. LU? A dui2 = —^2 A ^3 A 0^4 , we have PROPOSITION 6.3 The restriction of uj^ to each fibre of f is a contact form, and -so / : C —> C is a logarithmic contact bundle. D We note that a similar phenomenon can never occur for a function with isolated singularity, in any dimension. For if / : (C 2n , 0) —* (C,0) has isolated singularity at

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199

and so the restriction to the fibre of any logarithmic 1-form u> is actually the restriction of a regular form. Hence for any logarithmic 1-form a;, df/\uj/\(duj}A'n~'1 C d/Af&>~ 2 ; since ujf/df A fi5?2n+i has positive dimension r(D) if / is singular, no element of df A £22n can generate a;/. We do not know whether this property of A, or the fact that it is a free divisor, can be deduced from the singularity theory in which it is embedded.

REFERENCES [1] J.W. Bruce, R.M. Roberts. Critical points of functions on analytic varities. Topology 27 1:57-90 1988. [2] F. Castro, D. Mond, L. Narvaez. Cohomology of the complement of a free divisor. Trans. Amer. Math. Soc. 348 8:3037-3049, 1996. [3] T. Cooper, D. Mond. Complex monodromy and changing real pictures. J. London Math. Soc. 57 2:599-608, 1998. [4] T. Cooper, D. Mond, R. Wik Atlque. Vanishing topology of codimension 1 germs over R and C. Compositio Math, to appear. [5] J.N. Damon. The unfolding and determinacy theorems for subgroups of A and JC. A.M.S. Memoirs 306, Providence R.I. 1984. [6] J.N. Damon, ^-equivalence and equivalence of sections of images and discriminants. In: Singularity Theory and Applications, eds. D. Mond, J. Montaldi. Warwick 1989, Part I. Lecture Notes in Maths. 1462, Springer Verlag, 1991, pp. 93-121. [7] J.N. Mather. Stability of C°° mappings IV, Classification of stable germs by R-algebras. Publ. Math. I.H.E.S. 37:223-248, 1969. [8] J.N. Mather, Stability of C°° mappings VI, The nice dimensions. In: Singularities, ed. C.T.C. Wall. Liverpool 1971, Lecture Notes in Maths. 192, Springer Verlag, 1972, pp. 207-253. [9] D. Mond. How good are real pictures? In: Algebraic and Analytic Geometry, La Rabida, eds. A. Campillo, L.Narvaez, Progress in Math. 134, Birkhauser Verlag, 1996, pp.259-276. [10] D. Mond. Differential forms on free and almost free divisors. Proc. London Math. Soc. 81 3:587-617, 2000. [11] K. Saito. Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo Sect, Math. 27:265-291, 1980. [12] D. Siersma. Vanishing cycles and special fibres, In: Singularity Theory and Applications, eds. D. Mond, J. Montaldi, Warwick 1989, Part I. Lecture Notes in Math 1462, Springer Verlag 1991, pp.292-301.

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[13] R. Wik Atique. On the classification of multi-germs of maps from C2 —> C3 under ^-equivalence, In: Real and Complex Singularities, eds. J.W. Bruce, F. Taii. Research Notes in Math. Chapman and Hall/CRC 412, 2000, pp.119-133.

On the Topology of Hypersurface Singularities JOSE SEADE Institute de Matematicas, Unidad Cuernavaca, Universidad National Autonoma de Mexico, Apartado Postal 273-3, Cuernavaca, Morelos, 62251, Mexico. E-mail: [email protected]

1

INTRODUCTION

The topology of complex isolated hypersurface singularities has been long studied by many authors, as for instance Milnor [3], Le Dung Trang [2] and many more. There is a beautiful and well developed theory to this respect, though there are still many things to be understood. This note is concerned with the real counterpart of this theory and it was inspired by [3, 4]. The motivation was to consider a class of real hypersurface singularities that arise naturally from complex singularities. We study the topology of the real hypersurfaces that arise when we consider a C-valued holomorphic function H on an open set U in Cn and we compose this function with the projection onto a real line through the origin in C. We show that, for a fixed function //, all these hypersurfaces are homeomorphic and the link of the singularity is the double of the Milnor fibre of the holomorphic function H. It would be interesting to make a similar study for isolated complete intersection complex singularities with codimension higher than 1. This general case is certainly more interesting, and more difficult, than the one envisaged in this note, since the type of the real singularities so obtained, by composing the complex map with the projection onto a real line, will vary as the corresponding line intersects the discriminant of the original function. So this article should be regarded as the first approach to that general, and hopefully interesting, problem. 201

202

2

Seade

THE THEOREMS

Let U C Cn+l be a connected open set, 0 <E U, and let H : ( W , 0 ) - > (C,0), be a continuous function. For each real line Co C C passing through the origin with an angle 0, f) E [0, rc\, we let TT^ : C —> £# be the orthogonal projection; set he = HQ o H , so that h.Q and //.a are, respectively, the real and the imaginary parts of //. We set Me = h y l ( Q ) and V = H ~ l ( Q ) . We define the map 0 : U- V -> S1 by 4>(z) = p^y. For each eie € S1, we set E& = 4 > ~ l ( e i 0 ) . We prove the following results: LEMMA 2.1 One has :

U = UA/0 and V = rWe , for 9 E [0, TT[ . /Uso. /or ear:/;, # 6 [0, TT[ one /?,a.s : Me = E0 U V U E0+7T . Similarly, if Se is a. sphere embedded in IA with centre at 0, one has,

Sr = \j(M0 n Se), v n SF = r\(M0 n se), and

(M n 6V) = (EO n 5e) u (V n s?) u (E0+7r n 5e) . THEOREM 2.2 // // in holomorpic, let he = TTO o H be as before and let

be the usual Milnor fibration ['3], where K = V n S£ is the link of 0. Then : i) Each MQ = h^l(0) is a real analytic, hyper-surface in U. whose singular set is the singular set of V . ii) Each pair of antipodal fibers Fg and F#+7r of 6 are naturally glued together along K, forming a real analytic variety isornorphic to N@ = MQ D S£ . This theorem is telling us that the link NO of MQ is, in some sense, the double of the Milnor fibre of H, but Ng is singular if K is singular. We notice that even in this case, there is also a fibration [2, 3]. H : H-l(S$)nBF

-> S],

where 5$ is a small circle centered at 0 E C and B£ is a small open ball in C n+1 . This fibration is equivalent to the Milnor fibration. The closure of each fibre EO — H~l(el&) in this fibration is a compact non-singular variety with boundary K, while the closure of the Milnor fibre F is the union of F with the link K, which may be singular. I thank Le Dung Trang for explaining me that there is a natural contraction function from K onto K; it would be interesting to study the relation of this function with our construction. Equivalently, one may study the relation between the double of the

Topology of Hypersurface Singularities

203

Milnor fibre, which is a closed manifold, and the link NQ of the real analytic variety M0 = hJl(Q). We notice that MO and MIL are, respectively, the sets of points where ReH(z] = 0 and/m#(z)=0. If the singular set of V consists of an isolated point, one has the following theorem: THEOREM 2.3 IfQeU is an isolated critical point of H, the MQ'S are non-singular away from 0, and they are all homeomorphic. The link NQ = M0r\Se is diffeomorphic to the double of the Milnor fibre F of H, hence Ng is (n— 1) -connected; Ng is always stably parallelizable, and it is actually parallelizable if and only if n is odd and the Milnor number /j, of H is 1. We recall that a manifold M is parallelizable if its tangent bundle TM is trivial; M is stably parallelizable if TM0 (k] is trivial, where (k) is a trivial bundle over M, or equivalently, if M can be embedded in some sphere SN with trivial normal bundle. Since the topology of the Milnor fibre of H and the link K are well understood, these theorems determine the topology of the real hypersurfaces MQ. EXAMPLE 2.4 Consider the complex polynomial /(zi, z 2 ) = z? + z\ , with p,q > 1. The topology of this singularity is well understood. Its link K is a torus link (or knot if p, q are relative prime) and its Milnor number // is (p— l)(q— 1); we recall that the Milnor fibre F is in this case an oriented surface in the 3-sphere, with boundary K and the homotopy type of // circles. Now consider the real and the imaginary parts of/,

Both define real hypersurfaces in C 2 , which are cones over their link LI, L^. By the theorem above, L\ and L^ are homeomorphic, actually diffeomorphic, and they are the double of the Milnor fibre F. Hence LI and L% are closed, oriented surfaces of genus (p — l)(q — 1) in the 3-sphere S3 C C2. This can be interesting because it provides explicit analytic embeddings of surfaces of all genera in S'3 C C 2 . For instance, if p = 2 = g, then K is the Hopf link, F is a cylinder Sl x / and L\,L
3

THE PROOFS

Proof of The Lemma 2.1: If z 6 M#, then H(z) is contained in the line passing through 0 with an angle 0 ± f ; Hence z e MQI D Mg2 if and only if z 6 V where 6 1 = #2 + k-K , k 6 Z; We prove that V = nM
Seade

204

It is now clear that V C MQ. Let us prove that Eg C MQ . If z E EQ then

iH(z) \\H( which implies

eieiH(z) \H(< hence. =0.

Thus one has z G MQ. Similarly , if z G Eg+w, then

iH(z] so

which implies Ree lQ H ( z ] = 0. Hence one has Eg+n C MQ . Conversely, if z G A/0, then Ree~l°H(z} = 0; If H ( z ] = 0, one has z G V and there is nothing to prove. If //(z) ^ 0, then .. iU(-A

I -|

H(z}\\ so z is in EQ or in £"0+^, according to its sign, and we arrive to the formula above. To prove the second statement in the lemma we just restrict the previous discussion to the sphere S£. Proof of The Theorem 2.2: The second statement in Theorem 2.2 is a consequence of the lemma above, because the map 4> is c/> followed by the diffeomorphism z —-> iz of C. Hence the intersection of each A/0 with each sphere Se is a real analytic variety of dimension 2n; since MQ is a cone near 0, by [3], we know that MQ is a real hyper surface. Its singular set consists of the critical points of hg, i.e. the points where all the partial derivatives of hg vanish. This set does not change if we multiply b the number Thus, to prove the claim about the singularities of A/0, it is enough to consider the case 9 = 0, i.e. for the real part of //, / = ReH. One has,

where H is the complexe conjugate of H. Therefore, 2A/ =

•OH

OH

H

because the partial derivatives of // with respect to the z?; are all 0, the partial derivatives of H with respect to the z,- are all 0 and ^~ = ^- . Thus, the critical points of H are the critical points of /.

Topology of Hypersurface Singularities

205

Proof of The Theorem 2.3. The claims that the Me are regular away from 0, that they are all homeomorphic (actually diffeomorphic away from 0) and the link is diffeomorphic to the double of the Milnor fibre of /, are all consequences of Theorem 2.2, NO is connected because F is connected, by [3]. Furthermore, if n > 1, then K is connected and F is simply connected, by [3]. Hence Van Kampen's Theorem ([6], p. 151) implies that NQ is simply connected. Moreover, by [3], F is & wedge of n-spheres and K is (n — 2)-connected, thus Mayer- Vietoris (reduced if n=2) implies that one has #i(tf,;Z)S...S# n _ 1 (JVi,;Z)SO. Therefore Hurewicz's isomorphism ([6], p. 397) implies that Ng is (n — l)-connected. Finally, NO is stably parallelizable because it is a codimension 1, oriented submanifold of S£. Thus, ([1], Th.IX) , N0 is parallelizable if and only if its Euler-Poincare characteristic x(No) vanishes. One has:

where // > 0 is the Milnor number of H, since NT is the double of F, and F is a wedge of p, n-spheres. So x(-/V#) = 0 if and only if n is odd and \L = 1.

REFERENCES [1] M. Kervaire. Courbure integrate generalisee et homotopie. Math. Ann. 131:219252, 1956. [2] Le Dung Trang. Some remarks on the relative monodromy. Real and complex singularities, Sijthoff and Noordhoff, Oslo, 1976, pp. 397-403 [3] J. Milnor. Singular points of complex hypersur faces, Annals of Maths. Studies, 61, Princeton Univ. Press, 1968. [4] L. Paunescu. The topology of the real part of a holomorphic function. Math. Nadir. 174:265-272, 1995. [5] J.A. Seade. Fibred links and a construction of real singularities via complex geometry. Bol. Soc. Bras. Mat. 27:199-215, 1996. [6] E. Spanier. Algebraic Topology. Springer Verlag, 1966.

Polar Multiplicities and Equisingularity of Map Germs from C3 to C4 VICTOR HUGO JORGE PEREZ Universidade Estadual de Maringa, Departamento de Mate- matica, Av. Colombo 5790, Campus Universitario, CEP 87020-900, Maringa (PR), Brazil. E-mail: [email protected]

1

INTRODUCTION

In this work we consider the following problem: given a 1-parameter family of map germs, find analytic invariants whose constancy in the family implies the family is topologically trivial or Whitney equisingular. Gaffney in [6] describes this problem for families of finitely determined map germs of discrete stable type, in which the topological triviality of a 1-parameter family of a class of map germs is shown to be consequence of the constancy of finitely many analytic invariants associate to a member of that family, the number of invariants depending upon the particular dimensions of source and of target space being considered. The necessary invariants are the zero-stable invariants and the polar multiplicities defined by Teissier [17] and Gaffney in [6]. Gaffney in [6] and [3] uses this result to analyze mapping from the plane to plane and the plane to space. The author in [8] applies the results of Gaffney for mapping from 3-space to 3-space, in this case the number of invariants necessary has been reduced from 21 to 7. More recently Gaffney's approach has been used by Vohra [18], to study map germs from n-space (n > 3) to the plane. In particular, the number of invariants required for Whitney equisingularity in each of these situations was shown to be smaller than the a priori number given by the general result of Gaffney [6]. In this paper we deal with the case of map germs from 3-space to 4-space. According to Gaffney's result, for a family ft : C 3 ,0 —> C 4 ,0 to be Whitney equisingular it is needed the constancy of 20 invariants. We reduce this number to 8 for corank 207

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Perez

1 germs. We do this by finding relations among the invariants and using the fact that these are upper semi-continuous. In section 2 we recall some results of [6] and prove our main result in section 3. We apply this result in section 4 to compute some invariants for normal forms of map germs given in [2].

2

BASIC DEFINITIONS AND RESULTS

We denote a holomorphic map germ / : (Cn, 0) —> (C p ,0) by writing / G O(n,p). Our interest is primarily in finitely determined map-germs; a standard reference in this context is [19], although the material in section 1 of [6] is sufficient for our purposes. We denote by 1Z the group of diffeomorphisms of the source (C",0), and by £ the group of diffeomorphisms of the target (Cp, 0). The action of the product A := 13, x £ leads to ^-equivalence of map germs: /, g 6 O(n,p] are ./4-equivalent if they are equivalent by smooth coordinate changes at source and target. Similarly the action of the semi direct product /C := (7£.C) gives rise to /C-equivalence of map germs. A germ is said to be A:-^4-determined if any g e O(n,p) with the same fc-jet as /, i.e. jkg = j k f , is ^4-equivalent to /. The germ / is said to be finitely ,4-determined if it is A>.4-determined for some k. A map-germ / : C n ,0 —> C p ,0 is stable if, up families of (bianalytic) diffeomorphisms in source and target, every deformation is trivial. That is, if ft is a 1-parameter family with /o = /, then there should exist 1-parameter families ipt and ij.)t of diffeomorphisms of source and target such that and ?/v ° / ° Vt — ft- Stable type is the ^4-equivaleiice class of stable germs. A finitely determined germ / has discrete stable type if there exist a versal unfolding of / in which only a finite number of stable types occur. If the numbers (n,p) are in Mather's "nice dimensions" (which is true for (3,4), our focus here) or on the boundary thereof, then every finitely determined germ / <E O(n,p) has discrete stable type.

2.1

Colength and multiplicities of ideals

The colength of a given ideal / in a complex analytic ring R, is defined as dimc(jR//); it may or not be finite. The multiplicity of an ideal / is an integer invariant denoted by e(7) that is defined whenever / has finite colength. If R =< /?, m > is local and Cohen-Macaulay, if / is ra-primary and a complete intersection, then the multiplicity of / is just its colength.

2.2

Finite maps and degree

A smooth map germ / : (A", x) —» (Y, y] is said to be finite if the dimension of its G local algebra is finite, i.e. if the number rri(f) :— dime f*(m}$ < °°- Note that for f to be finite, it is necessary that dimX < dimF. In the context of complex analytic geometry, we have the following important result for finite maps. Let / : (X, x} —> (Y,y) be an analytic map of analytic spaces of the same dimension, such that f ( X ) is Zariski-dense in Y. Suppose f ( x ) is a smooth point of Y, and {x} a component of the fiber f ~ l ( f ( x ) ) . For open neighborhoods U C X of x, and V C Y and a closed analytic subset B C Y such that: (i) V \ B is connected;

Map Germs from C3 to C4

(ii) f(U] C V, f \ u is proper, f~l(f(x))

209

= {x}- and

(iii) f \ u \ f - i ( B ) smooth, then the number of pre-images in U, counted with multiplicity, of any point y 6 V\B, is called the degree of / at x, denoted deg(/), for a proof see 3.12 in [15]. In particular if X C (Cn, 0) is an analytic space germ defined as the zero set of germs #1,..., (ft with dimo(^) = d, and if Ox is Cohen-Macaulay, then we can often use a projection TT : (C n ,0) —» (C d ,0) such that deg(7r|x) is the colength of the ideal fa, ...,7r d ) in Ox, i.e. dimc (Tlt ... i ^ 1< ... tgt) .

2.3

Unfoldings

A 1-parameter unfolding F of / is a good unfolding if there exist neighborhoods U and W of the origin in C x Cn and in C x C p , respectively, such that F~1(W) = U, F maps (U D S(/)\C x 0) to W\C x 0 (£(/) is the critical set of /), and if (t0, yo) e W\C x 0, with 5 := F-l(t0,y0) n S(F), then the germ /to : (C n ,S) -> (C3,^) is a stable germ. We say that a good unfolding F of / is excellent if / is a finitely determined germ of discrete stable type, and the 0-stable invariants, see section 3 of [6], are constant in F, if n = p, then an additional requirement for excellence is that the degree of / be constant in the unfolding.

2.4

Whitney equisingularity and stratification of maps

A Whitney stratification of a given space is a stratification such that for any pair of strata S, 5", with Sf C S, the big stratum S is Whitney regular along Sr. The local topological type remains constant along each stratum of a Whitney stratification of a given space. Note that a Whitney stratification always exists in the local complex analytic space, see section 1 of [5]. We say a given space X is Whitney equisingular along Y if there is a Whitney stratification of X with y as a stratum. Recall that if F : Cn -»• Cp is a morphism, and A C C n , A' C Cp subsets such that F(A) C A', then a stratification of F : A —> A' is a pair (A, A'} of stratifications of A and A' respectively, such that F maps strata submersively to strata. A given stratification (A, A') of F is a regular stratification if A, A' satisfy the Whitney regularity conditions, and all pairs of incident strata in the source satisfy Thorn's Af condition: Let U be an open subset of some affine space, / : [ / — > C be an analytic function and M be a subrnanifold of U. Thorn's Af condition is satisfied between £/-£(/) andM if, wheneverPi e C7-£(/),/>» -> p 6 M, and TpiV(f - f (pi)} -+T, then TPM C T. If .F = ft is an unfolding with parameter axis T, then a regular stratification (A, A'} of F is said to be a Whitney equisingular along T if T is a stratum of A arid of A', (A and A' are Whitney equisingular along T). We also say in this case that F is a Whitney equisingular map. The polar multiplicities of the polar varieties (defined by Teissier in [17]) of the stable types are the invariants needed to show the Whitney equisingularity of unfoldings. DEFINITION 2.1 Suppose f : (X,0) -> (5,0) is a flat map smooth which fibers at every point of X — sing(X). Let p : Cn —* Cd~k+l be a linear projection such that her p = Dd-k+i where D^-k+i is a linear subspace of (C n ,0) of dimension k. For

210

Perez

x £ X — fiing(X), the fiber- X ( f ( x ) ) is non-singular at x contained in {f(x)} x Cn and one denote* by TTX : X ( f ( x ) ) -»• Cd~k+1 the restriction of p to X ( f ( x ) ) . Let P k ( f , p ) be the closure of points x EX — sing(X] such that x £ E(7r, T ). one calls the closed analytic sub space Pk(/,p) of X , the relative polar variety of codirnension k associate to Dd_k+i. If f is the constant map, we denote this by P^X), called absolute polar variety. The key invariant 0/-Pfc (/,/>) is its polar multiplicity which we denote by mie(X^ /), if f is the constant map, we denote this by mQ(Pjt(X)) or Gaffney in !6j page 195, defines a new invariant as following: Take a versal unfolding F : C" x Cs ->• Cs x C n+1 of /. Specify a stable singularity type or stratum £>(/) in source or target such that dimD(/) > 1. Select D\ a linear subspace of (C n+1 ,0) of dimension 1 and form P(i(D(F}} the polar variety on T>(F) with the projection (p, TT S ) : Cn x Cs -» C x Cs where d = dim(D(F}) - s. The d-th stable multiplicity of / of type £>(/), denoted w d (£>(/)), is the multiplicity ofm s C 0 o in Using the polar multiplicities of the stable types and Thorn's condition A/, Gaffney showed the following principal result. THEOREM 2.2 ([6] pp. 206-207) Suppose that F : C x C n , (0,0) -> C x C7,(0,0) is an excellent unfolding of a finitely determined, germ f £ O(n,p). Also suppose that the polar invariants of all the stable types defined in the discriminant A(/), in S(/) and, in /~ 1 (A(/)) — £(/) are constant at the origin for ft. Then the unfolding is Whitney equisingular. REMARK 2.3 1. The theorem also implies that such unfolding is topologicaly trivial; for the proof of this result Gaffney uses Thorn's second isotopy lemma for complex analytic mappings, see [6] page 204. 2. ([18]) The theorem remains valid if we replace the term "an excellent unfolding" in the hypothesis by ua 1-parameter unfolding which, when stratified by stable types and by the parameter axis T, has only the parameter axis T as 1-dimensional stratum at the origin" . In the next section we shall need the following definition and results, these results are a key tool in finding relations among our invariants. THEOREM 2.4 (Le-Greuel, [10], [13]) Let Xi be a complete intersection with isolated singularity at 0 6 C" ( an ICIS ). Let X be an ICIS defined, in Xi by fk = 0, and let /i,...,/fc_i be the generators of the ideal that defines X\ at 0 in C". Then °n ( f l i •••,fk-l,J(f\,

••••> fk))

REMARK 2.5 In the extreme case of a zero-dimensional ICIS we can appeal to the simpler formula: Let / : (C fc ,0) —> (C fc ,0) be a germ such that X = /~ 1 (0) is an ICIS. Then fj,(X, 0) = fi(f) - 1, where <$(/) = dimc f,(^)on ' see [121 Pa§e 78' Other elementary result that we appeal is: Let / : (C n ,0) —> (C n+1 ,0) be a finitely determined germ. Then (/ : C n ,0) -> f(Cn) C (C n+1 ,0) is bimeromorphic; see [4] page 138.

Map Germs from C3 to C4

3

211

EQUISINGULARITY OF MAP GERMS IN C$4)

3.1

The stable types in (9(3,4)

As highlighted in the introduction, our aim is to minimize the number of invariants denned in the stable types of / whose constancy in the family ft implies the family is Whitney equisingular (therefore topologically trivial). The strategy is to apply Theorem 2.2 and the techniques used by Gafrney in [6], that is, stratify the source and the target by the stable types and establish relations among the invariants on the strata. As these invariants are upper semi-continuous, the relations will allow us to reduce the number of invariants required in Gaffney's theorem. For this we first give the following preliminary definition. Given a continuous mapping / : X —» Y on analytic spaces, we define the ktk multiple point space of / as r / t 7~) (\J) f \ — "\ £1 -, \1— "\ 4V^i* T* -X^ -/- T*T» JL • <j•» ^ -^i* LJ f. JL/ — /""InciTroJ V-'IiJoUJ." |f T*-« I JU \^T*^» JL^^ .. .^T»,JLfa } \z\ -/V • .f J( rIX J — ...——f J(<~rI iX^ J 1L>1 I ^-

If 0 : Cn -> C is a function then we define Vf(g) : Cn+k~l -> C to be L

zi

•••

z{~1

g(x,zi)

Zk

•••

4

1

d(x,Zk]

z\+l

•••

zk~l

1

-

1

zk

~ fc ~ 1 •••

zk-l

Suppose / :Cn, 0 —» Cp, 0, with p > n, is of corank 1 and is given in the form THEOREM 3.1 ([14]) D k ( f ) is defined in Cn+k~l by the ideal I k ( f ) generated by Vk(hj(x, z)) for all i = 1, ..., k — 1 and j = 1, ..., j> — n + 1. In what follows we will take coordinates on C n+fc ~ 1 = Cn~l x Cfc to be (x,z) = (#1, . . . , X n _ i , Z i , . . . , Zk).

EXAMPLE 3.2 For a corank 1 map-germ / : C n , 0 -» Cp, 0, D2(f) is defined by the ideal generated by the system The main result of [14] is the theorem 2. 14 where a description of Dk (f) is obtained for a finitely ^-determined corank 1 map germ. There, it is shown that the multiple point spaces of / are ICIS. More precisely, / is finitely-determined if and only if for each fc, with p — k(p — n) > 0, D k ( f ) is ICIS of dimension p — k(p — n) or empty, and for those k with p — k(p — n) < 0, D k ( f ) = {0}. Furthermore, f is stable if and only if the spaces are non-singular or empty. DEFINITION 3.3 LetP = (n,r 2 , ...,r m ) be a partition ofk, i.e., ri+r2+...+rm = k, with k = 1, ...,4. Let T(P] be the ideal in On_\+k generated by the k — rn elements Vi - Ui+i for ri + r<2 + ... + r,,-_i + 1 < i < ri + r2 + ... + TJ - I , I < j < m, and let I f P , 3? are two partitions of k, we say P < 3? ifT(P) $1 Z(3?). We define a generic point of A("P) for any partition 3? of k with P < K.

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Define

Given a partition P = (r\, ..., r m ) of fc, define projections ^ i ( P ] '• C n " 1+fc —> C n , for 1 < / < m, by 7T 7 ('P)(.r, Z T , ..., z fc ) = (.r,2v 1 + ... + r ._. 1 + 1 ). The geometric significance of Dk (/, T3) is given in Lemma 2.7 by Marar and Mond [14] page 559. LEMMA 3.4 Let P = (n , ...,r m ) be a partition of k; at a generic point (x,z) of A(P) we have: Ik(f,P) = {fj ° TTi(P) - /7 ° ^.(P)

J = 1, —P - « + 1,2 < « < m} in C? n _i +fc , (x, 2).

In the corollary 2.15 of [14] page 562 it is shown the following result. If / is finitely determined then for each partition P — (r\,..., r m ) of k satisfying/>—k(p—n+l)+m > 0, the germ of D fc (/, P) at 0 is either an ICIS of dimension p — k(p - n 4-1) + m, or is empty. Moreover, those D k ( f 1 P ] for P not satisfying the inequality consist at most of the single point 0. With the definitions above, for any finitely determined germ / e 0(3,4) we denote by D l ( f , P ] and Df (/), with k = 2,3, the projections of D 2 ( f , P ) and £>*(/) respectively, to the (:r, ?/, ^jj-space. The stratifications in the source and target are as follows. In the source: The set of critical points £(/) = C3, the set of double points D'i(f], the set of triple points D'i(f), the family of cross caps Z)^(/,2) and the zerodimensional set DI(/, (2,1)). In the target: The discriminant of /, A(/) = /(C 3 ), the image of the double points set /(£> 2 (/)), the image of the triples points set /(£> 3 (/)), the image f ( D 2 ( f , (2))) and the zero-dimensional stable types. These are, normal crossing of a hyperplane with the family of cross caps, denoted by (A^C] and the set of quadruple points (Q). There exist k + l polar invariants associated to afc-dimensionalvariety. As /(C 3 ) is of dimension 3, the dimension of D 3 (/), /(D 3 (/)), Z)2(/, (2)) and /(D 2 (/, (2))) is 1, and the dimension of D2(f) and f ( D 2 ( f ) } is 2, there are 18 polar invariants defined on these sets. We also have 2 multiplicities of the zero-dimensional stable types. These are, the number of normal crossing of a hyperplane with the family of cross caps, denoted by ($A\C(f}} and the number of quadruple points ()}Q(/)). Therefore to apply Theorem 2.2 to germs in (9(3,4) we needed the constancy of 20 invariants, we reduce this number to 8 invariants.

3.2

Relations among the invariants of the stable types in the target

We start analyzing the discriminant /(C 3 ). We have the following general result.

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THEOREM 3.5 Let f e (9(3,4) be a finitely determined germ. Then -m 3 (/(C 3 )) + m 2 (/(C 3 )) - mx(/(C 3 )) +m 0 (/(C 3 )) = 1. Proof: We choose a generic linear projection p3 : C4 —>• C3 such that the degree 3 of p3|/(C ) is equal to the multiplicity of /(C 3 ) at the origin, and the multiplicity of the polar variety ^(/(C 3 )) = S(p3|/(C3)°) is (by definition) ra^C3)). By the result of Teissier [16], Pi(/(C 3 )) has dimension 2. Then let p2 : C3 -> C2 be a generic linear projection such that the degree of p2 °P3|-Pi(/(C 3 )) is mi(/(C 3 )). We can suppose that the projection p2 o p3 is also generic and defines the polar variety P 2 (/(C 3 )) = S(p2 °P3|/(C 3 )°). Choose p1 : C2 -»• C generic linear projection so that the degree of pi o p2 op 3 |P 2 (/(C 3 )) is ra2(/(C3)). Suppose that the projection Pi °P2°P3 is also generic and defines the multiplicity polar m 3 (/(C 3 )). Let

Xi = V(p2 ops o/), X = V(P3of}. As these varieties are ICIS, we apply Le-Greuel theorem and obtain

2 0 p 3 o / , J[p3o/])'

Applying Le-Greuel again to X% = V(p\ o p2 o p3 o /) and Xi, we have M (X!)

+/.(X 2 ) = dimc

(P10P20P30 ^ J(P20P30/ ))

= rn 2 (/(C 3 )).

As X is a 0-dimensional ICIS , we have n(X) = deg(p3 o /) - 1. As / : C3 —> C4 is bimeromorphic onto its image, p,(X] = rrio(/(C 3 )) — 1 and as / is bimeromorphic we have that

dim

Therefore m 2 (/(C 3 )) - n(X2) +mo(/(C 3 )) - 1 = mx(/(C 3 )). But ^(X 2 ) = m 3 (/(C 3 )). We take a versal unfolding F : C3 x Cs -> C4 x Cs of / and consider (j», TTS) : C4 x Cs —> C x Cs a projection in target such that, F : V(J(pl op2op3 oj)) C C3 x Cs -> F3(^_(C3 x C s )) C C3 x Cs is bimeromorphic, we have that the variety V(J(p\ o p2 o p3 o /)) has dimension s, since u is a generic parameter value, the deg(7rs|F3(JF(C3 x C s ))) = deg(7rs|F(J(pi o P2 o P3 o /))) = M*2)=m 3 (/(C 3 )). D

REMARK 3.6 (1). Note that in the kernel rank 1 case, m 2 (/(C 3 )) = 0, since = 0, m 3 (/(C 3 )) = 0, then m 0 (/(C 3 )) - 6(f) - 1.

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(2). In the kernel rank 2 case m 3 (/(C 3 )) = 0, so the constancy of mi(/(C 3 )) implies the constancy of m 2 (/(C 3 )),mo(/(C' 3 )) since all these invariants are upper semi-continuous. (3). The alternating sum of polar multiplicities in the statement is reminiscent of an Euler obstruction for singular complex analytic spaces, there is a relationship between the Euler obstruction on stable types / : C" —>• Cp and the the polar multiplicities; this result is in preparation by the author j9| . The other strata in the target are f ( D k ( f ) ) , k = 2, 3. These have dimensions 2 and 1. Their polar multiplicities are 7//o(/(D' 2 (/))), wi(/(D 2 (/))), m 2 (/(D 2 (/))), m 3 (/(D 2 (/)}) and •m 0 (/(D 3 (/))), //M (/(^ 3 (/)))• To compute /?/ 2 (/(D 2 (/))). we take a versal unfolding F : C3 x Ca -> C4 x Cs of / and consider (/>. 7r. s ) ; C4 x C s - > C3 x Cs a projection in the source such that F o (p,7r s ) : D'2(F) C £l x C;! - F(D'2(F)] C C4 x C*. We know that Fo (p.7vs}\D2(F}° is a 2 fold cover of F(l)'2(F})". Choose pi o f>2 '• C } •--» C, a generic linear projection for ( / V X D 2 ( F ) ) , (/>i o p 2 , 7 r s ) ) . To work directly with

However, it is much easier to work with E((pi op 2 ,7T s ) o Fo (p, 7r s )|D 2 (F)) which is

where T2(F) is the ideal that defines I)'2(F}, we include the singular set of D'2(F) in the critical set of (/^ o/> 2 ,7r s ) o Fo (p ; 7r s ) D2(F}. This set has two advantages. It is in the source, and its equations are computable. Our strategy is to extract an invariant from it which will be simply related to / / > 2 ( / ( / J 2 ( / ) ) ) and in fact which will control it. The variety V has dimension .s, since u is a generic parameter value, the degree of 7is\V(T2(F], J;r((pi o />>. TT.S) ° F o (p. 7r. s ),T 2 (/ r ))) is just the colength of rns in the local ring of the source at (0,0) and this is just o P2 o / o p,I

where T2(f) defines D 2 (/). We then have the following relation between c
anc

^ other irivai'iarits.

THEOREM 3.7 LcA, f & (9(3,4) be. a finildy determined germ of kernel rank 1. Then

where c,\ , c2

Map Germs from C3 to C4

215

Proof: The components of the variety V (J2 (F), Jx((p\ op2, TTS) oFo (p, TT S ), J2 (F))) are the closure of the set F-l(P2(D2(F)), the set F~l(Q}, and the set F~l(AiC). These have dimension at least s. For a generic projection pi, the dimension of V is s. As the multiplicity of V is the sum of the multiplicities of these components, it is enough to calculate the contribution of the degree of TTS restricted to each component, where TTS is the projection ofC 4 xC s toC s . We choose neighborhoods U\ of 0 in Cs and C72 of 0 in C4 x Cs such that at each point in t/i, TTS has e D 2(j)(/) pre-images in V fl C/2 counting multiplicity. If u 6 Cs is a generic parameter close to the origin we have e

D2(f)(f)

=

Z/zeS dime

(ms,T*(F),Jx((Pl,aTrs)oFo(p,7ra),T*(F)))

Ex£ where S = 717 ^ n V . We can compute this last number using normal forms. Hence the singularities of type Q and A\C contributes with ci$Q(f) and C2$AiC(f) to eD-2(f)(f], where ci,c 2 G Z + . A similar argument shows that F~l(P2(F(D2(F)))) contributes with rri2(f(D2(f)}). Since F restricted to each type of component is bimeromorphic and finite, and the generic point of each component is reduced, we have that deg(7r,y) =

D PROPOSITION 3.8 Let f € C?(3,4) be a finitely determined germ of corank 1. Then

Proof: The stratum in the target is /(D 2 (/)), it has dimension 2 so there are 3 polar invariants m 0 (/(D 2 (/))), m l ( f ( D 2 ( f } ) ) and m 2 (/(£> 2 (/))). Since/ £>?(/)-{()} is a2-fold cover of /(L> 2 (/))-{0} it follows that e((/*(m 4 ))C' D 2 ( / ) ,o)=2mo(/(L> 2 (/))). To compute mi(/( J D 2 (/))) choose p2 : C4 —> C 2 , a generic linear projection for f ( D 2 ( f ) ) such that the polar variety P 1 (/( J D 2 (/))) = £(p 2 |/(L> 2 (/))°) and the multiplicity of P l ( f ( D 2 ( f ) ) ) is m i ( f ( D 2 ( f ) ) ) . Let pi : C2 —> C be a linear projection such that the degree of pi op 2 |P 1 (/( J D 2 (/))) is 777,1 (/(D 2 (/))). We also require that p\ o p2 be a generic projection that defines

These spaces are ICIS, where J 2 (/) is the ideal that defined D2(f) and p : £> 2 (/) C C4 —> C3 a projection of D2(f) to the source of /. Then by Le-Greuel theorem we have = dimc

2 2 (Pi °P2 ° f ° P , Z ri(t\7( ( f ) , J(P2 o /1 op,T-n(t\\\ (/)))

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By our genericity assumption on p\ and p<2, the right-hand side of (3.2) is just 2mi(/(D 2 (/))). Applying Le-Greuel theorem again to the following spaces that are ICIS

We get f i ( X i } + fj.(X2) = dimc -—Then ^(Xi) = dime ^f^j(p^f___ In (3.2), we obtain

- //,(D 2 (/)).

Since .X" is ICIS of dimension zero we have that /J-(X) = deg(p2 o / op, J 2 (/)) — 1. As OD2(f) is a Cohen Macaulay ring = dime --T

(P2 o / o p,

- 1 = 2<m 0 (/(L> 2 (/))) - 1.

Since / : C3 —»• C4 is bimeromorphic -

2m0(f(D2(f))) D

We now deal with the stratum /(D 3 (/)). This has dimension 1 so there are two invariants m 0 (/(I> 3 (/))) and r n l ( f ( D 3 ( f ) ) ) . Since p o f \ D 3 ( f ) ° is a 3-fold cover of /(D 3 (/))° it follows that e(/*(m 4 )0 D 3 (/) . 0 ) = 3!m 0 (/(D 3 (/))). To compute w,i(f(D*(/))), we take a versal unfolding F : C3 x Cs -^ C4 x Cs and consider (p, TTS) : C° x Cs —> C3 x Cs a projection in the source, such that Fo (p, TTS) : D 3 (F) C C5 x Cs -* F(D 3 (F)) C C4 x C*. We know also that F o (p,7r s )|D 3 (F)° is a 3 fold cover F(D 3 (F))°. Choose pi : C 4 —* C, a generic linear projection for (F(D 3 (F)), (PI,TT S )). To work with P 1 (F(Z) 3 (F)),(p 1 ,-7r s ),D 1 x C s ) where Dj x Cs is the kernel of (PI,TT S ) we must work with

However, it is much easier to work with S((pi, TTS) o F o (p, 7rs)|.D3(F)) which is y(J 3 (F),J x (( P l ,7r s )oFo(p,7r s ),J 3 (F))) where TZ(F] is the ideal that defines D 3 (F), we include the singular set of D3(F) in the critical set of (PI,TT S ) o F o (p,7r s )|D 3 (F).

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217

So V has dimension s, since u is a generic parameter value, the degree of

is just the colength of ms in the local ring of the source at (0, 0) and this is

= dim, where J 3 (/) defines D 3 (/). We then have the following relation between e£>3(j)(/) and the invariants of the zero stable types. THEOREM 3.9 Let f e (9(3,4) be a finitely determined germ of corank 1. Then

where b\ , 62 6 Z+ Proof:

Using (3.3) and as in theorem above, we obtain the result.

D

THEOREM 3.10 Let f e 0(3,4) be a finitely determined germ of corank 1. Then n(D3(f))+3\m0(f(D3(f)))

- 1= c

Proof: Since / is finitely determined, the ideal I3(F] of a versa! unfolding F of /, has the property that J 3 (/o) = 0 is the defining equation of D3(f] C C5 with the reduced structure. Hence Le-Greuel theorem applies to the following varieties

that are ICIS, where pi : C4 —> C is a generic projection that defines eDa.(/)(/), and mi(/(D 3 (/))) and p : C5 -»• C3 is a projection of D 3 (/) to the source of /. So we have /n_ = dimC 7^77

As A" has dimension zero, it follows by 2.5 that

Since 0£>3(/-\ is a Cohen Macaulay ring, we obtain

and as / is bimeromorphic and finite

and the result follows.

D

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By the definition of D2(f,2) we consider it as a subset of C3, then the stratum in the target is /(D 2 (/, 2)). This has dimension 1 so there are two invariants <m 0 (/(£> 2 (/,2))) and mi(/(Z> 2 (/,2))). Since / D 2 ( f , 2 ) ° is a 1-fold cover of /(D 2 (/,2))° it follows that e(/*(m 4 )0 D 2 ( /,2).o) = m 0 (/(/>(/, 2))). To compute m l ( f ( D 2 ( f , 2 } ) ) , we take a versa! unfolding F : C3 x Cs -> C4 x Cs of /, We know also that f\D2(F, 2)° is a 1-fold cover F(D 2 (F, 2))°. Choose />! : C 4 —> C, a generic linear projection for (F(D 2 (F,2)), (PI,TT S )). To work with

where DI x Cs is the kernel of (PI,TT S ) we must work with £((pi,7r a )|F(I>2(F,2)) 0 ). However, it is much easier to work with H((PI,TTS) o f\D2(F, 2)) which is K(J2(F, 2), J r ((p!,7r,) o F, T 2 (F, 2))) where J2 (F, 2) is the ideal that defines D2 (F, 2) , we include the singular set of D2 (F, 2) in the critical set of (pi,7r s ) o /|D 2 (F, 2). This set is in the source and its equations are computable. We follow the same strategy in the case of the set of double points to extract m i ( f ( D ' 2 ( f , 2 ) ) ) . The variety V has dimension ,s, since .s is a generic parameter value, the degree of TTS\V(I2(F, 2), J x ( ( p i , TTS) o F, J2(F, 2))) is just the colength of ms in the local ring of the source at (0,0) and this is eDW](f]

= dime

(J2(/, 2) ,, 4(( ^ 0 ;, J 2 ( / , 2 ) ) )

where J 2 (/, 2) defines D 2 (/, 2). We then have the following relation between eD 2 (/,2)(/) zero stables types.

(3-4) an(

l the invariants of the

THEOREM 3.11 Let f e 0(3,4) be a /mitefy determined germ of corank 1. Then Proof: To understand the geometric meaning of the right hand side of the equality (3.4) we choose a versa! unfolding F = (,s-,/(.s,x)) of / and consider the variety F( Z2 (F, 2) , Jx (pi o F, X2 (F, 2) ) ) . The components of this variety are the closure of the sets F-^Pi (D2(F, 2)), the set F~l(Q) and the set F"1 (AiC). These have dimension at least s. For a generic projection p1; the dimension of V is s. As the multiplicity of V is the sum of the multiplicities of these components, it is enough to calculate the contribution of the degree of TTS restricted to each component, where TTS is the projection from C3 x Cs to Cs. We choose neighborhoods U\ of 0 in Cs and f/2 of 0 in C3 x Cs such that at each point in C/i, TTS has eo2(f,2)(f] pre- images in V D C/2 counting multiplicity. If u 6 Cs is a generic parameter close to the origin we have 3

dlm

C

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219

where S = 7r~ 1 (0) fl V. To count the contribution of singularities of type A\C, we take fa = {(x,y, z,0);(:r,?/,z 2 ,a:z)}. Then, to calculate the contribution of the singularities of type A\ C in F(J2(F,2), JX((PI,KS] o F, J 2 (F,2))) it is enough to use the ideal l^(/,2) = (2z,z). Using the formula (3.5) we have

where p(x, y, , z, z\,Z"i) = ax -+- by -f cz is a generic projection (b ^ 0). This means that the points of type A\C appear in V with a contribution equals to 0. In the same way, to count the contribution of the points Q it is enough to choose the normal form fs = {(#, y, z, 0), (0, x, y, z), (z, 0, y, z), (z, y, 0, z)}. Then the contribution of Q in V is also 0. Since F restricted to each component is finite and bimeromorphic, we have deg(7r s |F)=m 1 (/( J D 2 (/,2))). The theorem follows now by joining all the above equalities.

D

THEOREM 3.12 Let f e (9(3,4) be a finitely determined germ of corank 1. Then

Proof: Since / is finitely determined of corank 1, the ideal T2(F^2) of a versal unfolding F of / has the property that J 2 (/o,2) = 0 is the defining equation of L> 2 (/,2) and we can consider I^/o^) C 03, therefore £> 2 (/,2) C C3 with the reduced structure. Hence Le-Greuel theorem applies for

which are ICIS, where p\ : C4 —» C a generic projection. So we have sn V>(Xi) + I*(X) = dimc ,--,, 0

As A" has dimension zero, it follows that Ai(X)=deg( P l o/,I 2 (/,2))-l. Since C)D2(/,2) is a Cohen Macaulay ring, we obtain

and as / is bimeromorphic and finite dimc ^(f^) result follows.

1of)

=m

o(/(^ 2 (/ ; 2))), and the D

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Relations among the invariants of the stable types in the source

In this section we stablish relations among the invariants of the stable types in the source. The strata in this case are the set of double points, the curve of triple points, the 0-dimensional stable types and the critical points set (C3), minus the previous strata. The situation is less difficult than in the case of the target because the set of double points and the curve of the triple points are ICIS of corank 1 germs . We know from [16] that the absolute polar multiplicities of a hypersurface X with isolated singularity are related to the Milnor numbers j.i^ of the plane sections (n(k)(X] = fi(X n H k ) ) by the following equalities

for 0 < k < d - 1, where d = dim(X). This result is also valid for ICIS (see [10], [7]). The absolute polar multiplicities are defined when the dimension of X is > 1. The multiplicity rnd(X) cannot be defined directly like the other rafc, 0 < k < d — 1, because the singularities of pi\X are isolated points. However, Gaffhey [7] defines this multiplicity for spaces that are ICIS as follows. DEFINITION 3.13 The d-th polar multiplicity of (Xd,G) d), denoted by md(Xd), is defined by md(Xd] = dimc

(Xd is ICIS of dimension

/n

where f : (C n ,0) -> (C n - d ,0), f ~ l ( Q ) = Xd and pl : Cn -* C is a generic linearprojection.

REMARK 3.14 As V(piJ) is ICIS, then by Le-Greuel theorem, we have

md(xd) = When / € (9(3,4) is finitely determined and of corank 1, the double points set D2(f) and the triple points curve D3(f] are ICIS. Therefore we can apply Definition 3.13 and all the properties above to obtain the following. PROPOSITION 3.15 Let f e 0(3,4) be a finitely determined germ of corank 1. Then

(i)

m2(D2(f))

- m!(£> 2 (/)) + m 0 (D 2 m 0 (D 3 (/)) =

These equalities in the above proposition are the same for D^(/), Df(/), and , (2)), therefore we can now deduce our main theorem. Using the results of subsections 3.2 and 3.3, we reduce the number of invariants in Gaffney's theorem 2.2 from 20 to 8 in the corank 1 case.

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THEOREM 3.16 Suppose that f € 0(3, 4) is a finitely determined germ of corank 1 and F = (i, ft) is a good l-parameter unfolding. Then F is Whitney equisingular along T = C x {0} if, and only if, m 0 (/ t (C 3 )), e jD2(/t)(/t) , m 0 ( f ( D 2 ( f t ) ) ) , e I) 3 (/t)(/t) , ml(D2(ft)}, m 0 (/t(£> 2 (/t,2))), m 0 ( f t ( D 3 ( f t } } ) , m0(D2(ft,2}} are constant fort close to the origin. THEOREM 3.17 Suppose that f G (9(3,4) is a finitely determined germ and F an excellent unfolding of f . Then F is Whitney equisingular along T = C x {0} if, and only if, mi(/(C 3 )), the polar multiplicities of the l-dimensional and the 2-dimensional stable types in the source and in the target are constant for t close to the origin.

4

POLAR MULTIPLICITIES OF GERMS IN

In [2] Houston and Kirk classified germs from C 3 ,0 —> C 4 ,0. The models for these germs are the following: Label

Singularity

-

(X,7/,2,0) 2 2

E7 AQ E8 Ak Bk Dk Ck E6 F4

(x,y,z ,z(z +x3 +xy3)) (x,y,z2,xz) (x,y,z2,z(z2+x3+y5}} (x,y,z2,z(z2 +x2 +yk+l) (x,y,z2,z(x2+y3 + z2k}) (x,y,z2,z(z2 +x2y + yk~1} (x,y,z21z(x2 +yz2+yk}} (x,y,z2,z(z2+x3+y4)) (x,y,z2,z(x2+y3 + z4}}

We apply in this section the results of the previous sections to calculate the polar multiplicities of the stable types /(C 3 ) and f ( D 2 ( f } } . PROPOSITION 4.1 Let f : C 3 ,0 -»• C 4 ,0 be one of the normal forms given above. Then we have the following: Label

m 0 (/(C 3 )) '/^(/(C3))

AQ

2

1

Ak Dk E6 E7 E8 Bk Ck F4

2 2 2 2 2 2 2 2

I l 1 1 1 1 1 1

eD*(f}

m0(f(D2(f)J)

ml(f(D2(f)J)

v(D2(f})

-

f

k+l c +2 9 9 10 4fc+l 4k -2 Q

I 1 1 1 1 2 2 2

2 3 4 3 3 2k + 1 3k 6

k k 6 7 8 2k k+l 6

Proof: Let f(x,y,z) = (x, y, z2,z(z2 + x2 + yk+l}} be a germ of type Ak. To calculate mi(/(C 3 )), i — 0, 1, 2, 3 according to Theorem 3.16 and Remark 3.6 we have to calculate only m 0 (/(C 3 )) and mi(/(C 3 )). We calculate first the polar multiplicity m 0 (/(C 3 )):

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m 0 (/(C 3 ))=dimc ( , ® l f t where ps is a generic projection. We take p $ ( x , y , z, w) = (x, y, z). Then m 0 (/(C 3 ))

= dime

o

°33

(*,?y,2 2 )

We compute now the multiplicity mi(/(C 3 )). As pz is generic, we choose the generic linear projection p%(x.y, z) = (x, y). Then m^/OC 3 ))

= dim = d dime *— i

3 /(x,y,z) \

In the same way, we can calculate wo(/(C 3 )) and m-i(/(C 3 )) for the other normal forms above. To calculate the multiplicity Wj(/(D 2 (/))), -i = 0,1, and e D 2(j)(/) we need to find the ideal that defines the curve D2(f). As 2mo(/(I> 2 (/))) = dime foc/T^f)) where pafoj/, z, «;) - (. K ,y) and !?(/) = z2 + :r2 + y*+i , then 2m 0 (/( J D 2 (/))) = dime t^,^^^^) = 2' We now calculate rrii(f(Dl (/))). We have =

dlm

=

dim

=

2.

C (Plop2o/^( (x,z2+x2 + yk + 1,J[x,y,z2 + x2+yk + 1 } )

As the Milnor number of D\(/) = z2 +x 2 +y /c+1 is k, it follows from Proposition 3.8, that e D 2 ( / ) (/) = A; + 1. Let f ( x , y , z ) ~ (x,y,z2,z(z2 + x2y + yk~l). We compute the multiplicities mi(f(D2(f))), i = 0,1, and f-D'2(f)(f}- f°r ^ ms we need to find the ideal that defines the curve D 2 (/). As 2m 0 (/(D 2 (/))) = dime (p20f °J2 (f)) where p2(x,y,z,w) = (x +y,x - y) and Jf (/) = z2 -f x2?/ + y^" 1 , then 2ni.0(f(D2(f))) — dimc r-r—-—?i 2 , We now calculate ?//i(/(D 2 (/))). Choosing p\(x,y,z) = :r + y which is generic, we have =

dime

<

As the Milnor number of D2(f) = z2 4- x2?/ + y^1 is fc, it follows from the Proposition 3.8 that eD*(f)(f) = k + 2. The invariants of the other normal forms are computed in the same way as above. D

Map Germs from C3 to C4

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223

APPLICATION

EXAMPLE 5.1 Let F(t, x, y, z] = (t, x, y, z3 + xz, yz2 + z5 + z6 + tz7}, unfolding of the germ, f ( x , y , z ) = (x,y,z3 + xz,yz2 + z5 -+- z6). Then the family f t ( x , y , z ) = F(t,x,y,z) is Whitnev equisingular for 0 < t < 1/2. According to Theorem 3.16 we have to calculate 7 invariants. We calculate first the multiplicity of the discriminant m0(ft(C3)): rn0(ft(C3)) = dimc 7-^ = dime --•-•—^- - 3, Vt, (P3 oft) (x,y,x * where ps(x, y, z) = (x, y) is a generic projection. Now we compute the invariants eD2(ft)(ft), mo(ft(D2(ft}}) and mi(Z? 2 (/ t )), for this we need the ideal that defines D2(ft), which is

with P4(zl,z2) = Ei=0z4l-kzl, P5(zl,z2) = E5k=0z5l-kzl and P 6 ( 2l , z 2 ) = EL0z16-fcz2fc. We compute now the multiplicity mi(D2(ft}}. As D2(ft) is ICIS and isomorphic to y(zi + z-2) + PI(ZI, z-2) 4- PS (-21, z-z) + tPQ(zi,z-z)}, then for the relation

we have /z( 2 )(D 2 (/ t )) = 1 and fj.W(D2(ft)) = 3 therefore, m l ( D 2 ( f t ) ) = 4 V*. Now to calculate m0(ft(D2(ft))} we apply the formula (p2°/0P,/2(/t))'

choosing the generic projections p(x,y,z\,z<2) = (x,y,z\) and P2(wiiw2t W3iw
~

i.

C

2

therefore m0(ft(D2(ft))) = 4. To calculate eD*(ft)(ft)' tion (p2 o / o p)(x, y, z\, z%) = y and obtain

we

choose the generic projec-

= dime (/22 (/t ),J( P o/op,/2(/ t ))) 2

= 13. In the same way, to calculate eD3 ^ t ), rno(D3(ft)) and mo(ft(D3(ft))), the ideal I 3 ( f t ) that defines D3(ft) is given by V3(gj) where « = 1,2 and j = 1, 2. The !//%) are easily calculated using Maple. Then by theorem 3.10 it is sufficient to compute 3 and »(D3(ft)), and D3(ft) is defined by mo(ft(D (ft))) (x- (ziz2 + z2z3 + ziz3), Zi+z2 + z3, Vf(yz2 + z5 + z6 +tz7),V$(yz2 + z5 + z6 +tz7)).

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So eliminating x and z^ we have, that D3(ft] is isomorphic to the hypersurface denned by -tziz$ + zi+3zlz$ + 2ziz$+2z%Z2-tz%z2+z$ inC 3 , therefore ^,(D3(ft)) = 7, V * ^6,1,1/2,5/4,3/2. Now, as 3\mo(ft(D3(ft)}) = dimc (j3 (/f) ^f l0/t0p ), we choose generic projections p(.T,y, 21,22,23) = (.r,y,2i) and jp(w>i, w 2 ,W3, ^4) = ";2, thus rn0(ft(Ds(ft})) = 1, W ^ 0,1,1/2, therefore, e D 3 ( / t ) (/ t ) = 12, * ^ 0,1,1/2, 5/4,3/2. The calculation for m 0 (//(£) 2 (/ f ,2))) = 4 and m0(D2(ft, 2)) = 2 are similar.

REFERENCES [1] J.W. Bruce, A.A. du Plessis, C.T.C. Wall. Determinacy and unipotency. Invent. Math. 88:521-554. 1987. [2] K. Houston, N.P. Kirk. On the classification and geometry of map-germs from 3-space to 4-space. In: Proceedings of the European Singularities Conference in honour of C.T.C. Wall on the occasion of his 60th Birthday, eds. J.W. Bruce, D. Mond, Singularity Theory, London Math. Soc. Lecture Note Series 263, Cambridge University Press, pp. 325-351, 1999. [3] T. Gaffney, D. Mond. Cusps and double folds germs of analytic maps C2 —> C2. J. London Math. Soc. 43 2:185-192, 1991. [4] T. Gaffney. Properties of finitely determined germs. Ph.D. Thesis, Brandeis University. 1975. [5] T. Gaffney, D. Massey. Trends in Equisingularity theory. In: Singularity Theory, eel. J.WT. Bruce. D. Morid. Cambridge University Press, 1999. [6] T. Gaffney. Polar multiplicities and equisingularity of map germs. Topology 32 1:185-223. 1993. [7] T. Gaffney. Multiplicities and equisingularity of ICIS germs. Invent. Math. 123:209-220, 1996. [8] V.H. Jorge Perez. Polar multiplicities and equisingularity of map germs from C3 to C 3 , Cadernos de Matematica ICMC 1 1:261-280. 1999. [9] V.H. Jorge Perez, D. Levcovitz. Euler obstruction and Polar multiplicities on stable types of map germs from Cn to C p , n < p. In Preparation. [10] L.D. Trang. Calculation of Milrior number of isolated singularity of complete intersection. Funktsional'ny: Analizi Ego Prilozheniya. 8 2:45-49, 1974. [11] L.D. Trang, C.P. Ramanujam. The invariance of Milnor's number implies the invariance of the topological type. Amer. J. Math. 98:67-78. 1976. [12] E.J.N. Looijenga. Isolated singular points on complete intersections. London Mathematical Soc. Lecture Note Series 77, 1984. [13] G.M. Greuel. Der Gauss-Manin Zusammenhang isolierter Singularitaten von vollstandigen Durchschiiitten. Dissertation. Gottingen, 1973.

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[14] W.L. Marar, D. Mond. Multiple point schemes for corank 1 maps. J. London Math. Soc. 39 2:553-567, 1989. [15] D. Mumford. Introduction to Algebraic Geometry. Springer Verlag, 1976. [16] B. Teissier. Cycles evanescents, sections planes et conditions de Whitney. Singularites a Cargese 1972. Asterisque, 7-8, 1973. [17] B." Teissier. Varietes polaires 2: Multiplicites polaires, sections planes, et conditions de Whitney. Actes de la conference de geometric algebique a la Rabida, Springer Lecture Notes, 961 pp. 314-491, 1981. [18] R. Vohra. Equisingularity of map germs from n-space,(n > 3), to the plane, PhD. Thesis, Northeastern University, 2000. [19] C.T.C. Wall. Finite determinacy of smooth map-germs. Bull. London Math. Soc. 13:481-539, 1981.

Topological Invariants of Stable Maps from a Surface to the Plane from a Global Viewpoint D. HACON Dep. de Matematica, PUC-RIO, Rio de Janeiro-RJ, Brazil E-mail: [email protected] C. MENDES de JESUS* Dep. de Matematica, PUC-RIO, Rio de Janeiro-RJ, Brazil E-mail: [email protected] M. C. ROMERO FUSTER** Dep. Geometria y Topologia, Facultad de Matematicas, Universitat de Valencia, Burjasot (Valencia), Spain E-mail: [email protected] We associate to any stable map of a surface M to the plane certain global invariants: the first is a weighted graph which describes the pair M, £/ (where £/ is the singular set of /). The other is a Vassiliev type invariant related to the number of connected components of S/. We describe this invariant in terms of Chincaro's classification singularities of codimension one of maps between surfaces.

1 INTRODUCTION Vassiliev introduced the concept of finite order invariants for isotopy classes of embeddings of closed curves in 3-space. His method [7] [8] can be applied to various types of maps and the definition of the corresponding invariants is based, in each case, on the study of the topology of a discriminant subset determined by the subspace of maps under consideration.

(*) Work partially supported by CAPES, BEX0294/99-3 and CNPq, 142650/97-8. (**) Work partially supported by DGCYT grant no.BFM 2000-1110

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In [2], for instance. Arnol'd defined the three first order basic invariants for stable maps of the circle to the plane. On the other hand. Goryunov [5] studied stable maps of surfaces into 3-space, obtaining all first order local invariants for such maps. More recently, Ohmoto [6], applying ideas of Vassiliev and Goryunov, has given a general definition of first order local invariants for isotopy classes of stable smooth maps (between surfaces). Here we consider the problem of defining global invariants of stable maps from a compact surface to the plane. We introduce invariants defined via the topological classification of the singular set of the map in question (which consists of a number of disjoint closed regular curves in the surface) and its image (which may contain cusps). These invariants depend on the isotopy classes of the singular set in the surface and its image in the plane. This leads to the definition of certain weighted graphs associated to collections of disjoint closed curves in a compact surface. These graphs provide global isotopy invariants of stable maps from a surface to the plane. Here we study some aspects of these invariants, in particular the question of which weighted graphs may be realized by stable maps. The number of edges of a given graph coincides with the number of connected components of the singular set of the corresponding map. This is, clearly, an isotopy invariant of the map. In section 5 we express this number in terms of certain (codimension one) transitions of the singular set.

2

STABLE MAPS FROM SURFACES TO THE PLANE

Let M be a compact, orientable surface with empty boundary. We say that two smooth maps / and g from M to the plane are equivalent (written / ~ g) if there are diffeomorphisms, / : M —> M and k : ]R2 —> JR 2 , such that kf — gl A smooth map / is said to be stable if all maps sufficiently close to /(in the Whitney C00 topology) are equivalent to /. Denote by E/ the singular set of /. We refer to its image /(E/) as the branch set (of /). It is well known that if / is stable then its singularities are folds and (finitely many) cusps ( see [4]). The singular set of / consists of (finitely many) disjoint embedded closed curves in M. On the other hand, the branch set consists of a finite number of immersed plane curves with a finite number of cusps and a finite number of transverse intersections and self-intersections (disjoint from the set of cusps). An isotopy is a smooth 1-parameter family Ft of stable maps, 0 < t < I. The stable maps FQ and F± are said to be isotopic. Isotopy classes correspond to path components of £(M, iti2), the subspace of C°°(M, JR 2 ) consisting of stable maps. Isotopy implies equivalence ([6]) but clearly not vice-versa. If / and g are isotopic then E/ and Eg are isotopic subsets of M (i.e. there is a diffeomorphism of M which is isotopic to the identity and which takes E/ into Eg) and similarly for /(S/) and g(Sg). Thus any isotopy invariant of the singular set or of the branch set of / will automatically be an isotopy invariant of /. In Section 4 we associate a weighted graph to E/ whose isomorphism class is an isotopy invariant of /. As for the branch set, one may, following Ohmoto, attempt to define isotopy invariants using Vassiliev-type invariants. The first global invariant of / that comes

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to mind is /z, the number of curves of £/. In Section 6 we describe /z in terms of local transitions of /(£/).

3

GRAPHS ASSOCIATED TO COLLECTIONS OF CLOSED CURVES ON SURFACES

Since the singular set E/ is an isotopy invariant of stable maps /, the number of connected components of £/ as well as the topological type of its complement in M are isotopy invariants. This information will be encoded in a weighted graph from which the pair M, £/ may be reconstructed (up to equivalence). The complement in M of a collection C of disjoint embedded curves is the disjoint union of connected regions. The frontier of each region consists of a number of (boundary) curves. The weighted graph Q(M,C] is defined as follows: There is a vertex for each connected component of M — C and an edge for each curve in C. The two (or one) vertices of an edge correspond to the two (or one) regions whose boundaries contain the curve in question. Loops can occur. For example, the graph corresponding to a surface with a single (non separating curve) has one vertex and one edge. The weight g(v) of a vertex v is the genus of the corresponding region. We recall that the genus of a surface with boundary is defined to be the genus of the closed surface obtained by adding a disk to each boundary component. For example, the genus of any planar region is zero. Figure (1) shows an example of such a weighted graph.

Figure 1: Weighted graph. Two sets of curves C and C' in a surface M are said to be isotopic if there is an isotopy of M which takes C into C' This implies that there is a diffeomorphism of M into itself taking C into C'. If C and C' are sets of curves in M and M' respectively then we say that the pairs M,C and M',C' are equivalent if there is a difFeomorphism between M and M' taking C to C'. Clearly, equivalent pairs give rise to isomorphic weighted graphs (isomorphism of graphs preserving weights). In fact this defines

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a bijection between equivalence classes of pairs and isomorphism classes of weighted graphs (3.1). THEOREM 3.1 Every weighted graph is isomorphic to the graph of a pair M,C and this pair is unique up to equivalence. The genus g(M) is given by

where a is the number of edges and v the number of vertices, and £ represents the sum of the weights of all the vertices of the graph. Proof: Just embed the graph in 3-space and let MO be the boundary of a suitable neighbourhood of the graph, with curves defined in the obvious way, one for each edge. M is then the connected sum of MO with surfaces of appropriate genus: for each vertex v we form the connected sum of the region of MO corresponding to v with a closed surface of genus g(v}. The weighted graph of the pair Af, C is just the original graph. Conversely, a given pair M, C may be viewed as the connected sum of a pair W, C with surfaces of genus g ( v ) , one for each vertex. The two pairs have the same graph, but in the latter case all the weights are zero. The surface W, being orientable, is diffeomorphic to the boundary of an appropriate neighbourhood of the graph embedded in 3-space. Therefore, by uniqueness of connected sum, the pair M, C is equivalent to the pair obtained by reconstructing the graph as above. From the above description of the pair as a connected sum, the genus of M is the sum of Y^,v g(v] and the genus of MO- But the latter is equal to a — V +1, whence the formula for g(M}. An immediate consequence of this is COROLLARY 3.2 For any pair M,C the surface M is a sphere if and only if the associated weighted graph is a tree all of whose weights are 0. We observe that the case of the sphere is special in the sense that two sets of curves in a sphere are isotopic if and only if they are equivalent. The reason is that any pair M, C can be taken into itself by an orientation reversing diffeomorphism so that if the diffeomorphism of the sphere taking one set of curves to another set of curves happens to be orientation-reversing then it can be replaced by an orientation preserving diffeomorphism which is then isotopic to the identity, so that the two sets of curves are isotopic in the sphere. For surfaces of higher genus, however, isotopy and equivalence are different conditions.

4

GRAPHS ASSOCIATED TO STABLE MAPS FROM A SURFACE TO THE PLANE

Any stable map / gives rise to a weighted graph G(M, £/). Clearly equivalent stable maps have isomorphic weighted graphs. A natural question is now "which sets of curves may be realized as singular sets of stable maps?" In the case of the sphere, any nonempty set of curves can be realized if one allows cusps (if cusps are not allowed the realization question seems to be much harder). Recall that for M to be a sphere the graph must be a tree with all weights

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Figure 2: Orthogonal projection of a bitorus. equal to 0. A tree with one edge is clearly realizable by orthogonal projection of a (round) sphere into the plane. Next consider a tree T with more than two edges. Choose an extreme edge (i.e. one which meets the rest of the tree in only one of its vertices, v say). By induction, the tree minus this edge can be realized. Inside the region corresponding to the vertex v we create a pair of cusps giving rise to a foldcurve (containing the two cusps) bounding a disk which is mapped to a "lips"-shaped region in the plane. This realises the tree T. Consequently we have the following THEOREM 4.1 Any tree all of whose weights are zero may be realized as the graph of a stable map of the sphere into the plane. As for the question of realizing graphs by simple maps (i.e. maps which have no cusps) we offer the following examples. First, it is clear that if T is a tree realized by a simple stable map of the sphere into the plane then the tree obtained by adding a two-edge tree to T along a vertex v of T may also be similarly realized. Modify / inside a small disk contained in the region corresponding to v by introducing two parallel fold curves (without cusps). This has the effect of adding on a two-edge tree at v. Clearly, any tree all of whose weights are 0 and which can be obtained from a one-edge tree by successively adding on two-edge trees as above can be realized without cusps. A second class of examples can be constructed by orthogonal projection of suitably embedded surfaces in 3-space. A simple example with weights 1 and 1 is given by a map of a surface of even genus with exactly one simple fold curve, as shown in figure 2.

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VASSILIEV TYPE INVARIANTS

C°° (M , JfP} is a stratified set. The codimension zero strata are the connected components of £(M, ff?2). The complement of £(M,1R2) (the discriminant) consists of strata of codimeiisions greater than zero. First order Vassiliev invariants are defined as follows. Each codimension one stratum Y receives a ''transition index" £y . Let Ft be a generic path in C°°(M, IR2), i.e. one whose end points FQ = f and F'i = g lie in 8(M,IR'2) and which crosses the discriminant transversely in a (finite) number of points in the codimension one strata. Choose a base point / in £(M,]R2). Since C°°(M,]R2) is path connected (being contract ible), any other g in £(M,1R2) may be reached from / by a generic paths C. Define A^ to be the sum of the transition indices, one for each time the path crosses a 1-stratum Y (multiplied by ±1 depending on which direction the path crosses F).The increment A^ depends in general, on the path C, but depends only on the end point g if and only if the transition indices £y satisfy a compatibility condition around the codimension two strata ( the 1-cocycle condition). The local transitions in the branch set have been described by E. Chincaro [3]. These are illustrated in figure 3. The arrows signify the normal direction in which the number of pre-images increases (by two). Using this classification Ohmoto proves THEOREM 5.1 ([6]) For C°° stable mappings M —>• I?2, the abelian group consisting of all local first, order Vassiliev- type invariants is freely generated modulo constant functions by the following three elements: A/ D = 2(AT° 4- AT1 4- AT 2 ) 4- 2(AC1 + AC 2 ) 4- AS,

A/c = 2(A£ 4- AL 4- A5), A/F = 2 (AT0 + AT 2 ) + AC1 4- AC2. We observe that, except for L and 51, all the transitions shown in figure 3 involve (locally) more than one branch. We shall distinguish here between the cases in which such branches correspond (via /) to one, two or three connected components of E/. The beaks transition is peculiar in that it alters not only the total number of cusps, but also the total number, / M , of connected components of E/. Accordingly, we denote by B+ the beaks transition corresponding to the birth of (a pair of) cusps where the number of components of E/ increases, and by J3_ the beaks transition where the number of components decreases. Figure (4) shows a B- followed by a B+. The remaining transitions, (T°, T1, T 2 , C1, (72, Q°, Q1) acquire a lower index denoting the number (1, 2 or 3) of connected components of S/ appearing' in the transition. Thus

AT1 = AT0 = AT? + AT2°, AT2 - AT? + AT22,

AC1 =

Invariants of Stable Maps from a Surface to the Plane

233

Lips

X S

A

:

Swallowtail

X

•T- 0

X X

T1 : T2

Beaks

:

•Tangency

A

C1:

Cusp crossing

A

2

'"/^\

Triple point Q

X

1

Figure 3: Local transitions of codimension one.

iT^ ^(O

—>

\^^ B + =1

B_ = Figure 4: Beaks.

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AC2 = AQ1 = AQ} + AQj + The main result of this section is the following expression for A/M in terms of transitions. THEOREM 5.2 The increment of /M along any path tranversal to the strata of codimension one in D is given by

A/u = AZ?+ — A£?_ + AL. Proof: It follows from the classification of transitions (see figure 1) that the only transitions altering the number of connected components of £/ are B and L. Now, a positive transition of type B+ increases this number by one, whereas a positive transition of type B_ decreases it by one. On the other hand, a positive transition through the L stratum increases 1^ by one. g REMARKS: 1) The invariants ID, Ic and /M satisfy the relation In -\

Ic 2

+ ID — (1 + #(-M')) mod 2.

This follows from the fact that the only transitions which affect / M , Ic and ID are 5", L and B. It is easily seen that each of these transitions leaves the sum 7M + ^ +/D unchanged mod 2. The value of JM -+- ^ + ID f°r a projection of a surface (embedded in JR3) to the plane is 1 -f p(M) whence the relation. 2) For simple maps, Ic is clearly zero. Furthermore ID is even for simple maps. For, if / is a simple map, the surface M is divided into two regions M+ and M_ whose interiors are immersed (by /) into the plane with opposite orientations. M+ and M_ meet in £/ which is the boundary of both. Let us write H for M+ say (so that d'R- = £/). The boundary 7£ is immersed into the plane with transverse selfintersections . Consider the plane as the boundary z — 0 of the half-space z > 0. From the above immersion of 7£ we may construct an immersion of 7£ into the half space which is equal to the original immersion on dH and meets the plane z — 0 transversely in d'R,. We may further modify the immersion of Tl (in the interior of TV) so that it intersects itself transversely. The self-intersection will be a compact 1-manifold (i.e. a disjoint union of circles and intervals, meeting the plane z = 0 transversely in its boundary which is exactly the set of self intersections of d'R,. Hence d'R, consists of an even number of points as required.

REFERENCES [1] V.I. Arnol'd. Plane curves, their invariants, perestroikas and classifications. Advances in Soviet Math. AMS, Providence, RJ, 21:33-91, 1994.

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[2] V.I. Arnol'd. Topological Invariants of Plane Curves and Caustics. University Lecture Series. AMS, Providence, RI, 5, 1994. [3] E.A. Chmcaro. Bifurcations of Whitney Maps. ICEx, UFMG, 1978 [4] M. Golubitsky, V Guillemin. Stable Mappings and Their Singularities. Springer Verlag, Berlin and New York 1973. •[5] V. Goryunov. Local invariants of mappings of surfaces into threespace. The Arnold-Gelfand mathematical seminars, 223-255, Birkhauser Boston, Boston, MA, 1997. [6] T. Ohmoto. Vassiliev type invariants of order one of generic mappings from a surface to the plane. Topology of real singularities and related topics 55-68 (Japanese) Kyoto, 1997. [7] V.A. Vassiliev. Cohomology of knot spaces. Advances in Soviet Math. AMS, Providence, RI, 1:23-69, 1990. [8] V.A. Vassiliev. Complements of discriminants of smooth maps. Topology and applications, AMS, Providence, RI, 1992.

Cubics in R and C RADMILA BULAJICH Departamento de Matematicas, Facultad de Ciencias, Universidad Autonoma del Edo. de Morelos 62210 Cuernavaca. Morelos, Mexico E-mail: [email protected] LEON KUSHNER Departamento de Matematicas, Facultad de Ciencias, Universidad Nacional Autonoma de Mexico 04510 Mexico, D.F., Mexico E-mail: [email protected] SANTIAGO LOPEZ DE MEDRANO Institute de Matematicas Universidad Nacional Autonoma de Mexico, 04510 Mexico, D.F., Mexico E-mail: [email protected]

1

INTRODUCTION

In this article we study the stratification of the space of complex cubic forms in two variables under the classification by linear equivalence. The corresponding description for the space of real cubics was carried out by E.G. Zeeman [3] in a geometric way. We mainly study the topological characterization of the strata consisting of degenerate cubics and give a geometrical description of them inside C 4 . To present our method we start by applying it to the simpler and known case of real cubics and then proceed to the complex case. The stratification of the real cubics plays an essential role in the theory of singularities of functions, as well as in many of its applications. The name "umbilic bracelet"given by E.G. Zeeman derives from the fact that it gives a geometric description of the relation between the umbilic catastrophes. (See [1],[2],[3]). 237

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CUBICS IN E

We consider the space of cubics in two variables over R, L 3 (R 2 , R) = {ax3 -f /3x2?/ 47x?/2 4- A?/3} (which we will identify with R4 by taking the coefficients as variables) and the group of linear R- isomorphisms acting on it on the right. There are exactly five orbits, the orbits of: 0, x3, x 2 //, x(x 2 ± t/2), which we will denote by 0, Oa, O2 and O± , respectively. Let 0.3 2 = ^3 U O2 • The orbit Os is the set of cubics of the form (ax+by)3 with a, b e R and a2+b2 ^ 0. and it can be simply parametrized by the pair (a, 6). Therefore it is diffeomorphic to Sl x R. The orbit O2 is the set of cubics of the form (ax + by)'2(ex + dy) with a, 6, c, d 6 R and ar- — bd ^ 0. Developing the product (ax 4- by)2 (ex 4- d?y) one sees that O2 is the image of the map h2 : {(«, ft, c, d) e R4 : ad - 6c ^ 0} -> R 4

4- a

One computes easily that this mapping has rank 3 everywhere, so it can not be one-to-one. If two points (a,fe,r, d) and (a1 ,b',c',d') have the same image under h, then they give two cubics with the same zeroes, and so (a, b) — t(a',b') and (c, d) — s(c'.d') , ,s%£ ^ 0 . From the equality of the coefficients we have s = t~2. So the fibers of h^ are parametrized by

t ~* (at, bt, c.r2, dt'2) ,

for t ^ 0.

In each fiber there are exactly two points closest to the origin. They correspond to the values

+,,2

and satisfy the equation a 2 4- b2 — 2(c2 + d2). 80/12 restricted to the set X<2 = { (a, b, c, d) G R4 : ad -be ^ 0, a 2 +b2 = 2(c2+d2)} is two-to-one. To get a one-to-one map we consider the quotient of X^ by the map (a, 6, c, a1) —* (—a, —6, c, d), which we denote by X%. Since /?,2 has rank 3 everywhere, the induced map h2 : Xi -> 02

is a diffeomorphism. To describe the union Os.2 of these two orbits, we extend the parametrization h2 to a parametrization h^^2 defined in the set {(a, 6, c, d) £ R4 : a2 + b2 ^0, c2 +d 2 ^ 0} by the same formula and again restrict to the set X$t2 = {(a, 6, c, d) 6 R 4 : a 2 + fr2 = 2 (c2 + d 2 ) 7^ 0} . When we pass to the quotient ^3,2 by the antipodal map we obtain a homeomorphism ^3,2 : ^3,2 ~^ ^3.2-

The restriction of ^2 to the subspace X^ where ad— be = 0 gives a diffeomorphism hs : Xs —> Os which differs from the previous parametrization by a constant factor.

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The map h^^ is not a diffeomorphism, because ^3,2 has rank less than 3 precisely at the points where ad — be = 0 . Now take a point p in this set and a vector v transversal to it at p which is in the kernel of Dhz^ and consider the straight line p+tv. Then h^^(p+tv] is a simple cusp (equivalent to t —>• (t2, t3,0,0)). Therefore, at the points of X3 , the pair (R 4 ,0 3i2 ) is locally diffeomorphic to the pair (R 2 , a) x R 2 , where a is the curve x\ + x\ = 0. To describe more precisely the topology of Os^ we consider the intersection X^p = -^3,2 n S3. Since Xz,2 is conical we have that ^3,2 is homeomorphic to ^3,2 x R. The set A"s,2 is given by a2 + b2 = | and c2 + d2 = |, so it is homeomorphic to the torus Sl x S1. The quotient is obtained by identifying antipodal points in the first factor S1 of the product, so it is again homeomorphic to the torus. It follows that Xz,2 , and therefore also Os,2, is homeomorphic to 51 x S1 x R. If we remove from X^,2 the points where ad — be = 0 we obtain two copies of S1 x R . It follows that O^ is diffeomorphic to S1 x R 2 . The beautiful geometry of these orbits inside R4 is well-known [3]. We give here our own description of this geometry: Consider the isometry T : R4 —> R4 given by the matrix

Consider also the orthogonal projection TT from R to the subspace of points of the form (a, 0,0,<5). Then we have: The restriction of TV to T(O2,s) is a fiber bundle over R2\{0} with fiber the deltoid and structural group Zs. In fact, every fiber is the image of the standard deltoid under an affine conformal transformation, and all the fibers over the points of a fixed circle centered at the origin are isometric. This assertion follows from the fact that the set T(O2,s) can be parametrized by the mapping sin.(0))/2,psin(0)). Therefore the fiber over each point (/?cos(0), psin(0)) with p ^ 0 is a deltoid which can be shown to be the image of the standard one (cos(— 2<^) + 2 cos (
3

CUBICS IN C.

Our aim now is to describe topologically the stratification of the space of complex cubic forms in two variables. We consider the space of cubics in two variables over C, L 3 (C 2 ,C) = {az3 +(3z2w + 7zw2 +6w3} , which we will identify with C 4 , and the group of linear C-isomorphisms acting on the right. There are exactly four orbits,

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the orbits of: 0, z3, z2w, z(z2 + w 2 ), which we will denote by 0, 0s, 02 and O\ , respectively. Let (93,2 = 03 U O2 . The orbit 03 is the set of cubics of the form (az+bw)3 with a, b 6 C and aa+bb ^ 0. The pair (a, b) does not in this case give a good parametrization, because for p a cubic root of unity, (a, 6) and (pa, />6) give the same cubic form. Nevertheless, once we divide by this action of Zs the parametrization becomes bijective. If we consider only the points (a, 6) in the unit sphere ,93, the quotient is the lens space £3 . Therefore O$ is diffeomorphic to LS x JR. We now describe 032 using the map ^3/2 : {(«, b, c, d) G C4 : aa + bb ^ 0, cc + dd ^ 0} —> C4 with the same formula as in the real case: h3/2(oj),c,d) = (a2 c, 2 a be + a2d,2abd + b2c,b2d). By the same argument as before, the fibers of this map are parametrized by r —>• (ar, br, c'r~ 2 , dr~2J where now T €E C\{0}. In each fiber, the set of points closest to the origin is a circle that corresponds to the values of T such that _ i + dd' \aa + bb

and they satisfy the equation aa + bb — 2(cc. + ddJ. So the restriction of ^3,2 to the subset Z3,2 = {(a, 6, c, d) e C4 : aa + 66 = 2 (cc + dd) 7^ 0} has compact fibers homeomorphic to Sl . To get a one-to-one map we need to identify each circle to a point, or, equivalently to take the quotient of Z^^ by the action of S1 given by gu(a,b,c,d) = (au,bu,cu~2,du~2}. Denoting this quotient by ^3,2, the induced map

is a homeomorphism. When we restrict ^3,2 to Zs — ^3,2 H {(a, 6, c, d) : ad — 6c = 0} and pass to the corresponding quotient we obtain a diffeomorphism

which differs from the previous parametrization by a constant factor. The restriction °f ^3,2 to the complementary Z<2 = Z^^ H {(a, fr, c, d) : ad — be ^ 0} again induces a diffeomorphism

The map h^^ is not a diffeomorphism, because ^3^2 has rank less than 3 precisely at the points where ad~bc = 0. Again, a complex line transverse to this subset whose defining vector is in the kernel of Dh^ 2 is sent by the map as a simple cusp (equivalent to T -» (r 2 ,r 3 ,0,0)). Therefore, the pair (C 4 ,0 3 ,2) is locally diffeomorphic at the points of 03 to the pair (C 2 ,a) x C 2 , where a is the curve z2 + z\ — 0 , which is well known to be a cone on the trefoil knot. Therefore the pair is not locally flat topologically.

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To describe more precisely the topology of 03,2 we intersect the conical set £3,2 with the unit sphere 57, thus obtaining a compact manifold Z3,2 such that ^3,2 = .£3,2 x R . This intersection is given by aa + bo = | and cc + dd — |, so it is diffeomorphic to the product S3 x S3. We claim that the quotient of £3,2 by the action gu is diffeomorphic to the product S2 x S3. This will follow from the case k = —2 of the following more general lemma: LEMMA 3.1 Let gu,k be the action of S1 on S3 x S3 given by gu,k(p,q) = (up,ukq}. Then for any integer k the action gUik is smoothly conjugate to the action gUiQ which is the Hopf action on the first factor. Therefore, for any integer k, the quotient of S3 x S3 by the action gu^k is diffeomorphic to S2 x S3. Proof: Define a map Fk : S3 -> S3 by

Fk(z,w) = for

Considering now S3 as the group of unit quaternions, F^ satisfies Fk(up) u Fk(p) for u € S1 and p e S3. Define the map $fc : S3 x S3 -> S3 x S3 by k

Then &k is a diffeomorphism which satisfies $k(9u,k(pi (f^qp~l~) sends this difference to S3 x S1 and under the induced action of S1 any point in the complement S3 x (S3\Sl) is equivalent to a unique point of the form (p,w + rj} where w is complex and r > 0. It follows that the quotient of Z-2 by the action of gu is diffeomorphic to the product of S3 with an open disk and that Z^ is diffeomorphic to 5"3 x R3. Thus we have shown: THEOREM 3.2 The topological type of the orbits of the degenerate complex cubics is as follows: (i) 0$$ is horneomorphic to the product S2 x S3 x R, (ii) Os is diffeomorphic to LS x R, (Hi) O-2 is diffeomorphic to S3 x R 3 , (iv) O^.2 is smooth at the points of O<2 while at the points of O% is not locally flat, being transversally diffeomorphic to the cone on the trefoil knot. The geometry of these orbits inside C4 cannot be as nice and simple as in the real case: The set 02,3 does not project as nicely to a complex plane of C 4 , since it necessarily intersects the orthogonal subspace. Moreover, the fibers cannot be compact. Actually, not much seems to be gained by applying first a rotation (as we did in the real case), so it is simpler to project directly:

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Consider the closure of 02,3 in C4, that is, (92,3 = ^2,3 U {0} and the projection of C4 onto the subspace of points of the form (a%0, 0, 6). Let TT be its restriction to 02.3. Let (f> : C* —> C 2 be given by , , ,2a3-fl a 3 + 2 „(„) = ( _ _ - _ ) . Its image is the complex deltoid A, the real one being given by taking a in the unit circle. Then we have: THEOREM 3.3 The .fiber* of the projection TT : <52.3 —> C 2 are as follows: (i) The generic fiber Tr*1 (a, fi) for aft ^ 0 is the -image of the complex deltoid A 6?y an element in the group o/ 2 x 2 diagonal complex matrices. Moreover, TT restricted to the set ^2,3 n {a6 7^ 0} is a /i&er bundle with fiber A ant? structural group Zs. (MJ T7?,e exceptional fiber TT~l(a. />) /or' «<*! = 0, a, <*> not bot/i zero, is /Ae union of a parabola and a double line tangent to it. (in) The exceptional fiber Tr" 1 ^, 0) is £/&e union of two double lines, (iv) The restriction of TT to O% is a 3-fold, branched, covering over C2\{0}, branched over the set ato = 0. Restricted, further to the unit sphere, it is a 3-fold branched covering of the lens space L% onto a S-sphere. Proof: (i) The set (^2,3 fl {aft •£ 0} can be reparametrized by setting a — l,c = a,b = £,,d — b/£2 in /?3,2 to obtain the one-to-one parametrization

£

'

Here 7r~ 1 («,^) is the curve given by the second and third coordinates, which resembles the complex deltoid. To see that it is a linear image of A it is better to use a second parametrization, by setting a = r, b = sa. c = r, d = s/a2 in h^^ to obtain - 1 ^ 2 (T + 2

. rs

3

, ,s ).

This is a nine-to-one parametrization of O-2.3 D [06 ^ 0} which has the advantage that now it is clear that the fiber over a point (r 3 , .s3) with {rs ^ 0} is the image of A under the linear map with matrix r2-s 0

0 ,s2r

It follows also that TT restricted to the set (92,3 H {a8 ^ 0} is a locally trivial fiber bundle with fiber A . When we follow a loop in the base space that winds around the set a — 0 the value of r is multiplied by a cubic root of unity p and the deltoid is rotated by the matrix ' p2 0 0 p The effect of a loop that winds around the set 6 — 0 is a rotation by the inverse of this matrix (which is of the same form) arid so the structural group of the bundle is the group of order 3 formed by those matrices.

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(ii) Here it is better to consider the equation of the set (92.3 given by setting the discriminant of the cubic equal to zero: = 0.

When a = 0 and 6 has a non zero fixed value, this equation reduces to =0,

which represents the union of a parabola and the double line ft — 0 tangent to it. The case a ^ 0 and 6 = 0 is analogous. (in) When a = 6 = 0 the equation reduces further to

/*V = o, which represents the union of two double lines. (iv) Observe that Os is the image by $ of the set where a is a cubic root of unity, but it is enough to consider the restriction of <& to the set a = 1, which is three-to-one. The map TT restricted to Os is also three-to-one and a local diffeomorphism, except at the set aS = 0 where it is one-to-one and is equivalent transversely to the map r —> r3. When restricted to the unit sphere these maps are respectively: the covering S3 —» L% and a three-to-one covering £3 —* S3, branched over two circles. D

REFERENCES [1] V.I. Arnol'd, A.N. Varchenko, S.M. Goussein-Zade. Singularites des Applications Differentiables, Editions Mir, Moscou, 1986. [2] R. Thorn. Structural Stability and Morphogenesis. An Outline of a General Theory of Models. W.A. Benjamin, Inc., 1975. [3] E.G. Zeeman. The Umbilic Bracelet and the Double Cusp Catastrophe. In: Catastrophe Theory, Selected Papers 1972-77, Addison- Wesley, 1977, pp. 563601.

Indices of Newton Non-Degenerate Vector Fields and a Conjecture of Loewner for Surfaces in R4 CARLOS GUTIERREZ Institute de Ciencias Matematicas e de Computagao, Universidade de Sao Paulo, Departamento de Matematica, Caixa Postal 668, 13560-970, Sao Carlos, SP, Brazil, E-mail: gutp©icmc.sc.usp.br MARIA APARECIDA SCARES RUAS Institute de Ciencias Matematicas e de Computagao, Universidade de Sao Paulo, Departamento de Matematica, Caixa Postal 668, 13560-970, Sao Carlos, SP, Brazil, E-mail: [email protected]

Abstract

We study the index of a vector field in M 2 , with isolated singularity, in terms of conditions on the Newton polyhedra associated to its coordinates. When the vector field is Newton non-degenerate, we show that its index is determined by the principal part of the Newton polyhedra. As a consequence we can prove that, under very mild conditions, the index of an isolated inflection point of a locally convex surface generically embedded in R4 is the same as the index of an umbilic point of a surface immersed in R3.

Research of the first author was partially supported by PRONEX/FINEP/MCT grant # 76.97.1080.00 and research of the second author was partially supported by CNPq-grant #300066/88-0 and FAPESP, grant # 97/10735-3. 245

246

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INTRODUCTION

This article has been motivated by the classical (local) Loewner's Conjecture which states that every umbilic of a smooth surface immersed in R3 must have index less than or equal to one. It was shown in [3] that the index of an isolated inflection point (in the sense of Little [8]) of a locally convex surface generically embedded in R4 is ±1/2. Our first result. Theorem 1, was inspired in the result of M. Brunella and M. Miari [2], in which they proved that a Newton non-degenerate vector field in the plane possessing characteristic orbit, is topologically equivalent to its principal part. Under weaker nondegeneracy conditions given in terms of the Newton Polyhedra associated to the coordinates of the vector field, we prove that its index is determined by its principal part. We believe that our conditions are better suited for the case of indices of vector fields. Also, as it should be, proofs are much shorter. In our second result we show that, under very mild conditions given in terms of Newton Polyhedra, the index of an isolated inflection point is the same as the index of an umbilic point of a surface immersed in R 3 . It may be possible that the Loewner Conjecture can be extended to the case of isolated inflection points of locally convex surfaces in R 4 . The authors wish to thank the referee and J. Llibre for their comments, which have been taken into account in this paper.

2

INDEX OF A VECTOR FIELD AT A SINGULARITY

Let £n be the set of smooth germs g : (R n ,0) —> R. To g e £n associate its formal Taylor series expansion at 0: g(x] = ^akxk. Define supp g = {k e Zn : ak ^ 0}. Given A C Z n , we define g & = ^CjtcA akxk• The Newton Polyhedron of g 6 En , denoted by F + (#), is the convex hull in R™ of the set U{k + v : k E supp g, v G R" }. The Newton Diagram of g is the union r(_g) of all compact faces and vertices of T+(g). Observe that if g =£ 0 and T+(g] has no compact faces, T(g) will be the only vertex of F+(.g); also, if g = 0, then

r+(.9) = 0.

Let X = (/,
Indices of Newton Non-Degenerate Vector Fields

247

face of F+ (/) (resp. of T+(g}) and let L be the only straight line containing A; if L n F+( (R 2 ,0) be the germ, of a smooth vector field in the plane. Let Xp = (f \ r ( f } i 9 \T(n} ) ^e ^e Principal part of the vector field X. If X is Newton non- degenerate, then, there exists a neighborhood U of (0,0) such that, V(a:,y)e[7\{(0,0)'} a n d V s e [0,1],

:= sf(x,y) + (1 - s) f

r(f)(x,y)\

+ \sg(x,y) + (1 -

s)g\r(g)(x,y)\

In particular (0, 0) is an isolated singularity of both X and Xp and Index (X,0) = Index (X r ,0) Proof: We shall only consider the case in which / ^ 0 and g ^ 0. The proof in the other case is similar. Let S+ = {(a,&) E R 2 : a 2 + b2 — 1, a > 0, b > 0}. Given (a, b) e 5| and h e {/,/ r ( / ) > ^ . 5 l r ( < y ) } » denote by A(a,6) = A h (a,6) = {(x,y) E T + (/i) : ax + by = inf{au + bv / (u,v) € F+(/i)}} As F + (/i) is a convex subset of E^ , we obtain: (la) A(a, 6) is either a vertex or a compact face of F(/i); (Ib) if (mo, no), (mi, ni) E A(a, 6), then (m 0 ,no) • (a, &) = (mi,ni) • (a, 6), where "•" denotes the usual inner product ofR 2 . (Ic) if (mo, no) E A(a, 6) and F + (/i — /iJA(a,6)) is n°t empty, then, there exists e(a, 6, /i) > 0 such that (mo, n0) • (o,b) -fe(a,6,/i) = inf{(m,n) • (a,6)/(m,n) E F + (/i - /I|A( O ,&))}-

This proof is organized as follows: We shall find, for each (a, 6) E 5^U{(0,1), (1,0)}, a small curved sector of R2 having its vertex at the origin and such that X restricted to this sector satisfies the theorem. By local compactness, we will be able to cover a neighborhood of (0,0) E R 2 with finitely many of these sectors. In this way X restricted to this neighborhood of (0,0) will satisfy the theorem. We shall accomplish this by studying, when h E {/,
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(i) A(a, 6) is a compact face of T(h) non-reduced to a point (and so, (a, ft) e S\.)\ (ii) (a, 6) 6 S]_ and A(a, 6) is a vertex of F(/i); and (iii) A(l, 0) and A(0, 1) are one-point-sets (vertices of F(/i)). To simplify matters, we shall continue by assuming that

(2) r + (/i Suppose first that A (a, b] is a segment and let (M, n) and (rn,N) be its endpoints. We shall assume 0 < m < M and 0 < n < N. Let t = f(u, ?;,/?) be defined in Bs(a,b) = {(M,?;) € Si : |(w - a, 7; - 6)| < £} by f (?/,, ?;, /A) = inf {(//, i/) • (w, ?;) : (//, i/) e T+ (/i - /iU(a,6))} ) • (7/,7;) : (//, z/) G A(a, 6)}. If <5 > 0 is small enough, then (.(u,v, h) depends continuously on (77,, v) and (3)

3 5 f.(u, v, h)^(- e(a, 6, /»,), - f (a, 6, /i)); This implies that, for all (u,v] e Bs(a,b), (4)

(3/4) f (a, 6, /;,)+77, • A:f + 7; • n < f.(u, 7;, /i) + 7.1 • Af + 7; • ?v, < f (u, 7;, /i) + sup{(/i, i/) • (u, 7;) : (/^, i/) e A(a, 6)} = inf {(/*,z/) • (71,7;) : (/y,, z/) 6 r+(/». - //J A (a,6))} Therefore, if <^ > 0 and a > 0 are small enough, there is a continuous function R(t,u,v,Ji) defined for all (t,u,v) e [0, a) x B6(a,b), such that: (5) /,,(r, ^) = 0.m,N tU-m+V'N + •••+ 0 M ,n f u ' M+t " n

Let /? = /?(?/) be defined in {7;, < a : (u, v) e -£?#(«, 6)} by the equation (6) u — a p(u)

Also, let k = A;(/, 77,, 7;) be defined in (0, a) x {(w, 7;) <E J5^(a, 6) : 77, < a} by the equation (7)

tv =ktu As p(?/) < 1 and 7; > 6, we obtain that 0 < k(t,u,v] < 1. Therefore, using (Ib)-(lc) and (5)-(7), we obtain that, for all (/-,w,?;) <E (0,a) x {(w, 7;) € Bg(a,b) : u < a},

Indices of Newton Non-Degenerate Vector Fields

249

(8)

This last expression shows that, under conditions above (in particular when t > 0 is small) ,

can only be zero nearby the real roots of the polynomial, in the variable K, a m> jv K

~n + • • • + a.M,n

Suppose now that A(a, b} = {am^} = {dM,n} '1S a one- point-set. Then, proceeding as above we shall obtain the following relation which corresponds to item (8) above under the same assumptions: (8')

= (8 h((t")*,k (t')b) + (1 - 3} h r(h) ((tP)°, k (*')*))

= (kn (tP)a-M+b-n) (aM,n + s t(l/2^a'b^R(t, w, v, h}

This expression shows us that, if 6 > 0 and a > 0 are small enough, then for all (t,u,v) € (0, a] x {(u,v) £ Bs(a,b) : u < a},

Therefore, using the fact that X is Newton non- degenerate and by extending the arguments above to {(u,v) E Bf(a,b) : v < 6}, we obtain, (9) If (a, 6) E -S+, then there are 6 > 0 and a > 0 such that, V(w, v) E 5$ (a, 6) and

*") + (1 - *)s|r(p)(*V)l > 0Now, if A(l,0) = (0, -/V), then, it may be seen that there exists p > 0 such that for all 0 < \y\ < p and -1 < r < 1,

h(r yN+\y) = a0,NyN + yN+lR(y, r} for some continuous real valued function R = R(y,r). It follows that \h\ restricted to a set of the form {(x,y) '• 0 < |y| < p, —pN+l < x < pN+l} is positive.

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Proceeding similarly when A(0, 1) = (M, 0) and using the fact that X is Newton non- degenerate, we may conclude that there exists p > 0 such that, for all .s E [0, Ij, the restriction of .s //, -f (1 — s) /i|r(/i) to the set £(/>), which is the union of

and

{(.r,y) : 0 < is positive. Using the fact that {(a, ft) E R2 : a2 + b2 = 1, a > 0, ft > 0} is compact and assuming that p > 0 is small enough, we may cover a neighborhood of (0,0) in { ( x , y ) E R 2 : x > 0,;;; > 0} with the union of (0,0) U S(p) with finitely many sectors of the form {(x, ? y) E R 2 : x = tu,y = tv , (M,V) E Bs(a,b),Q < t < a]

and for which (9) is true. Similarly, we may cover a neighborhood of (0, 0) in R 2 with the union of (0, 0) U E(p) with finitely many sectors of one of the forms

{ ( x , y) E R 2 : x = tu {(a:, y) E R 2 : x = ~tu {(x, y} E R 2 : x = -tu

, , ,

y = tv y = f; // = -tv

(M, v) E £6 (a, 6), (u, v] E 56(o, 6), (M, ?») E B6(a, 6),

, , ,

and for which the statement that corresponds to (9) is true. The theorem follows from this. D

LOCALLY STRICTLY CONVEX SURFACES OF E4 The asymptotic line fields associated to an embedding / of a surface M in studied in [3]. A generically embedded surface /(A'/) always has an open region at which there are defined two line fields V\ and Vi of asymptotic directions. These two fields eventually collapse onto a unique one over a curve in M. Their integral lines are called asymptotic lines and their singular points are the inflection points of the embedding as defined by Little in [8]. The geometry of a generically embedded surface in the neighborhood of an inflection point was studied in [9]. The differential equation of the asymptotic lines ([3]) of M is the following binary equation:

|

fix

hx

/3.T

/4 X

0

f l y

fly

/3y

/4 y

0

11 xx I hxy I Jlyy where (/i, /2, /s, /i) are the

= 0; f z x x hxx /4X.T dy hxy foxy f^xy ~dxdy J^yy J'^yy J^yy coordinate functions of the embedding.

(3.1)

In this section we study singular points of this equation when M is locally strictly convexlv embedded in R 4 .

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DEFINITION 3.1 A hyperplane H in R4 is a nonsingular support hyperplane to M at the, point p if it is tangent to M at p, M n H = {p} and p is a non-degenerate critical point of the linear projection -n : M —> LH, where LH is the line in R 4 , passing through p, orthogonal to H. DEFINITION 3.2 The embedding f : M —» R4 is locally strictly convex if every point admits a nonsingular support hyperplane. When the embedding / : M —-» R4 is locally strictly convex, the pair of transversal foliations on M induced by the asymptotic lines are -away from a discrete set of singularities- globally denned ([3]); their singular points are isolated and coincide with the inflection points of imaginary type. We shall study the indices of these inflection points with respect to either of the asymptotic foliations. The index does not depend on the specific foliation Aj because A\ and A-2 are transversal to each other in the complement of the inflection points. This section is motivated by a famous conjecture due to Loewner which states that -with respect to either of the foliations induced by the principal lines of curvature of an embedded surface in R3 - there are no umbilics of index bigger than one. This Loewner Conjecture has been asserted to be true for analytic surfaces by several authors among which H. Hamburger [6], G. Bol [1], T. Klotz [7] and C. J. Titus [10]. In the following we shall show that under very mild conditions, the index of an isolated inflection point - with respect to A\ - is the same as the index of an umbilic point of a surface immersed in R3. The following lemma gives a useful normal form for M in a neighborhood of an isolated inflection point of a strictly convexly embedded surface. LEMMA 3.3 Let M be locally strictly convexly embedded in R4 and p G M be an isolated inflection point. Then, up to a rigid motion of R 4 ; it can be assumed that p — (0,0,0,0) and that M, around p, admits a parametrization of the form a: (x,y) -> (x,y,-(x2+ y2) + F(x,y),G(x,y)), where J 2 F(0,0) = J 2 G(0,0) = 0, Proof:

We can assume p = (0,0,0,0) and that the embedding is in Monge's form a : (x,y) -> (x,y,/ 1 (a:,y),/2(a;,y)),

where J1 fi (0,0) = Jl fi (0,0) = 0. Since p is an inflection point of the embedding, the second fundamental form has rank 1 at p. Hence, the 2-jets of /i and /2 are linearly dependent at zero. By assumption the embedding is locally strictly convex, hence by a rigid motion in R 4 , we can assume that M has the desired normal form. D For the parametrization a of Lemma 3.1, the differential equation (3.1) takes the form dy2 J-1 4i -1Fxx

—dxdy -*Fxy

This equation may be rewritten as:

dx2 •*-1 -Ii rFyy

= 0;

(3.2)

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(Gxy + A] 2 (F, G))dx2 where

A A

+

t L1 (~i\ 1 2 \ ; */

—

t J? (~*\ 13 V ' /

A 2 3 ( F G)

— =

(Gin/ - Gxx + Ai 3 (F, G)dxdy (~i L"1 ^rxyrxx

(3 3)

'

C* T? ^xxrxyi

C1 C< _ (~* /? *xx^-'yy ^~*xx^yyi

GXyFyy

—

GyyFxy.

THEOREM 3.4 Let F,G : R 2 ,0 -»• (R,0) fte germs of smooth functions such that J 2 F(0) = J 2 G(0) = 0. Let A",X 1 ,X 2 6e ^erm.s of vector fields in R 2 ,0 de/irted by

G:rv + 2A12(F, G)), GX?/ + 2A23(F, G)); Suppose that X is Newton non-degenerate and that Xp — (^i)r = (^2)rThen (i) the index at 0, of (either of the two foliations induced by) equation (3.3), is the same as the index at 0 of (either of the two foliations induced by) equation Gxydx2 + (Gyy - Gxx)dxdy - GXydy2 = 0.

(3.4)

This common index is half of the index of X at 0. (ii) there exists a smooth surface in R3 such that the differential equation of its principal lines of curvature -in a suitable coordinate system- is precisely equation (3.4). Proof: By the same argument as that of the Proof of Theorem 2.1, there exists a neighborhood V of 0 E R 2 such that, for all -s £ [0, 1], and -when restricted to V- the vector fields

), 2G xy + 2A 12 (.sF,G)), *2a)

=

(3.5)

(Gr.x - Gyy ~ A13(«F, G), 2Gxy + 2A23(*F, G)) ,

have 0 as the only singularity. Therefore, for all -s £ [0, 1] and for every point of V\{0}.

(Gxx - Gyy + A13(-sF, G)) 2 + ( 2Gxy + A12(.s-F, G) )( 2Gxy + A23(*F, G) ) > 0. This and Hornotopy Theory [5] imply that, given ,s 6 [0,1], the index at 0 of equation (Gxy + A]2 (.s-F, G))dx2

+

(GVy - Gxx + A13 (.sF, G)dxdy

+(-Gxy - A 23 (.sF,G))dy 2 = 0,

,

}

(6

*>

does not depend on ,s. Taking .s = 0, 1, the first statement of item (i) follows. It is not difficult to check the second statement of (i) (see, for instance, [15]). The proof of (ii) can be found in [4] . D In the following corollary we show that the above result always holds when the germ G is quasi- homogeneous arid X is Newton non-degenerate. We recall that the

Indices of Newton Non-Degenerate Vector Fields

253

germ G : E2, 0 —> R, 0 is quasi-homogeneous with weights w = (wi.w^} 6 N2 and quasi-degree d € N, if G(XWlx, Xw*y) = XdG(x, y), for all A > 0. COROLLARY 3.5 Let G, j' 2 G(0,0) = 0, be a quasi-homogeneous polynomial defining the vector field X — (Gxx — Gyy, 2Gxy) which is Newton non-degenerate. Then, the index of equation (3.3) does not depend on F : R 2 ,0 —» R, 0, for any F such that j 2 F(0,0) = 0, and it is half of the index of the vector field X at 0 . Proof: Let us assume that G is quasi-homogeneous with weights (cn,/3), a < f3, and quasi-degree d. Then, while the function 2Gxy is clearly quasi-homogeneous of type (a,/?;d — a — /?), this is not always true for the function Gyy — Gxx. However, a direct analysis of the possibilities shows that in any case, the remainders Ai 2 (F, <7), Ai3(JF, G), A23(F, G) satisfy the hypothesis of Theorem 3.2. D

REFERENCES [1] G. Bol. Uber Nabelpunkte auf einer Eiflache. Math Z. 49:389-410 1943/1944. [2] M. Brunella, M. Miari. Topological equivalence of a plane vector field with its principal part denned through Newton Polyhedra. Jour. Diff. Eq. 85:338-366, 1990. [3] R.A. Garcia, D.K.H. Mochida, M.C. Romero Fuster, M.A.S. Ruas. Inflection points and topology of surfaces in 4—space. Trans. Amer. Math. Soc. 352:30293043, 2000. [4] C. Gutierrez, F. Mercury, F. Sanchez-Bringas. On a Caratheodorys Conjecture: analyticity versus smoothness. Experimental Mathematics, 5:133-38, 1996. [5] V. Guillemin, A. Pollack. Differential topology. Englewood Cliffs, N. J., PrenticeHall, 1974 [6] H. Hamburguer. Beweis einer Caratheodoryschen Vermutung. Ann. of Math. 41:63-68, 1940. II, III, Act a Math. 73:174-332, 1941. [7] T. Klotz. On Bol's proof of Caratheodory's Conjecture. Comm. Pure Appl. Math. 12:277-311, 1959. [8] J.A. Little. On singularities of submanifolds of higher dimensional euclidean spaces. Ann. Mat. Pura et Appl. 83 4A:261-336, 1969. [9] D.K.H. Mochida, M.C. Romero Fuster, M.A.S. Ruas. The geometry of surfaces in 4-space from a contact viewpoint. Geometriae Dedicata. 54:323-332, 1995. [10] C.J. Titus. A proof of a conjecture of Loewner and of the conjecture of Caratheodory on umbilic points. Acta Math. 131:43-77, 1973.

Generic Singularities of //-Directions LUIS FERNANDO O. MELLO Institute de Ciencias Escola Federal de Engenharia de Itajuba 37.500-000, Itajuba, MG, Brazil E-mail:[email protected] Dedicated to Professor Mauricio M. Peixoto on the occasion of his 80th birthday Abstract 4

On immersed surfaces in /f we can consider the H-lines, which are characterized by the condition that their tangent lines are carried over the directions of the mean curvature vector field. We call the singularities of these lines Hsingularities. In this paper we analyze the //-singularities and we show that their generic local configurations are analogous to Darbouxian umbilic points on surfaces immersed in R3.

I

INTRODUCTION

The local configurations of singularities of line fields on surfaces in R4 are known in only very few cases: the local configurations of the axiumbilic points, which are the singularities of the axial curvature direction fields, studied by Gutierrez, Guadalupe, Tribuzy and Guinez in [4] and by Garcia and Sotomayor in [3], and the local configurations of inflection points, which are the singularities of the asymptotic direction fields, studied by Garcia, Mochida, Fuster and Ruas in [5]. For the study of the theory of surfaces in R4 see the papers of Wong [8], Little [6] and Asperti [1]. In this paper we study the //-singularities on oriented smooth surface immersed in R4. They consist of the singularities of the //-lines, which are characterized by the condition that tangent lines are carried over the directions of the mean curvature vector field. This paper is organized as follows: * This work is supported in part by CAPES-Brazil. 255

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Mello

In section 2 we give the precise definitions of //-lines and their singularities, the //-singularities. We analyze the differential equation of the //-lines and we show that it fits in the class of quadratic or binary differential equations. In section 3 we characterize the //-singularities which are locally structurally stable. The characterization is expressed in a Monge chart. These singularities are analogous to Darbouxiari umbilic points on surfaces in /?3 and are called //-singularities of S-type.

2

DIFFERENTIAL EQUATION FOR #-LINES

Let a : M2 —> R4 be a Cr , r > 4, immersion of an oriented smooth surface into J?4, which is endowed with the euclidean inner product (-, •) and oriented by a once for all fixed orientation. Denote respectively by TM and NM the tangent and the normal bundles of a and by TPM and NPM the respective fibers, i.e., the tangent and the normal planes at p G M . Assume that (w, ?;) is a positive chart of M and that [au,av, N\ , N%} form a positive frame of R4 where {Ni , A^} is a frame of vector fields orthoriornial to a. In such a chart ( u , v ) the first fundamental form of a, /Q, is given by / = la = (da, da) = Edu2 + IFdudv + Gdv2 , where E = ( f * u , a u ) , F = (au,t*v) and G = (av,av}. The second fundamental form of a, //a, is defined in terms of the NM- valued quadratic form // = IIa = (d'2a, Nl)N1 + (d2a, N2)N2 = 7/i,a M + //2,«Ar2, where IIj — Ili.a — ??du + 'Ifjdudv + gjdv2, and ei = {a wn ,AV}, /,- = (auv,N,), and #,- = (avv,Ni), for i = 1,2. We have the following functions associated to a (for more details see [6]): 1. The mean curvature vector of a H = Ha = HiNi + ^2^2, where TT 1 -ff ~ ?:'a ~-

2(EG -

for i = 1.2;

2. The normal curvature of

a

(g2-e 2 )/i + ( e i EG~F2

(2.1)

Generic Singularities of //-Directions

257

3. The resultant A of //i,a and //2,a O f

A = A^ = - 62 2/2 0 0

ei e2

C\

.g2 2/i 2/2

0 gi .92

(2.3)

4. The normal curvature vector of a defined by r; : TM —» NM, where r/(p,v] = We have the following results: • The image of the unitary circle Sl of TpM by r)(p) : TpM —> 7VpM describes an ellipse in 7VpM called ellipse of curvature of a at p and denoted by £a(p}', • This ellipse may degenerate into a line segment, a circle or a point; • The center of the ellipse of curvature is the mean curvature vector H. Therefore the ellipse of curvature £a (p) can be expressed as (77 — H} + H] • The area of £a(p) is given by | \kN(p)\] • The map r/(p} restricted to Sl being quadratic, is a double covering of the ellipse of curvature ea(p). Thus every point of the ellipse corresponds to two diametrically opposed points of the unitary tangent circle; • The ellipse of curvature is invariant by rotations in the tangent and normal planes. A point p G M is called a minimal point of a if H(p) — 0 and it is called an inflection point of a if A(j>) = k^(p] = 0. It follows that p is an inflection point if and only if ei/2 - e 2 /i = ei02 - e 2 #i = /i.92 - hg\ = 0. Thus p G M is an inflection point if and only if the ellipse of curvature £a(p) is a radial line segment. Considering the above results we have the following remark: from any well-defined continous choice of points on the ellipse of curvature, continous tangent direction fields may be constructed on M. If the choice is not well-defined for special points of M we say that they are singular points of the direction field. Consider some examples of this remark. Asymptotic direction fields. Suppose that the origin of NpM does not belong to the region bounded by the ellipse of curvature £a(p},Vp G M. The two points on ea(p) at which the lines generated by the normal curvature vectors are tangents to ea(p] induce a pair of directions in TpM, which in general is not orthogonal. Making this construction for all p G M we define two tangent direction fields on M, called asymptotic direction fields (see figure 1). The singularities of these fields are the points where the ellipse of curvature becomes a radial line segment, i.e., the inflection points. Garcia, Mochida, Fuster and Ruas studied these direction fields and their singularities in [5].

258

Mello

yyw

Figure 1

Asymptotic direction fields.

Axial curvature direction fields. The two maxima and two minima of the norm of (ry — H)(p} determine four points on the ellipse of curvature £a(p) which are their vertices. This construction induces eight points on the unitary circle Sl C TpM which define four directions in the tangent plane. Making this construction for all p G M we define a field of 2-crossings on M, called axial curvature direction fields (see figure 2). This construction fails at the axiumbilic points, where the ellipse of curvature becomes a circle or a point. Gutierrez, Guadalupe, Tribuzi and Gufhez in [4] and Garcia and Sotomayor in [3] studied this case.

/'• s -,»"

Figure 2

Axial curvature direction fields.

//-direction fields. The line spanned by the mean curvature vector H(p] will meet the ellipse of curvature ea(p) in two points. This construction induces two orthogonal directions on TpM. Making this construction for all p € M we define two direction fields on M, called H-direction field* (see figure 3). The singularities of these fields, called H-singularilies, are the points where H = 0 (minimal points) or at which the ellipse of curvature becomes a radial line segment (inflection points). The set of //-singularities will be denoted by S(a).

Generic Singularities of //-Directions

259

Figure 3 //-direction fields.

An H-line is a regular curve ip : (a, 6) -> M such that at each of its points the tangent line is an //-direction and contains any regular curve with this property with intersects it. The differential equation of the //-lines is given by TI A H = 0,

(2.4)

which is a quadratic differential equation of the form

A(u, v)du2 + 2B(u, v)dudv + C(u, v)dv2 = 0,

(2.5)

A = A(u, v] = (el92 - e29l)E + 2(e2f1 - ei/2)F,

(2.6)

B = B(u,v) = (fl92 - f29l)E

- ei / 2 )G,

(2.7)

C = C(u,v] = 2(fl92 - f2

- eig2)G.

(2.8)

where

The //-singularities are determined by A = B = C = 0 in (2.5). But is immediate that the equation EC == 2FB - GA holds. We have established the following . PROPOSITION 2.1 Let a : M2 -> R4 be a Cr, r > 4, immersion of a smooth and oriented surface in R4. With the above notations we have: 1. The differential equation of the //-lines is given by (2.5); 2. The //-singularities of a are given by A = B - 0, where A and B are defined in (2.6) and (2.7) respectively.

260

Mello

In isothermic coordinates where E — G = A 2 and F = 0, equation (2.5) has the form Al (u, v}du2 + 2Bl (w, v)dudv - AI (u, v)dv2 = 0, (2.9) where AI =ei
(2.10)

Bi = /i(e 2 +.92) - h(e
(2.11)

and

LEMMA 2.2 A(p] = B(p] — 0 if and only if p is either a minimal point an inflection point of M. Proof:

It is enough to prove the lemma in the isothermic case. We make 0 = Aifi + Bigi = (ei + .9i)(/i.92 ~ hgi]

and

0 = Aifa 4- Big2 = (e2 + .92)(/i.92 - /2.9i)If /i .92 — /2.9i 7^ 0 then e.\ + g\ — 62 +.92 = 0 and this implies that //(p) = 0, i.e., p is a minimal point. If /^ - f<2gl = 0 then e.if2 - e 2 /i = e^2 - e.2gi = fiQi - /2,9i = 0 and this implies that p is an inflection point. The reciprocal is immediate. From the lemma 2.2 we have S(a} — Si(a) U 5*2(0;), where 5*1(0-) is the set of minimal points of M and 5*2(0;) is the set of inflection points of M.

3

//-SINGULARITIES OF £TYPE

We take the surface M in a Moiige chart, i.e., the surface M is the graph of the map a(u, v) = (u, /;, 5*(u, v),R(u, ?')), where 5 and R are C r , r > 4, functions defined on a neighborhood U C /?2 of (0,0) with the conditions 5(0,0) = /?(0,0) = S u (0,0) = /? u (0,0) = 5 W (0,0) = ^(0,0) - 0. For each point rv ( « , ? > ) E M the tangent plane Ta(u^M is generated by {au(u,v) = (l,0,5 t t (7i,7?),72 M (w,?;)),« u (?z,?;) = (0, 1, 5 V («, ?/), Rv(u, v}}} and the normal plane A r a ( n v )Af is generated by {Ni.N^}, where

and

Generic Singularities of//-Directions

261

Therefore E= (au,au},F = (au,av),G = (av,av), e

i

==

\CX-uui J - * i / i Ji

=

\auv5 ™i)i Qi = \(^vv>-^i)i

where

for i = 1,2. Write the Taylor expansion of the functions S and R near (0, 0)

u2 + anut, +

S(u,v) =

*(«, T,) =

V 4 rnuv +

Z

u3 4-

V+

V4

Z

0

4-

n«2 4-

+

^

v + uv* 4 ^

3

0

+O(4),

(3.1)

«3 4 O(4).

(3.2)

Thus the coefficients of the first and the second fundamental forms in the Monge chart are given by

E = I 4 0(2), F = O(2), G = 1 4 O(2), ei = «20 4- au + dv 4- 0(2), /i = sn 4 du 4 ^w + 0(2), gi = s02 4few4 cv 4 0(2), £2 = r20 + au + dv 4 0(2), /2 = r n 4 dw 4 6u + 0(2), g2 = r02 4-few4 ci; 4- 0(2).

We define J = 802^20 - 520^02, -^ = *'02« + r 2 ofe - S2ob - r02a, L = s02^ 4-

P = su(c 4- d) - rn(c + d) 4 (r20 4 r 02 )fe - («20 + «02)feThe differential equation of the /f-lines (2.5) in the Monge chart has the form C(u, v)dv2 4- 2B(u, v)dudv + A(u, v}du2 = 0,

(3.3)

C(u, v) = J 4- Ku 4 Lv 4- Qi(w, v), B(u,v) = M +Nu + Pv +Q2(u,v), A(u,v) = —J — Ku — Lv + Qs(u,v), Qi,Q2 and Qz are of order 0(2). At the point (0,0) we have a fi (0,0) = (l,0,0,0)^(0,0) = (0,l,0,0),^i(0,0) = (0,0,l,0),^ 2 (0,0) = (0,0,0,1), ,0) = 0(0,0) - 1, F(0,0) = 0,

262

Mello

e 2 (0,0) =r 2 0 ,/ 2 (0,0) = r n , 02(0,0) =r 0 2 . From (3.3), the condition for (0,0) to be an //-singularity is that J — •S'02r20 ~ -S'20r02 = 0

and M — .sn(r2o + H)2) - rn(.s2o + ^02) == 0. We put p = ^ and from (3.3) we consider the function T(u,v,p) = (J + Ku+Lv+Ql)p2+2(M-i-Nu+Pv+Q2)p-(J+Ku+Lv+Q3).

(3.4)

We define the surface W = T~l(Q). If (0,0) is an //-singularity then (0,0, p) e W, for all p. Furthermore the surface W is smooth in a neighborhood of the axis p if and only if (0, 0) is a transversal //-singularity, according to [7] and [2] . Suppose that (0, 0) is a transversal //-singularity. In this case we consider the following vector field on W X(u,v,p) = (Tp(w,?>,p),jpT p ('u,,'",p), -(T u (w,i>,p) +pTv(u,v,p))).

(3.5)

The vector field X has generically either one or three singularities on axis p, which are either saddles or nodes. Furthermore we have only one of the following three possibilities according to [7] and [2] : 1. The vector field X has one singularity of saddle type. In this case (0,0) is a transversal //-singularity of the type S\ (figure 4); 2. The vector field X has three singularities, one node and two saddles. In this case (0,0) is a transversal //-singularity of the type 62 (figure 4); 3. The vector field X has three singularities of saddle type. In this case (0, 0) is a transversal //-singularity of the type 83 (figure 4).

Type 5|

Type

ft

Type $

Figure 4 //-singularities of S'-type. REMARK 3.1 The index ? — 1,2,3 of Sj denotes the number of separatnces of the //-singularity. These are //-lines which approach the //-singularity arid which

Generic Singularities of //-Directions

263

separate regions of different patterns of approach to it. Figure 4 shows the local //-configuration of the three different types of //"-singularity denoted by Si, 82 and $3 and called H- singularity of S-type. The index is | in Si and 62, and — | in 83. Let Ik = Ik(M2, R4) be the space of Ck immersions of M into R4, where M is a smooth oriented surface. THEOREM 3.2 Let lkc c / fc be the space of Ck immersions of M into R4, where M is a compact smooth oriented surface, endowed with the Ck topology. Then the subset Tk of immersions for which all //-singularities are of the S-type is open and dense in Ik, for all k > 4. Proof: The proof of this theorem follows the same lines as the Darbouxian case treated in detail in [7] .

3.1

The case of minimal points

Suppose that p 6 M satisfies H(p) = 0 and kn(p) ^ 0. By a rotation in the normal plane it is possible to write s2o = «02 = 0. By a rotation in the wy-plane it is possible to write b = —a. Thus (3.1) and (3.2) can be written as S(u, v) = Auv + u3 + -u2v - °-w? 4-

,v} =

£

(u2 - v2) + Cuv + V 0

J + O(4),

+ u2v + ui? + '-v3 + 0(4). ^

Z*

0

(3.1.1)

(3.1.2)

This implies that

j = 0, K - 0, L = B(c N = A(a + and

P = A(c + d)We define

LEMMA 3.1.1 With the above construction p e M is a transversal //-singularity if and only if WiW2^0. (3.1.3) Proof: (3.1.3).

The condition of trans versality is given by NL ^ 0, which is equivalent to

THEOREM 3.1.2 Consider a transversal //-singularity for which H(p) = 0 and kN (p} / 0. Then we have:

264

Mello

SH [(£)'+•]• so the //-singularity is of type Si;

2. If

so the //-singularity is of type ,92; 3. If

— < 0,

(3.1.6)

so the //-singularity is of type 83. Proof:

The singularities of the vector field X (3.5) on the axis p are given by
But

0 = v(p] =p\D(c + d)r>2 - 2\C(c. + d)-~ A(c + d ) } p + 1A(a + b] - B(c + L ' V ) ' Therefore the vector field X has: 1. One singularity which is a saddle if and only if (3.1.4) holds; 2. Three singularities, namely one node and two saddles, if and only if (3.1.5) holds; 3. Three singularities which are all saddles if and only if (3.1.6) holds. The local configurations of the //-singularities of .9-type illustrated in Figure 4 are obtained by projection on the «?>-plane of the local configurations of the above singularities, according to [7]. 3.2

The case of inflection points

Suppose that p £ M is an inflection point and that H(p) ^ 0. By a rotation in the normal plane it is possible to write 7*20 — r'02 = t'u = 0. By a rotation in the ttv-plane it is possible to write h = a. Thus (3.1) and (3.2) can be written as (3.2.1)

Il(u. v) = V D

+ fa, + |m> 2 + V + 0(4). Zi

Zi

D

This implies that .7 = 0, K = 0, L = A(d - r), M = 0,

(3.2.2)

Generic Singularities of //-Directions

265

N = 2(aB - Ad] and

We define Wi =A(d-c], W2=aB- Ad

and

LEMMA 3.2.1 With the above construction p e M is a transversal //-singularity if and only if WiW2£0. Proof: (3.2.3).

(3.2.3)

The condition of transversality is given by NL ^ 0, which is equivalent to

THEOREM 3.2.2 Consider a transversal //-singularity which is an inflection point and H(p) ^ 0. Then we have: 1. If

then the //-singularity is of type Si; 2. If

then the //-singularity is of type ^2; 3. If Wo

wf < 0,

(3.2.6)

then the //-singularity is of type 83. Proof:

The proof follows the same lines as that of theorem 3.1.2.

Acknowledgment: The results here presented are part of my doctoral dissertation at Universidade de Sao Paulo (IME). I wish to thank my advisor, Professor Jorge Sotomayor, for helpful suggestions.

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Mello

REFERENCES [1] A.C. Asperti. Immersions of surfaces into 4-dimensional spaces with nonzero normal curvature. Ann. Mat. Pura Appl. 125:313-329, 1980. [2] J.W. Bruce, F. Tari. On binary differential equations. Nonlinearity 8:255-271, 1995. [3] R. Garcia, J. Sotomayor. Lines of axial curvature on surfaces immersed in R4. Diff. Geom. and its Applications 12:253-269, 2000. [4] C. Gutierrez, I. Guadalupe, R. Tribuzy, V. Guiiiez. Lines of curvature on surfaces immersed in R4. Bol. Soc. Bras. Mat. 28:233-251, 1997. [5] R. Garcia, D. Mochida, M.R. Fuster, M.A.S. Ruas. Inflection points and topology of surfaces in 4-space. Trans. Am. Math. Soc. 352:3029-3043, 2000. [6] J.A. Little. On singularities of submanifolds of a higher dimensional Euclidean space. Ann. Mat. Pura Appl. 83:261-335, 1969. [7] C. Gutierrez, J. Sotomayor. Lines of curvature and umbilical points. IMPA, Rio de Janeiro, 1991. [8] W.C. Wong. A new curvature theory for surfaces in Euclidean 4-space. Comm. Math. Helv. 26:152 170, 1952.

Vertices of Curves on Constant Curvature Manifolds CLAUDIA C. PANSONATO Depto Matematica - CCNE - UFSM 97119-900 Santa Maria-RS Brazil E-mail: [email protected]

SUELI I. R. COSTA Institute de Matematica - Unicamp 13083-970 Campinas-SP Brazil E-mail: [email protected]

Abstract

A vertex of a curve on a Riemannian manifold of any dimension can be denned as a critical point of its first curvature at which the second curvature also vanishes. We prove here that under conformal parametrization there is a correspondence between Euclidean and Riemannian vertices if and only if the manifold has constant curvature. A four-vertex theorem for spherical curves is then obtained as a corollary. Special results and a characterization in terms of singularities of the evolute surface for the three dimensional case are also presented.

1

INTRODUCTION

This paper is an extension of some concepts and results concerning higher order singularities and geometrical contact of curves to constant curvature manifolds. It is an intrisic Riemannian approach of curvatures, contact and vertices via conformal parametrizations.

267

268

Pansonato and Costa

The connection between higher order singularities of height and distance functions and geometrical contact of Euclidean curves have been studied in different ways by several authors since long ago. We may quote, for instance, works like [2], [3], [13], i l l , [7] and [11]. A vertex of a plane curve can be viewed as a critical point of its curvature and also as a contact point of order at least three (a 4-point contact in [4] notation) with the osculating circle. This characterization can be naturally extended to higher dimensional Riemannian manifold in terms of a higher order singularity of the geodesic curvature functions. On the other hand, it allows, for constant curvature manifolds, a confermal approach like the one done in [5]. Previous works more related to what is done here are [6] and [16]. This last one is linked to [21 arid concerning vertices of a curve, denned as higher contact with the osculating hypersphere, in the hyperbolic n-dimensional space (Euclidean approach). We may say that the approach here besides being Riemannian is "dual" to that one in the sense that we deal with singularities linked to higher order contact with osculating circles. We point out that, for a curve on a three-dimensional manifold, both types of vertices determine singularities on the evolute surface. In section 2 we deduce general expressions for Riemannian first and second curvatures of a curve on a n-dimensional manifold and specialize them for the constant curvature ones. Riemannian contact is discussed and it is stated that the vanishing of the first and the second curvature corresponds to contact of order two and three, respectively, with a totally geodesic submanifold (Proposition 2). The main result in section 3 is proved in Theorem 1 which states that under a conformal parametrization vertices of curves correspond to Euclidean vertices if and only if the manifold has constant curvature. This extends a result presented in [6] for two-dimensional manifolds. A Four-Vertex Theorem for spherical curves is then obtained as a corollary (Theorem 2). Section 4 is devoted to curves on three-dimensional manifolds. Particular results (Theorems 3 and 4 ), also linked to the second curvature changes of sign, are deduced and for spherical curves vertices are characterized as cuspidal edges of the evolute surface.

2

HIGHER ORDER CONTACT BETWEEN CURVES AND TOTALLY GEODESIC SUBMANIFOLDS

We consider here regular C4- curves, a : / —> M on a n-dimensional Riemannian manifold M, n > 3. We denote by kgi and kg2 the Riemannian first and second curvature functions of a. For a conformal parametrization ? / > : £ / — » M, where U C M n is an open set, we denote by g^ = f>ijG2 the metric coefficients. We start getting expressions for the Riemannian curvatures of a in terms of the Euclidean curvatures of its conformal pre-image ft = t}>~1 o a which are to be used in the following sections. Considering constant curvature manifolds we discuss relations between Riemannian contact and the vanishing of the first and the second curvatures.

Curves on Constant Curvature Manifolds

2.1

269

General Expressions for Riemannian Curvatures

In the expressions to be deduced we follow the notations established in [14]: s is the Riemannian arc length of a, se the Euclidean arc length of ft; k\,k<2 are the curvatures of ft; ( , ) is the inner product in M and (, ) e is the Euclidean inner product; Vj, i = 1, • • • ,71 are the Frenet frame vectors of a C M in the tangent space and v$, i = 1, • • • , n is the Frenet frame of ft C R n . We start determining an expression for the first Riemannian curvature of a in terms of the metric G and the Euclidean first curvature of its pre-image. Since iy ^r i

da

, where —;— is the covariant derivative of Vi = —, we must ds ds ds develop an appropriate expression for the covariant derivative. If a^(s), i = 1, are the coordinates of / 3 s , D'

ds

k=l

Calculating the Christoffel symbols F^?- of the metric gij — £>ijG2, we get F*- = 0 if the three indices are different, and G Q

F-, = --

F!, -

'

Then, we have

r

JV

ds

k=l

+2

ds

,

2

'

E

'f

Q

j, K

Xk

Q

dxk \ , I ds J

s

G-r •t'l• dcL

G d.3

Y^ ( dx i\ / I T ) f— ' \ as J

\* J 5xfc '

Let Oi, i — 1, • • • n, be the angles between a and the z'-th coordinate curve. Then: = ^ G-— ds

andi

• a ' = — sm^—— ds

ffr(XXj

ds

rl w Lt/O

Replacing these values in the above expression and taking tk =

2

X~»

// r ' /7 w U;O

-^

1 d , lead us to G dxk

D'Vi ds

where V ( — ) is the gradient and vi = —- = (cos^i, • • • , cos# n ). \G/ dse We can also express the Euclidean curvature k\ and vector v2 in terms of 0;.
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Pansonato and Costa

(2.2)

fei =G.

and (2.3)

(if fci ^ 0)

sin Oi ——, • • • , — sin On

So from (2.1) and (2.2) we get an extension of Liouville Formula [15] for ndimensional manifolds: 1/2

If ATI 7^ 0, an alternative expression for kgi can be given: 1/2

(2.5)

We will not deduce here an explicit expression for the second Rieniannian curvature. This is done (in the next Proposition 1) just for constant curvature manifolds. If kgi ^ 0, V~2 =

— and kg2 is denned by

kgi

ds

__

j

[

i_ D'2V1

l_ dkgi p'V1

A:j-

ds

ds

fc9l

ds2

(2.6) ,Vi

Calculating the covariant derivative, P_ Vi ds2

A: = l

+ cos^ fc ( -I

(2.7)

),Vi)c

/^f /

?

I

rC"

The two first terms of the last summation can be developed as follows: • If fci ^ 0,

(2-

„ dV'2

is the fc-th coordinate of

where 10

c / A-

ds.

271

Curves on Constant Curvature Manifolds

• If fei = 0,

ds*

where

^ i,s the fc-th coordinate of ds

6

Developing the part of

D 2/ V d (\ — which involves — V I — ) , we get as1 ds \G

ds2 fc=

sin flt ^ (V (i) , (i) - £ (i

+jj E

(2.10)

L i^k

+tt..

2.2

Constant Curvature Manifolds

For a manifold with constant curvature K it is always possible to find a conformal parametrization [14] such that the metric is given by 9V =

%

(2.11)

But this is equivalent to say (up to translation) that g^ = 6ijG2 satisfies

(2.12)

( dxl \G)

dx] \G

Then we have PROPOSITION 2.1. Let a : I —> M be a regular C4- curve on a n-dimensional constant curvature manifold M and kgi ($) ^ 0. Under the above conform,al parametrization, its second curvature, kg2, is given by 1. For k\ ^ 0, r

"92!

J

2

/

e

\

» *? fe + L

f

1 I/ KI i (/ V r? (-g) { l\ , V dv22 p91

V

e

\

2ll/2

(2.13)

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Pansonato and Costa

2. For ki = 0,

\k .92

Proof: tiating *

D'Vi ^ ds

We observe first that =

D'Vi

(2.14)

ds?

D'\

with respect to .s, we also have,

ds dkr ds

\

fc0l

(2.15)

ds ' ds2

Then, by (2.6), we get 1/2

1 \92

ds-

^'.9l

On the other hand, it follows from (2.10) and (2.12) that if M has constant curvature, ——-— is given by

d k \ ] - sin • a9 - cosaOk I { ® -pk ,

ds-

cos Ok

dBk

—k

If k\ ^ 0, we replace (2.8) in

ds2

V

Td'V!

t fe .

D^Vi ds2

D /2 V! vi ^—> j KI i av'2 \ i afi'i . = Z^ i ?721 ^T ~ TJj^ 77~S1 d3 :2~ 2 C 2 ds Gk ' f^[ I > \ e J k 'l d*t

2

4-—-\i
ds

Then, from (2.1), (2.15) and (2.16),

ds

6V'ds c / ' d*e\G

V \GJ'

or (2.17)

Summing up we finalize the proof of part 1. The proof of part 2 is done analogously utilizing (2.9) in -^— and noting that

, —'

— 0 when k\ =0.

D

We now will establish relations between Riemannian contact and the vanishing of curvatures. The following definition extends the classical contact definition between curves in the Euclidean space R" .

Curves on Constant Curvature Manifolds

273

DEFINITION 2.2. Let a : / -> M and 7 : / -» M fee router <7m+1- curves on a constant curvature n-manifold parametrized by their arc lengths s and t, respectively. We say a and 7 have Riemannian contact of order m (exactly) at a common point P if

=

7(*o) = P

da ds

dt t=tc

D'j' dt

D'a'

ds

t=t0

D'r

t=t0

and D'ma' dsm

t=t0

REMARK: This geometric definition extends the usual contact notion in R n , which can also be formulated as lim

s—'So

s

'

= 0 if and only if k < n, where d is the

Euclidean distance. However, we may point out that, like in the Euclidean case, if two regular curves have same derivatives with respect to a common parameter, the same condition must be satisfied for their arc length derivatives. This means that the previous definition can be re-written as: two curves have contact of order m (exactly) at a point P = a(to) = 7(^0) */ and only if for some parametrization we have D'r

but t=t0

t=t,0

D' dtm

t=t0

t=t0

For constant curvature manifolds the contact notion is preserved under conformal parametrization, namely PROPOSITION 2.3. Under the same conditions of definition 2.1, a and 7 have Riemannian contact of order m (exactly) at a common point P if and only if their conformal pre-images have same order of contact at t{.>~l(P).

Proof: If u(s) = (w l5 • • • , un] are the coordinates of a vector field defined on a, then using the metric expression (2.11) we get D'u ds

= E

dse 1 From this and reminding that —-^ = — , we can get through a recursive process ds G that equality between covariant derivatives (with respect to s) is equivalent to equality between common derivatives (with respect to se). D

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Pansonato and Costa

Proposition 2 allows us do not distinguish between Riemannian and Euclidean contact. So, from now one, we just say contact when referred to constant curvature manifold. DEFINITION 2.4. A curve a : I —>• M has contact of order rn (exactly) with a submanifold S C M at a point P if there exist* at least one curve 7 on S which has a contact of order m with a at P and there does 'not exist a curve on S which has a contact, of order greater than m with a at P. The next result establishes the relation between the vanishing of the first and the second Riemannian curvatures and the contact with totally geodesic submanifolds, ie, submanifolds N C M such that every geodesic in N with the induced metric is also a geodesic in M . PROPOSITION 2.5. Let a : I —>• M be a regular C4- curve on a n- dimensional constant curvature manifold and. P — CX(SQ). Then: 1. k giso) — 0 if and only if a has contact of order 2 (at least) with, a geodesic of M at P.

And,

ifkgi(So)^Qat.P,

2. kg2(so) — 0 if a,nd only if a has contact of order 3 (at least) with, a totally geodesic 2-submanifold of M. Proof: The proof is quite long and can be sketched as follows (see [10] for the extended version). Since M has constant curvature, its universal covering is isometric to R77 , S" or M n . We start showing that when M = Sn or M = HP, with the metric given in (2.11), the totally geodesic j-submanifolds of M are image either of the jplanes through the origin in R" or j-spheres contained in (j + l)-planes through the origin in R n whose center C and ray r satisfy the condition

The proof of the part 1 is then divided in two cases: first case, we get from (2.5) and (2.11) that

0 and ki = 0. For the 1/2

where ft — V'~ 1

-2k- (ft, Vl)l~ (ft, v 2 ) e 2 = 0 and (ft, v 2 ) e 2 = So, kg, = 0 ~GK The first condition is equivalent to say that the osculating (Euclidean) plane must pass through the origin. The second condition implies that the center C of the Euclidean osculating circle (C = ft H -- VP) satisfies

And this implies, by (2.18) and the first condition, that the image of the osculating circle is a geodesic of M. Since the osculating circle has contact of order 2 (at least) with the curve, this completes the proof of part 1 for k± ^ 0.

Curves on Constant Curvature Manifolds

275

If ki = 0, we can derive from (2.4) and (2.11) that

Then, kgi = 0 <£> ft = Avi, A G R, and this means that the image of the tangent line is a geodesic of M. Besides, from ki = 0 we have that the curve has contact of order 2 (at least) with the tangent line. The proof of part 2 follows similar steps but now considering three different cases (ki 7^ 0 and k2 ^ 0; fci ^ 0 and fc2 = 0; ki — 0). We utilize (2.11) and the expressions for kg2 given in Proposition 1. In each case we show that kg2 = 0 is equivalent to either the osculating plane must pass through the origin or the center C and the ray r of osculating sphere must satisfy (2.18). D

3

CORRELATING RIEMANNIAN TO EUCLIDEAN VERTICES

In this section we obtain an extension to the n-dimensional case of a result presented in [6] for a curve on a constant curvature manifold: its Riemannian vertices are correlated to the vertices of its conformal pre-image. For doing so we adapted a definition of vertex of an Euclidean curve from [5], which restricted to 2-dimensional manifolds or plane curves, is the standard one. DEFINITION 3.1. Let a : I —> M be a regular C4- curve on a n-dimensional Riemannian manifold with never vanishing first curvature. A vertex is a point where _ dkgi _ k9 ' *~ ds ~ U > Developing the calculations presented in 2.1, we get the following LEMMA 3.2. Under a conformal parametrization, Euclidean vertices of any curve in Rn correspond to Riemannian vertices of the curve on the manifold if and only if

(3.1)

~2 I G J

Ox2

Proof: Since the above equations are the same conditions of (2.12) and have, using (2.17) in (2.13), that

&

91

G4

Taking into account that

A 1

ds*

C/V2

dsp

+ V,

1

fdkgi ds

V G [vi,V2J J ~, we have:

0 we

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Pansonato and Costa

The reciprocal is not straightforward. We will show that if equations (3.1) are not satisfied, there are curves for which Riemannian vertices are riot correspondent to the Euclidean ones. Let us suppose, for instance, that

— f — ] ^ 0 at a point P. For any given ^

2, • ^ o n

\

/

unit vector Vi — (cosBI, • • • , cos 9n) in R and any positive number r there is a circle through P with tangent vector Vi and radius r. We choose such a circle in order to get kgi ^ 0 at P. Starting from directions Vi = (cos 9, sin 9,0, • • • ,0) and V2 — (—sin 9, cos 0,0, • • • ,0). 7T For 6 =• —, we have from (2.5) that

0

G Oxl \G

(

—

I

l

—

G

Varying the radius r (r = — ) of the circles contained in the plane x\x<2,, with tangent vector YI (and 9 = —) we set — 2 G

— ) ^ 0 and then k gi G

0.

Since for circle k,2 — —- = 0, from (2.8) we get

i

\ ds /

.

,

^--.—. .-

ft

»

2o

dx

x~f9 2 ^"-" " «:

G

«

"-.-.*. -s

ft:

<W dx

Using this expression in (2.10) and Vi = (0,1,0, • • - , 0) results £-Xl = -fc2 V j + I si t. Z>

yI

/J G

So, from (2.6) we get

arid if —— — 0. we can establish rf.S"

"92

Ox2(^x3 \G dk vertces

GA:,

0. This means there is no correspondence between Euclidean = 0 1 arid the Riemannian ones ( A:"0o^ =

dk gl , = 0). W ..

Curves on Constant Curvature Manifolds

277

Suppose now that I —^ — f — ) „ f — I I 7^ 0 at a point P. Like in the \dx{ \GJ dxl \GJ J previous case we consider circles trough P in the xi^-plane and tangent to Vi for 7T

(tr\

0 = —. Trough an analogous reasoning we can deduce that if —— = 0, then 4 ds

i

o

what means that k92 ^ 0 due our initial hypothesis.

D

We now point out that the function — satisfies the condition (1) if and only if G

where A, Bi and C are constant. Besides, this means that the manifold has constant curvature

1=1 As a direct consequence of the previous lemma and this remark, we get the following: THEOREM 3.3. Under u conformal parametrization of a n dimensional manifold M, vertices of C4-curves on M correspond to Euclidean vertices of their pre-images if and only if M has constant curvature. In the next section we show for the 3-dimensional case that this correspondence can be stated even in a stronger way. REMARK: A. Kneser, in his classical four-vertex theorem article [8], already pointed out the correspondence between vertices of a plane curve and the vanishing torsion points of its stereographic image. We remark that at those points of spherical curves, we have also the vanishing of the curvature derivative. Hence, simple closed spherical curves have at least four Euclidean vertices in our sense. We can establish an analogous property for curves on Riemannian manifolds. Instead of spheres we will consider curves a : I —> M on a totally umbilic 2-submanifold of M. If M has constant curvature, we may call such curves spherical. ^.Frorn the previous theorem we can extend the four-vertex theorem: THEOREM 3.4. (Four- Vertex Theorem) Let M be a n dimensional complete constant curvature manifold and a : I —>• M a closed simple null homotopic spherical curve. Then a has at least four vertices.

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Pansonato and Costa

Proof: We consider first n = 3. a is a spherical curve homotopic to zero what means it can be lifted, via the universal covering map, TT : M —> M, to a spherical curve a on M. Since M has constant curvature, its universal covering is isometric to R 3 , S3 or H3. The conformal pre-image of the totally umbilic 2-submanifold of §3 or H3 is contained in an Euclidean sphere or in a plane. This implies, by the previous remark and Theorem 3.3, that a must have at least four vertices. The local isometry between M and M completes the proof for n = 3. The extension for general n, is done by considering the identification of a totally geodesic 3-dimensional submanifold of the covering space which contains the umbilical 2-submanifold with R 3 , §3 or H3. D As we will see in the next section, in the 3-dimensional case, Theorem 3.4 can be specialized by asserting at those vertices the second curvature also changes its sign. We point out that some authors ([12], [7]) also define a vertex of a curve in R3 as a flattening point, that is, just a vanishing torsion point. In [12] it is shown that if a closed space curve is convex (that is, contained in the boundary of its convex hull) it has at least four vanishing torsion points. Next we make use of Theorem 3.3 to exhibit a family of convex curves in R3 having no vertex in our sense. We consider the family of curves on S3 given by &0(t) = (cosdcost,cos9sin£,sin#cos(nt),sin#sin(n£)),

# € (O, — 1

where n €E Z. Under the stereographic projection, /3g — 7r(a0) are curves on a standard torus in R 3 . If I — n2 6 |sin#L" < 1. we can assert that those curves are convex because their orthogonal projection on R2 are convex plane curves with nowhere vanishing curvature. Since a$ curves have constant nonvanishing curvatures, kgi and fcg2, it follows from Theorem 3.3 that j30 curves have no vertex.

Figure 1:

A convex curve with no vertex and its projection in xiX2-plane

Curves on Constant Curvature Manifolds

4

279

THREE-DIMENSIONAL CASE

Vertices of a curve / ? : / — > Mn are also points where it has a contact with its osculating circle of order at least three. An vertex is called ordinary if this contact

dki

is of order exactly three. In the 3-dimensional case, n = 3, this means k^ = - = 0 dse rf2fc dk n or —? ^i 0. n andi —ri ^/ 0 ds^ dse Considering the space curve 7 = (3 H -- V2 given by the centers of the osculating

ki

circles of /5, we have - = -- *• - V2 + — v3. We can check that the vertices of ft dse fc} dse KI are singular points and the ordinary vertices are cusps of 7. For constant curvature 3-manifolds we can state a correspondence between ordinary vertices. THEOREM 4.1. For a C4 -curve on a constant curvature ^-manifold ordinary vertices are correspondent to Euclidean ordinary vertices of its conformal pre-image. Proof: and

If ki ^ 0 and n — 3, we may replace the expressions - = —fcivi+ fc dse

in (2.13) and get

*•From (2.17) and replacing —— = — fcivi + ^vs, we have dse

Differentiating (4.2) and the above expression with respect to s at a Euclidean (and then Riemannian) vertex point P = a(«o) we get dk,92 ds

dse \G ' (4.3)

and

V[^),v3/ (4.4)

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Pansonato and Costa

From (4.1), (4.3) and (4.4):

what concludes the proof.

D

The second curvature A;0, = /

, Vs } of a curve on a 3-dimensional manifold /e has a sign. Since a convex curve in IR has at least four changes of its torsion sign [7] (at those points it stays at one side of its osculating plane) we may ask if a similar result concerning kg2 changes of sign still holds for a spherical curve on constant curvature manifold M3. Under the same reasoning done in the proof of Theorem 2 we consider a a local lifting to the universal covering of a spherical curve on M3 and its conformal preimage ft = tj.>~l(a), a spherical curve in R3. Hence k\ 7^ 0 and \

ds

3

' _c - - —

-i

where C and r are the center and the radius of the sphere and kg is the geodesic /1.2 ,2

curvature of ft C M 3 , kg = -^—-

T

[9]. For a spherical curve we also have ^i = -rk2kikg. a,S'e

(4.5)

From these and from (2.11) and (4.2), we get

- \ C e2 ~T 2 what implies kg2 and k% have the same sign. We finally point out that changes of sign of k% of a spherical curve in R3 correspond to cuspidal edges in the evolute surface of the curve, that is, the surface denned as a envelope of the family of the curve normal planes [4]. Considering the standard parametrization of the evolute surface of a space curve X(fie, A) = ft+ — V2 + Av3, we see that the singularities are given by the vertices and the centers of the osculating spheres. Using (4.5) we can conclude that, for a spherical curve, the points at which the torsion changes its sign generate at least four cuspidal edges on the evolute surface. k For A = —f'j^- and any ,s-e, the evolute surface pass through the center of the sphere — a degenerate singularity. Summing up we can state the following THEOREM 4.2. Let M3 be a constant curvature 3-manifold and a : I —* M a regularclosed spherical null hornotopic C4-curve. Then a has at least four points at which the second, curvature changes its sign. Those points correspond to cuspidal edges on the pre-image curve evolute surface. Figure 2 ilustrates the evolute surface of a spherical curve with just four (Riemannian/ Euclidean) vertices.

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281

Figure 2: The evolute surface S of a, a anti-stereographic projection of an ellipse. S has four cuspidal edges and one degenerate singularity.

REFERENCES [1] V. Arnol'd. On the number of flattening points on space curves. Amer. Math. Soc. Transl. 171 2:11-22, 1996. [2] M. Earner. Uber die Mindestanzahl stationarer Schmiegebenen bei geschlossenen streng-konvexen Raumkurven. Abh. Math. Sem. Univ. Hamburg 20:196-215, 1956. [3] J.W. Bruce, P.J. Giblin. Generic isotopies of space curves. Glasgow Math. J. 29:41-63, 1987. -. Curves and singularities 2nd ed. Cambridge University Press, Cambridge,

[4]

1992. [5] G. Cairns, R. Sharpe, L. Webb. Conformal invariants for curves and surfaces in three dimensional space forms. Rocky Mountain Journal of Mathematics. 24 3:933-959, 1994. [6] S.I.R. Costa, M. Firer. Four-or-more-vertex theorems for constant curvature manifolds. In: Real and Complex Singularities, Research in Mathematics Series, Ed. J. Bruce, F. Tari. Champman & Hall/CRC 412:164-172, 2000. [7] S.I.R. Costa, M.C. Romero-Fuster. Nowhere vanishing torsion closed curves alawys hide twice. Geom. Dedicata. 66:1-17, 1997.

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[8] A. Kneser. Bemerkunger iiber die Anzahi der Extrema der Kriimmung auf gescholosseneri Kurven und iiber verwandt Fragen in einer nicht-euklidischen Geometrie. Festschrift Heinrich Weber, Teubner 1912 pp. 170-180. [9] R.S. Millrnan, G.D. Parker. Elements of differential geometry. Prentice-Hall Inc., Englewood Cliffs, Xew Jersey, 1977. 110] C.C. Pansonato. Contact and vertices of curves on constant curvature manifolds. Ph.D. thesis, Institute de Matematica-Unicamp, 2001. ;11] M.C. Romero-Fuster, E. Sanabria-Codesal. Generalized evolutes, vertices and coiiformal invariants of curves in R n+1 . Indag. Mathem. X. S. 10 2:297-305, 1999. [12] V.D. Sedykh. Four vertices of a convex space curve. Bull. London Math. Soc. 26:177-180, 1994. 113]

. Connection of Lagrangian singularities with Legendrian singularities under stereographic projection. Russian Acad. Sci. Sb. Math. 83 2:533-540, 1995.

[14! M. Spivak. A comprehensive introduction to differential geometry. Publish or Perish, Inc.. Houston, 1979. [15] D.J. Struik. Lectures on classical differential geometry. Dover Publications Inc. Xew York, 1961. [16] R. Uribe-Vargas. On the (2A: + 2)-vertex and (2fc -i- 2)-flattening theorems in higher dimensional Lobatchevskian space. C. R. Acad. Sci. Paris. 325:505-510, 1997.

Projections of Hypersurfaces in R4 to Planes ANA CLAUDIA NABARRO* Institute de Ciencias Matematicas e de Computagao, Universidade de Sao Paulo, Departamento de Matematica, Caixa Postal 668, CEP 13560-970, Sao Carlos, SP, Brazil, E-mail: [email protected]

Abstract

We study in this paper orthogonal projections of generic embedded hypersurfaces in M4 to 2-spaces. We list all the generic local singularities of such projections and establish geometric criteria for recognition of the singularities of codimension < 1 and for their versal unfoldings. We also determine a stratification of the set of singular planes of projections.

1 INTRODUCTION We study in this paper the ^-classification of the singularities of the orthogonal projections of a generic embedded hypersurface M in E4 to a 2-dimensional plane. The singularities of such maps measure, for instance, the contact of M with 2-dimensional planes. The contact of smooth varieties with degenerate objects (lines, planes, circles, spheres, etc.) is measured locally by the AC-classification of the singularities of some map-germs [20], although in practice (right-left) *4-equivalence classes are sought as they yield a finer classification, a richer set of invariants, and more geometrical information. This approach allowed the discovery of beautiful geometric results of * Current address: Universidade Estadual de Maringa, Departamento de Matematica, Av. Colombo 5790, Campus Universitario, CEP 87020-900, Maringa, PR, Brazil.

283

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Nabarro

surfaces in R 3 (for example, the study of singularities of the Gauss map [1], the classification of the umbilic points and the structure of the focal set at such points, the flat geometry of the focal set, the ridge and subparabolic curves, the changes on these set, etc.; see [2] for references). These results are being extended to surfaces in R 4 in [5, 8, 17, 18/24]. The work in this paper is part of our investigation in [22] of the flat geometry of generic embedded hypersurfaces in R 4 . In [23], we study the contact of M with hyperplanes and lines, and here we deal with the contact with planes. There is a natural family of projections of M C R 4 to planes, parametrised by the Grassmanian of 2-planes in R 4 , G'(2,4). Let a and b form an orthonormal basis of the plane of projection u E G'(2,4) then the family of projections to 2-spaces is given

by II:

A/xG(2,4) (p, u]

^ i—>•

R2 (< p, a > , < / ? , b >).

Given u E (7(2, 4) the map IIM measures the contact between M and the plane orthogonal to u. the kernel of Tiu. (Xote that Hu is of corank 1.) The A-equivalence class of Hu does not depend on the choice of orthonormal basis a, b, so in effect H is a 4-parameter family [61. Therefore the generic ,4-equivalence classes of Iiu are those of .4e-codimension < 4. (In the presence of moduli, the generic cases are when the union of the orbits along the moduli parameters form a set of ^4e-codimension < 4.) This fact is a direct consequence of Montaldi's Theorem in [21], and follows from the fact that the family II defined on the ambient space is ^4-versal. So the first task is to determine the normal forms of the singularities of corank 1 and codimension < 4 of germs R 3 ,0 —> R 2 .0. Those that are .-4-equivalent to (x, f ( x , y) ± z 2 ) are classified by Rieger and Ruas in [26]. We classify in section 2 those germs of v4e-codimension < 4 that are not covered in [26]. The main tools used are the complete transversal technique in [3, 7] and the software "Transversal" developed by Neil Kirk [14]. In section 3, we give geometric criteria for recognition of the codimension < 1 singularities of germs R 3 ,0 —>• R 2 , 0 and in section 4 criteria for determining when a deformation of a codimension 1 singularity is versal. Finally, we determine in section 5 a partition of the set of singular planes by the singularity type of I1M at a fixed point on M.

2

CLASSIFICATION RESULTS

As highlighted in the introduction, the generic singularities of IT are those of Aecodimension (or codimension of the stratum) < 4. Before giving a complete list of such singularities we need some notation and preliminaries. (Although our interest is in germs R 3 ,0 —> R 2 . 0 we set these notation in more general terms.) Let / : R n , 0 —> R p , 0 be a smooth map-germ and denote the Lie group of right-left equivalences by A = D///(R n ,0) x Z)?'//(R P ,0) which acts on the space of smooth germs £(n,p] as follows: (/;,, k}.f — h o / o A:^ 1 , where (h,k) E A. We say that / is ^-equivalent to g if there is (h, k] E A such that y — h o f ok~l. Let Jk(n,p] denote the space of kth order Taylor polynomials without constant terms, and write jkf for the fc-jet of /. A germ / is said to be A:-determined if, for any g such that jkf = jky, it follows that / is equivalent to g.

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285

Denote by £(n) and £(p) the local rings of function-germs in source and target whose respective maximal ideals are m(n) and m(p). Let 9f denote the £(n)-module of vector fields over /, and set 9n — 0(i R7l ) and 9P = #(I RP ) where l R n and IRP denote respectively the identity mapping of Rn and W. One can define a £(n)homomorphism tf:9n -* 0f (f) H-> df o (p and a £(p)-homomorphism wf : 9P

->

Of

The tangent space to the A-orbit at / is then given by LA • f — tf(m(n).9n) + w f (m(p) .9p). The .4-codimension of / is given by cod(A, /) = dim^m(n} .9 f / LA • f and the ^4e-codimension by Ae—codim(f) = dim^(£(n,p)/LAe • /), where The space LAe • f is called the extended tangent space. The more recent advances in determinacy and classification theory are given in the articles by Bruce, Kirk, du Plessis and Wall [3, 7]. We state below a corollary of one of the key results in [7] that can be used to list singularities of map germs

R n , nU

—», irpp JK , n U.

Let Q be a subgroup of one of the Mather's group and define Qs to be the subgroup of Q whose elements have the s-jet equal to the identity. COROLLARY 2.1 [7] f is k - Gs-determined (Q = C or A, s > 1) if and only if mkn+1.9f C Lgs.f + mkn+l.r(mp}.

Finite determinacy means that the classification problem can be reduced to the space of fc-jets. Let Hk+l(n,p} be the space of germs of homogeneous polynomials of degree k + 1 in £(n,p). PROPOSITION 2.2 (Complete Transversal to jets [3]) Let Q be a Mather's group and denote the tangent space to the Jk+lQi-orbit of jk+l f by L(Jk+lQi}.jk+l f . Then given f 6 m(n).£(n,p) and T C Hk+l(n,p) such that

any (k + l)-jet jk+lg, with jkg = jkf, te T.

is in the same Qi-orbit of j fc+1 / + 1, for some

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The techniques developed in [3, 7] provide a very efficient, wide-ranging classification scheme involving algebraic calculations which may be reduced to finite dimensional symbolic problems. However, the calculations can become very intensive and repetitive. In [14], Kirk produced a package, called "Transversal", which consists of a collection of procedures which run under the symbolic algebra system Maple and deals with problems in classification and unfolding theory for these equivalence relations. Recent applications of "Transversal" [4, 10, 11, 12, 13, 29] represent some of the most extensive classifications carried out to date. In this paper we use "Transversal" to classify germs R 3 ,0 -> M 2 ,0. Germs K 3 , 0 —> R 2 ,0 of corank 1 that can be written in the form (x, /(x, y) ± z2) in some coordinate system are classified by Rieger and Ruas in [26]. THEOREM 2.3 [26] The germs R 3 ,0 -> R 2 ,0 of corank I and Ae-codimension (or codimension of the stratum) < 4, that are equivalent to one of the form (x,/(x,?y)±z 2 ) are given below, where all the classes are simple with the exception of 8,9,10,15,18 and 19. Name 1 2 (fold) 3 (cusp) 4fc (lips/beaks when k — 2) 5 (swallowtail) 6 7 8 9 10 llofc+l

12 13 15 16 17 18 19

Normal form

Ae-cod

(X y}2

0 0 0 fc-1 1 2

(x,y ± z 2 ) (x,xy + y3 ± z2} (x,?/ 4- (±l)k~lxky ± z 2 ) , 2 < k < 5 (x, xy 4- y4 ± z2) (x, xy 4- y° ± y7 ± z2) (x xy 4- y° 4~ z" ) (x, xy 4- y6 ± y8 4- ay9 ± z 2 ) (x, xy 4- y 6 4- y 9 ± z 2 )

(x, xy 4- y7 ± y9 4- ay10 4- by11 ± z 2 ) (x, xy 2 4- y4 4- y 2fc+1 ± z 2 ), 2 < fc < 4 (x, xy 2 4- y° 4- y6 ± z 2 ) (x, xy 2 4- y° ± y9 ± z2) (x, xy2 4- y6 4- y7 + ay9 ± z 2 ) (x, x 2 y 4- y4 ± y5 ± z 2 ) (x,x' 2 y 4- y4 ± z 2 ) (x, x 2 y 4- xy3 4- ay5 + y6 4- 6y7 ± z 2 ) (x, x 3 y 4- ax 2 y 2 + y4 4- x 3 y 2 ± z 2 )

3 4(3f) 4 6*(4f) fc 3 4 5(4f) 3 4 6*(4f) 5(4f)

f codimension of the stratum * excluding exceptional values of the moduli. We complete in this section the list in Theorem 2.3 and obtain the remaining germs of codimension < 4. These germs are those that can not be written in the form THEOREM 2.4 The A-dasses of the singularities R 3 , 0 -> R 2 , 0 of corank I and Aecodimension (or codimension, of the stratum in the presence of moduli) < 4 that are not listed in Theorem, 2.3 are given in Table 1.

4

Projections of Hypersurfaces in R to Planes

Name NI N2 N3 N4 N5 NQ

N7 N8 N9 Nw Nu Ni2

287

Normal form 3 3 5 2 (x, xy + y + ay z + z ± z ) a / 0, 4a ± 27 ^ 0 3 3 3 2 (x, xy + y + ay z + z ), a j^0, 4a + 27 ^ 0 3 3 (x, xy + y + z ± y3*} 2 (x, xy ±y z -f z 3 ± z 5 ) 5 3 (x, xy ±y 4- yz 2 Hhz ) 2 (x, xy ±y z + z 3 ) 4 3 3 (x, xy + y + z ± y z) 3 4 3 6 5 4 (x, xy + z ±y + y z + ay z -f- b(y + \y z)), b^G) 2

(x, xy + yz ±y 4 + z5 + ay6 ) 3 4 (x, xy + y2z + yz ± z + az 6 ) 7 3 (x, xy ±y + yz2 _hz ) 2 3 4 2 (x, xyz±y z + Z3 + ax y + 6x2 z + cyz + z )

A e-cod 3 (2t) 4 (3t) 3 3 3 4 4 6 (4t) 5 (4t) 5 (4t) 4 7* (4f)

<5 5 5 4 5 5 5 5 6 6 6 7 4

Table 1 where X is a constant, and 6 denotes the determinacy degree of the germs. f codimension of the stratum * 46 - 1 ^ 0 and 46 - 1 + 6ac ^ 0. Proof: We are interested in germs of corank 1, so we can write F(p,y,z) — (x,g(x,y,z)} Let j2g — x0(x, y, z) + i/j(y, z ) . Then the germs that can not be written in the form (X,/(X,T/) ± z2) are those with 4J = 0- Therefore to complete the list in Theorem 2.3, we need to consider the cases where j2g = x(f)(x,y, z). It is not difficult to show that we can change coordinates so that j2F is given by (x,xy) or (x,0). We now follow these germs and carry out the classification inductively on the jet level, using the complete transversal method [3] and the "Transversal" package [14]. We start by considering the 2-jet (X,XT/). We have Fx = (l,y), Fy = (0,x), Fz = (0,0). Therefore, # 3 3,2 C and by Proposition 2.2 any 3-jet with a 2-jet (x, xy) is equivalent to (x, xy + ay'2z + by3+cyz2+dz3) for some a, 6, c, d 6 R. We can now make linear changes of coordinates and obtain the following orbits in J3(3, 2):

(x,xy ± y2z + z 3 ), (x,xy + z3), (x,xy± y3 + yz2), (x,xy + yz2), (x,xy),

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where a in the first orbit is a smooth modulus. Using "Transversal" we show that the last 2 cases lead to germs of Ae- codimerision > 5 and therefore will not be followed. • The 3-jet (x, xy 4- y3 4- o/y2z 4- z3): Using "Transversal" we show that the 4- transversal is empty and the 5-transversal is given by R.{z5}. The orbits in,/ 5 (3, 2) are (x,xy+y3+ay2z+z3±z5) if 4a 3 ±27 ^ 0 and a ^ 0, arid ( x , x y + y3 + ay2z 4- z 3 ) if 4a3 4- 27 ^ 0 and a / 0. Both germs are 5-determined, the first (N\) has codimerision 3 (the codimension of the stratum is 2) and the second (N^) is of codimension 4 (the codimension of the stratum is 3). The exceptional case 4a3 ± 27 = 0 will not be followed as the current version of "Transversal" does deal with constant such as (27/4) 2 / 3 . We shall only consider the case o = 0. • The 3-jet (x, xy 4- y A 4-transversal is given by R.{y 3 z}. The orbits in J 4 (3, 2) are (x, xyjt-y'3+z3±y3z) and (x,xy + y3 4-z 3 ). The first case (N^) is 5-determined and has codimension 3. For the second case, a 5-transversal is given by R.{?/4z} so in J°(3, 2) we have two orbits, namely (x,xy + y3 + z3 ± y4z) and (x,xy + y3 4-z 3 ). The form Ay is 6-determined and has codimension 4. The later yields germs of codimension > 5. • The 3-jet (x, xy ± y2z 4-z 3 ): The 4-transversal is empty and a 5-transversal is given by R.{z°}. The orbits in J°(3, 2) are (x,xy±y2z + z3±z°) and (x,xy±y2z + z3}. Both germs are 5-determined with the first (A^) having codimension 3 and the second (A^) is of codimension 4. • The 3-jet (x,xy 4- z 3 ): A 4-transversal is given by R.-jj/4, y3z} so the orbits in ./4(3, 2) are (x,xy 4- z3 + 3 y z ± y4} and (x , xy + z3 + y3z] . A 5-transversal of the first germ is given by R. {y*z} so any germ in -7° (3, 2) with a 4-jet (x, xy + z3 + y3z ± y4} is equivalent to (x, xy + z3 4- r/3z ± r/4 4- «// 4 z) for some a 6 ]R. Here the parameter a is a smooth modulus. The orbits in -76(3, 2) are given by (.x, xy 4- z3 ± y4 +y3z + ay4z 4- b(y6 4- A?y°z)) where b is also a smooth modulus and A is a constant. These germs (As) are 6-determined and have codimension 6 (the codimension of the stratum is 4) provided 6 7 ^ 0 . The orbit in -7° (3, 2) whose 4-jet is equal to (x, xy + z3 4- y3z) are (x,xy 4- z3 43 y z ± y°) and (x. xy + z3 + y3z). These lead to germs of codimension greater or equal to 5. • The 3-jet (x, xy ± y3 + yz2): The 4-transversal is empty and a 5-transversal is given by R.{z°}, so the orbits in J5(3, 2) are (x, xy ± y3 + yz2 + z 5 ) and (.x, xy ± y3 + yz2). The first germ (A5) is 5determined and has codimension 3. The second germ has an empty 6-transversal but a 7-transversal is given by R.{z7}. In J7(3, 2), the orbit (7Vn) (x, xy ± y3 4- yz2 + z 7 ) is 7-deterrnined and has codimension 4. The remaining orbit leads to germs of higher codimensions. • The 3-jet (x, xy 4- y z 2 ) : A 4-transversal is given by R.jty 4 } and a 5-transversal of the orbit (x, xy+yz2±y4) is given by R.{z°}. The orbits in J5(3, 2) are (x,xy + yz2 ± y4 4- z 5 ) and (x,xy + yz2 ± y4). This second orbit generates germs of codimension > 5. A 6-transversal of the first germ is given by R.{?y6}, therefore the germs in J°(3, 2) with a 5- jet as above are equivalent to (x, xy + yz2 ± y4 4- z° 4- ayQ), for some o, 6 R. In this case

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289

a is a smooth modulus. These germs (Ng) are 6-determined and have codimension 5 (the codimension of the stratum is equal to 4). • The 3-jet (x, xy + y2z): A 4-transversal is given by R.{z4,yz3} and the orbits in J 4 (3,2) are (x,xy + 2 y z + yz3 ± z 4 ) and (x,xy + y2z ± z 4 ). The second orbit generates germs of higher codimension. A 5-transversal of the first orbit is empty and a 6-transversal is given by R.{z6}. Therefore any germ in J 6 (3,2) with a 5-jet as above is equivalent to (x, xy+y2z+yz3±z4+az6) for some a E R. Here a is a smooth modulus. These germs (NIQ) are 6-determined and have codimension 5 (the codimension of the stratum is equal to 4).

A 4-transversal is given by R.{xyz, z3,y2z, x2y,x2z}. There are various cases to consider here but the least degenerate case (we are seeking strata of codimension < 4) is (x,xyz ± y2z + z3 •+- ax2y + bx2z). Then a 5-transversal is given by R.{?/z3, z4}. The least degenerate case is given by (x, xyz ± y2z + z3 + ax2y -f bx2z + cyz3 + z4} (Ni2) which is 4-determined and has codimension 7 (the codimension of the stratum is 4), provided 46 — 1 ^ 0 and 46 — 1 + 6ac ^ 0. Any other case is part of (or leads to) a stratum with Xe-codimension greater or equal to 5. D We also use "Transversal" to obtain normal forms for the versal unfoldings of the germs in Table 1. PROPOSITION 2.5 Versal unfoldings of the germs listed in Table 1 are given in Table 2.

Name Ivi N2 HT

yVo

A/4 A^5 AT TVe N? Ng NQ A^io

Versal unfolding (x, xy + y3 4- ay'2z + z3 ± z5 + u\z + (x,xy + y3+ay2z + z3+uiz + u2y2 + (

( x XII 4" II

^

4" Z

^

I

^

i II' Z ~\~ Mi Z 4" WoWZ y z + z ± z5 + w x z + W2?/2 y3 + yz2 + z5 + u\z + u2y2 2

3

(x, xy ± (x, x?/ ± + u3z3} / i 9 ^ 9 4 \ (a;, a^?/ ±y z + z + u\z + u2y + u3y *} + u^y^z) 3 3 4 U 2 (x, xy + y + z ± y z + MIZ 4- ^^ + 3y z •+- u^y3z] (x, xy + z3 ± y4 4- t/3z 4- ay 4 z + 6y6 4- c|/5z 4- u\z + u2yz 4- u3y2 + u^y3} (x, xy + yz2 ± it/4 4- z5 + ay6 + u\z 4- ^2?/2 + U3y3 + ^z 3 ) (x, xy 4- £/ 2 z + yz3 ± z4 4- az6 + wiz + ^2Z 2 4- u3yz2 4- W4Z 3 ) (x, xy ± y3 4- yz 2 + z7 4- wiz 4- u2y2 + u3y3 4- ^y 5 ) (x, xyz ± t/ 2 z 4- z3 4- ax 2 y 4- 6x2z + cyz3 + z4 4- wiy 4- u2z + u3xy + Table 2

REMARK 2.1 It is of interest to calculate more invariants associated to the singularities in Table 1. This is part of future investigation.

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GEOMETRIC CRITERIA FOR RECOGNITION OF THE SINGULARITIES OF Ae-codim < 1

Geometric criteria for recognition of the singularities of »4e-codim < 1 of germs R 2 .0 -> E 2 ,0 are given in [15] and [28]. We adapt these criteria to germs F : R 3 ,0 -» R 2 ,0. When F has corank 1, we can change coordinates and write F(x, y, z) = (:r, /(x, y, z}}. The differential of F at (x, //, z) is then given by 1 -(• •

0

}

^-(r

0 \ ^-(r v

so that the critical set of F is given by

Let G : R 3 ,0 -» R 2 .0 such that G(x,y,z) = ( f v ( x , y , z ) , f z ( x , y , z ) ) . S = G^O) is locally a smooth curve if G1 is regular, that is, if G has maximal rank at (0,0,0). It is easy to show that if E is smooth, then F can be written in some coordinate systems in the form ( x , f ( x , y ) ± z 2 ). In this case S = {(.r,?y,0); f y ( x , y ) — 0}. Let 0 : /, 0 —» M 3 , 0 be a parametrization of S, where / is a neighbourhood of 0 in R. The order of contact of E with the kernel of L>F(0, 0, 0) (ker(DF(Q, 0,0))) is the order of the vanishing of the derivatives of F o 0 at 0. This order of contact is independent of the parametrization of E and is an ^-invariant of the map F. We shall use this order of contact to recognize geometrically the fold, cusp and swallowtail singularities in the list of Theorem 2.3. When the critical set is smooth, we can set F(.x, ?/, z) = (x, f ( x , y ] ± z2} as the order of contact is invariant. The following is well known: PROPOSITION 3.1 Let F : R3, 0 -> R 2 ,0 a singular germ, with a smooth critical set E and let

Proof:

Let F ( x ,' y , z ) = (x,/(x,?y) ± z 2 ) and 0 a local parametrization of S =

So 0(0 = (0i (0,0) with 0i(0) = 0 and §0(£) = (-f^-(0i(0), ^(0iW)> °)Then

(3.1)

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291

and at the origin &(F o 0)(0) = -0(0,0). ( ^

.} . Therefore &(F o 0)(0) ^ 0

VaxvU'W/

if and only if 0(0,0) ^ 0, if and only if F is a fold. Differentiating (3.1) we get:

If £ (Fo 0)(0) = 0, that is

(0,0) = 0, then

Hence ^(F o 0)(0) = 0 and |^(F o 0)(0) ^ 0 if and only if 0(0,0) = 0 and ^ (0, 0) ^ 0 and 0 (0, 0) ^ 0, if and only if F is a cusp.

D

The conditions in Proposition 3.1 reflect the order of contact of S with the kernel of DF(0,0,0). COROLLARY 3.2 A singular germ F : R 3 ,0 —* R 2 ,0 of corank 1 and with a smooth critical set is a fold if and only if S is transversal to the set ker(DF(0, 0, 0)) at the origin. It is a cusp if and only if S and ker(DF(Q, 0,0)) have 2-point contact at the origin.

We observe that the previous two results are the statement that the fold and cusp orbits coincide with Boardman strata S1'0 and E1'1'0, respectively. PROPOSITION 3.3 Let F : R 3 ,0 -»• R 2 ,0 be a singular germ with a smooth critical set E and let 0 be a parametrization of S. Then F is a swallowtail if and only if

= 0 and Proof:

F o 0

0.

From the proof of Proposition 3.1 we have ^(F o 0)(0) = jj^(F o 0)(0) = 0

if and only if 0(0,0) = 0(0, 0).|^(0,0) = 0. Differentiating the expression (3.2) and setting 0(0,0) = 0(0,0). ^-(0,0) = y y X V 0, yield

_ Then, f (F o )(0) = j^(F o )(0) = 0 and j^(F o 0)(0) ^ 0 if and only if 0(0,0) = 0(0,0) = 0, |^(0,0) ^ 0, and 0(0,0) ^ 0, if and only if F is a swallowtail. D Again the conditions in Proposition 3.3 express the order contact of the critical set S with the fcer(DF(0,0,0)).

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COROLLARY 3.4 A singular germ, F : R 3 ,0 ->• R 2 ,0 of corank 1 and with a smooth critical set is a swallowtail if and only if the critical set S has contact 3 with the kernel of DF(Q, 0, 0) at the origin. When F has a lips/beaks singularity, its critical set £ is singular. Then we need additional algebraic conditions for recognizing these singularities. PROPOSITION 3.5 A singular germ that can be written in the form F = (x, /(;r, y)± z2) ts a lips/beaks if and only if 0(0,0) = |^(0,0) - 0, 0(0,0) + 0 and

Proof: The proof follows by a direct calculation.

D

PROPOSITION 3.6 The germ f£ :R 2 ,0-^R,0 ?:.s a germ of a Morse function if and only »/ 0(0,0) = ^(0,0) = Qand (J^) 2 (0,0) - ^(0, 0).0(0, 0) ^ 0. Proof: The proof follows immediately by considering the Taylor expression of -^ at the origin,

D

As ,4-equivalence preserves the singularity of the critical set, we have the following result .

COROLLARY 3.7 Let F ( x , y , z ) = (x,f(x,y) ± z2)). If 0(0,0) ^ 0, then F is a lips/beaks if and only if the critical set has a nan- degenerate curve singularity. REMARK 3.1 The above geometric criteria do not extend to higher codimension singularities. The reason is that when S is smooth its order of contact k with the kernel of DF(Q, 0,0) determines the Ak-\ /C-class of the projection. When k > 4, there are several *4-orbits of the projection inside the Afc_i-orbit. One needs algebraic conditions to distinguish between these .4-orbits.

4

GEOMETRIC CRITERIA FOR RECOGNITION OF THE VERSAL UNFOLDINGS SINGULARITIES OF A-codim - 1

Geometric criteria for determining when a deformation of a lips/beaks or swallowtail singularity of a germ R 2 ,0 —» R 2 , 0 is versal are given in [27]. We extend these results to germs F : M 3 ,0 —•> E 2 ,0. We deal in more details with the lips/beaks case. Let F : E3 x R,0 -* R2 x R,0 be an unfolding of a germ F0(x,y,z,Q) = ( x , f o ( x , y ) ± z2) that has a lips/beaks singularity at the origin. Then the critical set of FQ can be considered as a plane curve with a Morse singularity and the unfolding F induces an unfolding of this Morse singularity.

Projections of Hypersurfaces in R4 to Planes

293

THEOREM 4.1 An unfolding F of a lips/beaks singularity of FQ is Ae-versal if and only if induces a K,-versal unfolding of the Morse singularity of the critical set of FQ. Proof: Let F be an unfolding of a lips/beaks and write F(x,y,z,u] =

(Fi(x,y,z,u),F2(x,y,z,u),u).

CLAIM: F is equivalent to an unfolding written in the form

Proof of the claim: Without loss of generality we can assume that ^j^- ^ 0 at the origin. Then the local diffeomorphism R3 x R

H :

-> R3 x R

makes the following diagram commute R3 x R (x, y, z, u)

F

;

R2 x R (Fi (x, y, z, u), F2(x, y, z, w), w) lid

ff F

R3 x R ((Fi (x, y, z, w), y, z, M) _,,

^

,

R2 x R (Fi (x, y, z, IT,), F2(x, y, z, w), w)

i

where F2(x, y, z, u) — F2(?ri o FI (x, y, z, n), y, z, w) and TTI is the projection to the first coordinate. Observe that wioFoH~l =TTI , therefore F o H~l = (x,F2(x,y,z,u),n) and hence F and F are equivalents unfoldings (see [16]). Therefore we can set F(x,y,z,w) = (x,^(x,y,z) 4- ug2(x, y,z, u), w). The germ (x, <7i(x,y, z)) has a lips/beaks singularity at the origin, consequently, changes of coordinates in the source and target reduce FQ to the form (x, y 3 ±x 2 y±z 2 ). In this new system of coordinates F(x y z u) — (x 11 i x 17 i z 4" 17/09fx 17 z w) w). Since a lips/beaks singularity is 3-determined, we can work in the J 3 (x, y, z, w), and write 1 F\X 11 Z U) ~~ (^C "U

i

X u zt ,£

~}~ Ot\X Z UL\UL ~\~ j3\Z UiUIJ "4" fti U1J

~\~ d^UXIJ

Ui

Now successive explicit changes of coordinates in the source and target reduce this 3-jet to the form j3F(x, y, z, u) = (x, y3 ± x 2 y ± z2 + V'( M )?/ 5 M )Then F is a versa! unfolding of FQ if and only if ^'(0) ^ 0 (see [16]). We have E — {(x,y,z, u) : z = 0 and
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Then f£(0) = V/(0), and hence F is versal if and only if fg(0) / 0, if and only if a is a versal unfolding of the Morse singularity of S. D Let S^ denotes the critical set of the restriction of F to E. Then proceeding as above, we show the following. THEOREM 4.2 Let F(x, y, z, u) be an unfolding of a swallowtail singularity. Then F is Ac-versal if and only if Up is a smooth curve. REMARK 4.1 The results in Sections 3 and 4 can be extended to ^-singularities of map-germs / : R", 0 —* M p , 0, n > p, with *4e-codimension 1. This can be done by following the same procedure as above and using the normal forms for *4e-codimension 1 singularities given by Goryunov in [9]. We are currently looking for similar characterizations for wider classes of singularities.

5

A STRATIFICATION OF THE SPACE OF SINGULAR PLANES

Recall from the introduction that there is a natural 4-parameter family of projections H:

M X (7(2,4) (p,u)

-> H-*

R2 (, < p , 6 > )

where a and b form an orthonormal basis of the plane of projection u G G(2,4) (II is a 4-parameter family because the ^-equivalence class of Hu does not depend on the choice of orthonormal basis a, b}. We start with an elementary lemma that provides a simple way of writing the family II. LEMMA 5.1 The map IT can be written locally in the form II:R3xR4,0

->

R 2 ,0

Proof: Given an embedding F : M —>• R 4 , we can put F in Monge form in a neighbourhood of any point p, that is we can consider F : R 3 ,0 —> R 4 ,0 with F(x, y, z) — (x, y, z, /(x, y, z ) } . We assume, without loss of generality, that the initial plane of projection u0 is generated by the vectors a = (1,0,0,0) and b — (0,0,0, 1). Then vectors (l,d r i,/?L, 71) and («2 5/^2? 72,1) generate all the planes near MO- We claim that there are always linearly independent vectors of the form (l,/3i, 71, 0) and (0, #2,72, 1) that generate these planes. Indeed, by setting £1 = £4 = 1/(1 - «27i), £2 = — 7i/(l ~ ^271), an(4 Cs = ~^2/(l — ^271), we have

, 71) +^4(tt2,#2,72,l) = (0, (#2 Since Tlu depends only on the plane w, we can construct an orthonormal basis of u from the vectors a = (1, #1,71,0) and b = (0, #2,72, !)• Then, by changing

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coordinates in the parameter space (ai,/3i,a2,$2) and by scaling, we can write II in the form given above. D Given a generic embedded hypersurface M in Monge form (x,y,z,f(x,y,z)) at some point p, we can find the algebraic conditions on the coefficients of the Taylor expansion of / for IIM to have one of the singularities in Theorems 2.3 and 2.4 and for II to be a versa! unfolding of these singularities (see [22]). These conditions, in turn, give infinitesimal information concerning the embedding of M at p. However, they are in general too complicated and difficult to interpret geometrically. But for the singularities of codimension 1, we have the following corollary of Theorems 4.1 and 4.2. We observe that using Lemma 5.1 we can show that in the lips/beaks case II always induces a K-versal unfolding of the Morse singularity of IIW and for the swallowtail singularity £n' is always smooth. COROLLARY 5.2 The family II is always a versed unfolding of the lips/beaks and swallowtail singularities of n u . If a generic embedded hypersurface M C R4 is given in Monge form w = /(x, ?/, z\ then Hu is singular if and only if the normal direction to M at the origin, (0,0,0,1), belongs to the plane u. So we can take a = (0,0,0,1) and b = (ai,a2,c*3,0), with H&H = 1, as the generators of the plane u. Therefore, the singular planes of projections can be identified with the sphere S2 — {b = (ai, c*2, «3,0), ||b|| = 1}. We shall stratify this sphere according to the singularity type of the projection Tlu. We rotate the coordinates axis so that j2f = a\x2 -+- a^y2 + a^z2, where a» = ^-, i — 1,2,3, with KI the principal curvatures of M at the origin. In Proposition 5.3, we obtain a geometric characterization of the singularities of the projections at elliptic and hyperbolic points of the embedding. PROPOSITION 5.3 Let M be a generic embedded hypersurface in R 4 7 p a point on M, and u a singular plane of the projection determined by a direction b — (a 1 ,a 2 ,a 3 ,0)e5 2 . (i) If p is an elliptic point, then the projection Tlu has a singularity of type fold for any direction on S2. (ii) If p is a hyperbolic point, then Tlu has a fold singularity for any direction on S2 except on two circles where the singularity is in general of cusp type. On these circles, and at most hyperbolic points on M, there are up to 2k directions, k — 0, • • • 6, where the singularities are of swallowtail type. At certain hyperbolic points on M these directions could yield singularities of higher codimensions (generically of type 6,7, 8 of Theorem 2.3, respectively on a surface in M, on a curve on this surface and at isolated points on this curve). Proof: Let b — (ai,a2, 013,0) G S2. Then, as before, the projection to the plane u generated by b and (0,0,0,1) is given by

Hu = (aix + a%y + a3z, f ( x , y, z } } . We can assume without loss of generality that c*i ^ 0. We can then change coordinates in the source and target and write IIU = (x,a2f((x — a^y — «3^)/«i,y, z), so that J2IIU — (x,

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Let q(y, z] = (a^n2 + «i« 2 )?/ 2 + laia^a^yz + (a\a2 + o,^a2}z2 be the quadratic form obtained by setting x = 0 in the second component of j'2IIu. This form is degenerate if and only if = 0.

5-1

At an elliptic point (case (i)), the coefficients a?: have the same sign, so equation (5.1) does not hold. Therefore Hu ~ (z, y2 ± z2) which is a fold singularity. At a hyperbolic point (case (ii)) equation (5.1) determines a cone. This cone intersects the sphere S2 in two circles (we only need to consider one of them as b and —b define the same projection). Away from these circles, the singularity of Uu is clearly of type fold. On the circles the quadratic q is degenerate. As the point in consideration is not parabolic, we can suppose without loss of generality that the coefficient of z'2 in q is different from zero. Then the change coordinates a2 Z = z-\ --Oai£*? yield a a a$y•' and further changes o of coordinates in the source and target o

As the point is hyperbolic, the coefficient of xy is different from zero (note that we assumed a\ ^ 0, and if a-2 — 0 then the coefficient of z2 in q must be zero, which is not the case from our assumption above.) We now need to analyze the coefficient of y3 in the second component of Hu. To simplify the calculations, we set ai — 1, so equation (5.1) determines a circle (a curve of degree 2). Then the coefficient of ?y3 is a polynomial of degree 6 in a-} and 0:3. Therefore this polynomial can vanish at 0,2,4,6,8,10, or 12 directions on the circle. Away from these directions the singularity of Hu is a cusp. At one of these directions we need to analyze the higher jets of the second component of Hu which depend only on the Taylor expansion of the function / giving the surface M in Monge form. (Xote that the values of a,, i = 1,2,3 are now fixed). Without imposing any condition on the surface M, the coefficient of y4 in YLU does not vanish and the singularity of TIU is of type swallowtail. When the coefficient of y4 vanishes (this determines a 2-dimensional surface on M) the singularity of Hu is generically of type 6 (Theorem 2.3). There might be a curve on this surface where the singularity becomes of type 7 and a point on this curve where it is of type 8. D With the same hypothesis of the previous Proposition, we can proceed as above to obtain the geometric characterization at parabolic and partial umbilic points of the embedding. PROPOSITION 5.4 (i) If p is a parabolic point (say ft2 = 0), then Hu is a fold except on the circle a2 — 0 where it has generically a singularity of type 4 2 . For 0 or 2 directions on this circle the singularity is generically of type 4$. These directions could yield singularities of type 4$ and 4s at certain points (respectively on a curve on the parabolic set and at isolated points on this curve). Also on the circle a 2 = 0, and for some curve on the parabolic set (labelled the 11-,-curve), the singularity is generically of type 11,5. For 0; 4 or 8 directions on the circle the singularity is of type lly; for 0 or 4 directions the singularity is of type 12; and for 0 or 2 directions it is of type 16. For these directions and at possible isolated

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points on the lls-curve the singularity of the projection becomes respectively of type 119, 13 or 17. Furthermore, if KIK^ < 0, there are two more circles (which intersect transversely the circle a-2 = 0) where the singularity is generically of type NI. At possible directions on these circles and on some curve on the parabolic set (resp. isolated points on this curve) the singularity becomes of type N%, • • • , N$ (resp. NQ, • • • , NH). (ii) Ifp is a partial umbilic point (say KI = K? = 0 and KS ^ Q), then the projection has generically a singularity of type 42- There are 0 or 2 (resp. 0, 1 or 3) circles where the singularity is of type 4s (resp. llsj. At isolated points on these circles the singularity becomes of type 44 (resp. ll?, 12 or 16). There is an unique direction (not on the above circles) where the singularity is of type Acknowledgements: This work is part of my Ph.D. thesis. I would like to thank FAPESP for financial support. I am very grateful to Neil Kirk for letting me use his software "Transversal" and especially to Farid Tari for his extremely stimulating supervision. I would also like to thank Maria A. S. Ruas and the referee for useful comments.

REFERENCES [1] T. Banchoff, T. Gaffney, C. McCrory. Cusps of Gauss Mappings. Pitman, New York, 1982. [2] J.W. Bruce, Generic geometry and duality. In: Singularities, ed. J.P. Brasselet, Lille 1991, London Math. Soc. Lecture Note Series 201, Cambridge University Press, pp. 29-60, 1994. [3] J.W. Bruce, N. Kirk, A.A. du Plessis. Complete Transversals and the classifications of singularities. Nonlinearity. 10 1:253-275, 1997. [4] J.W. Bruce, N.P. Kirk, J.M. West. Classification of maps-germs from surfaces to four-space. Preprint. University of Liverpool, 1995. [5] J.W. Bruce, A.C. Nogueira. Surfaces in R4 and duality, Quart. J. Math. Oxford Ser. 49 2:196:433-443, 1998. [6] J.W. Bruce, A.C. Nogueira. Surfaces in R4 and duality II. Preprint 2000. [7] J.W. Bruce, A.A. du Plessis, C.T.C. Wall. Determinacy and unipotency. Invent. Math. 88:521-554, 1987. [8] J.W. Bruce, F. Tari. Families of surfaces in R 4 . To appear. [9] V. V. Goryunov. Singularities of projections of full intersections. Journal of Soviet Mathematics. 27:2785-2811, 1984. [10] W. Hawes. Multi-dimensional motions of the plane and space. Ph.D. Thesis, University of Liverpool, 1994.

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[11] C.A. Hobbs, X. P. Kirk. On the classification and bifurcation of multigerms of maps from surfaces to 3-space. To appear in Mathematica Scandinavica. [12] K. Houston, X.P. Kirk. On the classification and geometry of map-germs from 3-space to 4-space. In: Proceedings of the European Singularities Conference in honour of C.T.C. Wall on the occasion of his 60th Birthday, eds. J.W. Bruce, D. Mond, Singularity Theory, London Math. Soc. Lecture Note Series 263, Cambridge University Press, pp.325-351, 1999. [13] X.P. Kirk. Computational aspects of singularity theory. Ph.D. Thesis, University of Liverpool, 1993. [14] X.P. Kirk. Transversal. A maple package for singularity theory, Version 3.1. University of Liverpool, 1998. [15] Yung-Chen Lu. Singularity theory and an introduction to catastrophe theory. Springer-Verlag, 1976. [16] J. Martinet. Singularities of Smooth Functions and Maps. London Math. Soc. Lecture Xote Series 58, Cambridge University Press, 1982. [17] D.K.H. Mochida. Geometria generica de subvariedades em codimensao maior que urn em R n . Ph.D. Thesis, University of Sao Paulo, ICMC - Sao Carlos, 1993. [18] D.K.H. Mochida, M. C. Romero Fuster, M. A. S. Ruas. The geometry of surfaces in 4-space from a contact viewpoint. Geom. Dedicata 54 3:323-332, 1995. [19] D. Mond. On the classification of germs of maps from R 2 to E3. Proc. London Math. Soc. 50 3:333-369, 1985. [20] J.A. Montaldi. On contact between submanifolds. Michigan Math. J. 33:195-199, 1986. [21] J.A. Montaldi. On generic composite of maps. Bull. Lond. Math. Soc. 23:81-85, 1991. [22] A.C. Xabarro. Sobre a geometria local de hipersuperffcies em E4. Ph.D. Thesis, University of Sao Paulo, ICMC - Sao Carlos, 2000. [23] A.C. Xabarro. Duality and contact of hyper surf aces in R4 with hyperplanes and lines. Preprint, 2002. [24] A.C. Xogueira. Superficies em 3R4 e dualidade. Ph.D. Thesis, University of Sao Paulo, ICMC - Sao Carlos, 1998. [25] J.H. Rieger. Families of maps from the plane to the plane. J. London Math. Soc. 36 2:351-369, 1987. [26] J.H. Rieger, M.A.S. Ruas. Classification of ^-simple germs from kn to fc2. Compositio Math. 79 1:99-108, 1991. [27] J.E. Rycroft. A geometrical investigation into the projections of surfaces and space curves. Ph.D. Thesis, University of Liverpool, 1992.

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[28] F. Tari. Some applications of singularity theory to the geometry of curves and surfaces. Ph.D. Thesis, University of Liverpool, 1990. [29] J.M. West. The differential geometry of the crosscap. Ph.D. thesis, University of Liverpool, 1995.

Frobenius Manifolds and Hypersurface Singularities GLAUS HERTLING Mathematisches Institut der Universitat Bonn Beringstrafie 1, 53115 Bonn, Germany E-mail: [email protected]

This is a survey article on Frobenius manifolds and hypersurface singularities. It covers the four lectures of a minicourse at the 6th Workshop on Real and Complex Singularities (17 - 21 July 2000, ICMC-USP, Sao Carlos, Brazil). 1st lecture: Multiplication on the tangent bundle 2nd lecture: Frobenius manifolds and their structure connections 3rd lecture: Construction in singularity theory 4th lecture: Global moduli spaces for hypersurface singularities A Frobenius manifold is a manifold with a metric and a multiplication on the tangent bundle which harmonize in the most natural way. This notion was defined by Dubrovin 1991, motivated by quantum cohomology, topological field theory and mirror symmetry. But the first big class of Frobenius manifolds had been constructed implicitly already in 1983 by K. Saito and M. Saito in singularity theory: the base space of a semiuniversal unfolding of an isolated hypersurface singularity can be equipped with the structure of a Frobenius manifold. For a long time this construction had not been taken up. Even now results which put it in a precise way into the picture of mirror symmetry have yet to be found. But recently I found an application within singularity theory: a construction of global moduli spaces for isolated hypersurface singularities. 301

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The four lectures will focus on the general theory of Frobenius manifolds (1st and 2nd), the construction in singularity theory (3rd) and this application (4th). More detailed comments about the contents of the lectures can be found at their beginnings. The 2nd lecture presupposes some familiarity with meromorphic connections, the 3rd and 4th also with hypersurface singularities. The material in this survey article is taken from [14] [15] and aims at the role of Frobenius manifolds in singularity theory. For many different aspects of Frobenius manifolds and their relations to meromorphic connections, topological field theory, quantum cohomology, and mirror symmetry we recommend the books [4] [20] [21]. We hope that this survey article will stimulate the search for other applications and for stronger links between singularity theory on the one hand and topological field theory and mirror symmetry on the other hand.

1

MULTIPLICATION ON THE TANGENT BUNDLE

The subject of the first lecture is the local structure of a complex manifold with a multiplication on the holomorphic tangent bundle. An extensive study of this has been made in [14]. In this lecture central results and examples from [14] are put together. First, the pointwise structure is discussed: the structure of the tangent spaces as finite dimensional C-algebras. Then we introduce the analytic spectrum of a manifold with multiplication, a subvariety of the cotangent bundle which determines the multiplication. If the multiplication satisfies a certain integrability condition then the manifold is called an F-manifold. This notion is most central and was defined first in [13] (cf. [20] I§5). It has a good local decomposition property. If the multiplication is generically semisimple then the analytic spectrum is a Lagrange variety if and only if the manifold is an F-manifold. This makes it possible to interpret many results in [8] as results on F-manifolds. Important classes of F-manifolds are the base spaces of semiuniversal unfoldings of hypersurface and boundary singularities (cf. the 3rd lecture) and the complex orbit spaces of finite Coxeter groups. One can distinguish the last ones in a way which extends the correspondence between certain Coxeter groups and the simple hypersurface or boundary singularities. A manifold with multiplication (M,o,e) will always mean following: M is a complex manifold, dim M — m > 1, TM its holomorphic tangent bundle, TM the sheaf of germs of holomorphic sections of TM, o : TM x TM -» TM , ( X , Y ) ^ X o Y

an C?M-bilinear commutative and associative multiplication. Then o : TM®TM ~* TM is a (2, l)-tensor. The multiplication is supposed to have a unit everywhere, and e 6 F(M,T M ) is the global unit field. Fix t 6 M. Then (TfM, o,e t ) is an algebra of dimension m. It has a unique

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decomposition (cf. e.g. [14] Lemma 1.1) into local algebras,

i(t) TtM = 0(TtM ) fc fc=i

with

k=l

(TtM, o, e fc )

a local algebra with unit e fc ,

(TtM}j o (TtM)k = 0

torj^k

This is the simultaneous eigenspace decomposition for the commuting endomorphisms Xo : TtM -» TtM for X e r t M. EXAMPLE 1.1 Consider a polynomial / e C[x0, ...,£„] such that / : C n+1 -> C has only isolated singularities. ox0 5 "••> dxn is an ideal sheaf, and (Jf)poi

C C[rr] denotes the corresponding ideal in C[x]. Then

C[x]

is the decomposition into local algebras. Let (M, o, e) be as above. For any k = 1, ..., /(£), the map A^ : TtM —> C defined by efc maximal ideal in (TtM}k

i-» —>

1, 0 ,

(T t M)j

->

0

foij^k

is an algebra homomorphism. For X G T^M one obviously has A fc pO = eigenvalue of Xo : (TtM)k -* (TtM)k. Then (cf. [14] Lemma 1.1)

So its points correspond one-to-one to the local subalgebras in the tangent spaces of M as algebras. The analytic spectrum of (M, o,e) is as a set L := Specan(TM,o) = |J Hom a ^(T t M,C) C T*M .

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It has a canonical complex structure (L,0£), which is locally defined as follows. Let e = <5i,...,<5 m be a local basis of vector fields, and y\,..., ym the induced fiberwise linear- functions on T*M. Setting 8% o 61 = ^kaii ' ^k w^h a^j e C?M> the analytic spectrum is denned by the ideal / = (yi - 1, yiyj - LV^ a^ • yk] fc

wirh(9 L :=0T*M/I\LThe analytic spectrum L has good properties ([14] Theorem 2.1): The projection TT : L —-> M is finite and flat of degree m = dim M . The map a : TM —> X H-» o

K*OL (by X induced function on T*M)| L 'mult.

is an isomorphism of free OM -modules of rank m, and of OM -algebras. Therefore the multiplication o on TM is determined by L. The function a(X) can be called the "eigenvalue function" of Xo because of a(X)(\k} — The caustic 1C := {t <E M | l(t) < generic value} C M is a hypersurface or empty (the proof in [14] (Prop. 2.4) for the case rn = generic value of l(t] works for any generic value of /(/)). The multiplication is called semisimple if l(t) = rn Vt (E M . Then there is locally a basis e l 5 ..., em of vector fields with ei o ej = S^e^. (M, o, e) is called massive if l(t) = m, for t e M — /C. DEFINITION 1.2 (7^, /20y 7§5, [Uj Def. 3.2) a) (M, o,e) is on F-manifold if the integrability condition holds:

V X , F e T M Lie XoY (o) =XoLie y (o)-f y oLiex(o) .

(1.1)

&,) Le^ (M, o, e) fee an F-manifold. Then E e TM ^ a^ Euler field of weight c E C i/ LieE(o)=c-o

(1.2)

(Euler field of weight 1 =: Euler field). (1.1) for X = y = e gives Lie e (o) = 2 • Lie e (o), so Lie e (o) = 0, and e is an Euler field of weight 0. (1.2) can be written with vector fields X, Y and Lie brackets as [E, XoY]-[E,X]oY-Xo

[E, Y] = c • X o Y .

For X — Y = e it yields [e, E} = c • e. One can write (1.1) analogously with four vector fields and nine Lie brackets. PROPOSITION 1.3 ([14] Prop. 4-1) 7/Mi,M 2 are F-manifolds, then MI x M2 is an F-manifold. If E\,E
Frobenius Manifolds and Hypersurface Singularities

THEOREM 1.4 ([14] Thm. 4.2) Let (M,o,e) be an F-manifold. decomposition

305

Fix t E M. The

k=l

extends to a decomposition (Af,t) = ^[(Mk,t) of (M, t) into a product of germs of F-manifolds with TtMk * (TtM}k . An Euler field of weight c decomposes accordingly. The proof of Theorem 1.4 in [14] uses (1.1) in a way which justifies the name integrability condition. THEOREM 1.5 ([14] Thm. 6.2) Let (M, o, e) be a manifold with massive multiplication. Then the complex structure (L^O^} is reduced everywhere. The following conditions are equivalent: i) (M, o, e) is an F-manifold. ii) The vector fields e\,... , em defined locally in M — 1C with e^osj = 6^-6$ satisfy [e^e-] = 0 .

in) L is a Lagrange variety. Locally each 1-dimensional F-manifold is isomorphic to (C, o, e) with e = ^, u a coordinate on C. This F-manifold or a germ of it is called A\. Let (M, o,e) be a massive F-manifold. By Theorem 1.4, a germ in M — /C is isomorphic to A™. The corresponding local coordinates MI, ...,w m satisfy d &i = -—

and

e^ o GJ = Oij • e^ .

They are the canonical coordinates (Dubrovin). There each Euler field of weight 1 takes the form E = y^(ui -H r,)ei with rt e C . Let (M, o, e, E) be a massive F-manifold with Euler field E. The discriminant of the Euler field E is V ^detEo-1 CM Z> is a free divisor with DerAf(log£>) = E o TM ([14] Thm. 13.1). T> is everywhere transverse to e (consider the development of T> in PT*M to make this precise). It is a rich geometric object. The pair (Z>,e) determines the multiplication in a simple explicit way ([14] Cor. 11.6), which is explained in the Examples 1.6 (2) and (3).

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EXAMPLES 1.6 1) The base spaces of semiuniversal unfoldings of hypersurface singularities and of boundary singularities are massive F-manifolds (cf. the 3rd lecture and [14] Chapters 16 and 17). 2) / 2 (m), m > 3: M := C2 with coordinates (ii,* 2 ) = i, f- '•= -jr

M

\ - —^% = 0} .

In M — {t | t-2 = 0} define two vector fields ei , 62 as in the picture: For t £ M — {t 0} the e-orbit of t intersects D in 2 points. One shifts the tangent hyperplanes to D at these points with the flow of e into TtM . There are unique vectors e\, e^ € T^M which are tangent to them and satisfy ei + e2 = e.

Figure 1

Define the multiplication by e^ o ej := ^e^ on M — {t t2 — 0}. Then one finds with some calculations ([14] Ch. 12): e = j- is the unit field, d

and (Af, o, e) is a massive F-manifold with Euler field

d i d E = ti— -\ t — oti m 2Tot-2 and caustic - {t | t2 = 0} .

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307

3) Let W be a finite irreducible Coxeter group acting on Rm and Cm .

I)

Cm

—>

Cm/W =:M^Cm

refl. hyper planes

—>•

T>

C(Xl,...,xm}w

—> C[*i,...,t ro ] with degti=di ,

h = d\ > d% > ... > dm-i > dm = 2 ,

(M, X>, e) determines a multiplication o in a way which generalizes example 2). For generic t one defines ei,...,e m as in 2): The e-orbit of t intersects T> in m points. One shifts the tangent hyperplanes to T> with e into TtM. There they are in general position. There are unique vectors ei, ...,em E TtM which are each tangent to the intersections of ra — 1 hyperplanes and which satisfy together ei 4- ••• -4- em = e. The multiplication is defined by ei o EJ :— <5^-ej on TtM '. It extends to TM. (M, o, e, E) is a massive F-manifold with Euler field E. All these are nontrivial statements which follow implicitly from [8] as well as from [4] (Lecture 4), cf. [14] (Ch. 18). Dubrovin proved, moreover, that the complex orbit spaces are massive Frobenius manifolds (cf. the next lecture). Givental characterized the corresponding analytic spectra in a way which can be translated into Theorem 1.7 a),b) (cf. [8] Thm. 14 and [14] Thm. 18.4). A massive F-manifold (M, o, e) is a called simple if it contains only finitely many isomorphism classes of germs of F-manifolds. THEOREM 1.7 a) (Givental) The F-manifold of a finite irreducible Coxeter group is simple with analytic spectrum L ^ C™-1 x {(ar,y) e C2 | x2 - yn = 0} .

n - 1: n =2: n =3 : n:

A fc , Dk, Ek , B fe , F4 , H3, H4 , J 2 (n + 2 ) .

b) (Givental) An irreducible germ (M, t) of a simple F-manifold with (L, A) = TT( plane curve singularities) is isomorphic to the germ (C.m/W, 0) for some finite irreducible Coxeter group W. c) ([14] Thm. 18.3) An irreducible germ (M,t) of a simple F-manifold with TtM a Frobenius algebra is isomorphic to the germ (C m /W,0) for some finite irreducible Coxeter group W.

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2

Hertling

FROBENIUS MANIFOLDS AND STRUCTURE CONNECTIONS

THEIR

The notion of a Frobenius manifold was introduced by Dubroviii [3] , motivated by topological field theory. It turns up now in many places, especially in quantum cohomology (cf. [20]), mirror symmetry, and in singularity theory (cf. the 3rd lecture), The first part of this lecture presents the definition arid elementary properties of Frobenius manifolds. A Frobenius manifold is an F-manifold with a metric satisfying certain conditions. The integrability condition of the F-manifold and another condition can be recast as the potentiality, which gives rise to a potential, the GromovWitten potential in the case of quantum cohomology. One distinguished class of Frobenius manifolds, which can be described in a relatively elementary way, are the complex orbit spaces of finite irreducible Coxeter groups. This description is due to Dubrovin ([4] Lecture 4). The metric had already been defined by K. Saito [22]. A conjecture of Dubrovin concerning them is proved in [14] (Thm. 19.3) using Theorem 1.7. Associated to a Frobenius manifold are two kinds of meromorphic connections: the 1st structure connections have irregular singularities along a smooth hypersurface, the 2nd structure connections have logarithmic singularities along a discriminant. In the singularity case some of them correspond to (rigidified, enriched versions of) oscillating integrals (the 1st) and the GauB-Manin connection (the 2nd). Therefore in Lecture 3 it will be very useful to be aware of the 2nd structure connections. The 2nd and 1st structure connections are treated in much more detail in [15] (Ch. 4) .The presentation there unifies results of K, Saito [23] [24], Dubrovin [4] (Lecture 3 -r Appendix G), and Manin [20] (II §§1,2). Some general material about meromorphic connections is collected in [15] (Ch. 3), and much more in [21]. Let (M , o, e) be a manifold with multiplication. Then o : TM ® 1~M —>• TM is a (2, l)-tensor. For any vector field X 6 TM, the Lie derivative Lie^(o) is also a (2, 1)tensor. If V is a connection on TM, then V^(o) is also a (2, 1) -tensor, and V(o) is a (3, l)-tensor. In the following, a metric g on M is a symmetric and nondegenerate (2,0)-tensor g : TM ® TM -> OM , g : TtM x TtM -> C .

Its Levi-Civita connection V is the unique connection which is torsion free, Vx^ — VYX = [X, Y], and satisfies V
.

THEOREM 2.1 ([14] Thm. 5.3) Let (M, o,e,_g) be a 'manifold with multiplication o, unit field e, and metric g, such that g is multiplication invariant, i.e. Y,Z} = g(X,YoZ). Define the (symmetric) (3,0) -tensor A by A(X,Y,Z)=g(XoY,Z),

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309

and let V be the Levi-Civita connection of g ande := g(., e}, a 1-form, be the coidentity (with e(X o Y) = g(X, Y}). Then the following conditions are equivalent: i) (M, o, e} is an F-manifold and £ is closed. ii) (Potentiality) The (4,0) -tensor V^4 is symmetric in all 4 arguments. in) The (3,1) -tens or V(o) is symmetric in all 3 arguments. DEFINITION 2.2 (Dubrovin) A Frobenius manifold is a tuple (M,o,e,E,g) with a) (M, o,e,g) as in Theorem 2.1 with i)-iii). b) g flat. c) e flat, i.e. Ve = 0.

d) Lie£;(o) = 1 • o ,

LieE(g) = D • g for some D e C.

REMARKS 2.3 i) a)+c) implies Liee(o) = 0 and Liee(#) = 0. ii) Suppose (M, o,e,^) is given with g multiplication-invariant and flat. Then

<=» •<=>

Potentiality X A(Y, Z, W] is symmetric in X, Y, Z, W for flat fields X, Y, Z, W V t € M 3 a potential $ e OM.t with XYZ $ = A(X, Y, Z} for flat fields X,Y,Z .

In the case of quantum cohomology, the Gromov-Witten invariants give rise to such a potential. One also has the metric. Then one constructs the tensor A and with it the multiplication. In the case of singularities (3rd talk), the potentials of the Frobenius manifolds do not play a prominent role up to now. iii) Let (M, o, e,E, g} be a Frobenius manifold. Define the (^M-linear map (a (1, l)-tensor)

V : TM -> TM , X H-» VXE - ^ - X . Then V is skew-symmetric with respect to g, i.e.

and V maps flat fields to flat fields, i.e. VV = 0 . Let c?i, ..., dm be the eigenvalues of V -f y • id = VE. They can be numbered such that di + d m+1 _,j = D and c^ = 1. The latter follows from e = [e, E1] = VgE1, which is a consequence of Lie£(o) = o and Ve = 0. iv) For (M,o,e,E,g) as above, define the C^M-linear map (a (1, l)-tensor)

U:TM ^TM , X ^ EoX .

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THEOREM 2.4 ([4] Lecture 4) Let W be a finite irreducible Coxeter group acting on ( R m , { , ) ) and (C m ,{.)), with {, } the standard metric.

M reflecting hyper-planes

—>

T>

{, }

—>

q a flat metric on M — P ,

w

->

C [:r !,..., :rm ] "

C [t i , . . . , tm]

with deg tj = d^ ,

/7 = d]_ > rf2 > ••• > rfm-1 > d m = 2 ,

(M, X>, e) determine* a multiplication o on TM fn/. L6'. .^j. Define g on M —V by g(X,

Y] := g(E o X,Y) .

Then g extends to M and is K. Saito 's [22] flat metric, (M, o , e , E , # ) is a massive Frobenius manifold with di — ^ and D — 1 + ^. The next theorem was been a conjecture of Dubrovin ([4] p 268). The proof uses Theorem 1.7. THEOREM 2.5 ([14] Thm. 19.3) Let ((M, t), o, e, E, g) be a germ of a massive Frobenius manifold with

/or1 TJ /?a£ coordinates, centered at t, and all di > 0. Then ^s isomorphic to a product of germs at 0 of Frobenius manifolds as in Theorem 2.4 for finite irreducible Coxeter groups Wi with the same Coxeter numbers hi = D_l . The 2nd structure connections of a Frobenius manifold (M, o, e, E,g) (K. Saito. Dubrovin, Manin, H.) are certain meromorphic connections which will be denned in 2.6. For that one needs the lifted tangent bundle pr*TM over P1 x M, pr*TM -> TM

P1 x M

pr

>M .

The tensors 5, o, V, W lift to pr*TM in a canonical way. V lifts to a flat connection on pr*TM with V 9 z F = 0 for Y e pr~lTM. Here z is a coordinate on C C P1, and

Frobenius Manifolds and Hypersurface Singularities

311

There is a discriminant V C P1 x M, t> := { ( z , t ) £ C x M | U - zid is not invertible on TtM} C P1 x M, The projection V —» M is finite and flat of degree m. Define M : = C x M -t> . Part a) of Definition 2.6 was given first (for the semisimple case) in [20] (II (1.26)+(1.27)), part b) in [4] (Lecture 3), the whole definition can be found in [15] (Ch. 4). DEFINITION 2.6 a) Fix s e C. The 2nd structure connection V (s) on pr*TM\^ is defined for X, Y G pr*TM (here consider X as a vector field on M and Y as a section in the bundle) by

V^V

=

l VdY+(v+\+s\(U-zr (Y} 2 V /

b) Define

c) Fix s G C. Define A, :

The following unifies and extends results in [24] (§5), [4] (Lecture 3), [20] (II §§1,2). THEOREM 2.7 ([15] Ch. 4) a) V (s) is flat. b) g is symmetric, nondegenerate, multiplication invariant. g(X,Y) is constant if X^ is V(~s)-flat and Y is V(s)-flat. Hence the connections V^ s ^ and V*-s^ are dual. V(°)|({,z} x M - f>) is the Levi-Civita connection of g\({z} x M - Z>). c) As is a homomorphism of flat vector bundles

It is an isomorphism iff di — ^j^- + s ^ 0 for all i = 1, ..., m. d) The pair (pr*7M?V^ s ^) has a logarithmic pole along {00} x M, i.e. pr*TM is invariant by Vx for all X 6 ^TpixM tangent to {00} x M (e.g. X G pr*?M and

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This yield* on pr*TM|{oo} X M a fla^ residual connection with respect to the coordinate i (see e.g. [15] Ch. 3.2 for the. definition) and the residue endomorphisrn By the isomorphism pr*TM

TM

they are mapped to V and

TrM

V.

flat residual connection w.r.t. —

e) Suppose (M,o,e) is massive. Then the pair (pr*7^,/, V^) /ia.s a logarithmic pole along T>, i.e. Vx leaves pr* invariant for X G DercxA/(l°g^)T/?,e residue endomorphism around a point in T>reg has eigenvalues (— (| — . For s — —\ it is 0 or has one 2 x 2 «s), 0, ..., 0). /£ is semisimple for s Jordan block. One can recover most of the structure of the Froberiius manifold from the 2nd structure connection V^: The discriminant f> C P1 x M determines the multiplication on TM in a similar way as the discriminant D C M in Examples 1.6 2)+3) ([15] Lemma 4.14). V^ around {00} x M determines the flat structure V and the endomophism V -f \ + A- by Theorem 2.7 d). The essential features of V^ are put together in the picture.

Second structure connection V1-^ on P1 x M log. pole

{00} x M

=> V + i 4- -s, V

multiplication

Figure 2

The first structure connections V^ are meromorphic connections, where roughly the information in the discriminant T> is replaced by a 2nd order pole of the connection along the smooth hypersurface {00} x M. Many of them are Fourier-Laplace

313

Frobenius Manifolds and Hypersurface Singularities

transforms of the 2nd structure connections. A huge amount of material about connections of this type is presented in [21]. Definition 2.8 and Theorem 2.9 are taken from [15] (Ch. 4), but can be found also at many other places ([4] [20] [21]). DEFINITION 2.8 M := C x M - {0} x M. Fix s e C. The 1st structure connection V^ onpr*TM\^ is defined for X,Y by

THEOREM 2.9 a) V^ is flat, the restrictions to {z} x M coincide for all s 6 C. b) The pair (pr*TM,V^) has a logarithmic pole along {0} x M,

residue endomorphism zVg z flat residual connection w.r.t. z

—> V + - 42 —> V .

c) The pair (pr*TM, V^) has a pole of order 2 along {00} x M.

First structure connection V^ s > on P1 x M irr. pole

{00} x M

=> multiplication

P1 x {0}

log. pole

{0} x M

Figure 3

=> V + \ + s, V

\M

314

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CONSTRUCTION IN SINGULARITY THEORY

The first big class of Frobenius manifolds was constructed implicitly in singularity theory 1983, long before Dubrovin's definition. K. Saito [23] [24] and M. Saito [25] showed that the base space of a seiniuni versa! unfolding of an isolated hypersurface singularity can be equipped with the structure of a Frobenius manifold. The structure of an F-manifold with Euler field is canonical and not so difficult to obtain. It will be explained below. The construction of a metric is much more difficult and not unique. It depends on the choice of a primitive form of K. Saito [23] [24]. The construction of such a form depends on the choice of an opposite filtration to a Hodge filtration ([25], see Theorem 3.2 below). The original proof applies results of Malgrange [18] [19] on deformations and normal forms of microdifferential systems to the Gaufi-Manin system of an unfolding. A simplified version is given in great detail in [15] (Ch. 6). It avoids Malgrange's results, stays largely within the framework of meromorphic connections, and is sufficiently explicit for applications. Still it uses polarized mixed Hodge structures and K. Saito's higher residue pairings. This lecture intends to give an outline of it. The details can be found in [15] (Ch. 6). Let / : (C n+1 ,0) —> (C,0) be holomorphic with an isolated singularity at 0.

n ( «df— , •••) ~,df— C'c"+1,o/i \dx dxn 0 is the Jacobi algebra, the Milnor number is //, = dimO/J/-. A semiuni versa! unfolding is a holomorphic function germ -* (C,0) -» F(x,t)

with F|(Cn+1 x (0},0) = / and such that fg (Cn+1 x (0},0), i = I,...,//, are representatives of a basis of O/J/. A good representative of F can be chosen as follows: Choose e > 6 > 9 > 0 and A := B] C C , M := B$ C C" , X := F -1 (A) n (5en+1 x M) C Cn+1 x C^ , Then F : X —> A is a good representative. It can be seen as a family of functions Ft:Xr\ (5"'+1 x {*}) -> A , t € M , parametrized by M . The function Ft has finitely many singularities, the sum of their Milnor numbers is //. The map ( :X

->

A x M

generalizes a Milnor fibration. It is a locally trivial C*°°-fiber bundle outside of a discriminant P C A x M.

Frobenius Manifolds and Hypersurface Singularities

315

The critical space C C X of (p is the zero set of (§jr, •••, -§£-}• It is the union of the singularities of all Ft, t (E M. The projection prc,M '• C —> M is finite and flat of degree //. The Kodaira-Spencer map BC is an isomorphism of free C^-modules of rank //,

&c : TM —* 3_

(prc,M}*Oc dF_

mult, o

mult. [1]

771 £/

f TTll [^ J .

It induces a vector field E := a^,1([F]) and a multiplication o on TM by X o Y :— THEOREM 3.1 ([14] Thm. 16.3) (M,o,e,£) is a massive F-manifold with Euler field E and caustic JC = {t \ \Sing(Ft)\ < //} , the eigenvalues of E o TtM are the critical values of Ft, (TtM,o,E\t)=

£jj

Jacobi algebra of (Ft, x}, mult. ,[Ft]

\xeSing(Ft)

J

O/Jf carries a multiplication and extends to the sheaf (prc,M}*@c- The p,dimensional space fi^+i 0/df A f2^ n+ i 0 carries a "metric" and extends to the sheaf

Its support is the critical space C. It is a free (pr^M^^c-module of rank 1 and a free OM-mO(lule of rank //. The Grothendieck residue pairing is [11] [23] [24]

JF : SIF x $1F

—>

OM ,

[^•-•^\^

(^r+LJr§f0-...-j^

with F := {(x,t) | ||-| = 7J C B"'+l x {t} for some small 7 > 0. It is symmetric and nondegenerate (Grothendieck). It is independent of the coordinates XQ, •••,xn (K. Saito). The choice of a volume form u> = unit ( x , t ) • dx induces an isomorphism

X metric

^

ac(X)

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With it one lifts JF to a metric g on M. For which uj is (M, o, e, E, g] a Frobenius manifold? For ui a primitive form of K. Saito. We will riot give here K. Saito's definition [23] [24]. But we will outline the simplified version in [15] (Ch. 6) of the construction in [25] of such a form. Let U := Eo. By Theorem 3.1, the critical values of Ft are the eigenvalues o f U \ t . Therefore the discriminant f) of