MULTIPLICITIES OF SOLUTIONS TO SOME ENUMERATIVE CONTACT PROBLEMS

Abstract:

Let G degree

be a scheme parametrizing a family of hypersurfaces of d in p-, N N>2. We define and study a subscheme of G parametrizing those hypersurfaces that touch fixed non-singular curves c 1 , ..• ,ck simultaneously. We give equations cutting out this subscheme in some cases, and we ^s~ow how such equations can in principle be found in any case.

When k

=

dim G, we expect the subscheme to have isolated points. We show how the equations determine the multiplicities of these isolated solutions to the contact problem. Thereby we find the local contributions to the total number of isolated solutions, as determined e.g. by Fulton's refined intersection products.

Instead of working with conormal varieties, we use the bundles of principal parts of first order associated to the

divisors in question. Hence a hypersurface with a singularity at a point of the curve C is said to touch C using our set-up.

We give some results, some of which are essentially well- known already. At last we'use our results to study particular

examples of plane curves.

Subject Classifications:

Primarily: 14Nl0

Key words and phrases: Enumerative contact problems, multipli- cities of isolated solutions.

- 1 -

§1 INTRODUCTION

In enumerative geometry a typical problem is to find how many varieties in a given p-parameter family that are simultaneously touching p fixed varieties. A classical example is to determine the number of reduced plane conics that are tangent to 5 fixed conics. One finds that when the 5 fixed conics are in general position, the number is 3264.

For_a problem like this denote by s the number of solutions when the fixed varieties are in general position with respect to

the given family. of varieties.

When the p fixed varieties are not in general position, the following may occur:

1) The set of solutions is infinite.

2) 'The set of solutions is finite, but there are less than s solutions set-theoretically.

An example of 1) is the problem with the conics in the case where 4 of the 5 fixed ones possess a common tangent line. Then the union of this line and any tangent line to the fifth conic is a solution.

If there is a conic touching all the 5 fixed conics, and two of the contact points coincide, we have an example of 2) (provided the set of solutions is finite). When the set of solutions is

finite, but there are less than s solutions, one would like to count the solutions with multiplicity in such a way that the total weighted number is s. The problem is: Is this possible, and how

should one count? This is the topic of our paper.

In principle the question is answered by

w.

Fulton and others, see e.g. [F], p. 187-193. One studies some parameter space associated

to the family, and represents the sub-family of varieties tangent to one of the fixed varieties as a divisor or hypersurface in the parameter space. Then one uses the socalled refined intersections to study the intersection product of the p hypersurfaces in the parameter space.

In this manner one associates an intersection number to each of the components of the intersection of the hypersurfaces.In particular one associates intersection numbers to isolated solu- tions. Hence the precise meaning of the phrase "how to count solu- tions" will be to find the intersection number in the sense of Fulton. In order to give geometric substance to the number, one must verify that when the fixed curves are in general position, each reduced solution counts with multiplicity one.

Again we refer to [F], p. 187-193 for details. See also [F-K-M]

and [F-M] and [H-S].

In this paper we will work over an algebraically closed field of characteristic zero.

We will restrict ourselves to a situation where all the fixed varieties are curves in PN for some N, and'our family of varie- ties will be a family of hypersurfaces in P • Hence our parameter ^N

(N+d)-l

space can be regarded as a subvariety of P d , where d is the degree of the hypersurfaces.

We will introduce a technique for how to determine the multi- plicities (or intersection numbers) of the solutions in practice.

We will not take up the question of how the global (total) number of solutions is determined.

Instead of using conormal varieties of the curves and hyper- surfaces in question, we will use the bundles of principal parts

f- 1

- 3 -

of first order associated to the divisors of the fixed curves in question.

The family of hypersurfaces will be interpreted as a family of sections of the bundle of principal parts for each fixed

curve.

Using this approach one says that the hypersurface touches the curve if there is a point where the usual well defined inter- section number of the hypersurface and the curve is at least 2.

When one of the two varieties is singular, this does not necessa- rily mean that their respective conormalvarieties meet.

Fix a point in the parameter variety representing a solution of our problem. The p conditions will normally represent p divisors in this parameter variety, giving rise to p power series in the parameters, locally at the point.

In §3 we will give some results concerning the leading forms of these power series. (Results 3.1, Result 3.2, Corollary 3.3).

These results are essentially well known [F-M,· H-8].

We reproduce the well-known conditions for when the multipli- city of the solution can be found simply as the product of the degrees of the p leading forms, i.e. the p divisors meet transversally at the point.

In Result 3.4 we give a complete description of the power series in question, in the case where there is no point where the hypersurface and any fixed curve have intersection multiplicity more than 2 (higher tangency). ~ve do however .allow several simple tangencies.

In the proof of our results we also show how in any case the computation of the power series reduces to an explicit application of the constructive proof of Weierstrass' Preparation Theorem, as given in [Z-8], p. 140-141, 145.

In Result 3.5 we study the situation where the parameter (N+d)-l

space is a linear subspace of P d , and where the isolated solution is a hypersurface making a single simple contact with each curve. If the p linear parts of the power series in

question have rank p-r, we expect the multiplicity of the solution to be 2 • We give a necessary and sufficient condition for the ^r multiplicity to be '2r in this situation.

In §2 we introduce the necessary technical devices. In §4 we prove our results. §5 is devoted to some applications of Result 3.4 for some explicit curves and families of hypersurfaces.

In §5 we also study osculating and bitangent circles to a plane curve. We consider some of the results in [G-B].

§2 DEFINITIONS

We work over an algebraically closed field K of characteristic zero. Let C be a fixed non-singular curve in PN, and let

(N+d)-l

G c P d be a scheme parametrizing an algebraic family of hypersurfaces of degree d in PN.

We denote by CG the affine cone over G. CG parametrizes the corresponding family of homogeneous polynomials of degree d in N+l variables. Let Pr(C) be the bundle of r'th order prin- cipal parts of the divisor class on C, corresponding to hypersur- faces of degree d, as defined e.g. in [L], p. 223-224, for r

=

0,1,2, . • . . Obviously each member of CG gives rise to a section

r · r

of P (C). Let F c CGxP (C) be the incidence correspondence thus defined, i.e. (cg,p(r)) E CGxPr(C) isrontained in F iff p(r) is contained in the section induced by cg.

We study the following diagram:

- 5 -

uj

F

q

/ul

^P

/ f_r

CG~ ~C

^{c Pr(C)}

where p and q are the natural projections from F to Pr(C) and CG respectively, and C is embedded in Pr(C) as the zero sesction, andf,r isdefinedas p-1 (c).

Den~te

^by Fm(q*(Jf..r) the m'th Fitting ideal of the cJCG-module q*Ofr. Let

v~

^{c CG} be defined by the ideal Fs- 1 (q*(iir). we see that Vs in an affine cone. From now on we will regard PVs as

r r

the subscheme of G parametrizing hypersurfaces in our family that have at least s r-fold contacts with C, counted with multi-

plicity. In particular PVs (or PVs in short) is the subscheme 1

parametrizing hypersurfaces making at least s contacts with c.

We have thus transformed the problem of hypersurfaces making s r-fold contacts with C to a problem of induced section of Pr(C) intersecting the zero section s times, in other words an s-secant problem. This enables us to use parts of the set-up in

[G-P] where questions of secancy are treated.

For any homogeneous ideal

~

ⁱⁿ

CJCG

denote by

V~

^and

PV~

the corresponding subschemes of CG and G, respectively.

Assume now that we have k fixed curves; c 1 , ..• ,ck. The sub- scheme of G parametrizing hypersurfaces touching c 1 , ... ,ck simultaneously is defined as PVJ' where J

= (F

⁰

(q*U~i),

^..•

is defined from C. as in the diagram above,

~

and all the left projections are denoted by q). For a hypersur- face D and a curve C in PN denote by I(P,CnD) the intersec- tion multiplicity of C and D at the point P. For a finite set M denote by card M the number of elements in M.

(N+d)-l Denote by TG(g) the embedded tangent space in P d of

(N+d) G at g, and by TCG(cg) the embedded tangent space in A d of CG at cg. We see that TCG(cg) is the affine cone over TG(g) when cg is in the fibre over g.

Remark 2.1

For each of the fixed curves C. in question, set

l

,

^{H. l}

Each H. parametrizes those hypersurfaces D in our family that

l

touch C. , i.e. those D such that there is a point P, such that

l

I ( P, c . nn) :> 2

l

In §4 (in the proof of Result 3.1) it will be clear that H.

l is

regular of dimension dim G-1 at g iff neither of the following 3 statements holds:

a) b)

c)

D touches

c.

in more than one point.

l

There is a point Q such that I(Q,C.nD) ^:> 3.

l

There is a point ^p such that D touches

c.

at P, and

l

such that all hypersurfaces corresponding to some point in the (N+d)_ 1

embedded tangent space to G in P d at g pass through P.

- 7 -

In order to find out whether H. is a reduced divisor on G, one

1

must check that each of the conditions a), b), and c) fail to hold for a generic point on ^{H .•}

1

Let us assume that dim G=p, and that we have p curves c 1 , .•• ,cp, and that H1 , ... ,HP

We are interested in the sum

are reduced divisors.

s = L i(g,H • •.• •H ,G)

1 p

in the sense of [F], see e.g. Example 7.1.10, p. 123. (The sum s is taken over all isolated points g of

H n ••. nH • )

1 p

We will study each local contribution i(g,H_{1 . . .} Hp,G) at isolated points. Locally at g each H.

1 is represented by a power series f. 1 in the completion

a:G

_,g

•

^Since _<JG,g ^~ is non-singular and therefore Cohen-Macauley, i t follows from Example 7. l . 10 in [F]

that i(g,H 1 •.• Hp,G) is equal to the K-rank of

~G

_,g

I (

fl' .•• ' f ) . _p This is also the multiplicity of the scheme PVJ at g, where J = (F⁰ (q*~fi), i=l, •.. ,p). In §3 we will give some results concerning such multiplicities.

We conclude that these results can be used to determine the local contributions i(g,H 1 ... Hn,G) to the global number s, wherever this number is defined in terms of the intersection products in

[F

J.

A local description in CG ^x PN

Assume dim G=p, and let cg be a point of CG in the fibre over a non-singular point g of G. Let {b , ... ,b} be a set of regu-

0 p

lar parameters of CG at cg, where b 0 corresponds to deforming (N+d) A d cg in CG along the line from cg to the origin in

f

f-

The parameters b ' .•. 'b

1 p correspond to deforming cg in CG along directions transversal to this line. Let P E P . We have ^N

where X., i=l, ..• ,N are coordinates of some affine space

1

containing P.

Let CI ^c CG ^x PN be the incidence variety consisting of those (cg,P) such that P is-contained in the hypersurface determined by cg. We have

&cr (

_{' cg,} P)

=

K [ [b0 , ••• , bP,

x

1 , ••• , X._]] _-"N /M (b 0 , ••• , bP,

x

1 , ••• ,

xn)

where is "the general polynomial paramet-

rized by a point in an infinitesimal neighbourhood of cg". We set

M(b0 , ••• ,b

,x

_{1 , •••}

,x) =

_{(l+b0 )M} + R(b 1 , ••• ,b

,x

_{1 , •••}

,x)

p n canst - p n

where M

canst is the polynomial corresponding to cg1 and

R(b 1 , ••• ,b ^,x1^{~··•1X) E K[X}1^{~···~X ][[b}1 , ••• 1b ]].

- p n n p When G is a

(N+d)-l

linear subspace of P d 1 we can take _-R(b 1 , •. ^{-~b ~x}_p 1^~···~x _n

)

to be linear in b1^~···~bp.

Choose X. =

I ~. .

•t j ¹ i=l1 ••• ¹ N ¹ as local parametrizations of

1 j>O 11)

C at P. We define:·

We also write this as: ( 2. 2)

where the ex • E ^K1 and the A . (b ^{1 • • • 1} b )

J J 0 p are power series without

(N+d)-l

"constant terms". When G is a linear subspace of P d ¹ the

I

I

- 9 -

A . (b 0 , ••• , b ) J p will be homogeneous linear in b 0 , ... ,bp whenever

B

(b0 , ••• , bp, x1 , ••• , xn) is linear in b0 , ••• , bp.

When the hypersurface in question touches C at P, we have

A local description in CGxPr(C)

We now regard C as the zero-section of P (C). We use the terms ^r P and t for the point on C and the local parameter for C at the point, also when C c P (C). ^r

Consider the point .!: = (cg,P) in

-er

^{~ F c} CGxP (C). A set ^r of local parameters for CGxPr(C) at P is

where are as before, and v 0 , ••. , v r are the coordi- nates corresponding to the r+l terms of trunkated Taylor series of order r.

We have:

1\

10r "" K [ [b0 , ••• , b , t, v0 , ••• , v

J J I

J

v{r,P p r - r

where

Jr

=

(v0 , ••. ,vr,v0-j((b 0 , ... ,bp,t),v1-

~tv\"(b 0 ^,

.•• ,bp,t),

orj(

... , r 1 v - - (b 0 , ••• , b , ^t))

r

at

P

The r+l first generators arise from the fact that ~r is the inverse image of the zero-section. The r+l last generators are due to the incidence describing F. We easily conclude:

, V: or

Ar. ·

""K[[b0 , ... ,b _P ^,tJJ/vf~b0^, •.• ,b ,t), .•. _P

,--tJr

otr (b0 , .•. ,b ,t)) P ( 2. 3) We see that

_a,

^{:f 0} iff I(P,CnD) ;> r+l. Assume that

U{r,.!:

I ( P . , c

no )

^~ r+ 1 , for

1 i=1, ... ,k, where k is finite. Denote by the ring where

We have

I).

-~ P. corresponds to P., for

1

k (): s-1

(J

VF (q* r), cg

t

~ ~CG /Fs-1 ( $ R. )

(}1 , cg i=₁ 1

A useful identity is:

k

i=1, ... ,k.

R. ~

Fs-1 ( $ R.) = . 1 1

j 1 ~ jk

!: F (R 1 )x ••• xF ^(~) ( 2. 4)

1= j 1 +. • • + jk =s -1 {See (G-P], p. 16.)

Denote by n.

1 the multiplicity of the ring R.

1 with respect to the maximal ideal of

k

When

I

^{n. = s,}

i=1 ¹

We always have:

§3 RESULTS

8-cG,cg·

Formula {2.4)

0 k

F ( $ R . ) = . 1 1 1=

reduces to:

k n. -1 •

I

F 1 (R. )

i=1 ¹

( 2 . 5 )

(2.6)

We will give some results in the case r=1. Assume that {N+d)_₁

dim G = p, where G is a non-singular variety in P d and that D is a hypersurface in PN corresponding to the point g E G. Let c 1 , •.. ,cp be ^p fixed non-singular curves in

will start with reproducing 3 results (Result 3.1, 3.2, and Corol- lary 3.3) that are essentially well known.

- 11 -

Result 3.1

Let J =

(F ⁰ (q*crt~),

i=1, •.. ,p) as in §2. Assume that g is an isolated point of PVJ. Then the multiplicity of PVJ at g is at least

p

M

=

IT (deg D•deg c.- card(C.nD))

g i=1 ~ ~

The multiplicity is equal to M iff the following is true:

g N p

For all (P. 1 , ..• ,P. ) ^E (P) such that I(P .. ,c.nn) > 2,

~ ~p ~J J

i=1, •• ,p, D is the only hypersurface, which is parametrized by some point of TG(g), and which passes through P. 1 , ••• ,P . • _{~ ~p}

Result 3.2

Assume g E G corresponds to the hypersurface D, and that C is a fixed non-singul~r curve. Let P 1 , .•• ,Pk be the points where D

k touches C, and set n. = I(P.,CnD)-1, i=1, ... ,k. Set

~ ~ s=

In.,

. 1 ~

~=

and s-1

(J

J

= F (

q*

f_l ) ·

Then the embedded tangent space of in at is the subspace of TG(g) parametrizing those hypersurfaces H such that I(P.,CnH) > n., i=1, .•. ,k.

~ ~

Corollary 3.3

Assume J is as in Result 3.1.

g

a) If the number M defined in Result 3.1 is 1, then the embed- g

ded tangent space of PVJ at g is the subspace of TG(g) parametrizing all hypersurfaces passing through the uniquely

determined points P., i=1, ..• ,p, such that

~

b) The multiplicity of PVJ at g is 1 iff

r (

P .

c .

nn ) = 2 .

~ ~

i) M _g

=

¹

ii) D is the only hypersurface parametrized by some point in TG(g) passing through the points P 1 , ... ,P described _p in a).

Comment:

Corollary 3.3 is an easy concequence of Results 3.1 and 3.2. We will prove these results in §4. Result 3.1 is essentially contained

in [F-M] and

[H-S]

and other papers.

Result 3.1 only gives information about the leading forms of the power series expansions of the H.

~ (see Remark 2.1) at g. We will now give a technical result, from which one can extract more detailed information' about the local nature of PVJ at certain isolated points.

A technical result

Let G,g,c,~l,D be as before. Assume'that there is no point Q such that I(Q,CnD) ) 3, and that P is a point where

I(P,CnD)

=

2. Let

L

(A . (b 0 , ••• , b , t) +a . ) • t j

j)Q J p J

be as described in §2, Formula (2.2).

We have:

Result 3.4

Under the assumptions above

~PVFo( (J

)'g"' K[[bl, ... ,bp]J/J q*

tl

where the ideal J is generated by

f--1

~·

I

- 13 -

II !p(b1 ^{I • • •} ,bp) over those P

such that I(P,CnD)

=

2

Each !p (b 1 , .. -., bp) is of the following form where A.= A.(O,b 1 , •.• ,b ): _{J J p}

where

where

s

= - I (-A_{1 )} j+1 ·B₀•Q.(B_{0 , •••} ,B.)

j)O J J

j)O

I

(-A ) j 2

for .R. ) 2.

1 0

0

3(A3+a3 ) ... U.+1 )(A.R.+1+a.R.+l) 1

0

Q.•(B0 , . . . ,B.) is homogeneous of degree

J J j in B0 , . . . ,Bj' and the

coefficient corresponding to the monomial

i₀ · i.

J .

B0 ... Bj 1s the number of paths in the following diagram with the property that i arrows

s jump s steps· to the right, for s=O, ••. ,j.

( 01 j)

I

y t

I

~-->­

X

(j,j-1)

(j,O)

One starts at (O,j) and

i ends at ( j, 0), each stop is a point in

Zt

on or above the line x+y = j.

In each jump the y-compo- nent decreases by one, and the x-component increases or remains constant.

a

_{0 = 11}

a

_{1 = s 1} ^I

o

_{2 =} s

0

^s

2

^{+s~ I}

0 4 = B6B

4

^+4~~B

1

^B

3

^+2B~B~+6B

0

^BfB

2

^+B~.

The first few terms of ~P(b11 ... 1bp) are:

A2 cx3 A2A2

3 A{A 2 A{A3 Ao - 1

+ 1

A~A

2

^{A3 1 2 +}

4cx 2

--- ^{- - -} _T6 - - -

4cx 2 8cx⁴ 1

2cx 3 ^a'+ 8cx 2

2 2 2 2 2

9 ^a2 3 A'+ + cx4 ^A'+

- Ejj4

- -

aS 1

16cx⁴ 1

2 2

Hypersurfaces touching each fixed curve in one point

Assume that J and D are as in Corollary 3.3 including the assumption M = 1. Then there are unique points

g

that I(P.IC.nD) = 21 i=l~···~P·

]. ].

We will use a small part of Result 3.41 i.e.

A2 1 3

- A_{0 -} 4 cx 2 mod(b 11 •.• ,bp) to show Result 3.5 below.

such

Let cg correspond to a homogeneous polynomial f₀1 and let f 11 ... 1fp be homogeneous polynomials such that {f₀1 •••1fr} span Tv (cg) (r~p) 1 and such that (f 0 , ... ,fp) span TCG(cg). Let f

J be in

For any quotient h of homogeneous polynomials in the coordinates of PN, denote by

order (h) P. ].

the order at P. of the restriction of h to

c ..

]. ].

By Result 3.2 we have

i=l ^{1 o o o I} P

' ~

i

since order _P. (f 0 ) = 2.

~

We define:

We have the following:

Result 3.5

- 15 -

Assume that D and J are as above (M =1), and that dim TPV (g)

g J

-- r. Then the multiplicity of PVJ at g is at least 2r.

(N+d)-l Assume in addition ahat G is a linear subspace of P d Then the multiplicity of PVJ at g is strictly larger than 2r

if and only if:

There is a polynomal f and a vector space Vf as above, such that there exists an element h in V f with

•

order _p, (h) > 1 , i=l, ••• ,p

~

§4 CALCULATIONS AND PROOFS

Assume we have a hypersurface D and a non-singular curve C in PN, such that I(P,CnD) > 2. Let the schemes CG, G, and f r be as in §2, and let g E G correspond to the hypersurface D.

We recall the formula:

where J - r

&

_{;br u}

1..'=--

"' K [ [b 0 , ••• , b , t ]]

I

^J

P - r

We will work with the case r=1, and we will find a K[[b 0 , ... ,bp]J- free resolution of We will use this resolution to find explicit descriptions of the K [ [b0 , .•. , bp]] -ideals

for s=1 , 2, .•• · .

From now on we drop the index r (=1) in J , tJ r, and we - r \.

I(P,CnD) = n+1.

"'

K [ [b0 , .•. , bp] ] . Hence

ff f,

P = R [ [ t ] ]

I;]_.

Then

J((

0, ..• , 0, t) is of order n+1 in denote by

.

R the ring

Assume

t, and

oJf

ot{O,O, ... ,O,t) is of order n in t. We use Weierstrass' Preparation Theorem to find a polynomial S in t, of the form

where s n-1'····⁸o are power series in b 0 , .•. ,bP, and s

oJf:

ates the same ideal as ot(bo, •.. ,bP,t) in R[[t]J.

Furthermore we use Weierstrass' Preparation Theorem to polynomial T in t of the form

where Tn_ 1 , .•. ,T0 are rates the same ideal as

power series in JYc'b 0 , ... ,bP,t)

(b 0 , •.. , bp), and T in R[[t]] modulo

gener-

find a

gene-

s.

We remark that the constructive proof of Weierstrass' Preparation Theorem in

[Z-S],

p. 140-141,145 gives an explicit algorithm for constructing S and T, using formula (2.2) in §2.

We also remark that the power series sn_1 , ... ,s 0 ,Tn_1 , ... ,T0 con- tain no constant terms.

It is now clear that

bf.,f :.

R[[t]]/(S,T) and.we have the following R-free resolution:

- 17 -

R[[t]J/S

*

R[[t]J/S _:; R[[t]J/(S,T)

~

0 ^{( 4. 1 )}

~ is multiplication by T, and • is the natural map. We find the matrix representation of ~ with respect to the basis

{1,t, •.• , t n-1 }.

The following is easy to verify:

Observation 4.1

( i) The entries in the first column of the matrix are To, ... ,Tn-1·

(ii) The entries in the other columns of the matrix are contained in the R-ideal (T0 , . . . ,Tn_ 1 ).

(iii) Modulo the R-ideal (b_{0 , ...} ,bp) 2 the matrix is

T n-1 ^T n-2

0

Observation 4.1 gives rise to the following:

Lemma 4.2

a) Fn-1<&f,P)

=

(To, .•. ,Tn-1).

In particular~ when n

=

1, Fb (

& -e,

^{p )}

=

^{(TO )}

= (

^{T ) .}

b) F 0(&f_,P) is generated by an element, which is congruent to

n ~1

T0 modulo (b0 , ... ,bp) .

1-

Lemma 4.2 reduces the proofs of Results 3.1, 3.2, 3.3, 3.4 to

explicit computations of T0 , . . . ,Tn_1 for each point P such that '

I

I(P,C.nD)>2, for each of the fixed curves C ..

~ ~

Proof of Result 3.1

We will compute

F

⁰

(q*0~>

locally at cg for each fixed curve

c.

We study one such curve. Let P1 , .•. ,Pk be the points where D touches

c.

Let n.

=

I(P.,CnD)>2, i

=

1, ..• ,k. Let R. be the

~ ~ ~

ring

(2.6) this is equal to

For each P. denote by

~

We must compute F ( ^{0 k} @ R. ) • . 1 ~

~=

( ') (') n.-1 (') T ~

=

Tn~-

1

^{t ~ + .•. +T}

0

^~

~

By formula

the polynomial defined by Weierstrass' Preparation Theorem as above.

By Lemma (T(i})ni

0

4.2, part b, a generator of n. +1 modulo (b 0 , ... ,bp) ~

F⁰(R.) is congruent to

~

Hence a generator of is congruent to

modulo

For each point

k (.) n.

IT (T ~ ) ~

i=1

°

P., we rephrase formula (2.2} as

~

I

(A . . (b 0 , ••• , b , t) +a . . ) • t j

j>O J,~ p J,~

where the A . . are functions in b0 , . . . ,bp. For a power series

J,~

f(b0 , . . . ,bp} denote by f lin (b0 , . . . ,bp) the linear part of

- 19 -

f (b 0 1 • • • 1 bp). We use the algorithm in the proof of Weierstrass Preparation Theorem in [Z-8]1 p 140-141 to find:

T(i)

0 - A0 . (b 0 ^{I • • • I} b )

11. p modulo

Hence we obtain that a generator of is congruent to

k 1 . n.

l.n l.

II [A0 . (b 0 ^{I • • • I} b )

J

i=l ,1. p

modulo

This means that the leading form has degree at least k

I

I(P.~cnD)-k = degD•degC-card(CnD).

i=l l.

k

I

^n.

=

i=l l.

This implies that the mu1tiplicity of PVJ at g is at least p

II [degD•degC.-card(C.nD)].

j=l J J

Furthermore the multiplicity is equal to this number if and only if the only values of (b0 ,b1 , ••. ¹bp) satisfying the equations

k lin ni

II A0 . (b0 ^{I • • • ,} b >

=

o

i=l ,1. p

for all curves c_{11 •••} ,cp simultaneously are those with

b ¹ = ^{. . .} = b p = 0. (The variable b 0 is irrelevant and does not occur in these equati.ons. )

This implies the conclusion o~ Result 3.1.

Proof of Result 3.2

Let {P1 , . . . ,Pk} be the set of points where D touches c. Let the rings R. be as before.

l. We will compute

k s-1 k

F ( ^@ R. ) , where

i=l ~ s

= I

^(n.-1),

i=l ~

and n. = I(P.,CnD), i = l, ••• ,k.

~ ~

By Formula (2.5) this is the same as k n.

IF

^~(R.).

i=l ~

By Lemma 4.2, part a) this is the same as

(1) (1) (k) (k)

(To , .•• ,Tn -1, . . . ,To , .•. 1Tn -1).

1 k

Temporarily we fix PE{P 11 •.• 1Pk} 1 and study the corresponding power series T 01 ••. 1Tn-l 1 where n = ^I(P~cnn).

We use Weierstrass' Preparation Theorem and find that the linear part of T0 is

In the same way we see that the linear part of T. is J

modulo the linear parts of (T 01 •.• ,Tj_1 ) 1 for j = 1 1 •.• 1n-l.

Hence the linear parts of (T 01 •.• ,Tj_1 ) generate the ideal

lin lin

(A0 (bo~···~bP)I ... IAn_ 1 (b0 1 •.• 1bP)).

Those values of (b 0 ) 1 b 11 .•• 1bp that simultaneously satisfy the equations

A . lin ( b 0 1 • • • 1 b ) = ⁰¹ j

=

^{0 I • • • I n -1}

J p

correspond to those polynomials defining hypersurfaces that intersect C at least n times at P. This gives Result 3.2.

I

I-I

'

- 21 -

About the proof of Result 3.4:

As usual let {P1 , •.• ,Pk} be the set of points where D touches

c.

By combining formula (2.6), and Lemma 4.2, part a) we see that k

F 0 ( e R.)

=

i=1 ~

k

II T(i)

=

i=1

°

k

II T(i).

i=1

Hence i t is enough to prove that each T(i) is of the form F (b 1 , ... ,b ), described in Result 3.4.

:-P p (We once again forget the

irrelevant variable b 0 .) This follows from the constructive proof of Weierstrass' Preparation Theorem on p. 140-141, 145 of

[z-s].

We skip the easy, but tedious calculations here. The assumption

I(P,CnD)~2 for all P, is only included to make the calculations more tractable. In principle one can use the same sort of calcu- lations in any case, using formula (2.6), the resolution (4.1), and the constructive algorithm in

[z-s].

Proof of Result 3.5.

Since PVJ is locally a complete intersection of divisors of G at the isolated point g, and since the tangent space dimension of PVJ at g is r, it follows immediately that the multiplicity of PVJ at g is at least 2r. See [F], p. 233, Example 12.4.10.

This conclusion can also be derived from the following discussion.

(N+d) _ 1

Assume that G is a linear subspace of P d Then all the A.(b0 , ••. ,b ) described in formula (2.2) can be taken to be _{J p}

linear.

Since M _g = 1 ' we know that D touches c 1, ... ,cP ^{in one} point each, say P1 , •.• ,Pp, and that I ( P. , c. nn >

~ ~ = 2, i = l , . . . ,p.

Let

cX<b 0 ,b 1 , ... _~

,bP,t~)

_~ ⁼ j)O

I

(A . . (b0 , ... ,b )+a: . . )•t? ),~ p ),~ ~

be the expressions corresponding to formula (2.2) for each pair (C. ,P. ), i

=

1, ••• ,p.

~ ~

Put A. .

=

A. . (b 0 , ••• , b ) • (When

)I~ ),~ p

not involve b0 , ^{for i}

=

l, . . . ,p.)

j

=

^{Orl, the A . .}

)I~ do

We use Result (3.4) and find that J

=

(T(l) , •.• ,T(p)), where

The

T(i) - A

0 . _{4a:2 .} ^{modulo for i}

=

1 , ..• ¹ p.

p

,1.

I 1.

linear equations A. ₀ = 01 i = 1, .•. , p, in the p

1.,

give rise to a pxp-coefficient matrix variables b 1 , ••. ,bp

J't=

^{(m .. ).} _{1.) ~=} ¹ _{I 0 0 0 I P} ^.

j=l, ••. ,p

By assumption there are exactly r independent. relations between the rows of

~

^• We may assume that p-r last rows of

J1.

generate the vector space generated by all p rows, and we can find constants

"-1, r+l' · • • '"-r+l

A O,r

=

^A. l,r+l O,r+l ^A

=

^A. r,r+l O,r+l ^A

such that:

+ . . • +

+ . . • +

A. A l,p O,p

A. A r,p O,p

This means that J/(b 0 , ••• ,bp)³ is generated by

ALl A2 A2

-

"-l,r+l

l,r+l

- ... -

"-1 'p

...b.E.

2 a:2,r+l a:

0:211 2, p

A2

.

A2 A2

___!LE. _a:

-

^A. _{r,r+l a:} ^l,r+l

- ... -

^A. _{r,p a:} ^__h.£

21r ^{2,r+l 2 ,p}

and

- 23 -

A + quadratic term 0, r+1

A O,p + quadratic term

Now it is clear that the multiplicity of PVJ at g is at least 2r.

By making a linear change of parameters we may assume

A. 0 • =b., j = r+1, •.• ,p. Then it is clear that the multiplicity .... , J J

of PVJ at g is 2r if and only if the only r-tuple (b 1 , .•• ,br) satisfying the equations

A21 .(b1 , ..• ,b ,0, ... ,0) _,J r _A A21 +1 (b1 , ... ,b ,0, •.. ,0) ,r r

a2,j j,r+1 a2,r+1

- . . . -

^{A .} J,p ^A1 ,p (b 1 , ••• , b , 0, ••• , 0) r a 2,p

simultaneously for j = 1 , •.. ,r, is the zero-tuple.

On the other hand, let f 0 , ... ^,f~ be as in the text preceeding Result 3.5, i.e. f 0 defines D, {f 0 , ... ,fr} span

span TCG(cg). We think of fi as the polynomial corresponding to the parameter b., i

=

1, .•. ,p.

1.

The condition that a polynomial being a linear combination

should define a hypersurface passing through P 1 , .•• ,Pp, gives p equations in c 1 , •.. ,cp with coefficient

matrix~.

The assumption A0 . =b., j = r+l, •.. ,p means that the _{, J J} subspace of Span(f 1 , •.. ,fp) consisting of polynomials defining hypersurface passing through P1 , •.. ,Pp' is Span(f1 , ..• ,fr) .

Let f be in Span(f1 , ... ,fr). The condition that an

.

expression

should have positive order at P., regarded as a function on

c.,

l l

i = 1 ^{I • • 0 I} P gives p equations in d 1 ^{1 o o o 1} d + 1 with a coeffi-

cient matrix J{f. The first p columns of J{f are the same as those of

Jt1..

The condition that an element of

should have positive order at P., regarded as function on C.,

l l

i

=

lt•••,PI is equivalent to

rk}i _f

=

^rk

J1 ⁼

^p-r.

This means that all relations between the rows of

J1

relations between the rows of

, J1f.

^Put

We have the following power series expansion of at P.:

l

since

(A0 .(c 1 , ... ,c )+A1 .(c11 ... ,c )•t.+ ... )2 _{11 r 11 r 1}

=

0:2 .t~+a:3 .t~+ ...

l l l l l l

A21 .(c 11 ... 1c ) _{,1 r}

+ t.•something 1 0:2 ' _{, l} l

A01 i(c 1 , ..• 1cr) ⁼

o. AJ

Hence the entries of the last column of v•tf are A,2 ' ( c 1 ^{I • • • I} c )

11 r

a:2 ' _{I l}

lift to

Comparing with the multiplicity condition already found, this gives our desired result.

- 25 -

§5. SOME APPLICATIONS OF RESULT 3.4

We will use the formula of Result 3.4 in some examples.

Moreover w~ will study osculating and bitangent circles associated to a plane curve.

Example 5.1.

We let the points (a 0 ,a1 ,a2 ) in G ⁼ p2 parametrize ~he members of the family of plane curves with equations

a x2 + a y2 + a 1Yz + a z 2

=

o.

0 0 2

The real picture is the family of circles with centers on the line X = 0.

Assume we have two fixed curves c 1 and c 2 , both passing through the point P = (-1,0,1 ). Let D be the "variable curve"

parametrized by (a0 ,a1 ,a2 )

=

^(1,0,-1)

=

^{g, i.e. D} is the curve with equation x2+y2

=

^z2. We also assume that

I(P,C 1nD)

=

I(P,c 2nD)

=

2,

and that D touches neither c 1 nor c 2 in any point but P.

Set J = (FO(q/Jf.1),FO(q*(J'(2 )). Vile will show that the multi- plicity of PVJ at g is equal to I(P,c 1 nc 2 ). The polynomial M(b0 ,b1 ,b2 ,X,Y,Z) defined in §2 may be taken to be

From now on we set b 0

=

0.

We parametrize c 1 and c 2 at P as follows:

c 1 :X

=

^-1+

I

^{y.Yj, y}

₌

Y,

z =

j:>2 J c 2 :X

=

^-1+

I

^T).Yj,

j:>2 J

y

=

^Y,

z =

^{1 •}

I ;-

Referring to formula (2.2), we obtain:

where AO,l

Ao .+Al

.+La . .

^{Yj, i}

=

,1 ,1 j:>2 ] , 1

j

1,2

where

I

nknJ'-k'

k=O

where no

=

-1, TJl

=

o.

~ve now use Result 3. 4, and_ we obtain that

where

(1) (2)

~ = K [ [bl, b2]]

I

(T IT ) ^I PVJ

b2

= b - 1

2 4a 2 . _,1 ,g

8a23 .

,1

L

^R.(a_{2 ., •••} ^{,a . .} _{)b 1}^{j, i =}

j:>4 J ,1 ] , 1

where the R.(a 2 ,, ... ,a . . ) are rational functions in _{J ,1 ] , 1} a 2 . , • • • , ^{a . . ,} such that

,1 ] , 1

-1 j

R.(a_{2 ., •••} ,a . . )

= (

₂ ^{) •a . . +}

J ,1 J , 1 a2 . ] , 1

,1

terms only involving (a2 ,, •..

,a.

^{1 .).}

I 1 ) - I 1

1,2,

Hence the power series T(l) and T( 2 ) are congruent modulo

(b~),

but not modulo

(b~+

¹

),

^where

m

=

^min.{a. ₁

*a.

_{2 };}

J J I J I

and the multiplicity of PVJ at g is m.

We also see that m is equal to m', where

m' = min . {y . *TJ . } • J J J

But m' is clearly equal to I(P,c 1nc2 ). Hence the multiplicity of at g is equal to r ( P, c 1 nc 2) under the assumptions given.

- 27 -

Example 5.2.

We work with the same family of curves as in Example -5.1. Let

c

₁

and

c

₂ be two fixed curves that pass through Q

=

(0,0,1 ), and that are non-singular at Q. Let D be the "variable curve" with equation

x2 + y2

=

^0, i.e. a union of two lines.

We assume I(Q,C.nD) ]. = 2, i = 1 ¹ 21 ^{and that} D touches neither c1 nor c2 at any point but Q. Let gEG be the point (1,010)1 in

'

other words g corresponds to D. Set J =

I

^pO(q

^(Jf__).

. 1

* .

l.= ].

Let C'

1 be the "mirror i-mage" of c1 with respect to the line X =

o.

We will show that the multiplicity of PVJ at g is equal to r ( o ^I < c 1 u c

i

^> nc 2 ) . We work in 3 steps:

Step 1:

Let t. be a parameter for C. at Q, i

=

11 2. We proceed as in

]. ].

Example 5.1, and find two power series and is easy to see that if touches neither

c

₁

then the multiplicity of PVJ at g is 2.

Step 2.

nor C' 1

Then i t at Ql

Assume touches

c,

^{or C'} ₁ ^{at Q1} and that the common tangent is not the line with equation X

=

0. Then we use X as a common parameter for c 1 ^and c 2 ^{at Q1} we proceed as in Example 5.11 and we find that the multiplicity of PVJ at g is

Step 3.

Assume the common tangent of and

c•

1 and is the line X

=

^0. Then we use Y as a common parameter for

c

₁ and

c

₂ at Q. We find that the multiplicity of PVJ at g is

Summing up, we find that in any case the multiplicity of g is

Example 5.3.

PV J at

We will use Result 3.4 to study a scheme PVJ at a non-isolated point. Let G

=

P⁵ parametrize the conics in P2 , and fix 5 lines

c

_{1 , •••}

,c

_{5 .} Let X,Y,Z be coordinates of p2, and let g corre- spond to the double line D with equation x2 ~ 0. Assume for simplicity that the line L with equation X

=

⁰ intersects

c

_{1 , •..}

,c

₅ in 5 distinct points P1 , .•• ,P 5 . It is clear that in this situation PVJ is equal to the Veronese-surface parametrizing double lines, locally at g. It would be nice to reproduce this fact using Result 3.4.

The polyomial M may be taken to be

As always, we set b₀

=

^{0. Near D} the memqers of the Veronese- scheme have equations

(X+cY+dz)2

=

x2 + 2cXY + c2y2 + 2dXZ + 2cdYZ + d2z2

= o.

Hence one expects rl'pv ^/).

J,g

to be isomorphic to

- 29 -

which is simply

Choose: Z

=

1, X= X, Y

=

K[[b 1 ,b 3 ]].

I

^{Y . .}

xj

j;>O J,1 as local parametrizations for

c.

at P., i = 1, ..• ,5. We have

1 1

J( _{. ( o-,}

_{b 1} I • • • I b ^I X) . = X 2 + b 1

I

^{y . . X j+l + b 2 (}

I

^{y . . X J ) .}

1 p j;>O ] , 1 j;>O ] , 1

+ b 3

x

_{+ b 4}

I

^{y. .}

xj

+ b 5 , i = 1 , •.• , 5, j;>O J,1

where the

rJ.f

₁ were described in formula (2.2).

Using the obvious terminology, we get:

Ao .

_,1

Al . _,1

~,i

a_{2 . ,1}

=

^{1, a . .} _{] , 1}

=

^{0, j=F2}

for i = 1 , ••• , 5 •

We now use Result 3.4A Modulo (b1 , ... ^,b5 )³ PVJ is cut out locally by the equations:

<Yo,ibl+ 2Yo,iY1,ib2+b3+y,,ib4)²

4 - 0

i = 1, .•. ,5. Take any 3 of these equations. Modulo (b 1 , •.. ,b 5 ) 2 they reduce to b 2 = b 4 =.b 5

=

0, since the

Hence the equations reduce to

Simply using Cramer's rule we obtain

Yo . _,.1 are distinct.

-I

!

' '

modulo (b1 , ..• ,b5 )^{3 •}

Calculating this way one should show that all the 5 equations reduce to the 3 equations above modulo (b 1 , •.• ,b 5 )j for any j.

We have checked i t up to j

=

5. To show it for any jEN one

should find some inductive argument, in connection ~ith Result 3.4.

Since we already know the conclusion, we·haven't found it worth- while to make this effort.

First of all this example serves as a good way to check that the formula in Result 3.4 is correct.

Example 5.4.

Let G

=

P³ parametrize all the "circles" in p2, i.e. all conics passing through "the circular points at infinity", (l,±i,O).

Set G

=

Proj k[a,b,c,d], where

is the equation of the general "circle". Consider a fixed non- singular curve C c p2, not containing a circle or a line as a

component. We consider the variety PVJ c G, where J

=

Fl(q*~~

1

^).

(PVJ

=

PVI in the terminology of §2). We see that PVJ paramet- rizes those circles that are either osculating circles to C, or bitangent, or both. In general PVJ will be a curve U in G.

Let ul be the closure of those points in u that parametrize circles that are bitangent, but not osculating, and let u2 be the closure of those points that are osculating but not bitangent. We expect u to consist of the two components ul and U2, where ul and u2 meet in a finite number of points. We will denote ul by the pre symmetry set, and u2 by the pre-evolute associated to

c.

L_ i

- 31 -

When the polynomial

a(X²+Y2 ) + bXZ + cYZ + dZ²

defines a real circle, its center in p2 is (-b,-c,2a). We define a map $:G

=

^p3 ~ p2 by

$(a,b,c,d)

=

{-b,-c,2a).

We de£ine.

w,w

₁

,w

₂ ^as t(U),t{U 1 ),$(U 2 ) respectively. $ is simply a projection from the point (0,0,0,1) in G, followed by a trivial change of coordinates. We call

w

₁ the symmetry set, and

w

₂ the evolute associated to

c.

The symmetry set and evolute of a plane curve were considered in [G-B], and we will show how some of the results in [G-B] can be proved or at least indicated by our methods. As in [G-B], we will say that {a generalized) circle makes A -contact with

m1 ^{I m}2 I • • • 1 ~

C if i t intersects C in

k

2•degC-

I

m.-k j=l ~

points with intersection multiplicities m1 + 1 ^{I 0 0 0} I~+ 1 ¹ 1 I • • • 1 1 ¹

where _{m. 2}

J) ^I j

=

l, ••• ,k, for some k>l. For convenience we set m1 >m2> ••• >~. We will consider the following statements inspired by, and for a great deal coinciding with those in [G-B]:

(Sl)

U and are nonsingular at the corresponding point

fJ

^{in G.}

Assume a circle makes A1^, 1^~contact with

c.

Then the curves

The corresponding branch of W and

w

₁ ^{at t(P)} is also non- singular.

i I

I

(82)

c.

Then the curves Assume a circle makes A2 -contact with

P

ⁱⁿ

and are non-singular at the corresponding point G.

The corresponding branch of

w

^{and at} is also non- singular.

(83)

Assume a circle makes A2 , 1-contact with

c.

Let

j)

be as before. Then u₂ and

w

₂ are as in the A2 -case.

and

u,

^and

u

the branch of

w

₁ in question will in general have ordinary cusps

at

P

^and

^4>

^("Y) respectively.

{84)

Assume a circle makes A2 , 2 -contact with C, and let

:P

^and

4>(p)

be as before. Then

u

₂ will have 2 distinct non-singular

branches at

JP,

with non-coinciding tangential directions. The corresponding branches of

w

₂ at

4>UP)

will also be non-singular.

Their tangential directions coincide iff 0 1 and 0 2 are diametrically opposite points.

In general u₁ will have 2 ordinary cuspidal branches. at

j) ,

with coinciding tangential directions. In general the same will be true for the corresponding branches of

w

₁ at

For more and better drawings, see [G-B].

\

_\

4> <1'> •

- 33 -

Proof of (Sl).

Let

o

_{1 ,}

a

₂ be the 2 points of tangency. Clearly

f>

^{is on}

u

_{1 ,}

but not on u and are locally the same. By Result 3.2 the embedded tangent space of U in G at

JP

parametrizes those circles passing through

a

₁ and

a

_{2 .} Clearly this tangent space is 1-dimensional (we exclude the possibility that 0 1 or 0 2 is on the line at infinity).

The kernel of the projection ~:P3+p2 at

j)

parametrizes a family of concentric circles. Since the family of circles passing through

a

₁ and does not consist of concentric circles, the projection

~

is a local isomorphism at}), and (Sl) follows.

Remark. If both

a

₁ ^and

a

₂ are on the line at infinity, U will be singular at

Y

The same is true if either

a

₁ or

o

₂ is a circular point at infinity, i.e. (l,±i,O)~

Proof of (82).

The proof is essentially the same as the proof of (Sl). Let QEC be the point of contact. Result 3.2 gives that the embedded

tangent space to U in G at

JP

parametrizes those circles that touch C at Q. This is a 1-dimensional family of non-concentric circles. This gives (82).

Remark. When Q is at infinity, but non-circular, we will obtain a 1-dimensional family of "concentric" line pairs. When Q is circular, we obtain a 2-dimensional family.

About the proof of (83).

Clearly the curves u 2 and

w

₂ will be as in (82), since the

circle is only osculating at one point, say

o

_{2 .} The ideal defining u 2 locally at

j)

^{will be} (T6 2 >,Ti 2 >), using the terminology of

§4.

We

~ill

examine the local properties of U at

}J .

^Recall

that the affine cone over U is defined by the Fitting ideal Fl(q*cr(_{1 ).} Let the circle touch

c

at

o

₁ and be osculating at

o

_{2 .} By Formula (2.4) of §2, we have

Fl(R

1

^~R

2

⁾

=

(FO(R 1 )•Fl(R2 ),Fl(R1 )•F⁰(R 2 ))

=

(FO (R1) •Fl (R2) ,Fa (R2))

where R1 corresponds to 0₁ and to

o

_{2 .} Let and T ( 2 ) • t +T ( 2 )

1 0 be the polynomials defined by Weierstrass' Preparation Theorem for

o

₁ ^and

o

₂ respectively (see the proof of Result 3.1). By Observation 4.1 in §4

.F⁰(R 1 )

=

^{T ( 1 )} ₀

F 1 (R2)

=

(T(2) T(2)) 0 ^I 1 F 0 ( R2)

=

^(T(2))2+f _{0 3}

where f 3 is contained in the third power of the maximal ideal, and where f 3

E(T6

²

>,Ti

²

>).

^Hence

Fl(q*(jt1 >

is locally

( T

~

^{1 ) • ( T}

~

^{2 )}

IT~

^{2 ) )}

IT~

2 ) +f 3) .

In general

Tb•(T~

²

) ,T~

²

))

defines locally the union of a non- singular surface

s

and a curve; the pre-evolute u2. Inter- secting

s

with the singular surface with equation T( 2 )+f 0 3 ⁼ 0 we will in general get an ordinary cusp. Since f E(T( 2 ) T( 2 ))

3 0 ^I 1 ^I we

- 35 -

end up with the union of the preevolute

u

₂ and a cusp curve, which is u_{1 .}

About the proof of (S4).

Let the circle be osculating at

o

₁

gives rise to 2 distinct branches of

and p2. Clearly this with equations (T6 1 ) ,Ti 1 )) and (T6 2 >,Ti 2 )) respectively. The intersection of the 2 tangent lines corresponds to all circles tangent to , C at 0 1 and

o

2 ._ Clearly this is only one circle, corresponding to the point

~

^{in G.} The projections of the 2 tangent lines by $

will intersect the circle diametrically. Hence

w

₂ will have an ordinary node at $(p) iff

o

₁ ^and

o

₂ are not diametrically opposite points (\IThatever that means when our "circle" is not real). Using the same method as in the proof of (S3) we see that '

the ideal

F ¹ (q*~{ 1 ⁾

is locally:

(FO(R 1 )•Fl(R2 ),Fl(R1 )•F 0 (R2 ))

=

( ( ( T 61 ) ) 2 +g 3) • ( T 6 2 ) IT

i

^{2) ) I ( T 61 ) IT}

i

1 ) ) • ( ( T 6 2) ) 2 +h 3 ) ) where and h 3 are contained in the third power of the maximal

ideal, and

This gives an intersection between a surface germ s 1 union a curve germ [ 1 and a surface germ s 2 uion a curve germ [ 2 such that [, ES2 ^I and [ 2 Es 1 • We end up with [ 1

u [

₂

u [

_{s 1 ns 2}

J u ( [

_{1 n [ 2 )}

where

[, u

[2 is the germ of U2, [ 1 n [ 2 is

{]>},

and s 1 ns 2 is cut out by (1) 2 (2) 2

( ( T 0 ) +g 3 ^IT 0 ) +h3) • In general

u,

corresponding to s 1 ns 2 will therefore be two cusps with the same cusp tangent.

· In general the picture will remain unaltered after projection by $.

1-