• Aucun résultat trouvé

A NOTE ON THE ONE-DIMENSIONAL SYSTEMS OF FORMAL EQUATIONS

N/A
N/A
Protected

Academic year: 2022

Partager "A NOTE ON THE ONE-DIMENSIONAL SYSTEMS OF FORMAL EQUATIONS"

Copied!
8
0
0

Texte intégral

(1)

A NOTE ON THE ONE-DIMENSIONAL SYSTEMS OF FORMAL EQUATIONS

by Juan ELIAS

To Joan

0. Introduction.

Let (X,0) be an algebroid singularity defined by the ideal I C k[[Xi, ...,JQv]]. J. Nash in [N] proposed to study (X,0) using the set of arcs Ax, i.e. the set of a. G k[[r]]^ such that a.(0) == 0, and /(a.) == 0 for all f C I . Let A^ be the set of n-th truncations of Ax: 7. € k[[T]]^

belongs to A^ if and only if deg(7^) < n for all i = 1,..., N and there exists a E Ax such that a - 7. € (^^^[[r]]^. Denote by TT^ : A^ -^ A^-1

the truncation map ^((E^O^^)^!,...,^) = (E^=o1 7}T^=i,...,^ so we have a projective system of sets {A^,7Tn}n>o and a isomorphism of sets Ax ^ lim^.A'x. Hence a way to study Ax is look into A^-. In the complex case from the existence of a non-singular model of (X,0) J. Nash deduces that A^ is constructible for all n ( see [N],[Le]), on the other hand J.C. Tougeron ( see [Le]) proves that A^ is constructible from the formal version of the approximation theorem of M. Artin ([A]) due to J. Wavrik ([Wl]). In particular from this result one can deduce that there exists a numerical function /3 : N x N —> N such that: 7. € A^ if and only if there exists 7. € k[[T]]^ such that /(z) € (r)^)k[[r]]^ for all / C I and

z - z e ^ ) ^

1

-

Key-words ; Artin approximation - Singularity - Near point - Multiplicity - Reduction number.

A.M.S. Classification : 14B05 - 14B12.

(2)

As far as we know only in a few cases we have an explicit deter- mination of /3: first case is due to J. Wavrik for X reduced plane curve taking non-singular arcs ([W2]), the second one is due to M. Lejeune for hypersurface singularities taking general arcs ([Le]).

In this paper we determine the function /3 in the case of one- dimensional singularities X, taking non-singular arcs, in terms of the Milnor number associated to Xreci- See [La] for other results on f3.

The paper is divided in two sections, in the first we give some preliminaires results about contact between curves. In the second one we define the numerical function /3 and we prove the main result of this paper (Theorem 2.1).

Throughout this paper R will be the power series ring k[[Xi,..., Xjv]], where k is an infinite field. We denote by M the maximal ideal of R.

A curve of (k^,0) = Spec(-R) is a one-dimensional, Cohen-Macaulay closed subcheme X of (k^,0), i.e. X = Spec{R/I) where I = I(X) is a perfect height N-l ideal of -R; we put Ox = R / I . A branch is an integral curve.

2. Contact of curves.

If X is a reduced curve of (k^,0) then we denote by 6(X) the dimension over k of the quotient O x / O x where Ox is the integral closure of Ox- It f is the number of branches of X then we define the Milnor number of X by p,(X) = 26 - r + 1.

Let X be a reduced curve and let Q be an infinitely near point of X, see [ECh], [VdW]. It is known that there exists a unique sequence {Q}i=o,...,s of infinitely near points of X such that QQ •= O,...,^ == 0, and that Qi-\-\ belongs to the first neighbourhood of Qi for i = 0,..., s — 1.

We denote by (X, Q) the union of the irreducible components through Q of the proper transform of X by the composition of the blowing-up centered at Qi for i = 0,...,5 - 1. We denote by P(X,Q)(T) = e(X,Q)T - p(X,Q) the Hilbert polynomial of the local ring 0(x,o)-

For the readers convenience we will summarize some properties of e(X, Q) and /o(X, Q) that we will use in the paper:

(1) e(X,Q) - 1 < p(X,Q), ([M] Proposition 12.14),

(2) e(X,Q) = 1 if and only ifp(X,Q) = 0, ([M] Proposition 12.16),

(3)

(3) e(X,Q) =2 if and only if p(X,Q) = 1, ([M] Proposition 12.17), (4) dim^R/I + M71) = J?(X,Q)(^) for all n > e(X, Q) - 1, ([K] Theorem 6, or [M] Proposition 12.11).

Let T(X) be the set of infinitely near point Q of X such that its multiplicity e(X, Q) is greater than one. From [Ca] we obtain that

6(X)= ^ p(X,Q).

Q€T(X)

Let X, V be curves of (k-^,0), without components in common, we denote by (X.Y) the number dim^R/I(X) + I(Y)) ([H]).

Let Z\ be a branch, for every reduced curve Z2 ? such that Z\ is not a component of Z^^ we define /(Zi, Za) as the number of non-singular points shared by Z\ and Z^.

PROPOSITION 1.1. — JfZi is a non-singular branch then

( Z l . Z 2 ) < ^ ( Z 2 ) + / ( Z l , ^ 2 ) + l .

Proof. — From [C] and [M], Proposition 12.16, we deduce (^1.^2)^ ^ (p(Z,UZ^Q)-p(Z^Q))

QeK

where K is the set of infinitely near points shared by Z\ and Z^.

Since (Zz,Q) is a curve of (k^,Q) ^ (k^.O), we put (Zz,Q) = Spec(JZ/J^o) for % = 1,2. Consider the projection

_____R______ R (JI,Q n J2,o) 4- M71 "^ J2,Q + M71

for all n > e(Z^,Q)-, from this and [K], Corollary 6, we get (e(Z2,0) + 1)^ - /?(^i U Z^Q) > e(Z^Q)n - p(Z^Q).

Therefore p(Zi U Z^Q} - p(Z^,Q) < e(Z^,Q), and hence (1) (Zi.^2)< ^e(Z^Q).

Q^K

Assume that Z^, is singular. Since e{Z^,Q) < p(Z^,0) + 1 , [M]

Proposition 12.14, and r < e(Z^,0) we deduce

(2) e ( Z 2 , 0 ) < ( 2 p ( Z 2 , 0 ) + l - r ) + l .

(4)

Let K* be the set of points belonging to K such that e(Z^,0) > 2.

From [M], Proposition 12.17, we obtain that for all Q C K*

(3) e(Z^Q)<^2p(Z^Q).

By (2) and (3) we get

^ e(Z^Q)< ( 2 ^ ^ , Q ) + l - r ) +1, Qe^* \ QCK* ) since p(Z^,Q) = 0 if and only if e(Z^, Q) = 1 we have

^ e ( Z 2 , Q ) < / ^ 2 ) + l . QEK^

Recall that e(Z2,Q) = 1 tor Q C ^ - K\ from (1) we obtain the claim.

PROPOSITION 1.2. — Let Zi = Spec(R/Ii) be curves, i = 1,2.

Assume that Zi is non-singular and 1^ + M^2^^ C h + M^2)-^1. Then we have

n<f(Z^Z^).

Proof. — From the hypothesis we deduce that

A + h C h + h + M^^2^^1 = A + M^^2^^1, so that

/^(Z2) + n + 1 < dimk(^/A + M^2^^) < (^.^2).

The claim follows from Proposition 1.1.

COROLLARY 1.3. — Ifn > 2 then there exists a non-singular branch Y ofZ'2 such that n < f{Z^Y).

Proof. — By Proposition 1.2 we get /(Zi,^) > n > 2, so there exists a branch Y of Z^ such that Z\ and Y share n non-singular infinitely near points. Since a non-singular branch and a singular branch cannot share two non-singular near points, we get that Y is non-singular.

The following result is well known :

PROPOSITION 1.4. — Let Zi, Z2 be non-singular branches, for all n the following inequalities are equivalent :

(1) ( Z i . Z 2 ) > ^ ,

(5)

(2) Z\ and Z^ share n infinitely near points, (3) for all parametrization ofZ\:

[ xl = t

1 : \Xi= Xi(t) f o r a l l i = 2 , . . . , N , there exists a parametrization of Z^:

f Xi = t

2 : \ Xi = Xi(t) for all i = 2,..., N, such that

Xi{t) - Xi(t) E= 0 modulo (^n, foralli=2,...,N.

2. The function /?.

DEFINITION. — We say that a system of formal equations {F = 0} = {FI = 0,...,Fs = 0}, Fi € R, is one-dimensional if and only if (F) = (Fi,..., Fs) is a height N - 1 ideal of R. We denote by T the set of one-dimensional systems of formal equations.

Let {F = 0} be a one-dimensional system of formal equations, we define the curve Z p = Spec(JZ/rad(F)), and the numbers /x({F = 0}) =

^(Zp) and m({F = 0}) = Min{n € N [ rad((F))71 C (F)}.

DEFINITION. — Let /3 : N x N —> N be the numerical function:

/3(n, {F = 0}) = m({F = 0})(2/.({F = 0}) + n + 1).

THEOREM 2.1. — Given a one-dimensional system of formal equa- tions {F •==- 0}, and a non-negative integer n > 0 if Zp is singular and n > 1 if Zp is non-singular. Let X^(Xi, ...,Xr) C k[[Xi, ...,X^]], l < r < - / V , % ' = r + 1,...,7V be a set of fbrmal power series such that for every G e (F) :

C?(Xl,. ..,Xr, Xr 4-1 (Xi,..., X y ) , . ..,A'7v(A'i,...,Xr)) ^ 0

moduio^i,...,^,)^71^^0^ . Then there exist X,(Xi, ...,Xy.) e k[[Xi, ...,X^]], z = r+1, ...,A^, such that:

(6)

( 1 ) G ( X i , . . , X , , X , + i , . . , X ^ ) = O f o r a U G e ( F ) ,

(2) Xi(X^...,Xr) - Xi(X^...,Xr) EE 0 modulo (Xi, ...,X^)71 for all i = r + l , . . , J v .

Proof. — First of all we will prove that r = 1. From now on we put /z({F = 0}) = ^(Zp) = ^ P(ZF,O) = p and e(Zp,0) = e.

Let J be the ideal of R generated by Xi - X,(Xi, ...,Xy.) for i = r +1,..., N. Notice that J is the kernel of the map (p : R -> k[[Xi,..., Xy.]]

defined by

(p(G) = G(Xi,...,Xr,X^+i(Xi,...,X7.),...,Xjv(Xi,...,X7.)) . From the hypothesis we deduce that

(F) C J modulo (Xi,...,;^)^^0^ so

(1) rad((F)) C J modulo (Xi, ..^X^4-1-^1 . Recall [C] that

6(Zp)= ^ p(Z^Q), Qer(z^)

by [M], Proposition 12.14, we obtain that 6(Zp) + 1 > e; from this we deduce p, > 6(Zp), so p, > p.

From [M], Proposition 12.11, we get

dimk (rad((F)yfM^^) = e(2/A + n + 1) - p- Since Spec{R/J) is non-singular, from (1) we have

e^+n+^-p^i2^^^ .

Assume that r > 2, then (2/A + n + l)(e - (/A + 1) - n/2) > p. Since /A > /? > e - 1 ([M], Proposition 12.14) we obtain: p < (2/^ + n + l)(-7i/2).

If ZF is singular then we get p < 0 , but from [M], Propositions 12.14 and 12.17, we have that p > 1, so r=l. If Zp is non-singular we get that p < 0, since p is a non-negative integer ([M], Propositions 12.14) we deduce r = 1.

Consider the non-singular branch Z\ which admits the parametriza- tion:

. f X i = t

1 ' \Xi= X ^ ) f o r a l H = 2 , . . . , 7 v .

(7)

Notice that the series Hi = Xi - X,Xi), i = 2, ...,A^, form a system of generators of the ideal Ji defining the curve Zi. If G C rad(F) then

C?(Xi,X2(Xi),..,X^(Xi))=0 modulo (Xi)^^ , thus

rad((F))cJi modulo (Xi)^-1-^.

From Propositions 1.2,1.3 and 1.4 we deduce the claim.

Remark. — (1) From the proof of the theorem it is easy to prove that for the systems of formal equations with r = 1 one can take

/?(n, {F = 0}) = m({F = 0})(^({F = 0}) + n + 1).

(2) If we consider reduced systems of formal equations, i.e. rad((F)) = (F), then we have

/?(n, {F = 0}) = 2/,({F = 0}) + n + 1.

Notice that the number 2p,({F = 0}) + 1 has the following property ([E]): the analytic type of Zp is determined by any of its truncations:

{Zp)n = Spec(J?/(F) 4- M71) for all n > 2/^({F = 0}) -h 1.

BIBLIOGRAPHY

[A] M. ARTIN, Algebraic approximation of structures over complete local rings, Publ.

Math. IHES, 36 (1969), 23-58.

[Ca] E. CASAS, Sobre el calculo efectivo del genero de las curvas algebraicas, Collect Math., 25 (1974), 3-11.

[E] J. ELIAS, On the analytic equivalence of curves, Proc. Camb. Phil. Soc., 100, 1(1986), 57-64.

[ECh] F. ENRIQUES and 0. CHISINI, Teoria geometrica delle equazione e delle funzione algebriche. Nicola Zanichelli, Bologna 1918.

[H] H. HIRONAKA, On the arithmetic genera and the effective genera of algebraic curves. Memoirs of the College of Sciences, Univ.Tokyo, Ser.A, Vol.XXX Math., n°2(1957).

[K] D. KIRBY, The reduction number of a one-dimensional local ring, J. London Math. Soc., (2) 10 (1975), 471-481.

[La] D. LASCAR, Caractere effectif des theroemes d' approximation d' Artin, CRAS, 287 (1978), 907-910.

[Le] M. LEJEUNE-JALABERT, Courbes tracees sur un germe d' hypersurface. Preprint.

[M] E. MATLIS, E.I-Dimensional Cohen-Macaulay Rings, Lecture Notes in Math n°327, Springer Verlag, 1977.

(8)

[N] J. NASH, Arc structure of singularities. Preprint.

[VdW]B. Van der Waerden, Infinitely near points, Indagationes Math.,12(1950), 401- 410.

[Wl] J.J. WAVRIK, A theorem on solutions of Analytic equations with applications to deformations of complex structures, Math. Ann., 216(1975), 127-142.

[W2] J.J. WAVRIK, Analytic equations and singularities of plane curves, Trans A M S 245(1978), 409-417.

Manuscrit recu Ie 23 janvier 1989.

Juan ELIAS,

Universitat de Barcelona Facultat de Matematiques

Departament d'Algebra i Geometria Gran Via 585

08007 BARCELONA (Espagne).

Références

Documents relatifs

conductance fluctuation (UCF) of the two-probe conductance in the regime 2.5t 6t in 1D disordered quantum systems in the absence of any inelastic scattering, 1-e-, at zero or a

We wish to make clear that our result, namely that Levinsons’ theorem is an index theorem, is quite different from a result encountered in supersymmetric quantum mechanics about

tions. The wave of advance of advantageous genes. [11] Harry Kesten and Vladas Sidoravicius. A shape theorem for the spread of an infection. The spread of a rumor or infection in

qu'il y a diffusion de l'énergie parmi les ions Gd3+ et piégeage par des impuretés à haute température et par des ions G d 3 + perturbés à basse

We recall that article [N3] gives, in particular, explicit asymptotic formulas for finding the complex-valued scattering am- plitude from appropriate phaseless scattering data for

We prove that there exists an equivalence of categories of coherent D-modules when you consider the arithmetic D-modules introduced by Berthelot on a projective smooth formal scheme X

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des

We shall also mention that estimating the observability constant in small times for the heat equation in the one-dimensional case is related to the uniform controllability of