A sharped bound for the number of generators of ideals defining space curve singularities

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UNIVERSITAT DE BARCELONA

FACULT

AT

DE MAT

EMÁTIQUES

A SHARPED BOUND FOR THE NUMBER OF

GENERATORS OF IDEALS DEFINING

SPACE CURVE SINGULARITIES

by

Joan Elias

PRE

-

PRINT N

.

0

₇

man

;

1982

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A SHARPED InJND FOR TifE NUMBER CF GrnERA'IDRS CF IDEl\lS Df:FININJ SPJ\CE aJRVE SlNGUIARITIES

Joan Elias

Abstract: In tJris paper we stablish a lx>tm<l for the numher of gcnerators of '}eanetric determinantal ideals of codimension tw::>. Aften,,ards \..e sh:lw that tJris lxlund is sharp for curve singularities.

1.Definitions. 'lhrouglDut this article I shall use

e

to denote the local ring of formal power series

_{k((x1, ... ,~]] ,}

or the local ring at the origin of k [

x

₁, ••• ,Xn ) \mere k is an algebraically

(Xl, ... ,~) closed field of characteristic zero.

An ideal of f) will be called determinantal if there is a ruan matrix 11 with entries in

e ,

such that I is generated by the nrr sul:xleterminants of 11 and the height of I is nrudnun, i.e. (n-r+l) (m-r+l).

A germ (X,O) will be called determinantal if it can be defined

by a determinantal ideal of 0 • In the case of cxxl:imension b..o we have a nx (n+l) matrix M, and I is generated by the nxn sul:xleterminants of M. ~

can also assume that any m:lnimal basis of I is obtained taking the maxi-mal sul:xleterm:lnants of a matrix M with entries in the r.iaximal ideal of

the ring

e

(see

(at]

l.

It is known that in cxxl:imension t\..o, to be deterMinantal is equivalent to be perfect ( see [lb-E] ,

(at]

l •

l"E dernte by l'(I) the nmtier of elements of a miniroal basis of I, h the height of I ande the nultiplicity of the local ring 9

I

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2.n-e oourrl. Let I be a perfect ideal of e with height t1,,o,then

2.1 Lema. J\fter a suitable linear change coordinates

=

may assu,e that the cossets

x

1, ..•

,~e~

₁ forrn a systern of para-meters of ~

I

Proof: We shall procee:1 by induction on the dimension of

e .

Fran lemna5 of

[z-s]

pag.287 there exists a superficial element of degieel. NcM

since for an elenent to be superficial depends only on the initial fo:tm,

\loe can assure that there exista a cosset

X

such that is a superficial eleoont of degree l. Set

X.=~ .

It is easy to prove that a superficial elem:,nt is a non-zero divisor, therefore it is suficient to consider the quotient

o

I + ~O which is iscr.orphic to

I + ~ O /

/ ¾ª

and this ring is C.ohen-Macaulay.

o

-

-By induction on

¾

₀ we have a regular sequence x₃, ...

•¾

in the ring

f ,

and fran theoran 16.B of [Mat]it follovs that the

e

above sequence fonn a systern of parameters of I

2.2 'n-eorern.Let I be a perfect ideal of 8 of height t1,,o then

l'(I) ( P (I)-1) f. 2e •

Proof: Fran corollary ho to theoran five of

[Bu]

we know that

2

J'-1 I Cm

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where m is the maximal ideal of

e .

By the ab:,ve lamia we may assurre that

x3' ...•

~

are a syste-:i of parameters of

t .

'111erefore

f.breover the cossets

~ ~

with a+b~ 1'-2 form a k-indepandcnt set, because if ,-ie had a linear relation

with ~a,b C k , then

and hence

,ic

1-o.tld get in the ring that

which yields

~

a,b = O. '11,e nmtier of cossets

~ ~

with a-+b~ l'-2

is

r(j

-l) , and consequentely 2. e

~

I' ( I' -1).

3. '!he exatlJlles of Macaulay and f.bh. It is a classical problen to find pr.úoo ideals of 0 which need at least n generators, for each n,;::,. l.

Macaulay constructed prime ideals that requires exactly n generators( [Mac] ¡;p. 36-37). J\bhyankar found that this construction was " rather mysterious today and ••• in need of pro:,f" ( [Ab] ) • fbwever

J\bhyanl:ar substitutes the original claim of " exactly n generators" by

" at least n generators". Now I will shol, lnw theoren 2. 2 allows us to

3

r"_,;1

UNfVBRSJT,n ••

~ BARCHONA

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recover Macaulay' s claim.

3.1 'llleoran, l.et Mn be Macaulay's prinE ideals for the ring

l'(Mn)=n. 1 Proof: It is known that the nultiplicity e of the ideal Mn is

₂

n (n-1). Fran thc paper of 1\1:hyankar we get r;;;.n, therefore by theoran 2,2

e= n(n-1)

r ( r

-1>

~ e

hencen=I'.

lbre recently T.T.lbh sh:::,.,.e:l (

[lb] )

that in the case of

the ring e=

k[[x

₁

,x

₂

,x

₃

JJ

there exist prirre ideals 1'

0 E: Spec( 0)

which have n+l as their mini.mal mr.lber of generators. Like the case

abolle we have

I'( pn )( r( pn )-1)

=

2e.

'Ihus we see that our bound is sharp for genns of algebraic

and algebroid space curves.

Notice

that

the exaJ'ft)les of fücaulay and M:>h have mininal nultiplicity for

r

fi.xed, or maximal f' for a fued nultiplicity.

4. other bounds. G. Valla in his paper ( [va] ) gives a general bound that in our hypothesis is

l ' ( I ) ~ e+3

r

-Now it is easy to sh:M that the bound of thepran 2.2 is

J ' ( l ) ' - 1 + ( " ; ; ; ;

'olbich for any e ~ 3 is better thatl Valla' s.

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_!lanark. I refer to Valla's pap,r for a caip,rrison of this bound with previously obtained bounds by BeckP.r, Sally, lloralynski-EiSCIOld-Rees.

REFERENCF.S

[Ab] ABHYJ\NKAR "On Macaulay's exarrples" Lecture Notes in Math. nº311, [ ~ llURCH "On ideals of finite lu1Ulogical d:imé>.nsion in local

rings" Proc. Cambridge Plúlo,Scx::. Vol.64 (1968).

[Ho--FJ

HXIISTER-E/1/XN "Cohen-Macaulay rings, invariant theory ••. " l\mer, Journal of Math, 93 (1971).

[1-bc)

M!ICAUIAY "'11le algebraic theory of m:x:lular systaTIS" Carri>ridge Tracts in Math. and Math, Physics (1916).

l}-1at] 1\1\TSlMJRA "Camutative algebra" seoond edition. Benjamin.

[aj

IOH "On ganerators of ideals" Proc. of the l\MS. Vol. 77 n°3 llece!tver 1979.

(va]

"Generators of ideals and nultiplicities"

Cam'lmicationes in Algehra 9(15), 1541-1549(1981).

[z-s]

ZARISKI-SAKJEL "Camutative Algebra" Vol II. Springer-Verlag.

V1\LlA

DEPARJ'M1FNI' DE GEX:METRIA I 'IDPOILGIA. FJ\CULTAT DE MA.Tl'MATIOOES, UNIVERSITAT DE BI\ICE:lOIA.

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