Rend. Sem. Mat. Univ. Pol. Torino Vol. 48, 4 (1990)

ACGA - 1990

M. Boratynsktf*)

ON T H E CURVES OF C O N T A C T ON SURFACES IN A P R O J E C T I V E SPACE II

Dedicated to Paolo Salmon on his 60th birthday

A b s t r a c t . The aim of this paper is to characterize the smooth curves

of contact on surfaces in F^ which pass through rationale double points or

ordinary singular points.

Introduction

In [1] we characterized the smooth curves of contact on surfaces in P3. The aim of this paper is to specialize our result to curves which pass through the points An, Dn, E&^ E7 (the rational double points). In particular we obtain a numerical formula which gives a necessary condition in order that a smooth curve C be a set-theoretic complete intersection on a surface F provided that all the singular points of F which are on C are the rational double points. The formula involves degC, </('C), degF and some data of local nature which reflects the way C passes through the rational double points. The above formula was already obtained by Gallarati in case of nodes (A\) and ordinary biplanary points (A2) ([8]).

In section 3. we characterize the curves of contact which pass through ordinary singular points. As a corollary we obtain that a smooth rational quartic in F3 is not an intersection of two surfaces provided that one of them admits only ordinary singularities and the quartic is not contained in its

(*)This research was supported by the funds of MURST.

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singular locus. The same is true for a general (smooth) rational curve of degree > 5.

0.

In this section we shall recall and reformulate some results obtained in

Suppose C is a smooth (connected) curve contained in a surface F C P3. (k-algebraically closed). Let I C O^s denote its ideal sheaf. For any i > 1 we define J,- C O^ putting J Z / 0 F 3 ( - F ) = the subsheaf of O^A/O^Z(-F) which is equal locally to the t-th symbolic power of 1/0^3(~F).

LEMMA 0.1. ([1], Lemma 1). Suppose that C is a smooth curve contained in a surface F C P3 such that C (jL Sing F— the singular locus ofF. Then

1. For i > 1 Ji/Ji+i is a rank 1 locally free OQ-module.

2. J2/I2 + O^s(-F) is a torsion sheaf supported on C C\ Sing F

3. J{ - Jj C Ji+j and therefore there is a natural pairing

DEFINITION: Let C he an irreducible curve on a surface F C IP3 and let t > 1. C is called a curve of contact of the t-th order on F if there exists a surface G C F3 such that (t + 1)C = F • G as cycles.

T H E O R E M 0.2. ([1], Theorem 1). Let C be a smooth curve on an

irreducible F C F 3 and let u> denote the canonical bundle ofC. Suppose that

C (fi Sing F. Then C is a curve of contact of the t-th order on F if and only

if

1. degtF\(t + 1) deg C.

2. J / + 1 is locally a complete intersection.

3. Jt/Jt+1 3 'u(4 -degF- (t + l)deg C/degF).

4. HlJt{deg F+((t+ l)deg C/deg F) - 4) = 0 .

In the sequel we shall always assume that C is a smooth curve contained

in a surface f c F 3 such that C <f_ Sing F. We put m = deg F.

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PROPOSITION 0.3. I/J2 = w ® - 1 ^ - 4) ® 0<7(-D) where D denotes the divisor (supported on C D Sing F ) associated to J2/I2 + 0p3(—F).

Proof: The result follows from the proof of Theorem 2 in [1].

PROPOSITION 0.4. For t > 1 «/*/J<+i £ (I/h)®1^) wiiere (£<) is an effective divisor supported on C D Sing F .

Proof: The pairings J I / J J + I ® Jj IJj+\ —• J%+j I J%+j+\ induce the map at : (I IJ2)®* —• Jt/Jt+i(Ji = / ) • In an open neighborhood of a point of C which is nonsingular on F I — ( / , x) where / is a local equation of F and x 6 / . It follows that J,- = ( / , a:1) for every i and a< is an isomorphism on C\(C n Sing F). Hence J</Jt+i * (I/h)®1^) where £< is a divisor corresponding to the coker at.

In the sequel we shall need the following reformulation of Theorem 0.2 which follows easily from Propositions 0.3 and 0.4.

T H E O R E M 0.5. Let C be a smooth curve on an irreducible surface F C IP3 and let LJ be a canonical bundle ofC. Suppose that C tf_ Sing F. Then C is a curve of contact of the t-th order on F if and only if

1. m\(t -f 1) degC where m = deg F.

2. Jt+i/Oya( — F) is a locally principal ideal sheafofO^a/O^—F).

3. Oc(Et-tD) ^u>®(t+1\-tm-m-n + 4t + 4) where n = ( t+1 ) degC/m.

4. HxJt{rn + n-4) = 0.

Proof It is enough to note that

Jt/Jt+i = (I/J2)^i(Ei) * u>®-\tm - 4/) ® Oc(-tD) ® Oc(Et) .

R E M A R K . It suffices to verify condition 2. at the points of CD Sing F.

Let / G / = (X ,Y) C k[[X,Y,Z]]. Then / = ] T / , • ( * , Y)Z* and l > 0

/,-(0,0) = 0. Put p = / / ( / ) G fc[[X,Y,Z]]/(/). We define J 2 G M I * , ^ ] ] putting J 2 / ( / ) r p(2). Obviously J2/I

2 + (•/) - P * 2 V P 2 .

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LEMMA 0.6. If f £ I2 then dim^p^/p2 = the smallest i such that fi(X,Y) contains a non-zero linear term.

Proof. Put io — the smallest % such that fi{X, Y) contains a non-zero t 'o—l

linear term. We have / = V , fi{X',Y)Zl + Zl° • g and / = (</, h) for some i=o

h G I.J2 D {9th2) since J2 is a minimal unmixed ideal containing I2 -f (/)• J2 = {g,h2) because J 2 + ( / ) C I2+{g) = {g,h2). So J2/I

2+{f) is generated by the image of g. We conclude that dim^p^/p2 = dim*; J 2 / / 2 + ( / ) = io since g is annihilated modulo I2 + ( / ) by Zl° and not by any lower power of Z.

With the hypotheses of Lemma 0.1 let D — ^ D{a) ' a be the aeCnS'mgF

divisor associated to J2/I2 + 0^{-F). Fix a G C D Singi71 and pick X, Y, Z

- the regular parameters in a G IP3 in such a way that X,Y are the local

equations of C. Then 5p3 a = k[[X,Y,Z]]. Let / G fc[[A",y,Z]] be a local

equation of F. Then / = Ei>0{fi{X,Y)Zi G 7 \ / 2 where / = (X, Y).

PROPOSITION 0.7. D{a) - the smallest i such that / 2 (X,Y) contains a non-zero linear term.

Proof. Without altering anything we can pass to the completion of Op3 a . The result follows from Lemma 0.6.

Let Et = 2_j Et{a)a be the divisor associated to coker at where aeCDS'mgF

at : (//J2)®* —* Jt/'Jt+i (see the proof of Proposition 0.4). Fix a G C and let X,Y,Z be as before the regular parameters in a G IP3 such that X, Y are the local equations of C. Let / G k[[X, Y, Z]] = 0^3 a be a local equation of F . Put / = (X, Y) and p = / / ( / ) G *[[*, ̂ Z]]IU)-'

PROPOSITION 0.8. £<(a) = dim^pW/p* -fp( / + 1)) .

Proof. It suffices to note that we can pass to the completion of Op3 a

without changing anything.

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1.

In this section R will denote a normal domain (unless stated otherwise) and for any (finitely generated) i2-module M* will denote Horn (M, R). There exists a canonical homomorphism (J>M : M —• M**, M is called reflexive if 4>M is an isomorphism. For any i2-module M M** is reflexive. <J>M has the following universal property: For any a : M -+ A where A is reflexive there exists a unique /3 : M** —> A such that a = /3<j>j^. Hence for any ideal I C R, I** can be (and will be) identified with an ideal containing / . If M is torsion free then M** can be characterized as a reflexive module containing M such that codim Supp M** jM > 2. It follows in particular that for any ideal / C R and a G R (al)** = dl**.

LEMMA 1.1. Let p and I be two ideals of R. Suppose p is a ht 1 prime ideal and I is reflexive. If pn C I and p" = Ip then I = p^nK

Proof. Obviously I C p^ and codim Supp p(n>/I > 2. It follows that I = p(n) since / is reflexive and p\n> is torsion free ([3]).

PROPOSITION 1.2. Let p be a ht 1 prime ideal of R. Then (pn)** = p(n\ ((p(*))*)** = pW a n d (p(*) . p(0)** = p(*+0.

Proof. Apply Lemma 1.1 to I - (pnj**, / = ((p{k)f)** and / = (j)(k) . p(-0)** respectively.

PROPOSITION 1.3. Let p and q be two ht 1 prime ideals of R. Suppose

ap =.bqW for some a,b G' R. Then.pW/p* + P{t+1) = aqW/afaW)*

+y*«+i)) a n d pW/pV*1) * a9(*')"/M*('+1))

Proof. We get a V ; = &'(«<*>)*. It follows that («V)** - (&V* 0 / ) **

and by Proposition 1.2. we get atp"' — & y >. Multiplying by a we

also get at+1pW = atfq^ '. In the same way as above we obtain that

at+ip(t+i) _ ^+i^(*(«+i)).e Moreover from the equality dp1 = ^(q^)1 we

infer that a * + y = ab^q^)1- pW/pl + p^1) £ a ^ + V W / a ^ 1 C P V < + 1 > )

= a ' + y ^ + y + a< +V ( < + 1 ) ' = atfqW/atfiqW)* + &*+y*(*+i))

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In the same way we prove that pM/pi**1) * aqW/b^W+W.

LEMMA 1.4. Let R be a local ring with dim R = 2. Ifp C R is a prime

ideal such that R/p and Rp are DVR (discrete valuation ring) then

1. for every i; > 1 p\l)/p(l+1) is a rank 1 free R/p-module,

2. the natural map is non zero,

3. for every t > 1 the modules pW/p* + p</+1) and p(<+1)/p • pM + p('+2)

are finite length R/p-modules and length (p(*+1)/p*+1 -f p(<+2))

= length (pM/p1 + p(*+1>)+ length (p( '+ 1)/p -pW +p<*+2)).

Proof. p( l)/p( l+1) is free over i2/p since it is torsion free and R/p is a

DVR. Moreover p(*)/p(*+1) is cyclic over Rp since Rp is a DVR. Thus pW/p ( , '+1) .

is a rank 1 free 7£/p-module. The map p/pW ® p(l)/pi1*1) —> p(?+1)/p(?+2) is

non-zero since the graded ring © p(l)/p\l+l) (p(°) — j£) is isomorphic to .a i>0

polynomial ring in one variable over the quotient field of R/p.

p(t) /pl -f p(*+i) a n ( j _ p(*+1)/p . p(0 -|- p(^+2) are the finite length i2/p-modules

since pW/p* -f p'*+1) is a cokernel of the non-zero map (P/P(2)fl - • p{t)/p{t+1) and p('+1)/p.p(*)+p('+2) is a cokernel of the non-zero map p/p(2) 0 p(')/p('+i) _> p(*+1)/p(*+2). Note that both maps are the maps

between rank 1 free i2/p-modules.

Let e,- denote a free generator of p(*)/p( ,+1).

Put rf,- = length (p(*)/p* + p ( , '+1))- Then ej = aird*et where a G (# /p)*

and 7r is a uniformizing parameter of R/p. Let

r = length (p ( ' + 1 ) /p *P( /) + P ( / + 2 ) ) . Then ei • et = /3vrei+1 where /? £(R/p)*.

It follows that c{+1 = ei • ej = airdt • ei • et = a/3irdt+r • e<+1. This finishes the

proof of Lemma 1.4. •

PROPOSITION 1.5. Let p he a prime ideal of a domain

R = k[[X,Y,Z]]/(f) (k-algebraically closed, f^O) such that R/p and Rp are

DVR. Ifp(t+1^ is principal then for every s > 1

d im A . (p^ t + a - 1 Vp a * + s " 1 +P{st+S)) = dimJfc(p(i)/p*-+i>^+1>)

+ (s - l )di .m i b(^+ 1>/p1 + 1 + p( t + 2>).

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Proof. Note that for any finite length R/p-modvde M length

M = dim*M. Let J{ C k[[X,Y,Z]] be an ideal such that J,-/(/) = p^.

If p(m+n) is principal then J m + n is a ht2 complete intersection ideal of

k[[X, lr, Z]], From the formal version of the result in [2] (see also Proposition

4 in [1]) we get that the map Jm/Jm+i ® Jn/Jn+i -> Jm+n/Jm+n+i is an

isomorphism. It follows that the map ^(m) /^(m+1) ® j>(»)/p(»+i)

-*• p(m+»)/p(m+n+1) is an isomorphism.

We induce on s. If s = 1 the result is obviously true. Let et- denote a

free generator of p(l)/p('l+1) and put d{ = dim(p^/pl + p( l + 1)) . p(st+s) is a

principal ideal since a power of an unmixed principal ideal is unmixed (we do

not assume that R is normal). It follows that e(s-iu.f(5_2) * et+i = oiest+s-i

for some a G (R/p)*. Moreover

(8-l)t+(s-2) _ n d(s_l)t+(s_2) •

and e\+1 = 77r t+1 • e/+i where /?, 7 € {R/p)* and 7r is uniformizing parameter

of i2/p? It follows that

e l * - 1 = c ( - W - » ) . e * 1 = />7^(.-i)«+(.- ' )+*+i • e(s_1)tHs_2) • e ,+ 1

= o / ? 7 T r f ( - i ) < + ( - 2 ) + ^ i . C r f + a _ 1 >

So d im(y / s < + s - 1 ) / y / / + s - 1 + p(^+*)) = dim(p« s-1) '+(5-2)) /p(*-1)*+^-2)

+p((*-i)*+(5-i))) + ( i i m (p(<+i)/p*+i +/<+2)). The application of the inductive

hypothesis to dim (p((«-i)«+(«-2))/p(s-i)*+(s-2) +p((«-i)<+(*-i))) finishes the

proof. (Note that pK'-W+C*-1)) = p((*-i)*+(*+i)) i s principal).

2.

In this section wre shall describe all the formal smooth curves p which pass through the rational double points and in each case we shall find the values of/ for which p" + 1 ) is principal. Moreover we shall calculate dim p(2'/p2 and dim (p(i>/pt+p(i'*'1') i.e. the local contributions of the divisors D and Et which were defined in Section 0.

Let / G k[[X, Y, Z]) (A;-algebraically closed, ch k = 0) be one of the

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following:

An : A r n + 1 - YZ, n > l .

Dn \Xn~x + XY2 + Z2, n> 4 .

£6 : x3 + y4 + z2.

E7 : x2 + y3 + yz3

£8 : X2 + y3 + Z5 .

In the sequel x,y,z will denote the images in R = k[[X,Y,Z]]/(f) of X, Y and Z respectively.

PROPOSITION 2.1. Let / be an An-singularity. Every p C R such that R/p is a DVR is one of the following:

a) (y-axk,z-(3xe) where a,'/? € k[[X]],afi = ! ,& + £ = n+l,k > 1,£> l.~

b) (a? - ayk, z - / ty ' j where a G Ar[[y]]* U {0}, an+1 = /?, fc(ra + 1) = * + 1,

M > I-c) (x-azk,z-Pzl) where a e k[[Z]]* U { 0 } , a n + 1 = 0, fc(rc + 1) = £ + 1,

M - > 1.

If p is as in a) then

1) p( /+1) is principal if and only if (n + l) | ( t + l)k.

2) dim pW/p2 = m i n ( M ) .

3) Let t > 1 be such that (n + l) | ( t + l ) m i n ( M ) - Then dim (pW/p* + P ( / + 1 ) ) = ( s - l ) m i n ( M ) where s = (t + l)mm(k,£)/(n + 1).

If p is as in b) then

1) p(t+1) is a principal ideal if and only if (n + l) | ( t -f 1).

2) dim pW/p2 = 1.

3) Le t t > 1 be such that (n + l) | ( t + l ) . Then dim (pM/p* +p(t+1)) = 5 - 1 where s = (t + l)/(ra + 1).

if p is as in c) then

1) p"*1)- is a principal ideal if and only if (n .-f- 1)|(£ -f 1).

2) dim pW/p2 = 1.

3J Let* > 1 be such that (n +1) | ( / + 1). Then dim (pW/p* + p(<+1)) •= 5 - 1 where s =. (t + l ) / (n + 1).

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Proof. Let p = / / ( / ) C R be an ideal such that R/p is DVR. Then

/ is a kernel of a surjective horaomorphism 4> : k[[X, F, Z]] —> k[[T]] such

that </>(/) = 0. We can assume that <j> of one of the variables X, Y or Z is

equal to T. Let (/>(X) = T. Then <f>(Y) = a(T) • Tk and 0(Z) = / ? (T) r '

where a(T),/3(T) e k[[T]]* U { 0 } , T n + 1 = a ( T ) • / ? ( r ) T * + / since <£(/) = 0.

Hence a(T) • (3(T) — 1 and k + £ = n + 1. One easily obtains that

I = (Y - aXk,Z - j3Xe) where a = a(X) and /? = /3(X). This proves

a). If <j>{Y) = T or <£(Z) = T one obtains b) and c) respectively. Note that c)

is obtained from b) permuting y and z. So it suffices to consider the cases a)

and b) since / remains invariant when one interchanges Y with Z.

a) Put p = (y — ax ,z — /fa ). Then zp — (yz — axkz,z(z — /fa )

= (xn+1 - axkz, z(z - /fa*)) = (xk(xe - az), z(z - /fa*)) = (z - /3xe)(xk,z).

So p = (x , z) = q( > where q — (x,z). It is well known (and easy to prove)

that C£{R)— the divisor class group of R is Zn+\ and is generated by q.

It follows that p(<+1) is principal if and only if (n + 1)|(* + 1)&. Note that

(n + l) |( i + l)k if and only if (n + 1)|(* + l)£.

Let X' = Y- aXk, Y' = Z - /3X£, Z* = X be the new variables. Then

/ = X'Y' + P(Z')X'(Z'Y + a(Z')Y'(Z')k. It follows from Lemma 0.6 that

dimpW/p2 ~ min (&,£).

Let / > 1 be such that (n + 1)|(2 -f l)min (fc, t). We can assume that

k < £ since k and £ play the symmetric role. So (t -f 1)& = sk -\- s£ and

Jfc = (s - l)k + 5^ = 0 - l)(w + 1) + L Put as before q = (x,z). Then by

Proposition 1.3. q(ik) = ( ^ (^H*- 1 ) ) . ^* ) )** = zs~l{x^z) since q(n+1) = (z).

Moreover (qW)* = (xk,z)1 and qW+V) = q(s(n+i)) = ^sy W e h a v e a l r e a d y

proved that ap = bq(k) with a — z and b = z — /fa*. By Proposition 1.3

p(*)/p< + p (Hi) * fl9(*')/a(^))' + 69(*(<+D)

- (AVfl+1)/(<s*,*)*,2*+1 - / faV)

and pW/pV*1) £ (xlzs,zs+x)I(zs+x - f3x£zs). By Lemma 1.4 pO/pC^ 1 ) is

a rank 1 free i?/p-module and the image of 2 5 + 1 in pi^/pi1*1) is its free

generator. The elements xlkz3 with j > 1 and i + j = t + 1 generate z{xk\)z)i.

Let j > 3 + 1. Then xihz* = x^z^3'1 • ̂ + 1 = /?>- s"1 .a?»*+0-*-i)' . ^*+i

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in p(0/p(*+i). If 1 < j < S then ys~i • z8 = ys~i • zs~i • z> = a;(*+0(*-i) • zK

It follows that xikz> = x(t+1-tik >z> = xl'-W -x£ .s(*+')(*-J> •*>

= x(i~l)e-xe >ys~3 -zs = as~i •x(s~rtk+ti~1)e-xezs = as~^1 -x^s~^k+^~1^ -

z{*+*) in p(«)/p(«+i). We infer that dim (pW/p* + p<*+1))

= min <

A = min

i? = min

(/ + i-j)fc + ( j - s - i K 5 + 1 < j < t •+1

(s-j)k + (j-l)l l<j<s

since p^/p1 -f p^+ 1) is isomorphic to pM/p"*1) factored by the image of

z(xk, zY and the image of x is a uniformizing parameter of R/p. A — (t — s)fc

and B — (s — l)k since A; < /. We have 5// + 1 — 5 — kjl < 1.

It follows that (5 - l)k <(t- s)k and dim (pW/p* + p^+V) = (s - l)k

which was to proved.

b) Put p - (x - ayk,z - j3ye). Then yp = (y(x - ayk),xn+1 - /3y£+1)

= (y(x-ayk),x1l+1 - a n + 1 -yk(n+1)) = (y(x - ayk),(x - ayk)(xn - axn~1yk

+ . . . ± anykn)) = (x - ayk)(xn,y) = (x - ayk) • q^ where q = (x,y). It

follows that p(*+1) is principal if and only if (n -f 1)|(/ + 1) since q generates

C£(R).

Let X' = X - aYk, Y' = Z - /?y*, Z ' = Y be the new variables.

/ = £ ( " j 1 ) ^ ' ) 1 ' ( a (Z ' ) (Z ' )* f + 1 ~ ' - K 'Z ' . It follows from Lemma i<;<w+i

0.6 that dim pW/p2 = 1.

Suppose now that t is such that (n -+- 1)|/ + 1. Then / + 1 = s(n + 1) and

TZ* = (ns - l ) (n + 1) + 1. We get q ^ = yns''1(x,y), (qW)* = (ajn,y)* and

9(»(<+i)) = ^(»a(n+i)) = (ynsy W e h a v e a ] r e a dy proved that ap = ^ ( n ) with

a = y and b = x — ay . By Proposition 1.3

and

p(0/p(«+i) ^ (Xyn8,yn8+l)l(xyns - c*2/*+n5).

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Moreover the image of yns+l is a free generator of the free R/p-module p(*)/p(*+i)B

The elements xtny3 where i + j = t + 1 and j > 1 generate y(xn,y)t.

Suppose first that sn + 1 < j < t + 1. Then xinyi = a;(*+1-i)n • yi

min[(t + 1 - j)nfc + j - an - l]sn + l < j < t + l = a - l . If a = 0

then xinyi = y3'1 • ysn+x in p(*)/p(*+i) for j = * + 1 and a ' V = 0 for

sn+l< j <t+l. lfl<j<sn then ^ n " > • yns = zsn~i -ysn-i • yj

= a;(n+1)(an-J).yi. It follows that ar*V = a?(*+1-i)ny" = ^"-1 .a;-a:(n + 1)( a n-^-

y.i'= xi-1 • a? • z8n~i • 2/ns = xi-1 -zsn-i(xyns) = cv.^ - 1 . ^ 5 n - i • i/*"1 .2/na+1

= a J / ? a n " - ' • ^ - 1 ) * + ( a n - - » ) / + * - 1 - y n a + 1 inpW/p<*+1>. It follows that s ' V = 0

if a — 0. minfjfc + (sn — j)£ — 1] 1 < j < sn = snk — 1 since k < £.

dim p^'/p1 + p(<+1) = min[s — l,snk — 1] since the image of y is a uniformizing parameter of R/p. Thus dim (jft'/p1 + p(t+1') = 5 — 1 since k > 1 and zz > 1.

In what follows the ideals described in a) will be called the ideals of the type (k,£); we can assume that k < £ since y and z play the symmetric role. Note that 1), 2) and 3) of a) apply as well to the ideals described in b) and c) if one sets them to be of the type ( l , n ) , which we shall do in the sequel. Note in all the cases k + £ — n + 1.

PROPOSITION 2.2. Let f be a Dn-singularity. Then every p C R such

that R/p is a DVR is of order 2 or 4 in Cl{R).

a) lip is of order 2 in Ct(R) then

1) dim pW/p2 = k > 2, if k > 3 then n is even and n = 2k.

2) Lett > 1 be such t h a t 2 | t + l . Then dim (pW/P*+P ( '+ 1 )) = (s-l)k where s = (t + l ) /2 .

bj If /? is of order 4 in Ct(R) then n is odd and

1) dim pM/p2 = (n - l ) / 2 > 2.

2) Let t > 1 be such that 4|(t + 1). Then dim (p^/p1 + p(t+^) = s(t - 1) + (s - l)£ where £ = (n - l ) /2 and s = (t + l ) / 4 .

Proof. If n is even then C£(R) * Z2 X Z2 , if n is odd then « ( # ) = Z4

([7]). Therefore every p is of order 2 or 4 in C£(R). We proceed as in the proof

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of Proposition 2.1. Let <fii : fc[[X,Y, Z]] —• k[[T]] be a homomorphism such that <fa(X) = T, </>i(y) = aTk and <j>x{Z) = f3T£ where a,(3 G *[[r]]* U {0} and 0 i ( / ) = 0. Put p = ker & / ( / ) . We have T " " 1 + a2T2k+1 + /? 2 r 2 £ = 0. It follows that n = 2k -f 2 or n = 21+ 1. If n = 2fc -f 2 then an easy calculation shows that dim pM/p2 = jfe + 1 = n/2. Let now 02 : fc[[X,y, Z]] —• fc[pT]] be a homomorphism such that <f>2(X) = a T , ̂ 0 0 = 21 and <fe(Z) = /?T* where a,/? G Ar[[T]]* U {0} and fa(f) = 0. One proves easily that p = ker (fe/(/) is of order 2 in Cl(R) for every n > 4 and dim p(2>/p2 = 1. Let us consider again the homomorphism (j>\ and let us assume that n — 2t + I. The direct calculation shows that dim pW/p2 = £= (n —1)/2, dim (p(3) / p3 + pM) = £- 1 and dim (pW/p • P ( 3 ) + P(5)) = I- By Lemma 1.4. dim (pM/p4 + p (5 )) = ^ - 1 + ^ = 2 ^ - 1 . Suppose now that t + 1 = 45.

From Proposition 1.5 we infer that:

dim (pW/P* + p ( < + 1 )) = (* - 1) + (s - 1)(2* - 1) = s(£ - 1) 4- (5 - 1)£ .

If p is of order 2 in Cl(R) then p(3* •= p • p^ C p2 and dim (p(2)/p2+p(3)) = dim p(2)/p2 . Applying Proposition 1.5 a,gain one obtains 2) of a). Let us consider the homomorphism (fe : k[[X, Y, Z]] —> k[[T]] such that </>3(X) = aTk, fo(Y) = /?T' , <fe(Z) = T. We obtain the equation an-ir*(n-l) + ap2Tk+2l + ^ 2 = Q w h k h h a g n Q s o l u t j o n s a ? £ G fc[[r]]* (j

{0},fc,£ > 1. This finishes the proof of Proposition 2.2.

We shall omit the proofs of the next three propositions since the methods used are the same as above.

PROPOSITION 2.3. Let f be an E6 singularity. Then every p C R such that R/p is a DVR is of order 3 in Ct(R) and dim pW/p2 = 2. If t > 1 is such that 3|/ + 1 then dim (p^/p1 + p ( / + 1 ) ) = 2(5 - 1) where s = (t + l ) / 3 .

PROPOSITION 2.4. Let / be an E-^-singularity. Then every p C R such that R/p is a DVR is of order 2 in Ct(R) and dim p{2)/p2 = 3. If t > 1 is such that 2|(t + 1) then dim (pW/p* + p<*+1)) = 3(5 - 1) where s = (t + l ) / 2 .

PROPOSITION 2.5. Let / be an Eg-singularity. Then there is nop C R such that R/p is a DVR.

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DEFINITION. Let a e C c F c f 3 where C is a smooth curve and a is an An-singularity of the surface F. C is said to he of the type (k,£) in a (k < t) if the ideal pc a C Op^a is of the type (k,£) where pc a is the ideal of Of a which corresponds to C.

DEFINITION. Let a E C c F C F 3 where C is a smooth curve and a is a D or E-singularity of the surface F. C is said to he of order 2,3 or 4 in a if the order ofpca

ln Ct{Opa) is 2, 3, or 4 respectively.

T H E O R E M 2.6. Let C he a smooth curve on an irreducible surfa,ce F C F 3 . Suppose that C D Sing F = {{Pi}, {Pq}, {Pr}, {Ps}}, C'is of the type (ki,d-i) in Pi, C is of order 2 in Pq with dim p(2)/p2 = aq, C is of order 4 in Pr with dim p^/p2 = br and C is of order 3 in Ps (with dim p^/p2 = 2). Then C is a curve of contact of the t-th order on F if and only if

1. m.\(t -f l)dcg C, where m — dog F.

2. ki + £i\(t + l)fc,-, 2|(t + 1), 4|(t + 1) and 3|(t + 1) .

q n / y - ( * + i ) M i p , v (*+i) P , ^ (*+i) (?h , up

h + t l \ v 2

+ Es - 4 ( / + 1 ) A ' ) = u®t+1(-tm - m - n + 4/ + 4).

4. HlJi(m + n — 4) = 0 where n = (t + l)deg C/m.

Proof. From the Propositions 2.1, 2.2, 2.3 and 2.4. we infer that

D = Y1 k*Pi + Y^a9p9 + Y,brPr + Jl2Ps

and

Et = ̂ > - i)kiPi + £ (^ti _ ̂ flffPff + Y. 2 ^ < » . - 1 )

• t^ - ) * t + i Pr+y.2 r-^-i )p, s

It follows that

452

To finish the proof it suffices to apply Theorem 0.5 and note that for any a 6 CflSing F the order of pc a m C^i^Fa)ls equal to the order of ^c a

mce(dFta).

R E M A R K . Extracting the degrees of both sides in 3. we obtain

N^ Wi 1 1 1 1 4 n n J A , d2

> T-L1- + -a+-b+-u+-v = 2-2g + dm-4d + £-*i hi 4- 9i 9 9 A Q y

i ki + £ 2 2 4 3 m

where # = ^/(^)? d = deg C, a = J \ aq, b = ^2r 6r, u is the number of points of C fl Sing F where C is of order 4 and v is the number of points of CD Sing F where C is of order 3. Note that the equation does not involve t and gives the necessary condition in order that C be a set-theoretic complete intersection on F.

3.

In this section we shall characterize the curves of contact which lie on surfaces which admit only ordinary singularities. Recall ([5]) that a surface F C ^(chk = 0) admits ordinary singularities if Sing F is a curve (possibly reducible) and for a G Sing F Opa is one of the following

1. For almost all a e Sing F dF)0L £ fc[[X, Y,Z]]/(XY)-ordinary double point.

2. dF^a * k[[XyY,Z]]/(XYZ)-oTd\iiaxy triple point.

3. dFia S* k[[X,Y,Z]]/(X2 - Y2Z) - pinch point.

In the sequel we shall use the following:

PROPOSITION 3.1. (Hunecke, [6]). Let C be a smooth curve on a surface F C IP3 such that C (£ Sing F. If C is a set-theoretic complete intersection on F then for every a 6 C fl Sing F Ojra is unibranch.

R E M A R K . It follows in particular that in case F admits only ordinary singularities and C C F is a smooth curve such that C $_ Sing F then C fl Sing F consists only of pinch points if C is a set-theoretic complete intersections on F.

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PROPOSITION 3.2. Let R = k[[X, Y, Z]]/(X2 - Y2Z). Then for every p C R such that p ^ (#, y) and R/p is a DVR the following hold

1. p^1^ is prin cipal if an d only if 21 (t + 1).

2. dim pW/p2 = 2.

3. If2\(t + 1) then dim (p^lp* + p('+1>) = t - 1.

Proof. Let p = / / ( X 2 - Y2Z) C R be an ideal such that R/p is a DVR.

Then / is a kernel of a surjective homomorphism <f> : k[[X, Y, Z]] —• k[[T]\

such that cj>(X2 - Y2Z) = 0. Let <f>(X) = aTk, <f>(Y) = T and c/>(Z) = /3T£

where a = a(T), /? = /?(T) G Ar[[T]]* U {0} and k,i > 1. We obtain

a2T2k _ pTi+2 = o since <£(X2 - y 2 Z ) = 0, Let X1 = X - a(Y)Yk,

Y' = Z - /3(Y)Ye, Z' = Y be the new variables. Note that X' and Y1

generate / = ker $.X2 - Y2Z = (X1)2 + 2aX'{Z')k - Y'(Z')2.(a = a(Z')).

From the equation a2T2k - f3Tc+2 = 0 it follows that k > 2 unless a = 0.

So after setting -Y1 + 2aX'(Zl)k~2 to be a new variable we can bring the

equation X2 -Y2Z = 0 to the form X2 - YZ2 = 0 and p will be equal to

(x,y). It follows easily that p^ = (y) and p(<+1) is principal if and only if

2\(t + 1). From Lemma 0.6 we infer that dim pW/p2 = 2. Suppose now that

2\(t + 1). It follows from Proposition 1.5. that dim (p(*> /pf + pV+V)

= 2[(t + l ) /2 — 1] = / — 1. Consider now the homomorphism

<t>: k[[X,Y, Z]] —+ fc[[T]] such that 0(A") = T, <t>{Y) = aTk and a(Z) = f3Te.

The equation T2 - a2(3T2 + = 0 which we obtain has no solutions in

k[[T]]. The last case to consider is <j> : fc[[X,Y,Z]] —• k[[T]} such that

<t>{X) = aTk, <t>{Y) = /3Te and 0(Z) = T. The corresponding equation

a2T2k - /32T2i+1 = 0 has integer solutions fc and £ only if a = (3 = 0. Then

p = ker <t>l(X2 — Y2) = (x,2/). This finishes the proof of Proposition 3.2.

THEOREM 3.3. Let C he a smooth curve on an irreducible surface F C IP3 which admits only ordinary singularities. Suppose that C (j- ^mS F-Then C is a curve of contact of the t-th order on F if and only if

1. m\(t + 1) degC where m =deg F.

2. 2|(* + 1). 3. C H Sing F consists only of pinch points.

454

4. Oc( Yl ~(* + 1 ) P ) =u®t+1(-tm-m-n + 4:t-\-4) where PeCnsmgF

n = (t+l) deg C/m.

5. H1Ji(m + n-4) = Q.

Proof. From the Remark following Proposition 3.1. we infer that C PI Sing F consists only of pinch points if C is a curve of contact on F. If this is the case then it follows from Proposition 3.2. that

D= ] T 2P and Et = ^ (/ - 1)P if 2|(* + 1 ) . PeCnsmgF PeCnsmgF

So

Et -tB= ]T -(* + x ) p • PeCnsingF

To finish the proof it suffices to invoke Theorem 0.5.

COROLLARY 3.4. Let C he a, smooth rational curve on an irreducible surface F C F3 which admits only the ordinary singularities. Suppose C <f_ SingF. If deg C — 4 or deg C > 5 and C is general then C is not a curve of contact on F.

Proof. Extracting the degrees of both sides of 4. we get -s(t + l) = (t+l)(2g-2) + d(4(t+l)-m(t+l)-n) where s = # ( C n S i n g JF), d — deg C and g = 9{C) - the genus of C. n = (t + l)d/m and # = 0, so we obtain that s = 2 + (m — A)d -f d2/m. If d = 4 or d > 5 and C is general then the normal bundle of C is semi stable ([4]). Thus

deg O c ( - m + D) < -deg J / / 2 . It follows that 2s = deg Z) < 1 + rf(m - 2).

So 2 + (m - 4)J + d2/m < 1/2 -f f/(m - 2) /2. After simplifying we obtain the followiung quadratic inequality with respect to d

(*) 2d2 + (m2 - 6m)d -f 3m < 0

It follows that 2d2 - 9d + 3m < 0 since m2 - 6m > 9 and d > 0. The discriminant A = 81 — 24m is negative for m > 4. So m < 3. It suffices to consider the case when m = 3 since the remaining cases m = 1,2 are trivial. For m = 3 the inequality (*) becomes 2d2 — 9d + 9 < 0 and the only integer solutions are d — 2 and d = 3. This finishes the proof of Corollary 3.4.

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REMARK. In [6] there is an example of a twisted cubic which is a curve of simple contact on a cubic surface x2z = y2w which admits only ordinary singularities.

REFERENCES

[1] M. BORATYNSKI, On the curves of contact on surfaces in a projective space; Algebraic K-theory, Commutative Algebra and Algebraic Geometry, Contemporary Mathematics 126.

[2] M. BORATYNSKI, Locally complete intersection multiple structures on smooth algebraic curves, (to appear in the Proc. Amer. Math. Soc.)

[3] N. BOURBAKI, Algebre commutative, Chap. 7, Hemann 1965.

[4] F . GHIONE, G. SACCHIERO, Normal bundles of rational curves in F 3 , Manuscr. Math. 33 (1980), 111-128.

[5] P H . GRIFFITHS, J. HARRIS, Principles of algebraic geometry, Wiley

Interscience, New York, 1978. [6] D.B. JAFFE, On the set-theoretic complete intersections in F ,Math. Ann.

285 (1989), 165-176. [7] J. LIPMAN, Rational singularities with applications to algebraic surfaces and

unique factorization, Publ. Math. HIES 36 (1969), 195-279.

[8] D. GALLARATI, Ricerche sul contatto di superficie algebriche lungo curve, Acad. Royale de Belgique, Memoires Coll. n. 8, Tome XXXII, Fasc. 3 (1960).

Maximilian BORATYNSKI, Dipartimento di Matematica, Campus Universitario,

Trav. 200 di Via Re David, 4, 70125 Bari, Italy.

Lavoro pervenuto in redazione il 10.10.1990.