Ann. Univ. Ferr~ra - Sez. VII - Sc. Mat. Vol. X X X I I - 93-107 (1986)

O-Dimensional Subschemes of P~: New Application of Castelnuovo's Function. E D W A R D D. D A V I S (*) (**)

O. -

Introduction.

Given :~ finite p o i n t s e t Z of P2(k), k ~- it, let H ( Z , t) denote the n u m b e r of independent conditionu imposed on [Op.(t) f b y the r e q u i r e m e n t of passage t h r o u g h Z. A p p a r e n t l y CASTEL~UOVO first recognized the u t i h t y of the differenec function of H ( Z , - - ) for s t u d y i n g tile linear s y s t e m s of c r a t e s containing Z. Thcr(,fore, ((Castelnuovo's F u n c t i o n ~: C(Z, t) ~ H ( Z , t) --- H ( Z , t - 1), or as more fashionably described n o w a d a y s , hl(TJz(t- 1 ) ) - - - h ' ( 3 z ( t ) ) . The p a p e r s [DGM], [DM], [D] are p r i m a r i l y studies of t h a t f u u c t i o n ~ a fi~et which began to enlerge only in the last section of the last of t h e m . The p u r p o s e s of this p a p e r are three. The first is to reduce the f u n d a m e n t a l ideas of [D, w4], which deals with a r b i t r a r y P~, to a f o r m :~nd t~ lan~o-uage m o r e congenial to the g e o m e t r y of P2. I n f a c t we find it mor(, economical 1o develop those ideas a l m o s t ab initio for P~, r a t h e r t h a n to beeolnc e n m e s h e d in the i m p l a c a b l y algebraic v i e w p o i n t a n d notation of [D]. This allows dr~stic simplification of ;trguments, and in some ease~, a strengthening of t h e r e s u l t s - - w h i c h p a y s off in applications. T h a t w o r k is p e r f o r m e d b y w167 1-2, which "~re organized so t h a t t h e reader can s t u d y the results '~nd their ancillary not:~tion without reading t h e i r proofs. w167 3-4 are :~ppheations : w3 inverting the B(~zout th(.orcm ; w4 settling the existence p r o b l e m for Z with H ( Z , - - ) and (~C a y l e y - B a c h a r a e h N u m b e r ~) prescribed. (For quite different applications sce [DG]], [1)G2].) This p a p e r replaces a (mercifully unpublished) version f r o m which it differs substantially. Infinite t h a n k s to th:~t u n k n o w n referee who insisted

(*) Indirizzo dell'autore: Math. Dept. SUNYA, Albany, NY 12222 (U.S.A.). (*) During much of this work the author enjoyed the financial support of CNR (Italy) and the hospitality of the Departments of Mathematics of the Universities of Ferrara and Geneva and tile Politecnico of Torino.

94

EDWARD

D.

DAVIS

u p o n a radical r e s t r u c t u r i n g of the original. The present version is derived d i r e c t l y f r o m the a u t h o r ' s lectures at F e r r a r a (Spring-Summer, ]986). M a n y t h a n k s to the <~lWerraresi ~>, n o t a b l y M. F~ORE~'~INI and A. ]~ASCU, for t h e i r c r i t i c a l a t t e n t i o n . T h a n k s also to P. :TV[AROSCIAand to L. CmA~TI~I: the surprisingly strong Bezout-inversion (3.1b) is a response to a question raised b y Maroscia, and w4 exists to de'fl with a question of Chiantini.

1. - N o t a t i o n a l c o n v e n t i o n s and r e c a l l o f basic t e c h n i c a l i t i e s .

CONVENTIONS. F i x a projective plane P, over an algebraically closed field k, a n d a homogeneous coordinate algebra R for P. <~Scheme ~>m e a n s <~closed s u b s c h e m e of P ~, a n d S will always denote such a scheme. I n t e r section a n d c o n t a i n m e n t of schemes are to be understood sehemewise. 3s denotes the ideal of S in 0 ~ 0~; and I ( S ) denotes the ideal of S in R, the unique homogeneous ide~fl of R which b o t h defines the sheaf Js "rod has no i r r e l e v a n t component. <~Curve ~>means <~nonzero effective divisor on P ~>; so if A is such, then A e lO(d)l, for a unique positive integer d - - - - d e g A. :NOTATION. d, m, n, t, u, arc always integers, d, m, n positiw~; ~ t ---- Io(t)I, and s ---- {A] e ~ t : S c z ] } . (Effective divisors ~rc b o t h divisors a n d s c h e m e s : we s p e a k of the divisors containing a given scheme, a n d of the s u m of curves; and the f o r m F in R defines the divisor A if ~nd only if F generates the ideal I ( A ) of the scheme A). Observe: 9 1 6 3 r I ( S ' ) . c I ( S ) . (necessarily the ease if S c S'); 9 c 9 r 9 c c

~,(,~)(t
F IXED CURVES. Let: S r 0; A e ~ , defined b y F c R ~ . I n case e v e r y m e m b e r of 9 contains A, equivalently, I ( S ) t c R t _ ~ ( t < u ) , t h e n we say A is a fixed curve of 9 if A is a g r e a t e s t c o m m o n divisor cf 9 (viewed as a set of divisors), equivalently, F is a gre%test c o m m o n divisor of the set of polynomials I ( S ) . , then A is the fixed curve of 9 Obviously 9 has no fixed curve if and only if I ( S ) . has no nontrivi'd c o m m o n divisors. HILBERT FUNCTIONS. F o r a n y finite-type, Z - g r a d e d k-Mgebra A, its H i l b e r t F u n c t i o n H ( A , - - ) is the function f r o m Z to N defined b y : H ( A , t) ~- dim k At. B y definition: H ( S , --) ~- H ( R / I ( S ) , --). Therefore: H ( S , t) = h~ -- h~ I f S has finite k-length l(S), t h e n H ( S , t) = = l ( S ) - h'(~s(t)). I n t h a t case then: H ( S , t ) < l ( S ) ( t e Z); H ( S , t) ~- l(S) r <=> hl(3s(t)) -~ 0; H ( S , t ) -~ l(S)(t >>O). l~ow if A is as a b o v e a n d x e A ~ i,:, At_d-regular, t h e n H ( A / x A , t) = H ( A , t) -- H ( A , t -- el). Consequently,

0 - D I M E N S I O N A L SUBSCHEM:ES OF p 2

95

:ETC.

for a n y A ~ ~ a , a n d all t: H(z], t) ----- H ( P , t ) So if M ~ ~

and N e ~

H ( M C~ N , t) ~- h~

H ( P , t -- d) -~ h~

-- h~

-- d)) .

h a v e no c u r v e in c o m m o n , t h e n for all t: -- h~

-- m)) -- h~

-- n)) ~- h~

-- m -- n)) .

Theft last f o r m u l a i m m e d i a t e l y yields (of. w3 below): Bezout

Theorem:

l(M N N) :

ran.

Jacobi T h e o r e m : H ( M ~ N , t) :

m n <::> t>~m + n -- 2.

Finally observe t h a t for S c S ' : H ( S , u ) < H ( S ' , u ) ; if s ---- s

equality if and only

Ir F o r a n y m a p / f r o m Z to Z, let ] denote the function defined b y : ](t) : / ( t ) ] ( t - 1). N o t e t h a t for S r S': H ( S , u) : H ( S ' , u) <==>H ( S , t) -~ H ( S ' , t ) ( t < u ) . T a k e L e R ~ a n d let x ~ - ~ denote the e~nonio.fl m a p / ~ - + R / L R . F o r I - - - - I ( S ) , e v e r y sufficiently gener:fl 15 is R / I - r e g u l a r ; ~o for all such Z ,~nd all t, TI(S, t ) - ~ H ( R / i , t). And moroover, if F is :~ g'rc:~test c o m m o n divisor of I~, t h e n for e v e r y sufficie~ltly general L, _F is also "~ gre,~test c o m m o n divisor of _/~. So for S c S ' : H ( S , t ) < H ( S ' , t); if S=/= 0 and / t ( S , t) ---- H ( S ' , t), t h e n e v e r y fixed c u r v e of ~ , ( S ' ) is also a fixed curve of ~ ( S ) . ]~ESIDUAL SCI-IE:~E. Th~ sequel m a k e s use of t h e scheme (~residual ~) to S (~ A in S - - b u t only in case l(S) is finite a n d A is a curve, in which case we denote t h a t scheme b y S - - A (really the set-theoretic c o m p l e m e n t if S is reduced). So if F defines A, then the ideal I ( S ) : 3 defines 3~_~. l~ecall: 1 ( 8 ) : / v = {x E R : xFcI(S)}. (1.1) PROPOSITION. L , t A es be defined b y the f o r m /L l(S) is finite ~md F is a n y curve. T h e n :

Suppose

(a) t(s) ---- z(s - A) + 1(s n z]). (b)

I(S--

A) ~ I ( S ) : F.

(c) I(S--z])~_~cI(S)~(teZ); if S=/= 0, t h e n A is a fixed c u r v e of ~D,(S) if a n d only if I ( S - - z])t_~2' -~ I(S)~(t<~u). (d) S - - ( A

+F)=(S--A)--F.

t(~ n (A + P))
96

EDWARD

D. DAVIS

PlCOOF. IJet I = I(S), J = I: F, the f o r m G define F. We m a y assume S r 0. )lore: (b) n.~d definitions formally imp]y (c); and since I : F G ---- J:G, (b) implies (d). So we subsume (d) under (b). PI~OOF OF: (a) -t-- (b) :::>-(e). ~(a c~ (A + r ) ) = t(a) - z ( a -

(~ + r ) ) = z(a) - z((a - A) - F )

= l(S) -- l ( S - - A) + l ( ( S - - A) n _r') < / ( S c3 A) + l(S c3 1"). PROOF OF (a). F o r a n y t we have the k-linear isomorphisms:

It q-- JCt_~F/It ~ Rt_dF/It ~ (Rt_d~) ~-- Rt_aF/Jt_~F ~ Rt_dJ,_~. Therefore, for all t: n ( n / I , t) = n ( n / ( I , F), t) -~- H ( R / J , t

-

-

d) .

B u t for sufficiently large t: I t ~-- I(S)~; (I, F)t = I ( S n A)t; Jt_~ = -~ I ( S - - A)t_~; H(S, t) -~ /(S); H ( S (~ A, t) = l(S ~ A); H ( S - - A, t - - d) ~= l ( S - A). PEOOF OF (b). l~egarding J~/J us an R-module, we must p r o v e : m 6Ass (R/J), m the irrelevant maximal ideal of R. Now given a n y ~ e Ass (R/J), there is x e R such t h a t x 6 J , xp c J . Therefore x F 6 I, x F p c I ; whence p is contained in a m e m b e r of Ass (R/I). Done: because m is not a m e m b e r of Ass (R/I).

2. -

Old and new

observations

on Castelnuovo's

function.

DEFINITIONS-0BSERVATIONS. R e m e m b e r : schemes are b y convention closed subschemes of our projective plane P, and Sis always such a scheme. We define: C(S, t) = H(S, t) -- H(S, t - - 1); i.e., in the notation of restriction-to-a-general-line (w1), C(S, --) H(S, --). Observe: 2: {C(S, t)lt< u } = = H(S, u). Suppose S c S'. I t is i m p o r t a n t to know: for all t, s c s and C(S, t)
0-DIMENSIONAL SUBSCHEMES OF p2 ETC.

97

intended is clear from context. The definitions: ~(S) = min {t: ~t(S) r 0}.

fl(S) = min {t: s

has no fixed curve}.

v(S) = min {t: H(S, t) = / ( S ) } = min {t: h~(~s(t) ) = 0}. I t is i m p o r t a n t to know: ~ < f l < v - ~ - 1 . The first inequality is clear from definitions; the second follows from the elementary fact t h a t I(S) is generated by forms of degree < 3 zr 1 (e.g., [DGM, (3.7))]. (2.1)

FUNDAMENTAL OLD OBSERVATIONS ON CASTELNUOVO'S FUNCTION.

For any 0-dimensional S: (a) C ( S , t ) > 0 ; C ( S , t ) r

r

(b) For t ~ 0 : C ( S , t ) < t -~ 1; C ( S , t ) = t -~- 1 ~

t<~-- i .

(c) For t > ~ : C(S,t)<~C(S,t-- 1). (d) For ~ + l>~t>~fl: C(S, t) < C(S, t - - 1). (e) For a n y u: X{C(S, t): t < u } < l ( S ) ; equality ~:~ u > ~ . PROOF. Note t h a t (e) follows immediately from (a), given t h a t H(S, u) -- l(S)(u >7 0). The other assertion can be p r o v e d - - q u i t e elementar i l y - - b y restriction-to-a-general-line (ef. [DGM, (2.6)]). (Aside: this form of (2.1) was known to Dubreil (c. 1933); see [It, (5.1)] for the form probably knowu to C~stelnuovo (e. 1885).) Our new observations consist of two fundamental observations, (2.2ab). and three supplementary observations, (2.3), (2.4), (2.5). The first of the fundamental observations, (2.2a), is easily proved by restriction-to-a-generalline ([D, (4.5)], b e t t e r : exercise); the others are proved below after all the statements and remarks. The heart of the proofs is: (1.1) and (2.1) above, and (2.7) below. ~]'OTATIONAL CONVENTIONS. Henceforth we reserve X to denote an arbitrary 0-dimensional subschcme of P ; and we use unadorned ~, fl~ to denote, respectively, ~(X), fl(X), r(X).

DEFINITION-OBSERVATION. Remember d is always positive. Given a n y map ] from Z to N, let ]d denote the truncation o]] at d: ]~(t) : min {](t), d}. We define: ld(X)-~ X{Ca(X, t): t e Z } . Regarding (2.1) as a description of the behavior of the graph of C ( X , - - ) makes the following observations completely transparent. (1) max {C(X, t): f e Z } = C(X, ~ - - 1) = ~.

EDWARD D. DAVIS

98

(2) Suppose d<~, t < d, C~(X, t) =

C(X,

t);

and let q : m a x {t: C(X,t)>d}. C~(X, t) = d ( d - - ]

For

t ~ q or


(3) v ~- 1 = l~(X) < 12(X) < ... < l~(X) -~ l(X) . (2.2) FUNDAMENTAL NEW OBSERVATIONS ON CASTELNUOV0~S FUNCTION

[D, w4]. (a) F o r zJ E ~ , d
a fixed curve of s

and u ~ a : d
=>l(X n A) ---- la(X). (2.3) FIRST S U P P L E M E N T A I ~ Y NEW OBSEI~VATION [D. (4.2)]. Suppose C ( X , u ) = C ( X , u - - 1 ) r Then fl~u~>:r and curve of ~ ( X ) has degree C(X, u).

the

fixed

(2.4) SECOND SUPPLEMENTARY NEW OBSERVATION (el. [DGM, (4.4)]). Suppose a < f l t~nd C(X, fl)~-- C(X, f l - - 1 ) - - I . Then the fixed curve of 9 has d(-grce C(X, fl-- 1). I~EMARK. The abow~ results are all valid over an ~rbitrary base field k ; restriction-to-a-general-line presents no problems because all hypotheses and conclusions are stable u n d e r a purely t r a n s c e n d e n t a l base field extension. DEFINITION-OBSERVATION (Cayley-B~charach P r o p e r t y and l~umber). X is said to have the p r o p e r t y CB(u) provided t h a t the following holds: .4~s162 CB(X) is the largest u such t h a t X has the p r o p e r t y CB(u). CB(X) is well-defined because: X does not have CB(v); X has CB(--1), also CB(O) if l(X)ee 1. The Cayley-Bacharach T h e o r e m says: X - - - - M V ~ N , M E ~ m , I Y ~ C B ( X ) ~ - m - k n - - 3 . (See [DM] or [DGO] for this level of generality; the elassict~l form of the t h e o r e m was first p r o v e d (albeit implicitly) b y Jacobi [J].) (2.5)

THIRD SUPPLEMENTARY NEW OBSERVATION (el.

Suppose f l > u > ~ > C(X, u) and a fix(~d curve of s C(X, u). Then CB(X) 4 u -- 2.

[D)I, (4.3)]). has degree

REMARKS ON (2.5). (1) [GH] i n a u g u r a t e d a renewal of interest in the Cayley-Baeharach P r o p e r t y : in connection with realizing the (reduced) 0-dimensional subschemes of _P as zero-schemes of sections of rank-2 vector bundles ([GH], [B], [Sr], [C]); and there, for(,, in cenne~',ion with c h a r a c t ( r izing complete intersectio1~s in P ([GH], [DM], [Sr]). (3.1) below, in p a r t

0 - D I M E N S I O N A L SUBSCItEMES

OF p 2

99

ETC.

a corollary of (2.5)~ contains p e r h a p s the b e s t possible eharacterize~tion of that type. (2) Using (2.3) a n d in t e r m s of C ( X ~ - - ) . scheme Z, such t h a t value? The unswer is the proof of our new

(2.5) one calculates an u p p e r b o u n d for C B ( X ) p u r e l y [C] raises t h e question: Is t h e r e always a reduced C ( Z , - - ) = C ( X , - - ) , with CB(Z) of t h a t m a x i m a l affirmative (see w4), the proof i n t i m a t e l y related to o b s e r v a t i o m : - - t o which we now pass.

(2.6) PROPOSITmN (the (~trivial~) cases). Rec:~ll m a x {C(X, t): t 9 Z} = C(X, ~ - - 1 ) ---- ~. ~5oreover: (a) C(X, u) ---- ~ :=>for a n y A e 9

that

by

(2.1abe),

I(X)~ = I ( A ) t ( t < u ) .

(b) Suppose u~>a a n d A eZ)~ is a fixed curve of 9 d = o: ::> C(X, u) = ~.

Then: d < ~ ;

(e) Suppose C(X, u ~- 1) = ~. T h e n C B ( X ) > u . (2.7) P~OPOSlTION (Cf. [D, (4.1)]). Suppose u ~ > ~ > d a n d A e Z ) ~ is a fixed c u r v e of 9 T h e n because ~ e ( X ) ~ 0~ X n A =/= X ; whence b y (1.1a), X - - A =/= 0. Moreover: (a) C ( X , t ) < C ( X - - A , t - - d ) - ~ d ( d < t < u + ] ) , equality <=> A is a fixed curve of s fl(X- A)
OF (2.6ab). Given a n y A 9 s the explicit f o r m u l a for H(zJ, t) t)----t~-l(0~d--1). IfC(X,u)-~ b y t h a t f o r m u l a a n d (2.1bc): C ( X ~ t ) = C ( A , t ) ( t < u ) for a n y H e n c e (a). Now the h y p o t h e s i s of (b) implies I ( X h c I ( A ) t ( t < u ) , d = a ::>I(A) c I ( X ) . H e n c e (b).

PROOF OF (2.6e). Clear if l(X) = 1 ; for t h e n u ~ -- 1 ~ C B ( X ) . l(X) r 1, a n d consider : Y c X , l ( : Y ) ~ - l ( X ) - 1. We m u s t show: = I(X)~. ~Tow C ( Y , t) < C(X, t)(t e Z); whence b y (2.1e), t h e r e is a such t h a t C ( Y , n ) ~ = C(X, n). D o n e if u < n ; so a s s u m e not. (2.1bc), C(:Y, n -~ 1) =~ C(X, n + 1 ) - - a b s u r d .

Assume I(Y)~-~ unique n Then by

100

EDWARD D. DAVIS

Ptt00F OF (2.7abo). Definitions and (1.1c) formally imply (a); (2.1a) and the c,~ses t ~ u, u ~- 1 of (a) give (b). So d < C ( X , u). Then by (a) and the definition of C~(X,--), the formula of (c) holds for t < u . B y (2.1a), C ( X -- A, t -- d) -~ 0 for t > ~(X -- d) + d; and b y the definition of Ca(X, --), in ease d > C ( X , u -{- 1), C(X, t) -~ C~(X, t ) ( t > u). Done: b y (b). PROOF OF (2.2b). B y (2.6ab), we m a y assume C(X, u ) > ~, ~nd a d o p t the n o t a t i o n of (2.7). So d < C ( X , u) b y (2.7b); a n d in case equ~Aity holds, b y (2.1e) and the definition of ld(X), summing the formula of (2.7c) over Z gives: l(X) = l ( X - - A) -}- l~(X). So b y (1.1a): l ( X N A) : l~(X). P ~ o o F oF (2.3). B y (2.1abd), f l > u>:c; and b y (2.6ab), wc m a y as~ume C(X, u) < ~ and a d o p t the notation of (2.7) with A the fixed curve of ~ ( X ) . B y (2.7a) : C ( X - - A, u - - d - - 1 )

= C(X, u - - 1 ) - - d = C(X, u ) - - d --- C ( X - - A, u - - d ) .

So: C ( X - - zJ, u - - d ) : C ( X - - LJ, u - - d - - 1 ) .
:Now b y (2.7a), f l ( X - - A)-~< ~(X--A)
PROOF OF (2.4). L e t u = f l - - 1 . B y (2.6ab), we m a y assume 1hat C(X, u) < ~, and use the notation of (2.7) just as in the prc:cedi~)g par,~.gr,%pll. B y (2.7a), (2.1v), and hypothesis: C(X, u ) - 1 = C(X, u § 1) < C(X -- A, u - d § 1) § d < < C(X -- A, u -- d) + d = C(X, u) . So: C ( X - - A , u - - d ) = C(X--A,u--d+I). I t t h e n follows, e x a c t l y ~,s in t h e preceding p a r a g r a p h , t h a t d = C(X, u). P ~ o o F of (2.7d). L e t v---- C B ( X - - A ) , v - - - - v + d . :Now e < ~ ( X - - A ) , a n d b y hypothesis, ~(X -- A) < u -- d; so v < u . Note also l ( X ) > l ; for otherwise, ~ = f l = l . :Now if l ( X - - A ) - - - - 1 , t h e n : c - - - - - - 1 ; v = d - - 1 ; b y (1.1a), I ( X N A ) : I ( X ) - - I . So, if I ( X - - A ) - - - - I : C B ( X ) < v - ~ - I ; and b y (2.6v)~ C B ( X ) > ~ - - 2 > v . Therefore, for the rest of the proof we ~ssume l ( X - - A ) > 1, in which case then, v > 0 . P ~ o o F oF: C B ( X ) < v . Take F e ~ + ~ , l ( ( X Then dcg(A -}- T') ---- v ~ 1, and b y (1.lad):

z ( x n (A + ; ) ) = t ( x ) -

A) -- 1.

~(x - (3 + r ) ) = = l(X)-

T h a t proves CB(X) < v.

A) n F) = l ( X -

l ( X -- A) + l ( ( X -- A) c~ F)

l(X)-

1.

0-DIMENSIONAL 8UBSCHE.~IES OF p 2 ETC.

I01

PROOF OF: C B ( X ) > v . Take YcX, l(Y) --=-l ( X ) - - 1. To p r o v e : I(X)~ = I(Y)~. Now, as in the p r o o f of (2.6c), t h e r e is a unique n such t h a t C(Y, n ) ~ C(X, n). Done if n > u , because v < u; so a.s~ume not. Because C(Y, u) ---- C(X, u), z] is also a fixed c u r v e of Q~(Y). So, t a k i n g F to be a f o r m defining A, b y (1.1e): I(X)~ = I ( X - - A ) ~ F ; I(Y)~ : I ( Y - - A ) ~ F . Consequently, the p r o b l e m is reduced to p r o v i n g : I ( Y - A)~ = I ( X - A)~. A s t r a i g h t f o r w a r d application of (1.1a) shows: l ( Y - - A ) > l ( X - - z ] ) - - l . So I ( Y - - z])~ -: I ( X -- A)r QED. PR00F OF (2.5). B y (2.7bd), CB(X) ~ C B ( X - - z]) ~ d, for z] e Q ~ the fixed curve of s ~ o w in a n y case, C B ( X - - A ) ~ ~ ( X - - A ) ; a n d in the case a t h a n d , b y (2.7b), r ( X - - A ) < u - - d . So C B ( X ) < u - - 2 .

3. - I n v e r t i n g t h e B e z o u t - J a c o b i t h e o r e m s .

DEFINITION-OBSERVATION. Recall: X is a l w a y a 0-dimensional subscheme of o u r p r o j e c t i v e plane P ; ~ =: a(X), fl -- fl(X), ~ ~ v(X); d, m, n, t, u always d e n o t e integers, d, m, n nlways positive; 9 ~ 10~(t)]. I f X = M c~ 2V, M e s N e s we say X is a CI(a, b), a -~ min {m, n}, b = m a x {m, n}. F o r a n y such X, a : a ~ud fl ~ b, a n d m o r e o v e r : l ( X ) ~ ab (Bezout); v ~-- a Jr b - - 2 (Jacobi); CB(X) - - a ~ b -- 3 (C2,yl~y-Bacharach-Jacobi). The goal of this section is: (3.1) THEOREM. Given .'~y positive integers a al~d b, with a ~ b , the following assertion,,~ are equivalent. (a) X is a CI(a,b).

(b) l(X) < ab, v >~a ~ b -- 2, a n d for e v e r y reduced, irreducible c u r v e A, l(X N z J ) < ( d e g A)b. (c) l(X).~.ab, r ~ a ~ - b - - 2 , a n d for e v e r y d ~ a , t h e following * condition of g e n e r a l i t y ,~ holds: A e s =~ l(X n ~) < d(a ~ b -- d). (d) l(X)a, and C B ( X ) > a - J r b - - 3 . RE~_AI~KS ON (3.1). (1) (a) r (b) ~:~ (c), ~nd (a) ==~ (d) are valid over a n y base field. (d) =~ (a) is intrinsically a k ~--~ result; for one c a n n o t p r o v e it unless t h e s u p p o r t of X consists of k-rational points. (2) (a) <=> (d) solves the p r o b l e m posed (implicitly) in [GH] of characterizing c o m p l e t e intersections b y the C a y l e y - B a c h a r a c h P r o p e r t y , a n d includes the p a r t i a l solutions f o u n d in [GH], [B], [DN[], [Sr]. Our solution is the best possible (in the plane); for it answers affirmatively the question

102

E D W A R D D. DAVIS

raised in [DM, w4] of whether the solution developed there for reduced X is t r u e in general. (3) See [DG1] and [DG2] for the use of (a) <:~ (c) (and its ])roof) for applications involving the subschemes of colength 1 in complete intersections. (4) (( Segre's T h e o r e m ~). Assume X --~ M (~ iV, M e 9 ~ ~ , ~ . Then : (i) T > m + n - - 3 ; (ii) for every curve A c M , l ( X ~ A ) < ( d e g ~ i ) n . Segre P r o b l e m : Given M E E ) , , ( X ) and n such t h u t (i) and (ii) hold, does there exist/Y r ~ ( X ) such t h a t X ~ M (5 N? (See [Sc], [Ga], [Go], [D] for discussions of t h a t problem and generalizations of it.) We m a y think of (a) <:> (b) as t h e ultimate solution of t h a t problem (in the plane); for %n affirm~tive answer follows almost i m m e d i a t e l y from t h a t chara.cterization. PROOF (assuming (b) ::~ (a)). L~t a : min {m, n}, b --- max {m, n}, a~nd let A be a n y reduced, irreducible curve. Then since X r~ A c A n M, it follows t h a t l ( X ~ A ) < ( d e g A ) b : b y (ii) if A c M; otherwise b y Bezout. And since l(X) (el, a fortiori. So, since (a) ~ (d) is among the well known results recalled above, it remains only to p r o v e (el ==~(a) and (d) => (a). We shall do so b y reducing the proofs to the special case of (c) => (a) p r o v e d in [DGM, w 4] and the special case of (d) =r (a) p r o v e d in [DM, w4]. Moreover (see (3.2) below), modulo the results of w2 above and the a~ppropriate notation, those special cases become h a r d l y more than remarks. First that notation. DEFINITION-OBSEI~VATION. Given positive integers u
C(a, b; t) ~ 0 ~

O
C(a, b; t) = C(a, b; a ~- b - - 2 --t) -~ t + l ( - - l < t < a - - 1 ) ; C(a, b; t) ~ a(a-- l < . t < b - - 1). Motivational aside: I f X is a CI(a, b), .+hen b y the explicit formula for H ( X , t) given in w1 (i.e., Jacobi's formu]:~), C(X, --) --~ C(a, b; --).

0 - D I M E N S I O N A L SUBSCtIEMES

OF p 2

103

ETC.

Technical suggestion /or studying and applying this /unction: graph. I n this w a y one easily sees:

Use its

(1) ~, {C(a, b; t): t ~ Z} -~ ab; d<-~a ::> C~(a, b; --) -- C(d, a + b -- d ; - - ) . And then ~dso using (1) and (2.1), in case C ( X , - - ) ~ - - C(a, b ; - - ) : (2) a = a, r ~ a + b - - 2, b
C(a, b ; - - ) :

(4) f l = b = > X is a CI(a,b). (Proof: Because a = a , if f l = b , contained in a. CI(a, b), say k'. So X ~- :Y because l(X) -- l(Y).)

X is

(3.2) LE)[MA (cf. [DGM, (4.4)], [DM, (4.1)]). A s s u m e C ( X , - - ) = C(a, b; --). T h e n X is a CI(a, b) if either C B ( X ) > a + b ~ 3 or the condition of g e n e r a l i t y s t a t e d in (3.1e) holds for e v e r y d < a. PROOF. Suppose X is not :L CI(a, b). T h e n ~ ~- a < b a + b -- 2; C(X, --) =/= C(a, b; --). Then there exists b < u ~ a + b - - 2 such t h a t : C ( X , t ) ~ C ( a , b ; t ) ( t > u ) ; C(X, u - - 1) = C(X, u) = C(a, b; u). PROOF. ~ o t e t h a t {t: C(X, t) < C(a, b; t)} r 0: otherwise, either C(X, --) ---- C(a, b ; - - ) , or b y (2.;le), l ( X ) > a b . Given t h a t , we define: u ~ - m a x { t : C ( X , t - - 1 ) ~ C ( a , b ; t - - 1 ) } . 1Wow: C ( X , a + + b - - 2 ) > 1 b y h y p o t h e s i s ; C(a, b; a + b -- 2) -~ 1; C(a, b; t) =- 0 (t > a + +b--2). Therefore u < a + b - - 2 . I f u ~ b , then: C ( X , b - - 1 ) ~ C ( a , b ; b -- 1) = a; whence (directly f r o m t h e definitions) C(a, b; t) ~ C~(X, t ) < > C(a, b; u - - 1 ) => C(X, u - - 1 ) . A n d so b y (2.1bc), C ( X , u ) < C ( X , u - - 1 ) . P u t t i n g this all t o g e t h e r finishes the proof: C(X, u -- ]) < C(a, b; u -- ]) -- 1 ~- C(a,b; u ) < C ( X , u ) < C ( X , u - - 1). PROOF OF (3.1): (e) ::> (a). Assure,' (c). Done, b y (3.2), in case C(X, --) = C(a, b ; - - ) ; so a s s u m e not. TIWAI b y (3.3), (2.3) a n d (2.2b), there exists

104

:EDWARD D. DAVIS

/ l E n a , d -- C(a, b; u) < a, u as in (3.3), such t h a t l ( X t 3 A ) = l ~ ( X ) . Note (choice of u and f o r m a l consequence of definitions): C~(X, t ) ~ Cd(a, b; t) (t e Z). Then (again formally from the definitions): ld(X)>d(a @ b -- d). D o n e : violation of (v). PROOF OF (3.1): (d) ~ (a). Assume (d). Done, b y (3.2), in case C(X, --) = --- C(a, b;--); so assume not. Again take u as given b y (3.3), and note: C(X, u) < a<
4. - Pointsets with prescribed postulation.

(2.7) above presents us with an (( o b v i o u s , device for treating existence problems for schemes X with both C ( X , - - ) and properties (~compatible ~) with t h a t function prescribed. (Cf. [DGM, w4], where t h a t is d o n e - - a l b e i t r a t h e r clumsily, and for a v e r y special type of C(X, --).) This section employs the simplest case of t h a t device to settle the existence problem resulting f r o m prescribing b o t h C ( X , - - ) and the Cayley-Bacharach :Number of X. CONVEZ~TIO.~S. X continues to denote an a r b i t r a r y 0-dimensional subscheme of our projective plane P. However, in this section we focus oll reduced X. Therefore, to avoid confusion (and tiresome repitition), Z replaces X whenever a r(~luced 0-dimensional scheme is intended for the scheme in question. So Z is a n o n e m p t y finite subset of P, and l(Z) is the cardinality of Z. Morover, given a n y curve A, Z - A is the h o n e s t settheoretic complement. Finally, we cancel the device of writing ~, fl, v to denote the values of those numerical characters for the scheme X ; for we have another use in mind for u n a d o r n e d ~, fl, 3. DEFINITION-OBSERVATION. A function C f r o m Z to N is said to be a Castelnuovo Function p r o v i d e d t h a t there exist integers ct(C) ~nd v(C) such t h a t : l < ~ ( C ) < v ( C ) q- 1; C(t) ~ 0 .r O < t < ~ ( C ) ; C(t) = t q- l ( O < t < ~(C)); C(t) < C(t -- 1) (a(C) < t ) . (Macaulay called such a C an (~e l e m e n t a r y Osequence ~----see [GMR] for t h a t story.) F o r a n y such C we let l ( C ) = = X {C(t): t e g}. I f C = C ( X , - - ) , then C is a Castelnuovo F u n c t i o n and: a(C) = ~(X); v ( C ) = r ( X ) ; l ( C ) = l(X). Given a Castelnuovo F u n c t i o n C, we define:

t~(r = rain { t > ~ ( c ) :

c(t) < :~(r

;

b(V) = rain {t>~(C): C(t) = C(t + 1)}.

0-DIMENSIONAL SUBSCItEME8 OF p 2 ETC.

105

Observe: b ( C ) ~ v ( C ) ~ 1; if C : = C ( X , - - ) , then fl(X)~fl(C); however, in view of (2.1d), f l ( X ) > f l ( C ) if b ( C ) # ~ ( C ) ~ 1. F u r t h e r m o r e : (4.1) TnEo]tE)[.

For C a C~stelnuovo Function, fl = fl(C), b - ~ b(C):

(a) C ~- C(X, --) ::*-fl-- 2 < C B ( X ) < b - (b) f l - - 2 ~ c ~ b - -

2.

2 ::*-3Z with C ( Z , - - ) ~ - C and CB(Z) = c.

t~EMAI~K ON (4.1b). (4.lb) answers a question raised in [C]: Is there :~lways a Z with C(Z, --) = C and CB(Z) =- b -- 2? T h a t question is answered in b o t h [C] a n d [Si] for the ease b = v(C) ~- 1. Our m e t h o d s reduce t h e proof of (4.1b) to t h a t special case; and in t h a t ease wc sketch a p r o o f using (~liaison ~) (see (4.4) below). S t a n d i n g n o t a t i o n : a - - a(C); ~ ~ ~(C); fl = fl(C); b == b(C). (4.2)

TRIVIAL OBSElCVATION.

(a) f l = b - (b) C(X) :

v +l C r

Assume

0r

Then:

-l(C). l(X) : v ~.- 1 a n d a(X) -- 1 .

(c) C(Z, --) -~ C ~ Z is a set of v ~- 1 collinear points. (Of course (( line ~) m e a n s (( c u r v e of degree 1 ~)). P ~ o o F oF (4.1a). The lower b o u n d is given b y {2.6c), a n d the u p p e r bound is clear in case b - - v ~ - l . Assume b < v + l . T h e n b y {2.3), ~b(X) has a fixed curve of degree C(b); a n d the definition of b gives b > ~ > > C(b). B y (2.5) then, C B ( X ) < b - - 2 . P~OOl~ oF (4.1b): reduction to a (known) special ease). P r o c e e d b y induetion on a, b y (4.1a) and (4.2), a s s u m i n g a > 1. Define a function C' so: C ' ( t - - 1 ) ~ C ( t ) - - C l ( t ) , C1 the t r u n c a t i o n of C at 1 (see w Let m= m a x { t : C ( t ) > 1}. The following observations arc s t r a i g h t f o r w a r d consequences of definitions: (1) C' a n d C1 are Castelnuovo F u n c t i o n s ; a(C1)-----1, r ( C I ) - - v , l(C~)= v - I - l ; ~ ( C ' ) = a - - 1 , ~(C') = m - - l ; l(C)=l(C')+l(r (2) f l ( C ' ) = f l - - ] ; b(C')=b--2 other cases b(C') = b - - 1.

if b = v + l

and C(v)-~l;

in all

L e t fl(C')-- 2 < e < b ( C ' ) - - 2. I n d u c t i o n h y p o t h e s i s gives Z ' with C ( Z ' , - - ) = C' a n d CB(Z') = c. T a k i n g a c c o u n t of (4.2), t h e r e exists Z = Z ' L / Z ~ , where: C ( Z I , - - ) - - : C~; Z ~ c A e ~ D I , with Z ' r ~ A = 0 . Now A is a fixed line of ~ t ( Z ) ( t < ~ ) (Bezout): whence, s t r a i g h t f o r w a r d l y f r o m

106

EDWARD

D. D A V I S

(2.7a) and definitions, C(Z,t)~--C(t)(t<:v). Then because l ( Z ) = l ( Z ' ) + -~ l(Z1), l(Z) ~- l(C') + l(C1) ~- l(C). Ther(fore: •

{c(z,

t): t < 3} = • {c(t): t < 3} = l ( c ) ;

2: { c ( z , t): t < r} - _r {c(t): t < 3} < t ( c ) . B y (2.1e) then, v(Z) = 3; whence by (2.1a) a n d the definition of C, C(Z, --) =: ---- C. Supposc~---- v + 1 . T h e n f l = b - ~ ; a u d b y ( 4 . 1 a ) , t h e r e i s n o t h i n g more to prove in this case. 5Tow assume ~ < ~ , a n d a p p l y (2.7) with u - v and A the line selected above, noting t h a t b y (1), z ( Z ' ) = - - m - - I < T 1. (2.7d) gives: C B ( Z ) = C B ( Z ' ) + I = e + 1. B y (2) then, t h a t completes the induction step in all cases but t h a t one exceptional case of (2). T h a t case is handled by (4.4) below; but iirst an a d d e n d u m to the a r g u m e n t just made. T h a t a r g u m e n t at least proves (cf. [GMR]): There always exists Z with C(Z, --) = C. Therefore, since 9 cuts out Z f o r u -- v -[- 1, for u ~ v + 1, two general members of 9 meet transversally, and do not meet on ~my prescribed curve, e x c e p t ~t points of Z. H e n c e : (4.3) LEMMA. Given a Castelnuovo F u n c t i o n C' with v ( C ' ) < a--1~ there exist zJE9 MEs ~E 9 such t h a t : l ( M ~ l V ) - ~ o t f l ; MV~IV is reduced; C(M n _IV n 2, --) = C'. Accepting (4.3) makes possible a short proof b y liaison in the case remaining open in the proof of (4.1b): (4.4) LEPTA. b ~

v+l

:=>3Z with C ( Z , - - ) - ~ C and C B ( Z ) ~ v - - 1 .

P~ooF. This proof uses the duality between the Castelnuovo F u n c t i o n s of (( linked ~>schemes [DGO, (3)], and it freely employs the notations CI(o~, fl) and C(~, fl; --) of w3. L e t T = ~ + fl -- 2. 5low in a n y case, C(t) = C(~, fl; t) (t < / ? ) ; and the hypothesis b = v + 1 implies C(t) <. C(ot, fl; t)(t>fl). Now for a n y CI(o,~ fl) Z: C(Z, --) ~- C(~,/5; --); v(Z) ~- T -~ CB(Z) + 1. So assume C=# C(~, fl;--), and take C' to be the function ((duals) to C with respect to C ( ~ , f l ; - - ) (in the sense of [DGO, (3)]): C ' ( t ) = C(o~,fl;t)--- C ( T - - t). (Since C(a, fl; t) = C(~, fl; T - - t), this duality is reflexive.) Observe: C' is a C~stelnuovo F u n c t i o n ; v(C') < a - - 1; ~(C') = T - - 3. :Now a p p l y (4.3) to C', a n d let: X = M n i v ; Z'-~Xc~A; Z=X--A. [DGO, (3)] asserts: C ( Z , - - ) is dual to C ( Z ' - - ) , i.e., C ( Z , - - ) = C. :Now suppose Fes and l(Z (~ / ' ) > / ( Z ) -- 1. Then l ( X n (A + F ) ) > l ( X ) -- 1; whence because deg(A + / ' ) ~ T--l= CB(X), and Z ~ A is e m p t y , Z c F. Done.

Pervenuto in Redazione il 5 novembre 1986.

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ETC.

107

SUMMARY This note first reformulates and sharpens certain newly discovered properties of the first differencc of the postulation of a 0-dimensional subscheme of p2, a function which it seems historically appropriate to call (, Castelnuovo's Function ,~. Those results are then applied in two ways: inverting Bezout's Theorem; settling the existence problem for finite pointsets in p2 with postulation and (~Cayley-Bacharach Number ~>prescribed.

REFERENCES [B] [C] [DGM] [DM] [D(;O] [D] [DG1]

[DG2] [GMR] [aa] [Go]

[GH] [HI [J]

[Sr] [Se] [Si]

J. BRUN, Les ]ibrds de rang deux sur P2 et lenrs sectio~ts, Bull. Soc. Math. 107 (1979), pp. 457-473. L. CHIANTINI, Vector bundles o] rat~k 2 on p2 (work, in progress, 1986). E. DAVIS - A. GERAMITA - P. MAROSCIA, Per]ect homogeneons ideals: Dubreil's theorems revisited, Bull. Sci. Math., (2), 108 (1984), pp. 143-185. E. DAVIS - P. MAROSCIA, Complete intersections in p2: Cayley-Bacharach characterizations, LNM, 1092 (1984), pp. 253-269. E . D A V I S - A . G E R A M I T A - F . O R E C C I I I A , Gorenstein algebras and the CayleyBacharaeh theorem, PAMS, 93 (1985), pp. 593-597. E. DAVIS, Complete intersections o/ codimension 2 in P~: the Bdzout-JacobiSegre theorem revisited, Rend. Sem. Mat. Torino, 93 (1985), pp. 333-353. E. DAVIS - A . G E R A M I T A , Bese's very ampleness theorem and punctured complete intersections, The Curves Seminar at Queen's, Volume IV: (~Queen's Papers in Pure and Appl. Math. (in press). E. DAVIS - A. G E R A M I T A , Biratio'nal morpbisms to P~: an ideal-theoretic perspective (to appear ~ 1987). A. G~RAMITA - P. MAROSCIA - L. ROBERt'S, The Hilbert ]unction o] a reduced K-algebra, J. Lond. Math. Soc., 28 (1983), pp. 443-452. A. GERA)IITA (cd.), The Curves Senlinar at Queen's, Volume I I I : (( Queen's Papers in Pure and Appl. Math., 67 (1984). S. GRECO, Alcune ossercazio~ti sui sottoscbemi di codimensione 1 d i u n a variet~ proiettiva, Semin~ri di Geometria 1985-86, Universits di Bologna (in press). P. GmFFITIIS - J. HARRIS, Residues a~d zero-cycles on algebraic varieties, Ann. Math., 108 (1978), pp. 461-505. R. HARTS~IOaNE, Stable cector bundles o] rank 2 on p3, Math. Ann., 238 (1978), pp. 229-280. C. JACOBI, Theoremata ~oca algebrica circa systema duarum aequationum inter duas variabiles propositarum, J. Reine Angew. Math., 14 (1835), pp. 281-288 (also Wcrke III, pp. 285-294). T. SAUER, A note on the Cayley-Bacharach property, Bull. Load. Math. Soc., 17 (1985), pp. 239-242. B. SEGR~, Sui teoremi di Bdzout, Jacobi e Reiss, J. Mat. Pura Appl., (4), 26 (1947), pp. 239-242. A. SOD~II, P h . D . dissertations, Qucea's University (1986).