Graded Betti numbers of some embedded rationaln-folds

Math. Ann. 301, 363-380 (1995)

Annam �9 Springer-Verla 8 1995

Graded Betti numbers of some embedded rational n-folds

Anthony V. Geramita t,a, Alessandro Gimigliano 2,b, Yves Pitteloud 3,c 1 Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, Canada (e-mail: [email protected]) 2 Universit~ di Bologna Bologna Italy (e-mail: [email protected]) 3 Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, Canada, (e-mail: [email protected])

Received: 13 October 1993 / In revised form: 21 April 1994

Mathematics Subject Classification (1991): 14M20, 14J40, 14M05, 14A05, 13D02

Introduction

In recent years questions about the equations defining projective varieties (e.g. their number and degree) and, more generally, questions about the higher order syzygies among these equations, have received considerable attention. It has been recognized that many classical results can be phrased in terms of features of the minimal free resolution of the ideal defining the embedded projective variety.

The inspiring work of Mumford [Mu] and the significant amplifications by St. Donat [St.D] and Fujita [F] were among the early important works in this area. More recently, the strong theorems and interesting conjectures of Green [Gr] have captured the imagination of a large number of investigators. The further innovative advances by Green and Lazarsfeld [Gr-La] and Ein and Lazarsfeld [E-L] have served to strengthen the belief that this point of view is significant and offers interesting questions and beautiful theorems (see also [E]). This work is our reaction to these influences and concentrates on the important class of rational n-folds. We find, in this restricted context, that some of the general results can be markedly strengthened.

In this paper we study blow-ups of IP n embedded as rational n-folds in projective N-space by complete very ample linear systems. We will be interested in the defining ideals of these embedded n-folds. Our approach is to use the 0-dimensional subscheme of n )n which defines the embedding to extract information about the embedded variety.

More precisely, let X = {Pt . . . . . Ps} be a set o fs distinct points in IP ~ = IP~ and let Xs be the n-fold obtained from IP" by blowing up the points of

a Supported, in part, by the Natural Sciences and Engineering Research Council of Canada b Supported, in part, by the CNR of Italy r Supported by the Swiss National Research Fund

364 A.V. Geramita et al.

X. Our approach to the study of the embeddings of Xs (as a linearly normal variety) is through "fat points" in IP n. I.e. if Pi corresponds to the ideal goi c_ R = k[xo,xl,... ,x~] (k an algebraically closed field) then fat points refer to subschemes Z of IP ~ defined by ideals of the type I = ~ ' J A. . . N m~ ~o s , i.e. intersections of powers of ideals of points. These are very particular complete ideals in R whose graded pieces correspond to complete linear systems on X~ which, when those linear systems are very ample, embed Xs in some projective N-space.

Our aim is to uncover information about the projective resolution of the defining ideal of such an embedded variety. Many classical invariants of a variety (e.g. postulational formulas) are directly computable from such a resolution. In fact, the resolution provides some finer invariants than those which were classically studied.

In Section 1 we establish our fundamental observations about the saturation of powers of ideals of zero-dimensional subschemes of F ~. These purely algebraic results underline all our subsequent considerations.

In Sect. 2 we apply our results from Sect. 1. Our main result gives very general conditions which imply that these embedded rational n-folds are projectively Cohen-Macaulay. This knowledge gives us the length of the projective resolution described above. This includes earlier results in [Gi-l]. Varieties of this type appear in [Do-Or] (who study their intrinsic properties, following in the line of Coble's work) and in [Ok], [A1] and [Io] as well as in [Li] (where they are studied as varieties with "small" invariants). Particular cases of such varieties, (cubicsurface in IP 3, Bordiga surface in IP 4, White surfaces in IP ~) have all received a great deal of attention in the classical literature. (For more recent contributions see also [D-H] and [Ge-Gi].)

We then turn to a study of the individual graded Betti numbers and for this we first investigate the Castelnuovo regularity of the ideal sheaf associated to the embedded variety. We make several observations about these numbers, e.g. we will see that, in almost all cases, our n-folds are contained in a quadric hypersurface (they are always contained in a cubic hypersurface of the enveloping projective space).

We then restrict our attention to the case of surfaces, i.e. we consider fat points in IP 2. In this case, the ideals of fat points are better undestood, thanks to the Hilbert-Burch theorem. Using this, we give the complete resolution for an (infinite) family of surfaces of the type we have been discussing, and also discuss how geometric properties of the scheme of fat points affect the geometry of the embedded rational surface. As an application of the theory, we can make some observations about line bundles on general trigonal curves, as well as some remarks on embedding of hyperelliptic curves.

1 Saturations of powers of ideals

Although w e shall eventually be interested in the ideals corresponding to schemes of fat points in IP ~, our initial observations hold for a much wider

Graded Betti numbers of some embedded rational n-folds 365

class of ideals. Consequently, but just for this section, instead of assuming that I is an ideal o f fat points we shall just assume that 1 is a homogeneous ideal in R = k[xo . . . . . x , ] for which the Krull dimension of R/I is 1.

We let reg(I ) denote the Castelnuovo regularity of I. This is the smallest integer r for which Tori(I,k)i+j = 0 for all i and for all j > r. We let n(I) denote the sa turat ion degree o f I, i.e. the smallest integer r such that I and its saturation/sat agree in all degrees > r. We have r e g ( I ) = max{n(I),reg(Isat)} (compare [E-Go]).

Our main result in this section deals with the regularity of powers of such ideals.

Theorem 1.1 Let I C R be a homogeneous ideal for which the Krull dimension of R/I = 1. Then,

reg(K) _< r �9 reg(1) , f o r every r .

Proof Throughout the proof, we let L denote a linear form in S which is not a zero divisor in R/(K) s"t. (This implies that multiplication by L is an injection (R/I)j ~ (R/I)j+l for all j >> 0.) It follows from [B-S, Theorem 1.10], that reg(K) is the smallest integer j such that the multiplication by L induces a surjection (R/Ir)j_! --~ (R/Ir)j and an isomorphism (R/Ir)j ~ (R/Ir)j+l .

Claim A. For j > r �9 reg(I) , multiplication by L induces a surjection

( R / I ~ ) j _ ~ ---, ( R / 1 ~)j .

We prove Claim A by induction on r, the case r = l being clear. Let F be an element of Rj; as j > reg(1), we can write F = LF' + G, with F ' in R j - I and G in Ij. We can write G = ~FiGi where the Fi are minimal generators of I . In particular, we have that degGi > ( n - 1)reg(1), and hence, by the induction hypothesis, each Gi c a n be written in the form LG[ + Hi, where Hi is in F - l . Putting all this together yields an expression of the form F = L F " + G ~, with G' in F , which is precisely what is needed to establish Claim A.

Claim B. For j -> r �9 reg(I) , multiplication by L induces an injection

(R/Ir)j ~ (R/K)j+I .

The proof o f Claim B is by induction on n, the case r = 1 being again clear. Let F be in Ry such that LF is in I r. We want to show that F itself must be in I r. By the induction hypothesis, F is in F -1, and so we can write F = Y~FiHi, where the Fi are minimal generators for F - t . On the other hand, as LF is in 1 r, we can write LF = ~FiGi where the Gi are in I. Thus, we obtain the following relation among the Fi,

~ ( L H ~ - G ) F ~ = O .


In other words, if we consider the free R-module F with basis elements ei corresponding to the minimal generators Fi of I ~-l, we have that the element Y ] ( L H i - Gi)ei is a syzygy; we can thus write this syzygy in the form ~k,i Pkakiei where the elements ~ i akiei are minimal generators for the module of syzygies.

By the induction hypothesis and Claim A, reg(F - l ) ~ ( r - 1)reg(1). This implies, after a short computation, that we have degPk > reg(I). We can thus write Pk = LQk + G~ with G~ in L

Fixing i and regrouping the terms involving L, we get

L(Hi - Y]~akiQk) = Gi + ~~akiG~ E I . k k

As the degree of Hi - ~a~iQk is > reg(1), we deduce that Hi - ~kakiQk is also in /, and we have that

F = ~HiFi = ~ ( H i - Y]a~iQk)Fi i k

is in I r which proves Claim B. This concludes the proof of Theorem 1.1. []

Example 1.2 The bound given in Theorem 1.1 is sometimes sharp. One need only consider the case of an ideal with a linear resolution. On the other hand, if one adds the extra hypothesis that all the generators of I are in degree < z = reg(I) - 1, and then proceeds as in Theorem 1.1, one gets much better bounds ( reg( I ' ) ) < kz + 1 under this extra hypothesis).

Karen Chandler has announced some results along these lines for arbitrary ideals, and indeed it was her announcement of these results which made us re- examine our proof of Theorem 1.1. We then noticed that, with the additional assumption on the generators, the proof worked (with obvious modifications) to give stronger results.

Remark 1.3 As pointed out by the referee, Theorem 1.1 can also be proved using non-exact complexes, as in [E-L] .

In the next section, we will need a special case of Theorem 1.1. Recall that, given the saturated ideal I of a zero-dimensional subscheme of IP n, the least integer for which the first difference of the Hilbert function AH(R/I,t) equals zero, is usually denoted by a(1). Clearly, a(1) = reg(I).

I f I = (F I , . . . , Fn) is a complete intersection ideal then I" is saturated for all r, thanks to a theorem of Macaulay. There are even some situations in which it is possible to prove that complete intersection ideals are the only examples with this property (see e.g. [Hu-UI]). But, in general, I" is not necessary saturated, even if I is.

Corollary 1.4 Let I be a saturated ideal. I f I (r) denotes the saturation of I" (the r-th symbolic power) then:

n(I r) < ra(I) and a(I (r)) < ra(I) .


This follows at once from the equality r e g ( 1 ) = max{n(1),reg(lSat)} and the fact that a(I (~)) = reg(I~)) .

In the next section, will use this corollary in the case where 1 is an ideal o f the type ga] "~ n . . . f 3 m~ ga s , where the fai are ideals o f points. We will also make tacit use of the following remark.

R e m a r k 1.5 I f I = fa l I N . . . f 3 m, ga s , where the ~o i are ideals of points, we have, thanks to a theorem of Macaulay

rml rms I (~) fa t n . . . n g a s .

2 P e r f e c t n - f o l d s

For the remainder o f the paper I will be a homogeneous ideal of the form:

I = fa~'~ n . . . n fa~s c_ R = k[x0 . . . . . x . ] .

We write IPn= ProjR. Then each fai is a homogeneous prime ideal which corresponds to a point Pi E ]pn, with Pi # P j for i=l=j. The scheme of fat points Z C_ ]P" associated to I will be denoted Z = (Pl . . . . . Ps;mt . . . . . ms). s( Z is a 0-dimensional subscheme of F ~ of degree Y~i=t m; + n 1 with

n / support X = {PI . . . . . Ps}. Let rc :Xs --' F ~ be the morphism associated to the blowing up o f IW at X. It is well-known (see [Ha, Chapt. V, Sect. 3] for the case o f a surface) that Pic(Xs) '~ Z s+l ~ = = < Eo,EI . . . . . Es >, where El . . . . . Es denote the classes of the exceptional divisors and E0 denotes the class of the pull-back of a hyperplane which misses the points of X.

It follows from a simple extension of the ideas in [Ha, Chap. V, 3.4] (see also [Gi]) that, i f J is the ideal sheaf in Or , which corresponds to I C R, we have:

_ s E h~ dEo Zi=lmi i) = h~ J ( d ) ) = dimk Ia for all d E 7l ;

$ hl(Xs,dEo - ~, i=lmiEi) = hl(Pn, J ( d ) ) = d e g Z - H ( Z , d ) for a l ld E 7/; (1")

hi(Xs, dEo - ~ = t m s E i ) = he(IP",J(d) )

= 0 for all i, 2 < i < n - 1, and all d E 71.

The following very-ampleness result will be crucial in the sequel.

Theorem 2.1 (cf. [C]) Let X , Z and I be as above and let X~ denote F n blown s

up at the points o f X . I f a = a( I ) and Da = aEo - ~'~i=lmiEi E CI(Xs) then (i) D t is very ample f o r t > a + 1,

(ii) Do is very ample co, f o r any line ~ o f l W , d e g ( . ~ n Z ) < a .

A stronger version of this theorem (but only for the case of F 2) was first proved in [Da-Ge]. We wish to thank M.Coppens for informing us of his result. Note that statement (i) is fairly easy to prove and that the heart of the matter is (ii).


Standin9 assumption

From now on we shall assume that we are dealing with subschemes Z of IP u which consist of fat points and such that no line of IW meets Z in more than a points (properly counted). (The standing assumption allows us to use Theorem 2.1.)

We denote by V:e (or sometimes simply V) the embedding of X~ in IP u defined by the very ample divisor Dr, where t > a. Since t > a, it is easy to

see that N + l = [ ( t + n ) - and that deg V' z = t" - deg Z" V =

H~ Dt ). Since Dt is very ample, we have an exact sequence

0 ~ Iv --, S = Symk( V ) ~ ~ H ~ rDt) ~ ~ H l ( l P N , J v ( r ) ) ~ O. r > O r ~ O

The image of ~b is the homogeneous coordinate ring of V C ffJN and H~ rDt) describes the complete linear system which contains the sections cut on V by the hypersurfaces in IP u of degree r. Notice that ~b is surjective if and only if ~Dr>oHl(lpU, j v ( n ) ) = 0 which is, by definition, the statement that the embedded variety V is projectively normal, i.e. that S/Iv is an integrally closed integral domain.

Proposition 2.2 Under the standin 9 assumption, and with the notation above."

S/I "~ ~ H~ ) v = ,rDt . r~O

i.e. the variety V is projectively normal.

Proof. This is obvious for r = 0 and, since V = H~ rDt), it is also clear f o r r = 1.

m 5 r Since (S/Iv), = [((go~ ~ N. . . M gas )t) It, and H~ = (~o~ r~ A . . . M r t a s ~s )rt and t > a(1), the equality is a direct consequence of Corollary

1.4. []

Remark 2.3 Notice that to get the conclusion of Proposition 2.2 one only needs the very ampleness of D~ and a good bound on the saturation index of ideals of the type (1)" = (gd~ 'j N . . . n gdm')" for any integer r. So, for example, if X = {PI . . . . . Ps} is a complete intersection set of points in IP n (say the complete intersection of forms Fi . . . . , F, of degrees, respectively, d l < d2 <

s E ... < dn) and Dt = tEo - ~-]i=t i, then Dt is very ample as soon as t > d n + 1 (a number which is, in general, significantly smaller that a(I)). Since n(1) r = 1 for every r, we get that Vt is projectively normal for such t. This remark should be compared to some of the results of [G-G-H], particularly Theorem IV of that paper.


Theorem 2.4 With the standin9 assumption and assuming further that char (k ) = O, we get that the rin9

S/Iv TM ~]~ H~ rDt ) r>O

is a Cohen-Macaulay domain.

Proof I t ' s enough to show that Hl(IPN, J v ( m ) ) = 0 for all m E 7/.. and HJ(IpN,(_gv(m)) = 0 f o r j = 1 . . . . . n -- 1 and for all m E Z.

We have already seen that H l ( I p U , J v ( m ) ) = 0 for all m > 0. Since, for m > O, Or(m) is an ample invertible sheaf on V we have Hoop s, 6 v ( - m ) ) = 0. Thus, Hl(IPU,~Cv(m)) = 0 for all m E Z.

It remains to consider the cohomology groups:

HJ(IpN,(_gv(m)) = HJ(Xs, mDt) for j = 1 . . . . . n - 1 .

I f m > 0 and I = ga~ '1 A . . . N ~-a~ 's recall that I (") (the saturation o f I m) =

go~ ' '1 N . . . N gas" m~. Let J and J ( " ) denote the corresponding ideal sheaves in

(9~,,. Then, by ( i )

HJ(Xs, mOt ) = HJ(IP n, J(m)(mt))

and, for j = 2 . . . . . n - 1 these cohomology groups are all zero. Since t > or(I) we have, by Corollary 1.4 that ~(I (m)) < rot. Thus, again by ( t ) , Hl(lW,~c(m)(mt)) = 0 for all m > 0. It remains to consider these groups in case m < 0.

Since c h a r ( k ) = 0 we may use the Kodaira vanishing theorem (see e.g. [Ha, p. 248]) to obtain those vanishings. That proves the theorem. []

Remark 2.5 In case n = 2 (i.e. for surfaces), Theorem 2.4 is valid with no restriction on the characteristic of the field. This can be checked as in [G-G-H, Proposition III.1]. We were unable to drop this characteristic assumption in general. Some recent correspondence with E. Ballico seems to indicate there are many cases in which the characteristic assumption can be omitted.

Since the homogeneous coordinate ring o f the n-fold V is a Cohen- Macaulay domain, the length o f a minimal free resolution o f Iv C_ k[xo .. . . ,XN] = S is exactly N - n. I.e. we have:

O---~ FN-, --* ... ~ F! ---~ Iv ~ O

with Fi = ~ l S ( - ~ i j ) , where we assume ~ij < otij+~ for fixed i and j , 1 < j <= il - 1. We now seek information about the graded Betti numbers ~ j .

Coro l la ry 2.6 The varieties V = Vt,z C IP N, o f the theorem, are all Hilbertian, and the Castelnuovo regularity o f J v is < n + 1.

Proof Recall that being Hilbertian simply means that the Hilbert function of the coordinate ring o f V coincides with its Hilbert polynomial. In fact, we will


show that Corollary 2.6 applies to any projectively Cohen-Macaulay rational n-fold V.

For such a V its Hilbert function is H(V,t) = h~ t v ( t ) ) and its Hilbert polynomial is P(V,t) = h~ (gv(t)) + (-l)nh~(V, t v ( t ) ) = H(V,t) + ( -1)~h ~ (V, K v ( - t ) ) where Kv denotes the dualizing sheaf. Since V is rational, h~ K v ( - t ) ) = 0 for all t > 0 and this establishes the first claim.

For the statement about the Castetnuovo regularity, one need only observe that the dual resolution of a Cohen-Macaulay variety (properly shifted) is the resolution of its canonical module @H~ Kv(t)), and that, as we just pointed out, this latter module begins in degree >_ 1. []

Our estimate of the Castelnuovo regularity of J v gives us an upper bound for the graded Betti numbers of V. Now we seek a lower bound.

Proposition 2.7 Under the standin 9 assumption and with the notation o f this

section we have h~ except when n = 2,s = ( d + 11 2 and Z \ /

is reduced with d = a(Z). In this case h~ Jv(3))+O.

Proof Recalling our standard notation, consider Dt for t = a ( I ) + m , m > 0. Since I is generated by forms of degree < a(I), we can always choose, among the generators of It, elements MIF1,MIF2,MvFI,M2F2 where MI,M2 are monomials of degree m and FbF2 are minimal generators for 1,~(1). Since (MIF2)(M2FI)- (MIFI)(M2F2) vanishes identically on IP n we automatically have Vt,z contained in quadrics. So, we are left only to consider t = a(1).

For this case, we first consider the possibility Z C ~", deg Z = ( d - l + n ) n

a(Iz) = d, and Dd = dEo - ~=lmiEi , so V = V d , z .

Suppose that Z is not reduced, i.e. that ml > 2. In this case, let L C_ IP n be a generic (n - 2)-dimensional space containing Pl. Then the strict transforms, on X=, of the hyperplanes containing L will form a pencil in IE0 - El I, say Hx, with 2 E IP 1. For the generic 2, we have that

H~ (~H).(Da ) ) = H~ n-I , 6F,-t (dEto - mtEtl ))

where the E~ are the classes of the restriction of Ei to HA. I f we restrict ~t,z to H~ then its image, denoted H(2), spans a IP~ C_ •u, where M =

n - 1 - 1 - 1 , since H t ( V , ( d - 1 ) E o - ( m t - 1 ) E 1

- ~ = 2 m i E i ) = 0 (use the usual restriction exact sequence). Hence, if we consider the variety ~ = U;. IP~ then (see [E-H, Theorem 2])

~ l i sa ra t i~176176176 176 ( d + n - 1 ) - ( m ' ; 2 - 2 ) 1

F N, which contains V. Since

= . ( d + n ) (mi+nl) ,. n


( d + n ) ( d + n - 1 ) 1 ( d + n - 1 ) n n - 1 - 1

we have that ~ e l P N since ml > 2.

It is well kn~ that r = deg ~t = N - dim ~l + l = ( ml ; --n - 2 and

that the ideal of 9~ is generated by ( 2 ) quadrics which must then also vanish

/

\ /

on V. Now suppose that Z is a reduced scheme and we are still restricting the

degrees we shall consider. When n = 2 these surfaces were studied in [Gi-1] and are called White surfaces. The ideal of such a White surface V = Vd2 is generated by cubics, which are the 3 • 3 minors of a 3 • d matrix of linear forms. So, h~ J v ( 2 ) ) = 0 but h~ Jv(3))r

Still with our restriction on degZ we suppose that n > 2. Note that it is possible to find sets of points in IP" satisfying the hypothesis and, in addition, have determinantal ideals, i.e. points for which the defining ideal is generated by the d • d minors of a d • (d + n - l ) matrix of linear forms. But in this case, the map ~d,z has its image in the Grassmanian G(d, d + n - l) C IP u. It is well known that the ideal of such a Grassmanian is generated by quadrics (the Plu~ker equations) and so we are done in this case.

But in view of ( t ) and Corollary 1.4, the number of quadrics in the ideal of V does not depend on how the points arise! This finishes the special case

~ degZ= ( d - l + n )

So, now suppose that d = a ( I ) and degZ < ( d + n - l ' ~ . We work by \ n J

descending induction on degZ. Consider the set of points Z p, obtained by adding a generic point to Z. By induction, the ideal of Va.z, C_ IP N-I will contain quadrics. We can regard Vae, as the projection of Va~ from a point, and consider the hyperplane section H of Val given by Va,z f3 1I, where 11 is the IP N- I containing Va,z,. Then H C Val,, hence the ideal of H (and therefore the ideal of Var does contain quadrics.

The only time this induction might not work is when n = 2 and degZ < ( d + l )

2 . But, in this case, [Gi-Lo] have shown that the ideal of V always

contains quadrics. []

Remark2"8 Let Z be a set ~ ( d+l )2 reducedp~ ~ in generic

position. As we have seen above, the embedded variety Val does NOT lie on any quadric (and this is in fact the only case where this occurs). We give another justification as to why these Va,z do not lie on any quadric.

Note that I, the ideal of Z, is an unmixed licci ideal (i.e. an ideal in the linkage class of a complete intersection), and is generically a complete intersection since Z is a set of reduced points. It follows from results of Huneke [Hu] that the ideal I is syzygetic, in other words the canonical map


Sym2(I ) ~ 12 is an isomorphism. On the other hand, as Id-i = 0, the canonical map Sym2(Id) ~ Sym2(I ) is easily seen to be injective, and this in turn implies that the canonical map Sym2(la)---, I2a is an isomorphism. This is clearly equivalent to the fact that the variety Vd,Z does not lie on any quadric.

Remark 2.9 We know that hO(jr(2))=lN+21_hO((gv(2))/ 'x where \ ] 2

h~ =h~ By Corollary 1.4, we can calculate this number explicitly. Unfortunately, the numbers involved are made up of alternating sums o f binomial coefficients, some of whose entries are themselves binomial coefficients, and we could see no easy way to describe when this number was 4= 0.

Green in [Gr] has given a great deal o f detailed information about the internal structure o f the resolution o f an ideal like Iv above. I f we use the notation o f Green ' s paper and let

0 ~ F N - ~ ~ "'" --* Ft "--*Iv - - ' 0

be the minimal resolution of Iz then, in the terminology of Green,

Fi = ~)Ki ,q (Dt ) | for i = 1 . . . . , N - n . q~O

The vanishing of the vector spaces l(p,q for various p and q gives the non- existence o f certain graded Betti numbers o f Iv and conversely. E.G. using 2.6 above we get that Kp,q=O for 1 < p < N - n and q > n + l . More generally, with the notation o f Corollary 2.6, we have Kp~ = 0 for 1 < p < N - n and q > a (V) . On the other hand, Proposition 2.7 gives that we always have Kl,i 4:0, with the only exception the case of White surfaces, for which Kt,i = 0 and KI ,2~0 .

In the case that Z is non-reduced, we have that V C ~ , where ~ is a ra-

ti~176176176176 ( d + n - 1 ) - ( m ' n + n - 2 ) i n F " T h e n - 1 1

re s~176176176 i t is l inearand~ ( mi+n-2)n-I

- 1; this implies that Kp, l(V)~eO for 1 < p < ~. We can make some other observations about the vanishing of the vector

spaces Kp,q and hence about the resolution of Iv.

Proposition 2.10 Let char k = O. Then, with our standin 9 assumption, we have Kp,n(Dt) = 0 when."

(a) 1 =< p =< N - n and t =< n, or (,,) (m) (b) t > n + l andp < N - n - + ~is=t i

M ~ 1 7 6 n


Proof. Let Kv denote the canonical divisor on V. Then from [Gr, 2.c.6], we have

K p,q "~ K N - n - p,n+ l -q ( Kg, Dt )

whenever

H~(V, (q - i )Dt) = H i ( V , ( q - i - 1)Dr) = 0 for all i = 1 . . . . . n - 1

(see [Gr] for any undefined notation). This condition is verified in our case,

since

Hi(V , mDt ) = Hi+l(IP",J;(z ' )(mt)) = O, Vm C Z and i = 1 . . . . . n - 1 .

On the other hand, by [Gr, 3.a.1] we have that

K N - n - p , n + l - q ( K v , D t ) = 0 when h~ K v + (n + 1 - q)Dt) < N - n - p . (*)

Since K v = - ( n + 1)Eo + ]~,i(n - i )Ei, we get:

$

Kv + (n + 1 - q)Dt = [(n + 1 ) ( t - 1 ) - qt]Eo - ~ [ ( n + 1 - q)mi - n + 1]Ei . i

So

h~ + (n + 1 - q )Dt ) = 0 when (n + l ) ( t - 1) - qt < 0

i.e. when q > n + l - ( ( n + 1 ) I t ) . (**)

Hence Kp,q = 0 i f q > n + 1, as we already noticed, while for q = n we have that (**) is true for t < n, so a) is proved.

In order to prove (b), set q = n and consider the divisor

$

K v + (n + 1 - q)Dt = Kv + D, = (t - n - 1 )Eo - ~ ( m i - n + 1)El �9 i=1

Since h l ( r v ( K v + Dt) ) = h n - I ~ v ( - D t ) ) = 0 , (by the Kodaira vanishing theorem) we have:

E(t') (:)J h = h~ + O r ) = max n - ~ ' ,0 i = I

and so, from (*),Kp,n = 0 when p < N - n - h. Note that h = dim KN-n,n and so when h 4:0 we get that the Castelnuovo

regularity o f J v is exactly n + 1. []

R e m a r k 2.11 Since the Castelnuovo regularity o f the ideai sheaf o f the variety Vt,z is always < n + 1, the "width" o f the resolution of the ideal sheaf is, at most, n. Moreover, at least for the case o f surfaces, the only way the resolution


can be "narrower" is for White surfaces, and in this case the resolution is concentrated in its top line.

In higher dimensions there are also instances where the resolutions are narrower than one might expect, but this happens rarely. We can be more precise when Z is reduced.

In this case, we are looking for the smallest positive integer q with Kp,q(Dt) = 0 for all p, i.e. for the smallest q with gN-n,q(Ot) - - - 0, but with KN-n,q-l:t = O. From Proposition 2.10, this is equivalent to finding the largest q' with H~ + qtDt) = 0, where q = n + 1 - q'.

As Z is reduced we have that

h ~ 1) whenq ' < n - 1 .

In other words, h~ + q'Dt) is zero as long as q't < n (for q' < n - 1). Thus, for a reduced set of points Z in IP n, we obtain:

KN . . . . . [n/t](Dt)=t=O while Kp,q(Dt) = 0 for all p when q > n + 1 - [n/t]

where, given r E ~ we denote by [r] the greatest integer smaller than r.

Thus, for example, if Z is a generic set of points in IP 3, then KN-3,3(D~) = 0 when a(Iz) < 3, i.e. when Z consists of at most 10 points. Since KN-3,2(D~) is always non-zero, for such reduced sets Z the free resolution of V~g has width two rather than width three.

For a set of reduced generic points Z in IP 4 we have that KN_4,4(D~)4=O unless tr ~ 4 i.e. unless [Z I ~ 35. Moreover, KN-4,3(D~):~O unless a < 2. But, D2 is never very ample, so KN-a,3(Da):~O. Thus, when IZl =< 35 and Z is a reduced set of generic points, the free resolution of V~z has width 3 rather than width 4 (but never has width 2).

Remark 2.12 Several themes emerge in the proof of Proposition 2.7 that bear further comment. One is the special role played by those n-folds corresponding

s ( ) to embeddings by D~(I) = e(I)Eo - ~i=lmiEi when degZ = a(I) - 1 + n n

Another is the realization that it is possible to see some of the quadrics containing our n-folds as quadrics defining classically studied varieties (e.g. rational normal scrolls and Grassmanians). A third is that information about the dis- crete invariants of our n-folds can be discovered by choosing the 0-scheme Z as a determinantal scheme of a particular type. This points to some interesting connections with recent work by Dolgachev and Kapranov (c.f. [D-K-l] and [D-K-2]) and we hope to return to a more ample discussion of this in another paper.

As for this last theme, recall that by a theorem of Hilbert and Burch (see e.g. [C-G-O]) every zero dimensional subscheme of IP 2 is determinantal. Thus, a discussion of the surfaces that arise from our considerations must occupy a somewhat special place. We now turn to a more detailed study of surfaces.


3 Bordiga-White surfaces

We continue with the notation we introduced in Sec. 1.

Definition 3.1 Let Z = (PI . . . . . Ps;ml . . . . . ms) C IP 2 be a scheme o f fa t points

(equivalently, a -- a(Iz) = d). Assume further that for any line L in Ip2,deg (LV1Z) < a (so that Theorem 2.1 applies to d E o - ~is+lmigi = Dd).

Then V = Vd,Z C IP d, is called a Bordiga-White surface (BW surface, for short).

For simplicity in the ensuing discussions it will be assumed that whenever we have a Bordiga-White surface V = Vd,Z C_ If 'd, the zero dimensional subscheme Z o f IP: that we are referring to is as given in the definition. The reason for this apparent pedantry is to be found in Remark 3.2(b) below.

Remark 3.2 (a) Bordiga-White surfaces exist! One need only take Z to be a

set o f ( d + l / Ip2" general 2 points in Classical examples are the cubic surface \ /

in IP 3 and the Bordiga surface in ~4. It is more delicate to decide if there exists schemes Z as above which are not reduced. Hirschowitz [Hi] has made significant contributions to the question of existence of such schemes. A survey of the conjectures and known results in this direction can be found in [Gi-2].

(b) It is not always obvious if a particular surface in lP N is a BW surface. Consider the Bordiga surface in ~ mentioned in (a). It comes from embedding IP 2, blown up at 10 general points, into IP 4, using the divisor D4 = 4 E 0 - E l - . . . - - E l 0 . I f we make a change of coordinates in the Picard group (see [De]), we can rewrite this divisor as D~ = 5 E ~ - 2E' I - 2E~ - 2E~ - E4 - . . . - El0. We are thinking of our abstract blow-up of ]p2 at the original 10 points as the same IP 2 blown up at some other 10 points. In this way our embedded variety can also justifiably be written as Vs~z, where Z ' = ( P I , P 2 ; P3,' P4 . . . . . . . ,P i0 ;2 ,2 ,2 ,1 , . 1) and degZ ' = 16. A very unlikely looking B W surface! Notice that for this Z ' ,5 = a(Iz,) - 1 = Z(Iz') and yet the divisor D~ is very ample and Vsg, is projectively Cohen-Macaulay.

There are some simple observations one can make about the resolution of the ideal o f a BW surface V C_ IP d. By our general observations in Sect. 2 we already know that:

(a) Kr,q(Da ) = 0 for p > d - 2. (since Iv is perfect); (b) Kv,q(Dd ) = 0 for q > 3 (Corollary 2.6). When Z is reduced we have complete information on the resolution: Kp: =

0 for all p and Kp,2 +- 0 for 1 < p < d - 2. The exact values of Kp,2 are then easily calculated (see e.g. [Gi-1]). Our interest then, naturally focuses on those BW surfaces for which Z is not reduced.

The following result can be obtained by arguments similar to those used in the proof o f Proposition 2.10.

3 7 6 A . V . G e r a m i t a e t a l .

Proposit ion 3.3 The Cohen-Macaulay type of the Bordioa White surface V = Vd2 (as in Definition 3.1) is."

l + mi - 2 d .

$ Notice that i f the number 1 + ( E i = l m i ) - 2d is small compared to d - 2 then, by Green 's vanishing theorem, [Gr, Theorem 3.a.1], (see also [E-K]) we have that Kp,2 will be 0 for small values of p. To be more precise, recall the following definition o f Green [Gr].

Definition 3.4 V has property Np i f Ki,q = 0 for all i <= p and for all q >= 2.

Using arguments as in Proposition 2.10 we obtain:

Proposit ion 3.5 Let V = Vd2 be a B W surface in IP d. Then V has property

l%if ( - K v ) . D d > p + 3 .

From now on we consider BW surfaces based on a scheme of fat points Z

having multiplicity "(d +12 \) and type Z ---(PI . . . . ,Pn+l ;d - k, 1 . . . . . 1 ), where \ /

k and n are described by:

Dd =dEo - (d - k )El - E2 - ... - En+l, and

2 - 2 = n .

Let Vk _C IP d be the surface we get in this case and let H be its generic

hyperplanesect i~176 ( d ) - ( d - k )

2 - 2 . Notice also that H is at least k-gonal since

the lines o f IP 2 which pass through Pl cut, outside of P~, a 91 k o n H . When k --- 1 there is not much to say. In that case Dd is not very ample.

Nevertheless /I1 is smooth and is a rational normal scroll o f degree d - 1 in IP a. The first interesting case occurs when k = 2, i.e. when Da = dEo - (d - 2)Et - E2 - . . . - E2a. In this case V2 has degree 2d - 3, sectional genus d - 2 and hyperelliptic hyperplane sections.

In this case, it is possible to give a very explicit description of the quadrics in the defining ideal Iv2 of V2, in terms of the entries in the Hilbert-Burch matrix of the fat points. In fact, Iv2 is generated by the 2 x 2 minors of a matrix

Y,,, Y,.2 "'" Ytd-2 QQ:) ( . , Y2,1 Y2,2 �9 �9 �9 Y2,d-2

where the Yij are linear forms, QI, Q2 are quadratic forms and both sets o f

forms are derived from the Hilbert-Burch matrix for the ideal I = ~0 d-2 n go2 n

�9 . . ["] ~ '02d.


This implies, in particular, that the entire resolution of the ideal Iv2 can be given as an appropriate Eagon-Northcott complex (see [E-N]).

Proposition 3.6 The resolution o f Iv2 is:

where

and

0 ---~Fd-2 --~ ...---~Fk --+ .. . ~ F 2 ---+Ft ---~Iv2 ---~0

F~ = ~ R ( - k - 2),F~' = ~ R ( - k - 1) .

From the description of the generators of Iv2 and from the proof o f Propo- sition 2.7, it follows that the quadrics in Iv2 generate a 3-dimensional rational normal scroll ~ . We now consider how the geometry of the rational normal scroll is influenced by the underlying set Z C IP 2.

For example, consider a scheme of fat points Z having multiplicity

2 and type Z = (P1 . . . . , P 2 a ; d - 2, 1 . . . . . 1) where P2 . . . . . P2d are on

a conic ~f, and P l r ~ . To see that there is a BW surface associated to such a scheme amounts to showing that ( I z ) e - i = (0), and that is easy.

Hence a (Z) = d and so Da is very ample on X2a and we get a BW surface V2 C_ II ~ . The image of the strict transform ~ of ~ is a line, L in n ~a. The surface V2 is thus a conic bundle on the rational normal curve ~'d-2 (which is the image of E l ) whose fibers are the images of the lines in l e o - E~I (with 2 d - 1 degenerate fibers). The scroll ~ , generated by the quadrics in Iv2, has planes as fibers and is spanned by the conics on /,'2.

Since (E0 - E l ) �9 (2E0 - E2 - . . . - E2d) = 2, each such conic meets L in two points. It follows that L is contained in every plane of the fibering of ~ , i.e. ~ is a cone with vertex L over the rational normal curve ~a-2 .

Using this construction we can prove the following result (for the reverse implication, one can use the Hilbert-Burch theorem).

Proposition 3.7 The surface 1"2 is contained in a rational normal cone whose vertex is a line i f and only i f P2 . . . . . P2a are contained in a conic.

Proposition 3.8 In the case described in Proposition 3.7 the surface V2 is a set theoretic complete intersection (s.t.c.i.).

Our proof relies on the fact that II2 is a divisor on the 3-fold ~ which, in turn, is a cone over a rational normal curve. Hence P i c ~ - Z and is generated by a fiber F ( -~ Ip2). The hyperplane sections of ~ are H = ( d - 2)F and hence V2 - (2d - 3 )F and (d - 2)V2 - (2d - 3)H. So (d - 2)V2 is cut on by a hypersurface of degree 2 d - 3. Since ~ is itself a s.t.c.i. (see [Ve] or [DC - G]) we are done. []

We can use Proposition 3.8 to prove the following result:


Proposition 3.9 Every hyperelliptic curve ~ of 9enus 9 > 2 can be embedded in IP g+l as a (projectively normal) s.t.c.i, curve ~ of degree 2g + 1.

Proof Consider a plane model c~, o f cg, where cg, has degree g + 2 and has a g-uple point as its only singularity. Cut cg, with a generic conic F not passing through its singular point P l , and let :~' . F be {P2,...,P20+5}.

Consider the blow-up X of IP 2 at PI,P2 .. . . . P20+4, and on it the linear system

[Do+zl = I(g + 2)E0 - gEt - E2 --. . . -- E2g+41 �9

Note that this system contains the strict transform c~ of cr We are thus in the situation described by Proposition 3.8 (with d = 9 + 2). Since Dg+2 is very ample on X, its image V C_ IP0 +2 is a s.t.c.i. BW surface of degree 29 + 1 and its hyperplane section ~ is the curve we want. []

Notice that this is the smallest degree possible for a hyperelliptic curve to be embedded as a projectively normal curve in IP g+1 (see [ACGH, p. 221, C-3]).

Remark 3.10 One could consider, on V2 smooth curves given by generic divisors in the classes [mDdl to find other s.t.c.i, curves in IP d. E.g., if :g E [2Dd[ = (2d ;2d - 4,22d-1), then ~ is a 4-gonal curve of genus g = 4d - 8 and degree 4 d - 6 = g + 2 .

We can also use the surfaces V2 to find projectively normal curves in IP d. It is known (see e.g. [ACGH]) that on a general curve C of genus g, the

general line bundle c~, o f degree [2 + (3/2)g] is normally generated (i.e. it embeds C as a projectively normal curve). We will see that the same kind of bound on the degree also works for a generic trigonal curve.

It should be noted that for a generic curve C the previous bound on the degree o f normally generated line bundles can be improved. Using a construction on White surfaces, see [Gi-3], it can be shown that a generic ~ ' on C of degree

=>9+risn~ ( r - l ) 2 < 9 < ( 2 ) "

Theorem 2.11 Let C be a 9eneric trigonal (smooth) curve of genus g > 3. Then the 9eneric line bundle cd on C with deg~,r => 2 + (3/2)9, if g is even, or degC~ > [3 + (3/2)9] i f9 is odd, is normally generated.

To establish this result, it is enough to exhibit (for every genus g > 3) a trigonal curve C o f genus g and a non-special line bundle .LP on C o f the prescribed degree. Such a curve can be found as a divisor on the Bordiga- White surface V2 (cf. [Gi-3] for a similar argument).

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[AI] J. Alexander, Surfaces rationalles non-speciales dans ~4. (Preprint) [ACGH] E. Arbarello, M. Cornalba, P. Griffiths, J. Harris, The Geometry of Algebraic

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Graded Betti numbers of some embedded rationaln-folds

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