The lazarsfeld-rao property on an arithmetically gorenstein variety

manuscrlpta math. 78, 347 - 368 (1993) manuscripta mathematica

Springer -~rlag 199~

The Lazarsfe ld-Rao property on an ar i thmet ical ly Gorenstein variety

Giorgio Bolondi and Juan Carlos Migliore

We show how to use all the machinery of liaison techniques for the study of two- codimensional subschemes of a smooth arithmetically Gorenstein subscheme of pn.

Introduction. Liaison techniques have been intensively used in the classification of curves in

p3 and, more generally, of two-codimensional subschemes of pn. But already since the

fundamental paper of Rao [22] liaison is considered in a more general setting, in particular

when the variety where we work is a smooth arithmetically Gorenstein subscheme X of p , .

The main purpose of this paper is to furnish in this relevant situation (e.g., hypersurfaces of

pn, the Grassman variety Gr(1,3), etc.) all the machinery of liaison in codimension two as

developed in the projective situation. So, many of the results here are just an adaptation of

results known in the projective case, and hence our discussion will sometimes simply describe

the connections with the known cases and sketch the proofs.

We define the notion of basic double link (using ideas from [11]), the notion of shift

and of minimal shift of an even liaison class, and we prove a structure theorem for even liaison

classes, which generalizes the Lazarsfeld-Rao property known in pn.

We devote w to the licci (linkage class of a complete intersection) liaison class,

characterizing the possible locally free resolution of licci (in X) two-codimensionat subschemes

of X and, when dimX _< 4, of the smooth ones. This section is inspired by work of Sauer [24]

and Chang [7], and indirectly by work of Huneke and Ulrich. We hope that this will help in

the study of licci subscheme of pn in codimension >2.

The LR-property gives a description of all possible locally free resolutions of elements

of a fixed liaison class, and also geometric consequences, such as deformation to reducible

subschemes with nice irreducible components.

In the last section we sketch some applications of these results to the Grassmann variety

Gr(1,3), in the spirit of the paper by Arrondo and Sols [2].

Much of this paper was done while the second author was a CNR visiting professor in

Italy, and he would like to thank the Dept. of Maths of the University of Trento for its

hospitality. Both authors are grateful to E. Ballico for many stimulating and useful discussions

related to this material.

348 BOLONDI-MIGLIORE

I. Generalities about liaison in codimension 2 Let k bc an algebraically closed field, let S bc the graded ring k[x 0 ..... x n] and let

pn = p~. If F i s a sheaf on X, we write F(t) for 9~Ox(t) and we denote by n i ( ~ ) the

direct sum (~Hi(F(t)). For a subscheme Z of pn, we denote by Jz/Pn its ideal sheaf mz

and by I Z its homogeneous ideal in S. For a subscheme Z of a smooth variety X in pn,

we denote by ~ its ideal sheaf in X.

We suppose from now on that X c pn is a smooth (connected) arithmetically

Gorenstein subscheme of pn of dimension r > 2; examples will bc X = Gr(1;3), the

Grassmanian of lines in p3, or if X = a smooth hypersurface of pn. As a consequence

of the Gorenstein property, we have hi(X, Ox(t)) = 0 Vt, for i = 1 ..... r-1. Also, X is

subcanonical ([13]): co x_= Ox(c) for some integer c. Following [22] Remark 1.9, our

linkage will be done with divisors on X coming from the subgroup G c Pie(X)

generated by Opn(1)lX. So, OX(1) = Opn(1)l x is a very anaple line bundle on X (we

don't distinguish between vector bundles and the corresponding locally free sheaves),

and all divisors on X forming part of our complete intersections come from powers of

OX(1). Hence it is clear what the "degree" of such a divisor is in our contest. This

approach is particularly satisfactory when Pic(X)=Z.

Two vector bundles Fand Gon x arc said to bc stably equivalent if 1" $

9r(~ i=~Ox(ai) .=. G(c) (~ i=~Ox(bi) for some c,r,s,ai,b i.

Notice that if Z is a subscheme of X then the total ideal of Z in X (as an ideal in

S(X) = S/I x) is isomorphic to IZ/I x. Indeed, this follows immediately from the exact

sequence

0--~ .gX/p~ ~ .TZ/p~ ~ Jz/x ~ 0. We consider the set C x of locally Cohen-Macaulay (I.C.M.), equidimensional

and two-codimensional subschemes of X. Two such subschcmes YI and Y2 are

algebraically linked in X via the complete intersection Z of two hypcrsurfaccs of X if

JYllX/Jzi x = Homox(OY2,0Z) and Jy2/x/Jzlx -- Homox(OYl,OZ). Equivalently, we have I7_jX:IY1/X = IY2/X and Iz/x:IY2/X = Iyl/X

This nodon generates an equivalence relation in C X, which we will call briefly

liaison. There is also a natural notion of geometrical linkage, generating the same

equivalence relation, for which we refer to [21] and [22].

Notice that i fY 1 and Y2 arc algebraically linked in X via a complete intersection

Z, it does not follow that they are linked by a complete intersection in pn unless X is

itself a complete intersection. However, Schenzel ([25]) has shown that many of the

results which hold for linkage with complete intersections condnue to hold for linkage

with Gorcnstein ideals. (For example see Proposition 1.1.) Since the ideal of a

BOLONDI-MIGLIORE 349

complete intersection on X is the restriction to X of a complete intersection (FI,F2) in

pn, and since as a subscheme of pn this has ideal I X + (FI,F 2) which is again a

Gorenstein ideal in S (see for instance Lemma 3.1), one can check that YI and Y2 are

linked in pn using Schenzel's more general linkage.

If Z ~ C x, then 5Z/X has a resolution, via locally free Ox-modules, of length

two. In fact, let S(X) be the homogeneous coordinate ring of X, S(Z) the homogeneous

coordinate ring of Z and let 0---) E.--->L ~ S(X) ---) S(Z) ~ 0 be a presentation

where L is a free graded S(X)-module, whose generators are sent by (x onto the

generators of the homogeneous ideal of Z in S(X), and E = Ker (x. By sheafifying one

obtains a resolution 0 ---> E ---) L ---) O X ---) O z ---) 0 where L is a direct sum

of line bundles on X. Since Z is 1.C.M., we claim that E is a locally free sheaf over X.

In fact, V y r Z, projdim Oz,y = dim X - depth OZ,y = 2

(since Ox,y is regular and Z is I.C.M.). Hence ~ is free (see for instance [18], th.l.1,

p.200 and [23] p. 15). Note moreover that HI(X,E(t)) = 0 for every t r Z. We will

call (following [17] with a slight abuse of notation) such a resolution an E-resolution of

Jz/x.

The relation between the locally free resolutions of two directly linked elements of

C X is given, as usual, by the mapping cone procedure ([20], prop.2.5): i fZ r C X has a

locally free resolution 0 ---) F1 "-> F2 --'LTZ/X -'-) 0 and Y is linked to Z via two

divisors of degrees a and b forming a complete intersection V, then Y has a locally free

resolution of the form

0 ---) F2V(-a-b) --> FlV(-a-b) (B Ox(-a) @ Ox(-b) ---> JY/x ''> 0.

Notice that the fact that X is subcanonical (which as we noted above follows from the

Gorenstein assumption) is used to get this result. Indeed, we have the following (see

[23] pages 38-40 for more details). Assume that COx __.- Ox(c). Then

J7_4xIJv/x = Homox( Oy,Ov) = Ext~Jx( Oy,Jv) = Ext~x(Oy, Ox(-a-b))

= Ext2(Oy, Ox(c))(-c-a-b) = COy(-c-a-b)

Now, given locally free resolutions for Jz/x (as above) and Jv/x (Koszul), take the

mapping cone obtained from the short exact sequence given by the above isomorphism.

Then duaiize this and twist by -a-b, and the result follows.

Starting from an E-resolution of Jy/x, one gets a resolution of J/_dx of the form

O -+ L -e N-->Jzlx -+0

where L is a direct sum of line bundles on X, and N is a vector bundle such that

Hr'l(X,9~t)) = 0 for every t e Z. We will call it (as before) a N-resolution of JT_dx.

The main theorem of [22] says that the even liaison classes of elements of Cx are

in bijective correspondence with the stable equivalence classes of vector bundles Fon X

with the property that I-ll(X,F(t)) = 0 for every t E Z and det(F)= Ox(a) for some ae Z.


(This latter condition is trivial if Pie(X) = Z.) The correspondence is ven by Z --) F,

where F i s the kernel of an ~_rresolution of 5Z/X.

So, for instance, we will denote by P the liaison class of elements of C X linked

to a complete intersections on X. In the literature these arc sometimes caUcd licci schemes

(for linkage class of a complete intersection). This class corresponds to the stable

equivalence classes of vector bundles splitting in a direct sum of line bundies. There is

not much connection between this and being licci in Pn:there may be elements of C X

which arc not licci on X, but as subschemes of pn they are (take a single point on X = a

quadric surface in p3.) And unless X is itself a complete intersection, there is in general

no reason to expect that being licci on X should imply that it is licci in pn.

Now we consider the deficiency modules. It is well-known the invariance (up to

shifting degrees and dualizing) of these modules under liaison, and that this necessary

condition is also sufficient in the case of curves of p3 ([21]), but not in higher projective

spaces. We denote by Mi(Z,X) the graded S(X)-module m~zHi(X,Jz/x(t)).

A particular case is when these modules are zero. Recall that a subscheme T of

pn is said to be arithmetically Cohen-Macaulay (a.C.M.) if Mi(T,P n) (which we will

denote briefly Mi(T)) is trivial for I < i < dim T. Now consider a two-codimensional

subscheme Z E Cx. Z is an a.C.M, subscheme of pn if and only if Mi(z ,x) = 0, for I

< i < dim Z. This follows easily from the sequence 0 -=) Jx/pn - ) 57Jpn --) JZ/X =") 0,

which indeed implies that Mi(Z,X) --- Mi(z ,P n) as S-modules. But notice that for any

homogeneous F e I X, F annihilates Mi(z ,P n) as an S-module (see for instance [15]

Remark 2.7 and use the same line of reasoning for the case 2 ~ i < dim Z). Hence we

may view Mi(z ,x) _= Mi(z,P n) as modules either over S or over S(X). Recall also that

in general Z is I.C.M. if and only if Mi(Z,P a) has finite length for all I ~ i < dim Z.

Hence in our situation we may assume that they have finite length.

Simple examples (later on we will see this fact more deeply) show that there may

be many different liaison classes whose elements have deficiency modules equal to zero;

hence the above condition is not sufficient in general, even for curves on a threefold.

Now we prove the necessary part. By the remark that schemes licci on X are

linked in pn by Gorenstein ideals, this result is a consequence of [25]. However, since

the proof is immediate in this case, we give a different proof following [16].

Proposi t ion 1.1. Let X c pn be a smooth arithmetically Gorenstein subscheme

of pn of dimension r > 3, and suppose that to X _= Ox(c). If Z,Y ~ C X are directly

linked via a complete intersection V (:: X, with V = F a n F b, deg F a = a, deg F b = b,

then M r ' i ' l f y , x ) -- (Mi(Z,X))V(-c-a-b) for 1 <: i < r-2.

If Z,W ~ C x are evenly linked then there exists some t ~ Z (not varying with i) such

that Mi(w,x) = Mi(Z,X)(t)for 1 < i < r-2.


Proof. We have a locally free resolution 0 ~ F2 ~ FI "-r ~ 0

with F1 direct sum of line bundles. From the mapping cone procedure we get the exact

sequence 0 ~ FlV(-a-b) --* F2V(-a-b)@Ox(-a)@Ox(-b) ">-~ix ~ O. Hence Hr-i- l ,x , v ~ _ Hr-i-Dx a" v t a b 'x - Hi+l(x,F2(a+b)@tOX) v Mr ' i ' I (y ,x ) = . t, ,-,'Y/X) = . t, ,.r 2 k- - ) ) =

= S~ (X,Jz/x| v _=_ H i(x,ffZ/x(c+a+b)) v= Mi(Z,X)V(-c-a-b).

The second assertion is a consequence of the first one. �9

Remark 1.2. Prop. 1.1 could be carried out in somewhat greater generality-- a similar

proof holds for Z and Y directly linked of codimension 2< d < r-1 in X, as in [25]. �9

2. The liaison class of a complete intersection In this section we denote by P the liaison class of a two-codimensional complete

intersection in X (the liaison class of licci subschemes). If Y~ P, then it is a.C.M., but

the converse is not true in general (it is true if x=pn). If X=p3, there are in the literature

at least three different ways for studying this class: namely 1) via a resolution of a

projection of Y~ P([12]); 2) via a direct study of a resolution of J u ([24]); 3) via a

resolution of Jyc~H, where H is a general plane in p3 ([19]).

The relation between these three ways is studied in [10]. In the general case, with

X satisfying our hypothesis, it seems better to follow 2 for bounding the range of

existence of licci subschemes. In order to prove the existence part, we use the Bertini-

type results by Chang ([7]). The purpose of this section is hence to show that this liaison

class, although not yet explicitly studied, is completely describable by using known

results, simply adapted, and that the liaison approach is still useful.

It follows from w 1 (Rao's theorem and the mapping cone) that Y~ P if and only

if it has a locally free resolution of the form

p-I j~=l 0 --r ~10x( -a i ) ~ = Ox('bJ ) --4 ffy/x ----> 0

where we assume that al<a2<...<ap.1, bl<b2<...<bp, and a i = . Our purpose is i=l j=l

to classify the elements of P, giving a numerical condition on the integers {ai} and {bj},

and, if possible, to classify the smooth elements.

By collecting all common addenda in the left and the middle term we get t-1 t

0 ~ i~10X(-O~i)~B "-~ j=~l OX('~J )~)B -') JY/X "~ 0

where {0ti}~{~j}=O , and we assume again CXl<...<o~t.1, I]l<...<~t . It is clear that the

(2t-1)-ple of integers (r I ..... czt-l; [31 ..... I]t) is uniquely determined by the sequence of

integers { h0(pn,jy/x(t) } t ~ , and vice versa. So the point of focusing on the oq, I]j is

that this sequence of integers does not depend on the locally free resolution chosen at the

beginning (see also Remark 2.1). Following [24], we denote

m(Y) --min {i I cxk< 13k+ i for some k=l ..... t-I} if such an i exists, + c o otherwise


Remark 2.1. We claim that m(Y) is also equal to min { i I a k < bk+ i for some 1 ~c.~p- 1 }.

Sauer's notation for this last number is m(bl<... .<b p ; al<...<ap.l), and we use this for

this remark. Let us prove this fact, by proving that �9 �9 ! r !

m(blg. . . .<b p , al~...<ap. 1) = m(b~<....< b~, al< . . . . ap.1),

where b~ .... b~ and a~ .... ai~. 1 are obtained from bl,. . ,b p and al,..,ap. 1 by deleting a

common term bj=ai; we suppose, by changing indexes, that bj. 1 < bj (or j= l ) and ai.l< a i

(or i=l). Hence ~ = b k if k<j, b~ = bk+ 1 if k2j and a k = a k if k<i a[r ak+ 1 if k_>.i.

Now, m ( b l < .... <bp ; al<. . .<ap_l) = m <=> Vk, a k > bk+m. 1 and 3 k 0 such that

ak0<bk0+m. We must prove that Vk, a[ > b~c+m_ l, and that 3 k I such that akl < b~l+m.

I f k < i, then a~ = a k > bk+m-1, and a k < a i. Hence bk+m. 1 < a i = bj, and hence

bk+m. 1 = bit+re. 1. So i f k < i, then ak > b~+m.l.

. fei therb[~+m > bib+m-1 k ~ I f k > i, a~ = ak+ 1 > bk+m= ~ or bk+m. 1 : in both cases we have a' > ~''

Now let k 0 be such that ak0 < bk0+m. If k 0 > i, then ak0 > a i, and bk0+m > ak0 > a i = bj

so k 0 + m >j . Hence ak.0.1 = ak0< bk0+m = b~,o+m. 1, and hence a~0.1 < bko.l+ m.

either b~o+m I f k 0 < i , then: a ~ = a k o < b k 0 + m = or bl~0+m_ 1 < bl~o+ m

Therefore in both cases we have a k 1 such that akl<b[l+m, and the remark is proved. #

In particular, re(Y) can be computed starting from any locally flee resolution of Jy/x.

Proposition 2.2. Let Y e P. Then m(Y) > 2. I f moreover re(Y) = 2, then Y is

reducible. In particular, unless dim X = 2, we have that Y is singular.

Proof. We may assume that

p-1 j~=l 0 ---> ~I Ox('ai) "-'> = Ox(-bJ ) "-> ffY/X ~ 0

is minimal, and that it is represented by a (p- l ) x p matrix A of homogeneous

polynomials restricted to X (X is a.C.M, by hypothesis). More precisely, A can be

viewed as the restriction to X of a matrix B of homogeneous polynomials giving an exact

sequence p-I B j--~l 0 -'> ~10pn( ' a i ) "-> = Opn(-bj) -~ -~Z "-> 0

where Z is two--codimensional in pn and Y = X n Z as schemes. Clearly re(Z) = re(Y). I f re(Z) ~ 1, then this means that there exists n o such that %0 < bno+l" But this

implies ai < bj for every i < n0 and for every j > n0+l.

Hence the matrix B looks as follows:

B=( MIM2 0 3 )


where M 1 is n o x n 0, M 2 is (p-n0-1) x n o , M 3 is (p-no-l) x (p-n0) and 0 is n0x(P-no). If

h = det M l, then B drops rank on the hypersurface defined by h = 0, contradicting the

hypothesis that Z is two-codimensional.

Let us suppose now that m(Y) = m(Z) = 2. This means that there exists an integer

n such that an<bn+ 2. Let nl,..,n t (t > 1) be all such integers, and set n00 and nt+ 1 = p-1

(we follow [24] in this description of the matrix B). Then B is as follows: I b l ' l B0 /

B = B i 0 o o o o o .

where B i is a square matrix of order ni+l-n i, for i = 0,1 ..... t.

Recall that, thanks to the Hilbert-Burch theorem, we know that the maximal

minors of B form a minimal generating set for the homogeneous ideal of Z. Then, as in

[24], Remark 1.1, one can conclude that Z is reducible. (This fact is also proved in [9],

and reconsidered in [10], where it is explicitly noted that if there is a zero on the first

subdiagonal of B (note that their ordering of the integers {ai} and {bj} is different; in our

situation these elcments are the b n n+2's, where n = n 1 ..... nt), then at least the first two

minimal generators of Iy/X (ordered by degree) have a common factor). But then Y = X

c~ Z is also reducible (or non-reduced). But in dimension ~ 1, the property of being

a.C.M, implies conncctedness. Hence if dim X > 3 Y is connected and reducible, hence

singular. �9

Now we come to the "existence" part. Since we are interested in proving the

existence of a suitable Y, we use the notation m(bl<.. .<bp; al<.. .<ap. 1) (given the

sequence of integers (a 1 ..... ap.1; bl,..,bp)) rather than m(Y). The existence of a lieci Y

contained in X for every (2p-1)-ple of integers (a 1 ..... ap.1; b 1 ..... bp) with

m(bl<..<bp;at<..._<ap. 1) > 2 comes (via restriction) from the existence of a licci two-

codimensional subscheme of pn with a resolution with the same numerical behaviour,

nevertheless, it is better for us to prove this fact directly (via the main result of [7]) since

this approach will also give directly results about the existence of smooth licci two-

codimensional subschemes of X, with hypothesis on the dimension of X and no

hypothesis on the dimension of pn. The results here, in particular Theorem 2.4, are

similar to Example 2.1 of [7], but more general. Let us recall Chang's theorem ([7]):

T h e o r e m 2 . 3 Let E m D E t D E t . I D . . . D E l a n d F n D F t ~ F t . I : : 9 . . . D F l b e

filtrations of vector bundles over a smooth variety X with rk E i = m i and rk F i = n i,

and let oq = ni-mi, tx -- n-m. Assume that tx i > ~ > O for all i and that the subbundle B

of EV| defined by B(x)={f x I fx(Ei(x)) is contained in Fi(x) for all i } is generated by

global sections. Then for a general fE HomfE,F) with degeneracy locus Y we have

i) codim Y > o~+1, and ii) codimySingY > min{2txj-2ct+l, ctj+2, ix+3}. *


Now, in our situation we consider the following filtrations:

E l = Ox(-a:), F-,2 = Ox(-at) (B Ox(-a2) ..... Ep. 2 = Ox(-al) (B Ox(-a2) (B ..(B Ox(-ap_2)

F 1 = Ox(-bl) (B Ox(-b2), F 2 = Ox(-b I) (B Ox(-b 2) (B Ox(-b 3) .....

Fp. 2 = Ox(-bl) f13 Ox(-b2) ~ ... @ Ox(-bp.1) Let r be an element of HomfE,F), viewed as a p- 1 x p matrix ($ij). Consider its

(" al-bl al-b 2 al-b 3 . . . al-bp_ I - a l - b p "~

. . . . . . . . . . . . . . . . . . 1 ~,al~l-bl ap.l-b2 ap_l-b3 ... ap_l-bp, l ap.l-bp ,]

In order to preserve the filtration as required in the theorem, we need (~ij = 0 for j > i + 2.

Then among such morphisms, the hypothesis of being globally generated means that

al>b2,..,ap_2>bp.1 and ap.l>b p. This is equitavalent to m(bl<..<bp;al<..<ap.1)>2. So

the above theorem implies that there exists a morphism

p-1 j~=l ~10x(-a i ) --'> = Ox(-bj) dropping rank in codimension 2.

Now suppose that dimX < 4, and that m(bl<...<bp;al<...<ap. 1) > 3. I f p = 2 or

3, this condition implies that HomfE,F) is globally generated, and hence there is a

morphism whose degeneracy locus Y is smooth (Y has dimension < 2 and non-singular

away from a subset of codimension > 4 in Y). (Notice that for p = 2,3,

m(bl<.. .<bp;al<.. .<ap.1) > 3 implies m(bl<..<bp;al<..<ap. 1) =+oo .) If p > 4, we

filter E with bundles of rank 1,..,p-3 and F with bundles of rank 3,..,p-I. Again,

m(bl<.--<bp; al-<...<ap 1) -> 3 implies the hypoihesis of the theorem and hence there p-1 P

exists a morphism i~10x( 'ai)-- '> j~l Ox( 'bJ ) with a degeneracy locus Y with

codimySingY > 3. But dimY < 2, and hence Y is smooth.

Similarly, suppose dimX = 5. If we assume that m(bl<...<bp; al<...<ap. 1) > 3

then for p = 2 or 3 we again get that Hom(E,F) is globally generated and hence there is a

morphism with smooth degeneracy locus. (Again, m(bl<...<bp; al<...<ap. 1) =+oo in

this case.) Now assume that m(bl<...<bp; al<...<ap. 1) > 4; then we get the same

conclusion for p = 2,3 or 4. Furthermore, in this case if p > 5 then we filter E with

bundles of rank 1,2 ..... p-4 and F with bundles of rank 4,5 ..... p- 1 to get the existence of

a morphism with degeneracy locus Y with codimySingY > 4, hence smooth.

The only remaining case if dimX = 5 is m(bl<...<bp; al<...<ap. 1) = 3 and p > 4.

Here filter E with bundles E i of rank 1,2,..,p-3 and F with F i of rank 3,4,..,p-1. Then

the same argument given by [7], Claim 2 (p. 218) works to show that codimySingY = 3.

(in our situation it is easier to show that the general (~i: Ei -'> Fi drops rank in

codimension 3 since F is free. The condition m(bl<..<bp;al<..<ap.1)=3 implies the


hypothesis of Theorem 2.3 for the vector bundles E i and F i with the obvious filtrations,

so by Theorem 2.3 the degeneracy locus of t~ i has codimension > 3, hence = 3.)

We can summarize our results as follows:

T h e o r e m 2.4. Let X be a smooth arithmetically Gorenstein subscheme of pn of

dimension r > 2. Then there exists a licci (in X) two-codimensional subscheme Y of X

with locaUy free resolution

p-1 j~=l 0 --~ ~ 1 0 x ( ' a i ) --~ = Ox('bJ ) -4 ..qYpZ ~ 0

if and only if m(bl<....<bp ; al<...<ap. 1) > 2 and ~"a i = ~ b j . i=l j=l

/ f 3 < dimX < 4,there exists a smooth Y if and only if

m(bl<..<bp;al<..<ap.1)>3 and ~ a i = z~b j . i=1 j=l

/ f dimX = 5, there exists a smooth Y if and only if

m(bl<...<bp;al<..<ap.1)>4 and ~ a i = ~ b j * i=l j=1

Remark 2.5. This result gives a complete description of all licci (in X) congruences in

X=Gr(1;3). More generally, it gives a complete characterization of smooth licci (in X)

curves on a threefold X, surfaces on a four-fold, etc. Note that in the case where X is

p2 viewed as a subvariety of p3, all zero-schemes are licci and indeed our results are not

quite right in this very special case. Chiantini and Orecchia ([8]) study this case.

If X is contained in pn, with n < 5, then the theorem actually says that,

numerically, these smooth varieties are exactly those obtained by intersecting X with the

smooth a.C.M, subschemes of pn. If n > 5 the result says that these smooth

subvarieties are exactly (numerically) those obtained by intersecting X with the a.C.M.

two-codimensional subschemes of pn with the lowest dimensional possible singular

locus (the intersection being done in such a way as to avoid singularities).

An interesting question is to try to remove the "numerically" from the above

discussion. That is, we conjecture that if Y is a smooth licci subscheme of X and if X is

contained in pn, with n < 4, then Y actually is the intersection of X with a smooth

a.C.M, two-codimensional subscheme of pn. (For example the case where Y is a

smooth arithemtically Cohen-Macaulay curve in p3.) In the other cases, when dimX <

5 and Y smooth and licci, then Y is the intersection of X with an a.C.M, subscheme of

pn which is "as smooth as possible". Note that when X = p2 viewed as a subvariety of

p3, this is exactly the result of Chiantini and Orecchia ([8]). ,


Proposi t ion 2.6. Let Y and Z be licci two-codimensional subschemes of X, with

resolutions 0 ---) ~ 1 0 x ( ' a i ) u...~ ~ Ox(-bj) ~ JY/X "-> 0

p-1 J~--1 0 ~ ~10x( -a i ) v_._~ = Ox(.bj ) _.~ ff7_/X "--> O.

Then Y and Z lie in the same irreducible component of the suitable Hilbert scheme of

subschemes of X.

Proof. Consider for general ~.E k, ~.u+(1-~.)w Horn( ~ 1 0 x ( ' a i ) ' = Ox(-bj) ) *

3. Liaison Addition

In this section we do not assume that our subschemes are two-codimensional in

X, nor that they are 1.C.M. or equidimensional. Our goal, motivated by the work of

Schwartau [26] is to define a construction which produces from a given set {Vii of

schemes a new scheme Z whose modules MJ(Z,X) are a direct sum of suitable twists of

the Mi(vi ,x) . Our generalization in X is modelled after that in [11] for subschemes of

pn, and our proof is a reduction to that case. We will get the notion of basic double

linkage of subschemes of X for any codimension, paralleling the situation in pn

described in [14] and [4]. This will allow us to define the LR-Property for subschemes

of X in the next section, and to prove it for codimension two.

Let V 1 ..... V r be closed subschemes of X c pn, with 2 < r < codimxV i < dim X, having homogeneous ideals Iv1 ..... IVr (as subschemes of pn). Say I X = (Fr+ 1 ..... Ft).

("3 be such lhat they form an S/Ix-regular sequence For 1 < i < r, let Fi ~ l~.j.~r Ivi

(recall that X is a.C.M.). Say deg F i = d i. Notice that {Fi},l < i < r, can always be found for d i >> 0. Consider the subscheme Z of pn defined by the ideal I = FIIv1 + ...

+ FrlVr + I X. Z is clearly a subscheme of X (since I x c I c Iz). Let V be the scheme-

theoretic intersection of X with the complete intersection defined by (F1,...,Fr).

L e m m a 3.1. I v = (F 1 ..... Fr,Fr+ 1 ..... Ft); i.e. I v = (F 1 ..... Fr) + I x as total ideals.

This is again a Gorenstein ideal in k[x 0 ..... Xn].

Proof. Consider the exact sequence

0 ---) I x n ( F 1 ) ~ I x ~ ( F 1) ~ I x + ( F 1 ) ~ 0 .

Since F 1 is a non-zero divisor in S/I X, I x c~ (F1) = FI .I x = Ix(-dl) . Also, I x ~ (FI) =

I X ~ S(-dl). Sheafifying and taking cohomology, and using the fact that X is a.C.M.

(so H~(fl X) -- 0), we get H0(sh[Ix + (FI)]) = I x + (F1), where sh[I x + (FI)] denotes the

sheafification of I X + (FI). Notice also that the scheme defined by I x + (F 1) is a.C.M.

(since it has dimension one less than that of X). Continuing inductively (replacing I X by

I X + (F i) etc.) we get H0(sh[Ix + (F 1 ..... Fr)]) = I X + (F 1 ..... Fr), as desired.


Now we show that the ideal I x + (F 1 ..... F r) is again a Gorenstein ideal if I X is.

This follows (inductively) from looking at minimal resolutions for I x ~ (F 1) _=. Ix(-d 1)

and for I X @ (F1), and using the mapping cone to obtain a free resolution for I X + (F1),

which one checks has Cohen-Macaulay type one since X is arithmetically Gorenstein

(since (I X + (F 1) defines a scheme of dimension one less than that of X.) *

Theorem 3.2 (a) I = FIlVl + ... + FrIVr + I x = I Z, i.e. I is the total ideal of Z in pn.

(b) Set-theoretically, Z = V 1 t.) ... uJ V r u V; hence in particular Z has codimension r

in X (since codim X V i > r = codim X V).

(c) MJ(Z,X) = MJ(v1,x) @ ... @ MJ(Vr, X)foral l 1 < j < dim Z.

(d) Z is l.C.M, and equidimensional if and only if the V i all have codimension r in X

and are all I.C.M. and equidimensional.

(e) deg Z = Z deg V i + dl...dr.deg X

Proof. Formally define schemes Vr+ 1 . . . . = V t = iZI, with defining ideal S. Notice that

for all i, 1 '~i < t, we have F i e ('~ I l<j~ Vi j,i

(recall that I x c Ivj for 1 < j < r). Hence we are in a position to apply [11] Theorem

1.3, once we check the following: if A1F 1 + ... + ArF r + Ar+lFr+ 1 + ... + AtF t = 0 then

A i e Ivi for all i, 1 < i < t. This is ttrue for i > r+l, so we need only check it for 1 < i <

r. For example, we prove it for i = 1. We have A1F 1 = -A2F2..-ArFr-Ar+IFr+I-..- AtF t.

But F 1 is not a zero divisor on S/[Ix+(F 2 ..... Fr)], hence A 1 ~ I X + (F 2 ..... Fr) c Iv1.

Now the rest follows from [1 l] once we recall that for any subscheme Y of X,

MJ(Y,X) = MJ(Y,P n) for all 1 < j < n-2. ,

Co ro l l a ry 3.3

(a) Iz/x = F I l v l / x + ... + FrlVr/X, where F'i is the image of F i in S/I X.

(b) We have an exact sequence

�9 " "~ 1,:,~.j<_rOX(-di'dj) ~ l~,~rflVi/x('di)"-')-~JX "-')0

where the left-hand part is the restriction to X of the Koszul resolution for the complete

intersection (F 1 ..... Fr).

Proof. (a) is just the restriction to S/I x, recalling that X is a.C.M, and using 3.2(a). (b)

comes, as in [ 11 ], from observing that the image of ~) in the Koszul resolution is in fact

in the direct sum of the ideals-- again use the observation in the proof of 3.2. *

Corol lary 3.4 Assume that V 1,V 2 are two-codimensional I.C.M. equidimensional

subschemes of X and let Z be as above (so r = 2). I f

O ~ Fl ~ P ~ fVl /x --)Oand O ~ F 2 ~ Q--~Jv2/x ~ 0


are T_,resolutions of V1,V 2 respectively, then an T_,resolution of Z is

0 ~ Fl(-dl)~BF2(-d2)eOx(-dl-d2) ~ ~-dl)~BQ~-d 2) ~ 5Z/X ~ 0

Proof. The proof is identical to that of Proposition 2.7 of [4]. Of course,there is an

analog for ~-rcsolutions �9

We can now define Basic Double Linkage in X. As in [11], this is done as a

special case of Theorem 3.2 by taking V I c X non-trivial of codimension at least r and

V 2 . . . . . V r = 0 . The resulting scheme has ideal I Z = FIlVl + (F 2 ..... Fr) + I X, where

F 2 ..... F r e Iv1 and such that (F 1 ..... Fr) form an S/Ix-regular sequence. (No other

condition on F1.) Passing to S/I x we get IZ/X = FIlVl/X + (F2 ..... Fr). If V 1 is 1.C.M.

and equidimensional of codimension r in X, Z can be obtained from V 1 as follows (see

also [11]). Choose a general G e IVl of sufficiently large degree so that (G,F 2 ..... F r)

forms an S/Ix-regular sequence. Note that (GF1,F 2 ..... Fr) is also an S/Ix-regular

sequence. Then on X, one checks that Z is obtained from V 1 by linking first with

(G,F2 ..... Fr) and then linking the result using (GF 1, F 2 ..... Fr). Note that these are not

links as subschemes of pn via complete intersections, although they are links by

Gorenstein ideals.

Remark 3.5. Suppose that Y is a two-codimensional subscheme of X, and Z is

obtained from Y via a basic double link with a divisor of degree d containing Y and a

sufficiently general divisor of degree f. We henceforth write Y:(d,f)---~Z for this

procedure. Then from an E,-resolution of Y 0 ~ ~r._.) p...~ 3y/x ~ 0 one gets an if.-

resolution of Z 0 ~ F ( - f ) ~ O x ( - d - f ) ~ A ---> P( - f )~Ox( -d )~A-- - ) JZ/X ~ 0

where A is a direct sum of line bundles (from corollary 3.4 or by applying the mapping

cone twice). Since the cohomology of Z and its shift in L, do not depend on the divisors

chosen but only on their degrees, we can always consider instead of Y : ( d , f ) ~ Z a

sequence of basic double links of the form Y:(d,1)--~ Yl:(d,1)---) ...... Yf.l:(d,1)---) Yf,

where Yf has the same cohomology as Z , and lies in the same shift of L

4 The Lazarsfeld-Rao Property for non-licci even liaison classes on X

The Lazarsfeld-Rao property is a structure common to all non-a.C.M, even

liaison classes of codimension two in projective space (cf. [3] and [14]). The first step

is to define the notion of shift, and in particular minirf~l shift. In the situation of non-

a.C.M, in projective space, this was done by considering the deficiency modules. Now

we would like to allow these modules to be zero. Hence, we will take the point of view

of vector bundles.


Let V be a non-licci (this will be understood from now on) two-codimensional subscheme of X. Let N b e a vector bundle on X with H2,(P~ = 0 and such that we have

the exact sequence 0 ~ F--~ N ' ~ J v / x ~ 0 with Ff ree . Let N = I - ~ . ( ~ . N is a

finitely generated A = S/I X module. Taking cohomology we have 0 --~ F --~ N --~ Ivp A

--~ 0 where F is a direct sum of copies of A with twists. Now suppose that there is a

scheme W satisfying 0 --~ G ~ ~ t )@F" "-'~Jwlx "* 0 with G and F ' free. Note that

this holds if and only if W is evenly linked to V on X.

L e m m a 4.1. t is bounded above.

Proof. Suppose otherwise, so say t >> 0. Then taking cohomology we get V

0 ~ G ~ N(t)@F" ---~Iw/X ~ 0

where G = @A(-dj). Since N is finitely generated and t >> 0, we can assume that the

minimal generators n 1 ..... n r of N(t) occur in negative degree. On the other hand, the

minimal generators of Iw/x occur in positive degree. Hence ~((ni,0)) = 0 for all i (since

is a degree zero homomorphism). Then N(t) c ker r = ~g(G) c N(t)@F'.

On the other hand, say (ni,0) = ~g(gi) for each i, where gi ~ G. Because the n i

are minimal generators for N(t), the gi can be extended to a set of minimal generators of

G. We claim that N(t) is free. Indeed, suppose 1~ Ain i = 0, where A i e A. Then 0 --

EAin i = EAi(ni,0) = Z Ai~(gi) = E ~g(Aigi) = ~ ( ~ Aigi). But ~ is injective, so

EAigi--0. Then since { gi } are part of a minimal generating set of G, which is free, we

have Ai= 0 for all i. But if N(t) is free, then W and V are licci: contradict ion,

Hence for a vector bundle Nwi th H2,(5~ = 0, there exists t o -- t o ( N ) such that

some W does exist with 0 ~ G ~ ~ t ) @ F ' ~-Twlx ~ 0 exact for t = t 0, with

G , F ' free and W two-codimensional, and that for all other two-codimensional W

having such an exact sequence we have t < to. Let us call

f_~ = {W I an exact sequence of type (1) exists with t = to}.

Now let M b e stably equivalent to N This means that M@ T= 5~c) @ R f o r some c

e Z and 's Rfree. In particular, t0(Wc)) = to(5~ - c and L ~ = L ~ e ).

Now we prove that to(PVf) = to(P~c)) and L O = L ~ c ). Indeed, let to = to(Pc/)

and suppose we have 0 ~ G ~ M(to)@F' "'~JvPA ~ O.

Then adding summands trivially gives

0 ~ G@T(to) ..~ M(to)@T(to)@F" ~ J v / x ~ O,

that is 0 ~ .C~gT(to) ~ P~c+to)@R(to)@F' ~ J v / x ~ 0.

This implies c+to(Pt, 0 < to(P~0, or to(M) < to(P~c)), and Z O c L ~ e ). The reverse

inequality and inclusion are proved similarly. Once we have the minimal shift, it is easy to define the h-th shift of L: if to, y ' ,

N G, are as above, and W e L has a resolution


0 - o G - o N( t0 -h )@F '~ Jw/x -o0 ,

with h > 0, we say that W e f_). Of course this notion does not depend on the choice of

the vector bundle N , that this notion coincides with the one given via shifting of the

deficiency modules if L is not a.C.M., and that the notion of shift could have been

defined via E-resolutions as well.

Remark 4.2. It is worthwhile to point out that, given W e L, its shift is uniquely defined; that is to say, Jw/x cannot have two resolutions

(1) 0 ~ G "-} N ( p ) ~ A --o Jw/x -> O,

(2) 0 ---> G ' ~ N(s)@A'-o Jw/x "--> 0

with G,G',.RA'direct sums of line bundles and p # s. Take Y directly linked to W and

apply the mapping cone procedure to (1) and (2):

0 --~ E(p')@C1 -o B 1 -~ JY/x --o 0

0 -~ E(s')@C2 ~ B2 --o JY/x -'-> 0 with HI(E(p')@CI) = 0 and HI,(E(s')@C2) = 0. Hence the associated cohomology

sequence in either case is a short exact sequence with the middle and right-most terms

being the beginning of a free resolution of Iy/X. But now take a minimal free resolution

of Iv/X, and then sheafify: 0 --> V---> B---> JYlx ~ O, with B direct sum of line

bundles and Vlocally free (and non trivial) with Hl('k~ = 0. By minimality, B I _=_

B ~ D I and B2=_ B ~ D 2 for some D I and D 2 direct sums of line bundles. Then an

elementary argument gives that E(p')@C1--- 'IA~DI and E(s')@C2 = r 2 (we axe

grateful to A.V. Geramita and H. Charalambous for showing us this argument).

Without loss of generality we may assume that V has no line bundles as direct

summands. So E(p')@CI~D2 =. qA~DIEBD 2--- E(s')@C2~D1, and the uniqueness

of decomposition (of.[1]) implies p'=s' and p=s. *

A crucial application of the notion of shift was in the possibility of performing

deformations. We can state this important fact in our situation (see [4], prop.3.1):

Proposition 4.3. Let Z, ~ Pbe an even liaison class of two-codimensional

subschemes of X, and let Y,Z e f~ such that

h0(X, Jy/x(t)) = h0(X, JTjx(t)) for all t.

Then there exists an irreducible flat family { Ys } sE S of two-codimensional subvschemes

of X to which both Y and Z belong. Moreover, S can be chosen so that for all s~ S,

ys e f_Ja and h0(X, Jy/x(t)) = h0(X, JYs/X(t)) for all t.

Proof (sketch). Start with an E-resolution of the ideal sheaf of Y

O"~ E - ~ A'--~ Jv/x -o0

and apply the mapping cone procedure for finding a resolution of the ideal sheaf of Z:


0 ---) E(d)@B ~ A @ C ~ JzlX ~ O.

Since Y and Z are in the same shift, then d--0, and hence (after having added trivial

addenda and the identity map) we have this situation :

O--) ~ B U~ D ~ Jyp z ~ 0

O ~ ~ B V-~ F ~ gz/x ~ 0

where B, D and F a r e direct sums of line bundles, and hl(X,E(t)) = 0 for all t. Then

thanks to the hypothesis of fixed postulation, we get that D = F. Then the result

follows,as in 2.6.

The same proof works if hr'l(X, Jypz(t)) = hr'l(X, ffZ/X(t)) for all t (r --dimX). �9

Here is the central definition of this paper.

Definition. Let L g P be an even liaison class of dimension p subschemesof X. We

say that L has the LR-property if the following conditions hold:

1) Given V 1 and V 2 in L 0, there is a deformation from one to the other through

subschemes all in L0;

2) Given V 0 e L 0 and V ~ L!~ (h>l), there exists a sequence of subschemes V 0,

V 1 ..... V t such that for all i, 1 g i < t, V i is a basic double link of Vi. 1 , and V is a

deformation of V t through subschemes all in fla.

We want to prove that in the two-codimensionai case every non-licci even liaison

class has the LR-property. There is no substantial difference between the projective space

and a general X (satisfying our hypothesis), once one has set the right framework and

checked that the needed instruments are available, so for most technical details we will

refer to [3]. We will point out some minor differences. Nevertheless, the fin'st lemma is

important and we must treat it carefully.

Lemma 4.4. Let E be a rank-(s + l ) vector bundle on X and $ $

~ : i@Ox(-ai) ---) E ,~ : @lOX(-bi) - - ~ . = E , al < a2<. . . <as, bl < b2<. . . <bs,

be morphisms whose degeneracy loci are two-codimensional. Then there exists a $

morphisra ~ : @lOX(-Ci) ~ Ewith ci = min{ai,bi} Vi,

whose degeneracy locus is two-codimensional.

Proof. Suppose that ai = bi if i < k (with 0 < k < s), and that ak+l < bk+l; we

will show that there exists a morphism i t

X: i~ Ox('ei) ~ E

with ci -- ai if i < k+l, and ci -- bi if i > k+l. After a finite number of steps the lemma

will be proved. We denote ~t : = ~li~tOx(-ai), and the same thing for ~t. By

semieontinuity,without loss of generality we may assume that ~k = ~k. If ak --- ak+l we


$

fix (arbitrarily) a splitting of i~ 0x('ai) and consequently of (h. (Notice that bk = bk+l is

impossible: otherwise, bk+l > ak+l > ak = bk = bk+l .)

As in the projective case, we can prove that the morphism $

H : (90x(-bi)(9Ox(-ak+l) -.--> E i=l

which restricted to the first s factors concides with ~ and restricted to the last factor

coincides with (h restricted to Ox(-ak+t) has rank s+l (the maximum possible) and hence

is an isomorphism outside a divisor S of X. In fact, if this is not the case, Im(H) is a

rank-s subsheaf of E which contains Im(~), and since ~ drops rank in codimension 2 we

get Codim(Supp(Im(H)/im(~))) > 2.

If now we take a general 1 dimensional subvariety L of X which does not

intersect Supp(Im(H)/Im(~)), and avoids the degeneracy locus of (h, and restrict the exact

sequence 0 ---> Im(~) ~ Im(H) --> Im(H)/im(~ ) --> 0

to L, we have that Im(~)lL = Im(H)IL, and hence an inclusion $

(~lOL(-ak+l) : OL(-ak+l) "--> i=(91"= OL(-bi).

If k = 0 this is clearly impossible; if k > 0 we should get

Im((~101L ~ Im(@lOL(-ak+l)) for general L .

This implies that ~nk(tPk+l) < k and hence a contradiction. $

Hence outside a divisor S, for every rank-s factor T of (90v(-bi)(9Ox(-ak+l) i=l ^

the morphism HIT is an embedding, and this is still true for every morphism with the

same image as H. So we will arrange our morphism in the following way: if Ix is H

restricted to Ox(-bk+l) and ~. is H restricted to Ox(-ak+l), we can replace Ix with

I.t+PX, where P is a general element of Hom[ Ox(-bk+l),Ox(-ak+l)].

Now we study our morphisms locally. Let us fix a point xp, l<p<t, in every

irreducible component of S, such that at every Xp we have rank((h) = rank(k) = s.

A~:= (Im(~)l(Xp} and A~:= (Im~)l(Xp} are s-dimensional subspaces of ~{Xp}=ks+l

and contain a common k-dimensional subspace Ak:=(Im(hk) l{Xp}.Thus

Ak+l:=flm((~k+l))l{Xp} has dimension k+l and there axe integers ml<..~ns.k.1 such that

[ImHI (9 Ox(.b m )]l{xp}+Ak+l has dimension s. 1,;:~s-k-1 t

If mt = k+l+t (Vt) the lemma is proved. If not, we can again modify the map H by adding to HiOx(.bk+2 ) . . . . . HIOx(.br ), the map HIOx(-bk+9 times suitable elements of

Hom[Ox(-bt),Ox(-bk+l) ] which do not vanish at Xp, until we obtain the map ~. ,

C o r o l l a r y 4.5. Let L v~ P be an even liaison class of two-codimensional

subschemes of X, and let Y,V ~ L O. Then hi(X, Jy/x(t)) = hi(X, Jv/x(t)) for all t

and i. I f moreover Y e L 0 and ZE s have locally free resolutions

O ~ P ~ F ~ J Y I X ~0


0 ~ ~ B --~ F @ A ~ JZJx(h) --> 0 $

(P, Band Adirect sums of line bundles and hr-l(X,F(t)) ) and ~ A = @ Ox(-ai ) , i=l

$

~ B = @ Ox(-bi), with al ~ a2 ~ .. ~ as, bl ~ b2 ~ ... ~ bs, then bi ~ aif.or every i. i=�94

Proof.. The proof is exactly as in [3], 2.2 and 2.3; just note that also here, 8 --

cl(Jz/x(8)) = c l (~tL~) - ~,c i = Y~(ci-a i) (we keep the same notation as in [3]). e

T h e o r e m 4.6. Let L be a non-licci even liaison class of two-codimensional

subschemes of X. Then L has the LR-property.

Proof. The proof is a formal consequence of the above corollary, the mapping cone

procedure and the definition of basic double link: these are the technical ingredients of the

proof that can be found in [3], 2.4 or in [14], p.278-279. *

Most of the properties known for the even two-codimensional liaison classes in

pn follow from the LR-property, the mapping cone procedure and the machinery of

liaison. But now we have all these instruments- hence we have the following facts:

Proposi t ion 4.6. Let L be an even liaison class of two-codimensional subschemes

of X, and let Me L 0, Ye L h. Then there exists a sequence of basic double links

M:(s,b)---> Yl:(g2,1)---> Y2:(g3,1)--~ .... --e Yp. l : (gp,1)~ Yp, where

a) s = ct(M) is the minimal degree of a divisor containing M

b) b > 0, s < g2 < .. < gp, and b+p-1 = h (possibly p = 1,i.e. there are no gi's)

c) Yp and Y have the same cohomology and are in the same shift of L

Moreover, the sequence (b;g2,g 3 ..... gp) is uniquely determined by the cohomology

and the shift of Y.

Proof. [4], 5.2 and 5.3. *

Proposition 4.8. Let IAbl<b2<...~bp be a non-decreasing sequence of integers.

Then there exists a licci two-codimensional subscheme of X, C = C(hl ..... bp), having

a locally free resolution

0 --> ;~Ox(-bi-p+i-1)@B --~ i~lOx(-bi-p+i)@B~Ox(-p)---> JC/X---~0

where B is a direct sum of line bundles. Moreover, C is contained in a divisor of

degree = min { bp,p }.

Proof. It is enough to check that the integers appearing in the above resolution satisfy m

> 2 (see Remark 2.1). If we set {cj}, cl < c2 < ....~Cp the negatives of the integers in

the left term (suitably reordered), then the corresponding integers in the middle term are

el-1 ..... cp-l ,p (we can forget the term B ) . We order these integers and call them

{dk}, 1 < k < p+l . Then one checks directly that Cn > dn+l, for every n.

Alternatively, one can repeat the proof of [4], prop. 5.10, by induction on p. *

364 BOLONDI-MIGHORE

Theorem 4.9. Let Y be a two-codimensional subscheme of X, and let Yk be the

result of applying any sequence of basic double links to Y. Then Yk specializes to the

liaison addition of Y with a suitable licci subscheme of X (possibly void).

Proof. (see [4], th. 5.11) We can suppose that Yk is obtained via

Y = Y0:(bl, 1) "r Yl:(b2,1) -'> ........ " ) Yk-l:(bk, 1) "-> Yk, where b i > bi. 1 ~' i.

If b I = ... = b k, then Yk is the liaison addition of Y and the void set via divisors

of degrees b 1 and k. Otherwise, if m = min{i I bi>bl }, and C = C(bm-b 1, bm+l-b I ....

bk-bl), then Yk has the same cohomology as the liaison addition of Y with C via

divisors of degrees b I and k. Then the result follows from 4.3. �9

These results give a "standard form" for computing the cohomology of Y; joint

with the deformation result 4.3 we get the following

Corollary 4,10. Let L be an even liaison class of two-codimensional subschemes of

X, and let ME L O. Then the elements of L are distributed in irreducible disjoint non-

void subsets s where b > 0, gp > gp-1 >.-.> g2 > ~(M), such that any two

elements of s lie in the same subset if and only if they have the same cohomology and

are in the same shift of s In every such family there is an element which is the liaison

addition of M with a suitable licci subscheme of X (possibly void).

Proof. It follows from 4.3, 4.7 and 4.9. r

As a consequence, every element of L specializes to the liaison addition of M and

some licci subseheme of X.

As in the case of pn, one can give a nice geometric modification of this "standard form", i.e. we can find in every family s a subscheme which is obtained by M

by adding the "simplest possible" two-codimensional subscheme of X. We follow here

basically the ideas of [5].

Definit ion. A linear configuration L in X is a 1.C.M. subscheme of X whose

irreducible components are of the form Xc~H, where H is a two-codimensional linear

subspaee of pn; that is to say, the irreducible components {Ci} of L are the simplest

(non-void) lieci subschemes of X: they have a locally free resolution of the form

0 ---> OX(-2) --* OX(-1)~ OX(-1) -4 JCi/X --r 0.

A linear configuration is said to be hyperplanar of degree p if it is contained in the

union of p hyperplane sections of X , and this union is a reduced subscheme of X, and

simply hyperplanar if it is hyperplanar in degree tx(L) = min{ t I h0(X,.]L/X(t))•0 }.

Stick-figures are a particular case of linear configurations when X = pn. �9


In [6] it is proved that, given a two-codimensional a.C.M, subschemes C of pn,

then there exists a hyperplanar linear configuration L in pn having the same postulation

(and hence a locally free resolution with the same integers), In other words, given

al<... < ap. 1 and b 1 < ... <: bp satisfying m(bl<b2<...~"bp;al<...<ap.l) > 2, then there

exists a hyperplanar linear configuration L having a locally free resolution P-I j~l 0 ~ �9 Opn(-ai) --~ Ol, n(-bj) ~ .7 L ~ 0. i=l '=

Hence W = L n X is a hyperplanar linear configuration in X (if L is chosen generally

enough) having a locally free resolution P-I j~l 0 ~ i=~l Ox(-ai) "~ .= Ox('bj) -'~ JW/X ~ 0.

In particular,

Proposition 4.11. Every licci subscheme of X specializes to a hyperplanar linear

configuration.. #

With these definitions one has

Proposit ion 4.12. Let L be an even liaison class of two-codimensional

subschemes of X, and suppose that there exists a hyperplanar linear configuration

Me L O. Then every Ye L specializes to a linear configuration.

Proof. Suppose that Y has the same cohomology as Yk where Yk is obtained via

M = Y0:(bl,1) ~ Yl:(b2,1) ~ ........ ~ Yk-l:(bk, 1) ~ Yk,

with b i > bi.l; then Y has the same cohomology as Z, the liaison addition of M with a

suitable licci subscheme W of X. But now W can be chosen to be a hyperplanar linear

configuration, and the divisors to be union of divisors of degree 1. Hence all the

irreducible components of the liaison addition of M and W (chosing things generally) are

intersections of X with two-codimensional linear subspaces of pn. ,

More generally, even if in the minimal shift of L there is no hyperplanar linear

configuration, the same proof shows that every YE L specializes to the union of ME L 0

and complete intersections ( X n Z i n H i ) or ( X n H k n L k ) , where the Zi's are

hypersuffaces of degree a(M) and the Hi's, Hk'S and Lk'S are hyperplanes.

A problem related to the LR-Propcrty is the question of which two-eodimensional

subsehemes of X are the scheme-theoretic intersection of X with a two-codimensional

subscheme of pn. Recall that the licci two-eodimensional subschemes of X are exactly

the scheme-theoretic intersection of X with licci two-codimensional (i.e. aCM)

subschemes of pn. Since basic double links are essentially the union of "arbitrary"

subschemes of X with licci subschemes of X, and since the licci schemes "lift" in this

sense, what about the lifting of non-licci subsehemes and can the LR-Property be used to

study this question? First, do all schemes "lift" to two-r subsehemes of pn?


Certainly not: there exist entire liaison classes which do not contain any such schemes

(e.g. an odd number of points on a quadric surface).

Suppose that an even liaison class has one such scheme; do all elements of that

class have the same property? Not necessarily: let X be p3 viewed as a Gorenstein

subvariety of p4, let M be two skew lines and let Y be a rational quartic. Then M and Y

are evenly linked, but Y lifts whereas M does not. (Recall that we require I.C.M.

equidimensional schemes "upstairs", so a cone will not do.)

So let us assume that we have an even liaison class and an element N which lifts

to pn. (A special case is when N is minimal. Then if N lifts to V, it follows in general

that V is minimal in its even liaison class.) Suppose we perform a basic double link of N

on X, arriving at a scheme Y which is the union of N with a licci scheme A. Even this is

not enough to guarantee that Y will lift, even though N and A do separately. For

example, take X to be a quadric surface in p3 and V to be a general set of 4 lines in p3.

The restriction of V to X is a set N of 8 points, and there is a "quadric" hypersurface Z

of X (that is, the complete intersection of X with a quadric in p3) containing N. Z lifts

to a quadric surface in p3, but not to one containing V. So a basic double link of N

using Z and a general plane restricted to X gives a set of 12 points evenly linked to N,

which does not lift to a curve evenly linked to V. (By the LR-Property for curves in p3,

there is no curve of degree 6 in the even liaison class of V.) Notice that in order to

guarantee that our basic double link will lift, it is enough to choose E so that it lifts to a

hypersurface containing V.

About deformations: suppose that N in X lifts to V in pn. Let Y be a subscheme

of X which is a deformation of N through subschemes all in the same shift of the even

liaison class o f N. Does Y then lift to a scheme which is a deformation of V through

subschemes of pn all in the same shift of the even liaison class of V? The answer is

probably "no," at least in general. The point is that the "liaison" families in pn restrict to

"liaison" families on X, but these restrictions are probably not maximal with respect to

this property.

We would thus like to know whether there is any condition which guarantees that

the answer to the last question is "yes." Further, are there any even liaison classes for

which all elements lift to elements of the same even liaison class of subschemes of pn?

And if Y, N and V arc as above, does Y lift to a scheme which is a deformation of V but

not necessarily in the same even liaison class as V?

5. An example: the Grassmannian G(I,3)

A nice situation where these techniques can help is when X is a hypersurface of

pn; in particular, let us consider the case of X=G=----Gr(I,3). In this case, we. arc dealing

with the classification of (smooth) congruences of lines. This study was completely


carried out, for smooth congruences of degree <8, by Arrondo and Sols [2], and then

continued by several authors.

For the existence part, they give a presentation for every smooth congruence (in

fact, an Nresolution), except for the geometrically ruled congruence of bidegree (3,3),

for which they give an E-resolution. We underline the fact that their approach is hence

liaison-oriented, and that from this point of view we know in general that all the

congruence having the same 9~-resolution from an irreducible, unirational subset of their

Hilbert scheme. All these Nresolutions have the following form

0 ---> POG(-r) ---> E ~ JC/G "--> 0

with E(r) globally generated and rankE---r+l.

The E's appearing in their list belong to 8 different classes of stably equivalent

bundles. If one wants to make their list longer, then one need i) other bundles and ii)

morphism starting from sums of line bundles not necessarily of the same degree.

For this second point, at least for licci subschemes this is completely solved,

thanks to remark 2.5 (note that here for the existence it is sufficient to note the result in

p5, since then one cuts with G: the numerical hypothesis is the same). That is to say, all

possible examples of smooth licci subschemes of G can be chosen of the form GnC, C a

smooth two-codimensional a.C.M, subscheme of p5. These are not the only a.C.M.

subschemes of G, since there are other liaison classes with trivial intermediate

cohomology (for instance, all liaison classes corresponding to direct sum of the spin

bundles). For the other liaison classes, one can use the same technique of w which

makes use of Chang's result, for getting at least partial results. This works particularly

well when in the stable equivalence class there is a vector bundle of rank 2.

For instance, in the even liaison class corresponding to the spin bundle E (we

follow the notation of [2]; see f.i. their w there is a smooth congruence Y having a

presentation 0 ---> K ---> F ~ JY/G "--> 0

where K=OG(-5)~OG(-7)~OG(-7) and F=E(-4)~OG(-4)~OG(-6).

This follows from Chang's theorem, following the ideas of w filter K starting with the

line bundle OG(-5 ) and F with the vector bundle E(-4)~OG(-4).

For a general morphism in Hom(K,F) the degeneracy locus Y is hence smooth,

with bidegree (16,15), but Hom(K,F) is not globally generated.

Note by the way that every congruence in this liaison class specializes to the

union of an tz-plane (an element in the minimal shift) and a certain number of quadrics

(intersection of G with a three-dimensional linear subspace of p5).

R E F E R E N C E S [1] M. Atiyah: On the Krull-Schmidt Theorem with Applicaitons to Sheaves. Bull.

Soc. Math. France 84 (1956), 307-317 [2] E. Arrondo and I. Sols: Classification of smooth congruences of low degree.

J.reine angew. Math. 393 (1989), 199-219


[3] E.Ballico, G.Bolondi and J.C.Migliore: The Lazarsfeld-Rao problem for two- codimensional subschemes of pn. Amer. J. Math. 113 (1991), 117-128

[4] G.Bolondi and J.C.Migliore: The structure of an even liaison class. Trans. Amer. Math. Soc. 316,1 (1989), 1-37

[5] G.Bolondi and J.C.Migliore: Configurations of linear projective subvarieties. In: Algebraic curves and projective geometry, Proceedings Trento 1988, Lecture Notes in Mathematics 1389; Springer-Verlag (1989), pp. 19-32

[6] G.Bolondi and R.M.Mir6-Roig: Deformations of Arithmetically Cohen-Macaulay subvarieties of pn. Manuscr. Math. 64 (1989), 205-211

[7] M.C.Chang: A filtered Bertini-type theorem. J.reine Angew. Math. 397 (1989), 214- 219

[8] L. Chiantini and F. Orecchia: Plane Sections of Arithmetically Normal Curves in p3. In: Algebraic Curves and Projective Geometry, Proceedings Trento 1988, Lecture Notes in Mathematics 1389, Springer-Verlag (1989), pp.32-42

[9] C.Ciliberto, A.V.Geramita and F.Orecchia: Remarks on a thereom of Hilbert- Burch Boll. U.M.I. (7), 2-B (1988), 463-483

[10] A.V.Geramita and J.C.Migliore: Hyperplane sections of a smooth curve in p3. Comm. in Alg. 17 (1989), 3129-3164

[ 11] A.V.Geramita and J.C.Migliore: Generalized Liaison Addition (preprint) [12] L.Gruson and Chr. Peskine: Genre des courbes de l'espace projectif (1). In:

Algebraic Geometry, Proceedings Tromsr pp.31-59. Springer Lect. Notes in Mat. 687; Springer

[13] R.Hartshorne and Ogus: On the factoriality of local rings of small embedding codimension. Comm. in Alg. (1974)

[14] R.Lazarsfeld and A.P.Rao: Linkage of general curves of large degree. In : Algebraic Geometry-Open problems. Proceedings Ravello 1982, pp.267-289 Springer Lect. Notes in Math, 997; Springer

[15] J.C.Migliore: Geometric invariants for liaison of space curves. J. Algebra 99-2 (1986), 548-572

[16] J.C.Migliore: Liaison of a union of skew lines in p4. Pac. J. Math. 130-1 (1987), 153-170

[17] M.Martin-Deschamps and D.Perrin: Sur la classification des courbes gauches. Astdrisque, 184-185

[18] S.MacLane: Homology. Springer-Velag, Berlin 1963 [19] R.Maggioni and A.Ragusa: The Hilbert function of generic plane sections of

curves in p3. Inv. Math. 91 (1988), 253-258 [20] Ch. Peskine and L. Szpiro: Liaison des varidtds algdbriques, I, Invent. Math. 26

(1974), 271-302 [21] A.P. Rao: L~aison among Curves in p3, Invent. Math. 50 (1979), 205-217 [22] A.P.Rao: Liaison equivalence classes. Math. Ann. 258 (1981) 169-173 [23] A.P.Rao: Notes of a seminar held in Catania, by S. Giuffrida and R. Maggioni. [24] T.Sauer: Smoothing projectively Cohen-Macaulay space curves. Math. Ann.

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University (1982)

Giorgio Bolondi- Dip. di Matematica- Univ. di Trento- 1 38050 POVO (Trento) Italy Juan Carlos Migliore- Dept. of Maths- Univ. of Notre Dame- IN 46556 USA

(Received January 23, 1992; in revised form November 17, 1992)

The lazarsfeld-rao property on an arithmetically gorenstein variety

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