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

ACGA - 1990

P. Maroscia - U. Nagel - W. Vogel

ON D E G R E E B O U N D S FOR THE SYZYGIES OF A FINITE SET OF P O I N T S IN IP"

A Paolo Salmon per il suo 60° compleanno

A b s t r a c t . In this paper we prove some new degree bounds for the syzygies of

certain classes of finite sets of points in F n and we describe the corresponding

graded minimal free resolutions. In particular, a crucial role in played by some

new invariants attached to a finite set of points in P n .

Introduction

There is apparently a great interest for the (graded) minimal free resolutions of finite sets of points and projective varieties (see, e.g., [2], [3], [4], [5], [7]>'[8],.[10], [14], [15]). The aim of this paper is to describe the (graded) minimal free resolutions of certain classes of finite sets of points in IP77, (see, in particular, Theorems 3.3, 3.4 and Proposition 3.5). In some sense, the heart of the paper is our Main Lemma 1.5, proved in Section 1, which gives some vanishing results concerning the graded R-module Torp(K,R:/L(X)), where I(X) C R := •K[Xoy.'..,)Xn] is the defining ideal of a finite set of points X spanning P"v.

Actually, this lemma enables us to prove some new degree bounds for the syzygies of finite sets of points in P n , which is done in Sections 2, 3. Our results also improve and extend some statements given, for example, in [10], [5], [7], [23], [24], [19]. Another key-result is Lemma 2.6, first proved in [19], which provides (under certain hypotheses) some new invariants attached to a finite set of points in P77, (see [19, Remark]). The paper ends with various

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examples and remarks, developed in Section 4. We denote by X a finite set of distinct points in IPn = IP -, with n > 2

and K any algebraically closed field, by |X| the cardinality of X (we will assume |X| > n -f 1) and by A = R/I(X) the homogeneous coordinate ring of X, where I(X) is the defining ideal of X in R := K[Xo,... ,Xn]. Also we denote by {h>x(t)}t>o the Hilbert function {rankj{At}i>o, a n ( i by r(X) the index of regularity of X, that is (see, e.g., [21, Definition 0.2.2]), r(X) = min{/ G N/hx(t) = \X\}. Finally, if M = 0 Afj is a graded module,

we denote by [M]{ the i-th graded piece of M, i.e., [M]i = M{ and we write M(j) for the graded module with [M(j)]{ = [M]i+j. Throughout the paper we adopt the convention that the binomial coefficient (^) = 0 if a < b or 6 < 0, and (°) = 1.

1. The Main Lemma

We start with the following definition:

DEFINITION 1.1. Let X be a finite set of points in ¥n and let d,p be positive integers with 1 < p < n. Then we will say that X satisfies [Njp], if [Torf(K, A)]j = 0 for all j>d+ i and i = 1 , . . . ,p.

REMARK 1.2. Our condition [NdiP] can be viewed as a generalization of the condition (N^p) stated in [5], which in turn extends property (Np) first introduced in [10]. It says that the i-th syzygies of the ideal I(X) are generated by elements of degree < d + i, for all i = 0 , . . . ,p — 1. In particular, a set X satisfies [Nj x] if and only if the ideal I(X) is generated by forms of degree < d. We also note that, if p > 2, then [N^^] implies [^>J!,_1].

We first state a simple characterization of property [Ndn].

LEMMA 1.3. With the notation as above, we get that X satisfies [iV^n] if and only if hgld - 1) = \X\.

Proof. We have the following relation (see, e.g., [21, Lemma 1.4.3]):

\X\-hx{t) = rankK[H},{A)]t,

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where Hp(A) denotes the first local cohomology module of A with support in the irrelevant maximal ideal P oi A. Then our statement follows from [4, Proposition (a), (b)].

Some useful characterizations of property [N:£p] are given by the following result which seems to be well known, since our graded K-algebra A is Cohen-Macaulay. For completeness' sake, we give a short proof.

LEMMA 1.4. Let d,p be positive integers with 1 < p < n. Then the

following are equivalent for a finite set of points X C IPW:

. (i) X satifies [NdfP];_

(ii) [Tor*(K, A)]j = 0 for all j>d + p.

Moreover, if hx(d) = |X| , then both (i) and (ii) are equivalent to the

following condition:

(Hi) [Tor*(K, A)]j = 0forj = d + p.

Proof. Consider a minimal free resolution of A, say:

O->0 R(-enj) -* , . . - > e R(-eXj) -> R -+ A -> 0 . 3 3

Since A is Cohen-Macaulay, we have: ExtlR(A,R) — 0 for i < n.

Therefore, applying the functor Hom.R( — ,R), we get the following minimal free resolution:

0 -> R ->© R(eij) -> . . . - ^ 0 R(enj) -> ExtnR(A,R) -+ 0 ;

3 3

hence, by construction, we have: max{e^} > max{ei_ij}. Now, since j j

rank/([Torf(K,A)]i is equal to the number of the e^-'s with e\j = t (see, e.g., [9, Theorem l.b.4]), we get the equivalence between (i) and (ii).

Finally, it follows from Lemma 1.3 that (iii) implies (ii), and so we are done.

The next result will play a central role in the whole paper.

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MAIN LEMMA 1.5. Let X C IPn be a finite set of points spanning Wn

and let d be an integer such that hx(d) = \X\.

Suppose that there exist a subset X\ of X consisting of n -f 1 points spanning ¥n and an integer p (1 < p < n) such that:

(*) hX\u(d ~1) = \X\ -in~P) , {or aU subsets U of'Xx with \U\ = n - p.

Then X satisfies [N^p],

Proof. In view of Lemma 1.4, it is enough to show that

[Totf(K,A)]i+p = 0.

Now, if we denote by K := K(£ i , . . . , Ai+i; R) the Koszul complex of R with respect to the regular sequence £i,..., tn+\ of linear forms of R, which provides a, minimal free resolution of K as an i2-module, we can compute Torp(K,A) by means of the p-th homology module of K (x) A. Thus we get \Torp{K,A)](i+v — 0 if and only if the following complex

A " + 1 ( i J " + 1 ) ( - p - 1)® [ A ] ^ ! -4+ A"(J i" + , ) ( -p) .® [A]d

JUA"-\Rn+1)(-p+l)®lA]d+1

is exact in the middle.

If we set X2 := X\Xi and A* := R/I(Xi). for i = 1,2, we get: hx(d) =.\X\,h-x1(d)-= \X\\ (in view of the assumptions on X and. Xi) and hx2{d) — \X2\ by (*)); therefore the monomorphisms [A]t <—• [A1^ 0 [A2]t are isomorphisms for all t > (/.Hence the homomorphism b in (1) breaks up into a direct sum b = b\ 0-&2, with the natural maps:

bi : A ' ( ^ + 1 ) ( - p ) ® [A{]d —» A'-\Rn+1)(-P + 1) ® [ % , ,

for i. = 1,2. On the other hand, we have the following homomorphisms, for i = l , 2 :

a,- : A ' + 1 ( i 2 n + 1 ) ( - p . - 1 ) ® [4]a-i —* A p ( £ n + 1 ) ( - p ) ® [A'-]rf .

Hence we can consider the following sequence:

AP+l{Rn+l)(_p _ 1 } 0 [ 4 ] ^ ^ L A»»(fl»+1)(-p) ® [/I2],,

-^A'-HJZ^K-P+I)®^]^! ,

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which is exact, because X2 satisfies [iVj n ] , in view of (*) and Lemma 1.3.

It follows that (1) is exact, if and only if, for any /?. G A'er(&i), there is an element a G Ap^1(Rn+1)(-p -. 1) ® [A]d-i such that a(a) = /? and a2(a): = -'0.

Now we put Xi = { P i , . . . , P n + i } and fix a basis { e i , . . . ,en-fi} of i 2 n + 1 . Also, we may assume that ii(Pj) = 8ij (Kronecker's symbols) for i,j = 1 , . . . , n + 1. Moreover, since hx^d) — n -f 1, there exist n -f 1 elements of [A1]^, say ( /1, . . . , ^ „ + i , such that qi(Pj) = &ij. Hence ker(b\) is generated by elements of the form

eh A - • AejP ® 9..J where i g { j i , . . . , j p } •

In the following we fix such an element, say /3. Then, in view of our assumption (*), we can choose a form P of degree d— 1 vanishing on X2 and the p points P ^ , . . . , P j p , but not at P t . Also, we may assume F(P{) = 1; hence, we get l{F(Pi) = 1 and £{F(P) = 0 for all P G X \ { P ; } , and ^ P ( P ) = 0 for all P G X and rG { j i , . . . , i p } .

We set a := e^j A . . . A €j A e; ® P , where P is the residue class of P in

A. It follows that ft(a) — (—l)2+1ey1 A . . . A e^ ® ^ P , and 02(a) = 0 because P vanishes on Ar

2. Also, since 9 1 , . . . , qn+i form a basis of the A'-vector space '[A1]^ and q.s(Pj) = Ssj for all s , j = 1 , . . . , n -f 1, we get that IjF module I(Xi) is equal to q^ which shows that (1) is exact.

This completes the proof of the Main Lemma.

R E M A R K 1.6. (i) With reference to the Main Lemma above, we point out that property [Ndp] (with 1 < p < n — 1) does not imply, in general, the existence of a subset Xi of X satisfying condition (*) for the same p (see Example 4.1).

(ii) It is worth observing that condition (*) in the Main Lemma implies

that hx\xAd ~ 1) — 1^1 ~ (n + !)• Yet> the converse is not true in general.

We will end this section by proving a few general consequences of the Main Lemma. First we need an elementary observation.

LEMMA 1.7. Let X C IPW be a finite set of points cut out by hypersurfaces of degree < t (i.e. such that the ideal I(X) is generated by forms of degree < t), and let P be a point of¥n outside X. Then we have:

.hxu{py(t) = hx(t).+ i-

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Proof. Suppose not; then ^ Y U { P } ( 0 = hx(i) for i = 0,l,. . . ,tf. It follows that all forms in the ideal I(X) also vanish at P : contradiction.

T H E O R E M 1.8. Let X C^n be a set of s + n - p points spanning P n , with s > 72,1 < p < n, and suppose that there is an integer d > 1 such that

(**) hv(d -1) = s, for all subsets V of X with \V\ = s .

Then we have:

(i) hx(d) = \X\;

(ii) X satisfies [N^p],

Proof. We proceed by descending induction on p(\ < p < n) in order to prove (i), the case p — n being clear, in view of Lemma 1.3. Let p < n and choose a point P e A" such that Y = A ' \{P} spans P n . By the induction hypothesis, we get: hy{d) =]Y\ = \X\ — 1, which implies (by Main Lemma •1.5) that Y satisfies [^(/) /)+1], hence Y is cut out by hypersurfaces of degree < d. This shows (i), in view of Lemma 1.7. At this point, (ii) follows from the Main Lemma 1.5 and we are done.

COROLLARY 1.9. Let X C P n be a finite set of points spanning P n . Suppose that X contains a subset X\ of n -\-l points spanning Fn such that, for some integer d > 1:

hX\{P](d - 1) = hx(d-1) = \X\ - I , forallPeXi.

Then X satisfies [N^n_i].

Proof. In view of our assumptions, we get from Lemma 1.7 that hx(d) = \X\. Now, our assertion follows from Main Lemma 1.5.

Finally, we want to note explicitly that the following useful fact is also true.

PROPOSITION 1.10. Let X C Pn be a finite set of points spanning ¥n. Then we have:

i •IT' -Rd' V/TSVWI J 0 ' forj>r(X) + n

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Proof. The result simply follows from the following equality:

rankK[Tor*(K, R/I(X))]t+n = M O - M * ~ 1) ,

which is a consequence of the general fact:

n

An^hx(t) = ^ ( - l ) ^ , where ait = rankK[Tdr^(K,R/I(X))]t .

2. Some consequences for certain classes of sets of points

(A) P O I N T S IN GENERAL POSITION

First we recall that a finite set of points X of F n is said to be in general position, if it spans ¥n and n o n + 1 points of X lie on a hyperplane. Then, the properties of such a set X , described in the next statement, are well known (see, e.g., [11], [17] or [1]):

LEMMA 2 .1 . Let X C IP71 be a finite set of points in general position.

Then, for any t > 0, we get:

(i) hx(t) > m i n { | X | , i n + l } ,

(ii) hx(t + l)>miii{\X\,hx(t) + n}.

Our first result, which extends [10, Theorem 1] and also improves [23, Theorem 1.1], easily follows from Theorem 1.8.

PROPOSITION 2.2. Let X C F" be a set ofdn-\-l-p points in general position ((/•> 2,1 < p < n). Then X satisfies [N^p], hence, in particular, the ideal I(X) is generated by forms of degree < d.

Proof. If we put s — (d - l)n + 1, then it follows from Lemma 2.1 that hy(d — 1) = s for all subsets V of X with |V| = s. Now apply Theorem 1.8.

Our next statement, which is an immediate consequence of the Main Lemma 1.5, was first noticed in [22] and essentially proved in [18]. However both proofs used [23, Theorem 1.1].

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PROPOSITION 2.3. Let X he a finite set of points in general position in Pn and suppose that X contains a subset of n -f 1 points, say X\ = {Pi , . . . , Pn+i}, such that for some integer d > 1:

h(X\Xi)u{PitPj}(d - 1) = 1*1 " O ~ 1) i

for all i,j with 1 < t < j < n -f 1. Then the ideal I(X) is generated by forms of degree < d.

(B) POINTS IN UNIFORM POSITION

We recall that a finite set of points X of Pn is said to be in uniform position (in Harris' sense, see, e.g., [13]), if it spans Pn and for all subsets U of X we have:

hv(t) = mm{\U\,hx{t)} , for all t > 0 .

Clearly, any set of points in uniform position is also in general position (the converse being false, in general).

Our first result on points in uniform position extends and improves [18, Corollary 2.2] and [20, Lemma 3]:

THEOREM 2.4. Let X C Pn he a finite set of points in uniform position and let d he an integer > 1 such that \X\ < hx(d— l)-fn— 1. Then X satisfies [Nd,p] where p = hx(d - 1) + n — \X\, and

for j = d. -f- n rani »••---"'•" ^M,.,•^^^t . . . ,kK[Tor^K,RII{X))]j=[^-h^d-1^

for j > d+ n

Proof. First we note that 1 < p < n. Let V C X be a set of \X\ — (n — p) points. Then, since X is in uniform position, we get:

hu{d- 1) = min{|X| - (n-p),hx(d- 1)} = \X\ - (n - p).

New we can apply Proposition 1.10 and so we are done.

Next we prove a theorem which extends [24, Theorem 1.1].

First we introduce some notation. Let V be a non-degenerate, irreducible and reduced subvariety of P ^ of degree s and codimension n. Let

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I C F n denote the general n-plane section of V", with Hilbert function, say hx(t) = : at(t > 0). Then we define the following integers > 0, for any t > 1:

mt := n

[at

i i . s — n

= the largest integer < at-IJ

at := min{i > 0 : s — n — ra<(a< — 1) < az} ,

8t := m<.* -f a< + 1 and p< := mt(at - 1) + aat + ^ - 5.

T H E O R E M 2.5. With the notation as above, let V be a non-degenerate, irreducible and reduced subva.riety of P ^ of degree s and codimension n. Assume that V is arithmetically Cohen-Macaulay and char(K) = 0.

Then the general n-plane section XofV satisfies property [N$hpt]. In particular, V is cut out by hypersurfaces of degree < St for all t > 1.

Proof In view of the well-known Uniform Position Lemma (see, e.g., [12] or [1]), the general n-plane section X of V is in uniform position. Then, applying the relation (see [13]):

hx(t1+t2)>'miiL{\Xlhx(t1) + hx(t2)-l},

we obtain:

hx(St - 1) = hx(m.t.t + at) > mt(at - 1) + a,at > s - n.,

whence the conclusion, by applying Theorem 2.4.

(C) F I N I T E SETS OF POINTS X WITH hx(d- 1) = \X\ - 1.

We start with a "key-result" proved in [19].

LEMMA 2.6, Let X be a Unite set of points in ¥n and let t be any positive integer. We set hx(t)=: \X\ - 8(6 >0).

If there is a subset X't =: X' C X such that:

(i) hxf(t) = \X'\ -6, and

(ii) hy(t) = \V\, for all subsets V of X' with \V\ = \X'\ - 6,

then X' is unique.

Moreover, if 6 = 1, then such a set X1 always exists (which is not true, in general, if 8 > 2).

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REMARK 2.7. With reference to Lemma 2,6, we point out that, if 6 — 1 and p denotes the dimension of the linear subspace of P n spanned by Ar/, then we get: p •== 1 if and only if X sontains / -f 2 collinear points.

The next result is inspired by some work done in [14]:

T H E O R E M 2.8. With-the notation of Lemma 2.6 above, let X be a finite set of points spanning Wn, with hx{d — 1) = \X\ - 1 for some integer d > 1, and let p be the dimension of the linear subspace of¥n spanned by X'•= X'd_r Then we get:

R (0, for j > d+i rankxlTori (K, R/I(X))]j = ^ ,n-p^ fof j = d + i ' where 1 ^ i ^ n -

In particular, X does not satisfy [Njp]; yet, for p > 1, X satisfies

Proof. It is clear that, under our hypotheses on X , we get: hx(d) = \X\. Thus, in view of Lemma 1.3, we have only to show that :

(***) rankK[Tor?(K, R/I(X))]d+i = ( £ > ) .

In order to prove that, we proceed by induction on ra := n — p > 0. We distinguish two cases:

(I) m =. 0. Now we induct on |A'\A~'| > 0, the case X1 = X being clear, in view of Theorem 1.8 and Proposition 1.10. Let A""' C X. Hence

there exists a subset of \X\ — 1 points of A , say W — A^\{P}, such that h\y(d — 1) = hx(d — 1) — 1. Therefore we obtain:

I(W)/I(X) ~ (I(W) + I(P))/I(P) ~ R/I(P)(-t) where t < d - 1 ,

which yields the exact sequence:

0 -+ R/I(P)(-t) -* R/I(X) - • R/I(W) -> 0 .

Hence, from the long exact sequence (for i >. 1):

. . . -> Tor?(K, R/I(P))(-t) - • Tor?(K,R/I(X))

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-+ Torf(K, R/I(W)) -> Toif_x(K, R/I(P))(-t) - > . . . , we get (***) ,

because Torf(K,R/I(P))(-t) ~ /«'(«)(-* - <), and

r a n ^ i r o r ^ J S T ^ / J C l ^ ) ) ] ^ . ^ ( - ° J by induction.

(II) 7?i = n — p > 0. Let V be a subset of X containing X1 and spanning a hyperplane of ¥n defined, say, by a linear form £, and moreover such that hv(d - 1) = |V| - 1*. We put S = R/tR. Since t • R/I(V) = 0, it follows from [14, Lemma 1 (ii)] that:

Tor?(K, R/I(V)) * Torf(K, R/I(V)) © Torf^K, R/I(V))(-1).

Then the induction hypothesis on m gives:

- w * * « - - CT.';*)+(;:;:0 - (;:;) • Now, if we proceed by induction on | X \ V | > 1, in a similar way as in

case (1) above, we get (***) again, which completes the proof of our Theorem.

COROLLARY 2.9. Let X be a. finite set of points spanning Fw such that hx(d-l) — \X\ — l for some integer d > 1. Then X is cut out byhypersurfaces of degree < d, if and only if, no subset of d -f 1 points of X lies on a line.

3 . The case of points in generic position

We will give here some applications of the results stated in the previous sections to sets of points in generic position.

First we recall that a finite set of points X C Fn is said to be in generic position, if X has the following Hilbert function:

hx(t) = min {|X|, T + J } , for all t > 0 .

We start with a result first proved in [5], which also follows from Theorem 1.8:

*Such a set V certainly exists. In fact, it is enough to adjoin to X , each at a time, n—l—p points of F n , in such a way that the dimension of the subspace spanned at each step increases exactly by one.

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PROPOSITION 3.1. Let X c IPn be a, set of (d *+ n ) + n-'p points (d > 2,1 < p < n) such that:

hv(d - 1) = (rf~*+n) for all subsets V C X of (rf~*+?l) points.

Then we get:

(i) X is in generic position;

(ii) [Torf{K, R/I(X))]j = 0 for a/1 i = 1 , . . . ,p and j ^ d - 1 + %.

Before stating the main result of this section, we need some general observations.

Let X C IPn be a finite set of points in generic position and let d be the integer (> 2) such that ( r f~^n) < |X| < (d+n)- Then, it follows from Lemma 1.3 and [6, Proposition 1.1] (see also [15]), that X has a minimal free resolution of the form:

0 '-> R0n(-d-n)eRan(-d-n+l) -> ' . . . -> ^ ( - ^ - l ) © ^ 1 (-d) -+ R -> A -+ 0

Also, we recall that the integer an -\- /3n gives precisely the Cohen-Macaulay type of X (see, e.g., [15, p. 112]). Moreover, we get from Proposition 1.10:

A.-W-CT"). On the other hand, using the dual resolution of X , the Hilbert function

of the canonical module of AT and Lemma 1.4 (see also [4, Theorem A.l]), we get the following result:

PROPOSITION 3.2. With the notation as above, X satisfies [J/V£/n_1] if

and only if an =(d~lY) - n - [3n.

We now turn to the proof of the result announced above, already stated in [19].

T H E O R E M 3.3. Let X C Pw be a set of (rf~^+w) + 1 points in generic position, with n,d > 2, and let p be the dimension of the linear subspace of

spanned by the subset X' = Xld_1 (defined in Lemma 2.6 above). w

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Then X has a minimal free resolution of the form:

0 - • R?n(-d - n) 0 RQn(-d - n + 1) - • . . . -> Rfr(-d-p)@Rai,(i-d - p + 1)

-+ RaP-l(-d -p+ 2 ) . -* . . . . -4 # a i ( - < 0 "+ * -> # / * ( * ) "^ 0 ,

where, for i = 1 , . . . , n:

d - 1 + n\ (i - 2 + 1 \d-\\% V rf- iO-G-0+C-;-0-*-C-";)-

Proof First of all, since X is in generic position, we have, in view of [16, Proposition 1.1] and Lemma 1.3 above:

[Tor?(K,A)]j = 0 fo r j ^ d + i - 1, d+i.

Also, it follows from Theorem 2.8 that:

Pi = rankK[Torf(K,A)]dfi = ("Zj ' f o r X ^ { ^ n '

Finally, the cv 's are easily computed, by using the llilbert function of X , and so we are done.

The next result extends [19, Corollary 1].

T H E O R E M 3.4. Let X be a set of (d~n+n) + 1 points in generic position in Wn(n,d > 2). Consider the following conditions:

(i) X has a minimal free resolution of the type given in Theorem 3.3 above,

with p — n;

(ii) no d -f 1 points of X lie on a hyperplane;

(Hi) there exists a subset X\ C X of n + l points spanning IP" such that for all P G X\. we have:

hX\{P}(d~1) = 1 * 1 - 1 -

Then (ii) => (i), the converse being true, in general, only for n = 2,

and (Hi) =>• (/).

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Proof, (ii) ==>> (i) Suppose not; then, in view of Theorem 3.3 above, the set X' = X'd_1 defined in Lemma 2.6 spans a linear subspace IP C IPn.

Since hx'(d - 1) > d, we get |A"'| > d + 1: contradiction. The converse is clearly true for n = 2 (see Remark 2.7); yet, it is false, in general, for n > 2 (see, e.g. Example 4.5).

( in) =>. (z) Corollary 1.9 gives the values of the fa's. Hence the a.;'s can be easily computed, since X is in generic position, which completes the proof of the theorem.

We end this section by proving a statement which extends a known result (see, e.g., [19, Proposition] or [7]) and moreover provides a partial solution to the Problem posed in [19].

PROPOSITION 3.5. Let X C Wn(n > 2) be a set of n + 3 points in generic position. Suppose that for n > 3 there exists the subset X! = A { (defined in Lemma, 2.6).

Then X has a minimal free resolution of the form:

0 -> tfin(-n— 2) © Ran(-n - 1) - . . . -+ R^~l{-p - 1) © R^-^-p)

• - • RaP-2(-p + 1) -> • • • - • ,R a i ( -2 ) -* £ -+ A -> 0 ,

where

_ J2 , for.n = 2 P ~ \ max{/?.V/(l) - 1,2} , for n > 3 '

and for i = . 1 , . . . , n :

A-(;:;+>fr-»(i:,\'l;

and

—'•(::. ,)-'-C-0+9-("7-; l)+"-<-;;

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Proof. First of all, we may assume n > 3 (the case n = 2 is easy, see, e.g., [15, III - Corollary 3.1]). Also we note that it is enough to compute the /Vs, since then the o^'s can be easily obtained by using the Hilbert function ofX.

We will proceed by induction on the integer n — p > 0:

(I) n— p — 0. This means that X' = X or equivalently X' is in general position. Hence, by Proposition 2.2, X satisfies [^2,71-2]- Now choose a point P e X and put Y = A ' \{P} . Since hx{2) = hy(2) + 1, we get:

I(Y)/I(X) ~ (/(y).+ I(P))/I(P) * R/I(P)(-2),

which gives the exact sequence:

. • 0 -+ R/I(P)(-2).-+ R/I(X) -> R/I(Y) -> 0 .

Then, if we write the long exact sequence (as in case (I) of the proof of Theorem 2.8) and the minimal free resolution of Y provided by Theorem 3.3, we are done, since X satisfies [iV"2,n-2]'

(II) n — p > 0. In this case, we choose a point P £ X such that Y := X\{P} contains X'; therefore Y is contained in a hyperplane of Wn

defined by a linear form, say £, We put: S •= RjtR. Since I • R/I(Y) = 0, it follows from [14, Lemma 1 (ii)] that

Tor?(K,R/I(Y))xTorf(k,R/I(Y))QT«r?_1(K,R/I(Y))(-l):

By induction hypothesis, we get for i = 1 , . . . , n:

r.ntK\r°r*(K,It/l(r)U>'l-(il~'^+(r-vfc'+l)

«•(;:;) ^ - < 7 - ' ; ' ) -'-(i:;:,1)+"-«>G-*;+'.) • Since hx(l) = hy(l) + 1, we obtain (as in case (I) above) the following

exact sequence:

0 -+ R/I(P)(-1) -> R/I(X) -H. R/I(Y) -> 0 .

Then the long exact sequence gives for i — 1 , . . . , n:

Pi = rankK[Tor?(K,R/I(X))]i+3 = rankK[Torf-(K,RII{Y))}i+2,

and so we are done.

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REMARK 3.6. (i) With reference to Proposition 3.5, we note that a set X of n -f 3 points in generic position in W1 contains a subset Xf = X[, if and only if, for some q{\ < q < n) X contains a subset Z of q-\- 3 points in general position in ¥g(l < q < n), in which case we get: X' = Z and q = hx'(l) — 1.

(ii) It is worth observing that the technique used in case (I) of the proof of Proposition 3.5 above can be used to get a simple alternative proof of [25, Proposition].

4. Miscellaneous examples and remarks

We start with an example which shows that the converse of our Main Lemma, 1.5 is not true in general (see Remark 1.6 (i)).

EXAMPLE .4 .1. Consider a "general" set A' of points in generic position

spanning Fn' with \X\ = (rf~*+n) + .2 (n,d > 2) and 2 • n < ( n ' j l7 2 ) . Then

A satisfies [Nj n _ i ] , but there is no subset X\. C X of n -f 1 points satisfying

condition (*) of Lemma 1.5 for p = n — 1.

In fact, we get: /3n — 2 (by Proposition 1.10) and an = max{0, ( n +l" 2 ) - n • /?„} (by [25, Theorem]).

Hence, in view of Proposition 3.2, X satisfies [Nj n _x] . Now, in order to prove our assertion, it is enough to observe that, for any point P 6 A , we have: hx\{p}(d — 1) < \X\ — 1, and also hx(d) = |A| .

Our next example shows, with reference to Main Lemma 1.5 again, that if there exists a subset Ai C X satisfying condition (*), then in general the same is not true for any subset of X consisting of n + 1 points spanning IPn.

EXAMPLE 4.2. Let A = {Pi,...,P7} be a set of 7 points in general position in F2 , with P j , . . . , P 6 lying on a conic C and Pj £ C. Then one easily checks that condition (*), with d — 3 and p — 1, is certainly satisfied by the subset X\ = { P i , P 2 , P 3 } , but not by the subset Y\ = {P i ,P2 ,P 7 } . (In any case, A satisfies [A^i], which could also be obtained from Corollary 2.9).

However, we have the following result:

THEOREM 4 .3 . Let X C Fn he a finite set of points spanning W>n such that hx(d — 1) — |J\T| — 1 for some integer d > 1. Then the following are equivalent:

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(i) X satisfies [N^^];

(ii) X' = X'd_x (in the sense of Lemma 2.6) spans P n ;

(Hi) there exists a subset Xi ofXofn + 1 points spanning ¥n such that

(*). hX\{P}(d- 1) = W ~ I for »# P€XL

Proof, (i) =$> (ii) follows from Theorem 2.8. Also, (ii) => (Hi) since, by our hypotheses on X', there exists a subset X\ of X of n-f-1 points spanning ¥n such that :

• hX'\{p}(d-1}=IX'\-1 for all P £ X1 r

which implies (*). Finally, (in) =>• (?') in view of Main Lemma 1.5.

We now give an example which shows, with reference to Theorem 3.3, that all possible values for the "invariant" p actually occur.

EXAMPLE 4.4. Consider the Hilbert function of 11 points in generic position in F3 :

(**) 1 4 10 11 . . . .

Hence, the only admissible values for p are 1,2,3:

(i) p = 1. Let P i , . . . ,.7*7 be distinct points lying on a plane ir in IP3 such that P i , . . . , P4 lie on a line I and P5, PQ, P7 ^ I and moreover are not collinear. Then, if Q\,..., Q4 are four points in general position in IP3

outside 7r, one easily checks that the set X = { P i , . . . , P7, Q\,..., QA) has the Hilbert function (**) with X' — { P i , . . . , P i } , whence our claim.

(ii) p = 2. We proceed as in case (i) above, the only difference given by the choice of P i , . . . , P Q on a nonsingular conic, say C, lying in a plane 7T, with P7 G TT\C. Then the set X = {Pu...,P7yQi,...,Q4} has the Hilbert function (**), with X' = { P i , . . . ,Pe} , which shows our claim.

(iii) p =• 3. Let P i , . . . , P10 be distinct points on a rational quartic curve in P3 , say C, and let Q be a point of P 3 \C . Then the set X = { P i , . . . , P10, Q} has the Hilbert function (**), with XI = { P i , . . . , P10} a n d so we are done.

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Example 4.4 also suggests a general construction of a set X of ( i + n ) + 1 points in generic position in Pn with any preassigned value for the "invariant" p considered in Theorem 3.3.

We can proceed as follows: start with a set Y of ( ~l^p) + 1 points of F n (with d > 2 and 1 < p < n) and assume that Y is in generic and uniform position in a linear subspace of IPn of dimension p; then, add to Y, one at a time, |A"| — \Y\ points of IPn in such a way that at each step the value of the Hilbert function in degree d — 1 increases exactly by 1.

The last example shows, with reference to Theorem 3.4, that (i) does not imply (ii), for n > 2.

EXAMPLE 4.5. Let P i , . . . , P s be distinct points on a twisted cubic in F3 . Then choose three more points Qi,Q2,Q3 in F 3 such that:

(a) Q\ lies in the plane spanned by Pi,P2?^3?

(b) there is exactly one quadric surface, say F, passing through P i , . . . ,P 8 ,Q 1 ,Q 2 ;

(c) Qs t F. Now it is clear that the set A' = { P i , . . . ,P8,Qi5Q2?Q3} is in generic

position in F3 and also X,:(~= A" ) = '{Pi,.... ,Ps}- Hence X satisfies condition (i) of Theorem 3.4, but it does not satisfy condition (ii), in view of property (a) above.

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Paolo MAROSCIA, Dip. Metodi e Modelli Matematici, Universita di Roma "La Sapienza",

Via A.Scarpa, 10, 1-00161 ROMA, Italy.

Uwe NAGEL, Gesamthochschule Paderborn, FB Mathematik und Informatik,

W-4790 Paderborn, FRG.

Wolfgang VOGEL, Martin-Luther-Universitat, Sektion Mathematik,

0-4010 Halle, FRG.

Lavoro pervenuto in redazione il 3.12.1990.