Higher Infinitesimal Neighbourhoods

205, 460]479 Ž1998. JA977393

JOURNAL OF ALGEBRA ARTICLE NO.

Higher Infinitesimal Neighbourhoods Karen A. Chandler Department of Mathematics, Uni¨ ersity of Notre Dame, Mail Distribution Center, Notre Dame, Indiana 46556-5683 E-mail: [email protected] Communicated by D. A. Buchsbaum Received February 17, 1997

We study the Hilbert function of the kth order neighbourhood of a collection G of points in linearly general position in P n , written hP nŽ G k , m.. Our main results are the following bounds on hP nŽ G k , k q 1.: If deg G G 2 n y k q 1 then hP nŽ G k , k q 1. G hP nŽ C k , k q 1., where C ; P n is a rational normal curve of degree n. If G is sufficiently general and s s deg G then hP n Ž G k , k q 1 .

¡Ý Ž s

~

G

js0

¢ž

y1 .

j

s jq1

ž /ž

nqky1y2j , n

/

nqkq1 , n

if s F if s G

/

and equality holds when k s 3.

Ž n q k q 1 .Ž n q k . 2 Ž k y 1 .Ž k q 1 . n Ž n y k q 2. 2k

q k y 1,

Q 1998 Academic Press

1. INTRODUCTION A Žfat. point of order k supported at the point p g P n is the scheme whose homogeneous ideal, I Ž p . k , consists of polynomials all of whose partial derivatives of orders F k y 1 vanish at p. Thus its degree is Ž n qk yk y1 1 .. We call this the kth order neighbourhood of p, or the Ž k y 1.th infinitesimal neighbourhood of p. If X ; P n we shall denote by X k its kth order neighbourhood in P n. In case X ; P ny 1 we will write X k < P ny 1 for its kth order neighbourhood in the hyperplane. 460 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

HIGHER INFINITESIMAL NEIGHBOURHOODS

461

We wish to consider the Hilbert function of a collection G k of fat points in P n, written hP n Ž G k , m . s dim H 0 Ž OP n Ž m . . y dim H 0 Ž I Gk m OP n Ž m . . . If X ; P n is a zero-dimensional subscheme we shall loosely refer to hP n Ž X, m. as the ‘‘number of conditions imposed by X on m-ics’’ and say that X is m-independent if it imposes deg X conditions on m-ics. We start by discussing the conjectured Hilbert functions, both in the case where G consists of sufficiently general points, and where G is an arbitrary collection of points in linearly general position. General Collections of Points. It is easy to see that the Hilbert function of a general collection G of s Žreduced. points is hP n Ž G, m. s minŽ s, Ž n qm m ..: given m and given s y 1 points, choose a point that is not in the zero locus of the m-ics through the given points. A first guess for the Hilbert function of a collection G k of s fat points of order k might be that hP n Ž G k , m. s minŽdeg G k s sŽ n qk yk y1 1 ., Ž n qm m ... This would say that a general collection of fat points either imposes independent conditions on m-ics or else does not lie on any m-ic. The following conjecture asserts that this should be true provided that s is large enough: Conjecture 1 ŽHirschowitz.. For each n G 1 there is a number cŽ n. so that for any s G cŽ n. and for any k, there is a collection G of s points in P n so that for all m, hP n Ž G k , m. s minŽ sŽ n qk yk y1 1 ., Ž n qm m ... Alexander and Hirschowitz have verified a weaker form of the conjecture in wAH4x; namely, that for each n, k G 1 there is a number cŽ n, k . so that for any s G cŽ n, k . there is a collection G of s points in P n so that for each m, hP n Ž G k , m. s minŽ sŽ n qk yk y1 1 ., Ž n qm m ... An important instance in which the latter is made explicit is the following: THEOREM 2 ŽAlexander]Hirschowitz.. Let G ; P n be a general collection of s points. If m G 3 then hP n Ž G 2 , m. s minŽ sŽ n q 1., Ž n qm m .. except in four cases: Ž n, m, s . s Ž2, 4, 5., Ž3, 4, 9., Ž4, 3, 7., and Ž4, 4, 14.. In general, it is necessary that m G 2 k y 1 in order that a collection of s G 2 kth order neighbourhoods be m-independent, since such a scheme has a subscheme of degree 2 k lying on a line. We explore this phenomenon in detail in Section 2. To find a candidate for the Hilbert function in degrees F 2 k y 2 we refer to a conjecture of Froberg to the effect that given a general set of ¨ homogeneous forms, all of the low-degree syzygies between them are the

462

KAREN A. CHANDLER

Koszul ones. When all forms have the same degree, this is expressed as follows: .. Define a1 , a2 , . . . by the power series expanConjecture 3 ŽFroberg ¨ sion

Ž1 y t j . Ž1 y t .

s

s

nq1

`

Ý

am t m .

ms0

Define F Ž s, j, n q 1 . m s

½

am , 0,

if a1 , . . . , a m G 0 otherwise.

Then if I s Ž H1 , . . . , Hs . is the ideal generated by a general collection of homogeneous polynomials of degree j in n q 1 variables in the polynomial ring S, we have dim Ž SrIm . s F Ž s, j, n q 1 . m . Iarrobino wIx makes the further conjecture that in most cases it suffices to take a general set of jth powers of linear forms, and shows that this in turn gives a conjectural Hilbert function for collections of fat points. Specifically, if I s Ž L1jq1, . . . , L sjq1 . and S ; P n is the collection of s points dual to the L1 , . . . , L s , then Macaulay duality yields hP n Ž S k , k q j . s dim Ikqj , which gives rise to the following: Conjecture 4 ŽIarrobino.. Define G Ž s, k , n q 1 . m s n q m y F Ž s, m y k q 1, n q 1 . m . m

ž

/

Let G be a general collection of s points in P n. For each m G 0, hP n Ž G k , m . s G Ž s, k , n q 1 . m , provided s G n q 5. Note that in case s F n q 1 the conclusion of Conjecture 4 Žand hence of Conjecture 3. holds, since for suitable choice of s linear forms, the ideal of their jth powers is a complete intersection. By Theorem 2, the result of Conjecture 4 holds for k s 2, except in the four cases.

HIGHER INFINITESIMAL NEIGHBOURHOODS

463

The result of Conjecture 3 has been shown to hold in the following cases Žsee wIx.: s s n q 2 ŽStanley.; n s 1 ŽFroberg .; n s 2 ŽAnick.; m s j q 1 ¨ ŽHochster and Laksov., j s 2 and n F 10, j s 3 and n F 7 ŽFroberg and ¨ Hollman.; certain m F 2 j ŽAubry.. Collections of Points in Linearly General Position. It is a classical result of Castelnuovo that a collection G ; P n of s reduced points in linearly general position satisfies hP n Ž G, m. G minŽ mn q 1, s .. This says that the Hilbert function of an arbitrary collection of s points in linearly general position is bounded below by that of s points on a rational normal curve. Moreover, if s G mn q 3 and hP n Ž G, m. s mn q 1 then G in fact lies on a rational normal curve Žsee, e.g., wACGHx.. The following asserts an analogue for fat points: Conjecture 5 ŽCatalisano]Gimigliano.. Let G be any collection of s points in linearly general position. Let G0 be a collection of s points on a rational normal curve. Then hP n Ž G k , m. G hP n Ž G0k , m. for all m. One might actually expect that a stronger statement should hold, namely that if s is sufficiently large and hP n Ž G k , m. agrees with the Hilbert function of fat points on a rational normal curve, then the reduced points of G lie on a rational normal curve. The following asserts that fat points of order k in linearly general position impose independent conditions on m-ics whenever m is at least the minimal value at which a collection of such fat points supported on a rational normal curve can be m-independent. THEOREM 6 ŽCatalisano]Trung]Valla.. Let G be a collection of s points in linearly general position in P n. If m G 2 k y 1 and ks F mn q 1 then hP n Ž G k , m. s deg G k . The result of Conjecture 5 holds in the case of s fat points of order 2. In fact, more is true: THEOREM 7 wC1x. Let G be a collection of s points in linearly general position in P n. Then hP n Ž G 2 , m. G minŽ mn q 1, s . q Ž n y 1.minŽŽ m y 1. n y 1, s .. Equality holds for some m with Ž m y 1. n y 1 - s if and only if equality holds for e¨ ery such m if and only if G lies on a rational normal cur¨ e. In this paper, we develop an approach for estimating Žor evaluating. the Hilbert function of a collection of fat points in P n, particularly its values in low degrees. We apply this to obtain lower bounds on such a Hilbert function in degree k q 1, both in the case of a general collection of points and in that of an arbitrary collection of points in linearly general position. In the second case, we verify that for any collection of s G 2 n y k q 1

464

KAREN A. CHANDLER

points in linearly general position, the Hilbert function of its kth order neighbourhood is bounded below by that of the kth order neighbourhood of a rational normal curve in degree k q 1, which is a special case of Conjecture 5. Next, we give ranges of s Ždepending on k and n. for which the Hilbert function of the kth order neighbourhood of a general collection of s points in P n is bounded below by the value predicted by the conjectures of Froberg and Iarrobino. Further, we point out the corre¨ sponding new results on Froberg’s conjecture in the case of quadratic ¨ forms. The paper is structured as follows: in Section 2 we make explicit the linear degeneracies that prevent a collection of fat points from imposing independent conditions on hypersurfaces of low degree. Moreover, we show that these yield an inductive approach for estimating the Hilbert function of such a scheme, using neighbourhoods of its secant varieties. In Section 3 examples are given to illustrate cases in which this approach actually computes the Hilbert function, and useful preliminary results are obtained. In Section 4 we apply the strategy to the Hilbert function of a kth order neighbourhood of a collection of points in P n in degree k q 1, thereby verifying special cases of the conjectures of Catalisano]Gimigliano, Froberg, and Iarrobino. ¨

´ 2. A BEZOUT LEMMA We give a Bezout-type lemma, which yields a procedure for estimating ´ Hilbert functions of fat points. Notation. If X : P n is a reduced Žbut not necessarily irreducible. subscheme, we write Sec j X for the union of the j-planes spanned by subsets of X. ŽIn particular, Sec 0 X s X.. LEMMA 8. Suppose F is a form of degree k q i ¨ anishing to order k at a set of j q 1 points. Let V be the projecti¨ e span of the points. Then F ¨ anishes to order k y ij on V. Proof. Let W be the span of some j of the points. By induction F vanishes to order k y iŽ j y 1. on W. Let L be any line between the remaining point and any point of W. Then F vanishes 2 k y iŽ j y 1. times along L, and any r th partial derivative of F vanishes 2 k y iŽ j y 1. y 2 r times along L. Such a partial derivative is of degree k q i y r, so if r - k y ij then k q i y r - 2 k y iŽ j y 1., so every r th partial derivative of F vanishes identically on L. COROLLARY 9. Let S : P n be a collection of Ž reduced. points. Choose a hyperplane P ny 1 : P n. Suppose S s S n j S ny1 , where S ny1 : P ny 1

465

HIGHER INFINITESIMAL NEIGHBOURHOODS

and S n l H s B. Then ky 1 hP n Ž S k , k q i . G hP n Ž S nk j S ny1 , k q i y 1.

q hP ny 1 Ž Sec 1 Ž S n . l P ny1 .

ž

ky i

k j S ny1 ,kqi .

/

Proof. It follows from Lemma 8 Žwith j s 1. that every Ž k q i .-ic vanishing on S k also vanishes to order k y i on the union Sec 1Ž S . of lines through pairs of points of S. Thus hP n Ž S k , k q i . s hP n Ž S k j Sec 1 Ž S .

ky i

, k q i..

Now let I s I Ž S . Let L be a linear form defining P ny 1. Then I : L s ky1 . I Ž S nk j S ny1 . So the result follows from Castelnuovo’s exact sequence k.

0 ª Ž I : L . Ž k q i y 1 . ª I Ž k q i . ª Ž I q L . rL Ž k q i . ª 0. COROLLARY 10. Let S be a collection of points in P n. Let P n = P ny 1 = ??? = P ny d be a flag of planes. Suppose S s S n j S ny1 j ??? j S nyd , where S i : P i for each n y d F i F n and S i l P iy1 s B for each i. Then dy1

hP n Ž S k , k q i . G

Ý hP

ž

ny n

ns0

D

Ž Sec j S nqjy n .

0FjFky1ri d

lP ny n j

D

ls n q1

q hP ny d

ž

ky ji

D

ky1 S ny l, kqiy1

Ž Sec j S nqjyd .

ky ji

/

l P nyd , k q i .

/

0FjFky1ri

Proof. Lemma 8 implies that every Ž k q i .-ics vanishing on S also vanishes on ŽSec j S . ky i j. Now it is an easy induction to show that for each rFd hP n Ž S k , k q i . ry1

G

Ý hP

ny n

ns0

ž

D

Ž Sec j S nqjy n .

0FjFky1ri

lP ny n j

d

D

ls n q1

q hP ny r

ž

ky ji

D

ky1 S ny l, kqiy1

Ž Sec j S nqjyr .

ky ji

/ d

l P nyr j

0FjFky1ri

the case r s 1 having been given by Corollary 9.

D

lsrq1

/

ky1 S nyl ,kqi ,

466

KAREN A. CHANDLER

Remark. We may reinterpret Iarrobino’s conjecture as predicting that the failure of fat points Žand their secant varieties. to impose independent conditions on hypersurfaces of low degrees is measured precisely by the linear degeneracies described in Lemma 8. First, let us write explicitly the terms given in the conjecture as GX Ž s, k , n q 1 . kq i min ŽŽ ky1riq1 . , nq1, s .

s

Ž y1.

Ý js0

j

s jq1

ž /ž

n q k y 1 y j Ž i q 1. , n

/

and G Ž s, k , n q 1 . kq i

¡ž n q k q i / ,

s

if GX Ž s1 , k , n q 1 . kqi

n

~

¢G Ž s, k, n q 1. X

nqkqi for some s1 F s kqi otherwise. G

kq i ,

ž

/

Thus Žwhen s is not too large. the predicted value of hP n Ž G k , k q i . when G is a general collection of s points is

Ý Ž y1. l ls0

ž

n q k y 1 y li deg Sec l G, k y 1 y li

/

which is consistent with the inequalities of Corollaries 9 and 10 all being equalities and the neighbourhoods of the secant varieties to G’s having the ‘‘best Žlinearly. possible’’ Hilbert functions. 3. FAT POINTS SUPPORTED AT n q 1 POINTS IN LINEARLY GENERAL POSITION Let G ; P n be a collection of n q 1 points in linearly general position. It is easy to see algebraically that G k has the Hilbert function predicted by the conjecture of Froberg and Iarrobino; one can simply compute the ideal ¨ using the coordinate points of P n. However, we shall now look at the Hilbert function of G k from the perspective of finding subschemes that impose independent conditions on hypersurfaces of given degree. By this we intend to give some geometric intuition for the conjecture of Froberg. ¨

467

HIGHER INFINITESIMAL NEIGHBOURHOODS

LEMMA 11. Let G s  p 0 , . . . , pn4 ; P n be a collection of n q 1 points in linearly general position. Then G k imposes Ž n qk k . conditions on hypersurfaces of degree k. Specifically, if Vi s span p 0 , . . . , pi 4 , then  p 0 4 k N V 0 j ???  pn4 k N V n is a k-independent subscheme of G k of degree Ž n qk k .. Proof. Let p 0 , . . . , pn be points in linearly general position. Let H s span po , . . . , pny14 . By induction, the kth order neighbourhoods of  p 0 , . . . , pny14 in H s P n impose Ž n q kk y 1 . conditions on k-ics in H, whereas no Ž k y 1.-ic of P n vanishes to order k on pn , so the kth order neighbourhoods impose at least Ž n q kk y 1 . q Ž n q kk y 1 . s Ž n qk k . conditions on k-ics. LEMMA 12. Let G ; P n be a collection of n q 1 points in linearly general q 1. position. Then G k imposes Ž n qk qk q1 1 . y Ž nk q 1 conditions on hypersurfaces of degree k q 1. Proof. Choose a general flag P n > P ny 1 > ??? > P ny kq1. We claim that: Ža. For each j - k, there are no k-ics of P ny j vanishing to order k y j on Sec j Ž G . l P ny j, and Žb. Sec ky 1Ž G . l P ny kq1 imposes independent conditions on k-ics in ny kq1 P . To see this, we have by induction on k that there are no Ž k y 1.-ics of P ny j vanishing to order Ž k y 1 y j . on Sec j G so Ža. follows. For Žb., write G s  p 0 , . . . , pn4 . Let ri1 , . . . , i k be the point of intersection in P ny kq1 l spanŽ pi1, . . . , pi k .. We shall construct a k-ic of P ny k that vanishes on each of the points riX1 , . . . , iXk / ri1 , . . . , i k but not at ri1 , . . . , i k itself. Take the hyperplanes M1 , . . . , Mk by Ml s span pk : k / i l 4 . Then by linear general position, ri1 , . . . , i k f Ml for all 1 F l F k. Now suppose riX1 , . . . , iXk / ri1 , . . . , i k. Then for some j, i j f  iX1 , . . . , iXk 4 , in which case riX1 , . . . , iXk g M j . Hence M1 j ??? j Mk contains all the points other than ri1 , . . . , i k; correspondingly we obtain a Žreducible. k-ic as desired. Thus by Corollary 10, ky1

hP n Ž G k , k q 1 . G

Ý hP Ž Sec j G kyj , k . ny j

js0 ky1

s

Ý js0

ž

nyjqk nqkq1 nq1 s y . kq1 kq1 k

/ ž

/ ž

/

468

KAREN A. CHANDLER

Remark. For the purpose of comparison with Froberg’s conjecture, ¨ note that nq1

Ý Ž y1. j

js0

ž

nq1 jq1

/ž

nqkq1 nq1 nqky1y2j s y . n kq1 kq1

/ ž

/ ž

/

EXAMPLES. Let G ; P n be a collection of n q 1 points in linearly general position. We will see in some examples that Corollary 9 actually computes the Hilbert function of G k . EXAMPLE.

k s 3. We have seen that hP n Ž G 3, 3. s Ž n q3 3 . and hP n Ž G 3 , 4.

s Ž n q4 4 . y Ž n q4 1 .. The next stage is hP n Ž G 3 , 5 . G hP n Ž G 3 , 4 . q hP n Ž Sec G l P ny 1 , 5 . G

ž

nq4 nq1 nq1 y q 4 4 2

s Žnq1

/ ž / ž .ž /

/

nq2 2

s deg G, thus each of the stated inequalities for hP n Ž G k , m. is an equality. EXAMPLE.

k s 4. Similarly, since hP n Ž G 4 , 5. s Ž n q5 5 . y Ž n q5 1 ., we have

hP n Ž G 4 , 6 . G

ž

2 nq5 nq1 y q hP ny 1 Ž Ž Sec G . l P ny1 , 6 . . 5 5

/ ž

/

Now hP ny 1 ŽŽSec G . 2 l P ny1 , 6. s nŽ n q2 1 . Žsee, e.g., wC2x.. Thus, hP n Ž G 4 , 6 . G Ž n q 1 .

ž

nq3 nq1 y . 3 2

/ ž

/

Finally, hP n Ž G 4 , 7 . G hP n Ž G 4 , 6 . q hP ny 1 Ž Sec G l P ny 1 , 7 . G Ž n q 1.

ž

nq3 s deg G 4 . 3

/

Again, each of the inequalities is an equality, and we obtain the Hilbert function of G 4 .

469

HIGHER INFINITESIMAL NEIGHBOURHOODS

4. DEGREE k q 1 We consider the first interesting value of the Hilbert function: that of degree k q 1. First, we verify the following special case of the conjecture of Catalisano and Gimigliano: PROPOSITION 13. Let G ; P n be a collection of 2 n y k q 1 points in linearly general position. Then G k imposes at least Ž n qk qk q1 1 . y Ž n yk qk q1 1 . conditions on hypersurfaces of degree k q 1. Ž If G lies on a rational normal cur¨ e then equality occurs.. Proof. Write G s S j F where deg S s n q 1, deg F s n y k q 1. Let K be the Ž n y k .-plane spanned by F, and choose a general flag K s P ny k ; P ny kq1 ; ??? ; P n . We have seen that q 1. Ža. F k < P ny k imposes Ž nk q Ž n yk qk q1 1 . conditions on Ž k q 1.-ics in 1 y ny k KsP , and Žb. for each 0 F j F k y 1, no k-ic in P ny j vanishes to order k y j at Sec j Ž S . l P ny j.

These imply that ky1

hP n Ž G k , k q 1 . G

Ý hP js0 ky1

s

Ý js0

s

ž

ž

ny j

ž ŽSec

j

S.

ky j

l P nyj , k q hP ny k Ž F k , k q 1 .

/

nyjqk nq1 nykq1 q y kq1 kq1 k

/ ž

/ ž /

/

nqkq1 nykq1 y . kq1 kq1

/ ž

Observe that the value of the Hilbert function of the kth order neighbourhood of a rational normal curve in degree k q 1 is precisely Ž n qk qk q1 1 . y Ž n yk qk q1 1 .. To see this, if the rational normal curve C has ideal generated by the 2-by-2 minors of the matrix

ž

X0 X1

X1 X2

??? ???

X ny1 Xn

/

470

KAREN A. CHANDLER

then each of the Ž k q 1. = Ž k q 1. minors of



X0 .. .

X1 .. .

Xk

X kq1

??? ??? ???

X nyk .. . Xn

0

vanishes to order k on C. This gives Ž n yk qk q1 1 . linearly independent Ž k q 1.-forms in the Ž k q 1.st graded piece of the ideal of C k . On the other hand, by considering 2 n y k q 1 points on C, we see that there are at most Ž n yk qk q1 1 . such linearly independent Ž k q 1.-forms. Hence these Ž k q 1. = Ž k q 1. minors must generate the Ž k q 1.th graded piece. Remark. We speculate that we may in fact replace the number 2 n y k q 1 in Proposition 13 with 2Ž n y k q 1. q 1. For example, we have seen in wC1x that 2 n y 1 second-order points impose at least 2 n2 q 2 conditions on cubics. Counterexample. We illustrate that exceptions to the conclusion of Conjecture 4 do occur. Let S ; P 6 be any collection of 9 points. Then S must lie on a rational curve C, hence hP 6 Ž S 3, 4. F Ž 104 . y Ž 44 . s 209. On the other hand, GŽ9, 3, 7.4 s 9Ž 82 . y Ž 92 . s 216. That is, numerically S 3 is not predicted to lie on a quartic, but it does lie on the quartic containing C 3. Next, we observe a consequence of Proposition 13 for general collections of points. COROLLARY 14. Let m s wŽ n q 1rk .x. Write n q 1 s mk q e . Let G be 1. a general collection of s points in P n, where s G Ž m q 1. e q k Ž m q . Then 2 k Ž . G lies on no k q 1 -ics. Proof. ŽInduction on n.. Suppose n - k. Then a collection of n q 1 q 1. points of order k supported at general points imposes Ž n qk qk q1 1 . y Ž nk q 1 s Ž n qk qk q1 1 . conditions on Ž k q 1.-ics, as we have seen in Lemma 12. 1. Now suppose n ) k. We may assume that s s Ž m q 1. e q k Ž m q . By 2 semicontinuity it suffices to produce a collection G of s points in P n having the desired property. Write

m

ss

Ý max Ž n q 1 y ik, 0. is0 s

snq1q

Ý Ž n y 1. max Ž Ž n q 1 y k . y ik, 0. . is0

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HIGHER INFINITESIMAL NEIGHBOURHOODS

Assume that s y Ž n q 1. of the points lie in a P ny k and are general there. ny k q 1. Ž . By induction they impose Ž nk q . Thus by 1 conditions on k q 1 -ics in P the same argument as in Proposition 13, we have hP n Ž G k , h q 1. G q 1. Ž n qk qk q1 1 . y Ž nk q Žn q 1. 1 q kq1 . Remark. Note Žafter some arithmetic . that s G Ž nŽ n y k q 2.r2 k . q k y 1 is sufficient to ensure that the conclusion of the corollary holds. THEOREM 15. For any k G 2, n G k, and s F maxŽ n q 1, uŽŽ n q k q 1.Ž n q k .r2Ž k y 1.Ž k q 1..v. there is a set G ; P n of s points so that hP n Ž G k , k q 1. has at least the ¨ alue predicted by Froberg’s conjecture, except ¨ when k s 2, n s 4, and s s 7. Proof. We shall show the following: Let f Ž n, k . s uŽŽ n q k q 1.Ž n q k .r2Ž k y 1.Ž k q 1..v. Then for any k G 2, n G k, and d F f Ž n, k . there is a set G ; P n of degree s so that hP n Ž G k , k q 1. has at least the value predicted by Froberg’s conjecture, except when k s 2, n s 4, s s 7. ¨ To do this, we must use the facts

f Ž n, 2 . s

ž

nq3 3

/

Ž n q 1.

and

f Ž n y k, k . F f Ž n y f Ž n, k . q f Ž n y k , k . , k y 1 . whenever n G 1. To see that f Ž n, k . satisfies this recurrence, we verify the inequality ds

Ž n y f Ž n, k . q f Ž n y k , k . q k . Ž n y f Ž n, k . q f Ž n y k , k . q k y 1 . 2 Ž k y 2. k y

Ž n q 1. n G 0. 2 Ž k y 1 .Ž k q 1 .

We have

dG nq1y

ž

y s

kn

Ž k y 1. Ž k q 1.

Ž n q 1. n 2 Ž k y 1. Ž k q 1.

nŽ n q 1 q k 3 y k 2 y k . 2 2 Ž k q 1. Ž k y 1. k Ž k y 2.

/ž

ny

kn

/

1

Ž k y 1. Ž k q 1. 2 k Ž k y 2.

472

KAREN A. CHANDLER

so we conclude by noting that n q 1 q k 3 y k 3 y k 2 y k G 0 for all n G 1 and k G 2. By the theorem of Alexander and Hirschowitz in wAH3x, the conclusion of the theorem holds for k s 2. We shall assume by induction that the result holds for k y 1 and in the projective spaces of dimensions at most n y 1. Let s F f Ž n, k .. Call b s minŽ f Ž n, k . y f Ž n y k, k ., s . and c s minŽ f Ž n y k, k ., s y b .. Let F ; P ny k consist of c general points and S ; P n consist of b general points. Then ky1

hP n Ž S k j F k , k q 1 . G

Ý hP Ž Sec j S kyj l P nyj j F ky1 , k . ny j

js0

q hP ny k Ž F k , k q 1 . . First, since c F f Ž n y k, k . we have hP ny k Ž F k , k q 1 . s G Ž c, k, n q 1 y k . , by induction on n. Claim. For each j G 0 we have hP ny j Ž Sec j S ky j l P ny j j F ky 1 , k . G S Ž j; b, k, n q 1 y j . q G Ž c, k, n q 1 y j . k q H Ž j; b, c, k , n q 1 y j . , where b

Ý Ž y1. i

S Ž j; b, k , m q 1 . s

usjq1

ž

uy1 j

/ž

mqkyu m

/ ž ub /

and H Ž j; b, c, k , m q 1 . b

s

c

Ý Ý Ž y1. uy jqlq1 usjq1 ls1

ž ub / ž cl / ž u yj 1 / ž m q k ym 2 l y u / .

We verify the claim for j s 0: Write S s  p 0 , . . . , p by1 4 . We have by1

hP n Ž S k j F ky1 , k . G

Ý hP Ž  pi 4 k , k y 1 . q hP Ž F ky1 , k . ny i

ny b

is0 by1

G

Ý is0

ž

nyiqky1 q G Ž c, k y 1, n q 1 y b . k ky1

/

HIGHER INFINITESIMAL NEIGHBOURHOODS

473

since c F f Ž n y b, k y 1. by hypothesis. We just need to check that in case k s 3 we have avoided the exceptional case, i.e., that Ž n y b, c . / Ž4, 7.. Here, n y f Ž n, 3. q f Ž n y 3, 3. G uŽ3n y 6r4.v so n y b G 5 if n G 8. Then, computing Ž n y b, c . for n - 8 reveals that Ž n y b, c . / Ž4, 7. for all n. If j G 1, the claim evidently holds in case b s 0 by the induction hypothesis on n. We continue by induction on b. Observe that each k-ic through Sec j S ky j l P nyj actually contains spanŽ S l P ny j . ky j. Call M s spanŽ S .. We have hP ny j Ž Sec j S ky j l P ny j j F ky 1 , k . s hP ny j Ž M ky j l P ny j j F ky 1 , k . G hP ny j Ž M ky j l P ny j j F ky 2 , k y 1 . q hP ny jy 1 Ž M ky j l P ny i j F ky 1 , k . . Now Žby induction on k . hP ny j Ž M ky j l P ny j j F ky 2 , k y 1 . G S Ž j y 1; b y 1, k y 1, n q 1 y j . q G Ž c, k y 2, n q 1 y j . ky 1 q H Ž j y 1; b y 1, c, k y 1, n q 1 y j . since M ky j l P ny j contains Sec jy1 S 1ky j l P ny j for some b y 1 points S 1 ; M spanning M l P ny 1. Further Žby induction on n. hP ny jy 1 Ž M ky j l P ny jy1 j F ky 1 , k . G S Ž j; b, k , n y j . q G Ž c, k y 1, n y j . k q H Ž j; b, c, k , n y j . . We observe that S Ž j y 1; b y 1, k y 1, m q 1 . q S Ž j: b y 1, k, m . s S Ž j; b, k , m q 1 . , Ž 1. G Ž c, k y 2, m q 1 . ky 1 q G Ž c, k y 1, m . k s G Ž c, k y 1, m q 1 . k , Ž 2 . and H Ž j y 1; b y 1, c, k y 1, m q 1 . q H Ž j; b y 1, c, k, m . s H Ž j; b, c, k, m q 1 . , which establishes the claim.

Ž 3.

474

KAREN A. CHANDLER

ŽTo see this, Eq. Ž2. is immediate; while Eqs. Ž1. and Ž3. both follow directly from the elementary Lemma 18 below.. We have ky1

hP n Ž S k j F k , k q 1 . G

Ý Ž S Ž j; b, k, n q 1. q G Ž c, k y 1, n q 1 y j . k js0

qH Ž j; b, c, k , n . . q G Ž c, k , n q 1 y k . kq 1 . Now, ky1

Ý G Ž c, k y 1, n q 1 y j . k q G Ž c, k, n q 1 y k . kq1 s G Ž c, k, n q 1. k js0

and we shall show ky1

Ý S Ž j; b, k, n q 1 y j . s G Ž b, k, n q 1. kq 1

Ž 4.

js0

and ky1

Ý H Ž j; b, c, k, n . s G Ž b q c, k, n q 1. kq 1 js0

y G Ž b, k, n q 1 . kq 1 y G Ž c, k, n q 1 . kq1 Ž 5 . which will lead us to the desired conclusion, hP n Ž S k j F k , k q 1 . G G Ž b q c, k , n q 1 . kq 1 . For both Ž4. and Ž5. we need the following marvelous fact, proved later in Lemma 19, u

Ý Ž y1. i is0

u i

ž /ž

u nqtqi s Ž y1 . n q t . n nyuqi

/

ž

/

To verify Ž4., we have ky1

ky1

b

Ý S Ž j; b, k , n q 1. s Ý Ý Ž y1. i

js0

js0 usjq1 k

s

Ý us1 k

s

uy1

uy1 j

b u

/ž /ž

nyjqkyu nyj

/

ž / Ý Ž y1. ž u yi 1 / ž n qn kyyu 2qu 1qq1 iq i / b u

i

is0

u Ý Ž y1.

us1

ž

ž ub / ž n q k yn 2 u q 1 /

s G Ž k, b, n q 1 . kq 1 .

475

HIGHER INFINITESIMAL NEIGHBOURHOODS

For Ž5. we compute explicitly G Ž b, k , n q 1 . kq 1 q G Ž c, k, n q 1 . kq1 y G Ž b q c, k , n q 1 . kq1 ky1

Ý Ž y1. i

s

is1 ky1 iy1

s

b c bqc q y i i i

žž / ž / ž . ž /ž /ž

Ý Ý Ž y1 is2 js1

i

b j

c iyj

//ž

n q k y 1 y 2i n

n q k y 1 y 2 Ž i y 1. n

/

/

whereas ky1

Ý H Ž j; b, c, k , n . k js0 ky1

s

b

c

Ý Ý Ý Ž y1. uy jql

js0 usjq1 ls1 b

s

c

Ý Ý Ž y1. ly1 us1 ls1 uy1

= b

s

ž

Ý is0 c

ž

uy1 i

us1 ls1

ž ub / ž cl /

/ž

Ý Ý Ž y1. uq l

ž ub / ž cl / ž u yj 1 / ž n q k yn yj yj 2 l y u /

n q k y 1 y 2 Ž u q l y 1. q i n y Ž u y 1. q i

//

ž ub / ž cl / ž n q k y 1 yn2Ž u q l y 1. / .

Remark. We briefly discuss the optimality of the results given in Theorem 15 and Corollary 14. First, if n F Ž2 k y 3.Ž k q 1., Theorem 15 only returns information on s F n q 1, in which case we had already observed that the result of the conjecture of Froberg and Iarrobino does ¨ hold. However, when n F k y 1, the result for s s n q 1 is optimal, i.e., we have GŽ n q 1, k, n q 1. kq 1 s Ž n qk qk q1 1 ., when n F k y 1. Likewise, in case n F k y 1, Corollary 14 yields the optimal value s s n q 1, and if n s k the corollary gives the optimal value of s s n q 2. For larger n, observe that we have estimated hP n Ž S k , k q 1. in case deg S k is a little more than 12 Ž n qk qk q1 1 .; we still have GŽ s, k, n q 1. kq 1 Ž n qk qk q1 1 . when s f ŽŽ n q k q 1.Ž n q k .rŽ k q 1. k ..

476

KAREN A. CHANDLER

We would expect to improve the conclusion of Theorem 15 by using Corollary 10 Žrather than just Corollary 9. in the proof. However, the details of the proof would likely become rather more involved! We see this as the beginning of a first step toward verifying instances of Froberg and Iarrobino’s conjectures in greater generality. To illustrate a ¨ simple case of continuing this process we give the: COROLLARY 16. Let s F maxŽuŽŽ n q 4.Ž n q 3.r16.v, n q 1.. Let G ; P n be a general collection of s points. Then for all m hP n Ž G 3 , m . s G Ž s, 3, n q 1 . m . Proof. Again we write f Ž n, 3. s uŽŽ n q 4.Ž n q 3.r16.v. We show that if s F maxŽ f Ž n, 3., n q 1. we can find a collection G ; P n of s points satisfying hP n Ž G 3 , 5 . s s

ž

nq2 . 2

/

In case s s n q 1, we are already done, by our previous observations. Otherwise, assume by induction that the result holds in projective spaces of dimensions - n. Choose F ; P ny 2 consisting of c s minŽ f Ž n y 2, 3., s . points and choose S ; P ny 1 consisting of b s s y c points that are in a hyperplane H, with H l F s B. Then hP n Ž S 3 j F 3 , 5 . G hP n Ž S 3 j F 2 , 4 . q hP ny 1 Ž Ž Sec S l P ny 1 . j F 2 , 4 . q hP ny 2 Ž F 3 , 4 . . By induction on n we have hP ny 2 Ž F 3 , 4 . s c

n . 2

ž /

Next, hP ny 1 Ž Ž Sec S l P ny 1 . j F 2 , 4 . G hP ny 1 Ž Ž Sec S l P ny 1 . j F , 3 . q hP ny 2 Ž F 2 , 3 . G

b q c q c Ž n y 1. 2

ž/

and hP n Ž S 3 j F 2 , 4 . G hP n Ž S 2 j F 2 , 3 . q h H Ž S 3 , 4 . G Ž n q 1. s q

ž

nq1 b by 2 2

/ ž/

477

HIGHER INFINITESIMAL NEIGHBOURHOODS

for a total of hP n Ž S 3 j F 3 , 5 . G s

ž

nq2 . 2

/

Of course, since degŽ S 3 j F 3 . s sŽ n q2 2 ., the latter inequality must be an equality, and the inequality used from Theorem 15 must have been as well. Thus hP n Ž S 3 j F 3 , 4 . s s

ž

nq2 s y 2 2

/ ž/

and hP n Ž S 3 j F 3 , m . s s

ž

nq2 , 2

/

m G 5.

Next, we apply Macaulay duality to Theorem 15 and Corollary 14 to obtain the following result on quadratic forms: COROLLARY 17. Suppose Q1 , . . . , Q s is a general collection of quadratic forms Ž or e¨ en Q i s L2i , where L1 , . . . , L s are general linear forms. in n q 1 ¨ ariables, and let I s Ž Q1 , . . . , Q s .. Then

¡F Ž s, 2, n q 1.

dim Im G

~

m,

¢ž n qm m / ,

Ž n q m . Ž n q m y 1. 2 m Ž m y 2. n Ž n y m q 3. if s G q m y 2. 2 Ž m y 1. if s F

APPENDIX}ELEMENTARY COMPUTATIONS In this section we perform some of the computations used in the proof of Theorem 14. LEMMA 18.

Let b

T Ž j, b, p, m . s

Ý Ž y1. uy j usjq1

ž

uy1 j

b u

/ž /ž

pyu . m

/

478

KAREN A. CHANDLER

Then T Ž j, b, p, m . s T Ž j, b y 1, p y 1 . q T Ž j y 1, b y 1, p y 1, m . . Proof. Let d s T Ž j, b, p, m. y T Ž j, b y 1, p y 1, m y 1.. We have b

ds

Ý Ž y1. uy j

uy1 j

ž

usjq1

b y by1 u u

/ žžž / ž

/ / ž p ym u /

/ ž ž p ym u / y ž p ym yu y1 1 / / / ž u yj 1 / ž ž ub yy 11 / ž p ym u / q ž b yu 1 / ž p ymu y 1 / / q by1 u

ž

b

s

Ý Ž y1. uy j usjq1 b

s

Ý Ž y1. uy jq usj

ž b yu 1 / ž p ymu y 1 / ž ž uj / y ž u yj 1 / /

b

s

ž b yu 1 / ž uj yy 11 / ž p ymu y 1 /

Ý Ž y1. uy jq1 usj

s T Ž j y 1, b y 1, p y 1, m . .

LEMMA 19. u

Ý Ž y1. i is0

u i

ž /ž

u nqtqi s Ž y1 . n q t . n nyuqi

/

ž

/

Proof. The result certainly holds for u s 0 and all n, t. Fix n, t, and assume the result holds for u y 1. We have u

Ý Ž y1. i is0

u i

ž /ž

nqtqi nyuqi

/ Ý Ž . žž / ž ÝŽ . ž /žž ÝŽ .ž /ž u

s

y1

i

y1

i

is0 u

s

is0

uy1

sy

y1

is0

s Ž y1 .

u

uy1 uy1 q i iy1

uy1 i i

uy1 i

ž n qn t / .

nqtqi nyuqi

//ž / ž

/

nqtqi nqtqiq1 y nyuqi nyuqiq1 nqtqi nyuq1qi

/

//

HIGHER INFINITESIMAL NEIGHBOURHOODS

479

ACKNOWLEDGMENT I thank Andre ´ Hirschowitz for his helpful suggestions and encouragement.

REFERENCES wAx

J. Alexander, Singularites aux hypersurfaces de ´ imposables en position generale ´ ´ P n , Compositio Math. 68 Ž1988., 305]354. wAH1x J. Alexander and A. Hirschowitz, Polynomial interpolation in several variables, J. Alg. Geom. 4 Ž1995., 201]222. wAH2x J. Alexander and A. Hirschowitz, Une lemme d’Horace differentielle: Application ´ aux singularites ´ hyperquartiques de P 5, J. Alg. Geom. 1 Ž1992., 411]426. wAH3x J. Alexander and A. Hirschowitz, La methode d’Horace ´ eclatee: ´ ´ Application `a l’interpolation en degre ´ quatre, In¨ ent. Math. 107 Ž1992., 585]602. wAH4x J. Alexander and A. Hirschowitz, An asymptotic vanishing theorem for generic unions of multiple points, Duke e-print, March 1997. wACGHx E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, ‘‘Geometry of Algebraic Curves,’’ Vol. I, Springer-Verlag, New York, 1985. wC1x K. Chandler, Hilbert functions of dots in linear general position, in ‘‘Proceedings of Conference on Zero-Dimensional Schemes, Ravello, Italy, June 1992.’’ wC2x K. Chandler, Regularity of the powers of an ideal, preprint, 1994. wCGx M. V. Catalisano and A. Gimigliano, On the Hilbert function of fat points on a rational normal cubic, Duke e-print, alg-geomr9412016. wCTVx M. V. Catalisano, N. V. Trung, and G. Valla, A sharp bound for the regularity index of fat points in general position, Proc. Amer. Math. Soc. 118 Ž1993., 717]724. wHx A. Hirschowitz, La methode d’Horace pour l’interpolation ` a plusieurs variables, ´ Manuscripta Math. 50 Ž1985., 337]388. wIx A. Iarrobino, Inverse system of a symbolic power. III. Thin algebras and fat points, J. Algebra 174 Ž1995., 1091]1110.