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APPENDIX A

Divided Power Rings and Polynomial Rings

The aim of this appendix is to introduce the divided power ringsand prove the few elementary facts about them we use. For a morecomplete, coordinate free exposition the reader is referred to [Ei2,Section A2.4]. Let k be a field of arbitrary characteristic. Let R =

k[Xl,"" xr] = EBj>o R j . Let V be the graded dual of R, i.e.

V = EBHomk(Rj,k) = EBVj

We consider the vector space R; with basis Xl, .. . ,Xr and the leftaction of GLr(k) on R1 defined by AXi = 2:;=1 Ajixj. Since R =

Sym j R l this action extends to an action of GLr(k) on R. Byduality this action induces a left action of GLr(k) on V j . We

denote by xU = xt1 ••• lUI = Ul + ... + u; = j the standardmonomial basis of R j . Let Xl, ... .X; be the basis of VI dual to thebasis Xl, . . . ,Xr·

DEFINITION A.l. We denote by

X[U] = xiU 1] ••. XJurl (A.a.l)

the basis ofVj dual to the basis {xu: lUI = j}. We call these elementsdivided power monomials, or for short, DP-monomials. We call theelements of V j divided power forms, or for short, DP-forms and theelements of V divided power polynomials, or DP-polynomials. Weextend the definition of X[U] to multidegrees U = (Ul,'" ,ur ) withnegative components by letting X[U] = a if Ui < 0 for some i.

DEFINITION A.2. We define for every i, j a contraction map

R; x V j --> V j - i

as follows. For rjJ E R; and f E V j let rjJ 0 f E V j - i be 0 if j < i andbe the functional

('1/;, rjJ 0 f) = ('I/;rjJ, f) for 'I/; E Rj - i (A.a.2)

One extends these maps by linearity to a contraction map R x V ---t V.

266 ApPENDIX A. DIVIDED POWER RINGS AND POLYNOMIAL RINGS

PROPOSITION A.3. The contraction map has the following properties

i. It is GLr(k)equivariantii. One has

xt1 ... 0 XIJl] ... Xpr] = XIJI-Ul] ... Xpr-UrJ (A.O.3)

iii. It is a left action of R on V.iv. The Rmodule V is isomorphic to Macaulay's inverse system,

the Rmodule T = k[x1\ ... ,x;l] (see e.g. [Ei2, p.526]), by

the isomorphism sending XPI] ... xJjr] to x l j l ... «;':

PROOF. (i). The equality A(¢ 0 I) = (A¢) 0 (AI) for A E GLr(k)is clear from the definition of contraction (A.O.2).

(ii). This is obvious.(iii). Using formula (A.O.3) one verifies the equality (¢1jJ) 0 f =

¢o(1jJoI).(iv). This is immediate from (A.O.3). D

REMARK A.4. We have not used so far that k is a field. The sameconstruction holds for every commutative ring k, e.g. k = 2:. In thelatter case we denote by RIZ the ring 2:[Xl,'" ,xr ] and by VIZ thecorresponding RIZ-module constructed above.

EXAMPLE A.5. Suppose k is a field of characteristic O. Let R =k[Xl, ... ,Xr]. One considers the differentiation action of R =

k[Xl,'" ,xr ] on R given by

a a¢ 0 f = ¢( aX

l' ... , ax) f

It is GLr(k)-equivariant and yields a duality between R j and R j forevery j 2: O. The basis of R j dual to the monomial basis xu, lUI = jis

X[U] = (A.O.4)

So, in this case the R-module V is isomorphic to the polynomial ringk[X1, ... ,Xr] with divided power monomials given by (A.O.4). Thisisomorphism is compatible with the action of GLr(k).

We now define a ring structure on V. It is modeled on the charac-teristic 0 case. One defines multiplication of monomials by the equality

X[U] . XlV] = (U + V)! X[U+VjU!V!

(A.O.5)

ApPENDIX A. DIVIDED POWER RINGS AND POLYNOMIAL RINGS 267

where (u+V)! = (UI+VI)!···(ur+Vr)!. This is extended by linearity andU!V! UI!"'Ur!VI!"'Vr!

gives a structure of a k-algebra on V.

EXAMPLE A.6. If r = 1, char k = a, we may view x[n] E V asthe operator n!gxn on R. By (A.a.5), X[2] . X[2] = X[4] = 6X[4].Accordingly, we have

(X[2]. (X[2]) 0 x4 = 0 x4 = 6 = 6X[4] 0 x4.28x228x2

Thus, the multiplication introduced allows us to view V as the ringof (higher order) partial differential operators on R, when char k = a.We do not carry this viewpoint further (in which V acts on R), sincewe need primarily the action of R on the inverse system V. However,we will use the properties of the multiplication in V.

PROPOSITION A.7. The multiplication introduced above is commutative and associative. It is equivariant with respect to the action ofGLr(k):

AUg) = (AJ)(Ag) (A.a.6)

PROOF. It suffices to check commutativity and associativity formonomials, which one verifies immediately by the definition (A.a.5).For proving equivariance it again suffices to check equality (A.a.6)for monomials f = X[U], 9 = XlV]. Furthermore, since GLr(k) isgenerated by the subgroup of diagonal matrices and the transvectionstij = E + E ij , i I- j it is enough to prove (A.a.6) for A a diagonalmatrix or a transvection. From the definition of X[U] it is clear thatfor A = Diag(CI, ... .c;) one has A( X[U]) = c1u I

... c;ur X[U], hence(A.a.6) holds for diagonal matrices. For transvections it suffices tocheck that the formula holds in Vz (see Remark A.4). As we saw inExample A.5 Vz Q9 Q Q[XI , ... .x.]. This isomorphism preservesthe multiplication by (A.a.4) and (A.a.5). So, (A.a.6) reduces to astandard fact for the action of GLr(Q) on polynomials with rationalcoefficients. 0

DEFINITION A.8. Let L = aIXI + ... + a.X, E VI. The dividedpower LlJ] is defined as

" ajl ... ajrX[jl] ... XlJrl1 r 1 r

jl+·+jr=j

PROPOSITION A.9. Let LEVI be as above. Then

i. Lj = (j!)LlJ]

268 ApPENDIX A. DIVIDED POWER RINGS AND POLYNOMIAL RINGS

ii. The element L[j] E Dj = Horrn, (Rj, k) is equal to the functionaldefined by (¢, L[j]) = ¢(a).

111. A(L[j]) = (A(L))U] for every A E GLr(k).iv. L[i] . LU] = L[i+j].

2.-).

PROOF. Part (i) is easily proved by induction on j. Part (ii) isimmediate from the definition of L[j].

Part (iii). Let a = (al, ... ,ar) and let B = (tA)-l. Then therow vector of coordinates of A(L) is t(B(ta)). Recall that we definedthe action of GLr(k) on R, requiring that (-,.) is GLr(k)-invariant onR l x D l and then extending to R, = Symj(Rd. This implies that(A¢)(a) = ¢(t(B-l(ta))). Now, for every ¢ E R j we have

(¢, A(L[j])) = (A -l¢, L[j]) = (A -l¢)(a)

= ¢(t(B(ta))) = (¢, (A(L))[j])

Therefore A(LU]) = (A(L))U).Part (iv). This is by definition if L = Xl. The general case is

reduced to this using Part (iii). D

COROLLARY A.lO. COORDINATE-FREE DESCRIPTION OF THE VE-RONESE MAP. Let V = D l . Then the Veronese map Vj : JPl(V) -----;JPlHO(OIP'(v)(j))* can be identified with the map JPl(Dd -----; JPl(Dj ) givenby L I--t LU].

PROOF. We have V* R: and HO(OIP'(V)(j)) SymjRl = R j.Thus the target space of "i is JPl(Dj ) . The identification of Vj with themap L I--t L[j] follows from Proposition A.9(ii). D

REMARK A.n. The definitions of divided powers L[jJ and the prod-uct in D shows that the right-hand side in the definition of the DP-monomials X[U] (see (A.a. 1)) is not just a symbolic expression, buthas the meaning of a product of divided powers of Xi

We conclude the appendix by comparing divided power rings andpolynomial rings in char(k) > a.

PROPOSITION A.12. Lei R. = k[Yl, ... ,1";.] and lei D be the dividedpower ring. Let R = k[Xl,' .. , x r] act on D by contraction and on R.by differentiation (see Example A.S). Then

1. There is an isomorphism of k-algebras

ApPENDIX A. DIVIDED POWER RINGS AND POLYNOMIAL RINGS 269

given on monomials by

'P(XiU 1] ••• XJu r ] ) = 1 Yt1 ••• yUr (A.a.7)

Ul!"'Ur! r

and sending L[j] to Lj .J.

II. There is an isomorphism of R-modules

'PI : ttJi'5.jDi --. ttJi'5./Ri

given by the same formula as in (A.a.7)iii. Suppose char(k) = a. Then (A.a.7) defines an isomorphism

'P: D --t R which commutes with the action of R

PROOF. By Proposition A.9(i) the homomorphism {) : k[Yl , ... , Yr ]

--t D sending YlUl ... yrur to Xfl ... X;:r is an isomorphism if char(k) =

a, or an isomorphism in degrees :S j if char(k) > j. The statementabout L[j] follows from Proposition A.9(i). Part (ii) is clear from for-mula (A.a.3) 0

It follows that if char(k) > i, the highest degree of a polynomial weuse, or if char(k) = a, we may interchange in some cases the R-algebrasD and R.

APPENDIX B

Height Three Gorenstein Ideals

We collect a number of facts that we use concerning height threeGorenstein ideals, and we give proofs for several of them.

In Section B.1 we prove an expansion formula for Pfaffians of alternating matrices with zero diagonal blocks, analogous to the Laplaceformula for determinants. We use a particular case in the proofs of themain theorems of Section 5.3. We also state an interesting expansionformula for Pfaffians due to H. Srinivasan [SrI].

In Section B.2 we first state the BuchsbaumEisenbud structuretheorem, in the special case for graded Gorenstein ideals. We thenstate a variation of a KustinUlrich theorem concerning the resolutionof I" when I is a general enough Pfaffian ideal [KusU], in the casea = 2 (Theorem B.3). We use this result to bound above the dimensionof the tangent space to Gor(T), when r = 3 (Theorem 4.5B).

In Section B.3 we state several very useful results of M. Boij, connecting the coordinate ring of an ideal defining sets of points in IfDnand a related Gorenstein Artin algebras; these culminate in a resultdetermining the minimal resolution of the latter in terms of that forthe ideal of sets of points (Proposition B.lO).

In Section B.4 we first state and prove a nice criterion of A. Concaand G. Valla for determining the maximal Betti numbers consistentwith a height three Gorenstein sequence T, directly in terms of thesecond difference f).2(T) (Theorem B.13). We use this result in alternative proofs of Lemmas 5.29 and 5.37. A. Conca and G. Valla stateand prove a far more general version in [CoVI]. We prove the ConcaValla criterion as a convenience, and also to illustrate some methodsthat we have not greatly stressed in the main text.

Finally we state a result of S. J. Diesel concerning the number ofGorenstein sequences of height three, having given order (of definingideal) and socle degree. We use it in the proof of Theorem 7.15.

272 ApPENDIX B. HEIGHT THREE GORENSTEIN IDEALS

n.i. Pfaffian formulas

We use the definition of the Pfaffian of an alternating matrix according to Bourbaki [Bou]. Let S2n be the group of permutationsof {1,2, ... ,2n}. Let H C S2n be the subgroup consisting of permutations which transform every subset {2k 1, 2k} into a subset{2£ 1, 2£}. Clearly H is a semidirect product of Sn with (iZ2)n (thisis the Weyl group W(Cn ) ) . Let A be a commutative ring with identity and let X = (Xij) be a 2n x 2n alternating matrix with Xij E A,Xji = -Xij and Xii = 0 for i,] = 1, ... ,2n. The Pfaffian of X is

Pf(X) = L sgn(a)xa(1)0'(2)Xa(3)a(4)'" T a(2n- 1)a(2n)aES2n/H

1 '" . () (B.1.1)= '2n D SIgn a Xa(1)a(2) ... Xa(2n-1)a(2n)n.aES2n

when char(k) = O.

In this way the Pfaffian is normalized so that Pf(Xo) = 1, if Xo : Xij =] i if 1] i 1= 1, and Xij = 0 otherwise. The Pfaffians have thefollowing two basic properties [Bou, Ch, IX, §5]. If Y is an arbitrary2n x 2n matrix with coefficients in A then

Det(X) = Pf(X)2 and Pf(YXyt) = Det(Y) Pf(X) (B.1.2)

Suppose A = K is a field of characteristic O. Let w = L:7,j=1 Xijei 1\ ejbe the corresponding bivector. Then

nI\w = 2nn! Pf(X)e1 1\ e2 1\ ... 1\ e2n-1 1\ e2n'

Let us fix k columns of X with numbers j < ... < i«. For every subset1 c {1, ... ,2n}, #1 = k, 1 = {i1,'" ,id, i1 < ... < ik we denote byx5 the submatrix consisting of the entries in the rows i1, ... ,ik andthe columns ]1, ... ,]k. If 1 n J = 0we denote by XI,J the alternating(2n 2k) x (2n 2k) submatrix of X obtained by deleting the rowsand columns with numbers i1, .. · ,ik,]l" ·]k.

LEMMA B.1. EXPANSION OF PFAFFIANS BY MINORS. Let A bea commutative ring with identity and let X = (Xij) be a 2n x 2nalternating matrix with coefficients in A. Let 1 :s; ]1 < ... < ]k :s; 2nwhere k :s; n, Suppose the k x k block (Xj"j(3) l:50a,f3:50k equals O. Then,

§ E.1. PFAFFIAN FORMULAS

expanding along k columns,

273

Pf(X) =

L#1= k,InJ=0

(B.1.3)

(-1 )2::=1(i",+j",)-k+inv(il,jr, ... ,idk) Det (X}) Pf (XI,J)

where inv(i1,j1, ... ,ik,jk) is the number of inversions in (i1,j1, ... ,ik, jk)' If the rows {i1' ... , ik} = I are fixed and one assumes the k x kblock (Xi",ij3) = 0 then expanding along the k rows,

Pf(X) =

L#J=k,InJ=0

(B.1.4)

(-1)2::=1 (io+j",)-k+inv(il,jr, ... ,idk) Det (X}) Pf (XI,J)

PROOF. We give the proof for expansion along columns. The for-mula for expansion along rows is proved similarly. Both sides of theequality (B.1.3) are polynomials in Xij with integer coefficients, so itsuffices to prove it for the ring A = Z[Xij]i<j, and furthermore we mayreplace A by the field of fractions K of characteristic O. We sepa-rate the summands of w = L Xijei /\ ej which do not contain indicesj1, . . . .i; and using the assumption of the O-block we obtain

w 2<p +Wi, where

<p L L Xijei /\ eji!f;J jEJ

Wi L xuveu /\ ev·u,v!f;J

n nWe know /\w is proportional to e1 /\ ... /\ e2n' Expanding /\(2<p +Wi)

we see that there is only one summand which contains ejr /\ ... /\ ejkas a factor, and thus might be nonzero:

n (n) k k n-k/\w = k 2 (/\<p) /\ ( /\ Wi). (B.1.5)


Now,

kAe.p

= k! L L xO'(iIljl ... XO'(ik)jkeO'(i l) A eh A ... A eO'(ik) A ejk

InJ=00'ESk

= k! L(L sgn(a)XO'(il)h ... Xdik)jk) eil A ejl A ... A ei; A ejk

InJ=0 O'ESk

= k! L Det (X5) eil A ejl A ... A eik A ejk' (B.1.6)InJ=0

Each of the wedge products

(B.1.7)

n-kcancels summands of A io' which contain either ei, i E lor ej, j E J.So we can consider WI,J = '2:.u,vrf-IUJ xuveu A e v, Then (B.1.7) equals

eil A eh A··· A e.; A ejk A (n/\kWI,J) = (B.1.8)

= 2n - k (n - k)! Pf (XI,J) (ei" A ej,,) A (Aurf-UUJ)eu) .

It only remains to apply a standard lemma which counts the num-ber of inversions of the permutation i 1,j1,." ,ik,jk,U1, ,U2n-2k

with U1 < U2 < ... < U2n-2k and says it equals inv(i1,j1, ik,jk) +(ia + ja) - 2k(2;+1) [Kur, 1. §5]. This yields that the wedge

product in (B.1.8) equals

Combining (B.1.5), (B.1.6) and (B.1.8) we obtain the expansion for-mula (B.1.3) 0

Let us consider some particular cases of the formulas (B.1.3) and(B.IA). If we fix the i- th row, then Xii = 0 since X is alternatingmatrix and one obtains expansion of Pf(X) along a row:

2n

Pf(X) = L(-I)i+j-l+inv(i,j)xijPf(Xij)j=l

where inv(i,j) = 0 if i < j and inv(i,j) = 1 if i > j.

(B.1.9)

§ B.l. PFAFFIAN FORMULAS

Suppose X has the form

275

where A, Band C are square n x n matrices, A is alternating, C = - e:Then the condition of the lemma is satisfied with J = {n + 1, ... ,2n}and one obtains only one summand with I = {I, ... ,n}, x5 = B.One has inv(l, n+1, ... ,n, 2n) = (n-1)+(n-2)+o. ·+1 = n(n-1)/2.So

Pf(X) = (_1)n(n2- 1) Det(B)

Another case that we need is

(B.l.lO)

where A is an alternating (n+ 1) x (n+ 1) matrix, B is an (n+ 1) x (n-1)matrix, C = -Bt and D is a O-block of size (n - 1) x (n - 1). Letus expand along the last n - 1 columns. Then J = {n + 2, ... ,2n},the possible I are I = {1,2, ... ,n+ I} - {i,j}, 1:S; i < j:S; n+ l.

The matrix XI,J = xc/ ) . We let Bi j = X5, this is the matrix

obtained from B deleting rows i and j. Applying (B.l.3) we obtain

Pf(X) = L (_lr+j+n(n2- 3) X i j Det (Bij)

l::;i<j::;n+l

(B.l.ll)

We conclude this section by stating an interesting formula due toH. Srinivasan which may be used for an alternative proof of (B.l.ll).We need first some more notation. Let X be an alternating matrix oforder 2n. If 5 = (iI, ... ,iq ) denotes a length q increasing sequence ofintegers, with 1 :s; it :s; 2n for all t, then we denote by

the submatrix using the rows and columns specified by 5. We denotethe Pfaffian Pf(X(5)) by Pf(5). Srinivasan's formula [SrI, Corollary3.2J is the following. If I, J are increasing subsequences of 5, and


(B.2.1)

1I 1= p, I J 1= q satisfy p < q, thenq-2

Pf(I, J) = 2::(_1)(t-q)/ 2+ 1 2:: sign 0- Pf(I,W) Pf(JjW),t=O We]

IWI=t(B.1.12)

where 0- is the permutation such that o-(W, J jW) = 1.

B.2. Resolutions of height 3 Gorenstein ideals and theirsquares

We henceforth suppose for simplicity that R = k[Xl,'" ,xr ] isthe polynomial ring over a field k, and denote by mn the maximalideal mn = (Xl, ... ,xr ) . Let us recall some standard notions fromcommutative algebra. If I is an ideal of R we denote by V(I) the setof primes in the support of RjI: the primes containing I, or occurringin a primary decomposition of I. IfM is an R-module the grade (I, M)is the length of any maximal M-regular sequence in I; the grade of Iis grade (I, R), and satisfies (see [Ei2, Lemma 18.1]' [BruH, Prop.1.2.10])

grade (I) = min grade (Ip, Rp).PEV(I)

The height (also called codimension) of I is, in the special case weconsider, the codimension in Ar of the algebraic variety defined by I.In general, for a Noetherian ring R, the height of a prime ideal P isthe maximum length k of a chain P = Po =:l Pi =:l ... =:l Pk of primeideals of R containing P, or, equivalently, the dimension of Rp. Theheight of I satisfies (see [BruH, Prop. 1.2.14])

height (1) = min height (P);PEV(I)

height (1) 2': grade (I, R). (B.2.2)

Since R = k[Xl,'" ,xr ] is Cohen-Macaulay the grade and height ofeach ideal are equal (see [BruH, Cor. 2.1.4]' [Ei2, Thm. 18.7]).

We refer to [Ei2, 21.11] for the definition of a (possibly non-Artinian) Gorenstein algebra RjI. The ideal I is by definition Goren-stein if RjI is Gorenstein. The socle degree of a Gorenstein algebraof dimension greater than zero is that of its Artinian reduction (seepage 67).

We can now state the Buchsbaum-Eisenbud structure theorem, inthe special case for graded Gorenstein ideals. The main applicationsof this theorem in the book are for the case R = k[Xl' X2, X3], when

§ B.2. RESOLUTIONS OF ... GORENSTEIN IDEALS 277

f E 1)j is a DP-form, and I = Ann(f). Then R/I is Artinian, so theheight (= grade) of I is 3, and R/I is Gorenstein by Lemma 2.14. Notethat if the algebra is to be nonzero, the degrees di of the generatorsare at least one, and the relation degrees ei are at least two; the degreej = d - 3 below is the socle degree of R/I.

THEOREM B.2. (D. Buchsbaum and D. Eisenbud [BE2, BE3])STRUCTURE OF HEIGHT THREE GORENSTEIN IDEALS, GRADED CASE.Let g 3 be an odd integer, and d1 :::; ... :::; dg be a sequence of positiveintegers; set d = g':'l(d 1 + ... + dg) and suppose this is an integer, let

ei = d - di, and j = d - 3, and we suppose 1 :::; d1, dg :::; j + 1 (soei 2).

Let \If be an alternating g x g matrix with entries from the ring R,such that the entry 'l/Jij is homogeneous of degree e, - dj if ei > dj andzero otherwise (so the entries belong to the maximal ideal mR). Let \If ibe the (g - 1) x (g - 1) alternating matrix obtained by deleting the i-throw and column of \If. Then Pf(\lfi) is homogeneous of degree di. Let Ibe the ideal Pf(\If) generated by Pf(\If i), i = 1, . . . ,g. Then I has grade(height) :::; 3 in R. If I has grade 3, then I is a graded Corensieinideal of height three, and the socle degree of R/I is j = d - 3.

Let A be the column vector with entries Ai = (-1)ipf(\lfi).

1. Suppose I has the maximal possible grade 3. Then I has mini-mal resolution

o-? R(-d) L R(-ei) L R(-di) I -? O.

o-? R(-d) -? G* -? G -? I -? O.

(B.2.3)

(B.2.4)

ii. Conversely, if I i- R is a height three graded Corensieui idealof R, there is an alternating matrix \If as above, such that I =Pf(\If) .

NOTE ON PROOF. That the minimal number of generators of aheight three Gorenstein ideal is odd, was shown by J. Watanabe [Wal].The structure theorem above was proven in [BE2]; the original versionof their paper took a more naive approach, and appeared somewhatlater in [BE3]. See also [BruH, §3.4] for an excellent brief account.A typo in the description of conditions on the sequences (D, E) thatare possible for generator-relation degrees in [BE2] is corrected in[Di, HeTV, Hari3] (see also Theorem 5.25(v).)

We only give the proof of two simple facts. First that Pf(\lfi) ishomogeneous of degree di , and second that (B.2.3) is a complex.


Acting by the group Gm = k* on R = k[X1,'" ,xr ] multiplyingeach Xi by tx, we want to prove that to Pf(\[!i) = tdi pf(\[!i)' Eachentry '!/Juv of \[! is multiplied by t eu - dv (including the cases eu :S dvwhen '!/Jij = 0 by assumption). So Det(\[!i) is multiplied by

Now, by assumption

(g - l)d = L:)eu + du ) = 2(d1 + ... + dg ) .u#i

This yields 2:#i eu - 2:V#i d; = 2di . Thus Det(\[!i) is multiplied byt2di and Pf(\[!i) by ±tdi . The sign should be + since Pf(\[!i) is a poly-nomial in the coefficients of \[! i. This proves Pf (\[!i) is homogeneous ofdegree a..

Now, let us prove that (B.2.3) is a complex. For every 1 :S i :S 9consider the following alternating (g+ 1) x (g+ 1) matrix.

(

X = .

-'!/Jig

Subtracting the (i + 1)-th row of X from the first we see that Det(X)is a multiple of Det(\[!), thus equals O. Hence Pf(X) = O. Expandingalong the first row according to (B.1.9) we obtain

g

2)-l)j'!/Jij Pf(\[! j) = O.j=1

So if we let A be the column matrix with entries Aj = (-l)j Pf(\[!j),we obtain \[!A = O. Transposing we get AT\[!T = _AT\[! = O. Thisproves that (B.2.3) is a complex. D

We next give a result that is an application of the Kustin-Ulrichtheorem concerning powers of a Gorenstein height three ideal [KusU,Theorem 6.17]. The source is a letter from A. Kustin", to whomwe are grateful. Using the result, we give an example showing thatthe tangent space of a component Gor(T) of the catalecticant vari-ety V s (u, u; 3) may be larger for "smaller" T than for the maximumT = H(s,j; 3),j = 2u, such that Gor(T) lies in Vs(u, u; 3).

1Letter of A. Kustin, October 1993.

§ B.2. RESOLUTIONS OF ... GORENSTEIN IDEALS 279

°-+ L R(-(ei + ej)) -----+

lS,i<jS,g

THEOREM B.3 (A. Kustin and B. Ulrich). MINIMAL RESOLUTIONOF [2, FOR I A HEIGHT 3 ARTINIAN GORENSTEIN IDEAL.

Under the same hypotheses as Theorem B.2(i)) [2 has minimalresolution

lS,i,jS,g

(i,j);£(l,l)

L R( -(di + dj ) ) -t [2 -t 0,

0-+ A2G* -t G ® G* /rl-----+ S2G -+ [2 -+ 0.

Here TJ = b1 ® bi + " .+ bg ® b;, where b1, ... bg is a basis of G) andbi, ... ,b; is the dual basis of G*.

PROOF. A. Kustin refers to parts c. and d. of Theorem 6.17in [KusU], with q = 2, r = 9 - 2. He observes that '11 satisfiesSPCg_ 2 because, from page 52 of [KusU], the grade of the Pfaffianideal PFg_1(X) = 3 9 - (g - 1) + 2 (here X = '11); furthermore,9 - 1 is the only even integer t in the range r + 1 -:; t -:; 9 - 1. Thehypothesis SPCg- 2 implies the others needed. 0

EXAMPLE B.4. TANGENT SPACES FOR POINTS OF A SQUARE CA-

TALECTICANT. The determinantal locus V120(15, 15; 3) containsGor(Tr), Tl = H(120, 30, 3), and Gor(T2) , T2 = (1,3, ... ,105,115,120,115,105, ... ,3,1). Let h be a general point of Gor(T1) and h a gen-eral point of Gor(T2)' We claim that dirrn, Ttl = 360, and dirrn, Tt2 =365, while T2 -:; T1. In the light of J. O. Kleppe's smoothness result(see [KI2] or Theorem 4.21), these calculations of tangent spaces givethe dimensions of Gor(T1) and Gor(T2 ) . Further such examples oflarge components of Vs(t, t; 3) have been found by Y. Cho and B. Jung- see [ChoJ2] and also Example 4.28 in Chapter 4.

PROOF OF CLAIM. The third difference sequence of T2 containsthe sequence (-5, -1, -5, 5,1,5) in degrees (t - 1, ... ,t + 4), t = 15.We truncate the Kustin-Ulrich resolution, using degrees no greaterthan j = 2t = 30. It becomes,

°-t °-+ °-+ Sym2(G)s,j

-+ (I2)s,j -+ 0,

which yields


Thus, the dimension of (12)j = 6 . 15 + 3 . 5 + 26 = 131, out of N =

dim; R30 = e22) = 496; thus dirrn, 'TJ2 = 365, as claimed. D

B.3. Resolutions of annihilating ideals of power sums

In this section we state some important results of M. Boij from[Bol] connecting the minimal resolutions for ideals defining point setsX in JIDn and certain Gorenstein algebras A, for which X is an annihilating scheme. In particular he studies the case X smooth, but hismain results apply to X any tight annihilating scheme for I, whendeg(f) is sufficiently large (Corollaries B.9 and B.ll). Some relatedresults appear in the articles of M. Kreuzer, T. Harima, and others[Kr, Hari3, GPS]. We use his result concerning smooth X in thesecond half of Section 5.3, for Gorenstein sequences T not containing(s, s, s) (see Lemma 5.36).

We let r 2 and let X = {PI, ... ,Ps} be a set of s distinct pointsof JIDr1; we denote by A(X) or the quotient ring R/I(X) where asusual R = k[X1' ... ,Xr ].

LEMMA B.5. [Bol, Proposition 2.3] Let V be any codimension1 subspace of A(X)e, where c T(X). Then there are constantsAI, A2' ... ,As E k not all zero, such that

s

V = {g E A(X)e IL Aig(Pi) = O}.i=l

(B.3.1)

If B is a graded algebra, and V c Be we denote by V the largestideal of B such that V n (Be) = (V), the ideal generated by V (thisV is a relative version of the ancestor ideal of Equation (2.3.3)). Wehave V = ffii2:0(V : B)i where

(V: B)i = {a E Bd Iab E V, for all b E Be-d}.

The next result is related to our Lemma 6.1(a), in the case (ii); butis more precise, in specifying the condition needed on A (thus, onf = v-), for A = A f to have the symmetrized Hilbert function ofH(R/Iz).

LEMMA B.6. [Bol, Proposition 2.4] Assume c 2T(X) 1 andlet V be the hyperplane in A(X)e defined by AI, A2, . . . ,As E k wherefor each i, Ai i= o. Let A = A(X)/V be the Gorenstein Artin al-gebra quotient determined by V. Then the Hilbert function of A is

§ B.3. RESOLUTIONS OF ANNIHILATING IDEALS OF POWER SUMS 281

Sym(H(A(X)),j): thus

H(A)i = {H(A(X))i,H(A(X))c-i

if a ::; i ::; c/2

if c/2 ::; i ::; c.(B.3.2)

THEOREM B.7. [Bol, Theorem 3.5J Assume that c 2:: 2T(X) - 1and V be a hyperplane of A(X)c as in Lemma B.6. Then J = V isisomorphic to the canonical module wX .

Suppose that B is a graded Cohen-Macaulay algebra of Krull di-mension d. We denote by T(B) the integer satisfying, T(B)+r-d is thehighest degree of k). When B is a Gorenstein Artin algebraof socle degree c, then T(B) = c; when B = A(X) for X a punctualscheme, then T(B) = T(X). Given the ideal J of B we denote byA = B / J the quotient. We denote by WB the canonical module of B(see [BruH, §3.6]). M. Boij shows

THEOREM B.8. [BolJ!Theorem 3.3). Let I be a homogeneous idealin R, let B = R/I be a Cohen-Macaulay algebra of dimension d, andlet J c B be a homogeneous ideal of initial degree at least T(B) + 2such that A = B / J is Gorenstein of dimension d - 1. Then there isan isomorphism J ExtR-d(B, R) = WB, which is homogeneous ofdegree -T(A) - r + d - 1.

Note that the condition on degree in the following Corollary isequivalent to f E Gor(T), where T (s,s), and s = deg(X). We donot suppose that X is smooth.

COROLLARY B.g. Suppose that X is a zero-dimensional subschemeof lPn, that is a tight annihilating scheme of f E Dj , in the sense of Def-inition 5.1. Suppose further that j 2:: 2T(X)+ 1. Then J = Ann(f)/Ixis isomorphic to the canonical module of R/Ix.

PROOF. Here d = 1. The hypothesis that X is a tight annihi-lating scheme of f implies that H(R/ Ann(f)) = Sym(Hx,j); thatj 2:: 2T(X)+1 guarantees that the initial degree of J is at least T(X)+2,satisfying the hypothesis of Theorem B.8. D

We now state M. Boij's result whose special case B = A(X), d = 1connects the minimal resolution of the Gorenstein ideal V with thatof Ix. We suppose B = R/I is a dimension-d Cohen-Macaulay ring,and denote by B, k) Ji the vector space dimension of the degree-isummand of the h-th module in the minimal resolution of B as R-module. So for h = 1 this is the number of generators of degree iof I, for h = 2 this is the number of relations of degree i among the


generators of I etc. Using the exact sequence of Tor associated to°--t J --t B --t A --t 0, Boij shows

PROPOSITION B.lO. [Bol, Proposition 3.7] Let B = RjI be aCohen-Macaulay algebra of dimension d and let J c B be an idealof initial degree at least r(B) + 1 isomorphic to the canonical moduleWB, and defining the quotient A = B j J. Then

k)]i = k)]i EB k)]T(A)+r-d+l-i,(B.3.3)

for h = 0, 1,2, ... r - d and i E Z.

COROLLARY B.11. Suppose that X is a zero-dimensional subschemeof jplr-l, that is a tight annihilating scheme for f E 1)j, and thatj 2': 2r(X) + 1. Then the minimal resolution for A = Rj Ann(J)satisfies (B.3.3), where B = RjIx and d = 1.

We now suppose that B = A(X) where X a set of s pointsPI, ... .p, in jpl2, (so r = 3) and that L1, ... .L; are the correspondinglinear forms in D 1, determined up to nonzero-constant multiple. We

consider a divided power sum f = AuLHJ where each Au i= 0, welet I = Ann(J). We consider a minimal resolution of I , and denote byVi the number of generators of I in degree i, and by ui, the number ofrelations among the generators in degree i; we let Vi(Ix ) and ui, (Ix )denote the corresponding numbers for I(X).

COROLLARY B.12. Let B = A(X) = RjIx, where X = {PI, ... ,

Ps} is a subset ofjpl2, suppose j 2': 2r(X), and let f E (Ix)/- satisfy

f = AuLHJ, where each Au i= 0. Let I = Ann(J). Then we have

W·2

vi(Ix) +Wj+3-i(Ix)

ui, (Ix ) +Vj+3-i(Ix).(B.3.4)

PROOF. By the Apolarity Lemma 1.15 the space V consideredin Lemma B.5, Lemma B.6 and Theorem B.7 is, setting c = j, thehyperplane V = Ann(J)jj(Ix)j c A(X)j and V = Ann(J)jIx cA(X) (d. Lemma 2.15). Thus the condition on f being a length-ssum of divided powers is equivalent to the condition on V of LemmaB.6 and Theorem B.7 where c = j. The condition on j implies byTheorem B.7 that J = I jI(X) is isomorphic to the canonical module,and also by (B.3.2) that J has initial degree at least r(X) + 1. ThusProposition B.lO applies with r = 3, d = 1. 0

§ B.4. MAXIMUM BETTI NUMBERS, GIVEN T 283

B.4. Maximum Betti numbers, given T

In this section we first state and prove the Conca-Valla GeneratorTheorem, that shows how to find the degrees of a maximal generat-ing set of a Gorenstein ideal I having Hilbert function H(R/1) = T,directly from T. A different way of finding the - same - maxi-mal generating set from T had been given by Diesel [Di, §3], and byT. Harima [Hari3]. A. Conca and G. Valla show a rather more generalresult in [CoVI]. We then state and prove a result of S. J. Diesel thatcounts the height three Gorenstein sequences T having given order andsocle degree, by associating them with certain partitions P(T): thesepartitions are determined directly from the maximum set of generatordegrees for T, or, more simply, from the first difference sequence of T(Lemma B.15).

The proof we give next of the Conca-Valla Lemma follows the ap-proach of several authors including M. Boij, and A. Geramita, M. Pucci,and Y. S. Shin to related questions [Bol, GPS]. We include it largelyto give an introduction to their methods, which is to compare gener-ators for ideals defining smooth point subschemes of I1D2 and idealsI E I1DGor(T).2 We assume the first part of Equation (5.3.2), givingthe maximum number of generators v = 2v(T) + 1 for Dmax (T), aresult of S. J. Diesel and also of T. Harima.

We do not assume the main results of Section 5.3, such as Theorem5.31: these are shown independently of the Conca-Valla result, and socould give an alternate - but harder - route to the proof. The resultsof Section 5.3, show more, namely, that there is a mapping from theparameter space of ideals 1= Ann(f), f E Gor(T) (Theorem 5.31), orf E Gorsch(T) (Theorem 5.39) to HilbS (IID2) , rather than a map forcertain ideals I lying over smooth points in I1D2, as here.

We now state the Conca-Valla Theorem. Recall from Theorem5.25 that, given a height three Gorenstein sequence T, we denoteby Dmax(T),Emax(T), respectively, the (nondecreasing) maximum or-dered set of generator degrees consistent with T (or, nonincreasingmaximum ordered set of relation degrees); and we let Cmax(T) =Emax(T) - Dmax(T). Recall that .6.(T)i = T; - Ti- 1 , and that j isthe socle degree of T; also v(T) = min{ilTi i- rd is the order of T,which we will denote by m in the proof, and in the following discussion.

2This proof is by the first author, and is his sole responsibility


THEOREM B.13. (A. Conca and G. Valla [CoVl]) The numberDmax(T)i of generators of degree i in Dmax(T) satisfies,

if v(T) < i j + 2 - v(T);

if i = v(T);

otherwise - if b..2(T)i = 1.

(BA.1)

PROOF. Recall first from Definition 4.30 the sequence 5 = 5(T),namely 5 = b..(T)-::;'j/2' augmented by zeroes in higher degrees,

5(T) =(1,2, ... , v(T), hv, ,ht , 0, ... ), with

hv(T)-l = v 2 h; 2 h[j/2]' (BA.2)

By Theorem 5.25(i), 5 = (5(0),5(1), ,0) is the Hilbert functionof an Artin quotient of k[x,y]. We order the degree-i monomials inx, y lexicographically, xi > xi-ly > ... > yi, and we now recall thedefinition in this case of the lex-initial monomial ideal Ms of Hilbertfunction H(RjMs) = 5 (see (4.4.15), and also Appendix C.1, Defini-tion C.1). Let m = v(T), t = min{i I T; = s}, and for m i tlet /-li = xS(i)yi-S(i). If i 2 m, then (MS)i is the span of the firsti + 1 - 5(i) monomials in x, y in the lexicographic order, so,

(MS)i = (xi, xi-ly, ... ,xS(i)yi-S(i)).

If /-l is a monomial other than yi, in the following formula we let /-ll =y/-l : x denote the successor monomial to /-l.

Evidently, a minimal generating set G for Ms satisfies

!{xm, ... ,/-lm} of length 1-b..(5)m, ifi=m,

Gi= (Y/-li-d, ... ,/-ld of length -b..(5)i if m < i ::; t + 1

and b..(5)i < 0,

o in degree i, otherwise.(BA.3)

The total number of generators of Ms is

Hl

1+L b..(5)i = m + 1.i=m

It is easy to check that for m i t + 1 there are

-(b..5)i degree i + 1 relations among the generators of 5, (BAA)

and none other, for a total of m relations: this is one less than thenumber of generators of Ms, as required by the Hilbert-Burch theorem[Ei2, Theorem 20.15].

§ BA. MAXIMUM BETTI NUMBERS, GIVEN T 285

By a result of A. Geramita, D. Gregory and L. Roberts [GGR],a refinement of a special case of [Harl], any Artin algebra C definedby a monomial ideal in a polynomial ring can be smoothed to a zerodimensional subscheme X, whose coordinate ring Oz = R/Ix has alinear nonzero divisor, such that the quotient is C. It follows that Aand Oz have identical Betti information degrees and numbers ofgenerators, relation and that H(R/Iz) = H(T), the sum functionof S. We apply this construction to Ms, finding a smooth schemeX=Xs.

Choosing a general element f E (Ix)/-, we find that X is a tightannihilating scheme of f (Lemma 6.1). By Proposition B.10 the minimal resolution for I = AnnU) consists of that for Ix, patched to itsdual. In particular, we have

CLAIM B.14. A minimal generating set for I consists of two parts:

(i). First, the m + 1 generators of Ix c I, in degrees s: t + 1;

(ii). Second, m generators, beginning in degree i = j + 1 t,

corresponding by Vi = wj+3-i(Ix) of (B.3.4) to the relations of Ix.

In either case, the number of such generators of degree i is given bythe equation in (B.4.1).

PROOF OF CLAIM. Both (i) and (ii) follow directly from CorollaryB.12 and Equation BA.3. Since by (BAA) there are -b.Si relations ofdegree i + 1 for Ms, and hence for Ix, for m s: i s: t + 1, it follows by(B.3.4) that there are -b.Si corresponding generators for I in degreej + 3 (i + 1) = j + 2 i. But b.2(T)j+2_i = b.2(Tk If T :) (s, s)then we have

(B.4.5)

This completes the proof of the Claim if T:) (s, s). When T does notcontain (s, s), then there is there is equality in (B.4.5) for i s: t, but(b. 2Th+1 = 2(b.S)t+1: in this case j = 2t + 2, and the two parts ofthe generating set overlap in degree t + 1. This completes the proof ofthe Claim. D

Since there are 2m + 1 = 2v(T) + 1 generators for I, and this is themaximum number, given T from [Di, Theorem 3.3] (see (5.3.2)), theproof of the Claim completes the proof of Theorem B.13. D

A second way to show Theorem B.13 is to follow Diesel's determinationof Cmax(T) and Dmax(T) via saturation in [DiD, and then compare thepartition P(T) derived from Dmax(T)with the monomial ideal Ms (see


(B.4.6)

p. 123 and also the last paragraph of the proof of Lemma B.15 below).We leave this to the interested reader.

We now state a result of of S. J. Diesel, which is used in Theorem 7.15 ([Di, Proposition 3.9]). Let

Dmax(T) = (d1, ... ,d2m+I) , d1:S: d2:S: ... :s: d2m+1,denote the maximum set of generator degrees for T, (see Theorems 7.15and 5.25). We denote by P(T), and Q(T) the partitions (e.g., nonincreasing sequence of integers, possibly zero),

P(T) = (d2 m, , dm+1 m);

Q(T) = (d2 m, ,d2m+1 - m).

The partition P(T) is simply related to the "alignment character" ofT, by equation (4.4.16). Given a pair of integers, (m,j),j :.::: 2m, welet M = M(m, j) denote the blockpartition (whose Ferrer's graph is arectangle) having 2m equal parts, each of length m' = j - 2m + 2, andB = B(m, j) denote the block partition with m parts, each of lengtht + 1 m, where t = [j /2]. The first parts of the following Lemma arefrom [Di]; the last statement is from [19, Section III]. Recall that thesequence S is defined as f:1(T)<t in (B.4.2). We regard Sm :.::: ... :.::: S,as a partition (possibly with zero parts) having t + 1 m parts, eachof length no greater than m; the dual partition is obtained as usual byswitching rows and columns of its Ferrer's graph.

LEMMA B.15. The partition Q(T) = P(T) U PC(T), where PC(T)denotes the complement of P in an m bym' block. Also,

Cmax(T) = (m' + 1, m' + 1 2Q(Th, ... ,m' + 1 2Q(Thm).(B.4.7)

The partition P(T) is the dual partition to the partitionSm, Sm+l, ... ,St.

PROOF IDEA. Diesel's proof of the first statement relies on a studyof Cmax(T), as the unique saturation of Cmin(T) obtained by addingsuccessive pairs (a, -a) until the condition C, + Cv+ 2- i = 2 of (5.3.2)of Theorem 5.25(v.f) is satisfied for Cmax(T). She then shows that themaximum number of generators must be v = 2m + 1. Alternatively,once we have identified the generating set Dmax as in Claim B.14above (this requires knowing v(T) = 2m + 1, for which we relied on[Di, Theorem 3.3]), we obtain that Q(T) = P(T)uPC(T) from (B.4.4).Then equation (B.4.7) arises from the basic equation di+ei = d = j+3of Theorem B.2: the ith row of M has length m'; it is partitioned

§ B.4. MAXIMUM BETTI NUMBERS, GIVEN T 287

into di+1 - m, the i-th part Pi of P, and a complement P{ of lengthm' - (di+1 - m) = j - 2m + 2 - (di+1 - m) = ei+1 - m - 1. So fori 1, we have

(Cmax ) i+1 = ei+1 - di+1 = [m' - (di+1 - m + (m + 1)] - di+1

= m' + 1 - 2Q(T)i+1'

That P(T) is the dual partition of (Sm, Sm+1,' .. ,St) follows fromClaim B.14, and the description of the generators of M s in (B.4.3):from these and the definition of P(T) we have that for m :::; i :::; t,there are = Si-1 - S, parts of P(T) having length i - m. Thisshows that P(T) is indeed the dual partition as claimed. 0

EXAMPLE B.16. If T = (1,3,6,8,9,10,9,8,6,3,1) then S(T)(1,2,3,2,1,1), m = 3, t = 5,j = 10, m' = 6, P(T) = (2,1, l)V(3,1,0), and PC(T) = (6 - 3,6 - 1,6 - 0) = (3,5,6). SO Q(T)P(T) U PC(T) = (0,1,3,3,5,6), Dmax(T) = (3,3,4,6,6,8,9) (includ-ing an initial degree 3 = m), and Ms = (x 3, x 2y, xy3, y6). From (B.4.7)we have Cmax(T) = (7,7,5,1,1, -3, -5). From Theorem 5.25(ii),equation (5.3.1), Dmin(T) = (3,3,6,6,8) is read from andCmin(T) = (7,7,1,1, -3) is obtained by removing the (5, -5) pair fromCmax(T).

THEOREM B.17. [Di, §3.4] GORENSTEIN SEQUENCES WHENr = 3, AND PARTITIONS. The Gorenstein sequences T of order v(T) =m and socle degreej(T) = j correspond one to one via T --t P(T) tothe set P(B) of partitions of integers having no greater than m nonzeroparts, each part no greater than t +1- m. This is the set of partitionswhose Ferrer's graph fits in a m x (t + 1 - m) block B. There is like-'wise a one to one correspondence T --t Q(T) between such Gorensteinsequences T and the set SCP(M) of pairs (Q, M - Q), Q = M - Q ofself-complementary partitions inside the partition M = M (m, j).

The number of such sequences T is the same as the number ofelements of P(B), namely,

+ 1)-When r = 3, and j = 2t or j = 2t + 1, there are a total of 2t+1 - 1Gorenstein sequences of socle degree j.

PROOF. Evidently Dmax(T), hence T, is determined by the triple(P(T),m,j). Since P(T) is the dual partition to (Sm, ... St), whichmay be chosen arbitrarily in B V

, the rectangular partition with t+1-mparts of length no greater than m = v(T), by Theorem 5.25(i.), it


follows that P(T) runs through all partitions that fit into B. Thisestablishes the one-to-one map from the particular set of Gorensteinsequences to P(B). The set SCP(T) consists of partitions having theform Q = P u P", P E P(B), so is also in one-to-one correspondencewith the Gorenstein sequences of order m and socle degree j. Thecount of such partitions in P(B) is well known; we assume that m ?:1, so the total number of height-three Gorenstein sequences of socledegree j is 2t+1 - 1. 0

EXAMPLE B.18. If v(T) = 3 and j = 8, B has 3 rows of length 2,and there are (411

) = 10 partitions in P(B), from (2,2,2), (2, 2, 1), ...to (0,0,0). A simple way to reconstitute T from P(T) = (2,2,2), is toform PV(T) = (3,3), then S(T) = (1,2, ... ,m,pV(T)), here S(T) =

(1,2,3,3,3) then T5.t as the sum function JS(T), here (1,3,6,9,12),and symmetrize about j /2 to obtain T = (1,3,6,9,12,9,6,3,1).

For further discussion of the partition P(T), and its relation with thealignment character of T used by A. Geramita and others see pages122-124.

APPENDIX C

The Gotzmann Theorems and the HilbertScheme

by Anthony Iarrobino and Steven L. Kleiman

There are several excellent sources for results on O-sequences, andfor the Macaulay Theorem below describing the set of possible Hilbertfunctions (Theorem C.2) [Sp, Stl, BruH, St3, Gr2]. There are aswell several sources for the Gotzmann Persistence Theorem (TheoremC.17) [Gotl, Grl, Gr2, BruHF. However, the relationship of thePersistence Theorem to the Hilbert scheme is less well known, thoughit was an integral part of Gotzmann's work. We give here a briefsummary of these results. In particular, we show that the Castelnuovo--Mumford m-regularity a(P) of a Hilbert polynomial is the same asa combinatorial invariant rp(P), the Gotzmann number (PropositionC.24). We then state Gotzmann's Theorem describing the Hilbertscheme (Theorem C.29), and discuss its relationship to previous workof A. Grothendieck and D. Mumford. We also derive as a consequenceof the Gotzmann Theorem and an argument of Grothendieck, thatthe Hilbert scheme is given locally by simply described determinantalconditions (Proposition C.30), which were conjectured by D. Bayer[Ba, Chap.VI].

In Section C.5 we apply this Theorem and a related result ofG. Gotzmann from [Got4] to certain Gor(T), in a manner similar tothe applications in an article by A. Bigatti, A. Geramita and J. Migliore[BGM]. This leads to a better understanding of the structure of someGor(T) when T has a subsequence of maximal growth, and to a newexample of Gor(T) having several irreducible components when r = 4(Proposition C.33 and Example C.38).

1M. Green proved the theorem when the ground ring is a field in [Grl]. Seealso [BruH, 1998 ed. §4.3] and [Gr2] for this case

290 ApPENDIX C. THE GOTZMANN THEOREMS ...

C.l. Order sequences and Macaulay's Theorem on Hilbertfunctions

Let R = k[Xl, ... ,xr ] be a polynomial ring over an arbitrary fieldk. Given a monomial m = xfl ..... set exp(m) = (0'1, ... O'r)in f:JT. Recall that, in the lex order of monomials, ml > m2 if thefirst nonzero entry of (exp(ml) - exp(m2)) is positive. For example,xf > XI X2 > XIX3 > is in lex order.

DEFINITION C.l. LEX-INITIAL IDEAL AND O-SEQUENCES. Let Mbe an ideal in R that is generated by monomials. Then M is called alex-initial ideal, or segment ideal, if each homogeneous component M,has as basis the first dirm, M, monomials of degree i, those largest inthe lex order.f (Equivalently, M is an initial ideal in the graded lexorder of [CLO, p.57].)

A sequence S of integers is called an O-sequence if there is a mono-mial ideal M such that S = H(RjM).3

For example, (XI,XlX2,XlX3) is a lex-initial ideal.We'll see shortly that, because of Macaulay's Theorem, in the

defintion of O-sequence, M may be assumed to be lex-initial.

The following theorem was first proved by F. H. S. Macaulay. Itwas given a more elegant proof by F. Whipple, and was generalized byG. Clements and B. Lindstrom [Mac3, Wh, CIL]. For more detailson the theorem, see the reference [Bru'H], which is excellent.

THEOREM C.2. MACAULAY'S THEOREM ONHILBERT FUNCTIONS.Let H = H(I) be the Hilbert function of a graded ideal I in R; so Hi =dirrn, h Then there is a unique lex-initial (segment) ideal M = MHsuch that H = H(M).

COROLLARY C.3. Let H = (1, hI, ha, ... ) be a sequence of inte-gers. Then the following statements are equivalent:

A. H is the Hilbert function of a quotient S = RjI for some gradedideal I of R.

2The presentation in [BruH, pp.155-6] stresses the cobasis for I, so uses thereverse-lex order; their segment ideal is the last (smallest in reverse-lex order) tmonomials of degree d. By replacing the variables Xl, ... ,Xr by X r , . . . , Xl in thesegment ideal of [BruHl, we obtain our lex-initial ideal.

3The name O-sequence comes from order ideal, which is a subset M' of mono-mials closed under division. This is not an ideal in the usual sense, but rather M'is a basis of the graded complement of a monomial ideal M. Thus, an O-sequencegives the Hilbert function of the "order ideal" M', or, equivalently of the quotientRIM (see [St3, (2.1)l,[BruH, (4.2.1)]).

§ C.l. ORDER SEQUENCES AND MACAULAY'S THEOREM 291

B. H is an O-sequence.C. There is a lex-initial monomial idealM such that H = H (R/M).

One significant consequence of the Macaulay Hilbert-Function The-orem is that there is a minimum possible growth of the Hilbert func-tion H(I), where I is a graded ideal of R. A more precise statementis given in the following corollary. To state it, set ri = dimi, R; andfix 0 < t ri, Denote by V = V (t, i, r) the vector space spanned bythe first t monomials in R; in the lex order, and by (V(t, i, r))i+l thedegree-(i + 1) piece of the ideal generated by V(t, i, r). Evidently,

(V(t,i,r))i+l = Rl' V(t,i,r) = ({XIV, ... ,xrv} IV E V). (C.l.l)

Also, if Itt is the last monomial of V (t, i, r), then

Xrltt is the last monomial of R1 . V(t, i, r). (C.l.2)

COROLLARY CA. MINIMAL GROWTH OF AN IDEAL. Let I be agraded ideal of R, and set t = dirm, I, = H(Ik Then

H(I)i+l 2: dimi, (Rl . V(t, i, r)) .

EXAMPLE C.5. Suppose r = 3 and H(I)2 = 2. Then V(2, 2, 3) isequal to < xi,XlX2 >, and

(C.l.3)

So, by the corollary, dirrn,h 2: dim, R1 . V(2, 2, 3) = 5.If an ideal I' satisfies H(I'h = 3, then H(I'h 2: 6, since V(2, 3, 3)

is equal to < xr,xlx2,xlx3 >, and dimR lV(2,3,3) = 6.The vector space L =< xi,XlX2, x§ > is spanned by the first three

monomials of degree 2 in the reverse-lex order. Here dim R l . L = 7,not 6: the ideal (L) = R· L does not have minimal growth from degree2 to degree 3. Thus, we cannot use the initial monomials in reverse-lexorder to redefine the notion of segment ideal in Macaulay's Theorem.

We next consider the special case r = 2, which is used in the char-acterization of height-S Gorenstein sequences; these are the Hilbertfunctions of graded Artin Gorenstein algebras A = R/I where I is ofheight, or codimension, 3 (see [BE2, StI] and Theorem 5.25).

Let H = {hili 2: O} be a sequence, and set

v(H) = inf{i I hi < rd·Note that v(H) is the order of any ideal I such that H(Rj I) = H.

292 AP.PENDIX C. THE GOTZMANN THEOREMS ... ,

COROLLARY C.6. O-SEQUENCES WHEN r = 2. If hi = i + 1 fori < /./ and if ht+1 = 0 for some t., then H is an O-sequence for r = 2if and only if

t/ = hv - 1 -2: hv -2: ... ht -2: 0 and hi = 0 for i -2: t + 1. (C.1.4)

A graded ideal I c R = k[X1' X2] satisfies the condition of maximumgrowth H(Rj I)i = H(Rj1)i+1 = s if and only if there exists an hERssuch that I, = (h) n R i, and IH 1 = (h) n RH 1. If also I is generatedin degree i or less, then H(Rj I)i+k = 0 for k -2: O.

PROOF. First, suppose H is an O-sequence and H(Rj1) = H. Ifi -2: u, or equivalently, if I, i- 0, then by Corollary C.4, dimi, IH 1 >dimi, Ii, Since dirrn, R; = i + 1, it follows that

hi = dim, R; - dirrn, I, -2: dirru, R+1 - dimj, IH 1 = lu.;1·

Condition (C.1.4) follows immediately.Conversely, suppose that (C.1.4) holds. For i -2: u, form the

vector subspace Vi c R; spanned by the monomials in the interval[ i hi i-hi] Th 11 t . 1 f R TT' h, i+1-hi dxl' ... ,xl x 2 . e sma es monorrna 0 1 vi IS Xl x 2 , anit . t th hi +1 i+1-h i +1 b h > h S I TT1 IS grea er an Xl x 2 ecause i _ H1. et = EBi2:vVi.Then I is a lex-initial ideal such that H(Rj1) = H. Thus H is anO-sequence.

The last statements constitute a particular case of Part ii of Propo-sition C.32 below. In this case, there is a simpler proof, found in [11,p.56] and [Da, p.349]. For further generalizations to arbitrary r, see[Got4] and [BGM, pp. 219-221] and also Propositions C.33 andC.34. D

The next Corollary generalizes the inequality in the first part of Corol-lary C.6 to arbitrary r.

COROLLARY C.7. Let H be an O-sequence, i an integer. Ifi -2: Hi,then

(i) Hi -2: Hi+1, and

(ii) H is nonincreasing in degrees at least i. (C.1.5)

PROOF. Assume (i). Then i + 1 -2: i -2: Hi -2: Hi+1. So applying (i)to i + 1 yields Hi+1 -2: Hi+2 . Repeating yields (ii). Thus it remains toprove (i).

Let s = Hi. Since i -2: s, the last s monomials of degree i, in lexorder, are

(C.1.6)

§ C.2. MACAULAY AND GOTZMANN POLYNOMIALS 293

Let t = ri - s. Then the last (so t-th) monomial of the vectorspace V(t,i,r) is and the last monomial of Ru . V(t,i,r)is x:_I which immediately precedes the last s monomials ofR+u. It follows that cod(Ru · V(t,i,r)) = s in Ri+u. Taking u = 1,we have by Theorem C.2 that Hi+1 :::; s. This completes the proof of(i) and of the Corollary. D

C.2. Macaulay and Gotzmann polynomials

EXAMPLE C.8. MINIMAL GROWTH PERSISTS. Let V = V(t, i, r)and I = (V). So I is the ideal generated by the first t monomialsof degree i, say VI, ... , Vt. Then R I . V(t, i, r) is generated by themonomials in the set on the right in (C.1.1), and they clearly form aninterval in the lex order, starting with xi+ l . So, R I . V(t, i, r) is equalto V(t', i + 1, r) for some t'.

Continuing by induction, we conclude that, for every u 2': 1,

Ru . V(t, i, r) = V(t(u), i + u, r)

for some t(u). Thus, for I = (V), the growth of the Hilbert functionH(I) at each step is always the minimum allowed by Corollary CA.

It is not hard to show that the values of the Hilbert function H(I)zare given for z 2': i by a polynomial Q( t, i, r) in z with coefficients inthe rational numbers (Ql, and so the values of H(Rj I)z too are givenby a polynomial P(t, i, r); see [Gotl, (2.4)], [BruH, Corollary 4.2.9],and Remark C.ll below.

DEFINITION C.9. Given integers r, d, t with 0 < t :::; r«, we definepolynomials P(t, d, r) and Q(t, d, r)) in (Ql[z] by the formulas,

P(t,d,r)(z) = dimk(Rzj(Rz-d· V(t,d,r))),

Q(t, d, r)(z) = dimk(Rz-d· V(t, d, r)).

We say that P(z) and Q(z) in (Ql[z] are the Macaulay polynomial andthe Gotzmann polynomial of an r-variable lex-initial ideal if P =P(t, d, r) and Q = Q(t, d, r) for some d, t.

To describe these polynomials, we first determine the dimensionof V(t, d, r) in terms of the exponents in the t-th monomial of Rd inthe lex order. We denote by Ri the polynomial ring k[Xi' ... ,Xr ], soR I = R. Let t satisfy 0 < t < rd, and write the t-th monomial /1t,d ofRd as

(C.2.1)


(C.2.2)

for a unique sequence of integers ai with

o< ao ... ak d and 0 k r - 2.

Then V (t, d, r) satisfies

V(t d r ) = x ao . R ill xao-Ixa)-ao+l . R2, , I d-ao w I 2 d-a)rT\ ao-l a)-aO ai-2-ai- 1 ai-I-ai-2+1 n: rT\ •••w Xl X2 ... Xi- l . Xi . d-ai-I W

rT\ ao-l al-aO ak-ak-I+l Rk+l (C 2 )W xl X2 ... Xk+l . d-ak ..3

Since dimj, = C-:+U) , we have proved the existence part of the

following lemma.

LEMMA C.lO. [Got2, (2.3)] Suppose that d, r are fixed, and that tsatisfies 0 < t < rd. Then t has a unique "Macaulay representation,"

t = + ... + (C.2.4)

where the integers ao, . . . ak satisfy (C.2.2). The ai may be determinedfrom the t-th monomial J..lt,d by writing it in the form (C.2.1). Further-more, ifQ(t,d,r) = Q(t',d',r), then k = k' and ai = i k.

PROOF. Given a second representation of t determined by d'd, repeated application of Observation (C.1.2) yields =

whence = ai. 0

REMARK C.11. MACAULAY AND GOTZMANN POLYNOMIALS. Whatare the Macaulay and Gotzmann polynomials? Answers are given byEquations (C.2.6), (C.2.7) and (C.2.9) below. The preceding equa-tions (C.2.2)-(C.2.6) are well known (see [Gotl, Sp, Got4], whichare based on [Mac3, Wh]).

First, we determine the Gotzmann polynomials. By a similar ar-gument to that of Lemma C.lO ([Gotl, (2.4)], [Got2, 2.3]), we seethat if i d, then

dim(V(t d r))· = (i-ao+r-l) + ... + (i-ak+r-I-k). (C 2 5),,2 r-l r-I-k ..

Hence the polynomial Q(t, d, r ) E <Ql[z] is given by the formula,

Q(t, d, r)(z) = (z-arO!;-l) + ... + (C.2.6)

where the a; satisfy (C.2.2); here we expand the binomial coefficientsin (C.2.6) in the usual way. It follows from Lemma C.10, that the a;

are uniquely determined, given Q and r .

Note also that the right hand side of (C.2.6) depends only onthe sequence ao, ... , ai; and r ; but not on d and t; whence, so doesQ(t, d, r}. In particular, we may take d = ak: doing so is equivalent

§ C.2. MACAULAY AND GOTZMANN POLYNOMIALS 295

to assuming that f.1t,d is not divisible by x-, Since dimj, R; =

the Macaulay polynomial P(t, i, r) is given by the formula,

P(t,d,r) = - Q(t,d,r). (C.2.7)

In the preceding discussion, we followed Gotzmann's presentation,so (C.2.7) is not the "Macaulay representation" of P(t, i, r) that isfound in [BruH, Grl, Gr2, BGM]. We now explain the latter. Setc = r d - t, and expand c in its Macaulay representation,

c = + + ... + (k\ll)where k(d) > k(d - 1) > ... > k(l) O. (C.2.8)

Then we have

P(t d ) - (z+k(dl-d) ( z+k(d-l)-d ) (z+k(l)-d) (C 2 9), ,r - k(d)-d + k(d-l)-(d-l) + ... + k(l)-l' ..

In the above expressions, any terms at the end with k(i) < i vanish,and so are suppressed. It follows from [BruH, p.158] that if the t-thmonomial of Rd is written in the form,

m = Xj(1)Xj(2) ... Xj(d)' with 1 :S j(l) :S ... :S j(d),

then in (C.2.9) we have

k(i) = r + 1 - j (d+ 1 - i) + i - 2.

(C.2.1O)

(C.2.11)

The terms "Macaulay" and "Gotzmann" polynomials are not stan-dardized in the literature. For further discussion see the referencesalready mentioned, and as well [BaS2, Bla, Gasl, Gas2].

DEFINITION C.12. GOTZMANN NUMBER. Given a Gotzmann poly-nomial Q = Q(t, i, r), set CPr(Q) = ak where ak is the number de-fined by (C.2.6). Given a Macaulay polynomial P = P(t, i, r), soQ = - P, set CPr(P) = CPr(Q). In general, we will suppress thesubscript, and write cp(P) and cp(Q).

It is not hard to see from (C.2.10) that CPr(P) is the number ofterms in its Macaulay representation ([Grl]).

EXAMPLE C.13. If r = 4, then the 5-th monomial of degree 3is XIX§. SO by (C.2.1), we have (ao, aI) = (2,3). By (C.2.1O), wehave (k(I), k(2), k(3)) = (2,3,5); alternatively, the third Macaulayrepresentation of c = dirm, R3 - 5 is 15 = + + m, from whichwe can read off the k-sequence. Thus, we have

Q = Q(5,3,4) = + = (z!l) + (z;l),

P = P(5, 3, 4) = (z!2) + G) + (Z1 1) .


The Gotzmann number <p(P) = <p(Q) is al = 3, which is also thenumber of terms in the Macaulay representation of P.

EXAMPLE C.l4. Consider the Macaulay polynomial P = s (constant), when r 2': 2 and d 2': s. Let t = rd - s, and recall from theproof of Corollary C.7 that, in the vector space V(t, d, r), the last, sot-th, monomial is x:_l and that the last monomial of R; .V (t, d, r)is x:_l which immediately precedes the last s monomials ofRd+i. Thus, by definition, <p(P) ::; s; the opposite inequality can beseen directly by noting that, if d' = s land t' = rs-l s, thencodRi· V(t',d',r) = s+i, not s.

In the expression (C.2.l), we have ar - 2 = s, so k = r 2. Fromthis observation, or directly, we conclude that the Gotzmann number<p(P) = s. Also, if r 2': 3, then ao = ... ar - 3 = 1.

Alternatively, the Macaulay expansion (C.2.9) for P = s satisfiesk(i) = i for 1 ::; i ::; s, and all other terms vanish. So <p(P) = s,the number of terms. If d > s, then the expression (C.2.11) also givesk(i) = il for d 2': i > s, but the terms for which k(i) < i are omitted.

Let I c R be a graded ideal, and M the lexinitial ideal havingthe same Hilbert function; the existence of M is assured by Macaulay'sTheorem C.2. Let v(I) be the order of I; it is the smallest integer vsuch that L; 'I- O. For i 2': v(I), let f.li be the last monomial in M, in lexorder, and set Y = {f.lili 2': v(M)}. Pick simultaneously a degree <p(I)and a monomial f.lcp(I) E M as follows: let 0:1 be the smallest powerof Xl appearing in any element of Y; let 0:2 be the smallest power ofX2 appearing among those elements having Xl appear with power 0:1;and so on; finally, define

<p(I) = 0:1 + ... + O:r and f.lcp(I) = .....

Choose t so that f.lcp(I) is the tth monomial of degree <p(I).Macaulay's Theorem C.2 implies the following result.

COROLLARY C.l5. In degrees i at least <p(I), the Hilbert functionH(Rj I) is the following polynomial:

H(RjI) = H(RjV(t,<p(I),r)) = P(t, <p(I),r). (C.2.l2)

PROOF. Since M is a gradedlexinitial ideal, f.li+l ::; x; . f.li fori 2': v(I). Hence the power of Xl appearing in f.li is nonincreasing withi, so attains the minimum value 0:1, then is constant, say for i 2': ilwith il minimum. Similarly, for i 2': iI, the power of X2 appearing inf.li stabilizes at its minimum value 0:2, say for i 2': i2 with i2 minimum,and so on up to r 1. Clearly, i r - l = <p(I). Furthermore, if i 2': <p(I),

§ C.3. GOTZMANN'S PERSISTENCE THEOREM AND m-REGULARITY 297

then /-li+l = Xr/-li· Hence M n m'P(I) = (V(t, <p(I), r)). Therefore,H(Rj I) = P(t, <p(I), r) for i 2:: <p(I). 0

There is the minimum possible growth of H(I) from degree d tod + 1 if and only if, correspondingly, there is the maximum possiblegrowth of H(Rj I). The latter condition is given a concise combinato-rial treatment in [BruH, Theorem 4.2.10] or [St3].

As a matter of notation, if c = H(Rj I)d has d-th Macaulayrepresentation as in (C.2.8), and if P(t, d, r) is defined from it as in(C.2.9), set c(d) = P(t, d, r)(d + 1).

COROLLARY C.16. Under the above conditions,

H(RjI)d+l 2:: c(d). (C.2.13)

C.3. Gotzmann's Persistence Theorem and m-Regularity

We begin by stating G. Gotzmann's Persistence Theorem C.17.The persistence problem had been stated by D. Berman [Be] in thecontext of ideals in R; its answer is classical when r = 2 (see CorollaryC.6 above). Related theorems were proved recently by A. Aramova,J. Herzog, and T. Hibi [AHH] and by V. Gasharov [Gas2].

For u 2:: 0 and 0 ::; t ::; r«. define recursively N(t, d, r, 0) = t, and

N(t, d, r, u) = dimi, R1 . V(N(t, d, r, u - 1), d, r).

It is clear that N(t, d, r, u) = dirm, Ru .V(t, d, r) = Q(t, d, r)z=d+u; seeExample C.8.

H t < r d, then we have

dinu, R1 . V(t + 1, d, r) > dimi, R1 . V(t, d, r). (C.3.1)

Indeed, by Example C.8 both spaces are spanned by monomials form-ing an interval in the lex order with initial monomial By (C.1.2)the smallest monomial of R1 .V (t+ 1, d, r) is smaller than the smallestmonomial of R 1 . V(t, d, r). Hence (C.3.1) holds.

Note that (C.3.1) implies that N(t, d, r, u) is strictly monotonicallyincreasing as a function of t when 0 ::; t < rd.

Let A be an arbitrary Noetherian ring, and set S = A[Xl,'" x r ].Given a graded S-module M, denote by Md its piece of degree d.

THEOREM C.17. ([Gotl, Satz, p.61]) PERSISTENCE FOR IDEALSOF MINIMAL GROWTH. Let I be a homogeneous ideal of S, generated byId' and set M = S]I. If M, is A-fiat of rank P(t, d, r)i for i = d, d+ 1,then M, is so for all i 2:: d.

298 ApPENDIX C. THE GOTZMANN THEOREMS ... ,

In particular, take A to be a field k. Let Ie R be a graded ideal,and d > O. Set t = dim, Id , and assume dirrn, Id+1 = N(t, d, r, 1),which is the condition of minimal growth. Consider J = (Id), the idealgenerated by Id. Then dim; Jd+u = N(t, d, r, u) for all u 0; thatis, minimal growth persists. Such a vector space Id having minimalgrowth in degree d-l-l is called a "Gotzmann space" in [BruH, HePJ.

In the special case where A = k and I is monomial, the proof issomewhat easier than in the general case, and Gotzmann proved it in[GaU, (2.12)J using part of his version of the theory of Castelnuovo-Mumford m-regularity; see Section C.4 below. Gotzmann then derivedthe general case using Grothendieck's theory of the Hilbert scheme,and at the same time, he obtained a new description of the Hilbertscheme, which is discussed in the next section.

M. Green refined the proof of the Persistence Theorem in the spe-cial case A = k ([GrlJ, see also [BruH, Theorem 4.3.3, p.172, 1998ed.]). Green's key new ingredient is his Theorem 1 on p.77, whichgives a bound on the dimension of the quotient IdjtId- 1 for a generalelement t of R I . A generalization of M. Green's result to a generichomogeneous element of arbitrary degree when char k = 0 was shownby J. Herzog and D. Popescu; they apply this result to some con-jectures in higher Castelnuovo theory and Cayley-Bacharach theory[HePJ. V. Gasharov extended the Herzog-Popescu result to char k = P[Gas3J. In his 1996 notes, Green gives a slightly different proof of thespecial case using a "Crystallization Principle" based on D. Bayer andM. Stillman's criterion for m-regularity ([BaS2], [Gr2, p. 41,47]).

Let Z C jpJr-1 be the closed subscheme defined by the graded ideallz c R. Recall that there exists a polynomial Pz E Q[z] such that forj large enough,

in other words, Pz(z) is equal to the Hilbert function of RjIz forj »0. This polynomial Pz(z) is called the Hilbert polynomial of Z.Let We R; and u > O. Recall that, by definition,

W : u; = {h E u;.; IRuh c W}.

Recall also that the ancestor ideal Wv of Wv is defined by the formula,

Wv = L (Wv : Ru ) + (Wv ) ;

l:S:u:S:v

(C.3.2)


see Equation (2.3.3). Finally, given a graded ideal J of R, recall thatits saturation, denoted Sat(J), is defined by the formulas,

Sat(J) = lim Jv = {h I 3u 2': 0 S.t. Ruh c J}V->OO

= Uv2:u2:o(Jv : Ru ) .

COROLLARY C.18. Consider the ideal J = (Id) introduced afterTheorem C.l1. Its saturation Sat(J) is equal to the ancestor ideal Jd,and

Sat(J) n (Xl, .. . ,xr)d = J.

Furthermore, P(t, d, r) is the Hilbert polynomial of the subscheme Zjof rr- l defined by J.

PROOF. Note that, for u 2': 0 and i 2': d, we have

(Ru . J i ) : Ru = k (C.3.3)

Indeed, by descending induction on u, it suffices to prove for everyi 2': d that (Ji+l : R I) = k Since J = (Id), the growth from Jito Ji+l is always minimal by Theorem C.17. Set t = dim, J, andt' = dimk(Ji+1 : R I). Suppose t' > t. Then by Corollary CA,

dim; RI(Ji+1 : R I) 2': dim; R IV(t', i, r) = N(t', i, r, 1).

By the monotonicity proved above,

N(t', i, r, 1) > N(t, i, r, 1) = dim, Ji+l'

which is a contradiction. Thus (C.3.3) holds. Hence h E Sat(J) ifand only if hE (J) + Jd : R I + .... Therefore, Sat(J) is the ancestorideal Jd . Finally, it is immediate from the definition of P(t, d, r) andthe assumption on J = (Id) that P(t, d, r) is the Hilbert polynomialof z; 0

It is not hard to show that, if minimal growth happens from degreed to d+u, it happens from d to d-l-I (see [HeP, Lemma 4.6(ii)]).The following consequence of Macaulay's Theorem C.2 was proved byE. Sperner.

COROLLARY C.19. [Sp, p.161-163] Let Pz be the Hilbert polyno-mial of a closed subscheme Z of IPr - l . Then there is an integer i > 0such that, setting t = ri - Pz(i), we have Pz = P(t,i,r).

REMARK C.20. By Sperner's Corollary, the set of Hilbert polyno-mials of schemes in IPr -

1 is exactly the set of Macaulay polynomials


of Remark C.ll. This statement leaves open the question of recognizing which Hilbert polynomials occur for "good" schemes, for example CohenMacaulay schemes or Gorenstein schemes. For more, see[BruH, 1998 ed. §4.4].

Recall that a graded ideal I of R is called m-regular if

Hi (JPr 1 , i(m i)) = 0 for i > 0 (C.3.4)

where I is the sheaf associated to I. Recall that the Castelnuovo-Mumford regularity, or m-regularity, of I is the integer O"(I) defined bythe formula,

O"(I) = min{m I I is mregular}; (C.3.5)

see [Mum, p.99], and [BaM, §3]).D. Mumford proved that, if I is mregular, then it is tregular for

all t 2: m; moreover, its saturation is generated in degrees m and aboveby its piece of degree m.

PROPOSITION C.21. [Mum, p. 99] If i is m-regular and if t 2: m,then HO(i(t)) is spanned by HO(i(m)) ® HO(O(t m)), and t is t-regular, that is, u-ii« i)) = 0 for i > O.

The next Proposition is well known, and follows from the definitionof mregularity; for a proof see [Gotl, (1.2)].

PROPOSITION C.22. Let Z C JPrl be a closed subscheme, PitsHilbert polynomial, and Iz its saturated ideal. If Iz is m-regular,then H(RjIz)i = P(i) for i 2: m.

There has been much work on bounding the regularity degree ofIz for subschemes Z c JPn in terms, of, say, the degrees of generatorsof Iz. For arbitrary schemes, by an example of E. Mayr and A. Meyerthe bounds must be doubly exponential, but for "good" schemes thebounds are often even linear (see [MayM]; also [BaSI] for an effectiveregularity criterion, [BaM] for discussion and further references, andalso [BaS3, HoaM, Mall, Ma13, Ma14, MiV, Pa, SmSw, Cu]).

Given P(z) E Q[z], set

0"r (P) = inf{m lIz is m regular for every Z c JPr -1

with Hilbert polynomial P}.

We will sometimes suppress the subscript, and write O"(P).A. Grothendieck proved, using the finiteness theorem for Chow

coordinates, that O"r(P) < 00 [Grol, p.22l7], [ChW]. D. Mumford


introduced the theory of m-regularity, and showed that, given r, thereis a polynomial in the coefficients of P that gives an upper bound forO"r(P) [Mum, p. 101]. G. Gotzmann proved a significantly improvedupper bound for O"r(P), showing that a subscheme Z of JlDr-1 havingHilbert polynomial P, is cpr(P)-regular (see Definition C.12).

LEMMA C.23. GOTZMANN'S REGULARITY THEOREM[Gotl, (2.9)].If P is the Hilbert polynomial of a subscheme of JlDr - 1, then O"{P) :::;cp(P).

Although the following result is an immediate consequence, andwas known to experts, it appears not to have been stated explicitly inthis form in the literature. G. Gotzmann [priv. comm.] ascribes it toD. Bayer, who showed in [Ba] that the Castelnuovo-Mumford regu-larity O"(I) attains its maximum O"(P) at the lex-segment ideal, and inaddition gave a simple combinatorial expression for 0"(p).4 Our proofdepends on noting that when X r t fLt,d then the saturated lex-segmentideal determined by V(t, d, r) has Castelnuovo-Mumford regularity de-gree d. A lex-segment ideal is special in other ways, being a smoothpoint of the Hilbert scheme HilbP (JlDr-1), by a result of A. Reeves andM. Stillman [ReeS].

PROPOSITION C.24. Let P be a Hilbert polynomial. Then

O"(P) = cp(P).

PROOF. Set d = cp(P) and t = - P(d). Then the t-thmonomial fLt,d of Rd has no X r factor, and P = P (t, d, r); see RemarkC.ll just after (C.2.6). Set s = O"(P). Then s :::; d by Lemma C.23.Consider I = Sat(V(t, d, r)). Then H(I)d = t by Proposition C.22;hence, the last monomial of I d is fLt,d. Suppose s < d. Then I d =R1Id-1 by Proposition C.21. Let fL be the last monomial of I d- 1.Then z; fL is the last monomial of R1I d-1, and x; fL f. fLt,d. This is acontradiction. Hence, s = d, as asserted. 0

These regularity bounds are used, when constructing the Hilbertscheme, to show the finiteness of the construction. S. L. Kleiman's ar-ticle [Klel, Theorem 3] contains a generalization of Mumford's resulton regularity; his SGA VI article [Kle2] develops the theory further

4[Ba, §II.lO.l, II.lO.5]' cited also in [HaM, p.8]. G. Gotzmann mentioned ascompleting the proof of equality, a result in D. Bayer and M. Stillman's article[BaS2, Proposition 2.9]' (if char k = a and I is a Borel fixed monomial idealgenerated in degrees s; m and having a generator of degree-m then a(I) = m), andhis [Got2, (2.3)] (see (C.2.5) above). See also M. DeMazure's prenotes [DeM].


and uses it to prove a number of finiteness theorems for the Picardscheme, which had been announced by A. Grothendieck.

CA. The Hilbert scheme HilbP (II»r-l)

We discuss Gotzmann's description of the Hilbert scheme, whichhe obtained along with his persistence theorem, on the basis of Gro-thendieck's construction of the Hilbert scheme, and of his own versionof Mumford's theory of m-regularity [Gotl, Grol, MumJ.

Recall that the k-rational points of the Hilbert schemeHilbP (JIDr-1)represent the closed subschemes Z of JIDr-l whose Hilbert polynomialis P. In other words, there is a point PZ E HilbP (JIDr-1), whoseresidue class field is k, corresponding to a Z C JIDr-l if and only ifdimk(R/Iz)j = P(j) for j »0, where Iz denotes the saturated idealof Z. More generally, the k-maps T HilbP (JIDr-l) correspond bijec-tively to the T-flat closed subschemes of JIDr-l X T whose fibres haveHilbert polynomial P.

Consider the category C of Noetherian k-schemes, and its subcat-egory CA of affine schemes. Every scheme X defines a contravariantfunctor on C to the category of sets; it associates to a scheme T theset X(T) of maps T ----- X. Moreover, any functor on C restricts toa functor on the subcategory CA; for convenience, let X (A) stand forX(Spec(A)). Thus there are maps of categories,

C ----- ((functors on C)) ----- ((functors on CA))' (C.4.1)

The composition is fully-faithful; that is, given two schemes, everymap between their functors on CA arises from a unique map betweenthe schemes (see [EiH, Proposition IV-2J). In short, a scheme is deter-mined by its functor. A functor is said to be representable by a schemeif the functor is isomorphic to the one that arises from the scheme.

EXAMPLE C.25. Given 1 :::; P < r«, recall that the GrassmannianGrassP(Rd) is the projective variety that parameterizes the vectorsubspaces of Rd with codimension p. More precisely, GrassP(Rd) rep-resents the functor of summands of Rd with corank p. In other words,GrassP(Rd)(A) consists of the A-summands of F of 3d with corankp, where 3d = A Q9k Rd. (Note that the following conditions on anA-submodule F of 3d are equivalent: (1) F is a direct summand; (2)3d/F is projective; and (3) F is locally a direct summand. See [Ei2,pp. 615-16J.) Letting q = dim, Rd - p, we denote GrassP(Rd) also byGrass(q, Rd), which thus parametrizes subspaces of Rd having dimen-sion q.

§ C.4.THE HILBERT SCHEME HilbP(IP'r-l) 303

Grothendieck constructed the Hilbert scheme as a closed subschemeof a suitable Grassmannian. His main tool was his construction of the(relative) flattening stratification of a sheaf.

THEOREM C.26. [Grol] Given a polynomial P and integers randd, if d 2: (Jr(P), then the Grassmannian GrassP(d)(Rd) contains aclosed subscheme HilbP (lFr

- l ) that represents the functor of A-fiatsubschemes Z of whose fibers have Hilbert polynomial P. Asubscheme Z corresponds to the submodule (IZ)d of Sd.

The k-rational points of HilbP (IFr - l ) represent all the vector sub-spaces V C Rd such that the graded ring RI(V) has Hilbert polynomialP. More generally, the elements ofHilbP(IFr

-l )(A) represent all the

direct summands F of Sd such that the graded ring SI(F) is A-fiatwith Hilbert polynomial P.

To move on to Gotzmann's work, fix d and form the product,

G = GrassP(d)(Rd) x GrassP(d+l)(Rd+l)'

On CA, define a subfunctor W(A) of G(A) by

W(A) = {(F, G) E G(A) IF· SI = G}. (C.4.2)

REMARK C.27. G. Gotzmann assumed that d 2: 'Pr(P), and de-fined W(A) using the inclusion F . SI C G in place of the equationF . SI = G in (C.4.2). As he remarked, it is easy to show that,since d 2: 'Pr(P), the inclusion implies the equation [Gotl, (3.1),(3.2)].Here's how (priv. comm., Dec. 1998). Let M be a maximal ideal ofA, and set F' = FIMF and G' = GIMG. Then R IF' c G'sinceSIF c G. Now, dim R IF' = dim G' by Macaulay's Theorem. HenceR IF' = G'. So SIF + MG = G. Hence, by Nakayama's Lemma,the localizations at M of SIF and G are equal. Since M is arbitrary,SIF = G.

Thus it is a strong condition to require SIF to be contained ina direct summand G of Sd+l of the right rank. Furthermore, Gotz-mann's approach leads to local equations for the Hilbert scheme as asubscheme of the product G. Indeed, the condition that SIP C Gis equivalent to the condition that the map SIF -t H vanish, whereH = Sd+l/G. Thus the condition is realized by the subscheme whoseideal is the image of SIF Q9 H* in the structure sheaf.

In the proof of the next Proposition, we show that the inclusioncondition may be replaced by two rank conditions. They provide analternative approach, which leads to a proof of Proposition C.30.


The following result about W(A) is implicit in Grothendieck's construction of HUbP (pr-1).

PROPOSITION C.28. For any d, the subfunctor W(A) of G(A) isrepresentable by a subscheme W of G, and the first projection em-beds W in GrassP(d). If d 2: ar(P), then W contains the imageof HUbP (pr-1) under the (closed) diagonal embedding; on the levelof functors, an A-fiat subscheme Z of is carried to the pair((IZ)d' (IZ)d+l) E W(A).

PROOF. The question is local. So consider an affine open subscheme U of GrassP(d)(Rd)(A), and let A be its coordinate ring. LetP E GrassP(d)(Rd)(A) be the "universal" submodule of Sd, whichcorresponds to the inclusion map of U. For each variable Xi, considerthe map P t Sd+1 given by multiplication by Xi, and form the sum,

. pEBr Su. t d+1.

Then the image of u is simply F· Sl.Set q = P(d + 1) and qV = rd+1 q, and consider these two

conditions:

rk(u) ::; qV and rk(u) > qV - 1. (C.4.3)

The first condition is closed, and the second is open; formally, bothare given by the appropriate ideals of minors. So together, they definea subscherne V of U.

Let A' be an Aalgebra, (F', G') E W(A'). Then p' = A' (9 F;moreover, A' (9 u satisfies the conditions in (C.4.3) because this map'simage is equal to G'. So the first projection is an injection,

(3: W(A') '-t V(A').

We'll show (3 is a bijection.Let p' E V(A'). Then F' = A' (9AF. Set C' = Cok(u) (9 A'. Then

C' is a locally free A'module of rank q; indeed, keeping in mind Fitting's lemma [Ei2, 20.4, p. 493], see [Grol, pp. 2211516] or [Mum,8_1°, pp. 5557] or [Ei2, 20.8, p. 495]. Set S' = A'(9kR and G' =Then G' is a direct summand of of corank q since = C'and C' is locally free of rank q.

By construction, (F', G') E W(A'). Hence (3 is a bijection, andso V represents WIU. Now, V(A) W(A) c G(A); so the identitymap of V lifts to a map v: V t G. Then v is a section of the firstprojection of G; so v is an embedding. Thus the first assertion holds.

The second assertion follows directly from the theorem. D

§ C.4.THE HILBERT SCHEME HilbP(lpr-l) 305

In general, the two subschemes of G are certainly not equal, asit takes more than two degrees to determine the Hilbert polynomial.However, if d :2: 'Pr (P), then they are equal! Gotzmann proved it intwo steps. First, he proved that the embedding of HilbP (IPr -

1) in Wis open as well as closed. Then, inspired by Hartshorne's proof of theconnectedness of the Hilbert scheme, he degenerated an arbitrary pair(F, G) E W(A) into one in which F is spanned by monomials. So hecould now apply the special case of his Persistence Theorem, which heproved for this purpose.

This part of the argument can be simplified by using M. Green'slovely proof, which immediately yields the general case of the Persis-tence Theorem over a field [Grl]. Indeed, in Gotzmann's treatment,Corollary C.18 is derived from Theorem C.29. However, M. Greenproved the Persistence Theorem and so its Corollary C.18, directly forarbitrary graded ideals of R ([Grl, p.82], see also [BruH, 1998 ed.§4.3]' [Gr2, p. 47]). Finally, it is immediate from Corollary C.18 thatevery geometric point of W lies in HilbP (IPr -

1) .

THEOREM C.29. GOTZMANN'S HILBERT SCHEME THEOREM([Gatl, Satz,§(3.4),(3.7)]). If d > 'Pr(P), then the subschemesHilbP (IPr - 1) and W of G are equal.

In particular, Gotzmann's Theorem and the proof of PropositionC.28 imply that the first condition in (C.4.3), which is the conditionof minimal growth, already alone suffices to define HilbP (IPr - 1) asa subscheme of GrassP(d)(Rd).5 Hence we can describe this Hilbertscheme by local equations as follows.

Set p = P(d) and pV = rd-p. Use the set of all the monomials as abasis of Rd' and associate to each rd by pV matrix a subspace V of Rdof codimension p, namely, its column space. Choose a subset K of pmonomials of degree d, and form the affine space AK of those matriceswith the pV by pV identity matrix as thesubmatrix corresponding tothe monomials outside K. Then AK is an affine coordinate chart ofthe Grassmanian GrassP(Rd) , and as K varies, these charts cover.

PROPOSITION C.30. In the affine coordinate chart AK of the Grass-mannian GrassP(d) (Rd), the Hilbert scheme HilbP (IPr - 1) is defined by

5That the determinantal conditions suffice to give H ilb P crr- 1) as a reduced

algebraic set is shown in D. Bayer's thesis, where it is also explicitly conjecturedthat they give the scheme structure [Ba, p. 134]. This stronger statement waslater asserted, but not proved, in the first author's survey [15, Theorem, p.312], in[HaM, p.9], and also in [Cat, p. 584].


the ideal of all the (qV+ 1) by (qV + 1) minors of the rd+l by rpv matrix whose columns are the vectors Xi . "i E Rd+l for 1 :::; i :::; rand1 :::; j :::; p":

REMARK. We stated Proposition C.30 in local terms since thisform might be more useful for applications. It is equivalent to say thatHilbP (JPlr-l) is the determinantal scheme, in the sense of Definition5.19 (the definition there is valid over an arbitrary base), associatedto the following map of bundles:

JL : S @ £1 -----+ £d+l'

Here E, is the trivial sheaf R; @k OGrass, and S is the tautologicalsubsheaf of £d on GrassP(d) (Rd); also, JL is the induced multiplicationmap.

EXAMPLE C.31. If r = 4, d = 3, and JLt,d = X§X3' then from(C.2.1), ao = 1, al = 3, a2 = 3. So

Q(t, 3, 4) = (Z - + 3) + (Z - + 2) + (Z - + 1),and t = Q(t, 3, 4)z=3 = 12; whereas, c = 8 = (j) + G) + (i), so

p= (Z:I) + G) + =3z1,

of regularity degree O'(P) = 3. Then p = P(3) = 8 and q = P(4) =11; whereas, pV = Q(3) = 12 and qV = Q(4) = 24. The Hilbertscheme HilbP (JPl3) is defined locally on the affine coordinate charts ofGrass(12, R3) by the vanishing of the 25 x 25 minors of the 35 x 80matrix whose columns are Xl . Vj E R4 , as in Corollary C.30.

When r = 4 and P = 6z - 2, the Hilbert polynomial of a canonicalcurve, then O'(P) = 12 since P = (Zi l ) + G) + ... (Z14 ) + 6, so theMacaulay expansion of P will need 12 terms, implying O'(P) = 12.Here t = r12 - P(12) = 386: by writing 386 as in Lemma C.lO we haveQ = Q(386, 12,4) = (zj+3) + + C-\2+l) and J-L386,12 =

To obtain all components of HilbP (JPl3) one must work in degree 12,high compared to the Castelnuovo regularity 3 of a canonical curve([Cat, (2.23)]).

D. Bayer studied the action of Sl(r), and characterized the weightdecompositions of Hilbert points with respect to a maximal torus [Ba].This was further described by D. Bayer and 1. Morrison ([BaM], seealso [HaM, §4B]). Recently, R. Notari and M. L. Spreafico studiedthe stratification of HilbP (JPlr-l) by the locally closed subschemes

§ C.5. GORENSTEIN SEQUENCES WITH A CONSTANT SUBSEQUENCE... 307

HilbP (pr-l)M parametrizing those subschemes Z having a given mono-mial ideal M as initial ideal of Iz; they show that the strata are ei-ther affine schemes (when M is itself saturated), or locally closed sub-schemes of affine schemes, and they give equations for them [NotSp].

G. Gotzmann used his identification of the Hilbert scheme in fur-ther work, [Got2, Got5], the former on simply connected Hilbertschemes (see Proposition C.36 below, and also D. Mall's related [Mal2]).As well, F. Catanese used the Gotzmann Theorem to bound the com-plexity of the Hilbert scheme [Cat, p. 584].

There has been other recent work related to the connectednessor smoothness of the general Hilbert scheme: A. Reeves on "radius",K. Pardue and D. Mall concerning the connectedness of the Hilbertfunction strata H ilbH (pn) of the Hilbert scheme (generalizing G. Gotz-mann's proof of connectedness when ti = 2 in [Got3]), and J. Cheah'sstudy of smooth nested punctual Hilbert schemes [Che, Ree, ReeS,Pa, MaI4].

For a survey of earlier work on the punctual Hilbert scheme to 1986see [15]. For further references, and a discussion of the constructionof the Hilbert scheme, see the Lecture notes by E. Sernesi and byE. Sernesi and C. Ciliberto [Ser, CiS], and two Ergebnisse Volumes,the first by J. Kollar [Ko, §1.1], who also discusses the relation with theChow variety, and the second by E. Viehweg, who also treats geometricinvariant theory on the Hilbert scheme [Vi, Chapters 1,7].

C.5. Gorenstein sequences having a subsequence ofmaximal growth, and HilbP(pr-l)

Recall that a Gorenstein sequence T is a sequence that occurs asthe Hilbert function of a (graded) Artin Gorenstein quotient R/I. So Tis symmetric. One consequence of this symmetry and of the PersistenceTheorem C.17 is that the growth from T; to 1i+l rarely attains themaximum permitted by Corollary C.4 of Macaulay's CharacterizationTheorem C.2.

Indeed, set t = ri - Ti. Then Corollary C.4 says that

Ti+1 ::; ri+l - N(t, i, r, 1).

When equality occurs, then I is rather special. For one thing, thePersistence Theorem implies that Sat((Ii)) = Sat((Ii+l)) and thatthe Hilbert polynomial of R/(Ii) is P(t, i, r). So far though, we havenot used the symmetry of T. However, it restricts I further, as isillustrated by Proposition C.33 below.


A similar phenomenon was studied by A. Bigatti, A. Geramita, andJ. Migliore [BGM] in a different context. They considered a subvariety X of I!DTl whose Artin minimal reduction (RI (L)) IIx) + (L)),where L is a suitable vector space of linear forms, has a Hilbert functionH with a constant subsequence (Hd = S,Hd+1 = s) where s :::; d. IfX is a reduced (smooth) set of points satisfying a uniform positionproperty, they show that then X must lie on a reduced irreduciblecurve of degree s [BGM, Theorems 4.7]. They also study maximumgrowth in a wider context.

We are particularly interested in the case that the Hilbert polynomial P(t, i, r) is constant, and this case is treated in the followingProposition. The first two parts are immediate consequences of Gotzmann's work, and he treats Part (i) explicitly when r = 3 in his paper[Got3, p.544].

Recall that we denote by Crass" (Rd) or Grass(sv, R d) the Grassmanian parametrizing sdimensional quotients of Rd' or equivalently,sVdimensional subspaces, where SV = rd s.

PROPOSITION C.32. REGULARITY FOR THE CONSTANT POLYNOMIAL

i. Let P = s be the constant polynomial. Then a(P) = <p(P) = s.ii. Let I be a graded ideal of R such that, for some d 2': s,

dim; RdlId = dirrn, Rd+dId+ 1= s. (C.5.1)

Set Z = Proj RI(Id). Then Z is a subscheme of I!DTl of dimension 0 and degree s. Furthermore, the saturated ideal Iz ofZ satisfies (IZ)i = (Id)i for i 2': d and (Iz )d+l = Id+l.

iii. If d 2': s, then HilbS(I!DT1) is the subscheme of GrassS(Rd)determined by the condition dirrn, V . R 1 = rd+l s.

PROOF. Part (i) holds because <p(P) = s by Example C.14 andbecause a(P) = <p(P) by Proposition C.24. Alternatively, a(P) = sby Theorem 1.69.

For Part (ii), first note that, since d 2': s, by Example C.14, there isminimal growth in size from Id to Id+1 in the sense of Corollary C.4 ofMacaulay's theorem C.2. So since R1Id C Id+l, we have R1Id = Id+l'Part (ii) then follows from Gotzmann's Persistence Theorem C.17 andits Corollary C.18 because P(t, d, r) = s holds for t = rd s in view ofExample C.14.

Finally, Part (iii) follows from Part (i) and Proposition C.30. D

Recall that the socle degree of a Gorenstein sequence T is thelargest integer j such that Tj i= O.

§ C.5. GORENSTEIN SEQUENCES WITH A CONSTANT SUBSEQUENCE. . . 309

PROPOSITION C.33. Let s, d, j be integers satisfying 0 < s :s: d < jand let T = (Ti) be a Gorenstein sequence of socle degree j having aconstant subsequence (Td = S, Td+1 = s). Then the following assertions hold.

i. The sequence T is nonincreasing for i ?:: d and nondecreasingfor i :s: j - d. If d < j /2, then T is nondecreasing for i :s: d,constant at s for d :s: i :s: j - d, and nonincreasing for i ?:: j - d;in particular, then T is bounded above by s.

n. Let f E Gor(T). Set I = Ann(J) and Z = Proj R/(Id). ThenZ is a subscheme of JIDr-l of dimension 0 and degree s, andis an annihilating scheme of f. If also d :s: j /2, then Z isa tight annihilating scheme of f (see Definition 5.1). Always,Ii = (IZ)i for i:< max(d,j - d).

PROOF. To prove (i), we won't use the hypothesis Td = Td+1.

Indeed, Corollary C.7 implies that T is nonincreasing for i ?:: d. Now,T is symmetric about j /2; hence, T is also nondecreasing for i :s: j - d.If d :s: j /2, then the assertions about T follow formally. Thus (i) isproved.

Consider (ii). Let I be a Gorenstein ideal with H(R/1) = T, andlet J = Sat((Id)). Since s :s: d, part (ii) of Proposition C.32 impliesthat J, = (Id)i and H(R/ J)i = s for i ?:: d. It also says that Z is ofdimension 0 and degree s. Moreover, Corollary C.18 and Lemma 2.17imply that Iz c I; so Z is an annihilating scheme of f. If d :s: j /2,then £diff(J) = s by Part (i); since deg(Z) = s, therefore Z is a tightannihilating scheme by Definition 5.3. Since Jd = I d , Lemma 2.17 alsoimplies J, = I, for i :s: d. If d :s: j /2, then also H(R/ J)i = H(R/1)i = sfor d :s: i :s: j - d, implying that J, = I, for i :s: j - d in that case. D

More generally, we have the following result with a similar proof.

PROPOSITION C.34. Let P be an arbitrary Hilbert polynomial, letd, j be integers satisfying <p(P) :s: d < i, and let T be a Gorensteinsequence of socle degree j such that Td = P(d) and Td+l = P(d + 1).Then the following assertions hold.

1. The sequence T satisfies the bound T, :s: P( i) fOT i ?:: d. Ifd:S: j/2, then Tu:S: P(j - u) for u:S: d.

ii. Let f E Gor(T). Set 1= Ann(J) and Z = Proj R/(Id). Then Zis a subscheme of JIDr-l with Hilbert polynomial P. Furthermore,we have Iz c I, with equality in degrees d and d + 1.

EXAMPLE C.35. If r ?:: 3 and if T = (1, r, T 2 , ... ,T, 1) satisfiesj (T) = 9 and T4 = 4, then any f E GOT(T) has a degree-4 tight


annihilating scheme Z = Proj Rj(I4) where I = Ann(f). Indeed, thesymmetry of T implies that Ts = 4; so the assertion follows from Part(ii) of Proposition C.33.

Next we state a result of G. Gotzmann, which describes the structure of subschemes of pr-l with certain Hilbert polynomials P(t, d, r).The condition that k ::; r 3 is equivalent to the condition that, inthe lex order, the tth monomial /-It of Rd is not divisible by Xr-l' Wemay also take d to be the minimum possible; that is, if P(t, d, r) =P(t', d', r), then d' d. It is equivalent to assume that /-It is not divisible by z: see Remark C.1l. The integers ao ::; ... ::; ak appearin Gotzmann's expansion of the polynomial Q(t, d, r); see (C.2.2). Seta-I = I.

PROPOSITION C.36. ([Got4, Propositions 1,2]' [Got2, Satz 1])6Let P = -Q(t, d, r), and assume that the Gotzmann polynomialQ( t, d, r) has k ::; r 3, see (C.2.2). Then each saturated ideal I ofR with Hilbert polynomial P(Rj1) = P has, after a suitable lineartransformation of pr-l, a generating set of the form,

(C.5.2)

where deg fi = o.; ai-l for 0::; i ::; k - 1 and deg fk = ak ak-l + l.Furthermore, the Hilbert scheme HilbP (pn) is irreducible and simplyconnected, and under the action on it of the upper triangular group,Zo = ProjRj(V (t, d, r)) is the only fixed point.

EXAMPLE C.37. The Hilbert function T = (1,3,4,5,4,3,1) hasmaximum growth from degrees 2 to 3. So T corresponds to the gradedlexinitial ideal (V(2, 2, 3)), whose Hilbert polynomial is P(z) = z + 2.Part (ii) of Proposition C.36 implies that, if f E Gor(T), then I =Ann(f) is identical in degrees 2 and 3 with the saturated ideal Iz of asubscheme Z of p2 having Hilbert polynomial P = z + 2; namely, Z iseither a line union a disjoint point or a line with an embedded point.

It follows that h = h· V, where h E R I and V is the two dimensional subspace of R I defining the point in p2. This is the Hilbert function T2 of Example 3.6. Note that Q(z) = (z!2) P = C;l) + C1I),so (ao, al) = (2,2). Although k is r 2, not r 3 as is assumedin Proposition C.36, nevertheless Iz is of the form (C.5.2); indeed,Iz = (hx, hy), with h, x and y each of degree one.

6There is a typographical error in the degree of fk in Proposition 2 of [Gat4].Equation (C.5.2) agrees with the analogous Formula (2.3)(ii) of [Gat2].

§ C.5. GORENSTEIN SEQUENCES WITH A CONSTANT SUBSEQUENCE.. . 311

J 6 7 8 9 10dirrn,Tp 57 63 71 83 96dim(CH) 50 60 71 83 96

TABLE C.l. Dimension of the tangent space Tp at apoint of Gor(T) with T = Sym(H,j), and the corresponding value of dim(CH); see Example C.38.

Recall that Sym(Hz, j) denotes the symmetrization of the sequenceHz about j/2 (see p. 108).

EXAMPLE C.38. Let P = P(5, 3, 4) = (z!2)+G)+ Then thefifth monomial of R3 is XIX§, and Gotzmann's expansion has ao = 2and al = 3; see Example C.13. By Proposition C.36, any subschemeZ of jpJ3 with Hilbert polynomial P satisfies Iz = (Ix, fg), wheredeg f = 1 and deg x = 1 and deg 9 = 2. We claim that, if F is generalenough in (Iz)t, then the Gorenstein ideal I = Ann(F) contains Iz;furthermore, then

H(R/1) = (1,4,9,15,9,4,1) = Sym(Hz, 6),

and I, = (IZ)i for i = 2,3.This claim was confirmed by a calculation using the symbolic alge

bra program "Macaulay" in the special case Zo = ProjR/(V(5, 3, 4));the general case then follows by deforming from this special case.This construction determines a subfamily CH of Gor(T) where T =H(R/ I). The tangent space Tp is of dimension 57 at a general pointF lying over Zoo The construction can be compared with similar constructions in Chapter 6; the latter yielded irreducible subfamilies ofcertain Gor(T), consisting of forms having tight annihilating schemesof dimension 0, of a certain kind (as smooth, or "compressed"). Herethe "annihilating scheme" is of higher dimension, an idea introducedby M. Boij.

By choosing different values of i, we may construct in this wayother examples of Hilbert functions T = Sym(Hz,j). For j = 8,9,10,the dimension of the tangent space Tp is equal to the dimension ofthe subfamily CH; hence, CH is an irreducible component of Gor(T).By the argument in the proof of Theorem 6.27, which depends onthe result of P. Maroscia (see [Mar, Theorem 1.8], and Theorem5.21(A) above), we see that Gor(T) has at least two irreducible components: a second component lies over a family of smooth adimensionalsubschemes of jpJ3. The first such example has j = 8 and T =


(1,4,9,15,22,15,9,4,1); this example is similar to some examples thatwere constructed previously by M. Boij (see [Bo2] and Theorem 6.42).

Here we use the formula, dim CH = P(j) -l-dim HilbH (JID3); it holdssince F is chosen in (Iz)/, which is a vector space of dimensionP(j). To see that dim H ilbP (ITD3 ) = 11, note that the choice of Z isequivalent to the choice of two linear forms f, x and of a degree-2form 9 mod xR1, up to constant multiples; hence, dim H ilbP (JID3) =3+3+5. To produce Table C.1, we used the symbolic algebra program"Macaulay" to calculate the dimension of the tangent space Tp at apoint F E Gor(T) lying over Zoo

ACKNOWLEDGMENT. The authors of the Appendix would like toacknowledge the helpful comments of G. Gotzmann and V. Kanev.

Author addresses for Appendix C:

Anthony IarrobinoDepartment of Mathematics, 567 Lake HallNortheastern UniversityBoston, MA 02115USAe-mail address: [email protected]

Steven L. KleimanDepartment of MathematicsRoom 2-278, MIT77 Mass Ave, Cambridge, MA 02139-4307USAe-mail address: [email protected]

APPENDIX D

Examples of "Macaulay" Scripts

We used the D. Bayer, M. Stillman algebra program "Macaulay"[BaSI] to find the dimension of the tangent space to Gor(T) at thepoint parametrizing a given degree- j homogeneous form f: by The-orem 3.9 this is dimk(Rj/IJ) where I = Ann(f). If f E PS(s,j;r),

so f = L{ + ... + E R j , and L1, ... , L; are general enough ele-ments of R 1, and s ::; rt-l, t = l}/2J then by Lemma 3.16 it suffices

to calculate where Z is the vanishing ideal at thecorresponding points of JP>n. The following calculation shows that thehypthesis "general enough" is necessary in Lemma 3.16. In practice,to find a Gorenstein ideal I we replace R by the divided power ring

V, find f = + ... + then calculate the ideal I = Ann(f) inthe contraction action of R on V.

Warning: Using the contraction action - <Lfrom.dual ...in "Macaulay" - with ordinary powers, will give wrong answers. SeeAppendix A.

Dimension of the tangent space at a form F with annihi-lating scheme Z = 10 points on a rational normal curve. Wefind dirrn, Tp ; F = LI + ... + Lio E R 7, where £diff(F) = 10, and Fhas a tight annihilating scheme Z consisting of 10 points on a RNCin JP>4.<ring 5 v-z r<ideal L v v+w+x+y+z v+2w+4x+8y+16z v+3w+9x+27y+81z

v+5w+25x+125y+625z v+7w+49x+343y+2401z v-3w+9x-27y+81zv-2w+4x-8y+16z v-5w+25x-125y+625z v-7w+49x-343y+2401z

<pow_entry L 7 10 POWERS [This script forms ... , where

= v7 + v6w + ... ].

[F . li bi f L[7] L[7]]IS a mear com ination 0 l' ... , 10

[Annihilator in contraction action]

randoID 101MIDult POWERS M F<I_froID_dual F Istd I Ihilb I Ans: 1 5 9 10 10 9 5 1 (This is T)

314 ApPENDIX D. EXAMPLES OF "MACAULAY" SCRIPTS

pow 1 2 12std 12 12hilb 12 Ans: 1 5 15 35 50 51 50 44 26 13 9 5 1 = H(Rj12 ) .

Thus, the dimension of the tangent space to Gor (T) at F is dim, Tp= dimk(Rd1'1) = 44. Compare with Example 6.14, where j = 6 inplace of j = 7 here, but the tangent space has the same dimension.

Hilbert function of the square of the vanishing ideal at Z.The scheme Z c RNC C lpA corresponds to the linear forms Labove,and is defined by the ten points Z = {(I, 0, 0, 0, 0), (1, 1, 1, 1, 1), (1,2,4,8,16), ... ,(1,-7,49,-243,2401)}. By Lemma 6.I(c) Z is the uniquetight annihilating scheme for F. We find Htranspose L T [needed because of how "coef" works]coef T MONOM Mtranspose M P [P is now a 5 by 10 matrix with rows 1 a a2 a3 a4 ]<points P 12std 12 12 [vanishing ideal IZ = Iz]hilb 12 Ans: (1,4,4,1). [thus, H(RjIz) = (1,5,9,10,10, ... ) and

T(Z) = 3, so (Z, F) satisfy the hypotheses of Lemma 6.I(c)]pow 12 2 SQstd SQ SQhilb SQ Ans: 1 4 10 20 15 1 -1

Thus, (1,5,15,35,50,51,50,50, ... ), satisfying2: H(RjI2 ) , a consequence of I = Ann(F) c Iz. This

calculation is the line 5Q(5) in Table 6.1.

Hilbert function for the symbolic square I12)transpose L Tcoef T MONOM Mtranspose M P [P is now a 5 by 10 matrix with rows 1 a a2 a3 a4 ]<ideal Wx2 x2 x2 x2 x2 x2 x2 x2 x2 x2 [to get weights all 2]<points P J W

std J J [vanishing ideal J = I12)]hilb J Ans: 1 4 10 19 13 3 codimension = 4

Thus H(RjI(fl) = (1,5,15,34,47,50,50, ... ). This calculation isthe line 5YM5Q(5) of Table 6.1. See Example 6.17.

REMARK. Here dimj ]Rj(I12)h) = dim, (Rj h) = 50 > dirru, Tp= 44. When the linear forms L 1, ... , L lO determining the summandsLi of F are not sufficiently general- here they lie on a RNC - then the

ApPENDIX D. EXAMPLES OF "MACAULAY" SCRIPTS 315

dimension of the tangent space is not computed by dirrn,(R/(I12)h, asLemma 3.16 does not apply.

APPENDIX E

Concordance with the 1996 Version

We here give a comparison of the sections and chapters with thoseof the May, 1996 manuscript [IK]. We do this because the manuscriptwas circulated, and there are several references to it in the publishedliterature. Several of these are to the result describing the tangentspace to Gor(T), which is Theorem 3.9 here.

Almost all sections retained from the earlier version have manychanges and improvements; often the theorems stated here are stronger.We also tightened language, for example distinguishing between "family" of varieties and "parameter space". Conversion to LaTeX alloweda clearer notation, for example Gor(T) for scheme and Gor(T) for therelated variety or algebraic set. Many of the changes are related toour desire to make the book more userfriendly, and to include moreexposition and examples. This led to changes throughout, but is especially true of the added Introduction, Sections 1.2, 1.3, the two surveySections 4.4, 6.4, the added Sections 5.2, 5.4, 5.6, much of Appendix Band the new Appendix C. We outline below the main changes.

The Introduction is new, an informal preparation for the book.Chapter 1 is completely rewritten; and Sections 1.2, 1.3 are new.

Section 1.2, although an introduction to the cases j = 2,3 surveyssome new results. Section 1.3 concerns the binary forms case r = 2and Hankel matrices; it also assembles material not before collected inone place. Section 1.4 is updated from the old Detailed Summary.

Section 2.1 on the Waring problem is new, and contains a newresult in char p, the proof of Theorem 1.61 for charp t j. Sections 2.2and 2.3 are largely from the old Chapter 1, but contain improvements,especially in char k = p.

Chapter 3 is the old Chapter 2, with a new result Proposition3.14 for corank one catalecticant varieties.

Sections 4.1, 4.2 are the old Sections 3A, 3B, with some improvements and updating.

Section 4.3 is the old Appendix A2. This extends several of theresults of Section 4.2, but uses a different approach.

318 ApPENDIX E. CONCORDANCE WITH THE 1996 VERSION

Section 4.4 describes recent developments concerning Gor(T) whenr = 3, some inspired by the 1996 version of this book. This is rathermore narrow than the survey by the first author in [19], written in1997, but the narrow focus allows us to explain certain developmentsin more depth.

Chapter 5 is largely revised and rewritten from the old Section4A (new 5.1), 4E (new 5.3) and 4F (new 5.7). The new Section 5.2contains expository material on flat families of O-dimensional schemesand limit ideals, and a determinantal construction of the postulationHilbert scheme. Section 5.3 is greatly expanded, with added exposi-tion and details of proofs as well as additional results. Section 5.4,a summary on power sums, and Section 5.5, an application, are new.Section 5.6 is also new; it collects results concerning lengths of a formand the closure of PS(s, j; r), some not explicitly stated in the 1996manuscript.

Chapter 6 is mostly from the old Sections 4B-4D, with some im-provements. However, a new update Section 6.4 is added.

Chapter 7 is from the old Chapter 5, but with many revisions, andsome improvements.

Chapter 8 is based on the old Chapter 6.Chapter 9 on open problems has been extensively rewritten. A

few of the problems mentioned as open in 1996 have since been solved!(See, for example, Sections 4.4, 6.4).

Appendix A corresponds to the old Appendix A3, but is completelyrewritten and extended. Appendix B is new except for a part of Sec-tion B.2, which is from the old Appendix AI. The new Appendix C,by the first author and S. L. Kleiman, states basic results concern-ing extremal Hilbert functions, states Gotzmann's construction of theHilbert scheme, and as well shows a determinantal variation of thatconstruction, conjectured by D. Bayer; Section C.5 gives applications.Appendix D is the old Appendix A4.

There are rather more references. This is partly due to our de-cision to survey recent work in Sections 4.4 and 6.4, and to the newAppendix C; and partly due to our wish to give the reader the choiceto look further. We have added A. M. S. Math Review numbers afterseveral of the references, when we felt the review might be particularlyuseful. Because of the added introductory material, we decided to omita List of Theorems. However, we added substantially to the indexes,and have for the reader's convenience included short definitions in theIndex of Notation.

[ABW]

[AI]

[AlHl]

[AlH2]

[AlH3]

[AlH4]

[AnL]

[AHH]

[ACGH]

[AM]

[Bas][Ba]

[BaM]

[BaMo]

[BaSI]

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[Wa4]

[Weyl]

[WeI]

[We2]

[We3]

[We4]

[Wh]

[Ya1]

[Ya2]

REFERENCES

Wall C. T. C.: Determination of the semi-nice dimensions, Math. Proc.Camb. Philos. Soc. 81 (1977),351-364.Watanabe J.: A note on Gorenstein rings of embedding codimensionthree, Nagoya Math. J. 50 (1973), 227-232.___ : The Dilworth number of Artinian rings and finite posets withrank function Commutative Algebra and Combinatorics, AdvancedStudies in Pure Math. Vol. 11, Kinokuniya Co.j/North Holland, Am-sterdam, (1987) pp. 303-312.___ : The Dilworth number of Artin Gorenstein rings Adv. Math.76 (1989), no. 2, 194-199. MR90j:13023.___ : Hankel Matrices and Hankel Ideals (preprint,1985); 2nd rev. inQueen's Papers in Pure and Applied Mathematics 102, Curves Sem-inar at Queen's University, vol. 10, (A. Geramita, ed.), (1996), pp.351-363; 3rd. rev. Proc. School Sci. Tokai Univ. 32 (1997), 11-21. MR97g:13022,98c:13020.Weyl H.: The classical groups, 2nd ed., Princeton Univ. Press, Prince-ton, (1946).Weyman J.: Resolutions of the exterior and symmetric powers of a mod-ule, J. Algebra 58 (1979), 333-341.___ : On the equations of conjugacy classes of nilpotent matrices,Invent. Math. 98 (1989), 229-245.___ : On the Hilbert function of multiplicity ideals, J. Algebra 161(1993), 358-369.___ : Gordan ideals in the theory of binary forms, J. Algebra 161(1993), 370-391.Whipple F.: On a theorem due to F. S. Macaulay, J. London Math. Soc.28 (1928), 431-437.Yameogo J.: Decomposition cellulaire de uarietes parametrant desideaua: homoqenes de lC[[x, yJ]. Incidence des cellules I, CompositioMath. 90 (1994), 81-98.___ : Decomposition cellulaire de uarietes parametrani des ideauxhomoqenes de lC[[x, yJ]. Incidence des cellules II, J. reine angew. Math.bf 450 (1994), 123-137.

Index

ACM,134AD, see additive decompositionadditive decomposition, see decom-

position of formgeneralized, 138

Alexander-Hirschowitz Theorem, 40-41, 46, 58, 66, 96, 101

alignment character, of T whenr = 3, 122-124,286-288

annihilating scheme, 54,135-141,202,207-213, 308

a self-associated point set, 224and additive decomposition, 136conic, 208-211generalized, sequence of, 236locally Gorenstein, 227-234

compressed, 222compressed nonhomogeneous,

230, 234nonsmoothable, 221-226nonunique, 139not recoverable from any form,

141of higher dimension, 311points on a RNC, 217-220tight, 135, 202uniqueness of, 136, 208

apolar polynomials, 8apolarity, 4, 12Aronhold invariant of a plane cubic,

247Artinian Gorenstein algebra, see Goren-

stein Artin algebraassociated point, 142Auslander-Buchsbaum Theorem, 162

Bertini Theorem, 64Betti number strata

for f E Gor(T) when r = 3, 125,153-156, 160, 172, 173, 191

dimension of, 125for smooth aG curves in p4, 203of Gor(T), local closure of, 151,

191of the Hilbert scheme Hilb" (!P'2) ,

172, 189-196Buchsbaum-Eisenbud structure

theorem, 119, 126, 132, 160,199, 204, 255, 277

canonical curves, 215canonical form, xiv

of a binary form, 25Castelnuovo-Mumford regularity, 48,

300algebraic version, 49Gotzmann upper bound, 301

attained for lex-segment ideal,305

catalecticbinary form, xivline of verse, xiv

catalecticantdeterminant, irreducibility of, 80determinantal locus, 5

CI component, 76, 112-116connectedness of, 239high dimensional components,

when r = 3, 121irreducibility of in corank one,

81irreducible component of, 92,

102, 105, 112, 242-244reducible, 242reducible, r = 3, 76tangent space to, 74

336 Index

homomorphism, 4kernel and image of, 74

ideal, 5prime, 263prime, for V.(l,j - l;r) (Ja-

cobian), 20prime, in corank one, 81prime, when r = 2 (binary),

39matrix, 5

square, when r = 3, 95source of name, xivvariety, see catalecticant variety

catalecticant scheme, 5, 8, 73catalecticant variety, 5, 8

v,«, t; 3), 92V.(t, t; r), 105V.(u,v;r),102and Veronese, 21, 245-247, 264V.(l, l;r), and quadratic forms,

18V.(1,2;r),19V. (1.i - 1; r), 20

cr, see complete intersectionclosure

of PS(s,j;3), 199of PS(s,j;r), 238, 243of PGor(T) , 97, 241, 250

codimension, see heightcodimension of an ideal, 276complete intersection, 76

not in closure of PS(12, 9; 3),140not in the closure of PS(s,j;r),

146related component of Gar(T) , 108-

112related component of determinan-

tal locus, 76, 108-116component of A, see irreducible com-

ponent, Acompressed Gorenstein Artin algebra,

80, 111nonhomogeneous, 229

determining a component ofGor(T),T = II(s,j,r),r 2:5,230

nonhomogenous

dimension of parameter spacefor, 229

nonsmoothable, 221connectedness

of catalecticant determinantallo-cus, 239-241

=-:--=------0="'-of the closure P(Gor(T)), 240

contractionaction, 3pairing, 11

contraction map, 265curve

canonical, 215rational normal, 215

decomposition of a form, 10, 36, 62existence, 13generalized, 22normalized GAD of binary form,

22uniqueness, 62

when s r , 17depth

of a local ring, 47of a prime ideal P (= grade(P)),

205derivatives

dimensions of spaces of higherderivatives of a form, 8

derivatives of a form, see inverse sys-tem of Gorenstein ideal

determinantal loci, see catalecticant,determinantal locus

determinantal scheme, 147Dimension formulas

dimCompAlgp(II(t,n)),229dimIIilbH p 2

, 166, 180dim IIilbf5P2 , the Betti stratum,

193dim GarD (T), the Betti stratum,

125dimGorsch(T) when T (s,s),

172dimGorsch(T) when T (s

a,s,sa),177dimPGor(T),121

divided power LUI, 267divided power forms, 265divided power ring, 3, 265

Index

DP-forms, 265DP-polynomials, 265duality, Macaulay, 3

Eagon-Northcott complex, 40, 83equations

defining Vs (u, v; r), see catalec-ticant ideal

defining PS(s,j; 3),185,199defining PS(8,j;r), 10,61,102,

255defining the Hilbert scheme, 305of the multisecant varieties of a

RNC, 31of the Veronese variety, 16

extremal Gorenstein algebra, 43

form (homogeneous polynomial)and Gorenstein algebra, 8annihilating ideal of, 14annihilating scheme, see annihi-

lating schemeannihilating scheme of, 135

a self-associated point set, 217points on a RNC, 217

cubic, 19decomposition of, see decompo-

sition of a formhaving no tight annihilating scheme,

139-140, 245inverse system generated by, see

inverse systemquadratic, 18ternary quartic, 6uniqueness of power sum repre-

sentation, 42frontier property, 34

GAD, see decomposition of a form,generalized

General Position Theorem, 41, 64generalized additive decomposition, see

decomposition of a form, gen-eralized

generic nonsmoothable, 221Gor(T), scheme, 7

smoothness of, when r = 3, 116Gor(T), variety

closure of

337

r = 3, 97, 250r = 3, T = H(s,j, 3),92-99sequence of incomparable, 244

desingularization ofwhen r = 2,32when r = 3, 250

dimension of when r = 3, 119-122, 124-125

and T:> (8,8,8), 180having several components, 214,

233when r = 4,311when r = 5,6, 225when r ::::: 4 (Boij), 236when r ::::: 7, r i= 8, 223

intersection with P S (8, j; r), 238irreducibility of when r = 3, 51,

117morphism to the Hilbert scheme

not extending to the closure,201

when r = 3,159-176sequence of incomparable, 244smoothness of, when r = 3, 117

Gorenstein Artin algebra, 8compressed, 80, 229

nonhomogeneous, 229extremal, 43generic, 221strong Lefschetz, 109, 111weak Lefschetz, 179, 235

Gorenstein ideal, 8a CI when r = 2, 30, 39and annihilating ideal of a form,

8containing a C I, 111determined by a sum of powers

square of, and vanishing ideal,84

naming of, xviii, 8nonhomogeneous

dimension of parameter space,when r = 3, 126

square ofand tangent space to Gor(T),

79minimal resolution, 97

338 Index

minimal resolution, when r =3,276

when r = 3, s special, 93Gorenstein sequence, 8, 240

bounds on growth of, 308non-unimodal, 203, 239occuring for f E PS(s,j;r), 238unimodal, 153when r = 3, 153, 239

Gotzmann number of a polynomial,295

equality with Castelnuovo-Mumfordregularity, 301

Gotzmann polynomial, 293Gotzmann's persistence theorem, 297grade of an ideal grade (I, R), 276graded algebras of Hilbert function

T,249singularity of family of, when r =

3, 252Grassmann variety, 15, 249, 302

Hankel matrix, 6, 28, 74height of an ideal, 202, 276Hilbert function, 9

global, of local Gorenstein com-pressed scheme, 234

Macaulay's characterization of,290

of Gorenstein algebra, see Goren-stein sequence

and dimension of higher par-tial derivatives, 8

of square of Gorenstein ideal Ann(f),93

Hilbert polynomial, 298identical with Macaulay polyno-

mial,299Hilbert scheme, 42, 159, 212, 226

and Gor(T), 212subscheme of GrassP(Rd ) , 302

Hilbert scheme HilbP (Ipr-l), 302-307equations of, 306Gotzmann's structure theorem,

for certain P, 310Gotzmann's theorem, 305

Hilbert scheme of O-dimensional schemes,159, 308

Hilbert scheme, Betti number strata,see Betti number strata

Hilbert scheme, postulation strata, seepostulation H punctual Hilbertscheme

Hilbert-Burch Theorem, 39, 163

idealm-regular, 48, 300ancestor, of a vector space of forms,

68,299defining of scheme, see name of

schemelimit, 142, 143minimal growth of, 290, 291order ideal (O-ideal), 290order of, 43primitive, of a singularity, 101saturated, 46saturation, 298vanishing, at points

square of, 43, 100, 235square of, for 1P'2, 101square of, for IP'r-l, 104symbolic square mod square,

101initial ideal, see lex-initial idealinverse system of Gorenstein ideal, 4,

8and Hilbert function, 8dimension of graded components

of, 8inverse system

of locally Gorenstein ideal, 228irreducibility, see catalecticant,determinantal

locus, and Gor(T)of Gor(T) when r = 3, 51ofthe catalecticant determinant,

80irreducible components

of Vs(u, v; r), 53, 241-245of Gor(T), 53, 214-236

Jacobian ideal, 15-21Jordan Lemma, 24

latticeof height three Gorenstein sequences,

153, 286

Index

length of a form, 197-198for f a binary DP-form, 36when f is binary, 23differential length, 135scheme length, 135, 245

length of GAD, 22lex-initial ideal, 290liaison, 108, 205limit ideal, 142, 143limit scheme, 144

"Macaulay" , symbolic algebra program[BaS1], 96, 221, 223, 312

Macaulay or Mattis duality, 3, 4Macaulay polynomial, 293, 299

conditions defining, 295Macaulay representations of an inte-

ger, 294minimal resolution

of square of Pfaffain ideal, 97strata of Gor(T), when r = 3

dimension of, 125morphism

finite, 142proper, 142quasifinite, 142

m-regular ideal, 48, 300multiplication map, 94, 107, 113

kernel of, 104multisecant varieties, 10

of the Veronese, 245-247degree of, 247

nonsmoothable scheme, 221conic, Gorenstein, 54

normalized GAD, 22

order ideal, 290order of an ideal, 43O-sequence, 122, 124, 153, 290

when r = 2, 291

partitionasssociated to a height 3 Goren-

stein sequence, 122-124, 286-288

self-complementary, and height3 Gorenstein sequence, 286

persistence

339

Gotzmann's Theorem, 297, 305Pfaffians of an alternating matrix, 160,

204, 272-277IP'Gor(T), variety

closure of, 97, 250IP'Gor(T), see Gor(T)points

imposing independent conditionson a linear system, 12

vanishing ideal of, see vanishingideal

polar polyhedron, 11, 62postulation H punctual Hilbert scheme,

149dimension of Betti strata, when

r = 3, 189-196dimension of, when r = 3, 166,

180-181morphism to, from Gor(T), when

r = 3 and T:J (s,s,s), 159reducibility of Betti strata of, for

r large, 196power sum, 10

annihilator idealHilbert function of, 13

parameter space P S (s, j; r)closure of, 145, 213, 238closure of, and Gor(T) when

r = 3,200component of Gor(T), 102, 230-

233dimension of, 40, 59-61

representation, 17, 22, 29, 38,40, 182-189

uniqueness of, for a given form,62

primitive idealof a singularity, 101

principal system, 8P S (s, j; r) (parameter space for sums

of powers), 13punctual = zero-dimensional, xxpunctual Hilbert scheme, see Hilbert

scheme

quadratic form, 18

rational normal curve, 54, 215

340

points on, as annihilating scheme,217

regularity, see Castelnuovo--Mumfordregularity

saturated ideal, 46saturation, 47, 298saturation degree, 48scheme

generic, 221limit, 144nonsmoothable, 221smoothable, 145

schemesflat family of, 142

self-associated point set, 214-220as annihilating scheme, 217ideal of, 215

Serre's Criterion, 82small tangent space

to parameter space of compressedalgebras, 221, 224,230

smoothable scheme, 145socle degree, 67strong Lefschetz property, 109, 111sum of powers, see power sumSylvester's Theorem, 27

tangent spacesmall, to parameter space of com-

pressed algebras, 221to Gor(T), 42, 73, 79, 96, 97,

249and square of Gorenstein ideal,

79annihilating scheme 10 points

on RNC, 313when annihilating scheme is

a self-associated point set,217

when annihilating scheme IS

compressed, 222to lP'Gor(T), 96to determinantallocus of catalec-

ticant, 73, 113

reducible locus, 79to the punctual Hilbert scheme,

79, 97, 249

Index

Terracini's Lemma, xiii, 58tight annihilating scheme, 135tight subscheme

of higher dimension Gorensteinscheme, 202

unimodal, see Gorenstein sequence,unimodal

vanishing idealat points of lP'2

having no DP-form for whichit defines a tight annihila-tor scheme, 98

square of, 100square of, and symbolic square,

103at points of lP'r-l, 14

square of, and Gorenstein ideal,84

and Gorenstein ideal, 14square of, 104

square ofat points of jpA on RNC, 314

Veronese variety, 10, 16and minors of catalecticants, 21chordal variety of, 20generation of ideal of, 21multisecant variety of, 39, 245-

247rational normal curve, 39Terracini's Lemma, 58

Waring's problem for forms, xiii, xix,40, 57-62

Index of Names

J. Alexander, xiii, 40-41, 46, 96, 101,103

A. Aramova, 297

H. Bass, xviii, 8D. Bayer, 253, 301, 305-306, 313D. Berman, 297A. Bigatti, 308G. Boffi, 97M. Boij, 54, 118, 125, 151, 160-161,

168,172,177,179,182,190,193-194,220,235-236,258,280-283, 312

J. Briancon, 96, 123W. Bruns, 163, 289, 290, 298D. Buchsbaum, 119, 276-278

G. Campanella, 190-193F. Catanese, 306K. Chandler, 41, 46-50, 101-103J. Cheah, 307y. Cho, 44, 54, 122-125, 133, 150,

165, 226-234, 279C. Ciliberto, 307A. Clebsch, xx, 6, 60, 65G. Clements, 290A. Conca, 40, 120, 155, 236, 271, 283

S. J. Diesel, 44, 76, 95, 97, 124, 153-156, 263, 286-288

1. Dolgachev, 65, 215

J. Eagon, 40R. Ehrenborg, xiv, 58D. Eisenbud, 39,119,261,276-278G. Ellingsrud, 247, 260J. Emsalem, 57, 221

J. Fogarty, 164, 199

R. Froberg, 190, 197

V. Gasharov, 297, 298A. Geramita, 46-50, 54, 97,101-103,

116,123,182,204,225,238,263-264, 283, 285, 308

A. Gimigliano, 46-50, 101-103D. Gorenstein, xviiiG. Gotzmann, 149, 180, 290, 292-

298, 300-310M. Green, 290, 297, 298, 305D. Gregory, 285A. Grothendieck, 8, 300, 302, 303L. Gruson, 39, 75S. Gundelfinger, xiv

B. Harbourne, 46T. Harima, 123, 155, 156, 182, 280B. Hassett, 261J. Herzog, 203, 289, 290, 297, 298T. Hibi, 297A. Hirschowitz, xiii, 40-41, 46, 96,

101,103C. Huneke, 118

A. Iarrobino, 13,53, 57, 221, 226-234H. Ikeda (Sekiguchi), 236A. Iliev, 62

B. Jung, 44,122-125, 140,279

V. Kanev, 21, 40, 65, 264G. Kempf, 20A. King, 251S. L. Kleiman, 205, 289, 301H. Kleppe, 96J. O. Kleppe, 97, 116-120, 126-127,

150, 249

342 Index of Names

J. Kollar, 307M. Kreuzer, 280J. P. S. Kung, xivA. Kustin, 97-99, 276-279R. E. Kutz, 19

D. Laksov, 41, 64E. Lasker, 58R. Lazarsfeld, 58H. Levine, 261B. Lindstrom, 290

F. H. S. Macaulay, 3, 8, 30, 67, 290D. Mall, 306P. Maroscia, 54, 101, 149, 225, 238,

311E. Matlis, 3J. Migliore, 204-205, 308R. Miro-Roig, 117, 119-120, 1501. Morrison, 253, 307S. Mukai, 62D. Mumford, 300

D. G. Northcott, 40R. Notari, 253, 306

F. Orecchia, 54D. Ortland, 215

K. Pardue, 307C. Peskine, 39, 70, 75Y. Pitteloud, 46-50, 101-103D. Popescu, 298O. Porras, 19, 20, 263M. Pucci, 21, 97, 116, 182, 263, 283V. Puppe, 262

A. Ragusa, 166K. Ranestad, 62, 261P. Rao, 201J. Rathman, 64A. Reeves, 301, 307B. Reichstein, 62Z. Reichstein, 62B. Reznick, xivL. Roberts, 54, 225, 238, 285G.-C. Rota, xiv, 58

R. Sanchez, 97

H. Schenck, 163F.-O. Schreyer, 62, 261P. Schenzel, 43E. Sernesi, 307P. Serre, 30, 39B. Shapiro, 260Y. S. Shin, 97, 116, 123, 182, 283V. V. Shokurov, 216E. Sperner, 299M. L. Spreafico, 253, 306H. Srinivasan, 275R. Stanley, 108, 110M. Stillman, 301, 307, 313S. A. Stromme, 247, 260B. Sturmfels, 50J. Sylvester, xiv, 27

A. Terracini, 40, 58N. V. Trung, 203Y. Tschinkel, 261

B. Ulrich, 97-99, 205, 276

G. Valla, 120, 155, 203, 271, 283E. Viehweg, 253, 307

C. H. Walter, 251J.Watanabe,39,50, 75, 109-111, 163,

236, 277J. Weyman, 119,253F. Whipple, 290

J. Yameogo, 251

G. Zappala, 166

Index of Notation

A(Vj ) , affine space of dimension dim; V j ,

8A f , R-submodule R 0 J of R, 8ALCHAR(T) , alignment character,

122Ann(f), Gorenstein ideal determined

by J, 67

Bs 1£1, base locus of linear system, 17

Cf(U, v), catalecticant homomorphism,4

CcoMP(T), parameter space for formsJ E Gor(T) having com-pressed annihilating scheme,230

0, contraction action, 4CompAlgp(H(t, n)), parameter space

for (nonhomogeneous) com-pressed schemes at p, of lo-cal Hilbert function H(t, n),229

CsM(T), parameter space for formsJ E Gor(T) having smoothannihilating scheme, 230

CT, Cl component, 109

t::. kH, k-th difference function, 202t::.H(R(I), difference function, 49t::.k(T), k-th difference function, 151t::.(T), difference function, t::.(T)i =

Ti-Ti _ 1,122

depth(1, N) of an R-module N, com-mon length of maximal N-sequences in the ideal 1, 162

depth(I) of an ideal lof R, = depth(l, R),205

V f , divided power module RoJ (prin-ciple inverse system), 7

V, ring of divided powers, 3, 265V' , ring of divided powers in variables

X2, ... .X«, 208ds,r, related to regularity degree of s

points, 84

Gor(T), parameter variety for formsin V j having H, = T, 8

Gor(T), scheme whose closed pointsare Gor(T), open subscherneof Gor::;(T), 8

Gor::; (T), scheme defined by rank con-ditions on determinantal mi-nors of catalecticant matri-ces, 8

Gorsch(T), 168GradAIg(H), parameter space for graded

algebras, 118grade(J) (or grade(J, R)) = min{i I

Extk(R/J, R) =I O} = thecommon length of all max-imal R-regular sequences inJ, 163

Grass(r - s, Rd, Grassmannian, 15G(T), scheme parametrizing graded

algebra quotients of R hav-ing Hilbert function T, 97,249

height (1), = minpEv(I) height (P),276

H f , Hilbert function of V f , 7HilbHlP'n, postulation Hilbert scheme,

parametrizing subschemes Zof IP'n having Hilbert func-tion H, 149

344 Index of Notation

HilbP(lpr-1), Hilbert scheme parametriz-ing subschemes of IP'r-l hav-ing Hilbert polynomial P,302

Hilb" (IP'r-1), punctual Hilbert scheme,308

H (j, r), Hilbert function of compressedGorenstein Artin algebra, 43,80,229

H 2(R, A, A), obstruction space to de-formations, 117

H(8, j, r), Gorenstein sequence H(8, i, r)i =minf s, ri, rj-i), xvi, 92

Hs(r), Hilbert function of 8 genericpoints in P", Hs(r)i = minjr,;»},227,229

ls+1 (CatF(u, v; r )), ideal generated bythe (8+ 1) x (8+ 1) minorsof the catalecticant, 5

l J, the largest integer smaller thenxxvii

[ same as l J, 55r l, the smallest integer greater than

41

symbolic square of T», 45

symbolic power, 57Iz, defining ideal of scheme Z, 43l(Vs ) , same as Iz for Z = Vs , 19

t'difh(J), differential length of I: max-imum value of Hf , 198

t'(f), length of a form in the binarycase, 23, 36

t'(f), length of a form: smallest 8 suchthat I E PS(8,j; r), 198

La, linear form, 9L[j], divided power of linear form, 3,

9t'sch(f), smallest degree of an anni-

hilating scheme of I, 135,198

t'schsm(f), smallest degree of a smooth-able annihilating scheme of1,198

M, maximal ideal of R, 68

p,(T), minimum number of generatorspossible for an ideal 1 =Ann(!), I E Car(T), 151

n(I(a»), saturation degree, 48N(t, i, r, u), dimension of n; -V(t, i, r),

291v(I), order, or initial degree of an

ideal, = min{d I t, # O},43

v(T), order of ideals of Hilbert func-tion H(R/ I) = T, 153

IDIP'" - ,(j) I, linear system, 12

'P(Q), Gotzmann number of a poly-nomial,295

P S (8 , j; r), parameter space for sumsof j-th divided powers of 8

linear forms, 10r., same as PS(8,j; 2),25Iji, alternating matrix appearing in

Buchsbaum-Eisenbud reso-lution, 160

pet, d, r), Macaulay polynomial, Hilbertpolynomial of R/(V(t, d, r)),293

Q( t, d, r), Gotzmann polynomial, giv-ing dimk(V(t,d,r))i, 293

a, Macaulay's correspondence, 67a(I), Castelnuovo-Mumford regular-

ity,300a(Z), regularity degree, 300an (Q), regularity degree of Hilbert

poynomial,300

§" sequence (8,8, ... ), 1518 V, = dirrn, Rt - 8, 43Sat(J), saturation of the ideal J, 298Sec, (Vj (IP'r-1)), multisecant variety of

the j-th Veronese, 10SL, strong Lefschetz property, 1098l2, Lie algebra of special linear group,

109SmHlP'n, parametrizing finite sets Z

with Hilbert function H, 149SmslP'n, parametrizing sets of 8 points,

149

Index of Notation

S(T), first half of first difference ofT, r = 3, 122, 284

Sym(H (C), j), symmetrized Hilbertfunction of C I, 109

Sym(Hz,j), symmetrized Hilbert function of the scheme Z, 135

T, symmetric sequence of integers T =

(l,t1, ... ,tj = 1), 8T, related to regularity degree, 48Ts,r, related to regularity degree of s

points in !P'r1, 84TI , tangent space to Gor(T), 42T(j, r), sequence (1, r, 2r-l, 2r, ... , 2r, 2r-

l,r, 1),53,207

Us(u, v; r), catalecticant variety: rankexactly s, 5, 75

Us(u, v; r), catalecticant scheme: rankexactly s, 5

Vs (u, v; r), catalecticant variety: rank:s; s, 5

V s (u, v; r), catalecticant scheme: rank:s; s, 5

v(I), minimum number of generatorsof I, 151

Vi (1) (Vi (I): number of generators ofI, (Ann(l), respectively), having degree i, 167

V(t,i,r), vector space span of first tmonomials of Ri, in lex order, 291

WJ.., perpendicular space in V j to WER j,42

ZGor(T) , parameter space for nonhomogeneous Gorenstein ideals,126

Z(t), fiber of a family, 145

345

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