Hilbert functions on Cohen-Macaulay ideals with assigned generators' degrees

Hilbert Functions of Cohen-Macaulay Ideals with Assigned Generators’ Degrees.

ALFIO RAGUSA(*) - GIUSEPPE ZAPPALA` (**)

ABSTRACT- We give information on the Hilbert function of a Cohen-Macaulay ide- alIof the polynomial ringR4k[x₀,x₁,R, x_r] which is minimally generated by t forms of degrees d₁,R,d_t. Mainly we deal with the codimension two case in which we show that the Dubreil boundtGd₁11 is a necessary and sufficient condition to have such an ideal and we give a sharp upper bound and lower bound for the Hilbert function. In codimension greater than two we give a characterization for having such an ideal and in codimension 3 we find an Hilbert function which is maximal for these ideals with d₁4R4d_t4a and we produce a scheme which realizes such a Hilbert function.

Introduction.

Let R4k[x0,x1,R,xr] be the homogeneous polynomial ring over an algebraically closed fieldkand fixtpositive integersd₁,R,d_t. It is a very classical question, both of Commutative Algebra and Algebraic Geo- metry, to try to determinate the Hilbert function ofR/I, whereIis a ho- mogeneous ideal of R minimally generated by t forms of degrees d1,R,dt. Of course, one needs some further information on the ideal I.

For instance, one point of view can be to ask that the forms defining the ideal are generically chosen. Even in this strong context very few results are known. Clearly, in this case if tGr11 , we have htI4t and I is a complete intersection and then the Hilbert function ofR/Iis completely (*) Indirizzo dell’A.: Dip. di Matematica e Informatica, Università di Cata- nia, Viale A. Doria 6, 95125 Catania, Italy.

E-mail address: ragusaHdmi.unict.it

(**) Indirizzo dell’A.: Dip. di Matematica e Informatica, Università di Cata- nia, Viale A. Doria 6, 95125 Catania, Italy.

E-mail address: zappalagHdmi.unict.it

determined byd₁,R,d_t. But whentDr11 , we havehtI4r11 (soR/I is an Artinian ring) and very little is still known. By results of Stanley [St] and Watanabe [W] the Hilbert function is known ift4r12 , since a general Artinian complete intersection has the Strong Lefschetz proper- ty. Moreover, the case r41 and r42 was solved, respectively, by Fröberg [F] and by Anick [A]. Other authors studied the cased₁4R4 4d_tgiving information on some part of the Hilbert function or, with some further restriction, on the graded Betti numbers (see, for instance [HL], [Au], [FH], [MM]). In any case, it seems completely unexplored what can be the Hilbert function, or at least bounds for it, for an ideal of height r11 generated by anyt forms of degreesd₁,R,d_t. Now, from the Al- gebraic Geometry point of view, one is mainly interested with (saturat- ed) ideals of R of height Gr. So we can refrase the question of finding the possible Hilbert functions, or bounds for them, for Cohen-Macaulay homogeneous ideals I of R of fixed height h minimally generated by t forms of degrees d₁,R,d_t. Since, very little is known every result in this field seems interesting. We will deal essentially with the case r42 and in the caser43 whend₁4R4d_t4a. Precisely, we setH^(c)d₁,R,d_t4 4 ]H_R/I(R/IwhereHR/Iis the Hilbert function of any Artinian reduction of ac-codimensional Cohen-Macaulay idealIofRminimally generated byt forms of degrees d₁,R,d_t. We equip this set of an ordering, defining H_R/IGH_R/JifH_R/I(n)GH_R/J(n) for alln. In this paper we study first the casec42, for which, after observing thatH^(c)d₁,R,d_tis not empty if and only iftGd111 (the Dubreil inequality), we prove that as poset it has both a maximum and a minimum element (Proposition 2.2). Moreover, we produce a scheme which realizes such a maximum and compute explicitely the mini- mum in the case d₁,R,d_t4a. In the codimension 3 case we show that Ha;⁽³⁾t(i.e. in case whend₁,R,d_t4a) has a maximal element and again we produce a scheme which realizes such a maximal Hilbert function (Theo- rem 3.8). We still believe that, indeed, it is also a maximum.

The first section is dedicated to partial intersection schemes which will be used to produce schemes with the required Hilbert functions.

1. Partial intersections: definitions, properties and facts.

Throughout this paper k will denote an algebraically closed field,P^r the r-dimensional projective space over k, R4k[x₀,x₁,R,x_r]4 4_nZ

5

IfV%P^ris a subscheme,I_Vwill denote its defining ideal andH_V(n)4 4dim_kR_n2dim_k(I_V)_nits Hilbert function. Moreover, ifV%P^ris ac-codi- mensional aCM scheme with minimal free resolution

0K5R(2j)^acjRK5R(2j)^a2jK5R(2j)^a1jKI_VK0 then the integers ]a_ij(^j will denote the i-th graded Betti numbers.

In this section we recall the construction of thec-codimensional par- tial intersection schemes made in [RZ] and we collect from there the main facts that will be used in this paper.

Let (P^,G) be a poset. We denote, for every HP^, SH4 ]KP_N^K_E^H₍^, SH4 ]KP_N^K_G^H₍^.

DEFINITION 1.1. A subset A of the posetP is said to be a left seg- mentif for every HA^,SH’A. In particular, when P₄^Ncwith the or- der induced by the natural order onN, a finite left segment will be men- tioned as a c-left segment.

Note that everyc-left segmentAhas sets of generators but there is a unique minimal set of generators consisting of the maximal elements of A; we will denote it by G(A^).

Ifpi: N^cKNwill denote the projection to thei-th component, andA is a c-left segment, we set v(H)4

!

i41 c

pi(H) and ai4max]pi(H)NH A₍^{, for 1}_Gⁱ_G^{c. Thec-tupleT}₄^T(A⁾₄^(a1,R,a_c) will be called the size of A^.

A c-left segment is said to be degenerate if ai41 for some i. IfA^{is a}c-left segment,F(A) will denote the set of minimal elements of N^c0A^{, i.e.}

F(A⁾_{4 ]}^HNc0A_NSH’A₍^.

Note that, if H4(m₁,R,m_c)F(A^{) and m}iD1 , then H_i4 4(m₁,R,m_i21 ,R,m_c)A. Moreover, the elements

T₁4(a₁11 , 1 ,R, 1 ),R,T_c4( 1 , 1 ,R,a_c11 ) always belong to F(A), and we will call them canonical c-tuples.

In the sequel we denote thec-tuple ( 1 ,R, 1 ) byIand, for every sub- set Z of ST, we denote

CT(Z)4 ]T1I2HNHZ(.

Finally, for every c-left segment A ^{we define} A^*4C_T(ST0A^).

Observe that A* is a c-left segment.

PROPOSITION 1.2. If A is a c-left segment, then 1. F(A⁾₄CT(G(A^{* ) )}N]T1,R,Tc(,

2. F(A^{* )}4CT(G(A^{) )}N]T1*,R,Tc*(.

3. If T_i*cTi, for some i, then T_i*CT(G(A^{) ).}

PROOF. See Proposition 1.3 in [RZ]. r

Fix a c-left segment A and consider c families of hyperplanes of P^r, cGr,

]A_1j(¹GjGa₁, ]A_2j(1GjGa2,R, ]A_cj(1GjGac

sufficiently generic, in the sense thatA_1j1OROA_cjc are

»

i41 c

a_i pairwise distinct linear varieties of codimension c.

For every H4(j1,R,j_c)A, we denote by LH4_h

1

41 c

Ahj_h. With this notation we have the following

DEFINITION 1.3. The subscheme of P^r V4_H

0

A

L_H

will be called a c-partial intersection with respect to the hyperplanes ]A_ij( and support on the c-left segment A^.

THEOREM 1.4. Every c-partial intersection X of P^r is a reduced aCM subscheme consisting of a union of c-codimensional linear varieties.

PROOF. See Theorem 1.9 in [RZ]. r

Here are the main results on c-codimensional partial intersec- tions.

THEOREM 1.5. If V%P^r is a partial intersection of codimension c with support onA^{,then the(r}₂^c₁1 )-th difference of its Hilbert func- tion is

D^r2c11H_V(n)4N]HA_N^v(H)₄ⁿ₁^c₍N^. PROOF. See Theorem 2.1 in [RZ]. r

Now, ifXis a c-codimensional partial intersection with support onA and with respect to the families of hyperplanesAijwhose defining forms are f_ij, to every H4(m₁,R,m_c)A we associate the following form

P_H4

»

i41

c

»

j41 m_i21

f_ij.

THEOREM 1.6. Let V%P^rbe a partial intersection of codimension c with support A^. Then a minimal set of generators for I_V is

]P_HNHF(A⁾₍^. PROOF. See Theorem 3.1 in [RZ]. r

COROLLARY 1.7. Let V be as above then its first graded Betti num- bers depend only on A and they are the following integers

dH4v(H)2c (HF(A^).

And finally

THEOREM 1.8. Let V%P^rbe a partial intersection of codimension c with support A^. Then the last graded Betti numbers of V are

s_H4v(H) (HG(A^).

PROOF. See Theorem 3.4 in [RZ]. r

We conclude this section by discussing the question we want to deal with in this paper.

Let d₁GRGd_t be t positive integers and cGt; we denote by H^(c)d₁,R,d_t4 ]H_R/I(R/I

where I varies on the Artinian ideals of the polynomial ring R4

4k[x₁,R,x_c], which are minimally generated by t forms of degrees d₁,R,d_t, and H_R/I means the Hilbert function of R/I.

Note that the same setH^(c)d₁,R,d_tcan be obtained by using idealsIXof c-codimensional arithmetically Cohen Macaulay schemes of X%Pⁿ and D^n112cH_R/(IX), where D^n112c denotes the (n112c)-th difference of the Hilbert function of I_X.

To describeH^(c)d₁,R,d_tis a very hard task in this general setting. Nev- ertheless, many simpler (but still hard) questions can be posed. For in- stance, to establish if H^(c)d₁,R,d_t is empty or not or, more generally, to compute its cardinality in terms of the integers c;d₁Rd_t.

Moreover, since H^(c)d1,R,d_t can be ordered by defining H_R/IGH_R/J iff H_R/I(n)GH_R/J(n) for all nZ, (we call such an ordering the natural partial ordering) one can ask if H^(c)d₁,R,d_t has a maximum or a minimum element and when the answer is negative one can ask which are the max- imal and the minimal elements.

The previous questions will be studied in few particular cases: first in the codimension 2 case and for the codimension 3 case whend₁4R4d_t.

2. The codimension two case.

We first deal with the codimensionc42 for which most of the previ- ous questions can be answered.

In this caseH²d₁,R,d_tis not empty ifftGd₁11 (the Dubreil’s inequal- ity). Indeed, in this situation, it is easy to see thatH^{( 2 )}d₁,R,d_tis in 121 cor- respondence with the set

S^{( 2 )}d₁,R,d_t4

m

!

i

d_i4

!

i

s_i

n

i.e. the (t21 )-tuples which satisfy the Gaeta’s conditions. So, if we set s_i4d_i1x_i, where x_i is a positive integer, the Gaeta’s conditions are equivalent to say

!

i42 t

xi4d1, from which one gets thatS^{( 2 )}d₁,R,d_tis not emp- ty iff d₁Ft21 .

Of course all these information about 2-codimensional Cohen- Macaulay ideals can be deduced from the Hilbert-Burch theorem.

From now on a (t21 )-tuple (x2,R,xt) of positive integers such that

i

!

x_i4d₁, is said a (t21 )-partition of d₁. We are interested on the

(t21 )-partitions of d₁ which satisfy the condition (* ) x_i2x_i11Gd_i112d_i (i42 ,Rt.

Using the correspondence, which associates to s4(s₂,R,s_t)S^{( 2 )}d₁,R,d_t

the element H_sH^{( 2 )}d₁,R,d_t defined by H_s(n)4(n11 )2

!

diGn

(n112 2d_i)1

!

s_iGn

(n112s_i), one can induce an ordering onS^{( 2 )}d1,R,dt; precisely, s4(s₂,R,st)Gs8 4(s²8,R,s^t8) ` HsGHs8 `

!

siGn

(n112si)G G

!

si8Gn

(n112sⁱ8)(n.

We need first this simple technical lemma.

LEMMA 2.1. Let (M₂,R,M_t) be the maximum, by lexicographic ordering, in the set of (t21 )-partitions of d₁ satisfying condition (*).

Then, for every (t21 )-partition of d1 (x2,R,xt), satisfying condition (*), we have

i

!

M_iF

!

i42 h

x_i (h42 ,Rt. PROOF. If there is some h8 43 ,Rt such that

!

i42 h8

M_iE

!

i42 h8

x_i then there existjGh8such thatM_jEx_j; saymthe biggest one. On the other hand, since

!

i42 t

M_i4

!

i42 t

x_i there is some jDh8 such thatM_jDx_j; sayn the smallest one. Of course, by construction, mGn. Define, for every i42 ,R,t,

Mⁱ84

. /

´

M_i M_m11 M_n21

(icm,n i4m i4n.

We see that (M28,R,Mt8) is a (t21 )-partition ofd₁satisfying condition (*). Indeed, we need only to show that d_m1M^m8Gd_m111M^m118 and d_n211Mⁿ821Gd_n1Mⁿ8. Now

d_m1M^m8 4d_m 1M_m 11Gd_m 1x_mGd_m111x_m114d_m111 Mm11 4 dm11 1 M^m811 (if m11En, the case m114n is similar)

d_n211Mn28 14d_n211M_n214d_n211x_n21Gd_n1x_nGd_n1 M_n21 4 d_n 1 Mⁿ8 (if mEn21 , the case m4n21 is similar).

Now, since (M²8,R,M^t8) is lexicographically bigger than (M₂,R,M_t), we get a contradiction. r

PROPOSITION 2.2. Let d₁GRGd_t be t positive integers and H²d₁,R,d_tthe set of all Hilbert functions of Artinian ideals of the polyno- mial ring in two variables which are minimally generated by t forms of degrees d1,R,d_t.ThenH²d1,R,dt, as a poset, by the natural partial or- dering, has both a maximum and a minimum element. Precisely, if we denote by (m₂,R,m_t) and (M₂,R,M_t), respectively, the minimum and the maximum of the set of the(t21 )-partitions of d₁satisfying the condition (*), ordered by the lexicographic ordering, then in the poset S^{( 2 )}d₁,R,d_tthe element sⁱ84di1mi,for i42 ,Rt,is the maximum and the element sⁱ94d_i1M_i, for i42 ,Rt, is the minimum. Hence, the ele- ments Hs8(n)4(n11 )2

!

diGn

(n112di)1

!

s8iGn

(n112sⁱ8), Hs9(n)4 4(n11 )2

!

d_iGn

(n112d_i)1

!

si9Gn

(n112sⁱ9)are, respectively, the maxi- mum and the minimum element in H²d₁,R,d_t.

PROOF. Take any element (s₂,R,s_t) inS^{( 2 )}d₁,R,d_t and denotes_i4d_i1 1xi, fori42 ,R,t. Of course, (x2,R,xt) is a (t21 )-partition ofd1satis- fying the condition (*). For everynNwe get two integers i,jGt, de- fined bysⁱ9Gn11Esⁱ9¹¹ ands_jGn11Es_j11. For the minimum case, we need to prove that

!

h42 i

(n112dh2Mh)G

!

h42 j

(n112dh2xh). Now, if iGj, applying Lemma 2.1 we have

h42

!

i

(n112dh2Mh)G

!

h42 i

(n112dh2xh) hence

h42

!

i

(n112dh2Mh)G

!

h42 i

(n112dh2xh)1

!

h4i11 j

(n112dh2xh)4

4

!

h42 j

(n112d_h2x_h).

If iDj again by Lemma 2.1 we have

h

!

(n112d_h2M_h)G

!

h42 i

(n112d_h2x_h) hence

h

!

(n112d_h2M_h)G

!

h42 j

(n112d_h2x_h)1

!

h4j11 i

(n112d_h2x_h) ;

now, since(n112d_h2x_h)E0 for hDj, we get

h

!

(n112d_h2M_h)G

!

h42 j

(n112d_h2x_h) .

Finally, note that the minimum element in the set of the (t21 )-parti- tions ofd₁satisfying the condition (*), ordered by the lexicographic or- dering, is given bymi41 , for i42 ,R,t21 , andmt4d12t12 . This implies that for every (t21 )-partitions of d1satisfying the condition (*), (y₂,R,y_t), one has

i

!

m_iG

!

i42 h

y_i (h42 ,Rt,

i.e. we are in the same situation as in Lemma 2.1 (just reversing the or- der). Thus, repeating the same argument as in the minimum case, we get for every integer n

h4

!

(n112d_h2m_h)F

!

h42 j

(n112d_h2y_h) where i and j are defined as before. r

COROLLARY 2.3. The 2-partial intersection Xmax whose support is the2-left segment generated by the t21elements

g

^!

t

m_h,

!

h4i t

m_h

h

for i42 ,R,t, has the maximum Hilbert function in H²d1,R,dt; the 2- partial intersection X_min whose support is the 2-left segment generated by the t21 elements

g

^!

^!

h

minimum Hilbert function in H²d₁,R,d_t.

PROOF. The conclusion is a direct consequence of the previous theo- rem, Corollary 1.7 and Theorem 1.8. r

REMARK 2.4. We can get also the ideals of the partial intersections in the previous Corollary by lifting suitable Artinian monomial ideals.

More precisely every sequenced₁,R,d_t,s₂,R,s_t, satisfying the Gaeta conditions, is realized by the ideal of the maximal minors of the following

matrix

. `

` `

´

x^s22d1 0 Q Q Q

0 0

y^s22d2 x^s32d2

Q Q Q Q Q Q

Q Q Q y^s32d3

x^st212dt22 0

0 0

y^st212dt21 x^st2dt21

0 0

0 y^st2dt

ˆ `

` `

˜

.

COROLLARY 2.5. If d14R4dt4a the maximum and the mini- mum Hilbert function in H²a;t are given by, respectively,

H_max(n)4

. /

´

n11

2a112t2n 0

if nEa

if aGnE2a122t if nD2a122t

H_min(n)4

. /

´

n11

2a112t2n 0

if nEa

if aGnEa1q if nFa1q where a4(t21 )q1r, rEt21 .

PROOF. To get the conclusion it is enough to observe that in this case we have m_i41 for i42 ,Rt21 , m_t4a2t12 and M_i4q for i42 ,R,t2r and Mj4q11 for i4t2r11 ,R,t. r

3. The codimension greater than two: ideals generated in one degree.

In this section we approach the problem for codimension c greater than two. A first question is when the setH^(c)d₁,R,d_tis empty. Let us sup- pose that the integersd₁Gd₂GRd_tare assigned. In this case we denote p1Ep2EREps the distinct elements among the di’s and we set ai»4

»4N]d_j4p_iN1GjGt(N. Of course

!

i41 s

a_i4t. Moreover we set

b1»4

g

h

where the symbol a.b denotes the exponential of Macaulay (see the paper of Stanley [St1]).

PROPOSITION 3.1. Let d1Gd2GRGd_t t positive integers, with tF Fc. Then

H^(c)d1,R,dtc¯ ` a_iGb_i for 1GiGs. PROOF. Throughout this proof we set R»4k[x₁,R,x_c].

LetIH^(c)d1,R,d_t; we callJ_(i)the ideal generated by the homogeneous pieces ofIof degree less or equal top_i,Wi the Hilbert function of thek- algebra R/J_(i) and H the Hilbert function of R/I. Of course, we have Wi(pi)4Wi21(pi)2ai. Now we prove that Wi21(pi)Gbi for 2GiGs. Fori42 ,W₁(p₁)4b₁2a₁, and using Macaulay Theorem on the maxi- mal growth (see [St1]),

W₁(p₁11 )G(b₁2a₁)ap₁b,W₁(p₁12 )GW₁(p₁11 )^ap111bG

G(b₁2a1)^ap1bap111b,R,W1(p₂)G(b₁2a1)^{ap1bap111bRap221b}4b2. Now suppose that W_i21(p_i)Gb_i; then W_i(p_i)4W_i21(p_i)2a_iGb_i2a_i therefore by repeating the previous arguments we obtain that

Wi(p_i11)G(b_i2ai)^{apibapi11bRapi1121b}4bi11.

Now it is clear that a1Gb1 since b14dim_kRp₁; moreover for 2GiG Gs,

ai4dim_kR_pi2dim_kR₁I_pi212H(p_i)4Wi21(p_i)2H(p_i)Gbi2H(p_i)Gbi. Vice versa let us suppose thatd₁,R,d_tare integers such thataiGbi

for 1GiGs and t4

!

i41 s

aiFc (using the same notation). By [St1] there exists a lex-segment idealL%Rhaving ai minimal generators in degree p_i, for 1GiGs. LetMbe the set of the monomials minimally generating L. Then M4M_pNM_m where M_p is the subset of the elements of M which are powers of somex_iandM_m is the subset of the mixed monomi- als. We set k»4 NM_pN andu»4 NM_mN; NMN 4t4k1uFc, by hypothe- sis, so uFc2k; let M^m84 ]mk11,mk12,R,mc( be the subset of Mm

of the last c2k monomials in the lexicographic order; we set P4 ]x_i^degmiNk11GiGc(. Now let us consider the set of monomials M8»4(M0M^m8)NP. We claim that the monomial idealI, generated by M8, belongs to H^(c)d₁,R,d_t. First observe that height I4c since, by con-

struction, there is a power ofx_i belonging toI for 1GiGc. So, to con- clude the proof, we need to show thatM8is a set of minimal generators forI. Using again the lexicographic order, we set m»4max]mM8 Nm is not a power(; thenM8 4M¹8NM²8whereM¹84 ]mM8 NmGm( and M²84M80M¹8; since M¹8%M and M was a minimal set of generators, every element inM¹8cannot be in the shadow of the previous ones; on the other handM²8contains only powers, so, again, every element inM8can- not belong to the shadow of the previous ones. r

REMARK 3.2. WhenH^(c)d1,R,d_tc¯it is natural to guess that the mini- mal value for the Hilbert function is achieved by the ideal generated byt generic forms of degreesd1,R,dt. For istance the guess is true in the particular case d14R4dt4a with ntF

g

h

In this section we will use the notation about partial intersections in- troduced in section 1.

LetA4k[x,y,z] and letn,aN. We denote by Ia,nthe set of the homogeneous Artinian idealsI%A(i.e. A/Iis Artinian) where I is mini- mally generated by n forms of degree a.

We simply denote

Ha,n4 ]H_A/INIIa,n(.

Fixed two integersaF1 and 3GnG

g

h

Let n»4

g

h

0GiGa22 and b_a21»41 . Note that, since

g

h

^!

b_i,

there exist two integerskandh,22GkGa22 , 0GhGb_k1121 , such that

n4

!

i4 22 k

b_i1h.

We set s»4a222k; then 0GsGa. Moreover observe that n4

g

h

^!

a11

i2

!

i4 21 a22

b_i2h4

!

i41 a11

i2

!

i4a2k a

i2(a21 )2h4

4

!

i41 a2k21

i1(a11 )2(a21 )2h4

g

h

g

h

We set alsos»4a2s. Now let us consider the 3-left segment La,ngen- erated by the following elements:

(s,a,a), (s11 ,h,a) ;

](s11 ,y,s112y)Nh11GyGs(;

](x,y,a122x2y)Ns12GxGa, 1GyGa112x(.

LetX%P^r,rF3 , a partial intersection with support onLa,n. We would like to show that D^r22HX is the desired maximal element.

First of all we computeD^r22H_X. IfA^{is a}c-left segment andsGtare two positive integers we set Aj»4 ](HA_N^p1(H)4j(, A^(s,t)»₄

»4_j4s

0

»4s(A^(s,t) ). Of courseA^{(s,t) is a}c-left segment. IfXis a partial inter- section with support onA^{, letX(s,}t) be the subscheme ofXwith support on A^{(s,t). X(s,}t) is obviously a partial intersection with support on A^(s,t).

LEMMA 3.3. Let X%P^r a c-partial intersection with support on A^. Let t be the maximum of the first components of the elements ofA^,and let 1GsGt, so X4X₁NX₂ where X₁»4X( 1 ,s) and X₂»4X(s11 ,t).

Then D^r2c11H_X(i)4D^r2c11H_X1(i)1D^r2c11H_X2(i2s).

PROOF. By Theorem 1.5,

D^r2c11H_X(i)4N]HANv(H)4i1c(N4

4N]HA^{( 1 ,s)}Nv(H)4i1c(N1N]HA^(s11 ,t)Nv(H)4i1c(N4 4N]HA^{( 1 ,s)}_N^v(H)₄ⁱ₁^c₍N1N]HA^(s₁^{1 ,t)}_N^v(H)₄ⁱ₂^s₁^c₍N4

4D^r2c11HX₁(i)1D^r2c11H_X2(i2s). r

Proposition. 3.4. Let aF1 and 3GnG

g

h

D^r21HX(i)4

. `

` /

` `

´

i11 a112n s22(i2a11 ) 2s21

2s

i22a2s11 0

for 0GiGa21 for i4a

for a11GiGa1s21 for a1sGiGa1s1h21 for a1s1hGiG2a21 for 2aGiG2a1s22 for iF2a1s21 .

PROOF. We denotepj4 ](x,y,z)N³Nx4j(. We can decomposeX in an union of three disjoint partial intersections X₁,X₂,X₃.X₁ is sup- ported on a(s,a,a)b; X₂ on La,nOp_s11; X₃ supported on La,nO O

g

⁰

h

X₁ is a complete intersection of type (s,a,a). So we have

D^r21H_X1(i)4

. ` / `

´

i11 s

2a1s2222i 2s

i22a2s11 0

for 0GiGs21 for sGiGa21 for aGiGs1a21 for s1aGiG2a21 for 2aGiG2a1s22 for iF2a1s21 . X₂ is a plane partial intersection such that

D^r21H_X2(i)4

. ` / `

´

1 h2s 0 21 0

for 0GiGs21 for i4s

for s11GiGa21 for aGiGa1h21 for iFa1h.

X₃ is a partial intersection such that

D^r21H_X3(i)4

. /

´

i11 2

g

h

0

for 0GiGs22 for i4s21 for iFs.

By Lemma 3.3, D^r21HX(i)4D^r21HX₁(i)1D^r21H_X2(i2s)1 1D^r21H_X3(i2A^s21 ), so we are done. r

Now we compute the graded Betti numbers of a partial intersection with support on La,n.

PROPOSITION 3.5. Let aF1 and 3GnG

g

h

intersection with support onLa,n,and let R»4H⁰

*(OP^r).Then a graded minimal free resolution for I_X,the saturated homogeneous ideal of X,is of the type:

0KR(2(a12 ) )^n2s235R(2( 2a2s1h11 ) )5R(2( 3a2s) )K KR(2(a11 ) )^2n2s265R(2( 2a2s) )5R(2( 2a2s11 ) )5

5R(2( 2a2s1h) )5R(22a)KR(2a)ⁿKIXK0 PROOF. By [RZ], Proposition 3.4, we can compute the last graded Betti numbers ofIXdirectly from the generators ofLa,n. To compute the first graded Betti numbers, we have to find, using notation in section 1, the setF(La,n). An easy verification shows that the elements inF(La,n) are

(a11 , 1 , 1 ), ( 1 ,a11 , 1 ), ( 1 , 1 , a11 ) ; ](s11 ,y,s122y)Nh11GyGs11(;

](x,y,a132x2y)Ns12GxGa, 1GyGa122x(.

Finally we can compute the second graded Betti numbers, as it is well known, through the first and last ones and the Hilbert func- tion. r

If A is a 3-left segment, we set

m_i(A^)»₄^max_]pi(H)NHA₍^{; for 1}_Gⁱ_G^{3 ,}

andI»4( 1 , 1 , 1 ). Of coursem_i(La,n)4a, for 1GiG3 . Now we consid- erLa* ; then,n m₁(La,* )n 4a2s4sandm_i(La,* )n 4afori42 , 3 . So if we set T»4(a,a,a) and U»4(s,a,a) we have

La,**n4C_U

(

)

Now let s: Z³KZ³ be the transformation s(x,y,z)4(x2s,y,z).

LEMMA 3.6. With the previous notation La**,n4s(La,n)ON³.

PROOF. HLa,**n`U1I2HA<sup>S</sup>U0C<sub>T</sub>(S<sub>T</sub>0La,n)` U1I2HG GUand U1I2HC_T(S_T0La,n) ` HFIand T1I2(U1I2H) ST0La,n ` HN³ and H1T2UST0La,n ` HN<sup>3</sup> and H1 1(s, 0 , 0 )La,n ` Hs(La,n)ON³. r

LEMMA 3.7. Let X%P^r a scheme with support on La,n. Then X is CI-linked in two steps to a scheme Y whose(r22 )2th difference of the Hilbert function is

D^r22H_Y(i)4

. /

´

g

h

h

a1h212i

for 0GiGs21 for sGiGa21 for aGiGa1h22

through complete intersections of type (a,a,a)in the first step, and of type (s,a,a) in the second step.

PROOF. We link at firstXin a partial intersection complete intersec- tion of type (a,a,a) to a schemeY8, and then we linkY8in a partial in- tersection complete intersection of type (s,a,a) to a schemeY. Then Y is a partial intersection with support onLa** . So we can compute the (,n r2 22 )-th difference of the Hilbert function ofY, as in the proof of Proposi- tion 3.4, by consideringY4Y₁NY₂,Y₁with support onLa,**nOp₁andY₂ with support on La,**nO

g

⁰

s

p_j

h