some configurations of lines whose ideals are …the grids are 1-lifting of monomial ideals. in this...

International Journal of Pure and Applied Mathematics

Volume 92 No. 5 2014, 669-690ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.eudoi: http://dx.doi.org/10.12732/ijpam.v92i5.4

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SOME CONFIGURATIONS OF LINES WHOSE IDEALS

ARE LIFTINGS OF MONOMIAL IDEALS

Margherita Guida1, Ferruccio Orecchia1 §, Luciana Ramella2

1Dipartimento di Matematica e Appl.Universita di Napoli “Federico II”,Via Cintia, 80126 Napoli, ITALY

2Dipartimento di MatematicaUniversita di Genova

v. Dodecaneso 35, 16146 Genova, ITALY

Abstract: In this paper we consider some configurations of lines whose idealsare generated by a product of linear forms. We show that in general these idealsare 1-lifting or pseudo 1-lifting of monomial ideals. This proves also that theseideals are Arithmetically Cohen Macaulay.

AMS Subject Classification: 14Q05Key Words: lifting, lines

Introduction

The topic of ideals I generated by products of linear forms has raised interestin many authors (see for example [1],[9],[12]). It is clear that the ideals Iare union of linear varieties, but in general a union of linear varieties is notgenerated by products of linear forms (see [1], Proposition 5.7). It is also clear,by definition, that lifting and pseudo 1-lifting of monomial ideals (as definedby Migliore and Nagel [11]) are generated by products of linear forms andCohen Macaulay but the converse is not true (see Example 4.13). In [2] thethird author with Cioffi and Marinari studied geometric properties of unionof linear varieties defined by liftings and pseudo 1-liftings of monomial ideals.Furthermore in [7] the first two author introduced configurations of lines , called

Received: January 28, 2014 c© 2014 Academic Publications, Ltd.url: www.acadpubl.eu

§Correspondence author

670 M. Guida, F. Orecchia, L. Ramella

grids, which are generated by products of linear forms. It is natural to ask ifthe grids are 1-lifting of monomial ideals. In this paper we show that completegrids are 1-liftings of monomial ideals. We show also that incomplete gridscan be not pseudo 1-lifting. Moreover we prove that the complete grids of fatlines (introduced in [8]) are 1-lifting of monomial ideals (see Theorem 3.5) andthen we prove (Corollary 3.6) that these configurations of lines are A.C.M. inPn, thus generalizing this result proved in [8] in P3. Finally we describe somemore general configurations of lines in P3 (see Definition 4.2) that are A.C.M.(Proposition 4.5). The Cohen Macaulay property is proved by studying thebehaviour of their hyperplane section as in [7]. These configurations can be1-lifting, pseudo 1-lifting, and not pseudo 1-lifting.

1. Lifting Monomial Ideals

We recall and illustrate same definitions and results given in [11] and [2] aboutliftings and pseudoliftings of monomial ideals defining configurations of linearprojective varieties.

From now on K will be a field with characteristic 0, for n ∈ N∗ and t ∈N x1, . . . , xn, u1, . . . , ut will be indeterminates, S := K[x1, . . . , xn] and R :=K[x1, . . . , xn,u1, . . . , ut]. For t = 1 we put u1 = x0.

Let A = (Lj,i)1≤j≤n,1≤i≤r be a matrix whose entries Lj,i are linear forms inR and consider the following conditions (see Proposition 1.7 of [2]):

Condition (α). The n linear forms L1,i1 , . . . , Ln,in are linearly independentfor every 1 ≤ i1, . . . , in ≤ r.

Condition (α′). The vector spaces < L1,i1 , . . . , Ln,in > are n dimensional andpairwise distinct.

Definition 1.1. Consider an n × r matrix A of linear forms Lj,i ∈ Rsatisfying Condition (α). To a monomial m = xα1

1 . . . xαnn ∈ S, with αk ≤ r for

every 1 ≤ k ≤ r, we associate the homogeneous polynomial

m =n∏

j=1

( αj∏

i=1

Lj,i

)∈ R.

To a monomial ideal J = (m1, . . . ,ms) ⊂ S, where {m1, . . . ,ms} is theminimal system of monic monomial-generators of J and the largest power of xj

SOME CONFIGURATIONS OF LINES WHOSE IDEALS... 671

occurring as a factor of any of the minimal generators of J is lower or equal tor, we associate the homogeneous ideal I = (m1, . . . , ms) ⊂ R.

Then I is called a pseudo-t-lifting of J and A is called a pseudo-t-liftingmatrix ([11] Definition 2.22).

If the linear forms Lj,i of a pseudo-t-lifting matrix A actually belong toK[xj, u1,. . . , ut] ⊂ R and xj has non-zero coefficient, then the pseudo-t-lifting I of amonomial ideal J via A is called a t-lifting of J . In fact R/(I, u1, . . . , ut) ∼= S/J(see [11]).

Now we recall some properties about monomial ideals J and their orderideals N (J). The geometric representation of N (J) given in [10] was used in [2]to describe the irreducible components of the projective scheme V (I) ⊆ Pn+t−1

K

defined by a pseudo-t-lifting I of J .

Definition 1.2. A set N ⊆ Nn is an order ideal if and only if (a1, . . . , an) ∈N and (b1, . . . , bn) ∈ Nn with bj ≤ aj for every 1 ≤ j ≤ n implies (b1, . . . , bn) ∈N .

For every monomial ideal J ⊂ S the order ideal N (J) := {(a1, . . . , an) ∈Nn : xa11 · · · xann /∈ J} is called order ideal of J .

Proposition 1.3. (Proposition 2.5 of [2]) Let V (I) ⊆ Pn+t−1K be the

projective scheme defined by a pseudo-t-lifting I ⊂ R of a monomial idealJ ⊂ S induced by an n × r matrix A = (Lj,i) whose entries are linear formssatisfying Condition (α). Then we have:

(1) (Lj1,i1 , . . . , Ljk,ik) (with j1, . . . , jk distinct) gives an irreducible componentof V (I) iff {(a1, . . . , an) ∈ Nn : aj1 = i1 − 1, . . . , ajk = ik − 1} is anirreducible component of N (J) (here Nn is considered a subspace of An

Q

with the Zariski topology).

(2) If Condition (α′) holds, then V (I) is reduced and there is a bijectionbetween the irreducible components of V (I) and the ones of N (J).

For pseudo-1-liftings we put u1 = x0 and R = K[x0, x1, . . . , xn]. If I is anideal obtained by a pseudo-1-lifting of a monomial ideal of S, then we denoteby Va(I) the intersection of V (I) ⊂ Pn

K with the open affine set x0 6= 0.If we consider a monomial ideal J ⊂ S and the 1-lifting matrix A with

entries Lj,i = xj − (i− 1)x0, j = 1, . . . , n, then the order ideal N (J) ⊂ Nn of Jis Va(I) ∩ Nn, where I is the 1-lifting of J via A.

The pseudo-1-liftings of unmixed monomial ideals J ⊂ S with dim(S/J) = 1give examples of A.C.M. configurations of lines. We can prove the followingproposition by means of minimal resolutions as in [11].


Proposition 1.4. Let J ⊂ S be an unmixed monomial ideal with dim(S/J)= 1 and I ⊂ R be a pseudo-1-lifting of J . Then R/I is C.M..

Proof. The ring S/J is C.M. because the monomial ideal J is unmixed anddimS/J = 1. Moreover a monomial ideal J and a pseudo-t-lifting I of J havethe same Betti numbers (see e.g. [11]), then S/J e R/I have the same projectivedimension. By using the Auslander - Buchsbaum formula in the graduate case:

pdR(M) = depth(m, R)− depth(m,M)

where R is a graduate ring, m the irrilevant maximal ideal, M an R-module([3]), we can prove as in [11] Corollary 2.10 that R/I is C.M. if and only if S/Jis C.M..

Example 1.5. Consider in S = K[x1, x2] the ideals J = (x21, x22) (see

Figure 1) and J ′ = (x21x22) (see Figure 2).

Let A1 be the 1-lifting matrix satisfying Condition (α′)

A1 =

(x1 x1 − x0x2 x2 − x0

).

Then the 1-lifting of J via A1 is the ideal I1 = (x1(x1 − x0), x2(x2 − x0)) =(x1, x2)∩ (x1, x2 − x0)∩ (x1 − x0, x2)∩ (x1 − x0, x2 −x0) defining in P2

K the setof points {(1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)}. While the 1-lifting of J ′ via A1 isI ′1 = (x1(x1 − x0)x2(x2 − x0)) defining four lines (see Figure 3).

If A2 is the 1-lifting matrix not satisfying Condition (α′)

A2 =

(x1 − x0 2x1 − 2x0

x2 x2 − x0

),

then the 1-lifting of J via A2 is the ideal I2 = ((x1 − x0)2, x2(x2 − x0)) =

((x1 − x0)2, x2) ∩ ((x1 − x0)

2, x2 − x0) defining two double points. While the1-lifting of J ′ via A2 is I ′2 = ((x1 − x0)

2x2(x2 − x0)) defining two simple linesand a double line (see Figure 4).

Now consider the pseudo-1-lifting matrix A3 satisfying Condition (α′)

A3 =

(x1 + x2 + x0 x1 − x2 − x0

3x1 − x2 + 3x0 2x1 + x2 − 2x0

).

Then the pseudo-1-lifting of J via A3 is the ideal I3 = ((x1+x2+x0)(x1−x2−x0), (3x1 − x2 + 3x0)(2x1 + x2 − 2x0)) = (x1 + x2 + x0, 3x1 − x2 + 3x0)∩ (x1 +x2+x0, 2x1+x2−2x0)∩ (x1−x2−x0, 3x1−x2+3x0)∩ (x1−x2−x0, 2x1+x2−


Figure 1: The orderideal in N2 of J =(x21, x

22).

Figure 2: The orderideal in N2 of J ′ =(x21x

22).

Figure 3: The fourpoints Va(I1) andthe four lines Va(I

′1)

in A2K given by a re-

duced 1-lifting.

Figure 4: Va(I2) ⊂ A2K consisting

of two double points and Va(I′2) ⊂

A2K consisting of a double line with

two simple lines are non-reduced 1-liftings

Figure 5: The four points Va(I3) ⊂A2K and the four lines Va(I

′3) ⊂ A2

K

given by a reduced pseudo-1-liftingthat is not an 1-lifting.

2x0)) defining in P2K the set of points {(1,−1, 0), (1, 3,−4), (1,−2,−3), (1, 1, 0)}.

While the pseudo-1-lifting of J ′ via A3 is I ′3 = ((x1 + x2 + x0)(x1 − x2 −x0)(3x1 − x2 + 3x0)(2x1 + x2 − 2x0)) defining four lines (see Figure 5). Notethat (I3, x0)/(x0) 6= J and (I ′3, x0)/(x0) 6= J ′.


2. The Complete Grids of Lines are Liftings

Now we recall definitions given in [7] about lattice of points and complete gridsof lines.

Definition 2.1. Let {a11, . . . , a1ℓ1}, . . . , {an1, . . . , anℓn} be n finite subsetsof elements of K. The finite set

X := {(1, a1i1 , . . . , anin) : ik = 1, . . . , ℓk, k = 1, · · · , n} ⊂ PnK ,

consisting of ℓ1ℓ2 · · · ℓn points, is called a lattice of type (ℓ1, . . . , ℓn).If ℓ1 = . . . = ℓn = r, then X is called a cubic lattice of type r.

Remark 2.2. Consider, for k = 1, . . . n, the points P∞,k = (ξ0k, ξ1k, . . . , ξnk)∈ Pn

K , where ξik = 0 for i 6= k and ξkk = 1. Note that for every pointP = (1, a1i1 , . . . , anin) of a lattice X of type (ℓ1, . . . , ℓn), the line rP,k joint-ing the points P and P∞,k contains ℓk points of X whose coordinates are(1, a1i1 , . . . , akik , . . . , anin), with 1 ≤ ik ≤ ℓk.

Definition 2.3. Let X be a lattice of type (ℓ1, . . . , ℓn) of points of PnK .

The finite setY := {rP,k : P ∈ X, 1 ≤ k ≤ n}

of the (ℓ2ℓ3 · · · ℓn+· · ·+ℓ1 · · · ℓj−1ℓj+1 · · · ℓn+· · ·+ℓ1 · · · ℓn−1) lines rP,k jointinga point P ofX and one of the point P∞,k (see above Remark) is called a completegrid of type (ℓ1, . . . , ℓn) with basis X.

A grid line is a line of a complete grid.

Example 2.4. In Figure 3 the four points Va(I1) are a lattice and the fourlines Va(I

′1) are a grid of lines of type (2, 2). While in Figure 5 the four points

Va(I3) are not a lattice and the four lines Va(I′3) do not form a grid of lines.

The ideals defining lattices of points and complete grids of lines are calcu-lated in [7]. Now we prove that these ideals are 1-liftings of monomial ideals.

Lemma 2.5. Let X = {(1, a1i1 , . . . , anin) : 1 ≤ k ≤ n, 1 ≤ ik ≤ r} be acubic lattice of points in Pn

K of type r. Then Λ = (xj − ajix0)1≤j≤n,1≤i≤r is an1- lifting matrix satisfying Condition (α′).

Proof. The n-dimensional vector spaces < x1 − a1i1x0, . . . , xn − an,inx0 >are distinct because the points of the lattice X are distinct. So Condition (α′)is verified by the matrix Λ.

Theorem 2.6. Let X be a lattice of type (ℓ1, . . . , ℓn) of points of PnK

and put r = max{ℓ1, . . . , ℓn}. Then the lattice X can be completed to a cubic


lattice X of type r. Consider the 1-lifting matrix Λ determined by X as inLemma 2.5. Then the ideal IX defining X is the 1-lifting of the monomial idealJ = (xℓ11 , . . . , xℓnn ) via the matrix Λ.

Moreover, if Y is a complete grid of type (ℓ1, . . . , ℓn) with basis the latticeX of points, then the defining ideal IY of Y is the 1-lifting via the matrix Λ

of the monomial ideal J ′ = ∩nk=1(x

ℓ11 , . . . ,

ˆxℓkk , . . . , xℓnn ), which is generated by

{xℓjj xℓkk : 1 ≤ j < k ≤ n}.

Proof. Put Lj,i := xj − ajix0, j = 1, . . . , n, i = 1, . . . , r, with r ≥ ℓjfor every j. From [7] Lemma 3(see also [4]) the ideal defining X is IX =(∏ℓ1

i=1 L1,i, . . . ,∏ℓn

i=1 Ln,i), that is the 1-lifting of the monomial ideal J inducedby the 1-lifting matrix Λ.

If we prove that the ideal IY is generated by {(∏ℓji=1 Lj,i)(

∏ℓki=1 Lk,i) : j <

k, j, k ∈ {1, . . . , n}} then IY is the 1-lifting of the monomial ideal J ′ induced bythe matrix Λ, as claimed. By the definition of complete grid of type (ℓ1, . . . , ℓn)we have that IY is:

IY =

ℓ2⋂

i=1

(L2,i, L3,1, . . . , Ln,1)⋂

. . .

ℓ2⋂

i=1

(L2,i, L3,ℓ3 , . . . , Ln,ℓn)

ℓ1⋂

i=1

(L1,i, L3,1, . . . , Ln,1)

⋂. . .

ℓ1⋂

i=1

(L1,i, L3,ℓ3 , . . . , Ln,ℓn)⋂

. . .

ℓn−1⋂

i=1

(L1,ℓ1 , . . . , Ln−2,ℓn−2, Ln−1,i).

By applying the Lemma 7 of [7] we obtain that:

IY = (

ℓ2∏

i=1

L2,i, L3,1, . . . , Ln,1)⋂

. . .

⋂(

ℓ2∏

i=1

L2,i, L3,ℓ3 , . . . , Ln,ℓn)⋂

(

ℓ1∏

i=1

L1,i, L3,1, . . . , Ln,1)

⋂. . .

⋂(

ℓ1∏

i=1

L1,i, L3,ℓ3 , . . . , Ln,ℓn)⋂

. . .⋂

(L1,ℓ1 , . . . , Ln−2,ℓn−2,

ℓn−1∏

i=1

Ln−1,i).

By the modular law we have

IY = (

ℓ1∏

i=1

L1,i

ℓ2∏

j=1

L2,j, . . . ,

ℓn−1∏

i=1

Ln−1,i

ℓn∏

j=1

Ln,j).


Remark 2.7. By Proposition 1.3, the points of a lattice (resp. the linesof a complete grid) of type (ℓ1, . . . , ℓn) in Pn

K can be ”read” in Nn looking at

the order ideal of the artinian monomial ideal J = (xℓ11 , · · · , xℓnn ) (resp. of

the unmixed monomial ideal J ′ = (xℓ11 xℓ22 , xℓ11 xℓ33 , . . . , xℓn−1

n−1 xℓnn ). Note that, by

using results of [2], we can prove that every 1-lifting of J (resp. J ′) via an1-lifting matrix satisfying Condition (α′) is a lattice of points (resp. a completegrid of lines).

Definition 2.8. We say that J = (xℓ11 , · · · , xℓnn ) is the monomial ideal giv-

ing the lattices of points of type (ℓ1, . . . , ℓn) and J ′ = (xℓ11 xℓ22 , xℓ11 xℓ33 , . . . , xℓn−1

n−1 xℓnn )

is the monomial ideal giving the complete grids of lines of type (ℓ1, . . . , ℓn).

Example 2.9. A lattice of points and a complete grid of lines of type(2, 3, 2). Consider in S = K[x1, x2, x3] the ideals J = (x21, x

32, x

23) and J ′ =

(x21x32, x

21x

23, x

32x

23).

Let A be the 1-lifting matrix satisfying Condition (α′)

A =

x1 x1 − 3x0 x1 + a1,3x0x2 − x0 x2 − 4x0 x2 − 6x0x3 − 2x0 x3 − 5x0 x3 + a3,3x0

Then the 1-lifting of J via A is the ideal I = (x1(x1 − 3x0), (x2 − x0)(x2 −4x0)(x2 − 6x0), (x3 − 2x0)(x3 − 5x0)) defining in P3

K the set X of 12 pointsrepresented by Figure 6 in the open set x0 6= 0. While the 1-lifting of J ′ via Ais I ′ = (x1(x1 − 3x0)(x2 − x0)(x2 − 4x0)(x2 − 6x0), x1(x1 − 3x0)(x3 − 2x0)(x3 −5x0), (x2 − x0)(x2 − 4x0)(x2 − 6x0)(x3 − 2x0)(x3 − 5x0)) defining a set Y of 16lines (Figure 7).

Note that X is a lattice of points in P3K of type (2, 3, 2) and Y is a complete

grid with basis X.

Definition 2.10. A subset A of grid lines of a complete grid Y is calledincomplete grid.

In [7], [5] particular types of incomplete grids are considered. Here weexhibit a class of incomplete grids related to monomial ideals.

Proposition 2.11. Let J ⊂ S be a monomial unmixed ideal such thatdim(S/J) = 1 and n ≥ 2. Then any 1-lifting I of J induced by a matrixsatisfying Condition (α′) defines an incomplete grid V = V (I) ⊂ Pn

K .

Proof. Let I be a 1-lifting of J induced by a matrix satisfying Condition(α′). By definition, J = ∩n

k=1Jk, where, for k = 1, . . . , n, Jk is a mono-mial ideal whose minimal generators do not involve the indeterminate xk,


Figure 6: The lattice X in A3K(x0 6=

0) of type (2, 3, 2) consisting of 12points given by the 1-lifting of J =(x21, x

32, x

23) via the matrix A.

Figure 7: The trace in A3K(x0 6= 0)

of the complete grid Y with basisX given by the 1-lifting of J ′ =(x21x

32, x

21x

23, x

32x

23) via the matrix A.

i.e. Jk is the extension of a monomial artinian ideal in the n − 1 indeter-minates x1, . . . , xk, . . . , xn. Then, there exist positive integers ℓ1, . . . , ℓn such

that (xℓ11 , . . . , xℓkk , . . . , xℓnn ) ⊆ Jk for every k = 1, . . . , n. Thus

(xℓ11 xℓ22 , xℓ11 xℓ33 , . . . , xℓ11 xℓnn , . . . , xℓn−1

n−1 xℓnn ) = ∩n

k=1(xℓ11 , . . . , xℓkk , . . . , xℓnn )

⊆ ∩nk=1Jk = J.

Hence, V (I) is an incomplete grid in PnK .

Example 2.12. Incomplete grid with 8 lines. Configuration definedby a 1-lifting of a monomial ideal. Consider an incomplete grid constructed fromthe edges of a cube (of the euclidean space E3

R) corresponding to a completegrid of type (2, 2, 2) by eliminating four lines parallel to same coordinate axisfor example y. The projective closure of these 8 lines is given by the 1-liftingof the monomial ideal J = (x22, x

21x

23) induced by the matrix A:

A =

x1 x1 − x0x2 x2 − x0x3 x3 − x0

.

Remark 2.13. A complete grid of lines is A.C.M., this statement wasproved in [7] by studying the algebraic intersection with a hyperplane. Note


that the assertion follows also because a complete grid of lines is an 1-lifting(Proposition 2.6) of an unmixed monomial ideal satisfying the condition ofProposition 1.4.

Remark 2.14. In [7] Section 4, there are examples of incomplete gridsof lines that are not A.C.M., so these configurations of lines are not liftings ofmonomial ideals.

Remark 2.15. Note that the configurations of lines given by the first au-thor in [6] Theorem 3 are pseudo-1-liftings of the monomial ideals J ′ of Proposi-tion 2.6 giving complete grids of lines (Definition 2.8). Also these configurationof lines are A.C.M., by Theorem 4 of [6] or Proposition 1.4 and [11].

Remark 2.16. In [5] the first author computes the minimal free resolutionof a complete grid of P3

K of type (l,m, n) and poses a conjecture about theminimal free resolution of a complete grid of Pn

K of type (l1, . . . , ln).

3. The m-Fat Complete Grids of Lines are Liftings

Definition 3.1. Let m1, . . . ,mh be positive integers, a fat lattice (resp.fat complete grid) of type (ℓ1, . . . , ℓn) is the projective scheme

Proj(K[x0, . . . , xn]/(Pm1

1 ∩ · · · ∩ Pmh

h ),

whose support is a lattice of points (resp. a complete grid of lines) V (P1) ∪· · · ∪V (Ph) of type (ℓ1, . . . , ℓn), where h = ℓ1 . . . ℓn (resp. h = ℓ2ℓ3 · · · ℓn+ · · ·+ℓ1 · · · ℓj−1ℓj+1 · · · ℓn + · · ·+ ℓ1 · · · ℓn−1).

If m1 = · · ·mn = m, then the above projective schemes are called respec-tively m-fat lattice and m-fat complete grid.

At first we consider fat complete grid of lines in P3K and we recall the

following result of [8].

Theorem 3.2. (see [8]) Let Y ⊂ P3K be an m-fat complete grid of type

(ℓ1, ℓ2, ℓ3). Consider the linear forms Lj,i := xj − ajix0 as in Proposition 2.6and, for 0 ≤ q ≤ [m2 ], the polynomials

G1,q = Lq1,1 · · ·Lq

1,ℓ1Lm−q2,1 · · ·Lm−q

2,ℓ2Lm−q3,1 · · ·Lm−q

3,ℓ3,

G2,q = Lm−q1,1 · · ·Lm−q

1,ℓ1Lq2,1 · · ·L

q2,ℓ2

Lm−q3,1 · · ·Lm−q

3,ℓ3,

G3,q = Lm−q1,1 · · ·Lm−q

1,ℓ1Lm−q2,1 · · ·Lm−q

2,ℓ2Lq3,1 · · ·Lq

3,ℓ3.

If m is odd, then the defining ideal IY of Y is minimally generated by the32(m+ 1) polynomials G1,q, G2,q, G3,q, 0 ≤ q ≤ m−1

2 .


If m is even, then the defining ideal IY of Y is minimally generated by the32m + 1 polynomials G1,q, G2,q, G3,q, 0 ≤ q ≤ m−2

2 , and G := G1,m2

= G2,m2

=G3,m

2

.

Now we prove that the above ideal IY is an 1-lifting of a monomial ideal.

Theorem 3.3. Let Y ⊂ P3K be an m-fat complete grid of type (ℓ1, ℓ2, ℓ3)

and consider the linear forms Lj,i = xj − ajix0 of Proposition 2.6. Let A bean 1-lifting matrix of linear forms whose j-th row begins with the sequenceLj,1, Lj,2, · · · , Lj,ℓj repeated (1 + [m2 ])-times, j = 1, 2, 3.

Then the defining ideal IY of Y is the 1-lifting via the matrix A of themonomial ideal Γ = (xℓ11 , xℓ22 )m ∩ (xℓ11 , xℓ33 )m ∩ (xℓ22 , xℓ33 )m.

Proof. The scheme Y is supported on the complete grid Y (case m = 1) oftype (ℓ1, ℓ2, ℓ3) whose defining ideal IY is the 1-lifting of the monomial idealJ ′ = (xℓ11 xℓ22 , xℓ11 xℓ33 , xℓ22 xℓ33 ) = (xℓ11 , xℓ22 ) ∩ (xℓ11 , xℓ33 ) ∩ (xℓ22 , xℓ33 ) induced by thematrix Λ = (Lj,i) of Proposition 2.6.

Now, for m ≥ 2, we find a minimal system of monomial generators ofthe ideal Γ. They are the least common multiples l.c.m.(g1, g2, g3) with gkminimal monomial generator of

(xℓii , xℓjj )

m = (xmℓii , x

(m−1)ℓii x

ℓjj , x

(m−2)ℓii x

2ℓjj , . . . , xℓii x

(m−1)ℓjj , x

mℓjj ),

i, j, k ∈ {1, 2, 3}, k /∈ {i, j}.For 0 ≤ q ≤ [m2 ], put

γ1,q = xqℓ11 x(m−q)ℓ22 x

(m−q)ℓ33

γ2,q = x(m−q)ℓ11 xqℓ22 x

(m−q)ℓ33

γ3,q = x(m−q)ℓ11 x

(m−q)ℓ22 xqℓ33 .

If m is odd, then we can prove that Γ ⊂ K[x1, x2, x3] is generated by

{γ1,q, γ2,q, γ3,q : 0 ≤ q ≤ m−12 }.

If m is even, then Γ ⊂ K[x1, x2, x3] is generated by

{γ1,q, γ2,q, γ3,q : 0 ≤ q ≤ m−22 } ∪ {γ}, γ := γ1,m

2

= γ2,m2

= γ3,m2

.

Thus the defining ideal IY of Y is the 1-lifting of the monomial ideal Γinduced by the matrix A.

By means of 1-liftings of monomial ideals, we can describe m-fat latticesand m-fat complete grids in Pn

K also for n > 3.

Theorem 3.4. Let X ⊂ PnK be an m-fat lattice of type (ℓ1, . . . , ℓn) and

consider the linear forms Lj,i = xj − ajix0 as in Proposition 2.6 induced by thelattice X of points support of X. Let A be an 1-lifting matrix of linear forms


whose j-th row begins with the sequence Lj,1, Lj,2, · · · , Lj,ℓj repeated m-times,j = 1, . . . , n.

Then the defining ideal IX of X is the 1-lifting via the matrix A of themonomial ideal Jm, where J = (xℓ11 , xℓ22 , . . . , xℓnn ) is the monomial ideal givingthe lattice X.

Proof. The ideal IX defining the lattice X of points is the 1-lifting of J viathe matrix Λ = (Lj,i) of Proposition 2.6, we have

IX = ∩1≤ik≤ℓk,1≤k≤n(L1,i1 , L2,i2 , . . . , Ln,in).

Then the (saturated) ideal defining the m-fat lattice X is

IX = ∩1≤ik≤ℓk,1≤k≤n(L1,i1 , L2,i2 , . . . , Ln,in)m.

The above equality gives a minimal primary decomposition of IX. Note thatthe 1-lifting of Jm via the matrix A is just ImX = (L1,1 · · ·L1,ℓ1 , . . . , Ln,1 · · ·Ln,ℓn)

m.We have to prove that IX = ImX .

The lattice X of points is contained in the affine open set x0 6= 0. In thelocalized ring Rx0

the points of X are defined by coprime maximal ideals mk,k = 1, . . . , h and we have IXRx0

= ∩hk=1mk = Πh

k=1mk. We have also IXRx0

= ∩hk=1m

mk = Πh

k=1mmk = (IXRx0

)m = ImXRx0. Recall that for an ideal a of

R a minimal primary decomposition of aRx0is given by the primary ideals q

of a minimal primary decomposition of a such that the prime ideal√q does not

meet {xi0 : i ∈ N}. We could have ImX = IX ∩q, with √q = (x0, x1, . . . , xn), but

this case is not possible because the ideal ImX , being an 1-lifting of a monomialideal, is saturated (see [11] Corollary 2.10). So we have ImX = IX.

Theorem 3.5. Let Y ⊂ PnK be a complete grid of lines of type (ℓ1, . . . , ℓn)

and Y ⊆ PnK be the m-fat complete grid supported on Y . Consider the linear

forms Lj,i = xj − ajix0 as in Theorem 2.6 induced by the lattice X of pointsbasis of Y and an 1-lifting matrix A of linear forms whose j-th row begins withthe sequence Lj,1, Lj,2, · · · , Lj,ℓj repeated m-times, j = 1, . . . , n.

Then the defining ideal IY of Y is the 1-lifting via the matrix A of the

monomial ideal Γ = ∩nk=1J

mk , where Jk = (xℓ11 , . . . , xℓkk , . . . , xℓnn ).

Proof. The complete grid Y is the 1-lifting of the monomial ideal J ′ =∩nk=1Jk via the matrix Λ = (Lj,i) of Theorem 2.6 and we can write Y = ∪n

k=1Yk,where Yk denotes the union of the lines contained in Y passing through P∞,k.Note that Yk is defined by the (saturated) ideal 1-lifting via Λ of the monomial


ideal Jk. Then as in the proof of Theorem 3.4 we obtain that the scheme Yk

of the m-fat lines of Yk is just the 1-lifting of the monomial ideal Jmk via the

matrix A and the result is obtained.

Corollary 3.6. A m-fat complete grid of lines is A.C.M.

Proof. The m-fat complete grids of lines in PnK are 1-liftings of monomial

ideals and then they are A.C.M., by Proposition 1.4.

Remark 3.7. In [8] the statement that the m-fat complete grids of linesin the projective space P3

K are A.C.M. was proved by studying the algebraicintersection with a generic plane.

4. Generalization of Complete Grids in P3

K

In this section we consider an extension (see Definition 4.2) of complete grids inP3K . For configuration of lines we mean a (reduced) finite union of lines. We give

a class of configurations of lines that are A.C.M. and that are not constructedby pseudo-1-liftings of monomial ideals. We prove the A.C.M. property bystudying the algebraic intersection with hyperplanes as in [7] Theorem 10 forcomplete grids of lines. We apply the following Lemma

Lemma 4.1. (Lemma 9 of [7]) Let Z be a finite union of h lines in PnK

and H be a linear homogeneous form such that the hyperplane H = 0 doesnot contain any line of Z nor any points of Sing(Z), which are the pointsintersection of the lines of Z. Then Z is A.C.M. if and only if (IZ ,H) =∩hi=1(Pi,H), where Pi for i = 1, . . . , h denote the ideals defining the lines of Z

and IZ = ∩hi=1Pi is the ideal defining Z.

Definition 4.2. In P3K consider a point V , s distinct lines m1, . . . ,ms

passing through V , h planes σ1, . . . , σh do not passing trough V and do notcontaining the lines mi, i = 1, . . . , s. For i = 1, . . . , s, denote by λi the planedetermined by the edges mi and mi+1, putting ms+1 = m1. Suppose that theplanes λ1, . . . , λs are distincts. Denote by Pi,j the point intersection of the linemi with the plane σj and suppose that the set {Pi,j : 1 ≤ i ≤ s, 1 ≤ j ≤ h}consists of sh distinct points. For every 1 ≤ i ≤ s and 1 ≤ j ≤ h, denote by ℓi,jthe line of the plane σj jointing the points Pi,j and Pi+1,j , putting Ps+1,j = P1,j .

We call configuration Z the union of the s(h + 1) lines m1, . . . ,ms (callededges) and {ℓi,j : 1 ≤ i ≤ s, 1 ≤ j ≤ h}. The point V is called vertex of Z, theplanes σ1, . . . , σh are called base-faces of Z and the planes λ1, . . . , λs definedabove are called side-faces of Z.


The union of lines Z can assume various configurations: complete gridsin P3

K (see Remark 4.3), tetrahedron (see Example 4.7), parallelepiped (seeExample 4.9),pyramid (see Example 4.6 and 4.12), etc... We do not knowin general when a configuration of type Z is a 1-lifting or pseudo-1-lifting.In the following we provide serious examples that show the situation is verycomplicated, but all three situations can appear: 1-lifting , pseudo-1-lifting,not pseudo-1-lifting.

Remark 4.3. The Definition 4.2 in P3K generalizes the notion of complete

grid. In fact, a complete grid of P3K can be see as configuration Z with vertex

one of the points at infinity P∞,1, P∞,2, P∞,3.

Lemma 4.4. Consider L1, . . . , Ls, M1, . . . ,Mh, H linear homogeneousform in

K[x0, x1, x2, x3] and put L := L1 · · ·Ls.

1. If Li, Lj ,Mk are linearly independent for every k and i 6= j and Li,Mj ,Mk

are linearly independent for every i and j 6= k, then (L,M1) ∩ · · · ∩(L,Mh) = (L,M1 · · ·Mh).

2. If Li, Lj,Mk,H are linearly independent for every k and i 6= j andLi,Mj ,Mk,H are linearly independent for every i and j 6= k, then

(L,M1,H) ∩ · · · ∩ (L,Mh,H) = (L,M1 · · ·Mh,H).

Proof. Linearly independent linear forms generate a prime ideal. The proofis a computation based on properties of prime ideals (see [7] Lemma 7)

Proposition 4.5. The configuration Z ⊂ P3K described in Definition 4.2

is A.C.M.

Proof. We can suppose V = (0, 0, 0, 1) (so in the open affine set x0 6= 0, thelines m1, . . . ,ms are parallel to the third axe z = x3

x0). We calculate the ideal

IZ ⊂ R = K[x0, x1, x2, x3] defining Z.

On the plane x3 = 0, whose coordinate ring is K[x0, x1, x2], consider thepoints P1, . . . , Ps given by the intersection with the edges m1, . . . ,ms respec-tively and for every i = 1, . . . , s the line ℓi jointing the points Pi and Pi+1,putting Ps+1 = P1. The line ℓi in P3 can be defined by an ideal (Li, x3),where Li is a linear homogeneous form of K[x0, x1, x2], i = 1, . . . , s. Notethat in P3 the equation Li = 0 define the side-face λi. Let W be the setof points P1, . . . , Ps. Note that W in the plane x3 = 0 is defined by the ideal


IW = (L1, L2)∩(L2, L3)∩· · ·∩(Ls−1, Ls)∩(Ls, L1) ⊂ K[x0, x1, x2]. So the unionY of the edgesm1, . . . ,ms in P3

K is defined by the ideal IY = IWK[x0, x1, x2, x3].If the base-face σj ⊂ P3

K is defined by a linear homogeneous form Mj , thenthe edge ℓi,j ⊂ P3

K is defined by the ideal (Li,Mj) ⊂ K[x0, x1, x2, x3], 1 ≤ j ≤ h,1 ≤ i ≤ s.

The ideal IZ defining Z is :

IY ∩(L1,M1)∩· · ·∩(Ls,M1)∩(L1,M2)∩· · ·∩(Ls,M2)∩· · ·∩(L1,Mh)∩· · ·∩(Ls,Mh).

For i 6= j the linear forms Li, Lj ,Mk ∈ K[x0, x1, x2, x3] are linearly indepen-dent and for j 6= k the linear forms Li,Mj ,Mk ∈ K[x0, x1, x2, x3] are linearlyindependent, so by Lemma 4.4 we have IZ =IY ∩ (L1L2 · · ·Ls,M1) ∩ (L1L2 · · ·Ls,M2) ∩ · · · ∩ (L1L2 · · ·Ls,Mh) =IY ∩ (L1L2 · · ·Ls,M1M2 · · ·Mh).

Since L1L2 · · ·Ls ∈ IY , by the modular low we have

IZ = (L1L2 · · ·Ls) + ((M1M2 · · ·Mh) ∩ IY ).

Put IW = (f1, . . . , fr) ⊂ K[x0, x1, x2], since IWK[x0, x1, x2, x3] = IY is intersec-tion of prime ideals do not containing the linear form Mj , for every j = 1, . . . , h(i.e. every line mi is not contained in the plane σj), we obtain

IZ = (L1L2 · · ·Ls,M1M2 · · ·Mhf1, . . . ,M1M2 · · ·Mhfr).

Now we consider a linear homogeneous form H ∈ K[x0, x1, x2, x3] satisfyingconditions of Lemma 4.1. The configuration Y of lines is A.C.M., in fact itscoordinate ring (K[x0, x1, x2]/IW )[x3] is C.M. since W is a finite set of distinctpoints. Thus (IY ,H) = (L1, L2,H) ∩ · · · ∩ (Ls, L1,H).

Note that for i 6= j the linear forms Li, Lj ,Mk,H are linearly independentand for j 6= k the linear forms Li,Mj ,Mk,H are linearly independent, so byLemma 4.4 we have (L1,M1,H)∩· · · ∩ (Ls,Mh,H) = (L1 · · ·Ls,M1 · · ·Mh,H).

Since (L1 · · ·Ls,H) ⊂ (IY ,H), by the modular low we obtain(IY ,H) ∩ (L1 · · ·Ls,M1 · · ·Mh,H) = (L1 · · ·Ls,H) + ((M1 · · ·Mh) ∩ (IY ,H)).

Recall that Y is A.C.M., then (IY ,H) = (f1, . . . , fr,H) is the intersectionof the prime ideals defining the points intersection of Y with the plane H = 0.None of those points belongs to the planes σ1, . . . , σh. Thus the above ideal is(L1 · · ·Ls,H,M1 · · ·Mhf1, . . . ,M1 · · ·Mhfr,M1 · · ·MhH) = (IZ ,H). Then Z isA.C.M.

Now we describe some figures of the euclidean space E3R by using configu-

rations of lines in the real projective space P3R forming a configuration Z (Def-

inition 4.2). They generalize examples described in [6]. We see when they are


related to pseudo-1-liftings of monomial ideals of K[x1, x2, x3]. The coordinatesof E3

R are denoted as usual x = x1

x0, y = x2

x0, z = x3

x0.

Example 4.6. Pyramid. A configuration defined by a pseudo-1-lifting ofthe monomial ideal giving the complete grids of lines of type (2, 2, 1). Considerthe pyramid with vertices P1 = (0, 0, 0), P2 = (1, 0, 0), P3 = (0, 1, 0), P4 =(1, 1, 0), P5 = (0, 0, 1), whose faces are the planes x = 0, y = 0, x+z−1 = 0,y+ z − 1 = 0 and z = 0 (Figure 8). The projective closure of the configurationof the 8 edges of the pyramid is given by the pseudo-1-lifting of the monomialideal J = (x21x

22, x

21x3, x

22x3) (see Figure 9) induced by the matrix A, but is not a

1-lifting of J because the linear forms Lj,i of A do not belong to K[x0, xj ] ∀j =1, 2, 3, ∀i = 1, 2:

A =

x1 x1 + x3 − x0x2 x2 + x3 − x0x3 L3,2

.

Figure 8: The pyramid. Its projec-tive closure is a pseudo-1-lifting ofa monomial ideal considered in Def.2.8.

Figure 9: The order ideal in N3 ofJ = (x21x

22, x

21x3, x

22x3) giving the

pyramid.

To obtain a regular pyramid take the vertices P1, P2, P3, P4 as above and

P ′5 = (12 ,

12 ,

√22 ). Its projective closure in P3

R is the pseudo-1-lifting of themonomial ideal J induced by the matrix

A′ =

√2x1 − x3

√2x1 + x3 −

√2x0√

2x2 − x3√2x2 + x3 −

√2x0

x3 L3,2

.


Example 4.7. Tetrahedron. It is defined by a pseudo-1-lifting of amonomial ideal not giving complete grids of lines. Consider a tetrahedron withvertices P1 = (0, 0, 0), P2 = (1, 0, 0), P3 = (0, 1, 0), P4 = (0, 0, 1), whosefaces are the planes x = 0, y = 0, x + y + z − 1 = 0 and z = 0. LetZ be the configuration of the 6 edges of the tetrahedron (Figure 10). Thenprojective closure Z of Z is given by the pseudo-1-lifting I of the monomialideal J = (x21x2, x

21x3, x

22x3, x1x2x3) (see Figure 11) induced by the matrix B,

but is not a 1-lifting of J because the linear forms Lj,i of B do not belong toK[x0, xj ] ∀j = 1, 2, 3, ∀i = 1, 2:

B =

x1 x1 + x2 + x3 − x0x2 x2 + x3 − x0x3 L3,2

.

Figure 10: The tetrahedron. Itsprojective closure is a pseudo-1-lifting of a monomial ideal not sat-isfying Def. 2.8.

Figure 11: The order ideal in N3 ofJ = (x21x2, x

21x3, x

22x3, x1x2x3) giv-

ing the tetrahedron.

Note that the above matrix does not satisfy Condition (α′), however thepseudo-1-lifting I define a reduced scheme. For the linear forms Lj,i entries ofB we have that the vector spaces < L1,1, L2,2, L3,1 > and < L1,2, L2,2, L3,1 >are equal, but the plane L2,2 = 0 is not a face of the tetrahedron Z defined byI.

Remark 4.8. If we consider the pseudo-1-lifting of the above monomial


ideal J via the matrix

C =

x1 x1 +12 x2 + x3 − x0

x2 x2 + x3 − x0x3 L3,2

satisfying Condition (α′), we obtain a configuration Y of 6 lines that is neithera tetrahedron nor a configuration satisfying Theorem 3 of [6].

Example 4.9. Parallelepiped. Configuration defined by a pseudo-1-lifting of the monomial ideal giving the complete grids of lines of type (2, 2, 2).Consider in E3

R a solid with vertices Pi = (ai, bi, 0), Qi = (ai, bi, 1), i =1, 2, 3, 4(see Figure 12). The projective closure Z ⊂ P3

R of the 12 egdes ofthe solid is a configuration Z of Definition 4.2 with two quadrilaterals as basesand with vertex the point V = (0, 0, 0, 1) ∈ P3

R. Z is defined by the the pseudo-1-lifting of the monomial ideal J = (x21x

22, x

21x

23, x

22x

23) (see Figure 13) induced

by the matrix

M =

Λ1 Λ3

Λ2 Λ4

x3 x3 − x0

,

where, for 1 ≤ i ≤ 4, Λi = (bi+1 − bi)(x1 − aix0) − (ai+1 − ai)(x2 − bi), witha5 = a1, b5 = b1. Note that Λi = 0, i = 1, 2, 3, 4, give the 4 side-faces of thesolid, while x3 = 0, x3 − x0 = 0 give the base-faces of the solid. The set of the8 vertices of the solid {P1, . . . , P4, Q1, . . . , Q4} are given by the pseudo-1-liftingof (x21, x

22, x

23) induced by the above matrix M .

To obtain a parallelepiped in E3R we can consider a pseudo-1-matrix

M ′ =

ax1 + bx2 ax1 + bx2 + cx0bx1 − ax2 bx1 − ax2 + dx0

x3 x3 − x0

.

Note that for b = 0 we have that A′ is an 1-lifting matrix and we obtaina parallelepiped whose projective closure is a complete grid of lines of type(2, 2, 2).

Example 4.10. Configurations of lines with a regular exagon asbases. The configuration cannot be defined by pseudo-1-liftings of monomialideals. Consider in E3

R the solid with a regular exagon as bases, whose vertices

are P1 = (0, 0, 0), P2 = (12 ,√32 , 0), P3 = (32 ,

√32 , 0), P4 = (2, 0, 0), P5 =


Figure 12: The parallelepiped withrectangular bases. Its projectiveclosure is a complete grid of type(2, 2, 2) and an 1-lifting of a mono-mial ideal.

Figure 13: The order ideal in N3

of J = (x21x22, x

21x

23, x

22x

23) giving the

parallelepiped.

(32 ,−√32 , 0), P6 = (12 ,−

√32 , 0) and Q1 = (0, 0, 1), Q2 = (12 ,

√32 , 1), Q3 =

(32 ,√32 , 1), Q4 = (2, 0, 1), Q5 = (32 ,−

√32 , 1), Q6 = (12 ,−

√32 , 1) (Figure 14).

The 8 faces of the solid are the planes z = 0, z−1 = 0 and the 6 planes Li = 0passing through the vertices Pi, Pi+1, Qi, Qi+1, for 1 ≤ i ≤ 6, with P7 = P1 and

Q7 = Q1. We can calculate L1 =√3x−y, L2 = y−

√32 , L3 =

√3x+y−2

√3,

L4 =√3x−y−2

√3, L5 = y+

√32 , L6 =

√3x+y. The configuration Z of the

edges of that solid consists of 18 lines, its projective closure Z is a configurationZ of Definition 4.2 defined by the ideal I = (L1L2 · · · L6, f1x3(x3−x0), f2x3(x3−x0)), where Li = 0 is the projective closure of the plane Li = 0, for 1 ≤ i ≤ 6,f1 = (x1−x0)

2+x22−x20 and f2 = L1L3L5 (see proof of Proposition 4.5). ThenZ is contained in a unique minimal surface of degree 4 that cannot be definedby the product of linear forms, so the ideal I is not generated by products oflinear forms and it cannot be a pseudo-1-lifting of monomial ideals.

Remark 4.11. Consider the configuration Z of the above Example andthe monomial ideal J = (x31x

32, x

31x

23, x

32x

23) giving the complete grids of type

(3, 3, 2) (see Figure 15) and the matrix

Λ =

L1 L3 L5

L2 L4 L6

x3 x3 − x0 L3,3

.


Figure 14: Solid with regularexagon as base. It is a configurationof 18 lines whose projective closureis not a pseudo-1-liftings of mono-mial ideals.

Figure 15: The order ideal in N3

of J = (x31x32, x

31x

23, x

32x

23), giving by

means of a pseudo-1-lifting a con-figuration Y ⊂ P3

R of 21 lines suchthat its intersection with the openset x0 6= 0 is the above solid withexagon as base.

The configuration Y ⊂ P3R given by the pseudo-1-lifting of J induced by

the matrix Λ consists of 21 lines and in the open set x0 6= 0 the trace of Y isthe set Z of the 18 edges of the solid described in the above Example. On theother hand the projective closure Z of Z is obtained from Y by removing thelines defined by the ideals (L1, L4), (L3, L6), (L5, L2).

Example 4.12. Lines of a pyramid with base a regular polygon ofn ≥ 5 sides. This configuration of lines cannot be defined by pseudo-1-liftings ofmonomial ideals. Consider in E3

R the pyramid with base a regular polygon withn ≥ 5 sides, of vertices P1,1, . . . , Pn,1. The n side-faces λ1, . . . , λn of this pyramidare defined respectively by the linear homogeneous forms L1, . . . , Ln, and theonly base-face by the linear homogeneous form M . The configuration Z of theedges of that pyramid consists of 2n lines, its projective closure Z is a configu-ration Z of Definition 4.2 defined by the ideal I = (L1L2 · · · Ln, f1M, . . . , frM ),where Li = 0 is the projective closure of the plane Li = 0, for 1 ≤ i ≤ n, andM = 0 is the projective closure of the planeM = 0. Moreover, f1, . . . , fr are thegenerators of the ideal IY = (L1, L2)∩ . . .∩ (Ln−1, Ln) (see proof of Proposition4.5), but IY = IWK[x0, . . . , x3] where W is the set of points P1,1, . . . , Pn,1 thenIY contains a generator of degree 2 because P1,1, . . . , Pn,1 belong to a circumfer-


ence. Then Z is contained in a unique minimal surface of degree 3 that cannotbe defined by the product of linear forms, so the ideal I is not generated byproducts of linear forms and it cannot be a pseudo-1-lifting of monomial ideals.

Finally, we give an example of configuration of lines Y Cohen-Macaulaywhose defining ideal IY is generated by products of linear forms but is not apseudo-1-lifting nor a lifting of a monomial ideal. The following configurationY is obtained from a configuration Z (see Definition 4.2) consisting of the lines:m1,m2,m3, ℓ1,1, ℓ2,1, ℓ3,1 (where P1,1 = P1, P2,1 = P2, P3,1 = P3, V = P4) andeliminating the lines m2, ℓ3,1.

Example 4.13. Consider in P3R the following points: P1 = (7, 1, 5,−2), P2 =

(3, 5, 1,− 2), P3 = (7, 1,−3,−2), P4 = (3,−3, 1,−2) and we construct the lines throughthe points Pi and Pi+1, (i = 1, 2, 3, 4), where P5 = P1. The defining ideal IYthe configuration Y of these four lines is IY = ((x − z + t)(x + z + 2t), (x −y + 3t)(x + y + 4t)), hence IY is generated by products of linear forms. More-over since IY is a complete intersection it is A.C.M. But, IY is not a pseudo-1-lifting of a monomial ideal J , otherwise J must be unmixed of dimension1, the order ideal N (J) must consist of four lines: the three axes and aline parallel to an axis, hence a less of a change of coordinates we have thatJ = (x, y) ∩ (y, z) ∩ (x2, z) = (xz, yz, x2y), then J has a minimal generator ofdegree 3 (while IY is generated in degree 2). But IY and J must have the sameminimal resolution, contradiction.

Acknowledgments

Authors were supported by MIUR and GNSAGA (CNR).

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