The ACM property for unions of lines in P 1 × P 2
aa r X i v : . [ m a t h . A C ] S e p THE ACM PROPERTY FOR CODIMENSION TWO VARIETIES IN P × P GIUSEPPE FAVACCHIO AND JUAN MIGLIORE
Abstract.
This paper examines the Arithmetically Cohen-Macaulay (ACM) property for cer-tain codimension 2 varieties in P × P called sets of lines in P × P (not necessarily reduced).We discuss some obstacles to finding a general characterization. We then consider certain classesof such curves, and we address two questions. First, when are they themselves ACM? Second, ina non-ACM reduced configuration, is it possible to replace one component of a primary (prime)decomposition by a suitable power (i.e. to “fatten” one line) to make the resulting scheme ACM?Finally, for our classes of such curves, we characterize the locally Cohen-Macaulay property incombinatorial terms by introducing the definition of a fully v-connected configuration. We applysome of our results to give analogous ACM results for sets of lines in P . Introduction
It is still an open problem to determine a geometric characterization of the arithmeticallyCohen-Macaulay (ACM) property for varieties in a multiprojective space. While this problemhas strong connections to the analogous problem for varieties in a projective space, there are alsostriking differences. To give just two simple illustrations, any finite set of points in a projectivespace is automatically ACM, while this is not true in any multiprojective space. Indeed, a wholebook has been devoted to this topic [14] just for the case of P × P . Secondly, the structure ofa multihomogeneous ideal differs from that of a homogeneous ideal in important ways, and thiscontributes to the difficulty.In both the projective and the multiprojective settings, the problem can have a strong combi-natorial component in addition to geometric and algebraic ones. One specific problem that hasbeen studied in the projective setting in many different ways is the question of when a union oflines is ACM. Included in this is the intersection lattice of the set of lines, but simple examplesshow that even this is not enough to determine whether the union of lines is ACM or not. This isstill wide open in projective space, in general. In this paper we begin the study of this problemin a specific multiprojective space, namely P × P .Included in this problem (in either setting) is the question of what can be added to a non-ACM variety to make it become ACM, hopefully adding as little as possible. We propose a newvariation of this question (and solve it in our special setting): given a reduced configurationof lines that is not ACM, can one “fatten” one of the components and make the new schemeACM? Here, by “fatten” we mean that in a primary decomposition we replace a prime ideal p ,corresponding to one of the lines, by a power p k for suitable k . (Notice that p is a completeintersection, so p k is unmixed.) After solving this problem in the situation of this paper, we giveas a corollary the analogous result for unions of lines in P . (See Theorem 4.14 for P × P andCorollary 4.15 for P .) We also explore the property of being locally Cohen-Macaulay, giving acharacterization for our curves in P × P . Mathematics Subject Classification.
Key words and phrases.
Varieties in multiprojective spaces, Arithmetically Cohen-Macaulay, Configurationof lines.Version: September 8, 2020.
Many papers in the literature have studied the ACM property for different kinds of subvarietiesof multiprojective spaces, especially for sets of points. Despite the fact that in P × P thereare several characterization of the ACM property (again, see [14] for a detailed discussion of thetopic) only a few other results are known in general. In particular, a characterization was givenin [7] of the ACM property for sets of points in ( P ) r := P × · · · × P , and in [9] the authorsdescribed, under certain conditions, the ACM property for sets of points in P m × P n and, withmore details, in P × P n . See also [15] for some other results in this direction.A crucial difference in the study of the ACM property of a set of points in P × P and a setof points in any other multiprojective space is given by the codimension. A set of points hascodimension two in P × P , and strictly larger than two in a different multiprojective space.So, it is interesting to approach the study of the ACM property by looking at the codimensiontwo varieties more generally. The paper [8] investigates the ACM property for 2-codimensionalvarieties in P × P × P . Such varieties have a different nature and meaning from sets of pointsin ( P ) r , r ≥
2, but the characterization of the ACM property deeply uses a common fact. Allof them are defined by ideals generated by particular products of linear forms. In this case, thispeculiarity makes the description of the ACM property merely combinatorial.The vast nature of the problem draws in many standard tools and techniques from the ho-mogeneous setting. These include, just to cite some of them, hyperplane sections, basic doubleG-linkage, liaison addition, and liaison. On the other hand, the study of varieties in multipro-jective spaces plays an important role in several branches of mathematics, and it finds an appli-cation in different contexts; these include the study of monomial ideals (see for instance [3, 19]),scrolls ([6]), symbolic powers ([2, 12, 13]), tensor analysis ([5]) and virtual resolutions ([4, 10]),just to give a partial list.In this paper we call sets of lines of P × P certain codimension two varieties in P × P . Wewill also look at these varieties as unions of planes in P . This is possible since a bihomogeneouslinear form of k [ P × P ] also defines a hyperplane in P . Thus, a line in P × P can be viewedas a plane (the intersection of two hyperplanes) of P . This correspondence allows us to movethe study of several properties, included the Cohen-Macaulay property, from P × P to P . Theconverse is not always possible for two reasons due to the nature of the problem. First, a linearform in k [ P ] is not necessarily an element of k [ P × P ] (1 , or k [ P × P ] (0 , . Moreover, a planein P could be defined by an ideal that is, even if bihomogeneous, not saturated in P × P . From what has been observed above, we only have lines of two different types in P × P . The horizontal lines , given by the intersection of a hyperplane of degree (1 ,
0) with one of degree (0 , vertical lines , that are intersections of two hyperplanes of degree (0 , P since a set of vertical lines in P × P is the cone over a set ofpoints in P . We will give more details in Section 3, where we introduce this terminology.The main result in Section 2, Proposition 2.1, concerns sets of points in P , which will be usedlater in the paper. Given a set of (fat) points Y and a curve V ( F ) passing through some of them,we give a characterization, in terms of the h -vector of a certain subscheme of Y , to guarantee that I Y + ( F ) is a saturated ideal. In Section 3 we introduce the notation and examine some obstaclesto the characterization of the ACM property; see Remark 3.2 and Example 3.3. In Section 4we apply Proposition 2.1 to study the ACM property for sets of lines in P × P satisfyinga particular condition introduced in Notation 4.1. We begin with an additional assumption(see Theorem 4.4 and Remark 4.5) and then show that this can be extended (Proposition 4.6,Proposition 4.7 and Corollary 4.8). We then apply these results to a class of configurationsof lines in P × P where only one is non-reduced. In this situation we give a necessary andsufficient condition for ACMness in terms of the multiplicity of the non-reduced component. Asa corollary we give a result for P , as mentioned above. Our most general result (still with some CM PROPERTY FOR CODIMENSION TWO MULTIPROJECTIVE VARIETIES 3 assumptions) gives a characterization for when the configuration is locally Cohen-Macaulay (seeTheorem 4.17).
Acknowledgments.
Many of the results in this paper were inspired through computer exper-iments using CoCoA [1] and Macaulay2 [11]. This work was done while the first author waspartially supported by Universit`a degli studi di Catania, piano della ricerca PIACERI 2020/22linea intervento 2 and by the National Group for Algebraic and Geometrical Structures and theirApplications (GNSAGA-INdAM) and the second author was partially supported by a SimonsFoundation grant (
A preliminary result about saturation in P In this section we give a criterion, in terms of the h -vector, to establish if an ideal of fat pointsplus a form is saturated or not. In the next sections, we will relate it to the ACM property forsome sets of lines in P × P ; see Remark 4.5. There is an immediate connection with points in P since a set of vertical lines in P × P is the cone over a set of points in P . We start the sectionby recalling some standard notation. Given a finite set of n distinct points W = { P , . . . , P n } in P N , and m , . . . , m n positive integers, we write Y := m P + · · · + m n P n for the set of fatpoints defined by the saturated homogeneous ideal I Y := \ P i ∈ W ( I P i ) m i ⊆ S := k [ P N ] . The degree of Y is deg( Y ) = P (cid:0) m i + N − N − (cid:1) . Recall that for a zero-dimensional scheme Y ⊂ P N the Hilbert function of Y is defined as the numerical function H Y : N → N such that H Y ( i ) = dim k ( S/I Y ) i = dim k S i − dim k ( I Y ) i . Since H Y ( τ ) = deg( Y ) for τ large enough, the first difference of the Hilbert function ∆ H Y ( i ) := H Y ( i ) − H Y ( i −
1) is eventually zero. The h -vector of Y is h Y = h = (1 , h , . . . , h d )where h i = ∆ H Y ( i ) and d is the last index such that ∆ H Y ( i ) > Proposition 2.1.
Let Y := m P + m P + · · · m n P n be a set of (fat) points in P . Let C = V ( F ) ⊆ P be a curve defined by a form F of degree deg( F ) = d. For any i , assume that if F vanishes at P i ∈ Y , then the vanishing order is at least m i , so F ∈ [ I m i P i ] d . Let Y and Y be thetwo zero-dimensional schemes such that I Y = ( I Y + F ) sat and I Y = I Y : F. Then Y = Y ∪ Y ,where Y ∩ Y = ∅ , and I Y + ( F ) ⊆ k [ P ] is saturated if and only if, for each τ ≥ , we have h Y ( τ ) = h Y ( τ ) + h Y ( τ − d ) . Proof.
With these hypotheses we have Y := Y ∩ C and Y := Y \ C . Since I Y = I Y : F and I Y = ( I Y + F ) sat the result follows immediately from the following short exact sequence0 → RI Y : F ( − d ) · F −→ RI Y → RI Y + ( F ) → . (cid:3) Remark 2.2.
In Proposition 2.1, if Y is a reduced set of points then the condition about thevanishing order of F is automatically satisfied.In the next example we show different ways to add points, in a “non-saturated ideal case”, inorder to construct a saturated ideal. G. FAVACCHIO AND J. MIGLIORE
Example 2.3.
Denote by ℓ i the line of P defined by the linear form x + iy, ≤ i ≤
4. Let C be the curve given by the union of the lines ℓ , ℓ , ℓ , ℓ , and let F be a form defining C . Let Y be a set of five distinct points (no three on a line) such that only four of them belong to C . Sayfor instance P = (1 , − , P = (2 , − , P = (6 , − , P = (8 , − ,
1) and P = (1 , , .ℓ ℓ ℓ ℓ • P • P • P • P • P Figure 1.
Five points, four of them on a quartic.Using the notation in Proposition 2.1, we set Y := { P , P , P , P } and Y := { P } . So we have h Y h Y h Y F ) + I Y is not a saturated ideal. Now, by adding points to Y ,we show three different ways to make the resulting ideal ( F ) + I Y saturated. i ) We add to Y six points lying on C , consisting of two general points in ℓ and ℓ and onegeneral point in ℓ and ℓ . (Note that we have a similar situation occurs by adding atotal of 7, 8, 9 or 10 general points on ℓ , . . . , ℓ . ) ℓ ℓ ℓ ℓ • P • P • P • P • P ••• ••• So, we have h Y h Y h Y ii ) We add to Y the triple point defined by the ideal ( x, y ) . The vanishing order of F atthis point is 4 . ℓ ℓ ℓ ℓ • P • P • P • P • P • So, we get h Y h Y h Y CM PROPERTY FOR CODIMENSION TWO MULTIPROJECTIVE VARIETIES 5 iii ) Let ℓ be a line containing P , say for instance ℓ defined by the linear form y − z , andadd to Y the four points intersection of ℓ and C i.e. P = ( − , , , P = ( − , , ,P = ( − , , , P = ( − , , .ℓ ℓ ℓ ℓ • P • P • P • P • P •••• So, we have h Y h Y h Y Remark 2.4. • If I Y + F is saturated then F is a “separator” for Y . So, the ideal defining Y is obtained just adding F to I Y . • Let Y be a reduced complete intersection of two planar curves of degrees d ≥ d ≥
1. Assume that Y is a single point. Then a polynomial F as above cannot havedegree d + d −
3, because by the Cayley-Bacharach theorem, any F vanishing on Y must also vanish on Y , whereas we have seen that F has to be a separator. • Using liaison, this kind of remark can be strengthened. For example, if Y is a reducedcomplete intersection in linear general position and Y consists of three points then F cannot have degree d + d −
4, by a similar analysis.3.
Notation, terminology and examples
We work over a field of characteristic zero. For a product of two projective spaces we define V = P a × P a and π i : V → P a i to be the projection to the i -th component ( i = 1 , { e , e } be the standard basis of N . Let x i,j , with 1 ≤ i ≤ ≤ j ≤ a i for all i, j ,be the variables for P a and P a . Let R = k [ x , , . . . , x ,a , x , , . . . , x ,a n ] = k [ V ] , where the degree of x i,j is e i .A subscheme, X , of V is defined by a saturated ideal, I X , generated by a system of multihomo-geneous polynomials in R in the obvious way. We say that X is arithmetically Cohen-Macaulay(ACM) if R/I X is a Cohen-Macaulay ring.Let N = a + a + 2. Given a subscheme X of V together with its homogeneous ideal I X , wecan also consider the subscheme ¯ X of P N − defined by I X . Notice that if X is a zero-dimensionalsubscheme of V , then I X almost never defines a zero-dimensional subscheme of P N − .In the following, to shorten the notation, we write R := k [ s, t, x, y, z ] = k [ P × P ]. We willuse the letters “ A, A ′ , A i , . . . ” to denote the elements in R (1 , , and the letters “ B, B ′ , B j , . . . ”for elements in R (0 , . These bihomogeneous linear forms define hyperplanes in P × P . We calla line in P × P the intersection of two hyperplanes defined by a saturated ideal. Then, we onlyhave two different kind of lines, the horizontal lines V ( A, B ) and vertical lines V ( B, B ′ ). Notethat the ideal generated by two different forms A, A ′ is not saturated in P × P . (For instance, V ( s, t ) = ∅ .) Also note that two different vertical lines are disjoint in P × P . Whenever itis not necessary to make the type of line explicit, we just use the letter L. A set of lines X is a finite collection of lines. We will write either X = L + L + · · · + L n ⊆ P × P or X = { L , L , . . . , L n } ⊆ P × P . G. FAVACCHIO AND J. MIGLIORE
There is a natural partition on a set of lines X : we will write X = X ∪ X to denote thepartition of X into horizontal and vertical lines respectively. Example 3.1.
Let X be the set of lines in P × P X := V ( s, x ) ∪ V ( t, y ) ∪ V ( x, y ) . Then X consists of two horizontal lines, that are V ( s, x ) and V ( t, y ), and one vertical line V ( x, y ).In Figure 2 a representation of such set. V ( t, y ) V ( s, x ) V ( x, y ) Figure 2.
The above set X .Given a set of n lines X and positive integers m , . . . , m n , we let Z denote the subschemeof P × P defined by the saturated bihomogeneous ideal I Z := \ L i ∈ X ( I L i ) m i ⊆ R. We call Z a set of fat lines in P × P whose support is X = L + L + · · · + L n ⊆ P × P . Thescheme Z will be denoted by Z = m L + m L + · · · + m n L n . Remark 3.2.
The aim of this paper is to investigate which properties make a set of (fat) linesACM. In [7] and [9] the ACM property (or its failure) was studied for ACM sets of points.Although the ambient spaces in these two papers are different, and the descriptions of the ACMsets of points are certainly different, they have a property in common. Given a set of points ina multiprojective space, there is a complete intersection of points containing it. So its residualis again a set of points. An ACM set of lines in P × P is a codimension 2 scheme, henceviewed in P it is in the same liaison class as a complete intersection. One can hope that liaisontricks will continue to work here. One powerful trick is the result of Gaeta that says that if X ⊂ P N is ACM of codimension two then one can link in a finite number of steps, always usingcomplete intersections that are generated by minimal generators of the ideal, in such a way thatat each step the number of minimal generators drops by one, and the end result is a completeintersection. Conversely, if such a sequence of links exists then X is ACM. If X ⊂ P × P is viewed in P , this can still be done using homogeneous complete intersections. However, ifwe want to insist that our complete intersections are generated by bihomogeneous polynomials(so all varieties in the sequence of links lie in P × P ), it cannot necessarily be done. (See forinstance [17, 18].) This is illustrated in the following example. Example 3.3.
Let X = { V ( x, y ) , V ( s, x ) , V ( s, y ) , V ( t, x ) , V ( t, y ) , V ( t, z ) , V ( t, x + y + z ) , V ( s + t, y ) } . CM PROPERTY FOR CODIMENSION TWO MULTIPROJECTIVE VARIETIES 7 V ( s + t ) V ( t ) V ( s ) V ( x, y ) Figure 3.
A representation of the above set X .Then one can check using Macaulay2 or CoCoA that X is ACM, with four minimal generators: txy, sty, x yz + xy z + xyz , and s tx + st x. These have bidegree, respectively, (1,2), (2,1), (0,4) and (3,1). In P it is clear that one canlink X using a complete intersection consisting of homogeneous polynomials of degrees 3 and 4that are part of a minimal generating set. However, if we want to link using bihomogeneouspolynomials that are minimal generators, we claim that no such link exists. Indeed, note firstthat the given generators pairwise have common factors. A bihomogeneous minimal generatorof degree 3 has to be one of the two given generators, since any linear combination is no longerbihomogeneous. To get a bihomogeneous minimal generator of degree 4, the only possibility isa polynomial L · sty + ( s tx + st x ) for some linear form L of type (1,0). Note that st wouldbe a factor of any such bihomogeneous form. Thus any such bihomogeneous polynomial has afactor in common with both txy and sty , so no such link is possible.One could envision an approach wherein we begin with a set of lines in P × P and wesomehow link in P without regard to having the residual be viewable as a subvariety of P × P ,looking only to whether or not we can arrive at a complete intersection (hence X is ACM). Wehave not seen how to make such an approach work.4. ACM sets of lines in P × P : the starting case In this section we focus on the ACM property for a set of fat lines Z in P × P satisfyingextra conditions. From now on we will work under the following hypothesis. Notation 4.1.
Let Z = Z ∪ Z be the partition of a set Z of fat lines in P × P into horizontaland vertical lines respectively. Throughout this section we will assume that(a) Z is a non-empty set of horizontal reduced lines;(b) two different horizontal lines of Z are not contained in a hyperplane defined by a formof degree (0 , V ( A, B ) , V ( A ′ , B ) ∈ Z , then A = A ′ . Remark 4.2.
Note that a set of minimal generators of I Z is only in the variables x, y, z so I Z is the cone ideal of a set of (fat) points Y in P . G. FAVACCHIO AND J. MIGLIORE
Remark 4.3.
Let Z be as in Notation 4.1. If | π ( Z ) | = 1 then all the horizontal lines of Z arecontained in the same plane, say V ( A ) . Hence Z is a complete intersection of codimension 2.Indeed, say Z := V ( A, B ) ∪ · · · ∪ V ( A, B N ), we note that the ideal defining Z is I Z = ( A, F ) , where F := B B · · · B N . Since deg A = (1 ,
0) and deg F = (0 , N ) the ideal ( A, F ) is generatedby a regular sequence.The next result make evident what connection there is between the saturation problem studiedin Section 2 and the ACM property of Z . Theorem 4.4.
Let Z be as in Notation 4.1 and assume | π ( Z ) | = 1 . Let F be of the form inRemark 4.3. Then Z is ACM if and only if I Z + ( F ) is saturated in k [ x, y, z ] . Proof.
Notice that if I Z + ( F ) is artinian then it is not saturated, so the last condition includesthe statement that I Z + ( F ) has height 2 in k [ x, y, z ].We look at Z , Z and Z as unions of planes in P . Consider the short exact sequence(4.1) 0 → I Z → I Z ⊕ I Z → I Z + I Z → . Since | π ( Z ) | = 1, we have I Z = ( A, F ), a complete intersection (see Remark 4.3). In particular,note that Z and Z are ACM unions of planes in P (see Remark 4.2).Sheafifyng the exact sequence (4.1) and taking cohomology, we obtain the following exactdiagrams:(4.2) 0 → I Z → I Z ⊕ I Z −→ [ I Z + I Z ] sat → H ∗ ( I Z ) → ց ր I Z + I Z ր ց → H ∗ ( I Z + I Z ) → H ∗ ( I Z ) → . Recall that F ∈ k [ x, y, z ]. Recall also that Z is ACM if and only if H ∗ ( I Z ) = H ∗ ( I Z ) = 0. Notethat the form A is a regular element in RI Z + ( F ) and recall that I Z + I Z = ( A, F ) + I Z . Suppose first that F does not vanish on any component of Z . Then F + I Z is an unmixed (inparticular saturated) height 3 ideal in R , and ( A, F ) + I Z = I Z + I Z is a saturated ideal ofheight 4 in R . From (4.3) we know that H ∗ ( I Z ) = 0 if and only if H ∗ ( I Z + I Z ) = 0 . Since in the current situation I Z + I Z defines a zero-dimensional scheme, its first cohomologydoes not vanish. Thus Z has no hope of being ACM.What we have just shown is that if H ∗ ( I Z ) = 0 then I Z + ( F ) has height 2 (either in R or in k [ x, y, z ]). We claim that the converse is also true. Indeed, if I Z +( F ) has height 2 in k [ x, y, z ], itssaturation defines a zero-dimensional scheme in P , which is automatically ACM. Thus viewedin R , this saturation defines an ACM surface in P , so the (not necessarily saturated) ideal I Z + I Z defines the hyperplane section, which is also ACM. But the first cohomology of theideal sheaf does not depend on whether the original ideal was saturated or not, so it vanishesand hence H ∗ ( I Z ) = 0 by (4.3).Thus we can assume without loss of generality that I Z + ( F ) has height 2 (either in R orin k [ x, y, z ]) and that H ∗ ( I Z ) = 0, and we focus on H ∗ ( I Z ). From (4.2) we obtain CM PROPERTY FOR CODIMENSION TWO MULTIPROJECTIVE VARIETIES 9 H ∗ ( I Z ) = 0 if and only if I Z + I Z is saturated in R if and only if I Z + ( F ) is saturated in k [ x, y, z ].(As before, an ideal in k [ x, y, z ] of height 2 is Cohen-Macaulay if and only if it is saturated ifand only if it is unmixed.) (cid:3) Remark 4.5.
Theorem 4.4 shows that the ACM property of a set of (fat) lines Z = Z ∪ Z where | π ( Z ) | = 1 only depends on the saturation of the ideal ( F ) + I Z ⊆ k [ P ]. In P , theset Z can be viewed as a set of points and the form F defines a curve, C , that is a union oflines. So, if the hypotheses of Proposition 2.1 are satisfied, the ACM property only depends onthe h -vectors of I Z , I Y := ( F ) + I Z and I Y := I Z : I C . More precisely, Z is ACM if and onlyif h Z ( τ ) = h Y ( τ ) + h Y ( τ − d ) where d is the degree of F ∈ K [ P ] . We introduce the following definition.
Definition 4.6.
Let Z be a set of lines in P × P as in Notation 4.1. We defineˆ Z := { V ( s, B ) | V ( A, B ) ∈ Z for some A } ∪ Z . Note that in Definition 4.6 the set ˆ Z satisfies the hypotheses of Theorem 4.4; indeed wehave | π ( ˆ Z ) | = 1 . Proposition 4.7.
Let Z be as in Notation 4.1. If Z is ACM then ˆ Z is ACM, and Z and ˆ Z share the same multigraded homological invariants.Proof. Without loss of generality we can assume that the linear form t is a regular element in R/I Z . If we look at Z and ˆ Z as unions of planes in P , the hyperplane defined by t does notcontain any component either of Z or of ˆ Z , and so meets each such component in a line.Let us examine the effect on both kinds of components of Z . A “horizontal” component isdefined by an ideal of the form ( ℓ i , m i ) with ℓ i ∈ k [ s, t ] and m i ∈ k [ x, y, z ], so the hyperplanesection by t is defined by ( ℓ i , m i , t ) = ( s, t, m i ). A “vertical” component is defined by anideal of the form ( L j , M j ), with L j , M j ∈ k [ x, y, z ], so the hyperplane section by t is definedby ( L j , M j , t ).Since Z is ACM (viewed in P ), we have that I Z + ( t ) is the saturated ideal of the hyperplanesection (a union of lines), which is ACM. The entire hyperplane section by t then has thesaturated ideal I Z + ( t ) = \ i ( s, m i , t ) ∩ \ j ( L j , M j , t ) , since Z is ACM, and defines an ACM curve in P .On the other hand, up to saturation the latter is also the hyperplane section by t of ˆ Z . Sincethe curve is ACM, the union of planes ˆ Z must also be ACM by [16]. Thus I ˆ Z + ( t ) = \ i ( s, m i ) ∩ \ j ( L j , M j ) + ( t ) . Finally, since Z and ˆ Z are both ACM with the same hyperplane section, they must in fact havethe same homological invariants.Up to this point we have only shown that Z and ˆ Z have the same graded homological invari-ants, by viewing Z and ˆ Z as subschemes of P . But in fact they began in P × P , so I Z and I ˆ Z have multigraded Betti numbers and in particular multigraded minimal generators. When wereduce by the non-zerodivisor t , this preserves the multigrading. Hence we have the result. (cid:3) The following is an immediate consequence of Proposition 4.7. It gives us a necessary conditionfor the Cohen-Macaulayness of Z. Corollary 4.8.
Let Z be a set of lines in P × P as in Notation 4.1. If Z is ACM, then β , ( a,b ) ( I Z ) = 0 for each a > . Proof.
Indeed this is true for ˆ Z . (cid:3) Remark 4.9.
Corollary 4.8 seems very surprising, at first glance. But it assumes from thebeginning that Z is ACM, and then that it satisfies condition (b) of Notation 4.1. Neither ofthese is particularly restrictive by itself, but the point is that the combination is restrictive.Still, having both conditions does not by any means imply that Z = ˆ Z . For instance, considerExample 3.1. One might think at first that there is a minimal generator of type (2 ,
1) (e.g. stx ),but in fact one can check that the ideal has two minimal generators of bidegree (1 ,
1) and oneof bidegree (0 , Z that is ACM and satisfies Notation 4.1, it is easy to usebasic double linkage to produce a new Z ′ that is ACM and does not satisfy β , ( a,b ) ( I Z ) = 0for each a >
1, but we lose the property given in Notation 4.1 (b). For instance, returning toExample 3.1, one could form ( s + t ) · I Z + ( xy ), which is ACM and has a minimal generator ofbidegree (2,1); but now it does not satisfy Notation 4.1.We do not know if the converse of Corollary 4.8 is true. So we pose the following question: Let Z be as in Notation 4.1. Assume ˆ Z is ACM and β , ( a,b ) ( I Z ) = 0 for each a > . Is Z ACM?
However, the next example shows that, without the condition on β , ( a,b ) ( I Z ), ˆ Z ACM does notimply Z ACM.
Example 4.10.
Let Z the set of lines in P × P defined by the ideal I Z := ( t, x ) ∩ ( s, y ). Theset Z is clearly not ACM. However, ˆ Z , defined by I ˆ Z := ( s, x ) ∩ ( s, y ), is ACM. V ( s, y ) V ( t, x ) (a) The set Z . V ( s, y ) V ( s, x ) (b) The set ˆ Z . Figure 4.
Example 4.10.From Proposition 4.7 we know that the ACM property of ˆ Z is a necessary condition for theACM property of Z . Example 4.10 shows that it is not sufficient. The following lemma will giveus another necessary condition. Lemma 4.11.
Let Z be an ACM set of lines as in Notation 4.1. Let V ( A, B ) , V ( A ′ , B ′ ) ∈ Z betwo horizontal lines and A = A ′ . Then V ( B, B ′ ) ⊆ Z. Proof.
Look at Z as a set of planes in P . If V ( B, B ′ ) Z then Z is not locally Cohen-Macaulayat the point defined by the ideal p = ( A, A ′ , B, B ′ ), hence Z is not ACM. Contradiction. (cid:3) CM PROPERTY FOR CODIMENSION TWO MULTIPROJECTIVE VARIETIES 11
Then it is natural to give the next definition.
Definition 4.12.
Let Z be as in Notation 4.1. We say that Z is v-connected if, for each V ( A, B ) , V ( A ′ , B ′ ) ∈ Z where A = A ′ and B = B ′ , we have V ( B, B ′ ) ∈ Z .In the next example we show that this property is still not enough to ensure the ACM property. Example 4.13.
Let X be the set of lines in P × P defined by the ideal I X := ( s, x ) ∩ ( s, y ) ∩ ( t, x + y ) ∩ ( x, y ) . Note that X is v-connected. Moreover, from Theorem 4.4, the set ˆ X defined by I ˆ X := ( s, x ) ∩ ( s, y ) ∩ ( s, x + y ) ∩ ( x, y )is ACM. V ( t ) V ( s ) V ( x, y ) (a) The set X . V ( s ) V ( x, y ) (b) The set ˆ X . Figure 5.
Example 4.13.According to CoCoA, X is not ACM. A computer experiment shows that the set of fat lines, Z , whose ideal is I Z := ( s, x ) ∩ ( s, y ) ∩ ( t, x + y ) ∩ ( x, y ) is ACM. Thus that Z is a (non-reduced) ACM set of lines whose support X is not ACM. Thenext result explores this idea further. Theorem 4.14.
Let Z = V ( A , B ) + · · · + V ( A n , B n ) + m · V ( B , B ) be a set of lines of P × P such that(1) V ( A n − ) = V ( A n ) (i.e. not all the horizontal lines are in the same plane);(2) V ( B i ) = V ( B j ) for i = j , (i.e. B , · · · , B n define different planes);(3) V ( B , . . . , B n ) = V ( B , B ) , (i.e. B , . . . , B n define planes in the pencil containing V ( B , B ) ).Then Z is ACM if and only if m ≥ n − .Proof. First assume n = 2, i.e., Z = V ( A , B ) + V ( A , B ) + m · V ( B , B ).( ⇒ ) By Lemma 4.11, Z ACM implies m ≥ ⇐ ) On the other hand, take m ≥ → I Z → ( A , B ) ⊕ [( A , B ) ∩ ( B , B ) m ] → [( A , B ) + ( A , B ) ∩ ( B , B ) m ] → , and(4.5) 0 → ( A , B ) ∩ ( B , B ) m → ( A , B ) ⊕ ( B , B ) m → ( A , B ) + ( B , B ) m → . Look at (4.5) and note that ( A , B ) + ( B , B ) m = ( A , B , B m ). Thus, the ideal( A , B ) + ( B , B ) m defines an ACM scheme of height 3. Since the schemes defined by( A , B ) and ( B , B ) m are ACM of height 2, then a mapping cone argument gives thatthe ideal ( A , B ) ∩ ( B , B ) m has projective dimension 2. Now consider (4.4), and notethat ( A , B ) + [( A , B ) ∩ ( B , B ) m ] = ( A , B , B m ) , so it defines an ACM scheme of height 3. Moreover, the module in the middle of thesequence has projective dimension 2, hence Z is ACM.Now we proceed by induction, the base case having just been proven. Assume that the statementis true for n − ≥ n ≥
3. That is, assume that Z has n − Z is ACM if and only if m ≥ n −
2. Set Z ′ := V ( A , B ) + · · · + V ( A n , B n ) + m V ( B , B )and consider the following short exact sequence:(4.6) 0 → I Z → ( A , B ) ⊕ I Z ′ → ( A , B ) + I Z ′ → . ( ⇐ ) If m ≥ n − Z ′ is ACM. Thus, in order to prove that Z is ACM, by the sequence (4.6), it is enough to show that ( A , B ) + I Z ′ defines anACM variety of height 3. We show that ( A , B ) + I Z ′ = ( A , B , B m ) . • ( A , B ) + I Z ′ ⊆ ( A , B , B m ). Indeed, if F ∈ I Z ′ then F ∈ ( B , B ) m ⊆ ( B , B m ); • ( A , B ) + I Z ′ ⊇ ( A , B , B m ). We first claim that the form F = B m − n +22 · B · · · B n belongs to I Z ′ . By hypotheses 2) and 3), for j > B j = λ j B + µ j B ,where λ j , µ j = 0. Thus, F = F ′ B + aB m ∈ I Z ′ and therefore B m ∈ ( A , B ) + I Z ′ . ( ⇒ ) Assume that Z is ACM. View Z as a set of planes in P . By the sequence (4.6), theideal ( A , B ) + I Z ′ is saturated in k [ P ]. Without loss of generality we can assume thelinear form z to be a regular element for R/I Z , so that the intersection of Z with V ( z )is a proper hyperplane section of Z . Denoting by J the ideal defining Z ∩ V ( z ) in P , wehave that J ∼ = I Z + ( z ) / ( z ) and it is Cohen-Macaulay in k [ P ] ∼ = R/ ( z ). Let us denoteby Y the scheme defined by J in P . Hence Y is an ACM set of lines (one non-reduced)in P ; precisely J := ( A , ¯ B ) ∩ · · · ∩ ( A n , ¯ B n ) ∩ ( ¯ B , ¯ B ) m where V ( ¯ B j ) = V ( B j ) ∩ V ( z ). (Recall that A i ∈ k [ s, t ] while B j ∈ k [ x, y, z ], so abusingnotation we will view A i ∈ k [ P ].) We set J ′ := ( A , ¯ B ) ∩ · · · ∩ ( A n , ¯ B n ) ∩ ( ¯ B , ¯ B ) m and denote by Y ′ the scheme defined by J ′ and by λ the line defined by ( A , ¯ B ).Consider the short exact sequence(4.7) 0 → J → ( A , ¯ B ) ⊕ J ′ → ( A , ¯ B ) + J ′ → . Since J defines an ACM curve Y , sheafifying and taking the long exact sequence incohomology gives us that the ideal ( A , ¯ B ) + J ′ is saturated in k [ P ]; in particular ithas height 3, and(4.8) ( A , ¯ B ) + J ′ = ( A , ¯ B , ¯ B u ) . What is u ? Geometrically it is the intersection multiplicity of the line λ and the scheme Y ′ defined by J ′ . These two schemes meet only at the point P defined by ( A , B , B ).Since λ must meet the multiplicity m line and possibly some reduced components of Y ′ at P , in any case we have u ≥ m . CM PROPERTY FOR CODIMENSION TWO MULTIPROJECTIVE VARIETIES 13
But the fact that ( A , ¯ B ) + J ′ is saturated and that (4.8) holds gives us more. Theexponent u is determined by some form H ∈ J ′ that is not zero modulo ( A , ¯ B ). (Infact, it is the form of least degree with this property.) We first claim that without lossof generality we can take H to be of bidegree (0 , u ) (i.e. a product of ¯ B i ). By (4.8), H ∈ ( A , B , ¯ B u ). In particular, we have that H and ¯ B u are equal modulo ( A , ¯ B ), so¯ B u = A F + ¯ B G + H and we can assume without loss of generality that G does not have A as a factor andthat H does not have any terms involving A or ¯ B . Furthermore, since ( A , ¯ B ) + J ′ is bi-homogeneous, we can assume F, G, H to be bi-homogeneous. Note H = 0 since¯ B u / ∈ ( A , ¯ B ). We get that F = 0 since none of the other three polynomials involve A .Then without loss of generality, deg( H ) = deg( ¯ B u ) = (0 , u ).So we have that the value of u is the minimum b > , b ) is the degree ofan element F ∈ J ′ such that F / ∈ ( ¯ B ). Thus by hypothesis (2), any element of degree(0 , b ) in J ′ has ¯ B · · · ¯ B n as a factor. We conclude u ≥ n − . Now we assume by contradiction that m < n − . Consider the two forms F = ¯ B · · · ¯ B n − ¯ B n − A n and F = ¯ B · · · ¯ B n − A n − ¯ B n . Both F and F have degree (1 , n −
2) and both belong to J ′ , and so, F , F ∈ ( A , ¯ B ) + J ′ = ( A , ¯ B , ¯ B u ) . By hypothesis (2), F , F / ∈ ( B ) . By hypothesis (1), at least one among F and F doesnot belong to ( A ), say F . This means that F ∈ ( ¯ B , ¯ B u ) and hence ¯ B · · · ¯ B n − ∈ ( ¯ B , ¯ B u ), so u ≤ n − m ≥ n − (cid:3) As a corollary, Theorem 4.14 can be translated to a result about lines in P . Corollary 4.15.
Let λ and λ be two skew lines in P . Let α , . . . α n and β , . . . β n be planescontaining λ and λ respectively, such that • β i = β j for any i = j ; • α n − = α n . Set ℓ i := α i ∩ β i . Then, the set of lines Y := ℓ + · · · + ℓ n + m · λ is ACM if and only if m ≥ n − .Proof. We have k [ P ] = k [ s, t, x, y, z ] as usual. In Theorem 4.14, viewing the configuration as aunion of planes in P (one not reduced), the A i are hyperplanes vanishing on the 2-plane definedby s = t = 0 and (without loss of generality, after possibly a change of variables) the B i can betaken to be hyperplanes vanishing on the 2-plane defined by x = y = 0. In our current setting,without loss of generality (after possibly a change of variables) we can take k [ P ] = k [ s, t, x, y ],and set λ to be the line defined by s = t = 0 and set λ to be the line defined by x = y = 0.Note that z is a non-zerodivisor for R/I Z in Theorem 4.14. Then Y is the hyperplane sectionof Z cut out by the hyperplane defined by z . By [16, Proposition 2.1], Y is ACM if and only if Z is ACM. The result follows immediately. (cid:3) Theorem 4.14 leads us to introduce a new definition.
Definition 4.16.
Let Z be as in Notation 4.1. Assume that whenever V ( A, B ) , V ( A ′ , B ′ ) ∈ Z with A = A ′ and B = B ′ , the vertical line V ( B, B ′ ) meets n horizontal lines of Z (wherethe integer n ≥ B and B ′ ). We say that Z is fully v-connected if( n − V ( B, B ′ ) ⊆ Z , i.e., for each such B, B ′ the line V ( B, B ′ ) is contained in Z with multiplicityat least n − P × P is locally a completeintersection, and hence locally Cohen-Macaulay. In the next theorem we characterize the localCohen-Macaulay property of Z in terms of the fully v-connected property. Theorem 4.17.
Let Z = Z ∪ Z be a set of lines of P × P such that the assumptions ofNotation 4.1 hold, i.e.(a) Z is a non-empty set of reduced horizontal lines;(b) two different horizontal lines of Z are not contained in a hyperplane defined by a formof degree (0 , , i.e. if ( A, B ) , ( A ′ , B ) ∈ Z then A = A ′ . Then, Z is locally Cohen-Macaulay if and only if Z is fully v-connected.Proof. First assume that Z is locally Cohen-Macaulay. Let ( B, B ′ ) m define a vertical fat line mL in Z . It is not restrictive to assume that ( B, B ′ ) = ( x, y ), and also that z is a regular elementin R/I Z . Let V ( A , B ) , . . . , V ( A n , B n ) ∈ Z be the horizontal lines meeting L . Let P ∈ P be the point defined by I P = ( s, t, x, y ). In particular, ( x, y ) = ( B , . . . , B n ) , where B , . . . , B n define different hyperplanes. Let W = V ( A , B ) + · · · + V ( A n , B n ) + mL be the correspondingsubscheme of Z . Note that W satisfies the hypotheses of Theorem 4.14, so W is ACM if andonly if m ≥ n − W (viewed in P ) contain the point P , and no other componentof Z does. Thus since the components of W are linear, W is the cone in P with vertex P over a curve, C , in the hyperplane P defined by z = 0, and C is precisely of the form given inCorollary 4.15. Now, since Z is locally Cohen-Macaulay, in particular at P , we conclude that C is ACM, and hence by Corollary 4.15 we have that m ≥ n −
1. Since this holds for each verticalline (noting that m and n will change for different vertical lines), Z is fully v-connected.Conversely, assume that Z is fully v-connected. Certainly at any smooth point of Z (viewedin P ), Z is locally Cohen-Macaulay. Similarly, at the intersection of two or more horizontallines that do not have a vertical line through the point of intersection, locally it is a completeintersection, hence locally Cohen-Macaulay.Although the vertical (possibly fat) lines are disjoint in P × P , the corresponding (fat) planesdo meet in P . The intersection of any finite number of such planes is one-dimensional. Still,this is a cone over a set of (fat) points in P , so it is ACM and hence is locally Cohen-Macaulayalong this locus.Consider the intersection in P × P of a (fat) vertical line of multiplicity m and a collectionof n horizontal lines. Without loss of generality assume that the vertical line has supportgiven by ( x, y ). Let J := ( s, t, x, y ). Note that the ideal J defines a point P in P . Let W = V ( A , B )+ · · · + V ( A n , B n )+ mL be the subscheme of Z corresponding to the components,viewed in P , containing P . Note that W satisfies the hypotheses of Theorem 4.14, so W isACM if and only if m ≥ n − W are linear, W is the cone in P with vertex P over a curve, C ,in the hyperplane P defined by z = 0 (which is a non-zerodivisor), and C is precisely of theform given in Corollary 4.15. Since Z is fully v-connected, we know that m ≥ n −
1. Thus, byCorollary 4.15, C is ACM. Hence Z is locally Cohen-Macaulay at P . (cid:3) CM PROPERTY FOR CODIMENSION TWO MULTIPROJECTIVE VARIETIES 15
Remark 4.18.
Let Z = Z ∪ Z be a set of lines of P × P such that the assumptions ofNotation 4.1 hold. We wonder which hypotheses ensure Z to be ACM. Note that the fullyv-connected condition is not enough. Take for instance Z = V ( s, x ) ∪ V ( y, z ); it trivially satisfiesDefinition 4.16 and it is not ACM. So, as shown in Proposition 4.7, we at least need to ask ˆ Z tobe ACM. It would be interesting to show if these two conditions, Z fully v-connected togetherwith ˆ Z ACM, are sufficient to guarantee the ACM property for Z. References [1] J. Abbott, A. M. Bigatti and L. Robbiano.
CoCoA: a system for doing Computations in Commutative Algebra.
Available at http://cocoa.dima.unige.it .[2] S. Cooper, G. Fatabbi, E. Guardo, A. Lorenzini, J. Migliore, U. Nagel, A. Seceleanu, J. Szpond and A. VanTuyl.
Symbolic powers of codimension two Cohen-Macaulay ideals , Communications in Algebra, (2020).[3] A. Almousa, G. Fløystad and H. Lohne.
Polarizations of powers of graded maximal ideals. arXiv preprintarXiv:1912.03898, (2019).[4] C. Berkesch, D. Erman and G. G. Smith.
Virtual resolutions for a product of projective spaces . AlgebraicGeometry 7 (4), 460481, (2020).[5] L. Chiantini and D. Sacchi.
Segre functions in multiprojective spaces and tensor analysis.
From Classical toModern Algebraic Geometry. Birkhuser, Cham, 361-374, (2016).[6] D. Eisenbud and A. Sammartano.
Correspondence scrolls.
Acta Mathematica Vietnamica, 44(1):101–116,(2019).[7] G. Favacchio, E. Guardo and J. Migliore.
On the arithmetically cohen-macaulay property for sets of points inmultiprojective spaces.
Proceedings of the American Mathematical Society, 146(7):2811–2825, (2018).[8] G. Favacchio, E. Guardo and B. Picone.
Special arrangements of lines: Codimension 2 ACM varieties in P × P × P . Journal of Algebra and Its Applications, 18(04):1950073, (2019).[9] G. Favacchio and J. Migliore.
Multiprojective spaces and the arithmetically cohen–macaulay property.
Mathe-matical Proceedings of the Cambridge Philosophical Society 166(3):583–597, (2019).[10] J. Gao, Y. Li, M. C Loper and Amal Mattoo.
Virtual complete intersections in P × P . Journal of Pure andApplied Algebra. 225(1), (2021).[11] D. Grayson and M. Stillman.
Macaulay 2–a system for computation in algebraic geometry and commutativealgebra , 1997.[12] E. Guardo, B. Harbourne and A. Van Tuyl.
Fat lines in P : powers versus symbolic powers. Journal ofAlgebra, 390, 221-230, (2013).[13] E. Guardo, B. Harbourne and A. Van Tuyl.
Symbolic Powers Versus Regular Powers of Ideals of GeneralPoints in P × P . Canadian Journal of Mathematics, 65(4), 823-842, (2013).[14] E. Guardo and A. Van Tuyl. Arithmetically Cohen-Macaulay Sets of Points in P × P Springer, 2015.[15] E. Guardo and A. Van Tuyl.
ACM sets of points in multiprojective space.
Collectanea mathematica. 59(2),191–213, (2008)[16] C. Huneke and B. Ulrich.
General hyperplane sections of algebraic varieties.
J. Algebraic Geom. (1993),no. 3, 487–505.[17] J. Migliore. Introduction to liaison theory and deficiency modules , volume 165. Birkh¨auser, Progress in Math-ematics, 1998.[18] J. Migliore and U. Nagel.
Minimal Links and a Result of Gaeta . In: “Liaison, Schottky Problem and InvariantTheory: Remembering Federico Gaeta,” Progress in Mathematics, Vol. 280 (2010), 103–132, Birkh¨auser.[19] A. Nematbakhsh.
Linear strands of edge ideals of multipartite uniform clutters. preprintarXiv:1805.11432, (2018).
Dipartimento di Matematica e Informatica, Universit`a degli studi di Catania, Italy
E-mail address : [email protected] Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556
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