aa r X i v : . [ m a t h . A C ] O c t THE SYMBOLIC DEFECT OF AN IDEAL
FEDERICO GALETTO, ANTHONY V. GERAMITA † , YONG-SU SHIN ∗ , AND ADAM VAN TUYL ∗∗ Abstract.
Let I be a homogeneous ideal of k [ x , . . . , x n ]. To compare I ( m ) , the m -thsymbolic power of I , with I m , the regular m -th power, we introduce the m -th symbolicdefect of I , denoted sdefect( I, m ). Precisely, sdefect(
I, m ) is the minimal number of gen-erators of the R -module I ( m ) /I m , or equivalently, the minimal number of generators onemust add to I m to make I ( m ) . In this paper, we take the first step towards understandingthe symbolic defect by considering the case that I is either the defining ideal of a starconfiguration or the ideal associated to a finite set of points in P . We are specificallyinterested in identifying ideals I with sdefect( I,
2) = 1. Introduction
Let I be a homogeneous ideal of R = k [ x , . . . , x n ]. For any positive integer m , let I ( m ) denote the m -th symbolic power of I . In general, we have I m ⊆ I ( m ) , but equality mayfail. During the last decade, there has been interest in the so-called “ideal containmentproblem,” that is, for a fixed integer m , find the smallest integer r such that I ( r ) ⊆ I m .The papers [7, 8, 13, 16, 17, 26, 31, 40] are a small subset of the articles on this problem.In this note, we are also interested in comparing regular and symbolic powers of ideals,but we wish to investigate a relatively unexplored direction by measuring the “difference”between the two ideals I m and I ( m ) . More precisely, because I m ⊆ I ( m ) , the quotient I ( m ) /I m is a finitely generated graded R -module. For any R -module M , let µ ( M ) denotethe number of minimal generators of M . We then define the m -th symbolic defect of I tobe the invariant sdefect( I, m ) := µ ( I ( m ) /I m ) , that is, the minimal number of generators of I ( m ) /I m . We will call the sequence { sdefect( I, m ) } m ∈ N the symbolic defect sequence . Note that sdefect( I, m ) counts the number of generators weneed to add to I m to make I ( m ) ; this invariant can be viewed as a measure of the failureof I m to equal I ( m ) . For example, sdefect( I, m ) = 0 if and only if I m = I ( m ) .We know of only a few papers that have studied the module I ( m ) /I m . This list includes:Arsie and Vatne’s paper [3] which considers the Hilbert function of I ( m ) /I m ; Huneke’swork [32] which considers P (2) /P when P is a height two prime ideal in a local ring of Mathematics Subject Classification.
Key words and phrases. symbolic powers, regular powers, points, star configurations. † Deceased, June 22, 2016. ∗ This research was supported by a grant from Sungshin Women’s University. Corresponding author. ∗∗ This research was supported in part by NSERC Discovery Grant 2014-03898. dimension three; Herzog’s paper [28] which studies the same family of ideals as Hunekeusing tools from homological algebra; Herzog and Ulrich’s paper [29] and Vasconcelos’spaper [41] which also consider a similar situation to Huneke, but with the assumptionthat P is generated by three elements; and Schenzel’s work [38] which describes somefamilies of prime ideals P of monomial curves with the property that P (2) /P is cyclic(see the comment after [38, Theorem 2]).The introduction of the symbolic defect sequence raises a number of interesting ques-tions. For example, how large can sdefect( I, m ) be? how does sdefect(
I, m ) compare tosdefect(
I, m + 1)? and so on. In some sense, these questions are difficult since one needsto know both I ( m ) and I m . To gain some initial insight into the behavior of the symbolicdefect sequence, in this paper we focus on two cases: (1) I is the defining ideal of a starconfiguration, and (2) I is the homogeneous ideal associated to a set of points in P . Inboth cases, we can tap into the larger body of knowledge about these ideals.To provide some additional focus to our paper, we consider the following question: Question 1.1.
What homogeneous ideals I of k [ x , . . . , x n ] have sdefect( I,
2) = 1 ? Because one always has sdefect( I,
1) = 0, Question 1.1 is in some sense the first non-trivialcase to consider. Note that when sdefect( I,
2) = 1, then from an algebraic point of view,the ideal I is almost equal to I (2) except that it is missing exactly one generator.We now give an outline of the results of this paper. In Section 2, we provide the relevantbackground, and recall some useful tools about powers of ideals and their symbolic powers.In Sections 3 through 5, we study sdefect( I, m ) when I defines a star configuration.Note that in this paper, when we refer to star configurations, the forms that define thestar configurations are forms of any degree, not necessarily linear, which is required inother papers. Our main strategy to compute sdefect( I, m ) is to find an ideal J such that I ( m ) = J + I m , and then to show that all the minimal generators of J are required. Therecent techniques using matroid ideals developed by Geramita, Harbourne, Migliore, andNagel [21] will play a key role in our proofs. Our results will imply a similar decompositionfound by Lampa-Baczy´nska and Malara [35] which considers only star configurationsdefined using monomial ideals.In Section 3 we also compute some values of sdefect( I, m ) with m > I,
2) = 1 can force a geometric condition. Specifically, weshow that if X is a set of points in P with a linear graded resolution, and if sdefect( I,
2) =1, then I must be the ideal of a linear star configuration of points in P . In Section 5we apply our results of Section 3 to compute the graded minimal free resolution of I (2) when I defines a star configuration of codimension two in P n . This result gives a partialgeneralization of a result of Geramita, Harbourne, and Migliore [20] (see Remark 5.4).In Section 6, we turn our attention to general sets of points in P . Our main result is aclassification of the general sets of points whose defining ideals I X satisfy sdefect( I X ,
2) = 1.
Theorem (Theorem 6.3) . Let X be a set of s general points in P with defining ideal I X .Then ( i ) sdefect( I X ,
2) = 0 if and only if s = 1 , or . HE SYMBOLIC DEFECT OF AN IDEAL 3 ( ii ) sdefect( I X ,
2) = 1 if and only if s = 3 , , , or . ( iii ) sdefect( I X , > if and only if s = 6 or s > . Our proof relies on a deep result of Alexander-Hirschowitz [2] on the Hilbert functionsof general double points, and some results of Catalisano [11], Harbourne [25], and Id`a[34] on the graded minimal free resolutions of double points. We end this paper with anexample to show that the symbolic defect sequence is not monotonic by computing somevalues of sdefect( I X , m ) when X is eight general points in P (see Example 6.5). Acknowledgments.
Work on this project began in August 2015 when Y.S. Shin andA. Van Tuyl visited A.V. (Tony) Geramita at his house in Kingston, ON. F. Galettojoined this project in late September of the same year. Tony Geramita, however, becamequite ill in late December 2015 while in Vancouver, BC, and after a six month battlewith his illness, he passed away on June 22, 2016 in Kingston. During his illness, we (theremaining co-authors) kept Tony up-to-date of the status on the project, and when hishealth permitted, he would contribute ideas to this paper. He was looking forward toreturning to Kingston, and turning his attention to this paper. Unfortunately, this wasnot to be. Although Tony was not able to see the final version of this paper, we feel thathis contributions warrant an authorship. Those familiar with Tony’s work will hopefullyrecognize Tony’s interests and contributions to the topics in this paper. Tony is greatlymissed.We would also like to thank Brian Harbourne and Alexandra Seceleanu for their helpfulcomments. Part of this paper was written at the Fields Institute; the authors thank theinstitute for its hospitality. Finally, we would like to thank the referees for their helpfulsuggestions and corrections. 2.
Background results
We review the required background. We continue to use the notation of the introduc-tion. Let I be a homogeneous ideal of R = k [ x , . . . , x n ]. The m -th symbolic power of I ,denoted I ( m ) , is defined to be I ( m ) = \ P ∈ Ass( I ) ( I m R P ∩ R )where Ass( I ) denotes the set of associated primes of I and R P is the ring R localized atthe prime ideal P . Remark 2.1.
There is some ambiguity in the literature concerning the notion of symbolicpowers. The intersection in the definition is sometimes taken over all associated primesand sometimes just over the minimal primes of I . In general, these two possible definitionsyield different results. However, they agree in the case of radical ideals.In general, I m ⊆ I ( m ) , but the reverse containment may fail. If sdefect( I, m ) = s , thenthere exist s homogeneous forms F , . . . , F s of R such that I ( m ) /I m = h F + I m , . . . , F s + I m i ⊆ R/I m . HE SYMBOLIC DEFECT OF AN IDEAL 4
It follows that I ( m ) = h F , . . . , F s i + I m . Note that the ideal h F , . . . , F s i is not unique.Indeed, if G , . . . , G s is another set of coset representatives such that I ( m ) /I m = h G + I m , . . . , G s + I m i , we still have I ( m ) = h G , . . . , G s i + I m , but h F , . . . , F s i and h G , . . . , G s i may be different ideals.We state some simple facts about sdefect( I, m ). Lemma 2.2.
Let I be a homogeneous radical ideal of R .(i) sdefect( I,
1) = 0 .(ii) If I is a complete intersection, then sdefect( I, m ) = 0 for all m > .Proof. ( i ) This fact is trivial. ( ii ) This result follows from Zariski-Samuel [43, Appendix6, Lemma 5]. (cid:3) Recall that I is a generic complete intersection if the localization of I at any minimalassociated prime of I is a complete intersection. A result of [12, 39, 42] will prove useful: Theorem 2.3 ([12, Corollary 2.6],[39][42]) . Let I be a homogeneous ideal of k [ x , . . . , x n ] that is perfect, codimension two, and a generic complete intersection. If −→ F −→ G −→ I −→ is a graded minimal free resolution of I , then −→ ^ F −→ F ⊗ G → Sym G −→ I −→ is a graded minimal free resolution of I . Remark 2.4.
Weyman’s paper [42] gives the resolution of Sym ( I ). As shown in [12, 39],the hypotheses on I imply that Sym ( I ) ∼ = I .Many of our arguments make use of Hilbert functions. The Hilbert function of R/I ,denoted H R/I , is the numerical function H R/I : N → N defined by H R/I ( i ) := dim k R i − dim k I i where R i , respectively I i , denotes the i -th graded component of R , respectively I .Our primary focus is to understand sdefect( I, m ) when I defines either a star config-uration or a set of points in P . In the next section, we introduce star configurations inmore detail. For now, we review the relevant background about sets of points in P .Let X = { P , . . . , P s } be a set of distinct points in P . If I P i is the ideal associatedto P i in R = k [ x , x , x ], then the homogeneous ideal associated to X is the ideal I X = I P ∩ · · · ∩ I P s . The next lemma allows us to describe I ( m ) X ; although this result is well-known, we have included a proof for completeness. Lemma 2.5.
Let X = { P , . . . , P s } ⊆ P be a set of s distinct points with associated ideal I X = I P ∩ · · · ∩ I P s . Then for all m > , I ( m ) X = I mP ∩ · · · ∩ I mP s . HE SYMBOLIC DEFECT OF AN IDEAL 5
Proof.
The associated primes of I X are the ideals I P i with i = 1 , . . . , s . Because localizationcommutes with products, we have I m X R I Pi = ( I X R I Pi ) m = ( I P i R P i ) m = I mP i R P i . Note that the second equality follows from the fact that I P i is the only associated primeof I X contained in I P i . Since I mP i R P i ∩ R = I mP i , the result follows. (cid:3) For sets of points in P , the symbolic defect sequence will either be all zeroes, or allvalues of the sequence, except the first, will be nonzero. Moreover, we can completelyclassify when the symbolic defect sequence is all zeroes. Theorem 2.6.
Let X ⊆ P be any set of points. Then the following are equivalent: ( i ) I X is a complete intersection. ( ii ) sdefect( I X , m ) = 0 for all m > . ( iii ) sdefect( I X , m ) = 0 for some m > .Proof. Lemma 2.2 shows ( i ) ⇒ ( ii ), and ( ii ) ⇒ ( iii ) is immediate. For ( iii ) ⇒ ( i ), it wasnoted in [12, Remark 2.12(i)] that when X is not a complete intersection of points in P ,then I m X = I ( m ) X for all m >
2. This also follows from [33, Theorem 2.8] or [32, Corollary2.5]. (cid:3) Symbolic squares of star configurations
In this section, we will consider sdefect( I,
2) when I defines a star configuration. In fact,we prove a stronger result by finding an ideal J such that I (2) = J + I . It is interestingto note that the ideal J will also be a star configuration. For completeness, we begin withthe relevant background on star configurations. Definition 3.1.
Let n , c and s be positive integers with 1 c min { n, s } . Let F = { F , . . . , F s } be a set of forms in R = k [ x , x , . . . , x n ] with the property that all subsetsof F of cardinality c + 1 are regular sequences in R . Define an ideal of R by setting I c, F = \ i <...
A.V. Geramita is attributed with first coining the term star configurationto describe the variety defined by I c, F . The name is inspired by the fact that when n = c = 2, and s = 5, the placement of the five lines L = { L , . . . , L } that define alinear star configuration resembles a star. In this case, the locus of I c, L is a set of 10points corresponding to the intersections between these lines. It should be noted thatlinear star configurations were classically called l -laterals (e.g. see [15]). On the otherhand, our more general definition follows [21], where the geometric objects are calledhypersurface configurations. This more general definition of star configurations evolved HE SYMBOLIC DEFECT OF AN IDEAL 6 through a series of papers (see [1, 37, 21]); in particular, the codimension 2 case wasstudied before the general case. Star configurations have been shown to have many nicealgebraic properties, but at the same time, can be used to exhibit extremal properties.The references [7, 8, 9, 20, 23] form a small sample of papers that have studied the ideals I c, F . Remark 3.3.
Geometrically, the vanishing locus in P n of the ideal h F i , . . . , F i c i is a com-plete intersection of codimension c obtained by intersecting the hypersurfaces defined bythe forms F i , . . . , F i c . A star configuration is then a union of such complete intersections. Remark 3.4.
While the definition of a star configuration makes sense for s < n + 1, suchcases are less interesting (cf. [20, Remark 2.2]). Therefore we will always assume that s > n + 1. Theorem 3.5.
Let I c, F be the defining ideal of a star configuration in P n , with F = { F , . . . , F s } . Then { F i · · · F i s − c +1 | i < . . . < i s − c +1 s } is a minimal generating set of I c, F .Proof. See [37, Theorem 2.3] for generation (see also [21, Proposition 2.3 (4)]) and [37,Corollary 3.5] for minimality. (cid:3)
We will make use of the following decomposition of the m -th symbolic power; thisfollows from [21, Theorem 3.6 (i)]. Theorem 3.6.
Let I c, F be the defining ideal of a star configuration in P n , with F = { F , . . . , F s } . For all m > , we have I ( m ) c, F = \ i <...
Let I c, L be the defining ideal of a linear star configuration in P n , with L = { L , . . . , L s } . Suppose that |L| = n + 1. Then, up to a change of variables,we may assume that the hyperplanes forming the star configuration are defined by thecoordinate functions x , x , . . . , x n . By Theorem 3.6, we have I ( m ) c, L = \ i <... m for all 0 i < · · · < i c n. Let Supp( p ) denote the support of p , i.e., Supp( p ) = { x i | x i divides p } . HE SYMBOLIC DEFECT OF AN IDEAL 7
We are now able to describe an ideal M with the property that I ( m ) c, L = I mc, L + M . Theorem 3.7.
Let L = { x , . . . , x n } . Then I ( m ) c, L = I mc, L + M , where M is the idealgenerated by all monomials satisfying equation (3.1) whose support has cardinality atleast n − c + 3 .Proof. Clearly I ( m ) c, L ⊇ I mc, L + M . To show the other containment, consider a monomial p = x a x a . . . x a n n ∈ I ( m ) c, L . Since p ∈ I ( m ) c, L , we have p ∈ I c, L . Then | Supp( p ) | > n − c + 2by Theorem 3.5.If | Supp( p ) | = n − c + 2, then the complement of Supp( p ) in { x , x , . . . , x n } hascardinality c −
1. Therefore we can write { x , x , . . . , x n } \ Supp( p ) = { x j , . . . , x j c − } . For each x i ∈ Supp( p ), equation (3.1) implies that a i = a i + a j + . . . + a j c − > m. Thus p is a multiple of Y x i ∈ Supp( p ) x mi = Y x i ∈ Supp( p ) x i ! m which is the m -th power of a generator of I c, L by Theorem 3.5. Therefore p ∈ I mc, L .On the other hand, if | Supp( p ) | > n − c + 3, then p ∈ M by definition. (cid:3) For m = 2 and m = 3, we can improve upon the statement of Theorem 3.7. Corollary 3.8.
Let L = { x , . . . , x n } . We have I (2) c, L = I c − , L + I c, L .Proof. By [20, Lemma 2.13], we have I c − , L ⊆ I (2) c, L , which implies the containment I (2) c, L ⊇ I c − , L + I c, L (these containments hold for any linear star configuration ideal, not just amonomial star configuration ideal). To prove the other containment, we use the fact thatour ideals are monomial ideals.Consider a monomial p = x a x a . . . x a n n ∈ I (2) c, L . As observed in the proof of Theorem 3.7, | Supp( p ) | > n − c + 2 and, in the case of equality, p ∈ I c, L . Assume | Supp( p ) | > n − c + 3.Then p is divisible by one of the generators of I c − , L described in Theorem 3.5. Therefore p ∈ I c − , L . (cid:3) Remark 3.9.
The above result was first proved in [35, Corollary 3.7, Corollary 4.5] inthe special cases that n = c = 2, and n = c = 3. The above statement is also mentionedin [35, Remark 4.6], but no proof is given. Corollary 3.10.
Let L = { x , . . . , x n } . If c > , we have I (3) c, L = I c − , L + I c − , L I c, L + I c, L .Proof. We require c > I c − , L ⊆ I (3) c, L . Recall that I c − , L = h x i · · · x i n − c +4 | i < · · · < i n − c +4 n i . HE SYMBOLIC DEFECT OF AN IDEAL 8
Consider any subset A = { x i , . . . , x i c } of { x , x , . . . , x n } with | A | = c , and consider anygenerator m = x i · · · x i n − c +4 of I c − , L . Then at least three of the variables of A , say x i , x j , and x k , appear in Supp( m ) = { x i , . . . , x i n − c +4 } . Because x i x j x k ∈ h x i , . . . , x i c i , thismeans that m ∈ h x i , . . . , x i c i . But this implies that every generator m of I c − , L satisfies m ∈ \ i < ··· n − c + 2and, in the case of equality, p ∈ I c, L . Let | Supp( p ) | = n − c + 3. In this case, thecomplement of Supp( p ) in { x , x , . . . , x n } has cardinality c −
2, so we can write { x , x , . . . , x n } \ Supp( p ) = { x j , . . . , x j c − } . For each pair x i , x i ∈ Supp( p ), equation (3.1) implies that a i + a i = a i + a i + a j + . . . + a j c − > . Thus either a i > a i >
2. Repeating the same argument for all pairs x i , x i inSupp( p ), it follows that there are n − c + 2 elements x h ∈ Supp( p ) such that x h | p . Hence p is divisible by a monomial of the form x k x k . . . x k n − c +2 = ( x k x k . . . x k n − c +2 )( x k . . . x k n − c +2 ) , and therefore p ∈ I c − , L I c, L by Theorem 3.5. As in the previous proof, if | Supp( p ) | > n − c + 4, then p is divisible by a generator of I c − , L , which completes the proof. (cid:3) Theorem 3.11.
Let L = { x , . . . , x n } . We have sdefect( I c, L , m ) = 1 if and only if c = m = 2 .Proof. Let c = m = 2. By Theorem 3.5, I c − , L = I , L = h x x · · · x n i is a principal idealgenerated in degree n + 1. In contrast, I c, L is generated in degree n . Therefore, theequality I (2) c, L = I c − , L + I c, L of Corollary 3.8, implies that I (2) c, L /I c, L has a single minimalgenerator. Thus sdefect( I c, L , m ) = 1.Conversely, assume sdefect( I c, L , m ) = 1. By Theorem 3.7, I ( m ) c, L = I mc, L + M , where M isthe monomial ideal generated by all monomials satisfying equation (3.1) whose supporthas cardinality at least n − c + 3. Since sdefect( I c, L , m ) = 1, we deduce M = 0. Givenany monomial p ∈ M , we must have n + 1 > | Supp( p ) | > n − c + 3 . This implies c >
2. For any choice of indices 0 i < · · · < i n − c +3 n , the monomial(3.2) p = x i x m − i x m − i · · · x m − i n − c +3 HE SYMBOLIC DEFECT OF AN IDEAL 9 satisfies the condition in equation (3.1), and therefore p ∈ M . We claim that p is aminimal generator of M . If it was not, then we could divide p by a variable in its supportand obtain a new monomial still in M . However, if we divide p by any variable in itssupport, we either obtain a monomial whose support has cardinality less than n − c + 3or one that violates equation (3.1). Thus the claim holds. Note also that the degree of p is ( m − n − c + 2) + 1, and this is strictly smaller than the degree of a minimalgenerator of I mc, L , i.e., m ( n − c + 2). It follows that the residue class of p can be taken as aminimal generator of I ( m ) c, L /I mc, L . Hence each monomial of the same form as p contributes1 to sdefect( I c, L , m ). Now, if c > m >
2, the freedom in the choice of the indices i , . . . , i n − c +3 implies that sdefect( I c, L , m ) > (cid:3) The general case.
To extend the results of the monomial case to arbitrary starconfigurations, we recall a powerful theorem of Geramita, Harbourne, Migliore, and Nagel[21, Theorem 3.6 (i)].
Theorem 3.12.
Let I c, F be the defining ideal of a star configuration in P n , with F = { F , . . . , F s } ⊆ R = k [ x , x , . . . , x n ] . Let S = k [ y , . . . , y s ] and define a ring homomor-phism ϕ : S → R by setting ϕ ( y i ) = F i for i s . If I is an ideal of S , then we write ϕ ∗ ( I ) to the denote the ideal of R generated by ϕ ( I ) . Let L = { y , . . . , y s } . Then, foreach positive integer m , we have I ( m ) c, F = ϕ ∗ ( I c, L ) ( m ) = ϕ ∗ ( I ( m ) c, L ) . Since the operator ϕ ∗ commutes with ideal sums and products, Theorem 3.12 appliedto our results from the previous section gives the following more general statements. Theorem 3.13.
Let I c, F be the defining ideal of a star configuration in P n , with F = { F , . . . , F s } . Then I ( m ) c, F = I mc, F + M , where M is the ideal generated by all products F a · · · F a s s such that:(1) |{ i | a i > }| > s − c + 2 ;(2) ∀ i < . . . < i c n, a i + a i + . . . + a i c > m . Corollary 3.14.
We have I (2) c, F = I c − , F + I c, F . Corollary 3.15.
We have sdefect( I c, F , (cid:0) sc − (cid:1) . Furthermore, if F = L = { L , . . . , L s } ,that is, if I c, L , is a linear star configuration, then sdefect( I c, L ,
2) = (cid:0) sc − (cid:1) .Proof. By Corollary 3.14, I (2) c, F = I c − , F + I c, F . By Theorem 3.5, the ideal I c − , L is generatedby (cid:0) ss − c +2 (cid:1) = (cid:0) sc − (cid:1) minimal generators, so we need to add at most (cid:0) sc − (cid:1) generators to I c, F to generate I (2) c, L .If F = L , by Theorem 3.5 I c, L is generated by forms of degree 2( s − c + 1). On theother hand, again by Theorem 3.5, the ideal I c − , L is generated by generators of degree s − c + 2. Since s − c + 2 < s − c + 1), all the generators of I c − , L need to be added to I c, L to generate I (2) c, L , i.e., none of them are redundant. (cid:3) HE SYMBOLIC DEFECT OF AN IDEAL 10
Remark 3.16.
In the above proof, we appealed to the degrees of the elements of L tojustify why all the generators of I c − , L are required. In the general case, it may happenthat some of the minimal generators of I c − , F have degree larger than a minimal generatorof I c, F , thus preventing us from generalizing this argument.The following are also immediate consequences of results from the previous section. Corollary 3.17.
We have I (3) c, F = I c − , F + I c − , F I c, F + I c, F . In particular, sdefect( I c, F , (cid:18) sc − (cid:19) + (cid:18) sc − (cid:19)(cid:18) sc − (cid:19) . Theorem 3.18.
We have sdefect( I c, F , m ) = 1 if and only if c = m = 2 . Powers of codimension two linear star configurations.
We round out thissection by considering the higher m -th symbolic powers of the linear star configuration I , L in P . Note that in this case the linear star configuration defines a collection ofpoints in P . By applying [26, Corollary 3.9] of Harbourne and Huneke (and see also[12, Example 3.9] for additional details), we have the following relationship between theregular and symbolic powers of I , L in P . Theorem 3.19.
Suppose that I , L defines a linear star configuration in P . Then I (2 m )2 , L = ( I (2)2 , L ) m for all m > . We can then derive bounds on some of the values of the symbolic defect sequence.
Theorem 3.20.
Suppose that I , L defines a linear star configuration in P . Then sdefect( I , L , m ) |L| ( m − for all m > .Proof. Suppose that L = { L , . . . , L s } . By Corollary 3.14 we have I (2)2 , L = h L · · · L s i + I , L since I , L = h L · · · L s i . Let L = L · · · L s . It then follows by Theorem 3.19 that I (2 m )2 , L = (cid:2) h L i + I , L (cid:3) m = h L i m + h L i m − I , L + h L i m − I , L + · · · + h L i I m − , L + I m , L . Since I , L is generated by forms of degree ( s − h L i m + h L i m − I , L + h L i m − I , L + · · · + h L i I m − , L belongto I m , L .Define J a = h L a L ai | i = 1 , . . . , s i for a = 1 , . . . , m −
1. We claim that for 1 a m − h L i m + h L i m − I , L + · · · + h L i m − a +1 I a − , L + h L i m − a I a , L = h L i m + h L i m − J + · · · + h L i m − a +1 J a − + h L i m − a I a , L . Indeed, the ideal on the right is contained in the ideal on the left because each generatorof J a is a generator of I a , L . HE SYMBOLIC DEFECT OF AN IDEAL 11
For the reverse containment, we do induction on a . It is straightforward to check that h L i m + h L i m − I , L = h L i m + h L i m − J for the base case. Assume now that 2 a m − a , h L i m + h L i m − I , L + · · · + h L i m − a +1 I a − , L = h L i m + h L i m − J + · · · + h L i m − ( a − J a − . To finish the proof of the claim, we need to show that h L i m − a I a , L ⊆ h L i m + h L i m − J + · · · + h L i m − a +1 J a − + h L i m − a J a . Because of Theorem 3.5, I , L is generated by elements of the form F i = L/L i for some i = 1 , . . . , s . So, a generator of I a , L has the form F i F i · · · F i a where i , . . . , i a need notbe distinct. If i = · · · = i a = i , then the generator F i F i · · · F i a = L a L ai of I a , L is alsoa generator of J a , so L m − a F i F i · · · F i a ∈ h L i m − a J a . If at least two of i , . . . , i a aredistinct, say i = i , then F i F i · · · F i a = F i L i F i L i F i · · · F i a = L F i L i F i · · · F i a . But then L m − a F i F i · · · F i a = L m − a +1 F i L i F i · · · F i a ∈ h L i m − ( a − J a − . By induction, we then have L m − a F i F i · · · F i a ∈ h L i m + h L i m − J + · · · + h L i m − a +1 J a − + h L i m − a J a . This now verifies the claim.To complete the proof, note that to form I (2 m ) c, L , we can add all of the generators of h L i m + h L i m − J + · · · + h L i J m − to I m , L . This ideal has at most 1 + s ( m −
1) minimalgenerators (our generating set may not be minimal) since each ideal J a has s generators,so sdefect( I , L , m ) s ( m − (cid:3) A geometric consequence
By Theorem 3.18, if I c, L is a linear star configuration in P n of codimension two, thensdefect( I c, L ,
2) = 1 since c = 2. If n = 2, then the linear star configuration definedby I c, L is a collection of points in P , and thus, there exist sets of points X in P withsdefect( I X ,
2) = 1. In general, it would be interesting to classify all the ideals I X of setsof points X in P with sdefect( I X ,
2) = 1. In this section, we show under some additionalhypotheses, that if X is a set of points in P with sdefect( I X ,
2) = 1, then X must be alinear star configuration.We first recall some facts about the defining ideals of points in P ; many of these resultsare probably known to the experts, but for completeness, we include their proofs. Recallthat for any homogeneous ideal I ⊆ R , we let α ( I ) = min { i | I i = 0 } . Note that for any m > α ( I m ) = mα ( I ).The following is the so-called Dubreil’s inequality (see [10, 14]), but an elementary proof(which we now give) is also possible.
HE SYMBOLIC DEFECT OF AN IDEAL 12
Lemma 4.1.
Let X ⊆ P be a finite set of points. If α = α ( I X ) , then I X has at most α + 1 minimal generators of degree α .Proof. Because α = α ( I X ), the Hilbert function of X at α − H R/I X ( α −
1) = dim k R α − = (cid:0) α +12 (cid:1) . If I X has d > α + 1 generators of degree α , then H R/I X ( α ) = (cid:0) α +22 (cid:1) − d < (cid:0) α +22 (cid:1) − ( α + 1) = (cid:0) α +12 (cid:1) . In other words, H R/I X ( α − > H R/I X ( α ), contradicting the factthat the Hilbert functions of sets of points must be non-decreasing functions [22, cf. proofof Proposition 1.1 (2)]. (cid:3) The next lemma is a classification of those sets of points which have exactly α + 1minimal generators of degree α . Lemma 4.2.
Let X be a set of points of P . Then the following are equivalent: ( i ) The ideal I X has α + 1 minimal generators of degree α = α ( I X ) ; ( ii ) The set X is a set of (cid:0) α +12 (cid:1) points in P having generic Hilbert function, i.e., H R/I X ( i ) = min { dim k R i , | X |} for all i > ; and ( iii ) The ideal I X has a graded linear resolution.Proof. ( i ) ⇒ ( ii ) If I X has α + 1 minimal generators of degree α , it follows that (cid:18) α + 12 (cid:19) = H R/I X ( α −
1) = H R/I X ( α ) = (cid:18) α + 22 (cid:19) − (cid:18) α + 11 (cid:19) . Because the Hilbert function of a set of points in P is a strictly increasing function untilit reaches | X | , we have | X | = (cid:0) α +12 (cid:1) , and the Hilbert function of R/I X is given by H R/I X ( t ) = min (cid:26) dim k R t , (cid:18) α + 12 (cid:19)(cid:27) for all t > . ( ii ) ⇒ ( iii ) If R/I X has the generic Hilbert function, one can use Section 3 of [36] todeduce that the resolution is0 −→ R α ( − ( α + 1)) −→ R α +1 ( − α ) −→ R −→ R/I X −→ , i.e., the graded resolution is linear.( iii ) ⇒ ( i ) Assume that I X has a linear graded free resolution0 −→ R β − ( − ( α + 1)) −→ R β ( − α ) −→ R −→ R/I X −→ . Since H R/I X ( t ) = H R/I X ( t + 1) for t ≫
0, we get thatdim k R t − β dim k R t − α + ( β −
1) dim k R t − ( α +1) = dim k R t +1 − β dim k R ( t +1) − α + ( β −
1) dim k R ( t +1) − ( α +1) . This proves that β = α + 1, i.e., I X has α + 1 minimal generators of degree α . (cid:3) Lemma 4.3.
Let X be a set of points of P , and suppose that any of the three equivalentconditions of Lemma 4.2 holds. If I (2) X = I X + h F , . . . , F r i , i.e., the F i comprise a minimalset of homogeneous generators of I (2) X modulo I X , then deg( F i ) < α ( I X ) for all i =1 , . . . , r . HE SYMBOLIC DEFECT OF AN IDEAL 13
Proof.
We first observe that because I X is an ideal of points, then the saturation of I X is I (2) X . If d is the saturation degree of I X , i.e., the smallest integer d such that ( I (2) X ) t = ( I X ) t for all t > d , then it is known that reg( I X ) > d (see, for example, the introduction of [5]).Again, because I X is an ideal of points, we have2reg( I X ) > reg( I X ) > α ( I X ) = 2 α ( I X ) = 2reg( I X ) , where the first inequality follows from [18, Theorem 1.1], and the last equality holdsfrom the fact that I X has a linear resolution. Thus, we get that reg( I X ) = 2 α ( I X ), or inother words, I X and I (2) X agree in degrees > reg( I X ) = 2 α ( I X ). Therefore, any minimalgenerators of I (2) X have degrees less than 2 α ( I X ) = 2 α , as we wished. (cid:3) When I X is the homogeneous ideal of a finite set of points X in P , it is well knownthat I X is both perfect and has codimension two. In addition, I X is a generic completeintersection because I X is a radical ideal in a regular ring and the minimal associatedprimes of I X are simply the ideals of the points P ∈ X , and when we localize I X at I P ,we get the maximal ideal of k [ x , x , x ] localized at I P , which is a complete intersection.We can thus apply Theorem 2.3 to any homogeneous ideal of a finite set of points in P .In particular, we record this fact as a lemma. Lemma 4.4.
Let X ⊆ P be a finite set of points. Suppose that I X has d minimalgenerators of degree α = α ( I X ) . Then I X has (cid:0) d +12 (cid:1) minimal generators of degree α ( I X ) =2 α . In particular, H R/I X (2 α ) = (cid:18) α + 22 (cid:19) − (cid:18) d + 12 (cid:19) . Proof. If F , . . . , F d are the d minimal generators of degree α = α ( I X ), then by Theorem2.3, the elements of { F i F j | i j d } will all be minimal generators of I X . Eachgenerator will have degree α ( I X ) = 2 α and there are (cid:0) d +12 (cid:1) such generators. For the laststatement, since I X has no generators of degree < α , we have dim k ( I X ) α = (cid:0) d +12 (cid:1) . (cid:3) We also require a result of Bocci and Chiantini. Statement ( i ) can be found in theintroduction of [6], while ( ii ) is [6, Theorem 3.3]. Theorem 4.5.
Let X ⊆ P be a set of points. ( i ) Then α ( I (2) X ) > α ( I X ) + 1 . ( ii ) If α ( I (2) X ) = α ( I X ) + 1 , then X is a linear star configuration of points or a set ofcolinear points. We now come to the main result of this section.
Theorem 4.6.
Let X be a set of (cid:0) α +12 (cid:1) points of P with the generic Hilbert function. If sdefect( I X ,
2) = 1 , then X is a linear star configuration.Proof. Since sdefect( I X ,
2) = 1, there exits a form F such that I (2) X = h F i + I X . By Lemma4.3, deg F < α . HE SYMBOLIC DEFECT OF AN IDEAL 14
We now show that we must, in fact, have deg F α + 1. By Lemma 4.2, I X has α + 1generators of degree α . By Lemma 4.4, the ideal I X will have (cid:0) α +22 (cid:1) minimal generatorsof degree 2 α . Because I X ⊆ I (2) X , this means H R/I (2) X (2 α ) H R/I X (2 α ) = (cid:18) α + 22 (cid:19) − (cid:18) α + 22 (cid:19) = (2 α + 2)(2 α + 1) − ( α + 2)( α + 1)2 = 3 α + 3 α . Suppose that deg
F > α + 1. Because I X is generated by forms of degree 2 α or larger,we have ( I (2) X ) α − = [ h F i + I X ] α − = [ h F i ] α − , and consequently, H R/I (2) X (2 α −
1) = H R/ h F i (2 α −
1) = (cid:18) α + 12 (cid:19) − dim k h F i α − . If deg F = d , then h F i ∼ = R ( − d ) as graded R -modules, so dim k h F i α − = dim k R α − − d = (cid:0) α − d +12 (cid:1) . Because d > α + 2, we have H R/I (2) X (2 α −
1) = (cid:18) α + 12 (cid:19) − (cid:18) α − d + 12 (cid:19) > (cid:18) α + 12 (cid:19) − (cid:18) α − ( α + 2) + 12 (cid:19) = (2 α + 1)(2 α ) − ( α − α − α + 2 α − ( α − α + 2)2 = 3 α + 5 α − . Since sdefect( I X , = 0, X is not a complete intersection, and thus X cannot be a setof points on a line. Consequently, α >
2. But then we must have H R/I (2) X (2 α − > α + 5 α − > α + 3 α > H R/I (2) X (2 α ) . This is a contradiction, so deg F α + 1 as claimed.Because deg F > α by Theorem 4.5, we must have deg F = α + 1. Hence α ( I (2) X ) = α ( I X ) + 1. Theorem 4.5 then implies that X is a either a linear star configuration or a setof colinear points. If X was a set of colinear points, then Theorem 2.6 would imply thatsdefect( I X ,
2) = 0 because colinear points are a complete intersection. Thus X must be alinear star configuration in P . (cid:3) Remark 4.7.
As we will see in Section 6, there exist sets of points X in P withsdefect( I X ,
2) = 1, but X is not a linear star configuration. Remark 4.8.
It is natural to ask if a similar type of result holds for points in P n with n >
3, i.e., if sdefect( I X ,
2) = 1, along with some suitable hypotheses on X , implies that X must be a linear star configuration. However, this cannot happen. Indeed, if such a set ofpoints X existed, then X = V ( I n, L ) for some n > L , because X HE SYMBOLIC DEFECT OF AN IDEAL 15 is a zero-dimensional scheme. But then we would have sdefect( I n, L ,
2) = 1, contradictingTheorem 3.18.5.
Application: Resolutions of squares of star configurations in P n In this section, we use Corollary 3.14 to describe a minimal free resolution of thesymbolic square of the defining ideal I , F of a codimension two star configuration in P n . Lemma 5.1.
Let I , F be the defining ideal of a star configuration of codimension two in P n . Assume F = { F , . . . , F s } , where F , . . . , F s are forms of degrees d · · · d s ,and let d = d + · · · + d s . Then a graded minimal free resolution of I , F has the form → R ( s − )( − d ) → M i s R s − ( − (2 d − d i )) → M i,j s R ( − (2 d − ( d i + d j ))) → I , F → . Proof.
By [37, Theorem 3.4], the ideal I , F has a graded minimal free resolution of theform 0 → R s − ( − d ) → M i s R ( − ( d − d i )) → I , F → . Recall that I , F = \ i There exists a unique pair ( i, j ) such that h F i , F j i ⊆ P . Proof of Claim. The existence of the pair follows from [4, Prop. 1.11]. Assume h F α , F β i ⊆ P for some indices α, β with { α, β } 6 = { i, j } . Without loss of generality, we may assumethat α = i, j . Then F i , F j , F α ∈ P , which is a contradiction, since F i , F j , F α form a regularsequence of length 3 but P has height 2. (cid:3) It follows from the claim that( I , F ) P = \ k Let I , F be the defining ideal of a star configuration of codimension two in P n . Assume F = { F , . . . , F s } , and set F = F · · · F s . Then(i) [ I , F : F ] = (cid:10) F i · · · F i s − | i < · · · < i s − s (cid:11) = I , F ;(ii) I , F ∩ h F i = F [ I , F : F ] .Proof. ( i ) First, recall that I , F = (cid:28) F F i F j (cid:12)(cid:12)(cid:12)(cid:12) i j s (cid:29) . HE SYMBOLIC DEFECT OF AN IDEAL 16 Given indices 1 i < · · · < i s − s , let { i s − , i s } be the complement of { i , . . . , i s − } in { , . . . , s } . Then we have ( F i · · · F i s − ) F = F F i s − F i s ∈ I , F , and so F i · · · F i s − ∈ [ I , F : F ].Conversely, let G ∈ [ I , F : F ]. Since GF ∈ I , F , we have that(5.1) GF = X i s A i F F i + X i For every 1 i s , F i divides A i . Proof of Claim. For i = 1, GF = A F F + X i s A i F F i + X i Let I , F be the defining ideal of a star configuration of codimension two in P n . Assume F = { F , . . . , F s } , where F , . . . , F s are forms of degrees d · · · d s ,and let d = d + · · · + d s . Then a graded minimal free resolution of I (2)2 , F has the form → M i s R ( − (2 d − d i )) → M i s R ( − (2 d − d i )) ! ⊕ R ( − d ) → I (2)2 , F → . HE SYMBOLIC DEFECT OF AN IDEAL 17 Proof. Let F = F · · · F s . Thanks to Corollary 3.14, there is a short exact sequence0 → I , F ∩ h F i → I , F ⊕ h F i → I (2)2 , F → . We proceed to describe a minimal free resolution of the left term. By Lemma 5.2 ( i ),[ I , F : F ] = I , F . By [37, Theorem 3.4], a minimal free resolution of I , F has the form0 → R ( s − )( − d ) → M i s R s − ( − ( d − d i )) → M i Remark 5.4. If I , L defines a linear star configuration in P n , our formula agrees with theformula of [20, Theorem 3.2] with c = 2; thus Theorem 5.3 is a generalization of [20] inthe sense that the star configuration need not be linear.6. General sets of points In this section, we study general sets X of points in P . Specifically, we characterizewhen sdefect( I X , 2) = 1.Recall that a property holds for a general set of s points in P n if the subset of ( P n ) s for which it holds contains a nonempty open subset. If X ⊆ P n is a general set of points,then X has the generic Hilbert function , that is, H R/I X ( i ) = min { dim k R i , | X |} for all i > . The key ingredient that we require is the following famous result of Alexander andHirschowitz which computes the Hilbert function of R/I (2) X when X is a set of generalpoints in P n (we have specialized their result to P ). Roughly speaking, except if s = 2or 5, the Hilbert function of R/I (2) X is the generic Hilbert function of 3 | X | points. Theorem 6.1 ([2, Theorem 2]) . Let X be a set of s general points in P . If s = 2 , , then H R/I (2) X ( i ) = min { dim k R i , s } for all i > .If s = 5 , then H R/I (2) X ( i ) = ( min { dim k R i , s } i = 414 i = 4 . HE SYMBOLIC DEFECT OF AN IDEAL 19 In fact, the graded minimal free resolution of I X and I (2) X for s general points in P isknown. The resolution of I X and I (2) X is the cumulative work of many people. For sets X of simple points, the minimal resolution of I X was worked out by Geramita and Maroscia[22], Geramita, Gregory, and Roberts [19], and Lorenzini [36].For I (2) X , Catalisano’s work [11] determines the resolution of I (2) X for s 5, while Id`a[34] handles the case of s > s 9, and provinga conjecture of Harbourne [25, Conjecture 6.3] in the special case of m = 2). We recordonly the consequences we need. Lemma 6.2. Let X be a set of s general points in P . ( i ) If s = 5 , then the graded minimal free resolution of I X , respectively I (2) X , is −→ R ( − −→ R ( − ⊕ R ( − −→ I X −→ , respectively → R ( − ⊕ R ( − → R ( − ⊕ R ( − → I (2) X → . ( ii ) If s = 7 , then the graded minimal free resolution of I X , respectively I (2) X , is → R ( − ⊕ R ( − → R ( − → I X → , respectively → R ( − → R ( − → I (2) X → . ( iii ) If s = 8 , then the graded minimal free resolution of I X , respectively I (2) X , is → R ( − → R ( − ⊕ R ( − → I X → , respectively → R ( − → R ( − → I (2) X → . ( iv ) If s = 9 , then the graded minimal free resolution of I X , respectively I (2) X , is → R ( − → R ( − ⊕ R ( − → I X → , respectively → R ( − → R ( − ⊕ R ( − → I (2) X → . We now present the main result of this section. Theorem 6.3. Let X be a set of s general points in P . Then ( i ) sdefect( I X , 2) = 0 if and only if s = 1 , or . ( ii ) sdefect( I X , 2) = 1 if and only if s = 3 , , , or . ( iii ) sdefect( I X , > if and only if s = 6 or s > .Proof. By Theorem 2.6, sdefect( I X , 2) = 0 if and only if X is a complete intersection. Buta set of s general points is a complete intersection if and only if s = 1 , , or 4 (e.g., [27,Exercise 11.9]). This proves ( i ).We next consider the special cases of s = 3 , , , , , s = 3, then X is also a linear star configuration. Indeed, for each pair of points P i , P j with i = j , take the unique line L i,j through those two points. Then I X = I , L where L = { L , , L , , L , } . Then sdefect( I X , 2) = 1 by Theorem 3.18. HE SYMBOLIC DEFECT OF AN IDEAL 20 For the cases s = 5 , , , and 9, we first observe thatdim k ( I (2) X /I X ) α ( I (2) X /I X ) sdefect( I X , X t > dim k ( I (2) X /I X ) t . We can use Theorem 2.3 and Lemma 6.2 to find the Hilbert functions of R/I (2) X and R/I X for s = 5 , , , and 9. In these four cases, we will find that dim k ( I (2) X ) t = dim k ( I X ) t = 0 if t = α ( I (2) X /I X ), and consequently the above inequalities givedim k ( I (2) X /I X ) α ( I (2) X /I X ) = sdefect( I X , . Furthermore, we can use these Hilbert functions to compute the symbolic defect; precisely,sdefect( I X , 2) = ( s = 5 , 73 if s = 6 , . When s = 8, the Hilbert functions of I (2) X and I X disagree in two degrees, so the aboveapproach does not work. Instead, if s = 8, then Lemma 6.2 ( iii ) implies that α ( I X ) = 3and I X has two minimal generators of degree 3. So, I X has three minimal generators ofdegree 6. By Lemma 6.2 ( iii ), α ( I (2) X ) = 6 and I (2) X has four minimal generators of degree6. So, there exists a form F ∈ ( I (2) X ) \ ( I X ) . But since I (2) X is generated by these fourgenerators of degree 6, I (2) X = h F i + I X , that is, sdefect( I X , 2) = 1.Going forward, we now assume that s > 10. Our goal is to show that sdefect( I X , > I X , 2) = 1 and F is any homogeneous formsuch that I (2) X = h F i + I X , then the degree of F is restricted. Below, α = α ( I X ). Claim. If s > 10 and I (2) X = h F i + I X , then deg F > α − Proof of Claim. Suppose that d = deg F α − 2. Because s > H R/I (2) X ( d ) = H R/I (2) X ( d + 1) = 3 | X | by Theorem 6.1 . On the other hand, since I (2) X = h F i + I X and deg F α − 2, we havedim k ( I (2) X ) d = dim k h F i d = 1 and dim k ( I (2) X ) d +1 = dim k h F i d +1 = 3 . But this then means that (cid:18) d + 22 (cid:19) − H R/I (2) X ( d ) = 3 | X | = H R/I (2) X ( d + 1) = (cid:18) d + 32 (cid:19) − . So d , the degree of F , would have to satisfy (cid:18) d + 22 (cid:19) − (cid:18) d + 32 (cid:19) + 2 = 0 ⇔ d = 0 . But deg F > 0. So, deg F > α − (cid:3) Now suppose that s > 10 and sdefect( I X , 2) = 1. Consequently, there is a homogeneousform F such that I (2) X = h F i + I X , and furthermore, by the above claim, deg F > α − α = α ( I X ). We now consider the cases deg F = 2 α − 1, and deg F > α separately. Case 1. I (2) X = h F i + I X with deg F = 2 α − HE SYMBOLIC DEFECT OF AN IDEAL 21 If deg F = 2 α − 1, then we first claim that α 11. Indeed, by Theorem 6.1 H R/I (2) X (2 α − 2) = (cid:18) α (cid:19) | X | = H R/I (2) X (2 α − 1) = (cid:18) α + 12 (cid:19) − I (2) X has exactly one generator of degree2 α − 1. On the other hand, we know that | X | < (cid:0) α +22 (cid:1) since s general points have the genericHilbert function, so α is by definition the smallest number i such that (cid:0) i +22 (cid:1) > s = | X | .Combining these inequalities, we have (cid:18) α (cid:19) | X | < (cid:18) α + 22 (cid:19) . or equivalently, α must satisfy2 α (2 α − − α + 2)( α + 1)2 < ⇔ α − α − < . But the last inequality only holds if α 11. Since we are also assuming that | X | > α F = 2 α − 1, then 3 | X | = (cid:0) α +12 (cid:1) − (cid:0) α +12 (cid:1) − α 11 if andonly if α = 5 , , | X | = (cid:0) α +12 (cid:1) − α 11, then we have | X | = 54 / | X | =135 / | X | = 252 / I (2) X = h F i + I X .However, if | X | = 45, then α ( I X ) = 9, not 8. Also, if | X | = 84, then α ( I X ) = 12, not 11.If | X | = 18, then α ( I X ) = 5. So we need a separate argument to show that I (2) X = h F i + I X .So, let s = 18 with α = 5 and suppose that I (2) X = h F i + I X with deg F = 9. Then( I (2) X ) = ( F + I X ) . Now by Theorem 6.1, dim k ( I (2) X ) = 12. On the other hand, I X hasthree generators of degree α = 5, so by Lemma 4.4, I X has six generators of degree 2 α = 10,and no smaller generators. So dim k ( h F i + I X ) dim k ( h F i ) + dim k ( I X ) = 3 + 6 = 9.So, by a dimension count, we cannot have I (2) X = h F i + I X .To summarize this case, if s > 10, there is no set of s general points with I (2) X = h F i + I X with deg F = 2 α − Case 2. I (2) X = h F i + I X with deg F > α .If deg F > α , then we claim that α 7. Indeed, since I (2) X will be generated by forms ofdegree 2 α or larger, we have H R/I (2) X (2 α − 1) = (cid:18) α + 12 (cid:19) | X | . On the other hand, | X | < (cid:0) α +22 (cid:1) . Combining these two inequalities gives (cid:18) α + 12 (cid:19) | X | < (cid:18) α + 22 (cid:19) . HE SYMBOLIC DEFECT OF AN IDEAL 22 So, α must satisfy(2 α + 1)(2 α ) < α + 2)( α + 1) ⇔ α − α − < ⇔ α . Moreover, because s > 10, we have 4 α 7, or equivalently, 10 s = | X | d = (cid:0) α +22 (cid:1) − | X | , that is, d is the number of minimal generators of I X of degree α .If deg F = 2 α and I (2) X = h F i + I X , then I (2) X has (cid:0) d +12 (cid:1) + 1 minimal generators of degree2 α . If deg F > α and I (2) X = h F i + I X , then I (2) X has (cid:0) d +12 (cid:1) minimal generators of degree2 α . So, we will have H R/I (2) X (2 α ) = 3 | X | = (cid:18) α + 22 (cid:19) − (cid:18) d + 12 (cid:19) − F = 2 α, (cid:18) α + 22 (cid:19) − (cid:18) d + 12 (cid:19) if deg F > α. Thus, to summarize, if I (2) X = h F i + I X with deg F > α , then( a ) 10 | X | b ) (cid:0) α +12 (cid:1) | X | < (cid:0) α +22 (cid:1) , and( c ) either 3 | X | = (cid:0) α +22 (cid:1) − (cid:0) d +12 (cid:1) − | X | = (cid:0) α +22 (cid:1) − (cid:0) d +12 (cid:1) must hold with d = (cid:0) α +22 (cid:1) − | X | .A direct computation for each value 10 | X | 35 shows that no value of | X | satisfiesboth of ( b ) and ( c ). Table 1 explicitly verifies this statement; note that in the table, (T)denotes true and (F) denotes false.To summarize this case, if s > 10, there is no set of s general points with I (2) X = h F i + I X with deg F > α . Thus combining this case with the previous case, we see that if s > I X , > 1, thus completing the proof. (cid:3) Remark 6.4. The special case s = 6 in the Theorem 6.3 can also be explained byappealing to Theorem 4.6. 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