Demailly's Conjecture and the Containment Problem
Sankhaneel Bisui, Eloísa Grifo, Huy Tài Hà, Thái Thành Nguyên
aa r X i v : . [ m a t h . A C ] S e p DEMAILLY’S CONJECTURE AND THE CONTAINMENT PROBLEM
SANKHANEEL BISUI, ELO´ISA GRIFO, HUY T `AI H `A, AND TH ´AI TH `ANH NGUY˜ˆEN
Abstract.
We investigate Demailly’s Conjecture for a general set of sufficiently manypoints. Demailly’s Conjecture generalizes Chudnovsky’s Conjecture in providing a lowerbound for the Waldschmidt constant of a set of points in projective space. We also study acontainment between symbolic and ordinary powers conjectured by Harbourne and Hunekethat in particular implies Demailly’s bound, and prove that a general version of that con-tainment holds for generic determinantal ideals and defining ideals of star configurations. Introduction
Let k be a field, let N ∈ N be an integer, let R = k [ P N k ] be the homogeneous coordinatering of P N k , and let m be its maximal homogeneous ideal. For a homogeneous ideal I ⊆ R ,let α ( I ) denote the least degree of a homogeneous polynomial in I , and let I ( n ) := \ p ∈ Ass(
R/I ) I n R p ∩ R denote its n -th symbolic power . In studying the fundamental question of what the least degreeof a homogeneous polynomial vanishing at a given set of points in P N k with a prescribed ordercan be, Chudnovsky [Chu81] made the following conjecture. Conjecture 1.1 (Chudnovsky) . Suppose that k is an algebraically closed field of charac-teristic 0. Let I be the defining ideal of a set of points X ⊆ P N k . Then, for all n > α ( I ( n ) ) n > α ( I ) + N − N . (C)Chudnovsky’s Conjecture has been investigated extensively, for example in [EV83, BH10,HH13, GHM13, Dum15, DTG17, FMX18, BGHN20]. Particularly, the conjecture was provedfor a very general set of points [DTG17, FMX18] and for a general set of sufficiently manypoints [BGHN20]. The conjecture was also generalized by Demailly [Dem82] to the followingstatement.
Conjecture 1.2 (Demailly) . Suppose that k is an algebraically closed field of characteristic0. Let I be the defining ideal of a set of points X ⊆ P N k and let m ∈ N be any integer. Then,for all n > α ( I ( n ) ) n > α ( I ( m ) ) + N − m + N − . (D) Mathematics Subject Classification.
Key words and phrases.
Chudnovsky’s Conjecture, Waldschmidt constant, ideals of points, symbolic pow-ers, containment problem, Stable Harbourne–Huneke Conjecture. emailly’s Conjecture for N = 2 was proved by Esnault and Viehweg [EV83]. Recent workof Malara, Szemberg and Szpond [MSS18], extended by Chang and Jow [CJ20], showed thatfor a fixed integer m , Demailly’s Conjecture holds for a very general set of sufficiently manypoints. Specifically, it was shown that, given N > m ∈ N and s > ( m + 1) N , for each n > U n of the Hilbert scheme of s points in P N k such thatDemailly’s bound (D) for α ( I ( n ) ) holds for X ∈ U n . As a consequence, Demailly’s Conjectureholds for all X ∈ T ∞ n =1 U n . Chang and Jow [CJ20] further proved that if s = k N , for some k ∈ N , then one can take U n to be the same for all n >
1, i.e., Demailly’s Conjecture holdsfor a general set of k N points.In this paper, we establish Demailly’s Conjecture for a general set of sufficiently manypoints. More precisely, we show that given N > , m ∈ N and s > (2 m + 3) N , there existsan open dense subset U of the Hilbert scheme of s points in P N k such that Demailly’s bound(D) holds for X ∈ U and all n >
1. x
Theorem 2.8.
Suppose that char k = 0 and N >
3. For a fixed integer m >
1, let I be thedefining ideal of a general set of s > (2 m + 3) N points in P N k . For all n >
1, we have α ( I ( n ) ) n > α ( I ( m ) ) + N − m + N − . To prove Theorem 2.8 we use a similar method to the one we used in our previous work[BGHN20], where we proved Chudnovsky’s Conjecture for a general set of sufficiently manypoints. This is not, however, a routine generalization. In [BGHN20], Chudnovsky’s bound(C) was obtained via the (Stable) Harbourne–Huneke Containment, which states that for ahomogeneous radical ideal I ⊆ R of big height h we have I ( hr ) ⊆ m r ( h − I r for r ≫ . To achieve the Stable Harbourne–Huneke Containment, we showed that one particular con-tainment I ( hc − h ) ⊆ m c ( h − I c , for some value c ∈ N , would lead to the stable containment I ( hr − h ) ⊆ m r ( h − I r for r ≫
0. In a similar manner, Demailly’s bound (D) would followas a consequence of the following more general version of the (Stable) Harbourne–HunekeContainment: I ( r ( m + h − ⊆ m r ( h − ( I ( m ) ) r for r ≫ . (HH)Unfortunately, this is where the generalization of the arguments in [BGHN20] breaks down.We cannot prove that one such containment would lead to the stable containment. To over-come this obstacle, we show that the stronger containment I ( c ( m + h − − h +1) ⊆ m c ( h − ( I ( m ) ) c ,for some value c ∈ N , would imply I ( r ( m + h − ⊆ m r ( h − ( I ( m ) ) r for infinitely many values of r , and this turns out to be enough to obtain Demailly’s bound.It is an open problem whether, for a homogeneous radical ideal I , the general versionof the Stable Harbourne–Huneke Containment stated in (HH) holds; this problem is openeven in the case where I defines a set of points in P N k . In the second half of the paper,we investigate the general containment problem. We show that the containment holds for generic determinantal ideals and the defining ideals of star configurations in P N k . Our resultsare stated as follows. heorems 3.6 and 3.8. (1) Let I be the defining ideal of a codimension h star configuration in P N k , for h N .For any m, r, c >
1, we have I ( r ( m + h − − h + c ) ⊆ m ( r − h − c − ( I ( m ) ) r . (2) Let I = I t ( X ) be the ideal of t -minors of a matrix X of indeterminates, and let h denote its height in k [ X ]. For all m, r >
1, we have I ( r ( h + m − ⊆ m r ( h − (cid:0) I ( m ) (cid:1) r . Particularly, if I is the defining ideal of a star configuration or a generic determinantal idealthen I satisfies a Demailly-like bound, i.e., for all n > α ( I ( n ) ) n > α ( I ( m ) ) + h − m + h − . Determinantal ideals are classical objects in both commutative algebra and algebraic ge-ometry that have been studied extensively. The list of references is too large to be exhaustedhere; we refer the interested reader to [BV88] and references therein. In this paper, we areparticularly interested in generic determinantal ideals. Specifically, for a fixed pair of inte-gers p and q , let X be a p × q matrix of indeterminates and let R = k [ X ] be the correspondingpolynomial ring. For t min { p, q } , let I t ( X ) be the ideal in R generated by the t -minorsof X ; that is, I t ( X ) is generated by the determinants of all t × t submatrices of X . Itis a well-known fact that I t ( X ) is a prime, unmixed and Cohen-Macaulay ideal of height h = ( p − t + 1)( q − t + 1).Star configurations have also been much studied in the literature with various applications[CVT11, CGVT14, CGVT15, Toh15, PS15, Toh17, AGT17, BFGM20, CGS20]. They oftenprovide good examples and a starting point in investigating algebraic invariants and proper-ties of points in projective spaces; for instance, the minimal free resolution (cf. [AS12, RZ16]),weak Lefschetz property (cf. [Shi12, AS12, KS16]), and symbolic powers and containmentof powers (cf. [GHM13, HM18, Shi19, Man19]).We shall use the most general definition of a star configuration given in [Man19]. Let F = { F , . . . , F n } be a collection of homogeneous polynomials in R and let h < min { n, N } be an integer. Suppose that any ( h + 1) elements in F form a complete intersection . Thedefining ideal of the codimension h star configuration given by F is defined to be I h, F = \ i < ···
The second author thanks Jack Jeffries for helpful discussions. Thesecond author is supported by NSF grant DMS-2001445. The third author is partiallysupported by Louisiana Board of Regents (grant . Demailly’s Conjecture for general points
In this section, we establish Demailly’s Conjecture for a general set of sufficiently manypoints. Recall first that for a homogeneous ideal I ⊆ R , the Waldschmidt constant of I isdefined to be b α ( I ) = lim n →∞ α ( I ( n ) ) n . It is known (cf. [BH10, Lemma 2.3.1]) that the Waldschmidt constant of an ideal exists and b α ( I ) = inf n ∈ N α ( I ( n ) ) n . Thus, Demailly’s Conjecture can be equivalently stated as follows.
Conjecture 2.1 (Demailly) . Let k be an algebraically closed field of characteristic 0. Let I ⊆ k [ P N k ] be the defining ideal of a set of points in P N k and let m ∈ N be any integer. Then, b α ( I ) > α ( I ( m ) ) + N − m + N − . (D’)Demailly’s Conjecture for N = 2 follows from [EV83]. Thus, for the remaining of thepaper, we shall make the assumption that N >
3. We start by showing that Demailly’sbound (D’) follows from one appropriate containment between symbolic and ordinary powersof the given ideal. This result generalizes [BGHN20, Proposition 5.3].
Lemma 2.2.
Let I ⊆ R be an ideal of big height h and let m ∈ N . Suppose that for someconstant c ∈ N , we have I ( c ( h + m − − h +1) ⊆ m c ( h − (cid:0) I ( m ) (cid:1) c . Then, b α ( I ) > α (cid:0) I ( m ) (cid:1) + h − m + h − . Proof.
We will make use of a result of Ein–Lazarsfeld–Smith [ELS01, Theorem 2.2] andHochster–Huneke [HH02, Theorem 1.1 (a)], which says that I ( ht + kt ) ⊆ (cid:0) I ( k +1) (cid:1) t for all t > k >
0. We obtain that for all t ∈ N , I ( ct ( m + h − = I ( ht + t [ c ( m + h − − h ]) ⊆ (cid:0) I ( c ( m + h − − h +1) (cid:1) t ⊆ (cid:2) m c ( h − ( I ( m ) ) c (cid:3) t = m ct ( h − (cid:0) I ( m ) (cid:1) ct . Particularly, it follows that α (cid:0) I ( ct ( m + h − (cid:1) ct ( m + h − > ct ( h −
1) + ctα (cid:0) I ( m ) (cid:1) ct ( m + h −
1) = α ( I ( m ) ) + h − m + h − . By taking the limit as t → ∞ , it follows that b α ( I ) > α ( I ( m ) ) + h − m + h − . The assertion is proved. (cid:3) n light of Lemma 2.2, to prove Demailly’s Conjecture for the defining ideal I of a set ofpoints in P N k , the task at hand is to exhibit the containment I ( c ( h + m − − h +1) ⊆ m c ( h − (cid:0) I ( m ) (cid:1) c for a specific constant c . Our method is to use specialization techniques, in a similar mannerto what we have done in [BGHN20], to reduce the problem to the generic set of points in P N k ( z ) .We shall now recall the definition of specialization in the sense of Krull [Kru48]. Let z = ( z ij ) i s, j N be the collection of s ( N + 1) new indeterminates. Let P i ( z ) = [ z i : · · · : z iN ] ∈ P N k ( z ) and X ( z ) = { P ( z ) , . . . , P s ( z ) } . The set X ( z ) is the set of s generic points in P N k ( z ) . Given a = ( a ij ) i s, j N ∈ A s ( N +1) k , let P i ( a ) and X ( a ) be obtained from P i ( z ) and X ( z ) by setting z ij = a ij for all i, j . It is easy tosee that there exists an open dense subset W ⊆ A s ( N +1) k such that X ( a ) is a set of distinctpoints in P N k for all a ∈ W (and all subsets of s points in P N k arise in this way).The following result allows us to focus on open dense subsets of A s ( N +1) k when discussinggeneral sets of points in P N k . Lemma 2.3 ([FMX18, Lemma 2.3]) . Let W ⊆ A s ( N +1) k be an open dense subset such that aproperty P holds for X ( a ) whenever a ∈ W . Then, the property P holds for a general set of s points in P N k . To get the desired containment for the generic set of points in P N k ( z ) we shall need thefollowing combinatorial lemma, which is a generalization of [BGHN20, Lemma 4.4] and[MSS18, Lemma 3.1]. Lemma 2.4.
Suppose that N > and k > m + 2 . We have (cid:18) ( k − m + N −
1) + N − N (cid:19) > ( k + 1) N (cid:18) m + N − N (cid:19) . Proof.
We shall use induction on N . For N = 3, we need to show that (cid:18) ( k − m + 2) + 23 (cid:19) > ( k + 1) (cid:18) m + 23 (cid:19) , which is equivalent to( k − k − m + 2) + 2][( k − m + 2) + 1] > ( k + 1) ( m + 1) m. Set k ′ = k −
1. It follows that k ′ > m + 1 and we need to prove the following inequality k ′ [ k ′ ( m + 2) + 2][ k ′ ( m + 2) + 1] > ( k ′ + 2) ( m + 1) m, i.e.,(3 m + 4) k ′ − m + m − k ′ − (12 m + 12 m − k ′ − m + m ) > . By setting f ( k ′ ) to be the left hand side of this inequality, as a function in k ′ , it suffices toshow that f ( k ′ ) is an increasing function for k ′ > m + 1 and f (2 m + 1) > f (2 m + 1) = 4( m + 1) (2 m + 3) >
0. On the other hand, we have f ′ ( k ′ ) = 3(3 m + 4) k ′ − m + m − k ′ − (12 m + 12 m − . We will show that 2 m + 1 is greater than both roots of f ′ ( k ′ ). Indeed, the bigger root of f ′ ( k ′ ) is k ′ = 3(2 m + m −
2) + √ √ m + 48 m + 63 m + 33 m + 83(3 m + 4) . ince[(2 m + 1)3(3 m + 4) − m − m + 6] − m + 48 m + 63 m + 33 m + 8)= 3(36 m + 192 m + 381 m + 327 m + 100) > , we have 2 m + 1 > k . This establishes the desired inequality for N = 3.Suppose now that the desired inequality holds for N >
3, i.e., (cid:18) ( k − m + N −
1) + N − N (cid:19) > ( k + 1) N (cid:18) m + N − N (cid:19) . We shall prove that the inequality holds for N + 1 as well. That is, (cid:18) ( k − m + N ) + NN + 1 (cid:19) > ( k + 1) N +1 (cid:18) m + NN + 1 (cid:19) . Set x = ( k − m + N −
1) + N −
1. Then x + k = ( k − m + N ) + N , and we need toprove that, for k > m + 2, (cid:18) x + kN + 1 (cid:19) > ( k + 1) N +1 (cid:18) m + NN + 1 (cid:19) . (2.1)Indeed, by the induction hypothesis, we have (cid:18) x + kN + 1 (cid:19) = (cid:18) xN (cid:19) ( x + k ) . . . ( x + 1)( N + 1)( x − N + 1) . . . ( x − N + k − > ( k + 1) N +1 (cid:18) m + NN + 1 (cid:19) ( N + 1)( k + 1)( m + N ) · ( x + k ) . . . ( x + 1)( N + 1)( x − N + 1) . . . ( x − N + k − . Hence, it is enough to show that if k > m + 2 then( x + k )( x + k − . . . ( x + 1) > ( k + 1)( m + N )( x − N + 1) . . . ( x − N + k − . (2.2)Observe that x + i > x − N + i + 1. Thus, to prove (2.2), it suffices to show that( x + k )( x + k − > ( k + 1)( m + N )( x − N + 1). That is,[( k − m + N ) + N ][( k − m + N ) + N − > ( k + 1)( m + N )[( k − m + N ) − ( k − . This inequality, by setting k ′ = k −
1, is equivalent to( m + N )[ k ′ − (2 m − k ′ ] + N ( N − > . The last inequality clearly holds for k ′ > m + 1. Hence, (2.1) and (2.2) hold for k > m + 2.This completes the proof. (cid:3) Remark 2.5.
For N >
4, we can slightly improve the bound for k in Lemma 2.4 to be k > m + 1 or k > m if, in addition, m >
3. For N > m >
2, the bound is also k > m .In the next few lemmas, we establish a general version of the Stable Harbourne–HunekeContainment for the defining ideal of sufficiently many generic points in P N k ( z ) . Lemma 2.6 (Compare with [BGHN20, Lemma 4.5]) . Suppose that s > (2 m + 2) N and N > . Let I ( z ) be the defining ideal of s generic points in P N k ( z ) . For r ≫ , we have I ( z ) ( r ( m + N − − N +1) ⊆ ( I ( z ) ( m ) ) r . roof. For simplicity of notation, we shall write I for I ( z ) in this lemma. Let k > m + 2be the integer such that k N s < ( k + 1) N . It follows from [DTG17, Theorem 2] that b α ( I ) > ⌊ N √ s ⌋ = k . Particularly, we have α ( I ( r ( m + N − − N +1) ) > k [ r ( m + N − − N + 1] . By [TV95, Theorem 2.4] and Lemma 2.4, for r ≫
0, we havereg( I ( m ) ) m + [( k − m + N − − − m + ( k − m + N − − kr ( N − k ( m + N − − kr ( N − . This implies that r reg( I ( m ) ) α ( I ( r ( m + N − − N +1) ). Thus, we obtain the following inequalityfor the saturation degree of ( I ( m ) ) r :sat(( I ( m ) ) r ) r reg( I ( m ) ) α ( I ( r ( m + N − − N +1) ) . As a consequence, it follows that for t > α ( I ( r ( m + N − − N +1) ), (cid:2) ( I ( m ) ) ( r ) (cid:3) t = [( I ( m ) ) r ] t . It can easily be checked that I ( r ( m + N − − N +1) ⊆ I ( mr ) = (cid:0) I ( m ) (cid:1) ( r ) . Hence, we conclude that I ( r ( m + N − − N +1) ⊆ ( I ( m ) ) r . (cid:3) Lemma 2.7.
Suppose that s > (2 m + 2) N and N > . Let I ( z ) be the defining ideal of s generic points in P N k ( z ) . Let m z denote the maximal homogeneous ideal in k (cid:2) P N k ( z ) (cid:3) . For r ≫ , we have I ( z ) ( r ( m + N − − N +1) ⊆ m r ( N − z ( I ( z ) ( m ) ) r . Proof.
For simplicity of notation, we shall write I for I ( z ) and m for m z in this lemma.Let k > m + 2 be the integer such that k N s < ( k + 1) N . By Lemma 2.6, we have I ( r ( m + N − − N +1) ⊆ ( I ( m ) ) r for r ≫
0. Thus, it suffices to show that, for r ≫ α ( I ( r ( m + N − − N +1) ) > r reg( I m ) + r ( N − . (2.3)As before, it follows from [TV95, Theorem 2.4] and Lemma 2.4 that, for r ≫ I ( m ) ) m + [( k − m + N − − − m + ( k − m + N − − kr ( N − . That is, reg( I ( m ) ) + N − k ( m + N − − kr ( N − . Thus, for r ≫
0, we have r (reg( I ( m ) ) + N − rk ( m + N − − k ( N − . Furthermore, again by [DTG17, Theorem 2], we have b α ( I ) > ⌊ N √ s ⌋ = k . Particularly, itfollows that α ( I ( r ( m + N − − N +1) ) > k [ r ( m + N − − N + 1] = rk ( m + N − − k ( N − . Hence, (2.3) holds for r ≫
0, and the lemma is proved. (cid:3) e are now ready to state our first main result, which establishes Demailly’s Conjecturefor a general set of sufficiently many points in P N k for any field k of characteristic 0. Theorem 2.8.
Suppose that char k = 0 and N > . For a fixed integer m > , let I be thedefining ideal of a general set of s > (2 m + 2) N points in P N k . Then, b α ( I ) > α ( I ( m ) ) + N − m + N − . Proof.
Let I ( z ) be the defining ideal of s generic points in P N k ( z ) and let m z denote the maximalhomogeneous ideal of k (cid:2) P N k ( z ) (cid:3) . It follows from Lemma 2.7 that there exists a constant c ∈ N such that I ( z ) ( c ( m + N − − N +1) ⊆ m c ( N − z ( I ( z ) ( m ) ) c . This, together with [Kru48, Satz 2 and 3] (see also [BGHN20, Remark 2.10]), implies thatthere exists an open dense subset U ⊆ A s ( N +1) such that for all a ∈ U , we have I ( a ) ( c ( m + N − − N +1) ⊆ m c ( N − ( I ( a ) ( m ) ) c . The theorem now follows from Lemmas 2.2 and 2.3. (cid:3)
Remark 2.9.
By Remark 2.5, the bound for s in Theorem 2.8 can be improved slightlywhen N > s > (2 m + 1) N or s > (2 m ) N if, in addition, m >
3. For N > m > s > (2 m ) N . Remark 2.10.
When m = 1, Demailly’s Inequality (D’) coincides with Chudnovky’s Con-jecture, which we previously showed to hold for sufficiently many general points in P N in[BGHN20]. Theorem 2.8 is a generalization of [BGHN20, Theorem 5.1], extending Chud-novsky’s Conjecture for sufficiently many general points to Demailly’s Conjecture. In partic-ular, [BGHN20, Theorem 5.1] states that Chudnovsky’s Conjecture holds for s > N generalpoints in P N k , which is the result in Theorem 2.8 when m = 1. On the other hand, for N > s > (2 m + 1) N in Demailly’s Conjecture, when m = 1,agrees with the bound s > N given in [BGHN20, Remark 5.2] for Chudnovsky’s Conjecture.Another crucial difference is that in [BGHN20] we also showed that the containment I ( rN ) ⊆ m r ( N − I r holds for r ≫
0. Here the corresponding generalization would be I ( r ( N + m − ⊆ m r ( N − (cid:0) I ( m ) (cid:1) r for all r ≫ m >
1. Unfortunately, we have notbeen able to prove this stable containment for all r sufficiently large; we only show it forinfinitely many values of r .3. Harbourne–Huneke containment beyond points
In this section, we investigate a general containment between symbolic and ordinary powersof radical ideals, and show that this containment holds for generic determinantal ideals andthe defining ideals of star configurations. Specifically, we are interested in the followinggeneral version of the Harbourne–Huneke Containment for radical ideals.
Question 3.1.
Let I be either a radical ideal of big height h in a regular local ring ( R, m ), or ahomogeneous radical ideal of big height h in a polynomial ring R with maximal homogeneousideal m . Does the containment I ( r ( h + m − ⊆ m r ( h − (cid:0) I ( m ) (cid:1) r hold for all m, r > positive answer to Question 3.1 would in particular imply a Demailly-like bound forhomogeneous radical ideals in k [ P N ], i.e., an affirmative answer to the following question. Question 3.2 (Demailly-like bound) . Let R be a polynomial ring over k and let I be ahomogeneous radical ideal of big height h in R . Does the inequality α ( I ( n ) ) n > α ( I ( m ) ) + h − m + h − n, m > h appeared in[CEHH17, Conjecture 2.9]. The answer to both questions is yes for squarefree monomialideals by [CEHH17, Corollary 4.3], where in fact a stronger containment was established[CEHH17, Theorem 4.2]. Similar containment for the defining ideal of a general set ofpoints in P k were investigated in [BCH14]. Furthermore, by the same reasoning as in theprevious section, the left hand side of the inequality in Question 3.2 can be replaced by theWaldschmidt constant b α ( I ) of I . We refer the interested reader to [CHHVT20] for moreinformation about the Waldschmidt constant, containment and equality between symbolicand ordinary powers of ideals.Our goal in this section is show that Question 3.1 has a positive answer for special classesof ideals. In a natural approach to Question 3.1, one might hope to make use of the followinggeneral containment of [ELS01, HH02]: I ( r ( h + m − ⊆ (cid:0) I ( m ) (cid:1) r . Given this containment, to derive an affirmative answer to Question 3.1, one could aim tosimply show that α (cid:0) I ( r ( h + m − (cid:1) > r ( h −
1) + r ω (cid:0) I ( m ) (cid:1) , where, for a homogeneous ideal J , ω ( J ) denotes the maximal generating degree of J . Thisinequality, however, is often false, as illustrated in the following examples. Example 3.3.
Consider n >
3, a field k with char k = 2 containing n distinct roots ofunity, and R = k [ x, y, z ]. The symbolic powers of the ideal I = ( x ( y n − z n ) , y ( z n − x n ) , z ( x n − y n ))have an interesting behavior; in particular, I (3) * I [DSTG13, HS15], and in fact the case k = C and n = 3 was the first example ever found of an ideal of big height 2 with suchbehavior [DSTG13].By the proof of [DHN +
15, Theorem 2.1], α ( I (3 k ) ) = 3 nk ; moreover, by [NS16, Theorem3.6], ω ( I ( kn ) ) = k ( n + 1) for all k >
1. Therefore, we immediately see that α (cid:0) I (3( kn +1)) (cid:1) = 3( kn + 1) n > kn ( n + 1) = 3 + 3 ω ( I ( kn ) ) . In fact, Macaulay2 [GS] computations with n = 3 suggest that α (cid:0) I ( r ( m +1)) (cid:1) > r + r ω (cid:0) I ( m ) (cid:1) may never hold. However, this does not prevent the containment in Question 3.1, I ( r ( m +1)) ⊆ m r (cid:0) I ( m ) (cid:1) r , o hold — indeed Macaulay2 [GS] computations support this containment for small values of r and m when n = 3. If, in addition, char k = 0 then this containment holds a infinitely manyvalues of m . Indeed, we have I ( r ( mn +1)) = I ( rmn + r ) ⊆ m r I ( rmn ) = m r (cid:0) I ( n ) (cid:1) mr = m r (cid:0) I ( mn ) (cid:1) r .Note that Demailly’s bound can be checked in this case, at least for multiples of 3. Indeed,by the proof of [DHN +
15, Theorem 2.1], b α ( I ) = n and α ( I (3 m ) ) = 3 nm , so I satisfiesDemailly’s bound for all multiples of 3: b α ( I ) > α ( I (3 m ) ) + 13 m + 1 . Furthermore, if char k = 0 then Demailly’s bound can also be verified by taking powers ofthe form 3 m +2. Particularly, since I (3 m +3) ⊆ m I (3 m +2) , we have α ( I (3 m +3) ) > α ( I (3 m +2) )+1.Equivalently, we get (3 m + 3) n > α ( I (3 m +2) ) + 1, or b α ( I ) > α ( I (3 m +2) ) + 13 m + 3 . Example 3.4 (Generic determinantal ideals) . Fix some t q p , let X be an p × q matrix of indeterminates, and let R = k [ X ] be the corresponding polynomial ring over afield k . Consider the ideal I = I t ( X ) of t -minors of X , which is a prime in R of height h = ( p − t + 1)( q − t + 1). By [ELS01, HH02], I ( r ( h + m − ⊆ (cid:0) I ( m ) (cid:1) r for all m, r >
1. To show that I satisfies the containment proposed in Question 3.1, onemight attempt to check that for all m, r > α (cid:0) I ( r ( h + m − (cid:1) > ω (cid:16) m r ( h − (cid:0) I ( m ) (cid:1) r (cid:17) = r ( h −
1) + rω ( I ( m ) ) . However, this inequality does not always hold; for example, if I is the ideal of 2 × × p = q = 3 and t = 2, so h = 4) and we take r = 1, m = 5,it turns out that α (cid:0) I ( r ( h + m − (cid:1) = α (cid:0) I (8) (cid:1) = 12 <
13 = 3 + ω ( I (5) ) = r ( h −
1) + rω ( I ( m ) ) . Nevertheless, as we will show in Theorem 3.8 that the containment in Question 3.1 holds for I , and as a consequence so does the inequality in Question 3.2.Examples 3.3 and 3.4 demonstrate that the obvious approach to establish the containmentin Question 3.1 may not work. However, in the remaining of the paper, we shall see thatthis containment indeed holds for generic determinantal ideals and defining ideals of starconfigurations.3.1. Star configurations.
We start by recalling the construction of a star configuration ofhypersurfaces in P N k , following [Man19] (see also [GHM13]). Definition 3.5.
Let H = { H , . . . , H n } be a collection of s > P N k .Assume that these hypersurfaces meet properly ; that is, the intersection of any k of thesehypersurfaces either is empty or has codimension k . For 1 h min { n, N } , let V h, H be theunion of the codimension h subvarieties of P N k defined by all the intersections of h of thesehypersurfaces. That is, V h, H = [ i < ···
Let I be the defining ideal of a codimension h star configuration in P N k , forsome h N . For any m, r, c > , we have I ( r ( m + h − − h + c ) ⊆ m ( r − h − c − ( I ( m ) ) r . Proof.
For a complete intersection, symbolic and ordinary powers are equal. Thus, thestatement is trivial if h = n . Assume that h < n . Let F = { F , . . . , F n } be the collection ofhomogeneous forms which defines the given star configuration. By definition, we have I = \ i
1. Thus, it follows from (3.2)that, for each i = 1 , . . . , s , we have X j ∈ E i a j > r ( m + h − − h + c. Let d j ∈ Z > be such that d j r a j < ( d j + 1) r and set a ′ j = a j − d j r r − P j ∈ E i a ′ j ( r − h ). It can be seen that X j ∈ E i d j r = X j ∈ E i ( a j − a ′ j ) > r ( m + h − − h + c − ( r − h = r ( m −
1) + c. The left hand side of this inequality is divisible by r , so we deduce that P j ∈ E i d j r > rm. Particularly, we have P j ∈ E i d j > m. onsider the system of inequalities nP j ∈ E i d j > m (cid:12)(cid:12) i = 1 , . . . , s o . By successively reduc-ing the values of d j ’s we can choose 0 d ′ j d j such that the system of inequalities (X j ∈ E i d ′ j > m (cid:12)(cid:12)(cid:12) i = 1 , . . . , s ) is still satisfied, but for at least one value of 1 ℓ s we obtain the equality P j ∈ E ℓ d ′ j = m .Set f = Q nj =1 F d ′ j rj and g = Q nj =1 F a j − d ′ j rj . Then, M = f g . Also, it follows from (3.2) that Q nj =1 F d ′ j j ∈ I ( m ) . Thus, f ∈ ( I ( m ) ) r .On the other hand, it is easy to see thatdeg g > X j ∈ E ℓ ( a j − d ′ j r )= X j ∈ E ℓ a j − ( X j ∈ E ℓ d ′ j ) r > r ( m + h − − h + c − rm = ( r − h −
1) + c − . Therefore, g ∈ m ( r − h − c − . Hence, M ∈ m ( r − h − c − ( I ( m ) ) r , and the result follows. (cid:3) As a corollary of Theorem 3.6, we can show that Demailly’s bound holds for a star con-figuration in P N k . Corollary 3.7.
Let I be the defining ideal of a codimension h star configuration in P N k , forsome h N . For any m, r ∈ N , we have I ( r ( m + h − ⊆ m r ( h − ( I ( m ) ) r . In particular, Demailly-like bound holds for defining ideals of star configurations in P N k .Proof. The first statement is a consequence of Theorem 3.6 by setting c = h . The secondstatement follows immediately from the containment I ( r ( m + h − ⊆ m r ( h − ( I ( m ) ) r , whichimplies that α ( I ( r ( m + h − ) r ( m + h − > r ( h −
1) + rα ( I ( m ) ) r ( m + h −
1) = α ( I ( m ) ) + h − m + h − , and by taking the limit as r → ∞ . (cid:3) Generic determinantal ideals.
In this subsection, we to show that the Harbourne-Huneke Containment in Question 3.1 holds for generic determinantal ideals.
Theorem 3.8 (Generic determinantal ideals) . For fixed positive integers t min { p, q } , let X be a p × q matrix of indeterminates, let R = k [ X ] , and let I = I t ( X ) denote the ideal of t -minors of X . Let h = ( p − t + 1)( q − t + 1) be the height of I in R . For all m, r > , wehave I ( r ( h + m − ⊆ m r ( h − (cid:0) I ( m ) (cid:1) r . Particularly, Demailly-like bound holds for generic determinantal ideals. roof. After possibly replacing X with its transpose, we may assume that p > q . We will usethe explicit description of the symbolic powers of the ideals of minors of generic determinantalmatrices by Eisenbud, DeConcini, and Procesi [DEP80]. We point the reader to [BV88, 10.4]for more details.Given a product δ = δ · · · δ u , where δ i is an s i -minor of X , δ ∈ I ( k ) if and only if u X i =1 max { , s i − t + 1 } > k. Moreover, I ( k ) is generated by products of this form. Fix such a product δ = δ · · · δ u ∈ I ( r ( h + m − , and set s := s + · · · + s u . Note that by the rule above, whether or not δ is in aparticular symbolic power of I is unchanged if some s i < t , so we can assume without lossof generality that all s i > t . We thus have u X i =1 ( s i − t + 1) > r ( h + m − . We know that δ ∈ (cid:0) I ( m ) (cid:1) r , by [ELS01, HH02], but let’s explicitly write δ as a multiple ofan element in ( I ( m ) ) r . First, note that to write δ as a multiple of an element in I ( m ) , it isenough to find a subset of { δ , . . . , δ u } , say δ , . . . , δ v , such that v X i =1 ( s i − t + 1) > m. If we chose δ , . . . , δ v the best way possible, meaning no δ i can be deleted, then v − X i =1 ( s i − t + 1) m − , and since δ i is a minor of X , we must have s i q . Thus v X i =1 ( s i − t + 1) ( m −
1) + ( q − t + 1) = m + q − t. By repeating this process r times, we can extract δ , . . . , δ w (perhaps after reordering) suchthat δ · · · δ w ∈ (cid:0) I ( m ) (cid:1) r and w X i =1 ( s i − t + 1) r ( m + q − t ) . As a consequence, the remaining factors in δ satisfy u X i = w +1 ( s i − t + 1) > r ( h + m − − r ( m + q − t ) = r ( h − q + t − . We will show that δ ′ = δ w +1 · · · δ u ∈ m r ( h − , which proves that δ ∈ m r ( h − I ( r ( h + m − . Inorder to do that, all we need to show is that u X i = w +1 s i > r ( h − . o we need an estimate on u − w . Note that s i q , and thus( q − t + 1)( u − w ) > r ( h − q + t − , so u − w > r ( h − q + t − q − t + 1 , and degree( δ ′ ) = u X i = w +1 s i > r ( h − q + t −
1) + ( t − · r ( h − q + t − q − t + 1 . We just want to check that this is at least r ( h − r ( h − q + t −
1) + ( t − · r ( h − q + t − q − t + 1 > r ( h − t − h − q + t − q − t + 1 > q − t, and since h = ( q − t + 1)( p − t + 1), this becomes( t − p − t ) > q − t. Whenever t >
2, this holds because p > q by assumption. When t = 1, note that I is simply m , the ideal generated by all the variables in X , and thus a complete intersection — so thereis no question that the containment in Question 3.1 is satisfied.Finally, the last statement follows from the same proof as in Corollary 3.7. (cid:3) Remark 3.9.
Theorem 3.8 also holds for determinantal ideals of symmetric matrices. Let t p be integers, let Y be a p × p symmetric matrix of indeterminates, meaning Y ij = Y ji forall 1 i, j p , and let I = I t ( Y ) be the ideal of t -minors of Y in R = k [ Y ]. By [JMnV15,Proposition 4.3], given s i -minors δ i of Y , the product δ = δ · · · δ u is in I ( k ) if and only if P ui =1 max { , s i − t +1 } > k , and I ( k ) is generated by such elements. The element δ = δ · · · δ u has once again degree s + · · · + s u , and I is a prime of height h = (cid:0) p − t +22 (cid:1) = ( p − t +1)( p − t +2)2 .Fix r, m >
1. As we did in the case of generic matrices, we start with a product δ = δ · · · δ u ∈ I ( r ( h + m − of s i -minors δ i , and pick aside some of those terms in such a way thatwe can guarantee they live in (cid:0) I ( m ) (cid:1) r ; the remaining factors, say δ ′ , satisfy u X i = w +1 ( s i − t + 1) > r ( h + m − − r ( m + p − t ) = r ( h − p + t − . As before, we need to check that the degree of this remaining factor is sufficiently large,meaning u X i = w +1 s i > r ( h − , which amounts to checking that r ( h − p + t −
1) + ( t − · r ( h − p + t − p − t + 1 > r ( h − . n this case, the inequality ( t − h − p + t − p − t + 1 > p − t becomes ( t − p − t )2 > p − t, which holds for t >
3. Note that, as before, there is nothing to show when t = 1, since I = m . We have not been able to prove the same statement for t = 2. Remark 3.10.
Theorem 3.8 furthermore holds for pfaffian ideals of skew symmetric ma-trices. Let t p be integers, let Z be a p × p skew symmetric matrix of indeterminates,meaning Z ij = − Z ji for 1 i < j p and Z ii = 0 for all i , and let I = P t ( Z ) be the2 t -pfaffian ideal of Z . That is, I is generated by the square roots of the 2 t -minors of Z . Inthis case, the symbolic powers of I are generated by products of pfaffians; given 2 s i -pfaffians δ i , [DN96, Theorem 2.1] and [JMnV15, Proposition 4.5] tell us that δ · · · δ u ∈ I ( k ) if andonly if P ui =1 max { , s i − t + 1 } > k and s i (cid:4) p (cid:5) . Note also that a 2 s i -pfaffian δ i has degree s i , and the height of I is now h = ( p − t +1)( p − t +2)2 .Fix r, m >
1. This time, we start with a product δ = δ · · · δ u ∈ I ( r ( h + m − of 2 s i -pfaffians δ i , and after factoring out an element of (cid:0) I ( m ) (cid:1) r , the remaining pfaffians satisfy u X i = w +1 ( s i − t +1) > r ( h + m − − r (cid:16) m + p − t (cid:17) = r (cid:16) h − p t − (cid:17) = r ( p − t + 2)( p − t )2 . Unlike the case for symmetric matrices in Remark 3.9, in this case we get 2 s i p for all i .Particularly, it follows that (cid:16) p − t + 1 (cid:17) ( u − w ) > r ( p − t + 2)( p − t )2 . This implies that u − w > r ( p − t ) . Hence, in order to verify that u X i = w +1 s i > r ( h −
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Journalof Algebra , 176(1):182–209, 1995. ulane University, Department of Mathematics, 6823 St. Charles Ave., New Orleans,LA 70118, USA E-mail address : [email protected] University of California at Riverside, Department of Mathematics, 900 University Ave.,Riverside, CA 92521, USA
E-mail address : [email protected] Tulane University, Department of Mathematics, 6823 St. Charles Ave., New Orleans,LA 70118, USA
E-mail address : [email protected] Tulane University, Department of Mathematics, 6823 St. Charles Ave., New Orleans,LA 70118, USA
E-mail address : [email protected]@tulane.edu