Chudnovsky's Conjecture and the stable Harbourne-Huneke containment
Sankhaneel Bisui, Eloísa Grifo, Huy Tài Hà, Thái Thành Nguyên
aa r X i v : . [ m a t h . A C ] J un CHUDNOVSKY’S CONJECTURE AND THE STABLEHARBOURNE–HUNEKE CONTAINMENT
SANKHANEEL BISUI, ELO´ISA GRIFO, HUY T `AI H `A, AND TH ´AI TH `ANH NGUY˜ˆEN
Abstract.
We investigate containment statements between symbolic and ordinary powersand bounds on the Waldschmidt constant of defining ideals of points in projective spaces.We establish the stable Harbourne conjecture for the defining ideal of a general set of points.We also prove Chudnovsky’s Conjecture and the stable version of the Harbourne–Hunekecontainment conjectures for a general set of sufficiently many points. Introduction
Chudnovsky’s Conjecture suggests a lower bound for the answer to the following funda-mental question: given a set of distinct points X ⊆ P N k , where k is a field, and an integer m > , what is the least degree of a homogeneous polynomial in the homogeneous coordinatering k [ P N k ] that vanishes at each point in X of order at least m ? For a homogeneous ideal I ⊆ k [ P N k ], let α ( I ) be the least generating degree of I and let I ( m ) denote its m -th symbolicpower. By the Zariski–Nagata Theorem [Zar49, Nag62, EH79] (cf. [DDSG + k is perfect, the answer is α ( I ( m ) X ), where I X ⊆ k [ P N k ] is the defining ideal of X .When k = C , the following is equivalent to Chudnovsky’s Conjecture [Chu81]. Conjecture 1.1 (Chudnovsky) . Let I ⊆ k [ P N k ] be the defining ideal of a set of points in P N k . For any m ∈ N , α ( I ( m ) ) m > α ( I ) + N − N .
Chudnovsky himself proved the bound for any set of points in P C (see also [HH13]). Esnaultand Viehweg [EV83] showed moreover that for any set of points in P N C , α ( I ( m ) ) m > α ( I ) + 1 N .
These results extended the previous bound
N α ( I ( m ) ) > mα ( I ) by Waldschmidt and Skoda[Wal77, Sko77]. Decades later, Ein, Lazarsfeld and Smith [ELS01] and Hochster and Huneke[HH02] showed that I ( Nm ) ⊆ I m for all m ∈ N . More generally, given any radical ideal I of big height h in a regular ring, I ( hm ) ⊆ I m for all m ∈ N (see [MS17] for the mixedcharacteristic case). This particularly implies the Waldschmidt and Skoda degree bound.The containment result also gives more detailed information about the polynomials thatvanish to order m along X .In an effort to improve the containment between symbolic and ordinary powers, Harbourneand Huneke [HH13] conjectured that the defining ideal I ⊆ k [ P N k ] of any set of points in P N k Mathematics Subject Classification.
Key words and phrases.
Chudnovsky’s conjecture, Waldschmidt constant, ideals of points, symbolic pow-ers, containment problem, Stable Harbourne Conjecture. atisfies stronger containment, namely, I ( Nm ) ⊆ m m ( N − I m and I ( Nm − N +1) ⊆ m ( m − N − I m , for all m >
1, where m denotes the graded irrelevant ideal. Chudnovsky’s Conjecture, infact, follows as a corollary to the stable version of this conjecture, meaning it is enough tostudy the case when m ≫
0. More generally, we will study stable versions of the conjecturesof Harbourne and Huneke [HH13, Conjectures 2.1 and 4.1].
Conjecture 1.2 (Stable Harbourne–Huneke Containment) . Let I ⊆ k [ P N k ] be a homoge-neous radical ideal of big height h . For any r ≫
0, we have(1) I ( hr ) ⊆ m r ( h − I r and (2) I ( hr − h +1) ⊆ m ( r − h − I r . The Stable Harbourne–Huneke Containment enables us to obtain lower bounds on thedegrees of all symbolic powers by studying I ( m ) only for m ≫
0. By taking this approach,we prove Chudnovsky’s Conjecture for any set of sufficiently many general points in P N k . Theorems 4.6 and 5.1.
Suppose that char k = 0 and N > s > N then the stable containment I ( Nr ) ⊆ m r ( N − I r , for r ≫
0, holds when I isthe defining ideal of a general set of s points in P N k .(2) If s > (cid:0) N + NN (cid:1) then the stable containment I ( Nr − N +1) ⊆ m ( r − N − I r , for r ≫ I defines a general set of s points in P N k .In particular, Chudnovsky’s Conjecture holds for a general set of s > N points in P N k .Previously, Chudnovsky’s Conjecture had been shown for a general set of points in P k [Dum15], a set of at most N + 1 points in generic position in P N k [Dum15], a set of a binomialcoefficient number of points forming a star configuration [BH10, GHM13], a set of points in P N k lying on a quadric [FMX18], and a very general set of points in P N k [DTG17, FMX18].By saying that the conjecture holds for a very general set of points in P N k , we mean thatthere exist infinitely many open dense subsets U m , m ∈ N , of the Hilbert scheme of s pointsin P N k such that Chudnovsky’s Conjecture holds for all X ∈ T ∞ m =1 U m . We remove this infiniteintersection of open dense subsets to show that there exists one open dense subset U of theHilbert scheme of s points in P N k such that Chudnovsky’s Conjecture holds for all X ∈ U .In a different approach to tightening the containment I ( hm ) ⊆ I m , Harbourne’s Conjec-ture [BDRH +
09, Conjecture 8.4.2] asks if I ( hm − h +1) ⊆ I m holds for all m ∈ N . Whilethis conjecture can fail for various choices of m and I [DSTG13, HS15, BDRH +
16, MS18,CGM +
16, Dra17, DS20], the stable version of this conjecture, known as the
Stable HarbourneConjecture [Gri20, Conjecture 2.1], remains open.
Conjecture 1.3 (Stable Harbourne Conjecture) . Let I ⊆ k [ P N k ] be a radical ideal of bigheight h . For all r ≫
0, we have I ( hr − h +1) ⊆ I r . Adding more evidence towards the Stable Harbourne Conjecture [Gri20, GHM19, TX19],we prove this conjecture for any general set of points in P N k , by verifying a stronger conjectureon the resurgence of I , a notion that we shall describe in more details in Section 2. Theorem 4.2 and Corollary 4.3.
Suppose that char k = 0. The defining ideal I of ageneral set of points in P N k has expected resurgence. In particular, I satisfies the StableHarbourne Conjecture, and for any C , I ( rN − C ) ⊆ I r for all r ≫ . ur method in proving Theorems 4.2, 4.6 and, subsequently, Theorem 5.1 is to examinewhen we can obtain stable containment from one containment, and applying specialization .Techniques of specialization were already used in [FMX18] to establish Chudnovsky’s Con-jecture for a very general set of points. The main obstacle they faced in going from a verygeneral set of points to a general set of points is that specialization works for each m toproduce an open dense subset U m in the Hilbert scheme of s points in P N k where the corre-sponding containment holds, and to get a statement that holds for all m , one would needto take the intersection T ∞ m =1 U m . To overcome this difficulty and to apply specializationtechniques also to the stable containment problem, the novelty in our method is to applyspecialization for only one power and then show that one appropriate containment wouldlead to the stable containment. For example, to show Chudnovsky’s Conjecture for generalpoints, we show the following result: Corollary 3.2.
Suppose that for some value c ∈ N , I ( hc − h ) ⊆ m c ( h − I c . For all m ≫ I ( hm − h ) ⊆ m m ( h − I m . In fact, we prove a much more general statement (see Theorem 3.1 and the other results inSection 3). This gives a strategy to attack Chudnovsky’s Conjecture, and more generally theStable Harbourne–Huneke Containment, in other settings: rather than showing the contain-ment holds for all m large enough, it is sufficient to find one m for which the containmentholds. We use this idea in Theorems 4.2 and 4.6, by explicitly proving that one containmentis satisfied. This is achieved by using specialization to pass the problem to the generic pointsin P N k ( z ) , where the situation is better understood.Finally, note that Chudnovsky’s inequality, Harbourne’s conjecture and the Habourne–Huneke containment conjectures have all been proved for squarefree monomial ideals in[CEHH17]. For more related problems and questions, the interested reader is referred to, forinstance, [CHHVT20, DDSG + Acknowledgements.
The authors thank Grzegorz Malara and Halszka Tutaj-Gasi´nska fortheir suggestion to look at B¨or¨oczky configurations, and Alexandra Seceleanu for pointingto [DHN + . Preliminaries
In this section, we shall recall notations, terminology and results that will be used often inthe paper. Throughout the paper, let k be a field, and let k [ P N k ] represent the homogeneouscoordinate ring of P N k . Since most of the statements in this paper become rather straightfor-ward when N = 1, we shall assume that N >
2. Our work evolves around symbolic powersof ideals, so let us start with the definition of these objects.
Definition 2.1.
Let S be a commutative ring and let I ⊆ S be an ideal. For m ∈ N , the m -th symbolic power of I is defined to be I ( m ) = \ p ∈ Ass( I ) ( I m S p ∩ S ) . For an ideal I , the big height of I is defined to be the maximum height of a minimalassociated prime of I . The following result of Johnson [Joh14] is a generalization of theaforementioned containment of Ein–Lazarsfeld–Smith, Hochster–Huneke and Ma–Schwede[ELS01, HH02, MS17]. Theorem 2.2 (Johnson [Joh14]) . Let I ⊆ k [ P N k ] be a radical ideal of big height h . For all k > and a , . . . , a k > , we have I ( hk + a + ··· + a k ) ⊆ I ( a +1) · · · I ( a k +1) . Another result that we shall make use of is the following stable containment of the secondauthor of the present paper [Gri20, Theorem 2.5].
Theorem 2.3 (Grifo) . Let I ⊆ k [ P N k ] be a radical ideal of big height h . If I ( hc − h ) ⊆ I c forsome constant c ∈ N then for all r ≫ , we have I ( hr − h ) ⊆ I r . More concretely, this containment holds for all r > hc . As mentioned before, for a homogeneous ideal I ⊆ k [ P N k ], α ( I ) denotes its least generatingdegree. The Waldschmidt constant of I is defined to be b α ( I ) := lim m →∞ α ( I ( m ) ) m . The Waldschmidt constant of an ideal I is known to exist and, furthermore, we have b α ( I ) = inf m ∈ N α ( I ( m ) ) m . See, for example, [BH10, Lemma 2.3.1]. Hence, Chudnovsky’s Conjecture can be restated asin the following equivalent statement:
Conjecture 2.4 (Chudnovsky) . 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 . Then, b α ( I ) > α ( I ) + N − N . ymbolic powers are particularly easier to understand for the defining ideals of points.Let X = { P , . . . , P s } ⊆ P N k be a set of s > p i ⊆ k [ P N k ] be the definingideal of P i and let I X = p ∩ · · · ∩ p s be the defining ideal of X . Then, for all m ∈ N , it is awell-known fact that I ( m ) X = p m ∩ · · · ∩ p ms . Remark 2.5.
The set of all collections of s not necessarily distinct points in P N k is parame-terized by the Chow variety G (1 , s, N + 1) of 0-cycles of degree s in P N k (cf. [GKZ94]). Thus,a property P is said to hold for a general set of s points in P N k if there exists an open densesubset U ⊆ G (1 , s, N + 1) such that P holds for any X ∈ U .Let ( z ij ) i s, j N be s ( N + 1) new indeterminates. We shall use z and a to denote thecollections ( z ij ) i s, j N and ( a ij ) i s, j N , respectively. 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 often referred to as the set of s generic points in P N k ( z ) . For any a ∈ A s ( N +1) k ,let P i ( a ) and X ( a ) be obtained from P i ( z ) and X ( z ), respectively, by setting z ij = a ij forall i, j . 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 followingresult allows us to focus on open dense subsets of A s ( N +1) k when discussing general sets ofpoints in P N k . Lemma 2.6 ([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 . Another related notion about sets of points is that of being in generic position or, equiv-alently, having the generic (i.e., maximal) Hilbert function. For a set X ⊆ P N k with definingideal I X ⊆ k [ P N k ], let H ( X , t ) denote its Hilbert function. That is, H ( X , t ) = dim k (cid:0) k [ P N k ] /I X (cid:1) t . Definition 2.7.
Let X ⊆ P N k be a collection of s points. Then, X is said to be in genericposition (or equivalently, to have the generic Hilbert function) if for all t ∈ N , we have H ( X , t ) = min (cid:26) s, (cid:18) t + NN (cid:19)(cid:27) . The condition of being in generic position is an open condition so, particularly, the set ofgeneric points in P N k ( z ) and a general set of points in P N k are both in generic position.One of our main techniques is specialization. We recall this construction following [Kru48]. Definition 2.8 (Krull) . Let x represent the coordinates x , . . . , x N of P N k . Let a ∈ A s ( N +1) .The specialization at a is a map π a from the set of ideals in k ( z )[ x ] to the set of ideals in k [ x ], defined by π a ( I ) := { f ( a , x ) (cid:12)(cid:12) f ( z , x ) ∈ I ∩ k [ z , x ] } . Remark 2.9.
Let p i ( z ) and p i ( a ) be the defining ideals of P i ( z ) ∈ P N k ( z ) and P i ( a ) ∈ P N k ,respectively. It follows from [Kru48, Satz 1] that there exists an open dense subset W ⊆ ⊆ A s ( N +1) such that, for all a ∈ W and any 1 i s , we have π a ( p i ( z )) = p i ( a ) . We shall always assume that a ∈ W whenever we discuss specialization in this paper. Remark 2.10.
Observe that, by the definition and by [Kru48, Satz 2 and 3] (see also [NT99,Propositions 3.2 and 3.6]), for fixed m, r, t ∈ N , there exists an open dense subset U m,r,t ⊆ W such that for all a ∈ U m,r,t , we have π a (cid:0) I ( z ) ( m ) (cid:1) = I ( a ) ( m ) and π a (cid:0) m t z I ( z ) r (cid:1) = m t I ( a ) r . Here, we use m and m z to denote the maximal homogeneous ideals of k [ x ] and k ( z )[ x ],respectively. Note that m z is the extension of m in k ( z )[ x ]. We shall make use of this factoften.Finally, one invariant that plays an important role in the study of the containment problemis the resurgence, introduced by Bocci and Harbourne [BH10]. Definition 2.11 (Resurgence) . The resurgence of the ideal I is given by ρ ( I ) = sup n ab (cid:12)(cid:12)(cid:12) I ( a ) I b o . The resurgence always satisfies ρ ( I ) >
1, and over a regular ring, the resurgence of aradical ideal is always at most the big height h . As noted in [Gri20, Remark 2.7], the StableHarbourne Conjecture follows immediately whenever ρ ( I ) < h . In that case, we say that I has expected resurgence [GHM19]. Note that expected resurgence implies more than StableHarbourne; in fact, given any integer C , if I has expected resurgence then I ( hr − C ) ⊆ I r forall r ≫ From one containment to a stable containment
In this section, we prove our first main result, which establishes the stable containmentfrom one containment. Our theorem is in the same spirit as that of Theorem 2.3, but thecontainments we now study include an appropriate power of the maximal ideal on the righthand side. This result also forms an essential step in proving Chudnovsky’s Conjecture andthe stable Harbourne–Huneke containment for a general set of points, which we will coverin later sections.
Theorem 3.1.
Let ℓ ( x, y ) = ax + by + d ∈ Z [ x, y ] be a linear form. Let I ⊆ k [ P N k ] be anideal of big height h . Suppose that for some value c ∈ N , I ( hc − h ) ⊆ m hc − h − ℓ ( h,c ) I c . Then, forall r ≫ , provided that α ( I ) > h ( a + 1) + hd + 1 , we have I ( hr − h ) ⊆ m hr − h − ℓ ( h,r ) I r . Proof.
For any r ∈ N , write r = qhc + t , where q ∈ N and 0 t hc −
1. Applying Theorem2.2 for k = qh + q + t − a = · · · = a qh = hc − h − a qh +1 = · · · = a k = 0, we have I ( hr − h ) ⊆ (cid:0) I ( hc − h ) (cid:1) qh I q + t − ⊆ (cid:0) m hc − h − ℓ ( h,c ) I c (cid:1) qh I q + t − = m ( hc − h − ℓ ( h,c )) qh I q − I qhc + t ⊆ m ( hc − h − ℓ ( h,c )) qh m α ( I )( q − I r . hus, it suffices to prove that, for α ( I ) > h ( a + 1) + hd + 1 and r ≫ hc − h − ℓ ( h, c )) qh + α ( I )( q − > hr − h − ℓ ( h, r ) . This is equivalent to showing that α ( I )( q − − q [ h ( a + 1) + hd ] > ht − h − ah − bt − d. Note that the right hand side is bounded, so the inequality holds for α ( I ) > h ( a +1)+ hd +1and q ≫ (cid:3) Particular interesting consequences of Theorem 3.1 include the following choices of ℓ ( x, y ). Corollary 3.2.
Let b ∈ Z . Suppose that for some value c ∈ N , I ( hc − h ) ⊆ m c ( h − b ) I c . For all r ≫ , we have I ( hr − h ) ⊆ m r ( h − b ) I r . Proof.
Pick ℓ ( x, y ) = − x + by . Note that, in this case, the bound α ( I ) > h ( a + 1) + hd + 1in Theorem 3.1 is α ( I ) >
1, which is trivially satisfied. The assertion now follows directlyfrom Theorem 3.1. (cid:3)
Corollary 3.3.
Let b ∈ Z . Suppose that for some value c ∈ N , I ( hc − h ) ⊆ m ( c − h − b ) I c . Forall r ≫ and α ( I ) > h − hb + 1 , we have I ( hr − h ) ⊆ m ( r − h − b ) I r . Proof.
Pick ℓ ( x, y ) = by − b . The assertion follows from Theorem 3.1. (cid:3) Corollary 3.4.
Suppose that char k = 0 . If for some value c ∈ N and α ( I ) > h + 1 , I ( hc − h ) ⊆ m hc − h − c I c then for all r ≫ , we have I ( hr − h +1) ⊆ m ( r − h − I r . Proof.
By applying Theorem 3.1 with ℓ ( x, y ) = y , it follows that for all r ≫ α ( I ) > h + 1, we have I ( hr − h ) ⊆ m hr − h − r I r . Moreover, since char k = 0, we have I ( hr − h +1) ⊆ m I ( hr − h ) . The assertion now follows. (cid:3) Theorem 3.1 allows us to establish the stable containment for special configurations ofpoints for which the containment in low powers is known to fail.
Example 3.5 (Fermat configurations) . The ideal I = ( x ( y n − z n ) , y ( z n − x n ) , z ( x n − y n ))in k [ x, y, z ] corresponds to a Fermat configuration of n + 3 points in P , where k is a field ofcharacteristic not 2 and containing n > n -th roots of unity; see [Szp19] for moreon Fermat configurations. This was the first class found of ideals that fail I ( hn − h +1) ⊆ I n forsome n — more precisely, these ideals fail I (3) ⊆ I [DSTG13, HS15]. Using Corollary 3.2with b = 1, we can now establish the stable Harbourne–Huneke containment for these ideals: • I (2 r − ⊆ m r − I r for r ≫
0, and • I (2 r ) ⊆ m r I r for r ≫ I (2 r − ⊆ m r I r for all r ≫ +
15, Theorem 2.1], the resurgence of any of the Fermat ideals I above is ρ ( I ) = . his guarantees that I (2 c − ⊆ I c whenever c − c > or, equivalently, c >
5. Moreover, notethat I is generated by elements of degree n + 1, and for all m > I ( m ) = ( x n − y n , y n − z n ) m ∩ ( x, y ) m ∩ ( x, z ) m ∩ ( y, z ) m . In particular, α ( I (10) ) > α (cid:0) ( x n − y n , y n − z n ) (cid:1) = 10 n. Notice that α ( I ) = ω ( I ) = n + 1, where ω ( I ) is the maximal degree of a generator of I .Since n >
3, we have 10 n > n + 12 (which is the generating degree of m I ), and thus I (10) ⊆ m I . Hence, by Corollary 3.2, I (2 r − ⊆ m r I r for all r ≫ Example 3.6 (B¨or¨oczky configuration B ) . Let I be the defining ideal of the B¨or¨oczkyconfiguration B of 19 triple points in P , as described in [DHN +
15, Figure 1]. By [DHN + α ( I ) = ω ( I ) = 5 and ρ ( I ) = 3 /
2, which as before implies that I (2 c − ⊆ I c for c >
5. Moreover, in the proof of [DHN +
15, Theorem 2.2], it is shown that α ( I ( m ) ) > m for all m , which in particular implies that α ( I (8) ) >
32. Since ω ( I ) = 5 · ω ( I ) = 25, weconclude that I (8) ⊆ m I , and by Corollary 3.2 this guarantees that I (2 r − ⊆ m r I r for all r ≫
0. Once again, this establishes the stable Harbourne–Huneke containment for I : • I (2 r − ⊆ m r − I r for r ≫
0, and • I (2 r ) ⊆ m r I r for r ≫ Example 3.7.
Any finite group G generated by pseudoreflections determines an arrange-ment A ⊆ C rank( G ) of hyperplanes, where each hyperplane is fixed pointwise by one of thepseudoreflections in G . Given such an arrangement A , it turns out that the symbolic powersof the radical ideal I = J ( A ) determining the singular locus of A are very interesting. In fact,several of the counterexamples to Harbourne’s Conjecture that appear in the literature, suchas the Fermat [DSTG13, HS15], Klein and Wiman configurations [Kle78, Wim96, Sec15],turn out to arise in this form. Drabkin and Seceleanu studied this class of ideals [DS20], andin particular completely classified which I among these satisfy the containment I (3) ⊆ I .Fix such an ideal I , which has pure height 2, and assume that G is an irreducible reflectiongroup of rank three. By [DS20, Proposition 6.3], I (2 r − ⊆ I r for all r >
3. We claim that infact I (2 r − ⊆ m r I r for r ≫
0. To check that, we can make small adaptations to the proof of[DS20, Proposition 6.3]. First, note that the case when G = G ( m, m,
3) leads to the Fermatconfigurations, and we have shown our claim in Example 3.5. Otherwise, note that is enoughto show that α ( I (2 r − ) > reg( I r ) + r for r ≫
0, using [BH10, Lemma 2.3.4]. On the onehand, by the proof of [DS20, Proposition 6.3], α ( I (2 r − ) > (2 r − α ( I ) − r − I r ) = ( r + 1) α ( I ) − r >
2, by [NS16, Theorem 2.5]. We are thus donewhenever (2 r − α ( I ) − r − > ( r + 1) α ( I ) − r, or equivalently, α ( I ) > r − . As in [DS20, Proposition 6.3], this holds for r >
10 as long as
G / ∈ { A , D , B , G (3 , , , G } ,since in that case α ( I ) > G = A , I (8) ⊆ m I .(2) When G = D , I (8) ⊆ m I .
3) When G = B , I (4) ⊆ m I .(4) When G = G (3 , , I (4) ⊆ m I .(5) When G = G , I (4) ⊆ m I .By Corollary 3.2, we conclude that I (2 r − ⊆ m r I r for r ≫
0, which gives us the stableHarbourne–Huneke containment for I : • I (2 r − ⊆ m r − I r for r ≫
0, and • I (2 r ) ⊆ m r I r for r ≫ I (2 r − ⊆ m r I r for r ≫
0, and as we will see in Proposition5.3, that implies b α ( I ) > α ( I )+12 .4. Stable Harbourne and Harbourne–Huneke Containment for Points
In this section, we establish the stable Harbourne conjecture and the stable Harbourne–Huneke containment for a general set of points. We shall begin with the stable Harbourneconjecture, Conjecture 1.3. For the simplicity of statements, throughout the section, wemake the assumption that k has characteristic 0.We will show that general sets of points have expected resurgence. To prove that, we firstprove the following containment for generic sets of points. Lemma 4.1.
Assume that N > . Let I ( z ) be the defining ideal of s generic points in P N k ( z ) .For all r ≫ , we have I ( z ) ( Nr − N +1) ⊆ m I ( z ) r . Proof.
For simplicity of notation, we shall write I for I ( z ) in this proof. Let d be such that (cid:18) N + d − N (cid:19) s < (cid:18) N + dN (cid:19) . Then, α ( I ) = d and reg( I ) d + 1, by [MN01, Lemma 5.8] and [GM84, Corollary 1.6]. Now,by [TX19, Remark 2.3], we have I ( Nr − N +1) ⊆ I r for r ≫ . We will prove that α ( I ( Nr − N +1) ) > r reg( I ) + 1 , which will imply the asserted containment. It follows from [FMX18, Theorem 2.7] that α ( I ( Nr − N +1) ) > ( N r − N + 1) α ( I ) + N − N = ( N r − N + 1)( d + N − N .
We claim that if r > N + N ( N − d + N − N ( N − then( N r − N + 1) ( d + N − N > r ( d + 1) + 1 > r reg( I ) + 1 . uppose not. Then, ( N r − N + 1) ( d + N − N < r ( d + 1) + 1( N r − N + 1)( d + N − < rN ( d + 1) + N ( N − rN − ( N − d + N − < Nr < N + N ( N − d + N − N ( N − . This contradicts our assumption on r , and thus α ( I ( Nr − N +1) ) > r reg( I ) + 1, resulting in thecontainment, I ( Nr − N +1) ⊆ m I r for all r sufficiently large. (cid:3) Next we prove that an ideal defining a set of general points has expected resurgence, whichis both interesting in its own right and while we will also use to prove that general sets ofpoints satisfy the Stable Harbourne Conjecture.
Theorem 4.2.
Suppose that N > . Then the defining ideal of a general set of points in P N k has expected resurgence.Proof. Let I ( z ) be the defining ideal of s generic points in P N k ( z ) . By Lemma 4.1, there is aconstant c such that, I ( z ) ( Nc − N +1) ⊆ m I ( z ) c . It follows from [Kru48, Satz 2 and 3] that there exists an open dense subset U ⊆ A s ( N +1) such that for all a ∈ U , π a ( I ( z ) ( Nc − N ) ) = I ( a ) ( Nc − N ) and π a ( I ( z ) c ) = I ( a ) c . Thus, for all a ∈ U , we have I ( a ) ( Nc − N +1) ⊆ m I ( a ) c Hence, by [GHM19, Theorem 3.3], ρ ( I ( a )) < N . (cid:3) As a consequence, we conclude that a general set of points satisfies the Stable HarbourneConjecture and, particularly, we obtain the following stable containment.
Corollary 4.3.
Assume that N > . Let I denote either the ideal of s generic points in P N k ( z ) or the ideal of s general points in P N . For a given integer C and all r ≫ , we have I ( rN − C ) ⊆ I r . Proof.
In the case of generic points, Lemma 4.1 gives c such that I ( cN − N +1) ⊆ m I c , whichby [GHM19, Theorem 3.3] implies ρ ( I ) < N . In the case of general points, Theorem 4.2says that ρ ( I ) < N . The stable containment I ( rN − C ) ⊂ I r follows immediately (cf. [Gri20,Remark 2.7]). (cid:3) We shall now continue to establish the stable Harbourne–Huneke containment in Con-jecture 1.2. We shall need a few lemmas, whose underlying ideas come from those used in[DTG17, Lemma 3 and Theorem 4] (see also [MSS18, Lemma 3.1]). emma 4.4. The inequality k N (cid:18) N k − N − N (cid:19) holds in the following cases:(1) k > and N > ,(2) k > and N > , and(3) k > and N > .Proof. (1) The assertion is equivalent to( kN − N − . . . ( kN − N ) > k N N ! , i.e., (cid:18) N − N + 1 k (cid:19) . . . (cid:18) N − Nk (cid:19) > N ! . Since the left hand side increases with k , to prove (1) it suffices to establish the inequalitywhen k = 5. Equivalently, we need to show that (cid:0) N − N (cid:1) > N .Notice also that for N >
3, we have (cid:0) N +1) − N +1 (cid:1)(cid:0) N − N (cid:1) = (4 N + 3)(4 N + 2)(4 N + 1)(4 N )( N + 1)(3 N + 2)(3 N + 1)(3 N ) > (4 N + 3)(4 N )( N + 1)(3 N ) > . Thus, it is enough to show that (cid:0) N − N (cid:1) > N for N = 3, which is simply 165 = (cid:0) (cid:1) >
125 = 5 .(2) By the same line of arguments, it is enough to show that (cid:0) N − N (cid:1) > N for k > N >
4. This is indeed true since (cid:0) (cid:1) > and (cid:0) N +1) − N +1) (cid:1)(cid:0) N − N (cid:1) = (3 N + 2)(3 N + 1)(3 N )( N + 1)(2 N + 1)(2 N ) = 3(3 N + 2)(3 N + 1)2( N + 1)(2 N + 1) > . (3) Again, by the same line of arguments, it suffices to show that (cid:0) N − N (cid:1) > N for k > N >
9. This holds since (cid:0) (cid:1) > and (cid:0) N +1) − N +1) (cid:1)(cid:0) N − N (cid:1) = (2 N + 1)(2 N )( N + 1) N = 2(2 N + 1) N + 1 > . (cid:3) Lemma 4.5.
Let I ( z ) be the defining ideal of s generic points in P N k ( z ) . Suppose also thatone of the following holds:(1) N > and s > N ,(2) N > and s > N , or(3) N > and s > N .For r ≫ , we have I ( z ) ( Nr − N ) ⊆ m r ( N − z I ( z ) r . roof. For simplicity of notation, we shall write I for I ( z ) in this proof. Let k and d beintegers such that ( k − N s < k N and (cid:0) d + N − N (cid:1) < s (cid:0) d + NN (cid:1) .By [DTG17, Theorem 2], the Waldschmidt constant of a very general set of s points in P N k isbounded below by ⌊ N √ s ⌋ . This, by [FMX18, Theorem 2.4], implies that b α ( I ) > ⌊ N √ s ⌋ = k − I ) = d + 1.By the same proof as the one given in [DTG17, Theorem 4], we get N ( k − > N − d + 1) = N + d. (4.1)We claim that (4.1) is a strict inequality. Indeed, if N ( k −
1) = N + d then d = N k − N .This implies that (cid:0) ( Nk − N )+ N − N (cid:1) < s < k N . That is, (cid:0) Nk − N − N (cid:1) < k N , a contradiction toLemma 4.4. Thus, N ( k − > N + d . Particularly, we get N b α ( I ) > N − I ) . Thus, for r ≫
0, we have N b α ( I ) > rr − N − I )) . It follows that, for r ≫ α ( I ( Nr − N ) ) > ( N r − N ) b α ( I ) = ( r − N b α ( I ) > r ( N − I )) . (4.2)Now, by Corollary 4.3 and Theorem 2.3, we have that for r ≫ I ( Nr − N ) ⊆ I r . This,together with (4.2), implies that, for r ≫ I ( Nr − N ) ⊆ m r ( N − z I r . (cid:3) The stable Harbourne–Huneke containment for a general set of points in P k was alreadyproved in [HH13, Proposition 3.10]. We shall now establish a stronger version of the stableHarbourne–Huneke containment for a general set of sufficiently many points in P N k when N > Theorem 4.6.
Suppose that N > .(1) If s > N then the stable containment I ( Nr ) ⊆ m r ( N − I r , for r ≫ , holds when I isthe defining ideal of a general set of s points in P N k .(2) If s > (cid:0) N + NN (cid:1) then the stable containment I ( Nr − N +1) ⊆ m ( r − N − I r , for r ≫ ,holds when I defines a general set of s points in P N k .Proof. Let I ( z ) be the defining ideal of the set of s > N generic points in P N k . By Lemma4.5, there exists a constant c ∈ N such that I ( z ) ( Nc − N ) ⊆ m c ( N − z I ( z ) c . By [Kru48, Satz 2 and 3], there exists an open dense subset U ⊆ A s ( N +1) such that for all a ∈ U , π a ( I ( z )) ( Nc − N ) = I ( a ) ( Nc − N ) , π a ( I ( z ) c ) = I ( a ) c and π a ( m c ( N − z ) = m c ( N − . Thus, for all a ∈ U , we have I ( a ) ( Nc − N ) ⊆ m c ( N − I ( a ) c . (4.3)Applying Corollary 3.2 with b = 1 to (4.3), we get that, for a ∈ U and r ≫ I ( a ) ( Nr − N ) ⊆ m r ( N − I ( a ) r . his proves (1).Now, suppose that s > (cid:0) N + NN (cid:1) . This also gives that s > N since for N > (cid:0) N + NN (cid:1) > N . By [GO82, Proposition 6], α ( I ( a )) is the least integer d such that (cid:0) d + NN (cid:1) > s .This implies that α ( I ( a )) > N + 1. Furthermore, it follows from (4.3) that, for all a ∈ U , I ( a ) ( Nc − N ) ⊆ m cN − c − N I ( a ) c . Therefore, since α ( I ( a )) > N + 1, Corollary 3.4 applies to give I ( a ) ( Nr − N +1) ⊆ m ( r − N − I r for all a ∈ U and r ≫
0. This proves (2) and, hence, the theorem. (cid:3)
Remark 4.7.
By Lemma 4.5 and the same proof as that of Theorem 4.6, the stableHarbourne–Huneke containment I ( Nr ) ⊆ m r ( N − I r , for r ≫
0, is true for s > N generalpoints when N > s > N general points when N > Remark 4.8.
To prove Theorem 4.6, we in fact establish a stronger containment, namely, I ( Nr − N ) ⊆ m r ( N − I r for r ≫ , (4.4)when I is the defining ideal of a general set of (sufficiently many) points. The conditionthat we have a general set of points is necessary. The containment (4.4) is not true for anarbitrary set of points. For example, if I is the defining ideal of (cid:0) N + dN (cid:1) points forming a starconfiguration (see, for example, [GHM13] for more details on star configurations), then by[GHM13, Corollary 4.6], we have α ( I ( Nr − N ) ) = ( r − N + d ) < α ( m r ( N − I r ) = r ( N + d )for r ≫
0, and so the containment (4.4) fails. Note that the stable Harbourne-HunekeConjectures still hold for a star configuration.5.
Chudnovsky’s Conjecture
In this last section of the paper, we prove Chudnovsky’s Conjecture for a general set of atleast 4 N points in P N k . For an arbitrary number of points, we also show a weaker version ofChudnovsky’s Conjecture. We again make an umbrella assumption, throughout the section,that k has characteristic 0. Theorem 5.1.
Suppose that s > N . Chudnovsky’s Conjecture holds for a general set of s points in P N k .Proof. Chudnovsky’s Conjecture is known to hold for any set of points in P k (cf. [Chu81,HH13]). Suppose that N >
3. Let I be the defining ideal of a general set of points in P N k .By Theorem 4.6, for r ≫
0, we have I ( Nr ) ⊆ m r ( N − I r . This implies that α ( I ( Nr ) ) > r ( N −
1) + rα ( I ). Thus, α ( I ( Nr ) ) N r > α ( I ) + N − N .
By letting r → ∞ , we get b α ( I ) > α ( I ) + N − N .
The conjecture is proved, noting that b α ( I ) α ( I ( m ) ) m for all m ∈ N . (cid:3) emark 5.2. Following Remark 4.7, Chudnovsky’s Conjecture holds for s > N generalpoints when N > s > N general points when N > Proposition 5.3.
Let I be any ideal of big height h . Suppose that for some constant c ∈ N ,we have I ( hc − h +1) ⊆ m c ( h − I c . Then, b α ( I ) > α ( I ) + h − h . Proof.
We claim that for all t > I ( hct ) ⊆ m ( h − ct I ct . (5.1)This implies that α (cid:0) I ( hct ) (cid:1) > α (cid:0) m ( h − ct I ct (cid:1) = ct ( h −
1) + α ( I ) ct. Therefore, b α ( I ) = lim m →∞ α (cid:0) I ( m ) (cid:1) m = lim t →∞ α (cid:0) I ( hct ) (cid:1) hct > α ( I ) + h − h . Now we show (5.1): I ( hct ) = I ( ht + t ( hc − h )) ⊆ (cid:0) I ( hc − h +1) (cid:1) t by Theorem 2.2 ⊆ (cid:0) m ( h − c I c (cid:1) t by the hypothesis= m ( h − ct I ct . The result is proved. (cid:3)
Example 5.4 (Fermat configurations revisited) . As we saw in Example 3.5, the ideals I = ( x ( y n − z n ) , y ( z n − x n ) , z ( x n − y n )) in k [ x, y, z ] corresponding to a Fermat configurationof n + 3 points in P satisfies I (2 r − ⊆ I r for all r ≫
0. Similarly to what we did in Example3.5, one can show that I (5) ⊆ m I , which by Proposition 5.3 implies that b α ( I ) > n +12 . Infact, by [DHN +
15, Theorem 2.1], b α ( I ) = n . Example 5.5.
Let I be the defining ideal in R = C [ x, y, z ] of the curve ( t , t , t ), i.e., thekernel of the map x t , y t and z t , which is a prime of height 2. Macaulay2 [GS]computations show that I (3) ⊆ m I , and as in the proof of Proposition 5.3, this implies that I satisfies a Chudnovsky-like bound.It is natural to ask if such a bound holds when I is the defining ideal of the curve ( t a , t b , t c ),where a, b, c take any value. The symbolic powers of the ideals in this class are of considerableinterest, in particular because the symbolic Rees algebra of I can fail to be noetherian– although it is noetherian for ( t , t , t ) above. And yet, these Chudnovsky-like boundsalso hold in cases where the symbolic Rees algebra of I is not noetherian. For example,when I defines ( t , t , t ), which has a non-noetherian symbolic Rees algebra by [GNW94],Macaulay2 [GS] computations also give I (3) ⊆ m I .We shall now prove a slightly weaker version of Chudnovsky’s Conjecture holds for anygeneral set of arbitrary number of points in P N k . We first show that such a statement holdsfor any number of points if it holds for a binomial coefficient number of points. emma 5.6. Let f : N → Q be any numerical function. If the inequality α ( I ( m ) ) m > f ( α ( I )) holds for the defining ideal of a general set of (cid:0) d + NN (cid:1) points in P N k , for any d > , then theinequality holds for the defining ideal of a general set of arbitrary number of points in P N k .Proof. The proof goes in exactly the same line as that of [FMX18, Proposition 2.5(a)],replacing “set of generic points in P N k ( z ) ” by “general set of points in P N k ”. Note that boththe set of generic points in P N k ( z ) and a general set of points in P N k have the maximal ( generic )Hilbert function. (cid:3) The next lemma establishes one containment for a binomial coefficient number of genericpoints in P N k ( z ) . Lemma 5.7.
Let I ( z ) be the defining ideal of a set of s = (cid:0) d + NN (cid:1) generic points in P N k ( z ) , for N > . There exists a constant c ∈ N such that I ( z ) ( Nc − N ) ⊆ m c ( N − z I ( z ) c . Proof.
The proof mimics that of Lemma 4.5. By Corollary 4.3, for r ≫ I ( z ) ( Nr − N ) ⊆ I ( z ) r . It suffices to show now that for r ≫
0, we have α ( I ( z ) ( Nr − N ) ) > r ( N − I ( z ))) . Indeed, since s = (cid:0) d + NN (cid:1) , again as in Lemma 4.1, by [MN01, Lemma 5.8] and [GM84,Corollary 1.6], we have α ( I ( z )) = reg( I ( z )) = d + 1. Thus, r ( N − I ( z ))) = r ( N + d − . On the other hand, by [FMX18, Theorem 2.7], for all r we have α ( I ( z ) ( Nr − N ) ) > α ( I ( z )) + N − N ( N r − N ) = d + NN ( N r − N ) = ( d + N )( r − . Now, for r ≫
0, we have r − r > d + N − d + N , or equivalently, ( d + N )( r − > r ( d + N − α ( I ( z ) ( Nr − N ) ) > r ( d + N − (cid:3) We are now ready to show our general lower bound.
Theorem 5.8.
Let I be the defining ideal of a general set of points in P N k , for N > . Forall m ∈ N , we have α ( I ( m ) ) m > α ( I ) + N − N . roof. By Lemma 5.6, it suffices to prove the assertion for s = (cid:0) d + NN (cid:1) a binomial coefficientnumber.By Lemma 5.7, there exists a constant c ∈ N such that I ( z ) ( Nc − N ) ⊆ m c ( N − z I ( z ) c . It follows from [Kru48, Satz 2 and 3] again that there exists an open dense subset U ⊆ A s ( N +1) such that for all a ∈ U , we have π a ( I ( z ) ( Nc − N ) ) = I ( a ) ( Nc − N ) , π a ( m c ( N − z ) = m c ( N − and π a ( I ( z ) c ) = I ( a ) c . Now, applying Corollary 3.2 with b = 2, we have, for a ∈ U and r ≫ I ( a ) ( Nr ) ⊆ I ( a ) ( Nr − N ) ⊆ m r ( N − I ( a ) r . This implies that, for a ∈ U and r ≫ α ( I ( a ) ( Nr ) ) N r > α ( I ( a )) + N − N .
By letting r → ∞ , we get b α ( I ( a )) > α ( I ( a )) + N − N for all a ∈ U , and the assertionfollows. (cid:3) As a direct consequence of Theorem 5.8, we can show that the defining ideal of a fatpoint scheme over a general support and with equal multiplicity also satisfies Chudnovsky’sConjecture.
Corollary 5.9.
Let J be the defining ideal of a fat point scheme X = tP + · · · + tP s in P N k with a general support, for some t ∈ N . If t > then Chudnovsky’s Conjecture holds for J ,i.e., b α ( J ) > α ( J ) + N − N .
Proof.
Let I be the defining ideal of { P , . . . , P s } . By definition, J = I ( t ) . By [FMX18,Lemma 3.6], for t > max (cid:26) N − N ǫ , m (cid:27) , we have b α ( J ) > α ( J ) + N − N , where ǫ > b α ( I ) = α ( I ) N + ǫ and m is the integer such that( I ( m ) ) ( t ) = I ( mt ) for all t and m > m . Since I is the defining ideal of points, we have m = 1. Furthermore, by Theorem 5.8, ǫ > N − N (and ǫ > N − N if N = 2; see [Chu81]).Therefore, max (cid:26) N − N ǫ , m (cid:27) ( N − N − . We get the desired result. (cid:3)
References [BDRH +
09] Thomas Bauer, Sandra Di Rocco, Brian Harbourne, Micha l Kapustka, Andreas Knutsen,Wioletta Syzdek, and Tomasz Szemberg. A primer on Seshadri constants. In
Interactions ofclassical and numerical algebraic geometry , volume 496 of
Contemp. Math. , pages 33–70. Amer.Math. Soc., Providence, RI, 2009. BDRH +
16] Thomas Bauer, Sandra Di Rocco, Brian Harbourne, Jack Huizenga, Alexandra Seceleanu, andTomasz Szemberg. Negative curves on symmetric blowups of the projective plane, resurgencesand waldschmidt constants. arXiv preprint arXiv:1609.08648 , 2016.[BH10] Cristiano Bocci and Brian Harbourne. Comparing powers and symbolic powers of ideals.
J.Algebraic Geom. , 19(3):399–417, 2010.[CEHH17] Susan M. Cooper, Robert J. D. Embree, Huy T`ai H`a, and Andrew H. Hoefel. Symbolic powersof monomial ideals.
Proc. Edinb. Math. Soc. (2) , 60(1):39–55, 2017.[CGM +
16] Adam Czapli´nski, Agata G l´owka, Grzegorz Malara, Magdalena Lampa-Baczy´nska, Patrycja Luszcz-´Swidecka, Piotr Pokora, and Justyna Szpond. A counterexample to the containment I (3) ⊂ I over the reals. Adv. Geom. , 16(1):77–82, 2016.[CHHVT20] Enrico Carlini, Huy T`ai H`a, Brian Harbourne, and Adam Van Tuyl.
Ideals of powers andpowers of ideals: Intersecting Algebra, Geometry and Combinatorics , volume 27 of
LectureNotes of the Unione Matematica Italiana . Springer International Publishing, 2020.[Chu81] Gregory V. Chudnovsky. Singular points on complex hypersurfaces and multidimensionalschwarz lemma. In
S´eminaire de Th´eorie des Nombres, Paris 1979-80, S´eminaire Delange-Pisot-Poitou , volume 12 of
Progress in Math. , pages 29–69. Birkh¨auser, Boston, Sasel, Stut-gart, 1981.[DDSG + Singularities and foliations. geometry, topology and applications ,volume 222 of
Springer Proc. Math. Stat. , pages 387–432. Springer, Cham, 2018.[DDSG + Singularities and foliations. geometry, topology and applications ,volume 222 of
Springer Proc. Math. Stat. , pages 387–432. Springer, Cham, 2018.[DHN +
15] M. Dumnicki, B. Harbourne, U. Nagel, A. Seceleanu, T. Szemberg, and H. Tutaj-Gasi´nska.Resurgences for ideals of special point configurations in P N coming from hyperplane arrange-ments. J. Algebra , 443:383–394, 2015.[Dra17] Ben Drabkin. Configurations of linear spaces of codimension two and the containment problem,2017. arXiv:1704.07870.[DS20] Ben Drabkin and Alexandra Seceleanu. Singular loci of reflection arrangements and the con-tainment problem. arXiv:2002.05353 , 2020.[DSTG13] Marcin Dumnicki, Tomasz Szemberg, and Halszka Tutaj-Gasi´nska. Counterexamples to the I (3) ⊆ I containment. J. Alg , 393:24–29, 2013.[DTG17] Marcin Dumnicki and Halszka Tutaj-Gasi´nska. A containment result in P n and the Chud-novsky conjecture. Proc. Amer. Math. Soc. , 145(9):3689–3694, 2017.[Dum15] Marcin Dumnicki. Containments of symbolic powers of ideals of generic points in P . Proc.Amer. Math. Soc. , 143(2):513–530, 2015.[EH79] David Eisenbud and Melvin Hochster. A Nullstellensatz with nilpotents and Zariski’s mainlemma on holomorphic functions.
J. Algebra , 58(1):157–161, 1979.[ELS01] Lawrence Ein, Robert Lazarsfeld, and Karen E. Smith. Uniform bounds and symbolic powerson smooth varieties.
Invent. Math. , 144 (2):241–25, 2001.[EV83] H´el`ene Esnault and Eckart Viehweg. Sur une minoration du degr´e d’hypersurfaces s’annulanten certains points.
Math. Ann. , 263(1):75–86, 1983.[FMX18] Louiza Fouli, Paolo Mantero, and Yu Xie. Chudnovsky’s conjecture for very general points in P Nk . J. Algebra , 498:211–227, 2018.[GHM13] Anthony V. Geramita, Brian Harbourne, and Juan Migliore. Star configurations in P n . J.Algebra , 376:279–299, 2013.[GHM19] Elo´ısa Grifo, Craig Huneke, and Vivek Mukundan. Expected resurgences and symbolic powersof ideals. arXiv:1903.12122 , 2019.[GKZ94] Izrail’ M. Gel’fand, Mikhail M. Kapranov, and Andrey V. Zelevinsky.
Discriminants, resul-tants, and multidimensional determinants . Mathematics: Theory & Applications. Birkh¨auserBoston, Inc., Boston, MA, 1994.[GM84] Anthony V. Geramita and Paolo Maroscia. The ideal of forms vanishing at a finite set of pointsin P n . J. Algebra , 90(2):528–555, 1984. GNW94] Shiro Goto, Koji Nishida, and Keiichi Watanabe. Non-Cohen-Macaulay symbolic blow-ups forspace monomial curves and counterexamples to Cowsik’s question.
Proceedings of the AmericanMathematical Society , 120(2):383–392, 1994.[GO82] A. V. Geramita and F. Orecchia. Minimally generating ideals defining certain tangent cones.
J. Algebra , 78(1):36–57, 1982.[Gri20] Elo´ısa Grifo. A stable version of Harbourne’s Conjecture and the containment problem forspace monomial curves.
J. Pure Appl. Algebra , 224(12):106435, 2020.[GS] D. R. Grayson and M. E. Stillman.
Macaulay2, a software system for research in algebraicgeometry .[HH02] Melvin Hochster and Craig Huneke. Comparison of symbolic and ordinary powers of ideals.
Invent. Math. 147 (2002), no. 2, 349–369 , November 2002.[HH13] Brian Harbourne and Craig Huneke. Are symbolic powers highly evolved?
J. RamanujanMath. Soc. , 28A:247–266, 2013.[HS15] Brian Harbourne and Alexandra Seceleanu. Containment counterexamples for ideals of variousconfigurations of points in P N . J. Pure Appl. Algebra , 219(4):1062–1072, 2015.[Joh14] Mark R. Johnson. Containing symbolic powers in regular rings.
Communications in Algebra ,42(8):3552–3557, 2014.[Kle78] Felix Klein. Ueber die Transformation siebenter Ordnung der elliptischen Functionen.
Math.Ann. , 14(3):428–471, 1878.[Kru48] Wolfgang Krull. Parameterspezialisierung in Polynomringen.
Arch. Math. , 1:56–64, 1948.[MN01] J. C. Migliore and U. Nagel. Liaison and related topics: notes from the Torino workshop-school.volume 59, pages 59–126 (2003). 2001. Liaison and related topics (Turin, 2001).[MS17] Linquan Ma and Karl Schwede. Perfectoid multiplier/test ideals in regular rings and boundson symbolic powers. arXiv:1705.02300 , 2017.[MS18] Grzegorz Malara and Justyna Szpond. On codimension two flats in Fermat-type arrangements.In
Multigraded algebra and applications , volume 238 of
Springer Proc. Math. Stat. , pages 95–109. Springer, Cham, 2018.[MSS18] Grzegorz Malara, Tomasz Szemberg, and Justyna Szpond. On a conjecture of Demailly andnew bounds on Waldschmidt constants in P N . J. Number Theory , 189:211–219, 2018.[Nag62] Masayoshi Nagata.
Local rings . Interscience, 1962.[NS16] Uwe Nagel and Alexandra Seceleanu. Ordinary and symbolic Rees algebras for ideals of Fermatpoint configurations.
J. Algebra , 468:80–102, 2016.[NT99] Dam Van Nhi and Ngˆo Viˆet Trung. Specialization of modules.
Comm. Algebra , 27(6):2959–2978, 1999.[Sec15] Alexandra Seceleanu. A homological criterion for the containment between symbolic and ordi-nary powers of some ideals of points in P . Journal of Pure and Applied Algebra , 219(11):4857– 4871, 2015.[Sko77] H. Skoda. Estimations L pour l’op´erateur ∂ et applications arithm´etiques. In Journ´ees surles Fonctions Analytiques (Toulouse, 1976) , pages 314–323. Lecture Notes in Math., Vol. 578.1977.[SS16] T. Szemberg and J. Szpond. On the containment problem. axiv:1601.01308 , January 2016.[Szp19] Justyna Szpond. Fermat-type arrangements. arXiv:1909.04089 , 2019.[TX19] Stefan Tohaneanu and Yu Xie. On the containment problem for fat points ideals. arXiv:1903.10647 , 2019.[Wal77] Michel Waldschmidt. Propri´et´es arithm´etiques de fonctions de plusieurs variables. II. In
S´eminaire Pierre Lelong (Analyse) ann´ee 1975/76 , pages 108–135. Lecture Notes in Math.,Vol. 578. 1977.[Wim96] A. Wiman. Ueber eine einfache Gruppe von 360 ebenen Collineationen.
Math. Ann. , 47(4):531–556, 1896.[Zar49] Oscar Zariski. A fundamental lemma from the theory of holomorphic functions on an algebraicvariety.
Ann. Mat. Pura Appl. (4) , 29:187–198, 1949. 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