The regularity of points in multi-projective spaces

THE REGULARITY OF POINTS IN MULTI-PROJECTIVE SPACES arXiv:math/0211263v2 [math.AC] 9 Jul 2003 HUY TÀI HÀ AND ADAM VAN TUYL 1 s be the defining ideal of a scheme of fat points in Pn1 × Abstract. Let I = ℘m ∩ . . . ∩ ℘m s 1 nk · · · × P with support in generic position. When all the mi ’s are 1, we explicitly calculate the Castelnuovo-Mumford regularity of I. In general, if at least one mi ≥ 2, we give an upper bound for the regularity of I, which extends a result of Catalisano, Trung and Valla. Introduction In this paper, we study the Castelnuovo-Mumford regularity of defining ideals of sets of points (reduced and non-reduced) in a multi-projective space Pn1 × · · · × Pnk . If I ⊆ k[x0 , . . . , xn ] is the defining ideal of a projective variety X ⊆ Pn , then the CastelnuovoMumford regularity of I, denoted by reg(I), is a very important invariant associated to X. It has been the objective of many authors to estimate reg(I) since not only does it bound the degrees of a minimal set of defining equations for X, it also gives a uniform bound on the degrees of syzygies of I. The most fundamental situation is when X is a set of points. Examples of work on reg(I) in this case can be seen in [5, 7, 8, 15]. Recently, many authors (cf. [4, 9, 10, 11, 16]) have been interested in extending our understanding of points in Pn to sets of points in Pn1 × · · · × Pnk . We continue this trend by studying reg(I) when I defines a scheme of fat points in Pn1 × · · · × Pnk . In the context of N2 -graded rings, Aramova, Crona and De Negri [1] have introduced a finer notion of regularity that places bounds on each coordinate of the degree of a multi-graded syzygy. Extending the definition of regularity to multi-graded rings is also considered in [12, 13]. The usual notion of regularity could be treated as a bound on the total degree of the multi-graded syzygies. The Nk -graded ring R = k[x1,0 , . . . , x1,n1 , . . . , xk,0 , . . . , xk,nk ] where deg xi,j = ei , the ith basis vector of Nk , is the associated coordinate ring of Pn1 ×· · ·×Pnk . Let X = {P1 , . . . , Ps } be a set of distinct points in Pn1 ×· · ·×Pnk . The defining ideal of Pi is ℘i = (L1,1 , . . . , L1,n1 , . . . , Lk,1 , . . . , Lk,nk ) with deg Li,j = ei . If m1 , . . . , ms are positive integers, then we want to study regularity of ms 1 ideals of the form IZ = ℘m 1 ∩ · · · · · · ∩ ℘s . Such an ideal IZ defines a scheme of fat points n n Z = m1 P1 + . . . + ms Ps in P 1 × · · · × P k . The ideal IZ is both Nk -homogeneous, and homogeneous in the normal sense. Thus, when we refer to reg(IZ ), we shall mean its regularity as a homogeneous ideal in R, where R is viewed as an N1 -graded ring. if it A set of s points X = {P1 , . . . , Ps } ⊆ Pn1 × · · · × Pnk is said to be in generic position L has maximal Hilbert function HX (i) = min{dimk Ri , s} for all i ∈ Nk , where R = i Ri is the Nk -homogeneous decomposition of R. The existence of such sets is shown in [17]. Our main results consist of explicitly calculating reg(IZ ) when Z is in generic position and reduced (i.e. there is no multiplicity at each point), and giving a bound on reg(IZ ) in general. 2000 Mathematics Subject Classification. 13D02,13D40, 14Q99. Key words and phrases. regularity, points, fat points, multi-projective space. Version: June 9, 2003. 1 2 HUY TÀI HÀ AND ADAM VAN TUYL In the special case that each mi = 1 and the set of points is in generic position, we show n where di = min d ∈ N reg(IZ ) = max{d1 + 1, . . . , dk + 1} o d+ni
≥ s for each i = 1, . . . , k. To prove this we use the fact that d IZ is both Nk -homogeneous and N1 -homogeneous to obtain information about reg(IZ ). We also use the Bayer-Stillman criterion for detecting m-regularity [2]. We then show that if X is generic position, and if m1 ≥ m2 ≥ · · · ≥ ms with at least one mi ≥ 2, then   Ps   Ps  i=1 mi + n1 − 2 i=1 mi + nk − 2 reg(IZ ) ≤ max m1 + m2 − 1, ,..., + k. n1 nk Our strategy is to investigate the regularity index ri(R/IZ ) of R/IZ , considered as an N1 -graded ring, by extending the results of [5] for fat point schemes in Pn to Pn1 × · · · × Pnk , and then use the fact that reg(IZ ) ≤ ri(R/IZ ) + k. We have organized this papers as follows. In the first section we introduce the relevant information about regularity, the regularity index, and points in multi-projective spaces. In the second section we compute the regularity of a defining ideal of a set of points in generic position. In the last section we bound the regularity for a set of fat points with generic support. Acknowledgments. The authors would like to thank A. Conca, for originally poising this question and some helpful discussions, and E. Guardo, for her comments on an earlier version of this paper. This work was begun when the second author visited the Università di Genova, and he would like to thank them for their hospitality. The second author also acknowledges the financial support of NSERC and INDAM while working on this project. 1. Preliminaries Throughout this paper k denotes an algebraically closed field of characteristic zero. In this section, we recall the needed facts about the Castelnouvo-Mumford regularity, the regularity index, and points in multi-projective spaces. Let S = k[x0 , . . . , xn ] be a polynomial ring. Definition 1.1. A graded S-module M is m-regular if there exists a free resolution M M M 0 −→ S(−er,j ) −→ · · · −→ S(−e1,j ) −→ S(−e0,j ) −→ M −→ 0 j j j of M with ei,j − i ≤ m for all i, j. The Castelnuovo-Mumford regularity (or simply, regularity) of M , denoted reg(M ), is the least integer m for which M is m-regular. If I ⊆ S, then reg(I) = reg(S/I) + 1. The saturation I of the ideal I ⊆ S is the ideal I := {F ∈ S | for i = 1, . . . , n, there exists an r such that xri · F ∈ I}. I is said to be saturated if I = I. The regularity of a saturated ideal does not change if we add a non-zero divisor. In fact, Lemma 1.2 ([2, Lemma 1.8]). Let I ⊆ S be a saturated ideal, and suppose h is a non-zero divisor of S/I. Then I is m-regular if and only if (I, h) is m-regular. Thus, reg(I) = reg((I, h)). The following theorem provides a means to determine if an ideal is m-regular. Theorem 1.3 ([2, Theorem 1.10] Bayer-Stillman criterion for m-regularity). Let I ⊆ S be an ideal generated in degrees ≤ m. The following conditions are equivalent: THE REGULARITY OF POINTS IN MULTI-PROJECTIVE SPACES 3 (i) I is m-regular. (ii) There exists h1 , . . . , hj ∈ S1 for some j ≥ 0 so that (a) ((I, h1 , . . . , hi−1 ) : hi )m = (I, h1 , . . . , hi−1 )m for i = 1, . . . , j, and (b) (I, h1 , . . . , hj )m = Sm . The Hilbert function HM : N −→ N of a graded S-module M is defined by HM (t) := dimk Mt . It is well known (cf. [3, Theorem 4.1.3]) that there exists a unique polynomial HPM (t), called the Hilbert polynomial of M , such that HM (t) = HPM (t) for t ≫ 0. Definition 1.4. The regularity index of an S-module M , denoted ri(M ), is defined to be ri(M ) := min{t | HM (j) = HPM (j) for all j ≥ t}. The regularity and regularity index of an S-module are then related as follows. Lemma 1.5 ([14, Lemma 5.8]). If M is a graded S-module, then reg(M ) − dim M + 1 ≤ ri(M ) ≤ reg(M ) − depth M + 1. If M = S/I, then ri(S/I) ≤ reg(S/I) − depth S/I + 1 ≤ reg(I). Hence, we have Corollary 1.6. If I ⊆ S, then for all t ≥ reg(I), HS/I (t) = HPS/I (t). Our goal is to investigate reg(I) when I defines either a reduced or non-reduced set of points in Pn1 × · · · × Pnk whose support is in generic position. Let R = k[x1,0 , . . . , x1,n1 , . . . , xk,0 , . . . , xk,nk ], with deg xi,j = ei where ei is the ith basis vector of Nk , be the Nk -graded coordinate ring of Pn1 × · · · × Pnk . Let Rei = k[xi,0 , . . . , xi,ni ] be the graded coordinate ring of Pni for i = 1, . . . , k. If P ∈ Pn1 × · · · × Pnk is a point, then the ideal ℘ ⊆ R associated to P is the prime ideal ℘ = (L1,1 , . . . , L1,n1 , . . . , Lk,1 , . . . , Lk,nk ) with deg Li,j = ei . Suppose X = {P1 , . . . , Ps } is a set of distinct points in Pn1 × · · · × Pnk , and m1 , . . . , ms are s positive integers. Let m2 ms 1 IZ = ℘ m 1 ∩ ℘2 ∩ · · · ∩ ℘s where ℘i is the defining ideal of Pi , then IZ defines a scheme of fat points Z = m1 P1 + . . .+ ms Ps in Pn1 × · · · × Pnk with support X. When mi = 1 for all i, Z ≡ X is reduced, and we usually use IX instead of IZ . P Since ht(℘i ) = kj=1 nj for each i, it follows that K-dim R/IZ = k. Thus, by Lemma 1.5 we have reg(IZ ) ≤ ri(R/IZ ) + k. Note that we have equality if k = 1 because then depth R/IZ = 1. We shall find it useful to consider R/IZ as both an Nk -graded ring and as an N1 -graded ring. We shall, therefore, use HZ (t) to denote the multi-graded Hilbert function HZ (t) := dimk (R/IZ )t with t = (t1 , . . . , tk ) L ∈ Nk , and HZ (t) to denote the N1 -graded Hilbert function HZ := HR/IZ . Because (R/IZ )t = t1 +···+tk =t (R/IZ )t1 ,...,tk , we have the identity: X HZ (t) = HZ (t1 , . . . , tk ) for all t ∈ N. t1 +···+tk =t Definition 1.7. A set of s points X = {P1 , . . . , Ps } ⊆ Pn1 × · · · × Pnk is said to be in generic position if       tk + n k t1 + n 1 ··· ,s for all t ∈ Nk . HX (t) = min dimk Rt = n1 nk 4 HUY TÀI HÀ AND ADAM VAN TUYL Further results about points in Pn1 × · · · × Pnk can be found in [16, 17]. Remark 1.8. If I ⊆ R is an Nk -homogeneous ideal, then the Nk -graded minimal free resolution of I is 0 −→ Fr −→ Fr−1 −→ · · · −→ F0 −→ I −→ 0 L where Fi = j R(−di,j,1 , −di,j,2 , . . . , −di,j,k ). Since I is also homogeneous in the normal sense, the above resolution also gives a graded minimal free resolution of I: ′ 0 −→ Fr′ −→ Fr−1 −→ · · · −→ F0′ −→ I −→ 0 L 1 where Fi′ = j R(−di,j,1 − di,j,2 − · · · − di,j,k ) where we view R as N -graded. So if I is an Nk -homogeneous ideal with k ≥ 2, reg(I) can be interpreted as a crude invariant that bounds the total degree of the multi-graded syzygies. The following lemma, which generalizes [16, Lemma 3.3], enables us to find non-zero divisors of specific multi-degrees. Lemma 1.9. Suppose X = {P1 , . . . , Ps } is a set of distinct points in Pn1 × · · · × Pnk , ℘1 , . . . , ℘s are the defining ideals of P1 , . . . , Ps , respectively, and m1 , . . . , ms are positive integers. Set ms 1 IZ = ℘ m 1 ∩ · · · ∩ ℘s , and fix an i ∈ {1, . . . , k}. Then there exists a form L ∈ Rei such that L is a non-zero divisor in R/IZ . 2. The regularity of the defining ideal of points in generic position Let X ⊆ Pn1 × · · · × Pnk be a set of s reduced points in generic position. In this section we calculate the Castelnuovo-Mumford regularity of the defining ideal of X. o n d+ni
≥ s , and let D := max{d1 + 1, . . . , dk + 1}. For each i = 1, . . . , k, set di := min d d Note that if ni = min{n1 , . . . , nk }, then D = di + 1. Beginning with a combinatorial lemma, we use this notation to describe the some of the properties of points in generic position. Lemma 2.1. Let n ≥ 1. Then, for all a, b ≥ 1,      a+b+n a+n b+n ≤ . a+b a b Proof. Because     a+b+n (a + b + n) · · · (a + 1 + n) a + n = (a + b)(a + b − 1) · · · (a + 1) a a+b it is enough to show that the inequality   b+n (a + b + n)(a + b − 1 + n) · · · (a + 1 + n) ≤ (a + b) · · · (a + 1) b is true. This is equivalent to showing that (a + b + n)(a + b − 1 + n) · · · (a + 1 + n) (a + b)(a + b − 1) · · · (a + 1) ≤ . (b + n)(b − 1 + n) · · · (1 + n) b(b − 1) · · · 2 · 1 Rewriting the above expression, we see that we need to show that       h  h a a a a ai ai . 1+ 1+ ··· 1 + ≤ 1+ ··· 1 + 1+ b+n b−1+n 1+n b b−1 1 i h i h a a ≤ 1 + b−j for j = 0, . . . , b − 1 we are finished. But since 1 + b+n−j  THE REGULARITY OF POINTS IN MULTI-PROJECTIVE SPACES P n1 P nk be s points in generic position. If (t1 , . . . , tk ) ∈ Corollary 2.2. Let X ⊆ × ··· × such that t1 + · · · + tk = D − 1, then HX (t1 , . . . , tk ) = s. 5 Nk is Proof. Suppose that ni = min{n1 , . . . , nk }, and hence, D − 1 = di . Lemma 2.1 then gives             tk + n k tk + n i di + ni t2 + n 2 t2 + n i t1 + n 1 t1 + n i ··· ··· ≥ ≥ tk di t2 t2 t1 tk t1
i Since di d+n ≥ s, we have HX (t1 , . . . , tk ) = s.  i
Recall that if m ∈ N, then t+m denotes the polynomial m   t+m (t + m)(t + (m − 1)) · · · (t + 1) . = m! m Proposition 2.3. Let IX be the defining ideal of s points X ⊆ Pn1 ×· · ·×Pnk in generic position. (i) As an N1 -graded ideal, IX is generated by forms of degree ≤ D.
(ii) As an N1 -graded ring, R/IX has Hilbert polynomial HPR/IX (t) = s t+k−1 k−1 . (iii) Fix an i ∈ {1, . . . , k} and let L be the non-zero divisor of Lemma 1.9 of degree ei . If t = (t1 , . . . , tk ) ∈ Nk is such that t1 + . . . + tk ≥ D and ti > 0, then (IX , L)t = Rt . Proof. For (i) it suffices to show that for all t = (t1 , . . . , tk ) ∈ Nk with t1 + · · · + tk ≥ D + 1, (IX )t contains no new minimal generators. If t ∈ Nk is such a tuple, then there exists l, m ∈ {1, . . . , k}, not necessarily distinct, such that t − el − em ∈ Nk . By Corollary 2.2 it follows that HX (t − el − em ) = HX (t − el ) = s since t1 + · · · + tk − 2 ≥ D − 1. Now apply the results of [17] to conclude that (IX )t contains no minimal generators. Since X is in generic position, for t ≫ 0 we have X HX (t) = HX (t1 , . . . , tk ) = t1 +···+tk =t X t1 +···+tk   t+k−1 s=s . k−1 =t Since HPR/IX is the unique polynomial that agrees with HX for t ≫ 0, (ii) now follows. To prove (iii) we only consider the case i = 1. Since L is a non-zero divisor, the exact sequence ×L 0 −→ (R/IX )(−e1 ) −→ R/IX −→ R/(IX , L) −→ 0 implies that HR/(IX ,L) (t1 , . . . , tk ) = HX (t1 , . . . , tk ) − HX (t1 − 1, t2 , . . . , tk ) for all t ∈ Nk where HX (t1 −1, t2 , . . . , tk ) = 0 if t1 −1 < 0. Now suppose that t1 +. . .+tk ≥ D with t1 > 0. Since (t1 −1)+t2 +· · ·+tk ≥ D−1, by Corollary 2.2 we have HX (t1 , . . . , tk ) = HX (t1 −1, t2 , . . . , tk ) = s. Thus HR/(IX ,L) (t1 , . . . , tk ) = 0, or equivalently, (IX , L)t1 ,...,tk = Rt1 ,...,tk .  Theorem 2.4. Let IX be the defining ideal of s points X ⊆ Pn1 × · · · × Pnk in generic position. Then reg(IX ) = max{d1 + 1, . . . , dk + 1} n o
d+ni where di := min d ≥ s for i = 1, . . . , k. d 6 HUY TÀI HÀ AND ADAM VAN TUYL Proof. Without loss of generality, we assume that n1 ≥ n2 ≥ . . . ≥ nk ≥ 1. It thus suffices to show that reg(IX ) = dk + 1 = max{d1 + 1, . . . , dk + 1}. We first show that reg(IX ) > dk . By Lemma 1.9 there is a non-zero divisor L of R/IX with deg L = ek . As an N1 -homogeneous element of R, deg L = 1. Since IX is saturated, by Lemma 1.2 is it is enough to show reg(IX , L) > dk . From the short exact sequence ×L 0 −→ (R/IX )(−1) −→ R/IX −→ R/(IX , L) −→ 0. of N1 -graded rings, and from Proposition 2.3 (ii) we deduce that   t + (k − 2) HPR/(IX ,L) (t) = HPR/IX (t) − HPR/IX (t − 1) = s . k−2 If we can show that HPR/(IX ,L) (dk ) 6= HR/(IX ,L) (dk ), then by Corollary 1.6, we can conclude that reg(IX , L) > dk . So, write HR/(IX ,L) (dk ) = A + B where X X HR/(IX ,L) (t1 , . . . , tk−1 , 0) and B := HR/(IX ,L) (t1 , t2 , . . . , tk ). A := t1 +···+tk =dk , tk >0 t1 +···+tk−1 =dk From the short exact sequence ×L 0 −→ (R/IX )(−ek ) −→ R/IX −→ R/(IX , L) −→ 0 of Nk -graded rings, we have HR/(IX ,L) (t1 , . . . , tk ) = HR/IX (t1 , . . . , tk ) − HR/IX (t1 , . . . , tk−1 , tk − 1) where HR/IX (t1 , . . . , tk−1 , tk − 1) = 0 if tk = 0. Thus X A= HR/IX (t1 , . . . , tk−1 , 0). t1 +···+tk−1 =dk Since t1 + · · · + tk−1 = dk , by Corollary 2.2 we have HR/IX (t1 , . . . , tk−1 , 0) = s. Hence,   X d1 + k − 2 A= s=s = HPR/(IX ,L) (dk ). k−2 t1 +···+tk−1 =dk o n d+nk
≥ s On the other hand, because dk = min d d B ≥ HR/(IX ,L) (0, . . . , 0, dk ) = HX (0, . . . , 0, dk ) − HX (0, . . . , 0, dk − 1)   dk − 1 + nk = s− > 0. dk − 1 Thus, HR/(IX ,L) (dk ) = HPR/(IX ,L) (dk ) + B > HPR/(IX ,L) (dk ), as desired. We now show that reg(IX ) ≤ dk + 1 by demonstrating that IX is (dk + 1)-regular. By Proposition 2.3 (i), as an N1 -graded ideal IX is generated by elements of degree ≤ dk + 1. For each i ∈ {1, . . . , k}, by Lemma 1.9 there exists a non-zero divisor Li ∈ R/IX with deg Li = ei . After a change of variables in the x1,j ’s, a change of variables in the x2,j ’s, etc., we can assume that Li = xi,0 for i = 1, . . . , k. By the Bayer-Stillman criterion (Theorem 1.3), to show that IX is (dk +1)-regular, it is enough to prove: THE REGULARITY OF POINTS IN MULTI-PROJECTIVE SPACES 7 (a) ((IX , x1,0 , . . . , xj−1,0 ) : xj,0 )dk +1 = (IX , x1,0 , . . . , xj−1,0 )dk +1 for j = 1, . . . , k, (b) (IX , x1,0 , . . . , xk,0 )dk +1 = Rdk +1 . Proof of (a). We need to only show the non-trivial inclusion [(IX , x1,0 , . . . , xj−1,0 ) : xj,0 ]dk +1 ⊆ (IX , x1,0 , . . . , xj−1,0 )dk +1 for each j. If j = 1, then the statement holds because x1,0 is a non-zero divisor. So, suppose j > 1. Set J := [(IX , x1,0 , . . . , xj−1,0 ) : xj,0 ]. Because J is also Nk -homogeneous, if F ∈ Jdk +1 , then can assume that deg F = t = (t1 , . . . , tk ) with t1 + · · · + tk = dk + 1. There are now two cases to consider. In the first case, one of t1 , . . . , tj−1 > 0. Suppose tl > 0 with 1 ≤ l ≤ (j −1). Then by Proposition 2.3 (iii) we have F ∈ Rt ⊆ (IX , xl,0 )t ⊆ (IX , x1,0 , . . . , xj−1,0 )t . Since (IX , x1,0 , . . . , xj−1,0 )t ⊆ (IX , x1,0 , . . . , xj−1,0 )dk +1 (as vector spaces), we are finished. In the second case, t1 = t2 = · · · = tj−1 = 0. Then F xj,0 ∈ (IX , x1,0 , . . . , xj−1,0 )0,...,0,tj +1,...,tk . But since (IX , x1,0 , . . . , xj−1,0 )0,...,0,tj +1,...,tk = (IX )0,...,0,tj +1,...,tk , we have F xj,0 ∈ (IX )0,...,0,tj +1,...,tk . But because xj,0 is a non-zero divisor of R/IX , F ∈ (IX )0,...,0,tj ,...,tk ⊆ (IX , x1,0 , . . . , xj−1,0 )0,...,0,tj ,...,tk ⊆ (IX , x1,0 , . . . , xj−1,0 )dk +1 . L Proof of (b). Since Rdk +1 = t1 +···+tk =dk +1 Rt1 ,...,tk and because (IX , x1,0 , . . . , xk,0 ) is also Nk homogeneous, it is enough to show that Rt ⊆ (IX , x1,0 , . . . , xk,0 )t for all t = (t1 , . . . , tk ) ∈ Nk with t1 + · · · + tk = dk + 1. But for any t ∈ Nk with t1 + · · · + tk = dk + 1, there exists at least one tl > 0. Thus, by Proposition 2.3 (iii) we have Rt ⊆ (IX , xl,0 )t ⊆ (IX , x1,0 , . . . , xk,0 )t , thus completing the proof of (b). Since we have just shown dk < reg(IX ) ≤ dk + 1, the desired conclusion now follows.  Remark 2.5. If X is a set of s points in generic position in Pn , we recover the well known result l+n
that reg(IX ) = d + 1 where d = min{l | n ≥ s}. 3. Bounding the regularity of fat points in Pn1 × · · · × Pnk Let X = {P1 , . . . , Ps } ⊆ Pn1 × · · · × Pnk and m1 ≥ · · · ≥ ms ∈ N+ . Suppose ℘i is the defining ms 1 ideal of Pi for i = 1, . . . , s. Let I = IZ = ℘m 1 ∩ · · · ∩ ℘s . In this section, we give an upper bound for reg(I) when X is in generic position. If we consider R/I as an N1 -graded ring, then by Lemma 1.5 reg(I) = reg(R/I) + 1 ≤ ri(R/I) + dim R/I = ri(R/I) + k. To bound reg(I), it is therefore enough to bound ri(R/I). For convenience, we assume that n1 ≥ . . . ≥ nk . In the sequel, we shall also abuse notation by writing L for the form L ∈ k[xj,0 , . . . , xj,nj ], the hyperplane L in Pnj defined by L, and the subvariety of Pn1 × · · · × Pnk defined by L. Lemma 3.1. If ℘ is the defining ideal of point P ∈ Pn1 × · · · × Pnk , then ri(R/℘a ) = a − k for all a ≥ 1. P Proof. Since ℘ defines a complete intersection of height ki=1 ni , Lemma 1.5 gives ri(R/℘a ) = reg(R/℘a ) − k + 1. The conclusion follows since reg(℘a ) = a reg(℘) = a by [6, Theorem 3.1].  8 HUY TÀI HÀ AND ADAM VAN TUYL Lemma 3.2. Suppose P1 , . . . , Pr , P are points in generic position in Pn1 × · · · × Pnk , and let ℘i be the defining ideal of Pi and let ℘ be the defining ideal of P . Let m1 , . . . , mr and a be positive a mr 1 integers, J = ℘m 1 ∩ · · · ∩ ℘r , and I = J ∩ ℘ . Then ri(R/I) ≤ max {a − k, ri(R/J), ri(R/(J + ℘a ))} . Furthermore, R/(J + ℘a ) is artinian. Proof. The short exact sequence of N1 -graded rings 0 −→ R/I −→ R/J ⊕ R/℘a −→ R/(J + ℘a ) −→ 0 yields HR/I (t) = HR/J (t) + HR/℘a (t) − HR/(J+℘a ) (t). Combining this with Lemma 3.1 gives ri(R/I) ≤ max {a − k, ri(R/J), ri(R/(J + ℘a ))} . To show that R/(J + ℘a ) is artinian, we need to show that there exists b such that for all t = (t1 , . . . , tk ) ∈ Nk , if there is tj ≥ b, then (R/(J + ℘a ))t = 0. So, it suffices to show that there exists such a b so that for all t = (t1 , . . . , tk ) with tj ≥ b for some j, then all monomials of R of degree t are in (J + ℘a ). Suppose M is a monomial in R of degree t. Then M = N1 N2 · · · Nk where Nl are monomials in {xl,0 , . . . , xl,nl } and of degree tl . It is enough to show Nj ∈ (J + ℘a ). Let Q1 , . . . , Qr , Q be the projections of P1 , . . . , Pr , P in Pnj . Since the points are in generic position, the projections are distinct. Let Q1 , . . . , Qr and Q be the defining ideals of Q1 , . . . , Qr , Q a mr 1 in A = k[xj,0 , . . . , xj,nj ]. Then it is easy to see that A/(Qm 1 ∩ · · · ∩ Qr + Q ) is artinian. As m1 m1 a a a m m r r well, Q1 ∩ · · · ∩ Qr ⊆ J and Q ⊆ ℘ , and thus Q1 ∩ · · · ∩ Qr + Q ⊆ (J + ℘a ), and this is what needs to be shown.  From Lemma 3.2, to estimate ri(R/I) we need to estimate ri(R/(J + ℘a )), or equivalently, the least integer t such that (R/(J + ℘a ))t = 0, when this ring is consider as N1 -graded. Lemma 3.3. With the same hypotheses as in Lemma 3.2, and considering the N1 -gradation, we have   P i i+1 ) (i) HR/(J+℘a ) (t) = a−1 for all t ≥ 0. i=0 dimk (J + ℘ )/(J + ℘ t   (ii) If P = [1 : 0 : · · · : 0] × · · · × [1 : 0 : · · · : 0] then (J + ℘i )/(J + ℘i+1 ) t = 0 if and only if either i > t, or i < t and GM ∈ (J + ℘i+1 ) for every monomial M of degree i in {x1,1 , . . . , x1,n1 , . . . , xk,1 , . . . , xk,nk m }, and every monomial G of degree t − i in {x1,0 , x2,0 , . . . , xk,0 }. Proof. The first assertion follows from the short exact sequences: 0 −→ (J + ℘i )/(J + ℘i+1 ) −→ R/(J + ℘i+1 ) −→ R/(J + ℘i ) −→ 0 where i = 0, . . . , a − 1. To prove (ii), if i > t, then (J + ℘i )t = (J + ℘i+1 )t = Jt . So suppose i < t. We see that ℘ = (x1,1 , . . . , x1,n1 , . . . , xk,1 , . . . , xk,nk ). Thus ((J + ℘i )/(J + ℘i+1 ))t = 0 if and only if (℘i )t ⊆ (J + ℘i+1 )t if and only if F M ∈ (J + ℘i+1 ) for every monomial M of degree i in {x1,1 , . . . , x1,n1 , . . . , xk,1 , . . . , xk,nk } and every form F ∈ Rt−i . But because (J + ℘i+1 ) is Nk homogenous, we can take F to be Nk -homogeneous, and so F = G + H where G is a monomial of degree t − i in x1,0 , . . . , xk,0 and H ∈ ℘. Since HM ∈ ℘i+1 , we have ((J + ℘i )/(J + ℘i+1 ))t = 0 if and only if GM ∈ (J + ℘i+1 )t , as desired.  THE REGULARITY OF POINTS IN MULTI-PROJECTIVE SPACES 9 Pn1 ×· · ·×Pnk with n1 ≥ · · · ≥ nk , Lemma 3.4. Let P1 , . . . , Pr , P be points in generic position in m1 m r and let m1 ≥ · · · ≥ mr be positive J = ℘1 ∩ · · · ∩ ℘r . Suppose a = (a1 , . . . , ak ) ∈  Pintegers. SetP Pk r k k N is such that nk i=1 mi and i=1 ai ≥ m1 . Then we can find aj hyperplanes i=1 ai ≥ n j Lj,1 , . . . , Lj,aj in P , that is, Lj,l ∈ k[xj,0 , . . . , xj,nj ] for all l = 1, . . . , aj , such that ! aj k Y Y L= Lj,l ∈ J j=1 l=1 and L avoids P . Proof. If r ≤ nj for all j, then for each j we can find a linear form Lj ∈ k[xj,0 , . . . , xj,nj ] that passes through P1 , . . . , Pr and avoids P . If we take Lj,l = Lj for all j, we have L= k Y a |a| m1 mr Lj j ∈ ℘1 ∩ · · · ∩ ℘|a| r ⊆ ℘1 ∩ · · · ∩ ℘r = J, j=1 where |a| = Pk i=1 ai , since |a| ≥ m1 ≥ · · · ≥ mr . Moreover, L avoids P . k ≤ nk−1 ≤ · · · ≤ nl+1 < r ≤ nl ≤ · · · ≤ n1 . We shall use induction on PrSuppose now that nP r i=1 mi . Note that if i=1 mi ≤ nk then the conclusion follows since in this case r ≤ nk ≤ nj for all j. If ak = ak−1 = · · · = al+1 = 0, then the conclusion follows as in the case r ≤ nj for all j. Suppose there is p ∈ {l + 1, . . . , k} such that ap 6= 0. Choose a hyperplane L1 in Pnp (L1 ∈ P P k[xp,0 , . . . , xp,np ]) that avoids P and passes through P1 , . . . , Pnp . Since nk ( ki=1 ai ) ≥ ri=1 mi , we have ! r r k X X X mi − n p mi − n k ≥ ai − n k ≥ nk i=1 i=1 i=1 = (m1 − 1) + · · · (mnp − 1) + mnp +1 + · · · + mr . If we set (b1 , . . . , bp−1 , bp , bp+1 , . . . , bk ) = (a1 , . . . , ap−1 , ap − 1, ap+1 , . . . , ak ), then we have ! ! k k X X ai − nk ≥ (m1 − 1) + · · · (mnp − 1) + mnp +1 + · · · + mr . bi = nk nk i=1 i=1 By induction there exists Lj,1 , . . . , Lj,bj in Pnj for all j that avoids P such that   bj k Y Y mnp +1 mn −1 1 −1 r  Lj,l  ∈ ℘m ∩ · · · ∩ ℘m L= ∩ · · · ∩ ℘np p ∩ ℘np +1 r . 1 j=1 l=1 If we take L · L1 we have the conclusion since L1 ∈ ℘1 ∩ · · · ∩ ℘np (the ap hyperplanes in Pnp are  Lp,1 , . . . , Lp,bp and L1 ). Proposition 3.5. Let P1 , . . . , Pr , P be points in generic position in Pn1 × · · · × Pnk with n1 ≥ m1 ∩ · · · ∩ ℘mr . Let t be · · · ≥ nk . Suppose m1 ≥ · · · ≥ mP r ≥ a are positive integers. Set J = ℘ r r the least integer such that nk t ≥ i=1 mi + a − 1. Then ri(R/(J + ℘a )) ≤ max{m1 + a − 1, t}. Proof. Without loss of generality take P = [1 : 0 : · · · : 0] × · · · × [1 : 0 : · · · : 0]. Then ℘ = (x1,1 , . . . , x1,n1 , . . . , xk,1 , . . . , xk,nk ). If r ≤ nj for all j, then we can find a hyperplane Lj in Pnj , i.e., Lj ∈ k[xj,0 , . . . , xj,nj ], containing P1 , . . . , Pr and avoids P for each j. Then Lj ∈ ℘1 ∩ · · · ∩ ℘r for all j. 10 HUY TÀI HÀ AND ADAM VAN TUYL ak a1 Suppose G = x1,0 · · · xk,0 is a monomial of degree m1 in {x1,0 , . . . , xk,0 }. Then L := La11 · · · Lakk ∈ m1 ⊆ ℘m1 ∩ · · · ∩ ℘mr = J. We can rewrite L = x 1 ℘m j j,0 + Hj where Hj ∈ r 1 ∩ · · · ∩ ℘r 1 (xj,1 , . . . , xj,nj ) ⊆ ℘. Then L ∈ J implies G ∈ J + ℘. Thus, for any monomial M of degree i in ℘i for some 0 ≤ i ≤ a − 1, GM ∈ J + ℘i+1 . Since a − 1 ≥ i, this implies that for any monomial G of degree m1 + a − 1 − i in {x1,0 , . . . , xk,0 }, and any monomial M of degree i in ℘i , GM ∈ (J + ℘i+1 ) because G is divisible by a monomial of degree m1 . By Lemma 3.3, this implies that ri(R/(J + ℘a )) ≤ m1 + a − 1. Suppose now that r > nk . Since n1 ≥ . . . ≥ nk , by a change of coordinates we may assume that P1 = [0 : 1 : 0 : · · · : 0] × [0 : 1 : 0 : · · · : 0] × · · · × [0 : 1 : 0 : · · · : 0] .. . · · : 0} : 1 : 0 : · · · : 0] × · · · × [0 : · · · : 0 : 1] Pnk = [0| : ·{z · · : 0} : 1 : 0 : · · · : 0] × [0| : ·{z nk nk So for 0 ≤ j ≤ nk , ℘j = ({xl,q | l = 1, . . . , k, q 6= j}). 1 k Let h = max{m1 + a − 1, t} and 0 ≤ i ≤ a − 1. Suppose now that G = xa1,0 is a · · · xak,0 Qk Q cl,q monomial of degree h − i in {x1,0 , . . . , xk,0 }, and M = l=1 q6=0 xl,q is a monomial of degree i in ℘i . Because of Lemma 3.3 we need to show that GM ∈ (J + ℘i+1 ). It can be seen that M∈ i− ℘1 We also have, since i ≤ a − 1, k X Pk l=1 cl,1 ∩ Pk l=1 cl,2 ( ai = h − i ≥ m1 ≥ max m1 − i + i=1 and i− ℘2   k X aj  = nk (h − i) = nk h − ink nk  ∩ ··· ∩ k X i− ℘nk Pk l=1 cl,nk . cl,1 , . . . , mnk − i + k X i=1 i=1 cl,nk ) j=1 ≥ r X mj + a − 1 − ink ≥ ≥ j=1 mj + i − ink j=1 j=1 r X r X mj + nk k X X cl,q − ink l=1 q=1 = (m1 − i + k X cl,1 ) + · · · + (mnk − i + l=1 k X cl,nk ) + mnk +1 + · · · + mr . l=1 Using Lemma 3.4, there exists Lj,1 , . . . , Lj,aj ∈ k[xj,0 , . . . , xj,nj ] for each 1 ≤ j ≤ k such that   aj k P P Y Y mn −i+ cl,nk mnk +1 m −i+ cl,1 r  L= Lj,q  ∈ ℘1 1 ∩ ℘nk +1 ∩ · · · ∩ ℘m ∩ · · · ∩ ℘nk k r , j=1 q=1 and L avoids P . This implies that LM ∈ J. THE REGULARITY OF POINTS IN MULTI-PROJECTIVE SPACES 11 Since Lj,q avoids P we can write Lj,q = xj,0 + Hj,q where Hj,q ∈ (xj,1 , . . . , xj,nj ) ⊆ ℘. Then 1 k L = xa1,0 + N where N ∈ ℘. Thus, since LM ∈ J, then GM ∈ (J + ℘i+1 ) which is what · · · xak,0 we need to prove.  Theorem 3.6. Suppose P1 , . . . , Ps are points in generic position in Pn1 × · · · × Pnk (s ≥ 2 and ms 1 n1 ≥ . . . ≥ nk ), and m1 ≥ m2 ≥ · · · ≥ ms are positive integers. Set I = ℘m 1 ∩ · · · ∩ ℘s . Then    Ps i=1 mi + nk − 2 ri(R/I) ≤ max m1 + m2 − 1, nk where [q] denotes the floor function. Proof. Note that n1 ≥ · · · ≥ nk , so  Ps k   Ps i=1 mi + nk − 2 i=1 mi + nj − 2 = max nk nj j=1 i h P Also, min{t | nk t ≥ q} = q+nnkk −1 . So, if we take q = ri=1 mi + mr+1 − 1 and use Proposition 3.5 and induction successively, along with Lemma 3.2 we will have the conclusion.  We obtain an immediate corollary which gives a bound on the regularity of the defining ideal of a scheme of fat points in Pn1 × · · · × Pnk . Corollary 3.7. With the hypotheses as in Theorem 3.6 we have    Ps i=1 mi + nk − 2 + k. reg(I) ≤ max m1 + m2 − 1, nk Remark 3.8. When k = 1 we recover the result of [5] which was proved to be sharp. Thus, our bound in Corollary 3.7 is sharp. References [1] A. Aramova, K. Crona, E. 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