The border of the Hilbert function of a set of points in Pn1×⋯×Pnk

Tuyl, Adam Van

doi:10.1016/s0022-4049(02)00072-5

Cited by 21 publications

(38 citation statements)

References 18 publications

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“…, R i,t i } be the set of the distinct ith coordinates of the points that appear in X. If L R i,j is the form of degree e i that passes through the point R i,j , then [20,Proposition 4.6]). Because x i,0 is a nonzero divisor, the short exact sequence Now consider any i ∈ N r such that i (t 1 − 1, .…”

Section: Lemma 39mentioning

confidence: 99%

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ACM sets of points in multiprojective space

Guardo¹,

Tuyl

2008

Collect. Math.

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If X is a finite set of points in a multiprojective space P n 1 ×· · ·×P nr with r ≥ 2, then X may or may not be arithmetically Cohen-Macaulay (ACM). For sets of points in P 1 × P 1 there are several classifications of the ACM sets of points. In this paper we investigate the natural generalizations of these classifications to an arbitrary multiprojective space. We show that each classification for ACM points in P 1 × P 1 fails to extend to the general case. We also give some new necessary and sufficient conditions for a set of points to be ACM.

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Section: Lemma 39mentioning

confidence: 99%

“…1 We begin by recalling some relevant results about points in a multiprojective space. A more thorough introduction to points in a multiprojective space can be found in [20,21]. In this paper k denotes an algebraically closed field of characteristic zero.…”

Section: Introductionmentioning

confidence: 99%

ACM sets of points in multiprojective space

Guardo¹,

Tuyl

2008

Collect. Math.

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“…This fact compared to (5) gives us that j ≤ n − r 1 + 1, but by hypothesis j ≥ n − r 1 + 1 and so it must be j = n − r 1 + 1. This implies that the inequality in (7) is an equality, which means that M…”

Section: By Theorem 25 We Know Thatmentioning

confidence: 93%

“…A good reference for a general discussion on zero-dimensional schemes on P 1 × P 1 is [2] and further results about the Hilbert function of zero-dimensional schemes can be found in [4], [5] and [7].…”

Section: Introductionmentioning

confidence: 99%

On the Hilbert function of zero-dimensional schemes in $${{\mathbb P}^1 \times {\mathbb P}^1}$$

Bonacini

Marino

2010

Collect. Math.

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Let Q = P 1 × P 1 and let X ⊂ Q be a zero-dimensional scheme. The results in this paper give the possibility of computing, under certain hypotheses, the Hilbert function of a zero-dimensional scheme in Q that is not ACM. In particular we show how, under some conditions on X , its Hilbert function changes when we add points to X lying on a (1, 0) or (0, 1)-line. As a particular case we show also that if X is ACM this result holds without any additional hypothesis.

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“…Recently, many authors (cf. [4,[9][10][11]16]) have been interested in extending our understanding of points in P n to sets of points in P n1 × · · · × P n k . We continue this trend by studying reg(I ) when I deÿnes a scheme of fat points in P n1 × · · · × P n k .…”

Section: Introductionmentioning

confidence: 99%

The regularity of points in multi-projective spaces

Harbourne

Tuyl

2004

Journal of Pure and Applied Algebra

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Let I =˝m 1 1 ∩ · · · ∩˝m s s be the deÿning ideal of a scheme of fat points in P n 1 × · · · × P n k with support in generic position. When all the mi's are 1, we explicitly calculate the CastelnuovoMumford 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.In this paper, we study the Castelnuovo-Mumford regularity of deÿning ideals of sets of points (reduced and non-reduced) in a multi-projective space P n1 × · · · × P n k .If I ⊆ k[x 0 ; · · · ; x n ] is the deÿning ideal of a projective variety X ⊆ P n , then the Castelnuovo-Mumford 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 deÿning 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.

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The border of the Hilbert function of a set of points in Pn1×⋯×Pnk

Cited by 21 publications

References 18 publications

ACM sets of points in multiprojective space

ACM sets of points in multiprojective space

On the Hilbert function of zero-dimensional schemes in $${{\mathbb P}^1 \times {\mathbb P}^1}$$

The regularity of points in multi-projective spaces

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