2002
DOI: 10.1016/s0022-4049(02)00072-5
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The border of the Hilbert function of a set of points in Pn1×⋯×Pnk

Abstract: We describe the eventual behaviour of the Hilbert function of a set of distinct points in P n 1 × · · · × P n k . As a consequence of this result, we show that the Hilbert function of a set of points in P n 1 × · · · × P n k can be determined by computing the Hilbert function at only a finite number of values. Our result extends the result that the Hilbert function of a set of points in P n stabilizes at the cardinality of the set of points. Motivated by our result, we introduce the notion of the border of the… Show more

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Cited by 21 publications
(38 citation statements)
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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%
See 1 more Smart Citation
“…, 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%
“…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%
“…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%
“…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%