Resolutions of O-dimensional subschemes of a smooth quadric

Journal of Pure and Applied Algebra

2002

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 Hilbert function of a set of points. By using the Gale-Ryser Theorem, a classical result about (0, 1)-matrices, we characterize all the possible borders for the Hilbert function of a set of distinct points in P 1 × P 1 .

Section: Classifying the Borders Of Hilbert Functions Of Points Inmentioning

confidence: 92%

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

Journal of Pure and Applied Algebra

2002

“…In this situation there are several classifications. Giuffrida, Maggioni, and Ragusa [7], who helped to initiate the study of points in multiprojective spaces (see, for example [8,9,12,13,14,18,20,21,22] for more on these points), provided the first classification. They showed that ACM sets of points in P 1 × P 1 can be classified via their Hilbert functions.…”

Section: Introductionmentioning

confidence: 99%

ACM sets of points in multiprojective space

Guardo¹,

2008

Collect. Math.

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.

“…When k = 1, v(s; n) equals the expected value for ν(I X ) as predicted by the ideal generation conjecture. Furthermore, using [9,10], we show that ν(I X ) = v(s; 1, 1) for a sufficiently general set of s points in P 1 × P 1 .…”

Section: Introductionmentioning

confidence: 90%

“…Although such sets have been studied (cf. [9,10]) we could find no proof in the literature for the existence of such sets when k 2 (the case k = 1 is [5, Theorem 4]). We therefore begin by providing a proof of this 'folklore' result.…”

Section: Points In Generic Positionmentioning

confidence: 99%

“…In [9,10] Giuffrida, Maggioni and Ragusa computed the minimal free resolution of points in generic position in P 1 × P 1 , and in particular, showed that ν(I X ) = v on some non-empty open set of (P 1 × P 1 ) s . Since their notation and approach is different to ours, for the convenience of the reader we make this connection more transparent.…”

Section: The Expected Value Of ν(I X )mentioning

confidence: 99%

See 1 more Smart Citation

The Defining Ideal of a Set of Points in Multi-Projective Space

2005

J. London Math. Soc.

The defining ideal I X of a set of points X in P n 1 × . . . × P n k is investigated with a special emphasis on the case when X is in generic position, that is, X has the maximal Hilbert function. When X is in generic position, the degrees of the generators of the associated ideal I X are determined. ν(I X ) denotes the minimal number of generators of I X , and this description of the degrees is used to construct a function v(s; n 1 , . . . , n k ) with the property that ν(I X ) v(s; n 1 , . . . , n k ) always holds for s points in generic position in P n 1 × . . . × P n k . When k = 1, v(s; n 1 ) equals the expected value for ν(I X ) as predicted by the ideal generation conjecture. If k 2, it is shown that there are cases with ν(I X ) > v(s; n 1 , . . . , n k ). However, computational evidence suggests that in many cases ν(I X ) = v(s; n 1 , . . . , n k ).