Resolutions of generic points lying on a smooth quadric

Giuffrida, Sergio; Maggioni, Renato; Ragusa, Alfio

doi:10.1007/bf02567964

Cited by 25 publications

(28 citation statements)

References 2 publications

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“…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,

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: Introductionmentioning

confidence: 99%

ACM sets of points in multiprojective space

Guardo¹,

Tuyl

2008

Collect. Math.

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“…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: 89%

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

Tuyl

2005

J. London Math. Soc.

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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 ).

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“…Farkas, Mustaţǎ, and Popa proved in [9] that if X ⊂ P n is a canonical curve, then the Minimal Resolution Conjecture holds for a set of general points of sufficiently large degree, and that it fails always if X is a curve of sufficiently large degree. The conjecture was shown to be true for any general set of points on a smooth quadric surface in P 3 by Guiffrida et al (see [11]). …”

Section: Introductionmentioning

confidence: 91%