Geometric consequences of extremal behavior in a theorem of Macaulay

Bigatti, Anna Maria; Geramita, Anthony V.; Migliore, Juan

doi:10.1090/s0002-9947-1994-1272673-7

Cited by 62 publications

(147 citation statements)

References 10 publications

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“…, 1). Since, by assumption of non-unimodality, the second generator comes in degree ≤ [9] or [4]), it follows that (I, L 1 , L 2 )/(L 1 , L 2 ) has a GCD of degree 1 in degree e 2 + 1. We claim that by the genericity of L 1 and L 2 , this implies that I also has a GCD of degree 1 in the same degree.…”

Section: +2mentioning

confidence: 99%

A characterization of Gorenstein Hilbert functions in codimension four with small initial degree

Migliore

Nagel

Zanello

2008

Mathematical Research Letters

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Abstract. The main goal of this paper is to characterize the Hilbert functions of all (artinian) codimension 4 Gorenstein algebras that have at least two independent relations of degree four. This includes all codimension 4 Gorenstein algebras whose initial relation is of degree at most 3. Our result shows that those Hilbert functions are exactly the so-called SI-sequences starting with (1, 4, h 2 , h 3 , ...), where h 4 ≤ 33. In particular, these Hilbert functions are all unimodal.We also establish a more general unimodality result, which relies on the values of the Hilbert function not being too big, but is independent of the initial degree.

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Section: +2mentioning

confidence: 99%

A characterization of Gorenstein Hilbert functions in codimension four with small initial degree

Migliore

Nagel

Zanello

2008

Mathematical Research Letters

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“…But it is not, for example, known which functions arise as Hilbert functions of points taken with higher multiplicity, even if the induced reduced subscheme consists of generic points; see, for example, [5] and its bibliography, [11,20,21,24]. Other work has focused on what one knows as a consequence of knowing the Hilbert function; e.g., [1] shows how the growth of the Hilbert function of a set of points influences its geometry, [3] studies how the Hilbert function constrains the graded Betti numbers in case I has height 2, and [8] studies graded Betti numbers but more generally for graded modules M over R. Other work has focused on what one knows as a consequence of knowing the Hilbert function; e.g., [1] shows how the growth of the Hilbert function of a set of points influences its geometry, [3] studies how the Hilbert function constrains the graded Betti numbers in case I has height 2, and [8] studies graded Betti numbers but more generally for graded modules M over R.…”

Section: Introductionmentioning

confidence: 99%

Classifying Hilbert functions of fat point subschemes in ℙ2

2009

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The paper [10] raised the question of what the possible Hilbert functions are for fat point subschemes of the form 2p 1 +· · · + 2p r , for all possible choices of r distinct points in P 2 . We study this problem for r points in P 2 over an algebraically closed field k of arbitrary characteristic in case either r ≤ 8 or the points lie on a (possibly reducible) conic. In either case, it follows from [17,18] that there are only finitely many configuration types of points, where our notion of configuration type is a generalization of the notion of a representable combinatorial geometry, also known as a representable simple matroid. (We say p 1 , . . . , p r and p 1 , . . . , p r have the same configuration type if for all choices of nonnegative integers m i , Z = m 1 p 1 + · · · + m r p r and Z = m 1 p 1 + · · · + m r p r have 159 Collectanea Mathematica (electronic version): http://www.collectanea.ub.edu Geramita, Harbourne and Migliore the same Hilbert function.) Assuming either that 7 ≤ r ≤ 8 (see [12] for the cases r ≤ 6) or that the points p i lie on a conic, we explicitly determine all the configuration types, and show how the configuration type and the coefficients m i determine (in an explicitly computable way) the Hilbert function (and sometimes the graded Betti numbers) of Z = m 1 p 1 + · · · + m r p r . We demonstrate our results by explicitly listing all Hilbert functions for schemes of r ≤ 8 double points, and for each Hilbert function we state precisely how the points must be arranged (in terms of the configuration type) to obtain that Hilbert function.

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“…The 2-type vector (1,3,4) is not consecutive, as it fails to satisfy condition iÞ for l ¼ 1 and iiÞ for l ¼ 2. If T ¼ ðð2; 3Þ; ð1; 2; 3; 4ÞÞ, then T is consecutive, but is neither 2-consecutive nor 3-consecutive, as it fails to satisfy condition iiÞ for l ¼ 2; 3.…”

Section: Consecutivity In N-type Vectorsmentioning

confidence: 99%

“…(1,2,3,4), (1,2,3,4,5)) and its truncation at 28 À 10 ¼ 18 is H 0 = 1 4 10 18 !, which corresponds to the 1-consecutive 3-type vector T 0 = ((2),(1,2,3), (1,2,3,4)). …”

Section: N-type Vectors 3905mentioning

confidence: 99%