Reduced Gorenstein codimension three subschemes of projective space

Geramita, Anthony V.; Migliore, Juan

doi:10.1090/s0002-9939-97-03956-7

Cited by 25 publications

(13 citation statements)

References 22 publications

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“…It is well known, by the structure theorem of Buchsbaum and Eisenbud [6] and by the results of Diesel [9], what are all the possible sets of graded Betti numbers for Gorenstein Artinian ideals of height 3. Geramita and Migliore, in their paper [10], show that every minimal free resolution which occurs for a Gorenstein artinian ideal of codimension 3, also occurs for some reduced set of points in P d , a stick figure curve in P 4 and more generally a "generalized" stick figure in P n . In this case, the points in P d solving the problems can be found as the intersection of two nice curves (stick figures) which have good properties.…”

Section: Basic Facts On Hadamard Product Of Varietiesmentioning

confidence: 99%

Gorenstein points in $\mathbb{P}^3$ via Hadamard product of projective varieties

Bocci¹,

Capresi²,

Carrucoli³

2021

Preprint

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We show how to construct a stick figure of lines in P 3 using the Hadamard product of projective varieties. Then, applying the results of Migliore and Nagel, we use such stick figure to build a Gorenstein set of points with given h−vector h. Since the Hadamard product is a coordinatewise product, we show, at the end, how the coordinates of the points, in the Gorenstein set, can be directly determined.

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Section: Basic Facts On Hadamard Product Of Varietiesmentioning

confidence: 99%

Gorenstein points in $\mathbb{P}^3$ via Hadamard product of projective varieties

Bocci¹,

Capresi²,

Carrucoli³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The ideal I G := I C 1 + I C 2 is the saturated ideal of an arithmetically Gorenstein zero-dimensional scheme X 1 with the stated h-vector, which, of course, then lies on both C 1 and C 2 . By making general choices, we can arrange that X 1 , C 1 and C 2 be reduced [12]. By using liaison addition, we can add plane curves to C 1 to obtain an arithmetically Cohen-Macaulay curve C with the stated h-vector, and since C ⊃ C 1 ⊃ X 1 , we have the desired inclusion.…”

Section: Non-unimodalitymentioning

confidence: 99%

“…Then C 1 can be linked by a sufficiently general complete intersection C of type (3,4) to another curve, C 2 , of the same degree and genus. The intersection of C 1 and C 2 is a reduced, arithmetically Gorenstein set of points X 1 [12] with h-vector (1, 3, 6, 3, 1). On the other hand, X 1 lies on the complete intersection curve C. Taking a cubic hypersurface section of C, we obtain a complete intersection set of points X 2 .…”

Section: Proposition 52 There Exist Reduced Level Zero-dimensional mentioning

confidence: 99%

“…It is also not known if there is an Artinian level algebra of type 2 that fails to have WLP. 1 Of course, as a result it is not known if there is a reduced set of points whose Artinian reduction has these properties, although it is known [12] that for any set of graded Betti numbers that occurs for an Artinian Gorenstein algebra of codimension 3, there is a reduced set of points in P 3 with these graded Betti numbers. One can also ask, for any t 0 , what is the smallest socle degree for which there is an Artinian level algebra (resp.…”

Section: Open Questionsmentioning

confidence: 99%

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The Geometry of the Weak Lefschetz Property and Level Sets of Points

Migliore¹

2008

Can. j. math.

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Abstract. In a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property (WLP), and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its Artinian reduction from having WLP, which in itself is a very algebraic notion. More precisely, we produce level sets of points whose Artinian reductions have socle types 3 and 4 and arbitrary socle degree ≥ 12 (in the worst case), but fail to have WLP. We also produce a level set of points whose Artinian reduction fails to have unimodal Hilbert function; our example is based on Zanello's example. Finally, we show that a level set of points can have Artinian reduction that has WLP but fails to have the Strong Lefschetz Property. While our constructions are all based on basic double G-linkage, the implementations use very different methods.

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“…The ideal I G := I C 1 + I C 2 is the saturated ideal of an arithmetically Gorenstein zero-dimensional scheme X 1 with the stated h-vector, which of course then lies on both C 1 and C 2 . By making general choices, we can arrange that X 1 , C 1 and C 2 be reduced [11]. By using Liaison Addition we can add plane curves to C 1 to obtain an arithmetically Cohen-Macaulay curve C with the stated h-vector, and since C ⊃ C 1 ⊃ X 1 we have the desired inclusion.…”

Section: Non-unimodalitymentioning

confidence: 99%

The Geometry of the Weak Lefschetz Property and Level Sets of Points

Migliore¹

2008

Canad. J. Math.

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In a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property (WLP), and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its Artinian reduction from having WLP, which in itself is a very algebraic notion. More precisely, we produce level sets of points whose Artinian reductions have socle types 3 and 4 and arbitrary socle degree ≥ 12 (in the worst case), but fail to have WLP. We also produce a level set of points whose Artinian reduction fails to have unimodal Hilbert function; our example is based on Zanello's example. Finally, we show that a level set of points can have Artinian reduction that has WLP but fails to have the Strong Lefschetz Property. While our constructions are all based on basic double G-linkage, the implementations use very different methods.

show abstract

Reduced Gorenstein codimension three subschemes of projective space

Cited by 25 publications

References 22 publications

Gorenstein points in $\mathbb{P}^3$ via Hadamard product of projective varieties

Gorenstein points in $\mathbb{P}^3$ via Hadamard product of projective varieties

The Geometry of the Weak Lefschetz Property and Level Sets of Points

The Geometry of the Weak Lefschetz Property and Level Sets of Points

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