Extremal Point Sets and Gorenstein Ideals

Geramita, Anthony V.; Harima, Tadahito; Shin, Yong-Su

doi:10.1006/aima.1998.1889

Cited by 39 publications

(36 citation statements)

References 22 publications

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“…Any two k-configurations in P 2 of the same type have the same minimal free resolution, and so the same Hilbert function ( [2,3]). We recall that if X is a linear star-configuration in P 2 of type r with 3 ≤ r, then X is a k-configuration in P 2 of type T = (1, 2, .…”

Section: The Union Of Two K-configurations In Pmentioning

confidence: 99%

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SOME APPLICATION OF THE UNION OF TWO 𝕜-CONFIGURATIONS IN ℙ²

Shin¹

2014

Journal of the Chungcheong Mathematical Society

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Section: The Union Of Two K-configurations In Pmentioning

confidence: 99%

“…If I := I X is the ideal of a subscheme X in P n , then we denote the Hilbert function of X by H X (t) := H(R/I X , t) (see [2,3]). Let X be a set of s points in P 2 .…”

Section: Introductionmentioning

confidence: 99%

SOME APPLICATION OF THE UNION OF TWO 𝕜-CONFIGURATIONS IN ℙ²

Shin¹

2014

Journal of the Chungcheong Mathematical Society

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“…Although this appears to be a very challenging problem, it has attracted a great deal of attention. There are: complete results for reduced subschemes of P 2 (this follows from results of Campanella [10]); a sharp upper bound for any Hilbert function (in terms of the graded Betti numbers) by Bigatti [3], Hulett [41] and Pardue [53]; complete results in low codimension and under the assumption that the coordinate ring of the reduced scheme is (in some way) special-e.g., for codimension two and codimension 3 Gorenstein see, e.g., Diesel [21] and Geramita and Migliore [32], while for Gorenstein rings with the Weak Lefschetz Property see, e.g., Geramita, Harima and Shin [29], Migliore and Nagel [49], where there are sharp upper bounds.…”

Section: Introductionmentioning

confidence: 99%

On the first infinitesimal neighborhood of a linear configuration of points in P2

Geramita¹,

Migliore²,

Sabourin³

2006

Journal of Algebra

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We consider the following open questions. Fix a Hilbert function h, that occurs for a reduced zero-dimensional subscheme of P 2 . Among all subschemes, X, with Hilbert function h, what are the possible Hilbert functions and graded Betti numbers for the first infinitesimal neighborhood, Z, of X (i.e. the double point scheme supported on X)? Is there a minimum (h min ) and maximum (h max ) such function?The numerical information encoded in h translates to a type vector which allows us to find unions of points on lines, called linear configurations, with Hilbert function h. We give necessary and sufficient conditions for the Hilbert function and graded Betti numbers of the first infinitesimal neighborhoods of all such linear configurations to be the same. Even for those h for which the Hilbert functions or graded Betti numbers of the resulting double point schemes are not uniquely determined, we give one (depending only on h) that does occur. We prove the existence of h max , in general, and discuss h min . Our methods include liaison techniques.

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“…In particular if c = 2, it induces a complex (8) 0 Proposition 9), and its open subscheme of Gorenstein quotients coincides, at least topologically and infinitesimally, with PGor(H) (the corresponding scheme of forms with "catalecticant structure"; see Theorem 11 below and the material before it). B is called unobstructed as a graded R-algebra iff GradAlg H B (R) is smooth at (R → B).…”

Section: Moreover a Is Unobstructed As A Graded R-algebra If And Onlmentioning

confidence: 99%

“…It is standard to use sums of geometrically linked ideals to construct Gorenstein algebras (e.g. [8], [9], [10], [15]). By the corresponding more general construction of using geometric Gorenstein linkage, we also get Gorenstein algebras (e.g.…”

Section: Moreover a Is Unobstructed As A Graded R-algebra If And Onlmentioning

confidence: 99%

Maximal families of Gorenstein algebras

Kleppe

2006

Trans. Amer. Math. Soc.

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Abstract. The purpose of this paper is to study maximal irreducible families of Gorenstein quotients of a polynomial ring R. Let GradAlg H (R) be the scheme parametrizing graded quotients of R with Hilbert function H. We prove there is a close relationship between the irreducible components of GradAlg H (R), whose general member is a Gorenstein codimension (c + 1) quotient, and the irreducible components of GradAlg H (R), whose general member B is a codimension c Cohen-Macaulay algebra of Hilbert function H related to H. If the Castelnuovo-Mumford regularity of the Gorenstein quotient is large compared to the Castelnuovo-Mumford regularity of B, this relationship actually determines a well-defined injective mapping from such "Cohen-Macaulay" components of GradAlg H (R) to "Gorenstein" components of GradAlg H (R), in which generically smooth components correspond. Moreover the dimension of the "Gorenstein" components is computed in terms of the dimension of the corresponding "Cohen-Macaulay" component and a sum of two invariants of B. Using linkage by a complete intersection we show how to compute these invariants. Linkage also turns out to be quite effective in verifying the assumptions which appear in a generalization of the main theorem.

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Extremal Point Sets and Gorenstein Ideals

Cited by 39 publications

References 22 publications

SOME APPLICATION OF THE UNION OF TWO 𝕜-CONFIGURATIONS IN ℙ²

SOME APPLICATION OF THE UNION OF TWO 𝕜-CONFIGURATIONS IN ℙ²

On the first infinitesimal neighborhood of a linear configuration of points in P2

Maximal families of Gorenstein algebras

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Extremal Point Sets and Gorenstein Ideals

Cited by 39 publications

References 22 publications

SOME APPLICATION OF THE UNION OF TWO 𝕜-CONFIGURATIONS IN ℙ2

SOME APPLICATION OF THE UNION OF TWO 𝕜-CONFIGURATIONS IN ℙ2

On the first infinitesimal neighborhood of a linear configuration of points in P2

Maximal families of Gorenstein algebras

Contact Info

Product

Resources

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SOME APPLICATION OF THE UNION OF TWO 𝕜-CONFIGURATIONS IN ℙ²

SOME APPLICATION OF THE UNION OF TWO 𝕜-CONFIGURATIONS IN ℙ²