Integral arithmetically Buchsbaum curves in P3

Amasaki, Mutsumi

doi:10.2969/jmsj/04110001

Cited by 10 publications

(5 citation statements)

References 5 publications

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“…In this special case, this result can be deduced from [1] and is proved in a different way in [10]. In this special case, this result can be deduced from [1] and is proved in a different way in [10].…”

Section: V(7 K ) <mentioning

confidence: 83%

Submodules of the Deficiency Module

Migliore¹

1993

Journal of the London Mathematical Society

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Section: V(7 K ) <mentioning

confidence: 83%

Submodules of the Deficiency Module

Migliore¹

1993

Journal of the London Mathematical Society

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“…In the case where V is an arithmetically Buchsbaum curve, of course, K A is the entire deficiency module M(V) and v(K A ) is the dimension of this module as a k-vector space. In this special case, this result can be deduced from [1] and is proved in a different way in [10]. EXAMPLE 3.4.…”

Section: V(7 K ) <mentioning

confidence: 85%

Submodules of the Deficiency Module

Migliore¹

1998

Introduction to Liaison Theory and Deficiency Modules

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“…Thus the Buchsbaum case is particularly easy to calculate in examples. In this special case, the bound can be obtained from [1].…”

Section: ⊓ ⊔mentioning

confidence: 99%

“…Finally, in Section 3, we give an extension of a surprising result of Amasaki, [1], showing a lower bound for the least degree of a minimal generator of the ideal of a Buchsbaum subscheme. Originally, Amasaki gave a bound in the case of Buchsbaum curves in P 3 .…”

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confidence: 99%

Submodules of the Deficiency Modules and an Extension of Dubreil's Theorem

Martin

Migliore

1997

Journal of the London Mathematical Society

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In this paper, we consider a basic question in commutative algebra : if I and J are ideals of a commutative ring S, when does IJ l IEJ ? More precisely, our setting will be in a polynomial ring k[x ! , … , x n ], and the ideals I and J will define subschemes of the projective space n k over k. In this situation, we are able to relate the equality of product and intersection to the behavior of the cohomology modules of the subschemes defined by I and J. By doing this, we are able to prove several general algebraic results about the defining ideals of certain subschemes of projective space.Our main technique in this paper is a study of the deficiency modules of a subscheme V of n . These modules are important algebraic invariants of V, and reflect many of the properties of V, both geometric and algebraic. For instance, when V is equidimensional and dim V 1, the deficiency modules of V are invariant (up to a shift in grading) along the even liaison class of V [14, 11, 15, 7], although they do not in general completely determine the even liaison class, except in the case of curves in $ [14]. On the algebraic side, at least for curves in $, the deficiency modules have been shown to have connections to the number and degrees of generators of the saturated ideal defining V [12]. One of our main goals in this paper is to extend these results to subschemes of arbitrary codimension in any projective space n .We now describe the contents of this paper more precisely. In the first section, we set up our notation and give the basic definitions which we will use. Then we prove our main technical result : if I and J define subschemes V and Y, respectively, of n , we relate the quotient module (IEJ )\IJ to the cohomology of V, at least when V and Y meet properly. We are then able to give a different proof of a general statement due to Serre about when there is an equality of intersection and product.In the second section, we give an extension of Dubreil's Theorem on the number of generators of ideals in a polynomial ring. Specifically, our generalization works for an ideal I defining a locally Cohen-Macaulay, equidimensional subscheme V of any codimension in n , and relates the number of generators of the defining ideal to the length of certain Koszul homologies of the cohomology of V. The results in this section depend crucially on the identification done in Section 1 of the intersection modulo the product.Finally, in Section 3, we give an extension of a surprising result of Amasaki [1] showing a lower bound for the least degree of a minimal generator of the ideal of a Buchsbaum subscheme. Originally, Amasaki gave a bound in the case of Buchsbaum curves in $ (and later gave a natural extension to Buchsbaum codimension 2 subschemes of n [2]). Easier proofs were subsequently given by Geramita and

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Integral arithmetically Buchsbaum curves in P3

Cited by 10 publications

References 5 publications

Submodules of the Deficiency Module

Submodules of the Deficiency Module

Submodules of the Deficiency Module

Submodules of the Deficiency Modules and an Extension of Dubreil's Theorem

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