Sharp bounds on Castelnuovo-Mumford regularity

Miyazaki, Chikashi

doi:10.1090/s0002-9947-99-02380-6

Cited by 15 publications

(9 citation statements)

References 24 publications

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“…But their methods do not give a result as good as Theorem 4.8. Moreover, C. Miyazaki, see [13], has characterized those varieties for which the bound of 1.3 is attained.…”

Section: §2 Degree Bounds For the Generators Of Local Cohomology Modmentioning

confidence: 99%

Degree bounds for generators of cohomology modules and Castelnuovo-Mumford regularity

Nagel

Schenzel

1998

Nagoya Mathematical Journal

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Abstract. By extending Mumford's result on the generating by global sections there are estimates on the degree for generators of local cohomology modules. These arguments provide bounds on the Castelnuovo-Mumford regularity, in particular for Cohen-Macaulay varieties. As an application they imply a few more cases of varieties that satisfy a conjecture posed by Eisenbud and Goto. §1. Introduction Let T denote a coherent sheaf on the projective space F n = P^, K denotes an algebraically closed field. In Note that regT -rι(T) is called the Castelnuovo-Mumford regularity of) for all k > m. By Serre's vanishing result this is true for all m ^> 0. More precisely, Mumford's result, see loc. cit., says e^T) < regjF. Its extension is our first main result.

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“…But their methods do not give a result as good as Theorem 4.8. Moreover, C. Miyazaki, see [13], has characterized those varieties for which the bound of 1.3 is attained.…”

Section: §2 Degree Bounds For the Generators Of Local Cohomology Modmentioning

confidence: 99%

Degree bounds for generators of cohomology modules and Castelnuovo-Mumford regularity

Nagel

Schenzel

1998

Nagoya Mathematical Journal

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“…We calculate the regularity of the graded S-module K (n) in Section five. The interest in this topic is reflected by the existence of papers like [14] and Hoa's conjecture [13]. In Section six we show that the symbolic Rees ring of K is Noetherian.…”

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confidence: 93%

Divisors on rational normal scrolls

Kustin¹,

Polini²,

Ulrich³

2009

Journal of Algebra

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Let A be the homogeneous coordinate ring of a rational normal scroll. The ring A is equal to the quotient of a polynomial ring S by the ideal generated by the two by two minors of a scroll matrix ψ with two rows and catalecticant blocks. The class group of A is cyclic, and is infinite provided is at least two. One generator of the class group is [ J ], where J is the ideal of A generated by the entries of the first column of ψ. The positive powers of J are well-understood; if is at least two, then the nth ordinary power, the nth symmetric power, and the nth symbolic power coincide and therefore all three nth powers are resolved by a generalized Eagon-Northcott complex. The inverse of [ J ] in the class group of A is [K ], where K is the ideal generated by the entries of the first row of ψ. We study the positive powers of [K ].We obtain a minimal generating set and a Gröbner basis for the preimage in S of the symbolic power K (n) . We describe a filtration of K (n) in which all of the factors are Cohen-Macaulay S-modules resolved by generalized Eagon-Northcott complexes. We use this filtration to describe the modules in a finely graded resolution of K (n) by free S-modules. We calculate the regularity of the graded S-module K (n) and we show that the symbolic Rees ring of K is Noetherian.

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“…For example, it holds for smooth surfaces in characteristic zero ( [Laz87]), connected reduced curves ( [Gia05]), etc. Inspired by the conjecture, there are also many attempts to give an upper bound for the Castelnuovo-Mumford regularity for various types of algebraic and geometric structures ( [Stu95], [Kwa98], [Miy00], [DS02], etc).…”

Section: Introductionmentioning

confidence: 99%