Castelnuovo–Mumford regularity by approximation

Derksen, Harm; Sidman, Jessica

doi:10.1016/j.aim.2003.10.001

Cited by 22 publications

(21 citation statements)

References 17 publications

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“…A differential operator homogeneous of order 1 is nothing but a derivation. Hence D (1) (A) is the module of logarithmic derivations along A.…”

Section: The Modules Of Differential Operators For a Hyperplane Arran...mentioning

confidence: 99%

“…To prove that D (1) (A) is a free S-module, the Saito criterion ([8, Theorem 1.8 (ii)], see also [6,Theorem 4.19]) is very useful. Holm [4] generalized the Saito criterion to the one for D (m) (A).…”

Section: Saito-holm Criterionmentioning

confidence: 99%

“…In the derivation module case, when n ≥ 3, r > n, m < r − n + 1 with m = 1, Rose-Terao [7] and Yuzvinsky [11] independently gave a minimal free resolution of D (1) (A). In the course of the proof, Rose-Terao [7] gave minimal free resolutions of all modules of logarithmic differential forms with poles along A.…”

Section: Introductionmentioning

confidence: 99%

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The freeness and minimal free resolutions of modules of differential operators of a generic hyperplane arrangement

Nakashima¹,

Okuyama²,

Saito³

2012

Journal of Algebra

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Let A be a generic hyperplane arrangement composed of r hyperplanes in an n-dimensional vector space, and S the polynomial ring in n variables. We consider the S-submodule D(m)(A) of the nth Weyl algebra of homogeneous differential operators of order m preserving the defining ideal of A. We prove that if n ≥ 3, r > n, m > r - n + 1, then D(m)(A) is free (Holm's conjecture). Combining this with some results by Holm, we see that D(m)(A) is free unless n ≥ 3, r > n, m < r - n + 1. In the remaining case, we construct a minimal free resolution of D(m)(A) by generalizing Yuzvinsky's construction for m = 1. In addition, we construct a minimal free resolution of the transpose of the m-jet module, which generalizes a result by Rose and Terao for m = 1

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“…A differential operator homogeneous of order 1 is nothing but a derivation. Hence D (1) (A) is the module of logarithmic derivations along A.…”

Section: The Modules Of Differential Operators For a Hyperplane Arran...mentioning

confidence: 99%

Section: Saito-holm Criterionmentioning

confidence: 99%

See 1 more Smart Citation

The freeness and minimal free resolutions of modules of differential operators of a generic hyperplane arrangement

Nakashima¹,

Okuyama²,

Saito³

2012

Journal of Algebra

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“…For one proof of this, see Jiang and Feng [11], §4.2. Finally, we note that Derksen and Sidman [4] have recently obtained regularity bounds on D(A) for higher dimensional arrangements.…”

Section: Corollary 35 For An Arrangement On D Lines Reg(d) ≤ D − 2mentioning

confidence: 99%

Elementary modifications and line configurations in P^2

Schenck¹

2003

Comment. Math. Helv.

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Abstract. Associated to a projective arrangement of hyperplanes A ⊆ P n is the module D(A), which consists of derivations tangent to A. We study D(A) when A is a configuration of lines in P 2 . In this setting, we relate the deletion/restriction construction used in the study of hyperplane arrangements to elementary modifications of bundles. This allows us to obtain bounds on the Castelnuovo-Mumford regularity of D(A). We also give simple combinatorial conditions for the associated bundle to be stable, and describe its jump lines. These regularity bounds and stability considerations impose constraints on Terao's conjecture.

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“…Sidman and the author proved reg(I ) ≤ m in [3,4]. Easy considerations show that these regularity bounds imply the existence of polynomials h I (d) and h J (d) such that …”

Section: Subspace Arrangementsmentioning

confidence: 99%

Hilbert series of subspace arrangements

Derksen

2007

Journal of Pure and Applied Algebra

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The vanishing ideal I of a subspace arrangement V 1 ∪ V 2 ∪ · · · ∪ V m ⊆ V is an intersection I 1 ∩ I 2 ∩ · · · ∩ I m of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of the product ideal J = I 1 I 2 · · · I m without any assumptions about the subspace arrangement. It turns out that the Hilbert series of J is a combinatorial invariant of the subspace arrangement: it only depends on the intersection lattice and the dimension function. The graded Betti numbers of J are determined by the Hilbert series, so they are combinatorial invariants as well. We will also apply our results to generalized principal component analysis (GPCA), a tool that is useful for computer vision and image processing.

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Castelnuovo–Mumford regularity by approximation

Cited by 22 publications

References 17 publications

The freeness and minimal free resolutions of modules of differential operators of a generic hyperplane arrangement

The freeness and minimal free resolutions of modules of differential operators of a generic hyperplane arrangement

Elementary modifications and line configurations in P^2

Hilbert series of subspace arrangements

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