The Tangent Cone of a Local Ring of Codimension 2

Mandal, Mousumi; Rossi, Maria Evelina

doi:10.1007/s40306-014-0102-z

Cited by 4 publications

(4 citation statements)

References 14 publications

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“…A different proof of (ii) is given by M. Mandal and M.E. Rossi in [MR,Theorem 2.1]. The statement (iii) is obvious (see discussion after Lemma 8.1).…”

Section: Number Of Generators For a Single-block Partitionmentioning

confidence: 95%

Jordan types for graded Artinian algebras in height two

Altafi¹,

Iarrobino²,

Khatami³

et al. 2020

Preprint

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We let A = R/I be a standard graded Artinian algebra quotient of R = k[x, y], the polynomial ring in two variables over a field k by an ideal I, and let n be its vector space dimension. The Jordan type P ℓ of a linear form ℓ ∈ A 1 is the partition of n determining the Jordan block decomposition of the multiplication on A by ℓ -which is nilpotent. The first three authors previously determined which partitions of n = dim k A may occur as the Jordan type for some linear form ℓ on a graded complete intersection Artinian quotient A = R/(f, g) of R, and they counted the number of such partitions for each complete intersection Hilbert function T [AIK].We here consider the family G T of graded Artinian quotients A = R/I of R = k[x, y], having arbitrary Hilbert function H(A) = T . The cell V(E P ) corresponding to a partition P having diagonal lengths T is comprised of all ideals I in R whose initial ideal is the monomial ideal E P determined by P . These cells give a decomposition of the variety G T into affine spaces. We determine the generic number κ(P ) of generators for the ideals in each cell V(E P ), generalizing a result

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“…A different proof of (ii) is given by M. Mandal and M.E. Rossi in [MR,Theorem 2.1]. The statement (iii) is obvious (see discussion after Lemma 8.1).…”

Section: Number Of Generators For a Single-block Partitionmentioning

confidence: 95%

Jordan types for graded Artinian algebras in height two

Altafi¹,

Iarrobino²,

Khatami³

et al. 2020

Preprint

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“…This reduces the problem to codimension two, where we can take advantage of the rich literature (see [Ber09], [Bri77], [GHK06], [HW20], [Iar94], [MR15], [Mac1904]). We recall the following well-known result on the characterization of Hilbert functions of codimension two complete intersections which we will use frequently in the paper: the O-sequence h = (1, 2, h 2 , .…”

Section: Necessary Conditions For Gorenstein Sequencesmentioning

confidence: 99%

“…By Lemma 3.1 and Proposition 3.3, we have HF R/I (i) = HF S/J (i) for all i ≥ 2. As a consequence ∆(h) ≤ 2 since J ⊥ = F, G (see for instance [MR15]). Moreover, by (a) and (b) it follows that HF S/J (i) = HF S/ Ann S (F) (i) for all i except one position.…”

Section: Necessary Conditions For Gorenstein Sequencesmentioning

confidence: 99%

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On the Hilbert function of Artinian local complete intersections of codimension three

Jelisiejew¹,

Masuti²,

Rossi³

2022

Preprint

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In singularity theory or algebraic geometry, it is natural to investigate possible Hilbert functions for special algebras A such as local complete intersections or more generally Gorenstein algebras. The sequences that occur as the Hilbert functions of standard graded complete intersections are well understood classically thanks to Macaulay and Stanley. Very little is known in the local case except in codimension two. In this paper we characterise the Hilbert functions of quadratic Artinian complete intersections of codimension three. Interestingly we prove that a Hilbert function is admissible for such a Gorenstein ring if and only if is admissible for such a complete intersection. We provide an effective construction of a local complete intersection for a given Hilbert function. We prove that the symmetric decomposition of such a complete intersection ideal is determined by its Hilbert function.

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