On Prime Ideals with Generic Zero x i = t n i

Bresinsky, H.

doi:10.2307/2039739

Cited by 42 publications

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“…Our first aim is to give a complete description of the defining ideal I(C m ). From our computations in Macaulay [2] with particular values for m, we formulate a set of generators and prove that the set formulated is indeed a generator set for I(C m ) by applying the method Bresinsky used in [3] which depends on work of Herzog on semigroups [7].…”

Section: A Family Of Monomial Curves In 4-space Which Have CM Tangentmentioning

confidence: 99%

Cohen-Macaulayness of tangent cones

Arslan¹

1999

Proc. Amer. Math. Soc.

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Abstract. We give a criterion for checking the Cohen-Macaulayness of the tangent cone of a monomial curve by using the Gröbner basis. For a family of monomial curves, we give the full description of the defining ideal of the curve and its tangent cone at the origin. By using this family of curves and their extended versions to higher dimensions, we prove that the minimal number of generators of a Cohen-Macaulay tangent cone of a monomial curve in an affine l-space can be arbitrarily large for l ≥ 4 contrary to the l = 3 case shown by Robbiano and Valla. We also determine the Hilbert series of the associated graded ring of this family of curves and their extended versions.

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Section: A Family Of Monomial Curves In 4-space Which Have CM Tangentmentioning

confidence: 99%

Cohen-Macaulayness of tangent cones

Arslan¹

1999

Proc. Amer. Math. Soc.

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“…For this subclass we try to determine (see Proposition 3.3) the Cohen-Macaulay defect B(Γ ) using the explicit description (see Proposition 3.2) of the standard basis of Γ ; in particular, we prove that these balanced semigroups are 2-good and determine (see Theorem 3.4) when exactly G(Γ ) is Cohen-Macaulay. It is interesting to note that our subclass contains the special classes of semigroups considered by Kraft (see [8], [9]); Bresinsky considered a similar class of examples in [2]. Bresinsky and Kraft constructed these special classes to show that the defining ideals P(m 0 , m 1 , m 2 , m 3 ) of the monomial curves C(m 0 , m 1 , m 2 , m 3 ) ⊆ A 4 K with parametrization X 0 = T m 0 , X 1 = T m 1 , X 2 = T m 2 , X 3 = T m 3 and the type of R can be arbitrarily large, respectively (unless the semigroup is symmetric, see [3]).…”

Section: Introductionmentioning

confidence: 99%

CM defect and Hilbert functions of monomial curves

Patil

Tamone

2011

Journal of Pure and Applied Algebra

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a b s t r a c tIn this article we consider a semigroup ring R = K [[Γ ]] of a numerical semigroup Γ and study the Cohen-Macaulayness of the associated graded ring G(Γ ) := gr m (R) := ⊕ n∈N m n /m n+1 and the behaviour of the Hilbert function H R of R. We define a certain (finite) subset B(Γ ) ⊆ Γ and prove that G(Γ ) is Cohen-Macaulay if and only if B(Γ ) = ∅. Therefore the subset B(Γ ) is called the Cohen-Macaulay defect of G(Γ ). Further, we prove that if the degree sequence of elements of the standard basis of Γ is non-decreasing, then B(Γ ) = ∅ and hence G(Γ ) is Cohen-Macaulay. We consider a class of numerical semigroups Γ = ∑ 3 i=0 Nm i generated by 4 elements m 0 , m 1 , m 2 , m 3 such that m 1 + m 2 = m 0 + m 3 -so called ''balanced semigroups''. We study the structure of the Cohen-Macaulay defect B(Γ ) of Γ and particularly we give an estimate on the cardinality |B(Γ , r)| for every r ∈ N. We use these estimates to prove that the Hilbert function of R is nondecreasing. Further, we prove that every balanced ''unitary'' semigroup Γ is ''2-good'' and is not ''1-good'', in particular, in this case, G(Γ ) is not Cohen-Macaulay. We consider a certain special subclass of balanced semigroups Γ . For this subclass we try to determine the Cohen-Macaulay defect B(Γ ) using the explicit description of the standard basis of Γ ; in particular, we prove that these balanced semigroups are 2-good and determine when exactly G(Γ ) is Cohen-Macaulay.

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“…, m − 1} there exists a numerical semigroup, which is symmetric, has multiplicity m and the cardinality of its minimal presentations is equal to e(e − 1)/2 − 1. The second one is that in [2], Bresinsky gives a family of numerical semigroups with embedding dimension equal to 4 and with the cardinality of its minimal presentations arbitrarily large. This proves that the cardinality of a minimal presentation for a numerical semigroup cannot be bounded as a function of its embedding dimension.…”

Section: Introductionmentioning

confidence: 99%