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- W1591331750 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 <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=l> <mml:semantics> <mml:mi>l</mml:mi> <mml:annotation encoding=application/x-tex>l</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-space can be arbitrarily large for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=l greater-than-or-equal-to 4> <mml:semantics> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>l geq 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contrary to the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=l equals 3> <mml:semantics> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>l=3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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." @default.
- W1591331750 created "2016-06-24" @default.
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- W1591331750 date "1999-11-29" @default.
- W1591331750 modified "2023-09-26" @default.
- W1591331750 title "Cohen-Macaulayness of tangent cones" @default.
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- W1591331750 doi "https://doi.org/10.1090/s0002-9939-99-05229-6" @default.
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