On the defining equations of the tangent cone of a numerical semigroup ring

Jürgen Herzog; Dumitru I. Stamate

doi:10.1016/j.jalgebra.2014.07.008

paper

On the defining equations of the tangent cone of a numerical semigroup ring

arXiv:1308.4644 · doi:10.1016/j.jalgebra.2014.07.008

Abstract

Let $a = a_{1} < \dots < a_{r}$ be a sequence of positive integers, and let $H_{a}$ denote the semigroup generated by $a_{1}, \dots, a_{r}$ . For an integer $k \geq 0$ we denote by $a + k$ the shifted sequence $a_{1} + k, \dots, a_{r} + k$ . Fix a field $K$ . We show that for all $k ≫ 0$ the associated graded ring of the semigroup ring $K [H_{a + k}]$ is Cohen--Macaulay and that it has the same Betti numbers as $K [H_{a + k}]$ itself. As a consequence, we show that the number of defining equations of the tangent cone of a numerical semigroup ring is bounded by a value depending only on the width of the semigroup, where the width of a numerical semigroup is defined to be the difference of the largest and the smallest element in the minimal generating set of the semigroup. We also provide a conjectured upper bound of the above number of equations and we verify it in some cases.

18 pages. v2: added references to previous work of L. Sharifan and R. Zaare-Nahandi; v3: fixed some typos and extended Proposition 2.10. to appear in Journal of Algebra

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Minimal free resolutions for certain affine monomial curve

On the defining equations of the tangent cone of a numerical semigroup ring

Abstract

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