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

By Jürgen Herzog, Dumitru I. Stamate. 2013

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Abstract Let a = a 1 … a r be a sequence of positive integers, and let H a denote the semigroup generated by a 1 , … , a r . For an integer k ≥ 0 we denote by a + k the shifted sequence a 1 + k , … , a r + k . Fix a field K . We show that for all k ≫ 0 the tangent cone 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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8d3431209e677bfb52b9b0bc8a18fc96e0d5e8a4

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In Algebra and Number Theory

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

The trace of the canonical module (2016, background): “This goes in the same direction as a recent number of other results about eventually periodic properties in this shifted family, see [23], [39], [18].”