On the Cohen-Macaulay Property for Quadratic Tangent Cones
Let $H$ be an $n$-generated numerical semigroup such that its tangent cone $\operatorname{gr}_\mathfrak{m} K[H]$ is defined by quadratic relations. We show that if $n<5$ then $\operatorname{gr}_\mathfrak{m} K[H]$ is Cohen-Macaulay, and for $n=5$ we explicitly describe the semigroups $H$ such that $\operatorname{gr}_\mathfrak{m} K[H]$ is not Cohen-Macaulay. As an application we show that if the field $K$ is algebraically closed and of characteristic different from two, and $n\leq 5$ then $\operatorname{gr}_\mathfrak{m} K[H]$ is Koszul if and only if (possibly after a change of coordinates) its defining ideal has a quadratic Gröbner basis.
2018 ◽
Vol 99
(1)
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pp. 68-77
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2009 ◽
Vol 2
(4)
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pp. 601-634
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2012 ◽
Vol 47
(8)
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pp. 926-941
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1998 ◽
Vol 123
(1-3)
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pp. 275-283
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2013 ◽
Vol 24
(2)
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