4-generated pseudo symmetric monomial curves with not Cohen-Macaulay tangent cones

Authors: NİL ŞAHİN

Abstract: In this article, standard bases of some toric ideals associated to 4-generated pseudo symmetric semigroups with not Cohen-Macaulay tangent cones at the origin are computed. As the tangent cones are not Cohen-Macaulay, nondecreasingness of the Hilbert function of the local ring was not guaranteed. Therefore, using these standard bases, Hilbert functions are explicitly computed as a step towards the characterization of Hilbert function. In addition, when the smallest integer satisfying $k(\alpha_2+1)<(k-1)\alpha_1+(k+1)\alpha_{21}+\alpha_3$ is $1$, it is proved that the Hilbert function of the local ring is nondecreasing.

Keywords: Hilbert function, tangent cone, monomial curve, numerical semigroup, standard bases

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