4-Generated Pseudo Symmetric Monomial Curves with not Cohen-Macaulay Tangent Cones

TL;DR

This paper computes standard bases for certain toric ideals of 4-generated pseudo symmetric semigroups with non-Cohen-Macaulay tangent cones, explicitly determines their Hilbert functions, and establishes conditions for their non-decreasingness.

Contribution

It introduces explicit computation of Hilbert functions for these semigroups and proves non-decreasingness under specific numerical conditions, advancing understanding of their algebraic properties.

Findings

Standard bases for toric ideals are computed.

Hilbert functions are explicitly determined.

Non-decreasing Hilbert functions are proved under certain conditions.

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, non-decreasingness 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 non-decreasing.