Betti numbers for certain Cohen-Macaulay tangent cones

TL;DR

This paper computes Betti numbers for Cohen-Macaulay tangent cones of monomial curves in four-dimensional affine space linked to pseudo symmetric numerical semigroups, revealing an equivalence between homogeneity properties.

Contribution

It provides explicit Betti number calculations for these tangent cones and establishes the equivalence of homogeneity and being of homogeneous type for the semigroups.

Findings

Betti numbers for Cohen-Macaulay tangent cones are explicitly computed.

Homogeneity and being of homogeneous type are shown to be equivalent for the semigroups considered.

The results deepen understanding of the algebraic structure of monomial curves in affine space.

Abstract

In this article, we compute Betti numbers for a Cohen-Macaulay tangent cone of a monomial curve in the affine $4$-space corresponding to a pseudo symmetric numerical semigroup. As a byproduct, we also show that for these semigroups, being of homogeneous type and homogeneous are equivalent properties.