Homogeneous numerical semigroups

TL;DR

This paper introduces homogeneous numerical semigroups, classifies those of homogeneous type in embedding dimension three, and explores their properties under gluing and shifting, with many explicit examples.

Contribution

It defines homogeneous numerical semigroups, classifies those of homogeneous type in dimension three, and analyzes their behavior under specific constructions.

Findings

Homogeneous numerical semigroups with Cohen-Macaulay tangent cones are of homogeneous type.

Classification of all homogeneous type semigroups in embedding dimension three.

Construction of large families of homogeneous semigroups via gluing and shifting.

Abstract

We introduce the concept of homogeneous numerical semigroups and show that all homogeneous numerical semigroups with Cohen-Macaulay tangent cones are of homogeneous type. In embedding dimension three, we classify all numerical semigroups of homogeneous type in to numerical semigroups with complete intersection tangent cones and the homogeneous ones with Cohen-Macaulay tangent cones. We also study the behavior of the homogeneous property by gluing and shiftings to construct a large family of homogeneous numerical semigroups with Cohen-Macaulay tangent cones. In particular we show that these properties fulfill assymptotically in the shifting classes. Several explicit examples are provided along the paper to illustrate the property.