On faces of the Kunz cone and the numerical semigroups within them

TL;DR

This paper classifies faces of the Kunz cone associated with numerical semigroups and establishes bounds on their minimal generators based on face dimension, deepening understanding of their geometric and algebraic structure.

Contribution

It provides a classification of faces of the Kunz cone containing numerical semigroups and derives bounds on minimal generators related to face dimension.

Findings

Identified which faces of the Kunz cone contain numerical semigroup points.

Established sharp bounds on the number of minimal generators based on face dimension.

Connected geometric properties of the cone with algebraic properties of semigroups.

Abstract

A numerical semigroup is a cofinite subset of the non-negative integers that is closed under addition and contains 0. Each numerical semigroup $S$ with fixed smallest positive element $m$ corresponds to an integer point in a rational polyhedral cone $\mathcal C_m$, called the Kunz cone. Moreover, numerical semigroups corresponding to points in the same face $F \subseteq \mathcal C_m$ are known to share many properties, such as the number of minimal generators. In this work, we classify which faces of $\mathcal C_m$ contain points corresponding to numerical semigroups. Additionally, we obtain sharp bounds on the number of minimal generators of $S$ in terms of the dimension of the face of $\mathcal C_m$ containing the point corresponding to $S$.