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

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 $S$.