Numerical semigroups, polyhedra, and posets IV: walking the faces of the Kunz cone

TL;DR

The paper introduces the Kunz fan, a pure polyhedral cone complex obtained from projections of faces of the Kunz cone ${\mathcal{C}}_m$, and relates its structure to Kunz nilsemigroups. It develops a walking algorithm, analogous to Gröbner walks, to traverse the face lattice by leveraging the combinatorics of Kunz nilsemigroups, outer Betti elements, and Betti inequalities, providing both theoretical characterizations and computational gains. Through this framework, the authors prove results on minimal presentations, classify faces for embedding-dimension 3, and construct families (e.g., cup posets) that yield exponential growth in the number of rays, while connecting to open problems and Gröbner-fan concepts. The work combines deep combinatorial structure with practical enumeration techniques, yielding new insights into the relationships between multiplicity, embedding dimension, and algebraic properties of numerical semigroups.

Abstract

A numerical semigroup is a cofinite subset of $\mathbb Z_{\ge 0}$ containing $0$ and closed under addition. Each numerical semigroup $S$ with smallest positive element $m$ corresponds to an integer point in the Kunz cone $\mathcal C_m \subseteq \mathbb R^{m-1}$, and the face of $\mathcal C_m$ containing that integer point determines certain algebraic properties of $S$. In this paper, we introduce the Kunz fan, a pure, polyhedral cone complex comprised of a faithful projection of certain faces of $\mathcal C_m$. We characterize several aspects of the Kunz fan in terms of the combinatorics of Kunz nilsemigroups, which are known to index the faces of $\mathcal C_m$, and our results culminate in a method of "walking" the face lattice of the Kunz cone in a manner analogous to that of a Gröbner walk. We apply our results in several contexts, including a wealth of computational data obtained from the aforementioned "walks" and a proof of a recent conjecture concerning which numerical semigroups achieve the highest minimal presentation cardinality when one fixes the smallest positive element and the number of generators.

Figures (5)

• Figure 1: Kunz poset in Example \ref{['e:outerbettis']} with outer Betti elements in red.
• Figure 2: Posets from the chambers of the Kunz fan $\mathcal{G}(20; 6, 11)$.
• Figure 3: Posets from the chambers of the Kunz fan $\mathcal{G}(13; 1,2 ,9)$.
• Figure 4: Kunz posets from Examples \ref{['e:bettiequality']}, \ref{['e:nonkunznilsemigroup']}, and \ref{['e:innerfactsminimal']}
• Figure 5: Cup poset for $d = 4$, alongside the posets of three of its rays.

Theorems & Definitions (62)

• Theorem 2.1: kunzfaces3
• Example 2.2
• Theorem 2.3
• Definition 3.1
• Definition 3.2
• Example 3.3
• Proposition 3.4
• proof
• Remark 3.5
• Example 3.6