2024/01/11 by Brower, Cole, McDonough, Joseph, O'Neill, Christopher
#Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2401.06025
A numerical semigroup is a cofinite subset of \mathbb Z≥ 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 Cm ⊆ \mathbb Rm-1, and the face of \mathcal Cm 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 Cm. 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 Cm, 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.