Minimal free resolutions of numerical semigroup algebras via Ap\'ery specialization

TL;DR

This paper constructs minimal free resolutions for numerical semigroup algebras using Apéry specialization, revealing a uniform structure across semigroups in the same face of the Kunz cone and identifying conditions for minimality.

Contribution

It provides a unified construction of free resolutions for numerical semigroup algebras parametrized by the Kunz cone, generalizing previous results and characterizing minimality.

Findings

Resolution structure is identical for semigroups in the same face of the Kunz cone.

Minimality is achieved for semigroups with maximal embedding dimension.

Matrix entries are monomials parametrized by cone coordinates.

Abstract

Numerical semigroups with multiplicity $m$ are parameterized by integer points in a polyhedral cone $C_m$, according to Kunz. For the toric ideal of any such semigroup, the main result here constructs a free resolution whose overall structure is identical for all semigroups parametrized by the relative interior of a fixed face of $C_m$. The matrix entries of this resolution are monomials whose exponents are parametrized by the coordinates of the corresponding point in $C_m$, and minimality of the resolution is achieved when the semigroup is maximal embedding dimension, which is the case parametrized by the interior of $C_m$ itself.