Numerical semigroups with multiplicity m are parameterized by integer points in a polyhedral cone Cₘ, 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 Cₘ. The matrix entries of this resolution are monomials whose exponents are parametrized by the coordinates of the corresponding point in Cₘ, and minimality of the resolution is achieved when the semigroup is maximal embedding dimension, which is the case parametrized by the interior of Cₘ itself.
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Braun et al. (2023) studied this question.