The honest embedding dimension of a numerical semigroup

Abstract To a curve germ in d -space we can associate a numerical semigroup called its value semigroup. All numerical semigroups arise in this way. For a given semigroup we define its honest embedding dimension to be the minimal d for which there is a curve in d -space whose value semigroup is this semigroup. What are all the semigroups having honest embedding dimension d ? Bresinsky answered this question for $$d =2$$ d = 2 , i.e. for the case of planar curves. The question for $$d =3$$ d = 3 (space curves) is open. Refining the question by the multiplicity m of the semigroup simplifies the work, in part because we must have $$d \le m$$ d ≤ m . Our first Theorem classifies the numerical semigroups having multiplicity 4 and honest embedding dimension 3. Theorem 1 is a corollary of Theorem 2 which classifies the numerical semigroups for which $$d = m$$ d = m , i.e those semigroups whose multiplicity m and honest embedding dimensional are equal. Our main tool is the Kunz cone, a convex polyhedral cone in $$m-1$$ m - 1 dimensions which allows visualization of the space of semigroups of multiplicity m . The answer provided in Theorem 2 is phrased as the intersection of the Kunz cone by another convex polyhedral cone.