The Honest Embedding Dimension of a Numerical Semigroup

Attached to a singular analytic curve germ in d-space is a numerical semigroup: a subset S of the non-negative integers which is closed under addition and whose complement isfinite. Conversely, associated to any numerical semigroup S is a canonical mononial curve in e-space where e is the number of minimal generators of the semigroup. It may happen that d < e = e (S) where S is the semigroup of the curve in d-space. Define the minimal (or `honest') embedding of a numerical semigroup to be the smallest d such that S is realized by a curve in d-space. Problem: characterize the numerical semigroups having minimal embedding dimension d. The answer is known for the case d=2 of planar curves and reviewed in an Appendix to this paper. The case d =3 of the problem is open. Our main result is a characterization of the multiplicity 4 numerical semigroups whose minimal embedding dimension is 3. See figure 1. The motivation for this work came from thinking about Legendrian curve singularities.