2024/03/01 by Montgomery, Richard
#14B05 #14H20 - Singularities of curves #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #local rings 14H20
paper · doi:10.48550/arxiv.2403.00588
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.