The Honest Embedding Dimension of a Numerical Semigroup

Abstract

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.

Figures (1)

• Figure 1: The Kunz cone for multiplicity $4$ is the cone in $\mathbb R^3$ over the interior of this kite when placed on the plane $x_1 + x_2 + x_3 =1$. See equations (\ref{['Kunzcone']}). The set of lattice points in and on the boundary of the cone for which $x_i \equiv i$ (mod 4) are in bijection with multiplicity 4 numerical semigroups. The set of such points lying in the interior of the cone sweep out the semigroups with embedding dimension $4$. Those having minimal embedding dimension $3$ correspond to the shaded region of the kite and the interiors of its edges. Those having minimal embedding $2$ correspond to the left and right vertices of the kite. Thanks to Emily O'Sullivan for the figure.

Theorems & Definitions (18)

• Definition 1
• Example 1
• Definition 1.1
• Definition 1.2: Semigroup of a curve.
• Remark 1
• Definition 1.3
• Proposition 1.1
• Theorem 1.1
• Proposition 3.1
• Example 3.1