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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2403.00588 |
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| _version_ | 1866910353892311040 |
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| author | Montgomery, Richard |
| author_facet | Montgomery, Richard |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00588 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Honest Embedding Dimension of a Numerical Semigroup Montgomery, Richard Algebraic Geometry Combinatorics 14H20 - Singularities of curves, local rings 14H20, 14B05 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. |
| title | The Honest Embedding Dimension of a Numerical Semigroup |
| topic | Algebraic Geometry Combinatorics 14H20 - Singularities of curves, local rings 14H20, 14B05 |
| url | https://arxiv.org/abs/2403.00588 |