Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917675042603008 |
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| author | Fink, Alex Giansiracusa, Jeffrey Giansiracusa, Noah Mundinger, Joshua |
| author_facet | Fink, Alex Giansiracusa, Jeffrey Giansiracusa, Noah Mundinger, Joshua |
| contents | A "tropical ideal" is an ideal in the idempotent semiring of tropical polynomials that is also, degree by degree, a tropical linear space. We introduce a construction based on transversal matroids that canonically extends any principal ideal to a tropical ideal. We call this the Macaulay tropical ideal. It has a universal property: any other extension of the given principal ideal to a tropical ideal with the expected Hilbert function is a weak image of the Macaulay tropical ideal. For each $n\geq 2$ and $d\geq 1$ our construction yields a non-realizable degree $d$ hypersurface scheme in $\mathbb{P}^n$. Maclagan-Rincón produced a non-realizable line in $\mathbb{P}^n$ for each $n$, and for $(d,n)=(1,2)$ the two constructions agree. An appendix by Mundinger compares the Macaulay construction with another method for canonically extending ideals to tropical ideals. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_16338 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal Fink, Alex Giansiracusa, Jeffrey Giansiracusa, Noah Mundinger, Joshua Algebraic Geometry Combinatorics 14T10 A "tropical ideal" is an ideal in the idempotent semiring of tropical polynomials that is also, degree by degree, a tropical linear space. We introduce a construction based on transversal matroids that canonically extends any principal ideal to a tropical ideal. We call this the Macaulay tropical ideal. It has a universal property: any other extension of the given principal ideal to a tropical ideal with the expected Hilbert function is a weak image of the Macaulay tropical ideal. For each $n\geq 2$ and $d\geq 1$ our construction yields a non-realizable degree $d$ hypersurface scheme in $\mathbb{P}^n$. Maclagan-Rincón produced a non-realizable line in $\mathbb{P}^n$ for each $n$, and for $(d,n)=(1,2)$ the two constructions agree. An appendix by Mundinger compares the Macaulay construction with another method for canonically extending ideals to tropical ideals. |
| title | Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal |
| topic | Algebraic Geometry Combinatorics 14T10 |
| url | https://arxiv.org/abs/2405.16338 |