Tropical fans supporting a reduced 0-dimensional complete intersection
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908719381479424 |
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| author | Li, Linxuan |
| author_facet | Li, Linxuan |
| contents | An affine tropical fan is called regular if it supports a reduced 0-dimensional complete intersection. For some cases the classification of regular fans is already complete. It was proved by Fink that tropical varieties of degree 1 are exactly Bergman fans, and later Esterov and Gusev classified all lattice polytopes whose mixed volume equals 1. We introduce the notion of a gallery for tropical fans and use it to classify all one-dimensional regular fans, thereby obtaining a minimal model programme for such fans. In dimension 2 we prove a finiteness theorem: every regular fan that satisfies the given upper bound condition is precisely the support of a finite covering by two-dimensional galleries, and only finitely many such fans exist. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06694 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tropical fans supporting a reduced 0-dimensional complete intersection Li, Linxuan Combinatorics Algebraic Geometry 14T15, 14T20 An affine tropical fan is called regular if it supports a reduced 0-dimensional complete intersection. For some cases the classification of regular fans is already complete. It was proved by Fink that tropical varieties of degree 1 are exactly Bergman fans, and later Esterov and Gusev classified all lattice polytopes whose mixed volume equals 1. We introduce the notion of a gallery for tropical fans and use it to classify all one-dimensional regular fans, thereby obtaining a minimal model programme for such fans. In dimension 2 we prove a finiteness theorem: every regular fan that satisfies the given upper bound condition is precisely the support of a finite covering by two-dimensional galleries, and only finitely many such fans exist. |
| title | Tropical fans supporting a reduced 0-dimensional complete intersection |
| topic | Combinatorics Algebraic Geometry 14T15, 14T20 |
| url | https://arxiv.org/abs/2508.06694 |