Amalgams of matroids, fibre products and tropical graph correspondences
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913333597175808 |
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| author | Mineev, Dmitry |
| author_facet | Mineev, Dmitry |
| contents | We prove that the proper amalgam of matroids $M_1$ and $M_2$ along their common restriction $N$ exists if and only if the tropical fibre product of Bergman fans ${B(M_1) \times_{B(N)} B(M_2)}$ is positive. We introduce tropical correspondences between Bergman fans as tropical subcycles in their product, similar to correspondences in algebraic geometry, and define a "graph correspondence" of the map of lattices. We prove that graph construction is a functor for the "covering" maps of lattices, exploiting a generalization of Bergman fan which we call a "Flag fan". |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_18127 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Amalgams of matroids, fibre products and tropical graph correspondences Mineev, Dmitry Algebraic Geometry Combinatorics 14T15 (Primary), 05B35, 14C17, 52B40, 18A10 (Secondary) We prove that the proper amalgam of matroids $M_1$ and $M_2$ along their common restriction $N$ exists if and only if the tropical fibre product of Bergman fans ${B(M_1) \times_{B(N)} B(M_2)}$ is positive. We introduce tropical correspondences between Bergman fans as tropical subcycles in their product, similar to correspondences in algebraic geometry, and define a "graph correspondence" of the map of lattices. We prove that graph construction is a functor for the "covering" maps of lattices, exploiting a generalization of Bergman fan which we call a "Flag fan". |
| title | Amalgams of matroids, fibre products and tropical graph correspondences |
| topic | Algebraic Geometry Combinatorics 14T15 (Primary), 05B35, 14C17, 52B40, 18A10 (Secondary) |
| url | https://arxiv.org/abs/2404.18127 |