On Bands and Limit Theorems in Tropical Geometry
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866918489783009280 |
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| author | Kuhrs, Arne Méndez, Alejandro Martínez Souza, Pedro |
| author_facet | Kuhrs, Arne Méndez, Alejandro Martínez Souza, Pedro |
| contents | We review the basic theory of bands and band schemes introduced by Baker-Jin-Lorscheid, which is an algebraic framework for tropicalization, analytification, and $\mathbb{F}_1$-geometry. For an affine scheme $X$ over a non-Archimedean valued field $k$, one can associate to every affine embedding $ι$ of $X$ a naturally defined affine band scheme $Y_ι$ whose rational points over the tropical band $\mathbb{T}$ recover the tropicalization $Trop(X,ι)$. We prove that $X$ is the limit of the $Y_ι$ in the category of band schemes, thereby obtaining a scheme-theoretic enhancement of Payne's limit theorem. By taking $\mathbb{T}$-rational points, this recovers Payne's theorem for affine tropicalizations from the perspective of band scheme theory and the same method provides an analogous result in the real tropical setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_07991 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Bands and Limit Theorems in Tropical Geometry Kuhrs, Arne Méndez, Alejandro Martínez Souza, Pedro Algebraic Geometry We review the basic theory of bands and band schemes introduced by Baker-Jin-Lorscheid, which is an algebraic framework for tropicalization, analytification, and $\mathbb{F}_1$-geometry. For an affine scheme $X$ over a non-Archimedean valued field $k$, one can associate to every affine embedding $ι$ of $X$ a naturally defined affine band scheme $Y_ι$ whose rational points over the tropical band $\mathbb{T}$ recover the tropicalization $Trop(X,ι)$. We prove that $X$ is the limit of the $Y_ι$ in the category of band schemes, thereby obtaining a scheme-theoretic enhancement of Payne's limit theorem. By taking $\mathbb{T}$-rational points, this recovers Payne's theorem for affine tropicalizations from the perspective of band scheme theory and the same method provides an analogous result in the real tropical setting. |
| title | On Bands and Limit Theorems in Tropical Geometry |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2605.07991 |