Toric vector bundles, valuations and tropical geometry
| Autor: | Kaveh, Kiumars, Manon, Christopher |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | A toric vector bundle $\mathcal{E}$ is a torus equivariant vector bundle on a toric variety. We give a valuation theoretic and tropical point of view on toric vector bundles. We present three (equivalent) classifications of toric vector bundles, which should be regarded as repackagings of the Klyachko data of compatible $\mathbb{Z}$-filtrations of a toric vector bundle: (1) as piecewise linear maps to space of $\mathbb{Z}$-valued valuations, (2) as valuations with values in the semifield of piecewise linear functions, and (3) as points in tropical linear ideals over the semifield of piecewise linear functions. Moreover, we interpret the known criteria for ampleness and global generation of $\mathcal{E}$ as convexity conditions on its piecewise linear map in (1). Finally, using (2) we associate to $\mathcal{E}$ a collection of polytopes indexed by elements of a certain (representable) matroid encoding the dimensions of weight spaces of global sections of $\mathcal{E}$. This recovers and extends the Di Rocco-Jabbusch-Smith matriod and parliament of polytopes of $\mathcal{E}$. This is a follow up paper to arXiv:1806.05613. Comment: 28 pages, 3 figures. The paper is an expanded and thoroughly revised version of the second half of an early version of arXiv:1806.05613. Following the advice of the referee for arXiv:1806.05613, this second half was removed and made into the present paper |
| Databáze: | arXiv |
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