Valuative independence and cluster theta reciprocity
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913151406047232 |
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| author | Cheung, Man-Wai Magee, Timothy Mandel, Travis Muller, Greg |
| author_facet | Cheung, Man-Wai Magee, Timothy Mandel, Travis Muller, Greg |
| contents | We prove that theta functions constructed from positive scattering diagrams satisfy valuative independence. That is, for certain valuations $\operatorname{val}_{v}$, we have $\operatorname{val}_v(\sum_u c_u \vartheta_u)=\min_{c_u\neq 0} \operatorname{val}_v(\vartheta_u)$. As applications, we prove linear independence of theta functions with specialized coefficients and characterize when theta functions for cluster varieties are unchanged by the unfreezing of an index. This yields a general gluing result for theta functions from moduli of local systems on marked surfaces. We then prove that theta functions for cluster varieties satisfy a symmetry property called theta reciprocity: briefly, $\operatorname{val}_v(\vartheta_u)=\operatorname{val}_u(\vartheta_v)$. For this we utilize a new framework called a "seed datum" for understanding cluster-type varieties. One may apply valuative independence and theta reciprocity together to identify theta function bases for global sections of line bundles on partial compactifications of cluster varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09585 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Valuative independence and cluster theta reciprocity Cheung, Man-Wai Magee, Timothy Mandel, Travis Muller, Greg Algebraic Geometry Commutative Algebra Combinatorics Quantum Algebra Representation Theory 13F60 (Primary) 14J33 (Secondary) We prove that theta functions constructed from positive scattering diagrams satisfy valuative independence. That is, for certain valuations $\operatorname{val}_{v}$, we have $\operatorname{val}_v(\sum_u c_u \vartheta_u)=\min_{c_u\neq 0} \operatorname{val}_v(\vartheta_u)$. As applications, we prove linear independence of theta functions with specialized coefficients and characterize when theta functions for cluster varieties are unchanged by the unfreezing of an index. This yields a general gluing result for theta functions from moduli of local systems on marked surfaces. We then prove that theta functions for cluster varieties satisfy a symmetry property called theta reciprocity: briefly, $\operatorname{val}_v(\vartheta_u)=\operatorname{val}_u(\vartheta_v)$. For this we utilize a new framework called a "seed datum" for understanding cluster-type varieties. One may apply valuative independence and theta reciprocity together to identify theta function bases for global sections of line bundles on partial compactifications of cluster varieties. |
| title | Valuative independence and cluster theta reciprocity |
| topic | Algebraic Geometry Commutative Algebra Combinatorics Quantum Algebra Representation Theory 13F60 (Primary) 14J33 (Secondary) |
| url | https://arxiv.org/abs/2505.09585 |