Cluster structures on Cox rings
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909417498214400 |
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| author | Francone, Luca |
| author_facet | Francone, Luca |
| contents | We construct a graded cluster algebra structure on the Cox ring of a smooth complex variety $Z$, depending on a base cluster structure on the ring of regular functions of an open subset $Y$ of $Z$. After considering some elementary examples of our construction, including toric varieties, we discuss the two main applications. First: if $Z$ is a flag variety and $Y$ is the open Schubert cell, we prove that our results recover, by geometric methods, a well known construction of Geiss, Leclerc and Schröer. Second: we define the diagonal partial compactification of a (finite type) cluster variety, and prove that its Cox ring is a graded upper cluster algebra. Along the way, we explain how similar constructions can be done if we replace the Cox ring with a ring of global sections of a sheaf of divisorial algebras on $Z$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04173 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cluster structures on Cox rings Francone, Luca Algebraic Geometry Commutative Algebra Representation Theory We construct a graded cluster algebra structure on the Cox ring of a smooth complex variety $Z$, depending on a base cluster structure on the ring of regular functions of an open subset $Y$ of $Z$. After considering some elementary examples of our construction, including toric varieties, we discuss the two main applications. First: if $Z$ is a flag variety and $Y$ is the open Schubert cell, we prove that our results recover, by geometric methods, a well known construction of Geiss, Leclerc and Schröer. Second: we define the diagonal partial compactification of a (finite type) cluster variety, and prove that its Cox ring is a graded upper cluster algebra. Along the way, we explain how similar constructions can be done if we replace the Cox ring with a ring of global sections of a sheaf of divisorial algebras on $Z$. |
| title | Cluster structures on Cox rings |
| topic | Algebraic Geometry Commutative Algebra Representation Theory |
| url | https://arxiv.org/abs/2412.04173 |