Toric degenerations of cluster varieties and cluster duality
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2018
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| _version_ | 1866912023019782144 |
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| author | Bossinger, Lara Frías-Medina, Bosco Magee, Timothy Chávez, Alfredo Nájera |
| author_facet | Bossinger, Lara Frías-Medina, Bosco Magee, Timothy Chávez, Alfredo Nájera |
| contents | We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: a cluster $\mathcal{X}$-variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed cluster $\mathcal{X}$-variety $\widehat{\mathcal{X}}$ to the toric variety associated to its $\mathbf{g}$-fan. Moreover, we show that the fibers of this family are stratified in a natural way, with strata the specially completed $\mathcal{X}$-varieties encoded by $\mathrm{Star}(τ)$ for each cone $τ$ of the $\mathbf{g}$-fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to $\mathcal{A}_{\mathrm{prin}}$ of Gross-Hacking-Keel-Kontsevich, and the fibers cluster dual to $\mathcal{A}_t$. Finally, we give two applications. First, we use our construction to identify the Rietsch-Williams toric degeneration of Grassmannians with the Gross-Hacking-Keel-Kontsevich degeneration in the case of $\mathrm{Gr}_2(\mathbb{C}^5)$. Next, we use it to link cluster duality to Batyrev-Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1809_08369 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Toric degenerations of cluster varieties and cluster duality Bossinger, Lara Frías-Medina, Bosco Magee, Timothy Chávez, Alfredo Nájera Algebraic Geometry Combinatorics Rings and Algebras Representation Theory 13F60, 14M25, 14D06 We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: a cluster $\mathcal{X}$-variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed cluster $\mathcal{X}$-variety $\widehat{\mathcal{X}}$ to the toric variety associated to its $\mathbf{g}$-fan. Moreover, we show that the fibers of this family are stratified in a natural way, with strata the specially completed $\mathcal{X}$-varieties encoded by $\mathrm{Star}(τ)$ for each cone $τ$ of the $\mathbf{g}$-fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to $\mathcal{A}_{\mathrm{prin}}$ of Gross-Hacking-Keel-Kontsevich, and the fibers cluster dual to $\mathcal{A}_t$. Finally, we give two applications. First, we use our construction to identify the Rietsch-Williams toric degeneration of Grassmannians with the Gross-Hacking-Keel-Kontsevich degeneration in the case of $\mathrm{Gr}_2(\mathbb{C}^5)$. Next, we use it to link cluster duality to Batyrev-Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry. |
| title | Toric degenerations of cluster varieties and cluster duality |
| topic | Algebraic Geometry Combinatorics Rings and Algebras Representation Theory 13F60, 14M25, 14D06 |
| url | https://arxiv.org/abs/1809.08369 |