Toric degenerations of cluster varieties and cluster duality
Lara Bossinger, Bosco Frías-Medina, Timothy Magee, and Alfredo Nájera Chávez
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Submission date: 28. Sep. 2018
MSC-Numbers: 14J33, 13F60, 14M25
Keywords and phrases: Toric degeneration, cluster varieties
Link to arXiv:See the arXiv entry of this preprint.
We introduce the notion of a Y -pattern with coeﬃcients and its geometric counterpart: a cluster 𝒳-variety with coeﬃcients. We use these constructions to build a ﬂat degeneration of every skew-symmetrizable specially completed cluster 𝒳-variety to the toric variety associated to its g-fan. Moreover, we show that the ﬁbers of this family are stratiﬁed in a natural way, with each stratum encoded by Star(τ) for some cone τ of the g-fan. These strata degenerate to the associated toric strata of the central ﬁber. We further show that the family is cluster dual to 𝒜prin of Gross-Hacking-Keel-Kontsevich, and the ﬁbers cluster dual to 𝒜t. 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 Gr2(ℂ5). Next, we use it to link cluster duality to Batyrev-Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.