Toric degenerations of cluster varieties and cluster duality

@article{Bossinger2020ToricDO,
  title={Toric degenerations of cluster varieties and cluster duality},
  author={Lara Bossinger and Bosco Fr'ias-Medina and Timothy Magee and Alfredo N{\'a}jera Ch{\'a}vez},
  journal={Compositio Mathematica},
  year={2020},
  volume={156},
  pages={2149 - 2206}
}
We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: an $\mathcal {X}$-cluster variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed $\mathcal {X}$-cluster variety $\widehat {\mathcal {X} }$ 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 strata the specially completed $\mathcal {X}$-varieties… Expand
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