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Classification of rank 2 cluster varieties

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  • Travis Mandel, Denmark
We classify rank 2 cluster varieties (those whose corresponding skew-form has rank 2) according to the deformation type of a generic fiber U of their X-spaces, as defined by Fock and Goncharov. Our approach is based on the work of Gross, Hacking, and Keel for cluster varieties and log Calabi-Yau surfaces. We find, for example, that U is "positive" (i.e., nearly affine) and either finite-type or non-acyclic (in the cluster sense) if and only if the monodromy of the tropicalization of U is one of Kodaira's matrices for the monodromy of an ellpitic fibration. In the positive cases, we also describe the action of the cluster modular group on the tropicalization of U.
Original languageEnglish
Number of pages27
Publication statusPublished - 2014

    Research areas

  • Cluster Variety, Cluster Modular Group

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