q-Painlevé equations on cluster Poisson varieties via toric geometry

We provide a relation between the geometric framework for q-Painlevé equations and cluster Poisson varieties by using toric models of rational surfaces associated with q-Painlevé equations. We introduce the notion of seeds of q-Painlevé type by the negative semi-definiteness of symmetric bilinear forms associated with seeds, and classify the mutation equivalence classes of these seeds. This classification coincides with the classification of q-Painlevé equations given by Sakai. We realize q-Painlevé systems as automorphisms on cluster Poisson varieties associated with seeds of q-Painlevé type.