$q$-Painlev\'e equations on cluster Poisson varieties via toric geometry

We provide a relation between the geometric framework for $q$-Painlev\'{e} equations and cluster Poisson varieties by using toric models of rational surfaces associated with $q$-Painlev\'{e} equations. We introduce the notion of seeds of $q$-Painlev\'{e} 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 $q$-Painlev\'{e} equations given by Sakai. We realize $q$-Painlev\'{e} systems as automorphisms on cluster Poisson varieties associated with seeds of $q$-Painlev\'{e} type.