Newton--Okounkov bodies and minimal models for cluster varieties

Let $Y$ be a (partial) minimal model of a scheme $V$ with a cluster structure. Under natural assumptions, for every choice of seed we associate a Newton–Okounkov body to every divisor on $Y$ supported on $Y\setminus V$ and show that these Newton–Okounkov bodies are positive sets in the sense of Gross, Hacking, Keel and Kontsevich \cite{GHKK}. This construction essentially reverses the procedure in loc. cit. that generalizes the polytope construction of a toric variety to the framework of cluster varieties. In a closely related setting, we consider cases where $Y$ is a projective variety whose universal torsor ${\text{UT}}_Y$ is a partial minimal model of a scheme with a cluster structure of type $\mathcal{A}$. If the theta functions parametrized by the integral points of the associated superpotential cone form a basis of the ring of algebraic functions on ${\text{UT}}_Y$ and the action of the torus $T_{\text{Pic}(Y{)}^{\ast }}$ on ${\text{UT}}_Y$ is compatible with the cluster structure, then for every choice of seed we associate a Newton–Okounkov body to every line bundle on $Y$. We prove that any such Newton–Okounkov body is a positive set and that $Y$ is a minimal model of a quotient of a cluster $\mathcal{A}$-variety by the action of a torus. Our constructions lead to the notion of the intrinsic Newton–Okounkov body associated to a boundary divisor in a partial minimal model of a scheme with a cluster structure. This provides a wide class of examples of Newton-Okoukov bodies exhibiting a wall-crossing phenomenon in the sense of Escobar–Harada \cite{EH20}. This approach includes the partial flag varieties that arise as minimal models of cluster varieties. For the case of Grassmannians, our approach recovers, up to interesting unimodular equivalences, the Newton–Okounkov bodies constructed by Rietsch–Williams in \cite{RW}.