Gorenstein-Fano polytopes and compactifications of rank 2 polyptych lattices

The notion of polyptych lattices, introduced by Escobar, Harada, and Manon, wraps the data of a collection of lattices related by piecewise-linear bijections together into a single semi-algebraic object, equipped with its own notions of convexity and polyhedra. The main purpose of this manuscript is to construct an explicit family of polyptych lattices, and to illustrate via explicit computations the abstract theory introduced by Escobar-Harada-Manon. Specifically, we first construct a family of rank-$2$ polyptych lattices $\mathcal{M}_s$ with $2$ charts, compute their space of points, and prove that they are full and self-dual. We then give a concrete sample computation of a point-convex hull in $\mathcal{M}_s \otimes \mathbb{R}$ to illustrate that convex geometry in the polyptych lattice setting can exhibit phenomena not seen in the classical situation. We also give multiple examples of $2$-dimensional ``chart-Gorenstein-Fano'' polytopes, which give rise to pairs of mutation-related $2$-dimensional (classical) Gorenstein-Fano polytopes. Finally, we produce detropicalizations $(\mathcal{A}_s, \mathfrak{v}_s)$ of $\mathcal{M}_s$, and in the case $s=1$ where the detropicalization is a UFD, and with respect to a certain choice of PL polytope $\mathcal{P}$, we give an explicit generators-and-relations presentation of the (finitely generated) Cox ring of the compactification $X_{\mathcal{A}_s}(\mathcal{P})$ of $\mathrm{Spec}(\mathcal{A}_s)$ with respect to $\mathcal{P}$.