A construction of polyptych lattices via a pair of Gorenstein PL cones

Source abstract

The theory of polyptych lattices seeks to incorporate, into a single combinatorial framework, the piecewise-linear bijections (mutations) that appear in many theories that generalize toric geometry, such as Newton-Okounkov bodies, toric degenerations, cluster varieties, and other areas. The link to algebraic geometry comes from the accompanying notion of detropicalizations of polyptych lattices, and their associated compactifications. The main result of this manuscript gives a concrete construction of a strict dual pair $(\mathcal{E}, \mathcal{F})$ of polyptych lattices, which we call a Gorenstein PL cone extension. To define our construction, we introduce the notion of Gorenstein PL cones, which are a polyptych analogue of a Gorenstein cone in the classical setting. If the original polyptych lattices are detropicalizable, then the new polyptych lattices $\mathcal{E},\mathcal{F}$ are also detropicalizable, via a simple explicit formula. Our Gorenstein PL cone extensions give a rich source of examples of strict dual pairs of polyptych lattices. In particular, any pair of Gorenstein-Fano polytopes $Δ,Δ'$ of dimension $n,n'$ respectively, gives rise to a strict dual pair $(\mathcal{E}(Δ,Δ'), \mathcal{F}(Δ,Δ'))$ of detropicalizable polyptych lattices of rank $n+n'+1$. Other examples can be built from cluster data.