2024/08/03 by Laura Escobar, Escobar, Laura, Megumi Harada +3 · 1 citation
Engineering · #Advanced Theoretical and Applied Studies in Material Sciences and Geometry
paper · pdf · doi:10.48550/arxiv.2408.01785
We introduce the notion of a polyptych lattice, which encodes a collection of lattices related by piecewise linear bijections. We initiate a study of the new theory of convex geometry and polytopes associated to polyptych lattices. In certain situations, such a polytope associated to a polyptych lattice encodes a compactification of an affine variety whose coordinate ring can be equipped with a valuation into a certain semialgebra associated to the polyptych lattice. We show that aspects of the geometry of the compactification can be understood combinatorially; for instance, under some hypotheses, the resulting compactifications are arithmetically Cohen-Macaulay, and have finitely generated class group and finitely generated Cox rings.