Geometric families of degenerations from mutations of polytopes

TL;DR

This paper introduces polyptych lattices, a new combinatorial framework linking convex geometry, polytopes, and algebraic compactifications, revealing their geometric properties through combinatorial methods.

Contribution

It develops the theory of polyptych lattices and associated polytopes, connecting them to algebraic geometry and providing new insights into compactifications and their properties.

Findings

Polyptych lattices encode lattices related by piecewise linear bijections.

Associated polytopes can compactify affine varieties with desirable algebraic properties.

Certain compactifications are arithmetically Cohen-Macaulay with finitely generated class groups.

Abstract

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.