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arXiv · 2408.01785

Geometric families of degenerations from mutations of polytopes

Laura Escobar·Megumi Harada·Christopher Manon
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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.

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Laura Escobar, Megumi Harada, Christopher Manon. 2024-08-03. Geometric families of degenerations from mutations of polytopes. https://arxiv.org/abs/2408.01785

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