The many faces of a logarithmic scheme

TL;DR

This paper introduces a decomposition of coherent logarithmic schemes into fine faces, enabling the application of tropical geometry techniques to more general schemes and providing algorithms for their computation.

Contribution

It presents a novel decomposition into fine faces for coherent logarithmic schemes, extending tropical geometry methods beyond fineness assumptions.

Findings

Decomposition into fine faces facilitates tropical geometry applications.

Algorithms for computing fine faces are developed.

Applications demonstrated in enumerative geometry and moduli spaces.

Abstract

A standard assumption in the study of logarithmic structures is "fineness", but this assumption is not preserved by intersections, fiber products, and more general limits. We explain how a coherent logarithmic scheme $X$ has a natural decomposition into "fine faces" -- a collection of fine logarithmic subschemes. This allows for importation of the techniques of tropical geometry into the study of more general logarithmic schemes. We explain the relevance of the construction in enumerative geometry and moduli via a range of examples. Techniques developed in the study of binomial schemes lead to effective algorithms for computing the fine faces.