Polyhedra, lattice structures, and extensions of semigroups

TL;DR

This paper explores the relationship between polyhedral decompositions, semigroup extensions, and algebraic properties, providing explicit descriptions of initial free extensions and their connections to log geometry and toric singularities.

Contribution

It introduces a new explicit description of initial free extensions of semigroups associated with rational polyhedra, linking geometric decompositions to algebraic flatness.

Findings

Existence of initial objects in the category of free extensions

Explicit description of these initial free extensions

Connections to log geometry and deformation theory of toric singularities

Abstract

For an arbitrary rational polyhedron we consider its decompositions into Minkowski summands and, dual to this, the free extensions of the associated pair of semigroups. Being free for the pair of semigroups is equivalent to flatness for the corresponding algebras. Our main result is phrased in this dual setup: the category of free extensions always contains an initial object, which we describe explicitly. These objects seem to be related to unique liftings in log geometry. Further motivation comes from the deformation theory of the associated toric singularity.